Top Banner
OPTICS LAB MANUAL Surendra Singh with Raj Gupta, Ray Hughes, and Reeta Vyas Department of Physics, University of Arkansas 2005
110
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: Lab

OPTICS LAB MANUAL

Surendra Singh with Raj Gupta, Ray Hughes, and Reeta Vyas

Department of Physics, University of Arkansas2005

Page 2: Lab

How to use this manual

This manual is to help you perform the experiments described herein anddevelop an understanding of the basic principles of Optics. Refer to yourtextbook for additional discussion of these principles. You should familiarizeyourself with both the details of the procedure and the “physics” involved ineach of these experiments before you come to the laboratory.

We invite your comments to improve the manual.

ii

Page 3: Lab

Contents

How to use this manual ii

1 Thin Lenses 11.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Converging Lens . . . . . . . . . . . . . . . . . . . . . 31.2.2 Diverging Lens . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Optical Instruments 112.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 The Magnifying Lens . . . . . . . . . . . . . . . . . . . 112.1.2 The Telescope . . . . . . . . . . . . . . . . . . . . . . . 132.1.3 The Compound Microscope . . . . . . . . . . . . . . . 15

2.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . 162.2.1 The Magnifying Lens . . . . . . . . . . . . . . . . . . . 162.2.2 The Telescope . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Beam Expanders (optional) . . . . . . . . . . . . . . . . . . . 202.4 Other Instruments . . . . . . . . . . . . . . . . . . . . . . . . 202.5 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5.1 The Magnifying Lens . . . . . . . . . . . . . . . . . . . 212.5.2 The Telescope . . . . . . . . . . . . . . . . . . . . . . . 21

3 Measurement of Refractive Index 233.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 The Prism . . . . . . . . . . . . . . . . . . . . . . . . . 243.1.2 Inteferometric Refractometer . . . . . . . . . . . . . . . 25

3.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . 263.2.1 The Prism . . . . . . . . . . . . . . . . . . . . . . . . . 26

iii

Page 4: Lab

3.2.2 The Refractometer . . . . . . . . . . . . . . . . . . . . 313.3 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.1 The Prism . . . . . . . . . . . . . . . . . . . . . . . . . 333.3.2 The Interferometirc Refractometer . . . . . . . . . . . 34

4 Interference by the Division of the Wavefront 384.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1.1 Lloyd’s Mirror . . . . . . . . . . . . . . . . . . . . . . . 384.1.2 Fresnel’s Biprism . . . . . . . . . . . . . . . . . . . . . 40

4.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . 404.2.1 Determination of λ . . . . . . . . . . . . . . . . . . . . 42

4.3 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . 444.3.1 Lloyd’s Mirror . . . . . . . . . . . . . . . . . . . . . . . 444.3.2 Fresnel’s Biprism . . . . . . . . . . . . . . . . . . . . . 45

5 Michelson Interferometer 465.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.1.1 Coherenc Time and Coherence Length . . . . . . . . . 495.1.2 Polarization Dependence of Interference . . . . . . . . 50

5.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . 515.2.1 Wavelength of HeNe Laser . . . . . . . . . . . . . . . . 515.2.2 Wavelength of Sodium D Lines . . . . . . . . . . . . . 52

5.3 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . 53

6 Polarization of Light 556.1 States of Polarization . . . . . . . . . . . . . . . . . . . . . . . 56

6.1.1 Unpolarized light . . . . . . . . . . . . . . . . . . . . . 566.1.2 Polarized Light . . . . . . . . . . . . . . . . . . . . . . 57

6.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . 586.2.1 Linear Polarization . . . . . . . . . . . . . . . . . . . . 586.2.2 Brewster Angle (Optional) . . . . . . . . . . . . . . . . 606.2.3 Circular Polarization . . . . . . . . . . . . . . . . . . . 616.2.4 Optical Diode : Optical Isolator . . . . . . . . . . . . . 626.2.5 Polarization Rotator . . . . . . . . . . . . . . . . . . . 636.2.6 Elliptically Polarized Light . . . . . . . . . . . . . . . . 636.2.7 An Amusement . . . . . . . . . . . . . . . . . . . . . . 64

7 Resolving Power of a Grating and Limit of Resolution 677.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . 68

Page 5: Lab

7.2.1 Resolving Power of a Grating . . . . . . . . . . . . . . 687.2.2 Determination of the Grating Constant d . . . . . . . . 69

7.3 Limit of Resolution . . . . . . . . . . . . . . . . . . . . . . . . 697.4 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . 71

8 Image Formation and Spatial Filtering 738.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . 768.3 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . 79

9 Holography 809.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . 81

9.2.1 Transmission and Reflection Holograms . . . . . . . . . 829.2.2 Darkroom Chemicals Preparation . . . . . . . . . . . . 859.2.3 Viewing a Transmission Hologram . . . . . . . . . . . . 86

10 Photoelectric Effect 8910.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8910.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . 91

A Error Analysis 95A.1 Significant Figures . . . . . . . . . . . . . . . . . . . . . . . . 96A.2 Classification of Errors . . . . . . . . . . . . . . . . . . . . . . 98A.3 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . 98A.4 Sample Mean and Variance . . . . . . . . . . . . . . . . . . . 100A.5 Propagation of Errors . . . . . . . . . . . . . . . . . . . . . . . 101A.6 Standard Deviation of an Average . . . . . . . . . . . . . . . . 103A.7 Special Rules for Combining Errors . . . . . . . . . . . . . . . 104

A.7.1 Sum and difference: V = ax± by . . . . . . . . . . . . 104A.7.2 Product of various powers: V = xp yq . . . . . . . . . . 104A.7.3 Logarithm of a quantity: V = A ln x . . . . . . . . . . 105A.7.4 Exponential of a quantity: V = Ae±bx . . . . . . . . . . 105

Page 6: Lab

Chapter 1

Thin Lenses

1.1 Theory

The object and image distances, so and si, both measured from the centerof a thin lens, are related by the lens formula

1

so+

1

si=

1

f, (1.1)

where f is the focal length of the lens. The image distance si is positive fora real image and negative for a virtual image. Similarly, the object distanceso is positive for a real object and negative for a virtual object. A real imageis formed by light rays that converge after passing through the lens. A realimage can be projected onto a screen. A virtual image is formed by lightrays that diverge after passing through the lens. To the eye viewing theserefracted rays, they appear to come from an image located on the other sideof the lens. A virtual image cannot be projected onto a screen.

Q1 : What kind of rays are associated with real and virtualobjects?

Lenses can be divided into two categories : converging and diverging.A converging lens is thicker in the middle than at the edges. A diverginglens is thinner in the middle than at the edges.

Q2 : When are these two statements about converging anddiverging lenses false?

Light rays incident parallel to the axis of a converging lens converge to-ward a point on the axis after refraction by the lens1. This point is called

1For a diverging lens the refracted rays appear to diverge from a point.

1

Page 7: Lab

2 CHAPTER 1. THIN LENSES

Qty. Description

1 optical bench with stages

1 positive lens

1 negative lens

1 lens holder

1 laboratory telescope with a holder

1 circular frame with vertical wire attached

1 image screen with holder

1 plane mirror with holder

1 illuminated object (could use a desk lamp and a pin or aperture)

Table 1.1: Apparatus for the Thin Lenses experiment.

the image focal point of the lens. Thus the focal point is the image of anaxial object at infinity. There is another axial point which is such that therays starting from an object placed there emerge parallel to the axis of thelens2. This point is called the object focal point. The distance from the cen-ter of the lens to the focal point is called the focal length of the lens. For alens surrounded by the same medium on the two sides, the object and imagefocal points are located symmetrically on the opposite sides of a lens. Theirdistances from the lens have the same value and we will denote this lengthby f .

A converging lens has a positive focal length, f > 0, and is therefore alsocalled a positive lens. A diverging lens has a negative focal length, f < 0.For this reason a diverging lens is also called a negative lens. Algebraic signsshould be observed carefully in Eq. (1.1).

Q3 : Focal length f is one of the two characteristic parametersof a lens. What is the other parameter?

From Eq. (1.1), we see that the object and image locations are inter-changeable. This means if an object at a distance so produces an image atsi, then the same object placed at a distance si from the lens will producean image at at so. Two such points that an object at one produces an imageat the other, are called conjugate points.

2For a diverging lens the rays are incident on the lens converging toward the object

focal point and emerge parallel to the axis.

Page 8: Lab

1.2. EXPERIMENTAL PROCEDURE 3

1.2 Experimental Procedure

1.2.1 Converging Lens

In this portion of the experiment, you will determine the focal length of apositive lens. For every procedure described below, make at least threedeterminations of the focal length. Sample tables for recordingyour measurements have been provided at the end of this Chapter.Be sure to record the correct number of significant figures, forexample, 50.0cm± 0.1cm. Report your results along with uncertainties.

Focal length by parallel rays

Method IForm a sharp image of a distant object (10 m or farther) upon a screen andmeasure the distance from the center of the lens to the screen [ Fig 1.1(I)].This distance is equal to the focal length (f) of the lens. Draw a ray diagramfor this method. Make several independent measurements. Record yourmeasurements in a Table in INK. Take the average of your results and assignuncertainties.Method IIFocus a telescope on a distant object and without changing the adjustmentof the telescope, place it on an optical bench. Aim it toward the converginglens. Place an illuminated object on the other side of the lens [Fig 1.1 (II)].Keep the telescope, the lens, and the object at the same height above thebench. Look through the telescope and move the lens either toward or awayfrom the object until you get a sharp image of the object in the telescope.Measure the distance from the center of the lens to the object. This distanceis equal to the focal length of the lens. Make three or more independentmeasurements for the lens provided and take the average of your results.Assign uncertainty. Draw a ray diagram to explain this method.

Focal length by parallax

Place a vertical plane mirror just behind the lens [ Fig 1.1 (III)]. Set anilluminated object (a pin or short vertical wire) in front of the lens. Lookpast the illuminated object toward the lens. You should see an invertedimage of the object. Move the object toward or away from the lens untilits image appears to “float” directly above the object. When this happens,the object and its image are a focal length away from the lens. Draw a ray

Page 9: Lab

4 CHAPTER 1. THIN LENSES

Mirror Lens ObjectDistant Lens ImageObject on a screen

≈f

Distant Lens TelescopeObject

f

(I) (II) (III)

Image

Observer

Figure 1.1: Measurement of the focal length of a positive lens by (I) the

image of a distant object; (II) using a telescope focused for parallel rays;

(III) the method of parallax.

diagram and explain how this method works. Take several measurements,calculate the average of your measurements and assign errors.

Focal length by conjugate points

Place an illuminated object near one end of the optical bench. Place the lensbefore the object and put a screen on the other side of the lens to receivethe image of the object. Adjust the positions of both the lens and the screenuntil a sharp image appears on the screen. The object and its image definea pair of conjugate points (Fig. 1.2).

Record the object and image distances so and si. Measure the distance bbetween the object and its image.

Without moving the object or the screen, move the lens until anothersharp image is cast on the screen. Record the distance a between the newand the old positions of the lens. Also record the object and image distancess′o and s′i for the new position of the lens. In terms of a and b the focal lengthof the lens is given by

f =b2 − a2

4b. (1.2)

Q4 : Derive this relation in your report.Repeat the measurements described above for at least five values of b and

calculate the focal length in each case using Eq. (1.2). Take the average andassign uncertainty.

Use the data recorded above for object and image distances to plot 1/si

(y−axis ) against 1/so (x−axis). The points should lie on a straight line.

Page 10: Lab

1.2. EXPERIMENTAL PROCEDURE 5

so si

a

b

s'o s'i

O L I

F

O L I'

F

Figure 1.2: Focal length by conjugate points

Note that you should have at least 10 pairs of data from five measurementsof b. Intercept of this straight line with the x− or the y− axis is 1/f .Determine f from the intercept.

1.2.2 Diverging Lens

A diverging lens by itself does not form a real image. Therefore the methodsdescribed for positive lenses cannot be used to determine the focal length ofa diverging lens. In this part of the experiment, we will learn two methodsfor determining the focal length of a negative lens.

Focal length by virtual object

Set up an illuminated object, positive lens, and screen as in the Focal lengthby conjugate foci subsection of Sec. 1.2.1 and obtain a sharp image I1 onthe screen. Record the image distance from the positive lens as si1 in thetable provided at the end of this Chapter. Now place the diverging lensbetween the converging lens and the screen. The original image I1 serves asa virtual object (negative object distance) for the diverging lens. Withoutmoving the screen record the distance so2 with due care for the algebraicsigns. The diverging lens will cause the image to lie farther from the positivelens (see Figure 1.3). Now move the screen until the new image I2 is infocus. Record the distance of the screen from the negative lens as si2 in the

Page 11: Lab

6 CHAPTER 1. THIN LENSES

O

so1 si1

so2si2

L1 L2

F1

I1I2

Figure 1.3: Measurement of focal length of a negative lens.

table provided. This arrangement results in a real image if the converginglens focal length is shorter than the magnitude of the focal length of thediverging lens. The two positions of the screen recorded above are a pair ofconjugate points of the diverging lens. Compute the focal length of the lensusing Eq. (1.1). Make at least three measurements.

Focal length by parallax

Mount the diverging lens on the optical bench. On the side opposite theviewer place an illuminated object. Looking through the lens, determine thegeneral location of the image. At the approximate site of the image, mountthe image locator. The image locator is a short vertical wire attached to acircular frame (see Fig. 1.4). The image locator wire frame should be directlyvisible outside the field of view of the lens. Use the tip of the wire frame tolocate the image of the object. Adjust the position of the image locator wireframe viewed above the lens until its position coincides with the image of theobject formed by the lens. Measure the distances from the center of the lensto both the wire frame (di) and the object (do).

Q5 : How are di and do related to so and si?

Repeat this procedure three times for the given lens. Record the data inthe table provided and calculate the focal length of the lens.

Page 12: Lab

1.3. EXPERIMENTAL DATA 7

Image ofobject wire

Curved wire

Field of View

Figure 1.4: Special image locator; circular frame with vertical wire.

1.3 Experimental Data

Converging Lens

Q6 : In Method I of the Focal length by parallel rays subsectionof Sec. 1.2.1, why will the image formed lie in the principal focalplane of the lens?

trial 1 trial 2 trial 3 average std dev

focal length f

Table 1.2: Focal length by parallel rays, method I.

Q7 : In Method II of the Focal length by parallel rays, Sec. 1.2.1,why is the distance measured equal to the focal length of the lens ?

Q8 : In the Focal length by parallax,Sec. 1.2.1, why are theobject and its image a focal length away from the lens when the

Page 13: Lab

8 CHAPTER 1. THIN LENSES

trial 1 trial 2 trial 3 average std dev

focal length f

Table 1.3: Focal length by parallel rays, method II.

trial 1 trial 2 trial 3 average std dev

focal length f

Table 1.4: Focal length by parallax.

virtual image lies directly above the object? Draw a ray diagram.

Q9 : In the Focal length by conjugate points, Sec. 1.2.1, howdoes the second set of measurements ( s′o, s′i) compare with the firstset (so, si)?

Q10 : Compare the values of the focal length determined above.Which method gives more reliable result ? Give reasons.

Use equation (1) to make a plot of si versus so for a positive lens. Youmay find it convenient to plot so and si in units of the focal length of thelens. Remember that so and si can be positive or negative. Label regionsof graph by the type of image that would be formed (magnified/reduced,upright/inverted, real/virtual). Your data for the positive lens correspond toone branch (real objects and images) of this curve. There is another branchof this curve that corresponds to virtual objects and images.

Page 14: Lab

1.3. EXPERIMENTAL DATA 9

trial 1 trial 2 trial 3 trial 4 std dev

so

si

s′o

s′i

a

b

f

Table 1.5: Focal length by the method of conjugate points

Q11 : What arrangement would you use to explore this branch?

Diverging Lens

Page 15: Lab

10 CHAPTER 1. THIN LENSES

trial 1 trial 2 trial 3 std dev

so1

si1

so2

si2

f

focal length f =

Table 1.6: Focal length of a negative lens by virtual object.

trial 1 trial 2 trial 3 std dev

df

dv

f

focal length f =

Table 1.7: Focal length of a negative lens by parallax.

Page 16: Lab

Chapter 2

Optical Instruments

2.1 Theory

The simple magnifier, telescope, and microscope are examples of optical in-struments. Optical instruments aid the eye by increasing the angular sizeof the retinal image of an object. The angular size of the retinal image, inturn, is equal to the angle subtended at the eye by the object or the imageunder view. This angle increases as the object under view approaches theeye. The shortest object distance for which the eye can comfortably producea sharp retinal image is called the near point N of the eye. Thus with theunaided eye the largest retinal image is obtained when the object is at thenear point. For the normal eye N ≈ 25 cm. With an optical instrument, theeye sees the magnified image of the object formed by the instrument. It isthis magnifying power (M.P.) that makes an optical instrument useful. It isdefined as the ratio of the angle subtended at the eye by the image producedby the instrument to the angle subtended at the unaided eye by the objectwhen it is placed at the near point of the eye. In this experiment we studyhow two common instruments aid the eye.

2.1.1 The Magnifying Lens

Suppose a small object OO′ subtends an angle α at the unaided eye when itis placed at the near point as shown in Fig. 2.1(a). Consider now the sameobject placed just inside the principal focus F of a positive lens LL′ as shownin Fig. 2.1(b). The lens forms a virtual, upright, and magnified image II ′ ofthe object. The eye placed behind the lens sees the image II ′. If the eye is

11

Page 17: Lab

12 CHAPTER 2. OPTICAL INSTRUMENTS

O

O'

I'

I

L'

L

si = - q

so = p

β

(b)

F

O'

O

α

N = 25 cm

(a)

Figure 2.1: Magnifying power of a simple magnifier.

placed immediately behind the lens, the angle β in Fig. 2.1(b) is very nearlyequal to the angle subtended at the eye by the image II ′. Thus the lens hasaided the eye by effectively increasing the angle subtended at the eye by theobject from α to β.

The magnifying power of the lens is then given by

M.P. =β

α, (2.1)

which for small angles α and β (tan θ ≈ θ, sin θ ≈ θ), becomes [see Fig. 2.1]

M.P. =N

p. (2.2)

Using the relation between object location so ≡ p, image location si ≡ −q(notice that we have a virtual image), and the focal length f

1

p−

1

q=

1

f, (2.3)

we can express the magnifying power as

M.P. =N

p= N

f + q

fq. (2.4)

Since q lies in the interval N ≤ q < ∞, the magnifying power may vary inthe interval

N

f≤ M.P. ≤

N

f+ 1. (2.5)

Page 18: Lab

2.1. THEORY 13

It is clear that for the magnifying power to exceed unity the focal length fmust be smaller than N .

Note that, in general, magnifying power and ordinary magnification (ratioof image size to object size) are different. Also, since the eye is relaxedwhen looking at distant objects, it is better to focus the magnifying lensso that the virtual image appears far from it. In this case the magnifyingpower is M.P.= N/f . When the image is at N , the magnifying power is(M.P.= N/f + 1). This small gain in magnifying power is seldom worth theeye strain that accompanies it.

A simple magnifier described here (or some sophisticated version of itcalled an eyepiece) is part of many other instruments such as compound mi-croscopes and telescopes. For microscopic objects a simple magnifier doesnot have enough magnifying power. In such cases we use a compound mi-croscope. A compound microscope uses a short focal length lens, called theobjective, to produce a real enlraged image of the microscopic entity underinvestigation. The image formed by the objective is further enlarged by theeyepiece.

2.1.2 The Telescope

The telescope is an instrument for viewing objects hundreds of meters tomillions of kilometers away. Such objects subtend small angles at the eyeand cannot be brought closer to the eye for direct inspection. We shall seethat a telescope aides the eye in such cases.

A simple telescope consists of a large aperture objective lens Lo of longfocal length fo > 0, and an eyepiece lens Le of short focal length fe > 0.Figure 2.2 shows the optical paths of two sets of rays propagating through atelescope. Rays 1 and 2 originate from a point on the lower edge and rays 3and 4 originate from the upper edge of some distant object ( so À fo angularsize α ). The objective forms a real, inverted ( and minified) image PP ′ ofthe distant object in its focal plane. 1

Image PP ′ formed by the objective is observed through the eyepiece Le

which is placed such that the image PP ′ lies just inside the focal plane ofthe eyepiece. Rays 1 and 2 then emerge nearly parallel to each other fromthe eyepiece. An observer viewing these rays sees the final virtual image II ′

at infinity. Image II ′ subtends an angle β at the eye. Thus the telescope

1This image may be seen directly by placing the eye 25 cm (to maximize the angle

subtended at the eye) behind the image and looking toward the objective lens. The

magnifying power in this case is fo/25 which is greater than unity only if fo > 25 cm.

Page 19: Lab

14 CHAPTER 2. OPTICAL INSTRUMENTS

H

H'P'

eye piece

P

α

objectiveEntrance Pupildiameter D

β

1

2

3

4

to infinity

fo fe

34

12

Exit PupilDiameter d

LoLe

I

I'

Figure 2.2: Telescope; optical paths of two sets of rays.

has effectively increased the angle subtended by the object at the eye. Notethat the final image II ′ is inverted with respect to the distant object. Thisis a characteristic of the astronomical telescope, as distinguished from theterrestial telescope.

The magnifying power of the telescope is given by,

M.P. ≡β

α. (2.6)

These angles, greatly exaggerated in the figure, are quite small in the casesof practical interest. By tracing a ray incident through the first focal point ofthe objective you can show that, for relaxed viewing, the magnifying poweris also given by

M.P. =fo

fe. (2.7)

It might appear from Eq. 2.7 that eyepieces of smaller focal lengths fe wouldresult in bigger and better images, but this is not true because diffractionby the objective lens — which you may not have studied yet — ultimatelylimits the resolving power (ability to see details) of a telescope.

There is another way for determining the magnifyiong power of a tele-scope. An examination of Fig. 2.2 shows that all rays emerging from theeyepiece pass through a small circle at HH ′ called the eye ring (exit pupil)of the telescope. The eye ring is just the real image of the objective lens

Page 20: Lab

2.1. THEORY 15

formed by the eyepiece.2 The position and size of the eye ring may thenbe computed by treating the objective lens Lo as an object in front of theeyepiece. The diameter d of the eye ring HH ′ in Fig. 2.2 is then given by

d = D

(fe

fo

). (2.8)

Using this in Eq. (2.7) we can write,

M.P. =D

d. (2.9)

Thus the magnifying power of the telescope is also equal to the diameter of theobjective divided by the diameter of the eye ring. By properly illuminatingthe objective lens, its real image (the eye ring) may be projected onto ascreen and measured.

In a well designed optical instrument, the eye ring should match the pupilof the eye in order to gather all the rays from the object. A single eyepiecelens used in this experiment gives an eye ring that is too far from the lensfor convenient observing. To avoid this and other defects, an eyepiece isusually composed of several lenses, e.g., a Ramsden eyepiece. Equation (2.9)clearly indicates the need for an objective lens of large aperture and longfocal length if large magnifying power is demanded from a telescope. Also,a study of diffraction effects in telescopes reveals the desirability of havingan objective lens of large aperture in order to increase the resolving power(ability to show detail) of the instrument.

2.1.3 The Compound Microscope

Microscope is used for viewing small nearby objects. Just like a telescope, acompound microscope is consists of two lenses: objective and eyepiece (seeFig. ??). However, unlike a telescope, focal length of both the lenses in amiscroscope are much smaller then the distance L between the focal pointsof the objective and the eyepiece. Objective forms a real, inverted, andmagnified image. Thus magnification of the objective is given by (−L/fo).If the final image formed by eyepiece is at infinity and the standard nearpoint is choosen to be 25 cm, then magnifying power of the microscope isgiven by

2This can be seen by considering rays 1 and 3 as originating from the lower edge and

rays 2 and 4 from the upper edge of the objective.

Page 21: Lab

16 CHAPTER 2. OPTICAL INSTRUMENTS

Qty. Description

1 optical bench with stages

1 short (≈ 12.5 cm) focal length positive lens

1 long (≈ 50 cm) focal length positive lens

1 aluminum lens holder

1 spring-tension lens holder

1 short section of a translucent scale

1 photocopy of a meter stick posted on wall

wire suitable for use as object

Table 2.1: Apparatus for the Optical Instruments experiment.

2.2 Experimental Procedure

Several precautions taken during the performance of the experiments de-scribed in this Chapter will literally save you headaches. Align lenses andother optical components on the optical bench so that they define a goodoptic axis. Keep all optical surfaces clean. Ask your instructor for helpin cleaning the optics. Bad alignment and dirty optics produce distortedand poor quality images. These problems become especially acute for amulti-lens optical system such as the telescope. Lenses with very short fo-cal lengths (high magnifying power) will produce poor results in the eventof misalignment. When using the parallax method for locating images, lensdefects and/or poor alignment may result in misleading movement of theimages/reference object if the eye is moved too far. Adjustments of this sortshould be made with limited motion of the head.

Make at least three measurements of each quantity being measuredand then determine its average and the standard deviation.

2.2.1 The Magnifying Lens

We will use a short focal length converging lens as a magnifying lens and atransluscent scale as the object. We will study the magnifying power of thelens for two values of q (image location): 25 cm and 100 cm.

Determine the focal length of the lens and set it at a convenient marknear one end of the optical bench. Mount the transluscent scale horizontally

Page 22: Lab

2.2. EXPERIMENTAL PROCEDURE 17

F

Eye Lens Object Wire& the image

q

Figure 2.3: Determination of the magnifying power of a lens.

beyond the lens, near the principal focus of the lens. Place the eye justbehind the lens and focus the system by moving the object (scale) ratherthan the lens. You will notice that there is a wide range of object positionsfor which a sharp image of the object is obtained. This range corresponds tothe image distances (si = −q) 25 cm ≤ q < ∞.

To focus the system so that the image distance q = 25cm = −si, seta long vertical wire (or a pin) 25 cm from the lens on the optical bench.Focus the system by moving the object (the illuminated translucent scale)until the virtual image — as seen through the lens — coincides with the wire— as seen directly above the lens. When this happens, there should be noparallax (relative motion) between the wire and the image of the object (seeFig. 2.3). You can accomplish this as follows. With both eyes open, bringone eye as close to the lens as possible so that you can see the image throughthe lens. With your other (unaided) eye look, past both the lens and theobject, directly at the wire. Move the object so that the two images mergeinto one. This may require some patience, since your eyes are not used to thissort of maneuver. One of your eyes may fail to function properly initially,allowing you to see only one of the images. To overcome this, simply closethe eye that is accepting the image so that the other eye will form an image.Then try opening your other eye again. Eventually, you should be able to seeboth the wire and the object image simultaneously and remove any parallaxbetween them. When this happens the virtual image of the object appear atthe same location as the wire.

Remove the wire and replace it with a horizontal section of a referencescale of some sort (e.g., a meter stick). Match the virtual image of thetransluscent scale — as seen through the lens — with the reference scale justmounted — as seen above the lens. In this case the magnifying power ofthe lens is equal to the linear magnification. Determine the magnification bydirect observation and compare it with the value 25/p given by Eq. (2.5) forthe magnifying power. The value of p may be read directly from the optical

Page 23: Lab

18 CHAPTER 2. OPTICAL INSTRUMENTS

bench scale.

Next set the vertical wire at q = 100 cm, and focus the system so thatthe virtual image coincides with the position of the wire. Calculate themagnifying power and compare it with the theoretical value N/f (= 25/f )derived in section 2.1.1. Record the value of p (object distance) for this caseand calculate the magnifying power.

2.2.2 The Telescope

We now build a telescope and measure its magnifying power. Set the shortfocus lens, which you have just used as a magnifier, approximately 20 cmfrom one end of the optical bench. This lens will now be used as an eyepiecefor the telescope. On the other side of the lens, place a short vertical wire (asubstitute for cross hairs) at the principal focus of the lens. Point the opticalbench toward a scale several meters away in the laboratory. (You can use aphotocopy of a meter stick posted on the wall.) Focus the eyepiece on thewire by moving the wire toward the eyepiece until the virtual image of thewire — as seen through the lens — coincides in position with the the distantscale (see Fig. 2.4, Step 1). Use the method of parallax for this adjustment.In the use of a real telescope, this procedure is equivalent to focusing theeyepiece on the cross hairs of the telescope.

Determine the focal length of the objective and place it on the opticalbench in front of the wire, at a distance equal to its focal length. Adjustthe position of the objective lens so that a sharp image of the distant scaleis visible through the eyepiece. Continue this adjustment until there is noparallax between the image of the vertical wire and the image of the distantscale — viewed through the eyepiece. This procedure corresponds to focusingthe eyepiece and cross hairs on a distant object.

If the preceding adjustments have been properly made, then (i) the ob-jective forms a real inverted image of the distant scale at the location of thevertical wire and (ii) the eyepiece forms virtual images of both the wire andthe real image of the distant scale (formed by the objective) that coincidewith the location of the actual distant scale (see Fig. 2.4). Make sure thatyou understand this.

The magnifying power of the telescope can be determined experimentallyby comparing the distance between two scale marks — as seen through thetelescope and as seen by the unaided eye. 3

3This comparison may be done as follows (see Fig. 2.4). With both eyes open — one

viewing the distant scale directly and the other viewing it through the telescope — align

Page 24: Lab

2.2. EXPERIMENTAL PROCEDURE 19

Step 2: Focusing the eyepiece

Virtual imageof the wire

F

Eye EyepieceDistant scale & thevirtual image of thewire

Wire≈

Step 1: Focusing the eyepiece

F

Eye Eyepiece

Distant scale

Wire & the realinverted image ofthe distant scaleformed by the objective

Objective Virtual image formed bythe eyepiece of the realimage produced by theobjective

Figure 2.4: Determination of the magnifying power of a telescope.

We make another determination of the magnifying power of the telescope.Illuminate the objective of the telescope from the side and project its realimage formed by the eyepiece lens on a white screen placed behind the eye-piece. This real image is the eye ring HH ′ of the telescope (Fig. 2.2). Noticethe size and location of the eye ring. Measure the diameters of the eye ringand the objective lens. Compute the magnifying power by use of Eq. 2.9 andcompare it with that calculated using Eq. 2.7.

the telescope so that the images on the retina of your eye are superimposed (recall the

advice in subsection 2.2.1 if this proves difficult to accomplish). Eventually, obtain the

dual image and make a direct comparison of scale length; i.e., one scale division on the

virtual image scale equals X scale divisions on the real distant scale. Then, X = M.P..

If necessary, have your partner indicate divisions on the distant scale with a pointer and

then have him/her read the value of X for you. Compare this value of the magnifying

power with that given by Eq. 2.7.

Page 25: Lab

20 CHAPTER 2. OPTICAL INSTRUMENTS

fi fo

DiDo

Figure 2.5: A beam expander

2.3 Beam Expanders (optional)

In many optical experiments light beams with large cross-section are needed.Suppose a collimated optical beam of diameter Di is incident on the arrange-ment of lenses shown in Fig. 2.5.

Q1 : What is the diameter Do of the beam exiting the systemin terms of Di, fi and fo? When does this arrangement act as abeam expander? Consider both cases - fi positive and fi negative.What would you do to reduce beam size. You may use a laser beamto carry out this experiment.

2.4 Other Instruments

Telecope described here is a simple refractor astronomical telescope. In allcases the final image is inverted with respect to the object. This is of noconsequence for astronomical objects such as the planets, stars, and galax-ies. For terrestrial objects orientation of the image is important. Terrestrialtelescope have an image erecting system between the objective and the eye-piece. This arrangement usually leads to long draw tube. For this reasonbinocular telescopes utilize erecting prisms.

The refractor telescopes two drawbacks : the chromatic aberration andlong tube lengths, especially if large magnification is desired. Chromaticaberration is partly corrected by using multi-element achromatic lenses. Areflector telescope, where a spherical mirror is used as an objective, addressesboth the chromatic aberration and the long draw tube problems.

You may wish to assemble a slide projector or a compound microscope.Feel free to ask for help in assembling these intruments.

Page 26: Lab

2.5. EXPERIMENTAL DATA 21

2.5 Experimental Data

2.5.1 The Magnifying Lens

Q2 : In the The Magnifying Lens subsection of Sec. 2.2.1, show thatthe linear magnification for the case q = N equal to the magnifyingpower of the lens.

trial 1 trial 2 trial 3 average std dev

p

linear mag

Table 2.2: The magnifying lens for image at q = 25 cm.

trial 1 trial 2 trial 3 average std dev

p

lin mag

Table 2.3: The magnifying lens for image at q = 100 cm.

2.5.2 The Telescope

Q3 : Prove that the magnifying power of a telescope is given by

f0/fe. Also derive the relation

M.P. =fo

fe=

LL′

HH ′ =D

d,

Page 27: Lab

22 CHAPTER 2. OPTICAL INSTRUMENTS

where LL′, HH ′, D and d are defined in Section 2.1.2.

fofe

=

trial 1 trial 2 trial 3 average std dev

measured LL′ (D)

measured HH ′ (d)

Calculated M.P.

Table 2.4: Measurements of the telescope eye ring.

Page 28: Lab

Chapter 3

Measurement of Refractive

Index

3.1 Theory

The refractive index n of a medium is defined by

n =c

v, (3.1)

where c is the speed of light in free space and v is the speed of light insidethe medium. The refractive index n depends on the wavelength (strictlyspeaking, frequency) of light traversing the medium. For most transparentmaterials the dependence of n on wavelength is given by the Sellmeier’sformula

n(λ) = n0 +Bλ2λ2

0

λ2 − λ20

. (3.2)

The constant B is given by

B =Ne2

8π2ε0mec2, (3.3)

where N is the effective number of electrons per unit volume participatingin the polarization process, νo = c/λ0 is their characteristic frequency, and eand me are the electronic charge and mass. In the limit λ À λ0 Eq. (3.2)can be approximated as

n(λ) ≈ A +B′

λ2, (3.4)

23

Page 29: Lab

24 CHAPTER 3. MEASUREMENT OF REFRACTIVE INDEX

where constants A = no + Bλ2o and B′ = Bλ4

o. It follows from Eq. (3.2)that the refractive index increases as the wavelength of light decreases. Thisbehavior of the refractive index is referred to as normal dispersion. For adilute gas n0 ≈ 1 and the electron density N is proportional to the moleculardensity ρ. It follows that the refractive index is proportional to the densityρ of molecules,

n− 1 = const× ρ . (3.5)

From the ideal gas law, ρ = p/kBT , and it follows from Eq. (4) that atconstant temperature

n− 1 = C × p , (3.6)

where p is the absolute pressure of the gas, C is a constant of proportionality,and kB is the Boltzmann constant.

A knowledge of the refractive index of a material is important in manyapplications involving optical fabrication and design. For example in fabri-cating antirefelection coating on lenses for fine optical instruments requires aknowledge of the refractive indices of the materials of the substrate and thinfilms. Many studies involving phase changes are carried out using measure-ment of n and its relation to macroscopic properties. Here we will study twomethods for determining the refractive index. The first method is more suit-able for liquids and solids. The second method is an interferometric methodused for rarified media such as gases.

3.1.1 The Prism

A convenient method for determining the refractive index is to use the mate-rial in the form of a prism. Refractive index of the material of a prism usingthe method of “angle of minimum deviation” is given by

n(λ) =sin [(A + Dm(λ)) /2]

sin (A/2). (3.7)

Here A is the Prism angle and Dm(λ) is the angle of minimum deviation.Since n depends on the wavelength (strictly speaking, frequency), each wave-length has a slightly different angle of minimum deviation. We will use thisto study the variation of the refractive index as a function of wavelength.

Page 30: Lab

3.1. THEORY 25

M1

M2

BSdiffuse screen or a lens

*Source L

n

Figure 3.1: Setup of the refractometer experiment.

3.1.2 Inteferometric Refractometer

This refractometer is a variation of Rayleigh refractometer and uses two-beam interference of light to determine the refractive index of a gas. Thismethod is very sensitive and is excellent for measuring small changes in therefractive index of a material. We will use this method to study the pressuredependence of the refractive index of air. Refractive index of air at roomtempertaure and pressure is n ≈ 1.000. As shown in Fig. 3.1, a beam splitterdivides the incident light into two and directs them along separate arms ofan interferometer. One beam travels in air at room temperture and pressure.The second beam traverses a gas tube at room temperature, but where thepressure can be varied. Two beams are brought together at the beam splitter.The resulting interference pattern depends on the relative path differencebetween the two arms of the interferometer. If a laser is being used as thesource you may be able to project these fringes on a wall or screen. In ourexperiment we change the refractive index of the gas in the cell by alteringthe gas pressure inside the cell.

Suppose initially the cell is open to the atmosphere so that its pressureis the atmospheric pressure (p0) and then it is evacuated with a vacuumpump. As the pressure in the tube drops the refractive index of air in thecell decreases (cf Eq. (5)) from its initial value n0 and the interference fringeswill be observed to shift.

Let the refractive indices of air in the open arm and the cell be n0 andn, respectively. Then the difference in the optical paths of the two beams is

Page 31: Lab

26 CHAPTER 3. MEASUREMENT OF REFRACTIVE INDEX

(n0 − n) 2L where L is the cell length. When the refractive index of the airin the cell changes by ∆n, the total number of fringes m, that are shifted, isgiven by

m = ∆n2L

λ,

= C [∆p]2L

λ, (3.8)

where ∆p is the change in the pressure of the evacuated compartment fromits initial pressure p0. Equation (3.8) has the form of the equation of astraight line, where m would be plotted along the ordinate (vertical axis)and ∆p along the abscissa (horizontal axis). The slope of this straight line is(∂m/∂∆p) = 2CL/λ . Assuming we know the wavelength λ of the light andthe length of the cell L, we can determine the value of the proportionalityconstant C. Substituting this result into Eq. (3.6), we find that the refractiveindex of air is given by

n = 1 +

(∂m

∂∆p

2Lp . (3.9)

Absolute pressure p is given by p0−∆p. By counting the number of fringes mthat pass a reference point as the pressure changes by ∆p, we can determinen for any pressure p.

3.2 Experimental Procedure

Make at least three measurements of each quantity and then de-termine the average and its standard deviation.

3.2.1 The Prism

The spectrometer consists of a collimator, a dispersing element, and a tele-scope. Two types of student spectrometer are available. Their working issimilar. The locations of various controls are listed in Table 3.2.

Note that in order to use any of the fine adjust features the correspondinglock screw must be tightened. Do not force rotation of the instrument whenthe lock screw is tightened. Placed the prism upon the spectrometer tableand clamp it to the table. A built-in magnifying glass is used to read theGaertner vernier, while the Pasco spectroscope has a separate magnifyingglass. If you are unfamiliar with reading vernier scales ask your instructorfor help.

Page 32: Lab

3.2. EXPERIMENTAL PROCEDURE 27

Qty. Description

1 Pasco or Gaertner student spectrometer

1 glass prism

1 helium discharge tube

1 desklamp or other white-light source

Table 3.1: Apparatus for the Prism Experiment.

Control Gaertner Pasco

telescope rotation lock screw on telescope base directly beneath telescope

telescope rotation fine adjust directly beneath telescope beneath and to right of telescope

spectrometer table lock screw directly beneath table directly beneath table

spectrometer table height lock screw directly beneath table n/a

spectrometer table leveling screws underneath table underneath table

table rotation lock screw n/a base

table rotation fine adjust n/a base

collimator slit width adjust input of collimator input of collimator

telescope focus sliding eyepiece focus knob on telescope

collimator focus sliding lens focus knob on collimator

Table 3.2: Comparison of Gaertner and Pasco spectrometers.

Page 33: Lab

28 CHAPTER 3. MEASUREMENT OF REFRACTIVE INDEX

A2A

collimated beam

prism

Figure 3.2: Determination of the prism angle A.

Determination of Prism Angle

To determine the refractive index of a material in the form of a prism requiresthat we know the angle of the prism. A white light source is used for thismeasurement. Place the prism onto the spectroscope table so that it splitsthe collimated beam into two as in Fig. 3.2. With the telescope rotation lockscrew loosened, swing the telescope to view one of these beams. Tighten thetelescope rotation lock screw and use the telescope rotation fine adjust controlto place the crosshairs on one of the edges of the image of the collimator slitseen through the telescope. Focus the crosshairs if necessary. Record theangular position of the telescope using the degree scale and vernier, and thenloosen the lock screw and swing the telescope to the other side to receive thesecond beam. Using the technique described above, record the position ofthe telescope again. As shown in Fig. 3.2, the angle of the prism, A, is simplyhalf the angle between the two beams. Repeat this measurement three timesand calculate the average value of A and its standard deviation.

Determination of Refractive Index

Once the prism angle A is known the refractive index of the prism is deter-mined by measuring the angle of minimum deviation suffered by the lightbeam from the collimator in passing through the prism. When the prismis set for minimum deviation, the light passes symmetrically through theprism. Using the Helium light source, move the telescope and prism aboutuntil the spectrum of light appears as shown in Fig. 3.3(a). You will see dif-ferent colored images of the slit. The collimator, prism, and telescope shouldbe placed as shown in Fig. 3.3(b). Rotate the prism table and observe the

Page 34: Lab

3.2. EXPERIMENTAL PROCEDURE 29

D

colllimator

prism

angle ofdeviation

telescope telescope

Dangle ofdeviation

(b)

blue red

next order -2 -1 0 1 2 next order

(a)

Figure 3.3: (a) Spectrum as viewed through the spectroscope eyepiece. (b)

Measurement of angle of minimum deviation.

Page 35: Lab

30 CHAPTER 3. MEASUREMENT OF REFRACTIVE INDEX

Helium spectral lines

Color λ(nm)

red 706.5

red 667.8

yellow 587.5

green 500.1

blue-green 492.2

blue 471.3

violet 447.1

Table 3.3: Helium spectral lines.

motion of slit images. You will notice that as the prism table is rotated slitimages first move closer to the original undeflected beam direction directionand then move away from the original beam direction as we continue to ro-tate the prism table in the same direction. This means that the angle ofdeviation D first decreases and then increases. At some point the angle ofdeviation D (see Fig. 3.3(b)) is minimized. This happens at different posi-tions for different colored images. While viewing a particular color image ofthe slit through the telescope, and using the cross hairs measure the angle ofminimum deviation D you can produce.

Rotate the prism and the telescope so that the angle of deviation changessign. Measure the angle of minimum deviation again. Note that D is mea-sured from the undeviated beam from the collimator. Repeat this measure-ment for a total of three readings for each of the six to eight colors of thespectrum. Calculate averages and standard deviations as usual.

From measured A and Ds calculate the refractive index at different wave-lengths by using Eq. (3.7). The wavelengths for each spectral line may beobtained from the Table 3.3. Plot a n vs λ curve by using the data collectedabove drawing error bars for each data point. Fit a smooth curve of the formgiven by Eq. (2) through these points. What are the values of B, λ0 and n0.

Page 36: Lab

3.2. EXPERIMENTAL PROCEDURE 31

Qty. Description

1 optical bench with translating stage for eyepiece

1 Helium-Neon laser (λ ≈ 632.8nm)

1 laser mount that will attach to optical bench

1 beam expander to provide spatially broad beam of light

1 double slit with large slit separation

1 gas cell

1 vacuum pump for evacuating tube compartment

1 pressure gauge

Table 3.4: Apparatus for the Refractometer experiment.

3.2.2 The Refractometer

Determination of Refractive Index

The He:Ne laser, beam expander or diffuser, interferometer, and eyepieceare placed on the optical bench so that they are aligned vertically and hor-izontally. Adjust the interferometer arm lengths to be approximately thesame. Measure the tube length L, recording the data in the place providedin Sec. 3.3.2.

When the system is properly aligned, the interference fringes should ap-pear similar to those seen in the Michelson interferometer.

Connect the vacuum pump and the gauge to the cell. Adjust the rate ofchange of pressure in the compartment for ease of counting fringes. One wayto achieve this is to restrict the evacuation valve. Another method is to firstevacuate the compartment, record the gauge reading, and then turn off thevacuum pump, allowing air to leak slowly back into the compartment. Oneperson should count the fringes, while the other reads the gauge. Recordfringe count m and ∆p in the table provided. Note that m recorded in thedata table is the running count of the fringes passing the reference point. Aconvenient pressure interval to record m is 2cm of mercury, although you mayuse any interval that is convenient. If fringe shift is being recorded duringevacuation the first nonzero table entry will be for ∆p = 2cm, followed by∆p = 4cm, etc. If m is being recorded while the air is leaking back into thetube the first reading should be the maximum gauge pressure pm. The firstnonzero entry will be pm − 2cm, followed by pm − 4cm etc. It is suggested

Page 37: Lab

32 CHAPTER 3. MEASUREMENT OF REFRACTIVE INDEX

that you convert all pressure readings from cm of mercury to Torr usingthat 76.0cm of Hg = 760Torr. Take three sets of data, as indicated by thedata-table headings mtrial1, mtrial2, and mtrial3. Then find the average andstandard deviation for each ∆p, entering these values in the table, too. Plotthe averages of these readings and from the slope of the resulting straightline C

(2Lλ

)determine the value of C, which should be on the order of 10−7

(λ ≈ 632.8nm). Please note that you are to plot ∆p and not p.The absolute pressure within the evacuated compartment of the refrac-

tometer is needed for the final portion of this lab. This may be obtainedfrom the gauge pressure as follows (all units are cm of mercury):

pabs = patm − pgauge , (3.10)

where pabs is the absolute pressure of the compartment, patm is atmostphericpressure — which we’ll assume is 76.0cm of mercury or 760 Torr, pgauge isthe gauge pressure. Using this relationship and (n − 1) = Cpabs, calculate(n − 1) for various absolute tube pressures (in Torr) that were encounteredduring your experiment. A table has been provided for this purpose. Notethat n should increase with pressure.

Now plot (n − 1) versus pabs and extrapolate the resulting straight lineto pabs ≈ 760 Torr. Record the extrapolated value in the table referred to inthe preceding paragraph.

Page 38: Lab

3.3. EXPERIMENTAL DATA 33

3.3 Experimental Data

3.3.1 The Prism

trial 1 trial 2 trial 3 average std dev

left beam

right beam

prism angle A

Table 3.5: Determination of prism angle A.

λ (nm) trial 1 trial 2 trial 3 average std dev

706.5

667.8

587.5

500.1

492.2

471.3

447.1

Table 3.6: Determination of Dλ.

Plot n as a function of λ

Page 39: Lab

34 CHAPTER 3. MEASUREMENT OF REFRACTIVE INDEX

3.3.2 The Interferometirc Refractometer

Plot of m as a function of ∆p (Torr)

Page 40: Lab

3.3. EXPERIMENTAL DATA 35

Plot of (n− 1) versus pabs (Torr)

Page 41: Lab

36 CHAPTER 3. MEASUREMENT OF REFRACTIVE INDEX

∆p m statistics

cm Torr mtrial1 mtrial2 mtrial3 < m > σm

0 0 0 0 0 0 0

2 20

4 40

6 60

8 80

10 100

12 120

14 140

16 160

18 180

20 200

22 220

24 240

26 260

28 280

30 300

Table 3.7: Determination of m. L = .

Page 42: Lab

3.3. EXPERIMENTAL DATA 37

pabs (Torr) (n− 1)

760

Table 3.8: Absolute pressure versus refractive index.

Page 43: Lab

Chapter 4

Interference by the Division of

the Wavefront

4.1 Theory

There are two methods for producing two coherent sources. One methodinvolves the division of the wavefront as in the Young’s double slit experi-ment and the second method involves the division of the amplitude as in theMicheslson interferometer. In this experiemnt we study two arrangementsfor producing inteference by wavefront division and use these to determinethe wavelength of light. Refer to Chapter 13 of Jenkins and White or Section9.3 of Hecht for a discussion of Lloyd’s Mirror and Fresnel’s Biprism.

4.1.1 Lloyd’s Mirror

Lloyd’s mirror is a mirror or a glass plate used to produce interference fringesby superposing the light from a slit source with the light from the virtualimage of that slit source, as shown in Fig. (4.1). A part of the incident wave-front from source S1 is intercepted by the mirror and is reflected back. Thereflected wavefront overlaps the wavefront coming directly from the source S1

in the region shaded region and produces interference. The reflected wave-front appears to come from the virtual image S2 of S1. Thus in the shadedregion we have two source interference. Line EF is the perpendicular bisec-tor of S1S2. F is therefore equidistant from S1 and S2. If the distance y ofpoint P from F is small compared to D the angle θ is small and will be given

38

Page 44: Lab

4.1. THEORY 39

S2

P

glass plateor mirror

θθ

FEa

D∆

yS1

Figure 4.1: Formation of fringes in the Lloyd’s mirror setup.

by

θ =∆

a=

y

D, (4.1)

or

∆ = y( a

D

). (4.2)

Since the reflected beam suffers a phase change of π upon reflection fromthe mirror the location of the bright fringe of order m ( corresponding to2π∆/λ + π = 2πm) on the screen EF will be

ym =

(m− 1

2

a

D, (4.3)

where a is the separation between the sources S1 and S2, and D is thesource to screen distance. We can determine the wavelength of light frommeasurements of the fringe separation. If ym and yn denote the locations offringes of order m and n, respectively,

(2π ∆n

λ

)+ π = 2π n , (4.4)

(2π ∆m

λ

)+ π = 2π m , (4.5)

Page 45: Lab

40 CHAPTER 4. INTERFERENCE BY THE DIVISION OF THE WAVEFRONT

where λ is the wavelength of the light. Rearranging and subtracting theseequations, we find that

λ =∆n −∆m

n−m=

(yn − ym

n−m

)a

D, (4.6)

or

λ = (fringe width)( a

D

). (4.7)

Hence, to determine λ, we simply measure the “width” of an interfer-ence fringe (center-to-center distance between two successive fringes), theseparation a between S1 and S2, and the distance D.

The extra phase change of π experienced by the reflected beam may beverified by placing a screen flush with the end of the mirror or the glass plate.Since the real slit S1 and its virtual image S2 are equidistant from the linewhere the mirror and receptor touch, we might expect to see a bright fringedue to constructive interference. The dark fringe one sees instead verifies therelative π phase change suffered by the beam reflected by the mirror.

4.1.2 Fresnel’s Biprism

The arrangement from Fresnel biprism is shown in Fig. (4.2). The biprismconsists of two small angle prisms joined together at their bases. When awavefront from the source S is incident, the upper portion of the wavefront isrefracted downward and the lower portion is refracted upward. The refractedwavefronts appear to come from virtual sources S1 and S2. They overlapin the shaded region and produce inteference fringes. If a be the separationbeteween S1 and S2 and d+D be the source to screen separation, the locationof the m-th bright fringe is given by

ym = mλ

(D + d

a

), (4.8)

where y is measured from the center of the interference pattern. Just as inthe Lloyd’s mirror experiment, the wavelength of light can be determinedfrom measurements of fringe separation in the interference pattern.

4.2 Experimental Procedure

Make at least three measurements of each quantity and then de-termine its average and the standard deviation.

Page 46: Lab

4.2. EXPERIMENTAL PROCEDURE 41

S1

S2

S

d D

θ

α

a

a = d θ

yP

θ

Figure 4.2: An outline of Fresnel’s biprism setup.

Qty. Description

1 optical bench

1 mirror or glass plate to serve as Lloyd’s mirror

1 eyepiece with micrometer-motion crosshairs

1 one translating stage to mount eyepiece on bench

1 adjustable, rotating slit to accomodate alignment with mirror

1 mercury discharge tube with green filter (Pasco 9113)

1 sodium discharge tube

1 positive lens used to measure a only

Table 4.1: Apparatus for the LLoyd’s Mirror experiment.

Page 47: Lab

42 CHAPTER 4. INTERFERENCE BY THE DIVISION OF THE WAVEFRONT

Mercurylamp

filter

slit

Lloyd's mirror

green

eye piece

lens (for measurement of d only )

≈ 100 cm

Figure 4.3: Outline of the experimental setup.

4.2.1 Determination of λ

Arrange the apparatus on the optical bench as shown in Fig. 4.3. Rotate theslit in its mount so that it is parallel to the plane of the mirror. Study theeffect of changing the slit width and its orientation on fringe contrast andadjust the slit width and orientation for maximum clarity. Note that becauseof diffraction at the edge of the mirror, the fringe width varies a little nearthe mirror — do not use the first three or four fringes for measurements.D is the distance from the real slit to the cross hairs of the eyepiece andmay be measured directly on the optical bench. The cross hairs inside themicrometer eyepiece are located at the front face of the plate to which theeyepiece is attached. To determine the source separation a use a positivelens to form images of S and S ′ in the focal plane of the eyepiece. Thedistance between the two slit images viewed through the eyepiece can then bemeasured with the eyepiece micrometer. (Generally five turns of the eyepiecemicrometer is equal to 1.25mm of travel. Confirm this by calibrating theeyepiece micrometer.) If we call this distance R, we see that

a

R=

so

si, (4.9)

where so is the object distance (the distance from the real slit to the positivelens), and si is the image distance (the distance from the positive lens tothe focal plane of the eyepiece). Obviously, D = so + si. Determine a bothbefore and after measuring the interference fringe width, to ensure the setuphas not changed. The positive lens is to be placed on the bench only whenmeasuring a. Remove the positive lens from the bench when measuring theinterference fringe width.

Make one determination of the wavelength of the green line of mercury.

Page 48: Lab

4.2. EXPERIMENTAL PROCEDURE 43

α

d D

Primarysource

slit

Filter

eye piece

Figure 4.4: Outline of the experimental setup.

As always, make at least three measurements of the fringe width for eachwavelength. Be sure to note the uncertainty in all of your measurements,especially that of measuring the fringe width. Record these measurements inTable 4.2. You will find it convenient to count several fringes at a time whilemoving the cross hairs across the field of view. Then divide the distancetraveled by the number of fringes counted to obtain fringe separation. Theaccepted value for the green line of mercury is λHg ≈ 546.1 nm.

Also study colored interference fringes in this experiment by removing thegreen filter. Notice the order of colors in interference fringes and commenton their visibility.

Experimental arrangement for Fresnel’s biprism is shown in Fig. 4.4. Re-place the mercury source with a sodium lamp. Adjust slit orientation andwidth to obtain sharp fringes. Use micrometer to determine fringe separa-tion. Distances d and D are directly measurable. It remains now to measurethe linear separation a between the virtual sources S1 and S2. Since theprism angle α is small (∼ 1o) source separation is given by

a ≈ θ d. (4.10)

The angular separation θ between the virtual sources can be measured usinga spectrometer. If parallel light from the collimator illuminates both halves ofthe biprisms, two images are produced and the angular separation θ betweenthem is easily measured. Virtual source separation is also given by

a ≈ 2dα(n− 1) , (4.11)

where n is the refractive index of the prism. Prism angle α can again bedetermined by using a spectrometer. Use one of these methods to deter-mine a. The accepted value of wavelength for the yellow line of sodium isλNa ≈ 589.3 nm. Record your data in Table 4.3

Page 49: Lab

44 CHAPTER 4. INTERFERENCE BY THE DIVISION OF THE WAVEFRONT

4.3 Experimental Data

4.3.1 Lloyd’s Mirror

trial 1 trial 2 trial 3 average std dev

Rbefore

so,before

si,before

abefore

# of fringes

distance traveled

fringe width

Rafter

so,after

si,after

aafter

λHg

Table 4.2: Determination of the wavelength of mercury green line. D = .

Page 50: Lab

4.3. EXPERIMENTAL DATA 45

Draw a sketch and explain the technique used to determine a.

4.3.2 Fresnel’s Biprism

trial 1 trial 2 trial 3 average std dev

# of fringes

distance traveled

fringe width

a

λNa

Table 4.3: Determination of wavelength by Fresnel’s biprism. D = ; d = .

Page 51: Lab

Chapter 5

Michelson Interferometer

5.1 Theory

Michelson interferometer is the best known example of amplitude splittinginterferometers. An outline of this interferometer is shown in Fig. 5.1. Lightfrom the source S is divided into two beams of nearly equal amplitudes bya thin aluminum coating on the back side of the plane parallel plate M .The two beams are reflected back by two highly reflecting mirrors M1 andM2 and return to beam splitter M . The transmitted portion of the beamfrom M1 and the reflected portion of the beam from M2 overlap and produceinterference fringes. The compensator plate C is an exact replica (otherthan that it is not coated) of the beam splitter M and serves to equalizethe optical paths in the two arms for all wavelengths. This is not essentialfor producing fringes with monochromatic light, but it is indispensible whena white light source ( or some other source with short coherence length) isused. The mirror M1 translates on straight, parallel ways by turning a screwwhose pitch is quite accurately 1 mm. The mirror M2 is provided with tiltadjustment screws to make its image in M either parallel to M1 or at a smallangle with M1. The coated surfaces of the plates and mirrors must never betouched, as they cannot be cleaned without damaging them.

The interferometer is aligned by first making the optical paths MM1

and MM2 approximately equal with the aid of dividers or a ruler. Lookinginto the interferometer, one sees several images of a wire placed betweenthe diffuser screen and M . The strongest of these images will be due toreflections originating at the aluminized surface of M . The tilt screws on M2

are adjusted until the strong images coincide in the field of view. It should

46

Page 52: Lab

5.1. THEORY 47

M1

M2

BS CS

*lightsource

2d

dM1

M2

S1

S

S2

θ

2d cos θ

*

(a) (b)

Figure 5.1: (a) Ray paths in the Michelson interferometer ; (b) Two-source

interference arrangement equivalent to the Michelson interferometer.

then be possible to see interference bands. On refining the orientation ofM2 one obtains a set of concentric circles (Haidinger’s fringes) which donot expand or contract as the position of the eye is changed sideways orvertically. As M1 is translated, the interference rings expand or contract.Let d be the difference in the optical lengths of the two arms. Then the pathdifference between the two interfering beams at the center of the interferencepattern will be 2d because both beams traverse the inteferometer twice. Asd decreases interference rings shrink and each time d decreases by λ/2 a ringdisappears at the center. As d increases interference rings expand and eachtime d increases by λ/2 a ring grows out from the center. This provides asimple method for determining the wavelength of monochromatic light. If Nrings appear or disappear at the center as d changes by D, the wavelengthof light will be given by

λ =2D

N. (5.1)

Michelson inteferometer can also be used to determine the wavelength oftwo closely spaced spectral lines emitted by a source. If the incident lightcontains two wavelengths, each wavelength will produce its own interference

Page 53: Lab

48 CHAPTER 5. MICHELSON INTERFEROMETER

L

λ2

2

λ1

2

Figure 5.2: Relative locations of the fringes at the center for the two wave-

lengths λ1 and λ2 as d is varied. λ2 is assumed to be smaller of the two

wavelengths. The minima in the two patterns (or the maxima) coincide at

the center each time d changes by L.

pattern. As d is varied the two interference patterns will shift relative to oneanother. The intensity at the center will be simultaneously a minimum forboth patterns when

2d0 = p1λ1 , (5.2)

2d0 = p2λ2 , (5.3)

where p1 and p2 are two integers, and 2d0 is the optical path difference atthe center of the pattern. As d is increased the minima of the two interfer-ence patters will at first shift farther apart and then come closer togetheragain. The visibility of the total fringe pattern accordingly falls off and thenincreases to a second maximum. Figure 5.2 shows the relative positions ofthe minima (dark bands) in the two interference patterns as the position ofM1 is varied. The two patterns are seen to resemble a vernier and a scale.When the minima of the two patterns at the center coincide, the fringeshave high visibility; half way between the positions of maximum visibility,fringe visibility falls to a minimum and if both lines have equal intensity onesees a uniform illumination across the field of view because the minimum ofone fringe pattern coincides with the maximum of the other. At the nextoccurrence of maximum visibility,

2 (d0 + L) = (p1 + N )λ1 , (5.4)

2 (d0 + L) = (p2 + N + 1)λ2 , (5.5)

Page 54: Lab

5.1. THEORY 49

where L represents the displacement of M1 between successive visibility max-ima, and N is the number of fringes λ1 that pass by during the displacementof M1. By using Eqs. (2)-(5) we find that

2L = Nλ1 , (5.6)

2L = (N + 1)λ2 . (5.7)

From these equations we find that the ratio of the wavelengths is

λ1

λ2=

N + 1

N. (5.8)

If one wavelength is known, the other can be determined with the help of thisequation, in principle. Even when the two wavelengths cannot be determinedthe difference between the wavelengths can always be determined. From Eqs.(6) and (7) the wavelength difference is

λ1 − λ2 =2L

N(N + 1). (5.9)

Usually L can be measured with good accuracy. It is easier to locate visibilityminima than the maxima. The value of L is, therefore, best determinedby observing at least two successive minima of visibility and obtaining theaverage of many such observations. It is usually too tedious and not veryaccurate to determine N by direct count. For example for sodium D-linestypically 1000 fringes must be counted. We can cast Eq. (9) in a form thatdoes not involve N . Using Eqs.(5) and (6) on the right hand side of Eq.(9)to eliminate N and N + 1 we find

1

N(N + 1)=

λ1λ1

(2L)2≈

λ2rmav

(2L)2, (5.10)

where λrmav is the average wavelength. It can be determined by adjustingthe Michelson interferometer for maximum fringe visibility and counting thenumber of fringes that disappear or appear at the center as the path differenceis varied. Thus one may obtain a value for a wave-length interval which isnot easily resolved.

5.1.1 Coherenc Time and Coherence Length

Michelson interferometer can also be used to illustrate several other con-cepts important for interference. In all experiments involving interference we

Page 55: Lab

50 CHAPTER 5. MICHELSON INTERFEROMETER

have considered, the two interfering beams are always derived from the samesource. We find by experiments that it is impossible to produce interferencefringes from two separate sources, such as two incandescent filaments side byside. This is due to the fact that the light from any one source is not an in-finite train of waves. There are sudden changes in phase occurring on a veryshort time scale of the order of 10−8 s. Thus although interference fringesmay exist for such short intervals, they shift their position each time there isa phase change, with the result that no fringes at all will be seen. In essencethese sources produce wavetrains of average duration 10−8 s. This time iscalled the coherence time τc of the source. The average length, `c = cτc, ofthe wavetrains emitted by the source, is called the coherence length of thesource. Successive wavetrains even from the same source, in general, haveno definite phase relation with one another. Special arrangements are nec-essary to produce sources of light that have a definite phase relationship toone another. In Young’s experiment, Lloyd’s mirror, Fresnel’s biprism, andMichelson interferometer the two sources always have a point to point phasecorrespondence since they are both derived from the same source. If thephase of the light from a point in S1 suddenly changes, that of the light fromthe corresponding point in S2 will shift simultaneously. The result is thatthe difference in phase between any pair of points in the two sources remainsconstant, and so the interference pattern remains stationary. Sources thathave this point to point phase relation are called coherent sources.

Even when there is point to point phase correspondence between sourcesone further condition must be satisfied if stationary fringes are to be observed.The path difference between the light coming from the two sources must notexceed the coherence length (average length of the wavetrains emitted by thesources) of the light emitted by the sources. If the path difference exceedsthe coherence length the two interfering waves are derived from differentwavetrains that have no definite phase relation to one another. Cohertencelength of lasers can be several meters. On the other hand for mercury andsodium lamp it is only about 10−2 m. Using Michelson interferometer we canactually measure the length of the wavetrains emitted by a source !

5.1.2 Polarization Dependence of Interference

Interference of light also depends on the vector nature (polarization) of light.Thus two beams having orthogonal polarization will not produce interference.This is easily illustrated by placing two linear polarizers in the two arms of

Page 56: Lab

5.2. EXPERIMENTAL PROCEDURE 51

Qty. Description

1 Michelson interferometer

1 sodium discharge tube

1 Helium-Neon laser

1 white-light source

Table 5.1: Apparatus for the Michelson interferometer experiment.

the interferometer.1 Keep one polarizer fixed and rotate the other polarizer.It will be seen that no interference fringes occur when the transmission axesof the polarizers are orthogonal. On the other hand, sharp fringes with goodcontrast are produced when the transmission axes are parllel.

5.2 Experimental Procedure

Make at least three measurements of each quantity and then deter-mine the average and the standard deviation of that quantity.

A simple technique for adjusting the angles of mirrors M1 and M2 is tolook into the output port of the interferometer while holding a thin objectlike a mechanical pencil between a white-light or sodium light source and theinput port of the interferometer. “Ghost” images of the object will be seenunless the mirrors are aligned correctly. Be sure to place the tip of the objectin the middle of the field of view.

5.2.1 Wavelength of HeNe Laser

In this step of the experiment, we use a HeNe laser as the light source, castingthe fringes formed by the interferometer onto the wall or perhaps a sheet ofsome sort. DO NOT LOOK DIRECTLY INTO THE INTERFER-OMETER ITSELF.

These fringes are distinct enough that they may be counted as the pathlength of the Michelson interferometer is changed. While one person carefullyturns the micrometer to move mirror M1 a distance DHeNe, the other personcounts the number of fringes NHeNe that either collapse into the central bright

1If the source produces polarized light you may need only one polarizer.

Page 57: Lab

52 CHAPTER 5. MICHELSON INTERFEROMETER

spot or “grow” out from it. These data are recorded in Table 5.2, whichincludes entries for the initial and final locations of the mirror M1.

Insert a linear polarizer in each arm. Keep one polarizer fixed and rotatethe other polarizer until a fringe pattern with good contrast is obtained.Record this position of the polarizer. Rotate this polarizer slowly and noticethe brightness of the fringe pattern. When the fringe pattern vanishes recordthe polarizer reading. Compute the angular separation between the twopositions of the polarizer.

Explain your observations. Estimate the coherence length ofthe laser by moving one mirror and finding the distance at whichfringes disappear.

5.2.2 Wavelength of Sodium D Lines

Next we use the sodium discharge lamp as a light source. Light from sodiumlamp consists of two closely spaced wavelengths. We will determine thedifference between the two wavelengths. First we determine the averagewavelength. Translate the movable mirror until you get fringes with goodcontrast. This happens when the minimum (or the maximum) of the twofringe patterns coincide at the center. Determine the average wavelengthby counting, say 50, fringes that disappear at the center and recording thecorresponding mirror displacement. Wavelength is then given by Eq. (1). Todetermine the wavelength difference between the two sodium D lines measurethe distance LNa between the fringe visibility minima as described in Sec. 5.1.Note that the field of view will be evenly illuminated when the visibility ofthe fringes is at a minimum. Fringe visibility maxima correspond to themost distinct fringes. It is easier to determine when the field of view is freeof fringes than it is to decide when the fringes are most distinct.

While looking into the output port of the interferometer, carefully rotatethe micrometer until you recognize a fringe visibility minimum. Record themicrometer setting in Table 5.4 and then translate mirror M1 until you arriveat a second minimum, record that mirror position, and repeat for a thirdand final time. Now you may determine a value for LNa using these threemirror positions. Repeat this procedure for a total of three trials, and thendetermine the average and standard deviation of LNa. Using the formulaegiven in Sec. 5.1, determine λD2 and compare it with the accepted value 589.0nm.

Estimate the coherence length of the sodium lamp.

Page 58: Lab

5.3. EXPERIMENTAL DATA 53

5.3 Experimental Data

trial 1 trial 2 trial 3 average std dev

initial M1

final M1

dHeNe

NHeNe

λHeNe

Table 5.2: Wavelength of HeNe laser. The number of fringes N that “pass

by” when mirror M1 is displaced by d allows a determination the wavelength

via λ = 2d/N .

trial 1 trial 2 trial 3 average std dev

Initial setting Pi

Final setting Pf

Angular separation |Pf − Pi|

Table 5.3: Polarization dependence of interference.

The coherence length of the laser was found to be `c ≈

Page 59: Lab

54 CHAPTER 5. MICHELSON INTERFEROMETER

trial 1 trial 2 trial 3 average std dev

min1

min2

min3

dNa

NNa

λD2

Table 5.4: Wavelength of Sodium D lines. Distances dNa separating fringe

visibility minima are recorded. Then NNa = 2LNa/λD1.

Page 60: Lab

Chapter 6

Polarization of Light

Light waves are transverse electromagnetic waves. This means the electricand magnetic field vectors ~E (V/m) and ~B (T) lie in a plane perpendicular tothe direction of propagation characterized by the energy flux density vector(Poynting vector) ~S (W/m2). The relative orientation of ~E, ~B and ~S is givenby

~S =~E × ~B

µ0. (6.1)

Given the direction of propagation and the direction of the electric vector,the direction of the magnetic vector is determined. Polarization of light isrelated to the orientation of the electric field. If the electric vector pointsalong a fixed direction, light is said to be linearly polarized. Many otherpossibilities exist. If the electric vector rotates as a function of time we

E

B

S

Direction of propagation

Figure 6.1: Relative orientation of the electric and magnetic fields and the

Poynting vector for an electromagnetic wave.

55

Page 61: Lab

56 CHAPTER 6. POLARIZATION OF LIGHT

have elliptically polarized wave. If the orientation of the electric field vectorchanges at random, the wave is said to be unpolarized. What state of polar-ization is produced depends on the source. In laboratory, of course, we canproduce any state of polarization. In this experiment we will study differentstates of polarization and how they are produced and detected.

6.1 States of Polarization

Consider a plane wave propagating in the z direction. Then the electric fieldvector lies in the x− y plane. We can then write its x and y components inthe form

Ex = E01 cos(kz − ωt + φ1), (6.2)

Ey = E02 cos(kz − ωt + φ2). (6.3)

Here ω is the angular frequency of the wave, k = nω/c is the wave number,n is the refractive index of the medium and φ1 and φ2 are the initial phasesfor the two components of the wave. Different states of polarization arerealized for different choices of the phase difference φ = φ2 − φ1 and therelative magnitudes of E01 and E02. At all times the relation ~E × ~B = µ0

~Sis satisfied.

6.1.1 Unpolarized light

Light from natural sources, such as the sun or an incandescent solid, is unpo-larized. For natural light φ1 and φ2 vary over a time scale of the order of 10−8

s in an uncorrelated manner. Therefore, the phase difference φ also variesat random. There are fluctuations in the amplitude as well. The electricfield takes all possible orientations in the x − y plane and on average, theamplitudes of the x− and y− components are equal E01 = E02 ≡ E0. Sucha wave can be represented by two orthogonal linear polarizations of equalamplitude but random phases.

Ex = E0 cos[kz − ωt + φ1(t)], (6.4)

Ey = E0 cos[kz − ωt + φ2(t)] , (6.5)

where φ1(t) and φ2(t) are random functions of time.

Page 62: Lab

6.1. STATES OF POLARIZATION 57

6.1.2 Polarized Light

Even when φ1 and φ2 vary at random, the phase difference φ = φ2(t)−φ2(t)could still be constant provided that φ1 and φ2 vary in a completely correlatedmanner. If φ = φ2(t) − φ1(t) is a constant we can write the electric field,without the loss of generality, as

E1 = E01 cos(kz − ωt), (6.6)

E2 = E02 cos(kz − ωt + φ). (6.7)

Let us consider some special values of the phase difference φ in the interval[−π, π]. For φ = 0 we have a wave linearly polarized in a direction makingan angle θ = tan−1(E02/E01) with the x− axis. For φ = π we have a linearlypolarized wave in a direction making an angle θ = − tan−1(E02/E01) ≡π − tan−1(E02/E01) with the x− axis. For E02 = 0 we have a wave

~E1 = xE01 cos(kz − ωt) , (6.8)

polarized in the x-direction and for E01 = 0 we have a wave

~E2 = yE02 cos(kz − ωt) , (6.9)

polarized in the y− direction. Waves in Eqs. (6.8) and (6.9) are said to have

orthogonal polarizations because ~E1 · ~E2 = 0. In general, a wave of amplitudeE0 linearly polarized in a direction making an angle θ with the x−axis canbe expressed as a superposition of two orthogonal linearly polarized waves as

~E = (xE0 cos θ + yE0 sin θ) cos(kz − ωt). (6.10)

If the wave amplitudes are equal, E01 = E02 = E0, and φ = ±π/2 ≡ ±90o

the electric field components can be written as

E1 = E0 cos(kz − ωt) (6.11)

E2 = ∓ E0 sin(kz − ωt) . (6.12)

In this case the x− and y− components of the field satisfy

E12 + E2

2 = E20 (6.13)

which is the equation of a circle with radius E0. Thus the tip of the electricfield vector traces a circle with angular frequency ω. The rotation of theelectric field vector is counterclockwise for φ = π/2 and clockwise for φ =

Page 63: Lab

58 CHAPTER 6. POLARIZATION OF LIGHT

−π/2 for an approaching wave. The former is referred to as left circularlypolarized (positive helicity) and the latter as right circularly polarized wave(negative helicity).

For values of φ and E01 and E02 other than those discussed in the preced-ing two paragraphs we have an elliptically polarized wave because the tip ofthe electric field vector traces an ellipse in the x− y plane. For example, forE01 > E02 and φ = ±π/2 the electric field vector rotates in an ellipse withmajor axis along the x− direction and minor axis along the y− direction.Similarly for E01 = E02 = E0 and φ = π/4 we have a left elliptically polarizedlight with the axes of the ellipse rotated 45o with respect to the co-ordinateaxes. Since straight line and circle are special cases of an ellipse, ellipticalpolarization is the most general state of polarization of light. We now studyhow these states are produced and detected.

6.2 Experimental Procedure

CAUTION : Handle all optical elements by their frames. Do not touchany optical surfaces. Linear polarizers are marked by the direction of theirtransmission axis and wave plates ( also referred to as retarders) by their fastaxis.

6.2.1 Linear Polarization

Mount a laser close to one end of the optical table so that the beam istraveling parallel to and on top of a line of tapped holes in the table. Ifnecessary, use a beam expander (3×) to obtain a beam of 2-3 mm diameter.Mount two mirrors M1 and M2 at the two far corners of the table as shownin the diagram.1 Adjust mirror heights until the beam is incident very nearlyat the centers of the mirrors and it is traveling parallel to the surface of theoptical table.

Place the detector about 100 cm from the second mirror M2 and makesure that the detectors intercepts the whole laser beam. Insert a polarizer P1

between M2 and the detector so that the beam is incident normally on thepolarizer. Rotate the polarizer and observe the variation of light intensityas recorded by the detector. From your observations what can you concludeabout the polarization of the light from the laser? Leave this polarizer P1

1These mirrors are not essential; they do provide better control over beam height and

beam pointing.

Page 64: Lab

6.2. EXPERIMENTAL PROCEDURE 59

Laser

BeamExpander

M1

M2

P1 P2

Detector/Meter( Power Meter )

Figure 6.2: Experimental setup for producing and analyzing linear polariza-

tion.

with the notch in the polarizer frame pointing vertically up. Mount a secondpolarizer P2 (it will be called the analyzer) in a rotation stage and place itin line with the laser beam between the first polarizer P1 and the detector.Rotate the analyzer P2 and monitor the intensity falling on the photodetec-tor until a maximum in intensity is reached. Record both the intensity andthe angular position of P2. This angular position of P2 will be the referencefor measuring angles. Rotate the analyzer in 5o increments between 0o and180o. For each setting of P2, record the angular displacement θ from thereference position, and the output of the detector as measured by the volt-meter. The output of the detector is proportional to the irradiance of thelight (Watts/m2) falling on the detector which in turn is proportional to thesquare of the amplitude of the electric field transmitted by P2.Maximum intensity I(θ = 0) ≡ I0

Plot the results of your measurements and compare them to the Law ofMalus

I(θ) = I0 cos2 θ. (6.14)

Q1 : Derive this in your report. In your comparison you may find itconvenient to plot I(θ)/I0. You may have to adjust your plots to make thecomparison, but you should justify any adjustment in your notebook.

Q2 : What does this tell you about the nature of light ? Whatis the state of polarization of light after it emerges from the firstpolarizer? Record your comments on the comparison in your laboratorynotebook.

Page 65: Lab

60 CHAPTER 6. POLARIZATION OF LIGHT

Laser

BeamExpander

M1

M2

P1

Rotation Stage

Detector/Meter( Power Meter )

Figure 6.3: Polarization in reflection.

6.2.2 Brewster Angle (Optional)

A polarizer is one way of producing linearly polarized light from an unpo-larized beam. Reflection can also produce polarized light. The degree ofpolarization depends on the angle of incidence. At one particular angle,called Brewster angle (polarization angle), reflected light is completely po-larized perpendicular to the plane of incidence. At this angle of incidence,the reflection coefficient is zero for light polarized parallel to the plane ofincidence.

To study polarization in the process of reflection, remove the analyzerP2 and mount a lens holder in a rotation stage. Attach the rotation stageto the table such that the laser beam passes through the center of the lensholder. Tape a microscope glass slide to the lens holder so that the slide isheld firmly in place and the beam does not pass through the tape.

Rotate the lens holder so that the beam reflected from the slide is sentback along the input beam. Adjust the screws at the back of the lens holderif necessary. You may want to use an index card with a hole in it to set thebeam. Set the rotation stage to 0o and tighten the screw on the lens holderto fix it in the holder.

Rotate P1 so that its transmission axis is horizontal (notch horizontal).This means the light incident on the microscope slide is polarized horizontallyand the electric vector lies in the plane of incidence. Remove the detectorand attach an index card to it since you are going to have to follow the beamon the table.

Page 66: Lab

6.2. EXPERIMENTAL PROCEDURE 61

Turn the rotation stage away from 0o and record the reflection. Observehow the intensity depends on the angle of incidence. Locate accurately anymaximum or minimum. By successive approximations, bring the stage toproduce the extremum. (You may find that you can improve the minimumby slightly tweaking the input polarizer by a small amount) Record the angleof the rotation stage and determine the angle between the beam and thenormal to the reflecting surface.

What is the refractive index of the material of the microscopeslide ? Make a plot of the irradiance as a function of the angle ofincidence.

Rotate the input polarizer P1 to the orthogonal position so that the elec-tric vector is perpendicular to the plane of incidence. Observe the angulardependence of the irradiance of the reflected beam. Make a plot of irradianceas a function of the angle of incidence. Compare the plots for parallel andperpendicular polarizations and comment on the similarities and differences.

Q3 : When is the reflected light completely polarized? Whatis the state of polarization of the transmitted light then? Makepredictions and check them experimentally.

6.2.3 Circular Polarization

Polarization of light can be used to control the passage of light through anoptical system and to impress information on a light wave modulating theamount of light passing through a birefringent material.

In this project a birefringent material will be used to change the polar-ization of the laser. Using a quarter-wave plate and a polarizer we will buildan optical isolator. When we add a second quarter-wave plate, a polarizationrotator results.

Mount the beamsplitter (BS) in a lens holder and place the unit about10 cm to the right from mirror M2. Orient the beamsplitter such that thebeamsplitter surface is inclined at 45o to the beam.

Place a white index card to monitor the reflection (of the return beam)from the beam splitter. Mount a polarizer P1 on the optical breadboard inline with the laser beam and several centimeter to the right of the beam-splitter (BS). Set the polarizer with its transmission axis up. Adjust thepolarizer mount slightly so that the reflection off this polarizer can be seenon the index card.

Mount a second polarizer P2 into a rotation stage about 15 cm to the rightof the first polarizer. Rotate the second polarizer P2 until it completely blocksthe light from the first polarizer (i.e. Polarization axes crossed). Adjust

Page 67: Lab

62 CHAPTER 6. POLARIZATION OF LIGHT

Laser

BeamExpander

M1

M2

P1 P2BS QWPDetector/Meter( Power Meter )

Index Card

Figure 6.4: Experimental arrangement for producing circular polarization.

the second polarizer so that the reflection from this can be recognized as aseparate beam on the index card.

Insert a quarter-wave plate QWP into a rotating mount and place theassembly between the two polarizers P1 and P2. Slowly rotate the quarter-wave plate until the output through the second polarizer is a maximum.

Rotate the second polarizer and observe the variation of the output irra-diance.

Q4 : What state of polarization have you produced ? Whatare the states of polarization at the output of P1, QWP, and P2 ?How would you distinguish between this state and an unpolarizedbeam? The quarter-wave plate has converted the linear input beam to acircularly polarized beam.

6.2.4 Optical Diode : Optical Isolator

Remove P2 and replace it with a mirror M3 that reflects light back on toitself (Fig. 9-2). Observe the light reflections on the index card. There willbe surface reflections off the surfaces of the polarizer and the quarter-waveplate, but there will be no strong reflection from the mirror because themirror reverses the circularly polarized light and on second passage throughthe quarter-wave plate, the beam is again linearly polarized but at rightangles to the original polarization. When the beam reflected by M3 hitsthe polarizer P1 again, it is absorbed. This is equivalent to saying that theoutgoing beam has been “isolated” from reflections after the quarter-waveplate.

Page 68: Lab

6.2. EXPERIMENTAL PROCEDURE 63

Laser

BeamExpander

M1

M2

P1 M3BS QWP

Index Card

Figure 6.5: Experimental arrangement for an optical diode (optical isolator).

Q5 : How would you test that this is indeed what is happening?

6.2.5 Polarization Rotator

Start with the arrangement for producing circularly polarized light with asingle quarter-wave plate. Hold a second quarter-wave plate between thecrossed polarizers without disturbing the orientation of the first. Rotate thesecond quarter-wave plate until the light passing through the second polarizerP2 is a maximum. You have now created a half-wave plate which rotates theinput polarization by 90o. Instead of two QWP’s you may also use a singlehalf wave plate (HWP).

Q6 : How would you check this out? The device that you haveconstructed is a polarization rotator.

6.2.6 Elliptically Polarized Light

Elliptically polarized light is the most general type of polarized light. Con-sider a set of crossed polarizers. A QWP is introduced between them withits optic axis at an angle θ to the transmission axis of the polarizer. We canresolve the light incident on the QWP into components parallel and perpen-dicular to the optic axis of the QWP. After passing through the QWP thetwo components have a relative phase difference of π/2. Light incident on

Page 69: Lab

64 CHAPTER 6. POLARIZATION OF LIGHT

Laser

BeamExpander

M1

M2

P1 P2BS

QWP1 QWP2 or

a Single HWP

Index Card

Detector/Meter( Power Meter )

Figure 6.6: Experimental arrangement for polarization rotator.

the analyzer is described by

Ee = A cos θ cos ωt (6.15)

Eo = A sin θ(cos ωt− π/2) (6.16)

Show that this represents an elliptically polarized light. Determine the prin-cipal axes of the ellipse by rotating the analyzer. Start with the QWP opticaxis aligned with the transmission axis of the polarizer. Increment the anglebetween the QWP axis and P1 in steps of 10o and record how much do youhave to rotate the analyzer from its crossed position to find the major andminor axes of the polarization ellipse.

Q7 : How would you determine the direction of rotation of theelectric vector?

Finally take the unknown retarder and place it between the crossed po-larizers with its axis making angles of 10o, 20o, 30o, 40o · · · 80o with P1.Determine the maximum or minimum by rotating the analyzer. Find theretardation introduced by the unknown plate.

6.2.7 An Amusement

Starting with the arrangement in the previous section, rotate the first po-larizer by some specific angle, say 10o, and then determine the amount bywhich the second polarizer must be rotated to extinguish the beam.

Page 70: Lab

6.2. EXPERIMENTAL PROCEDURE 65

Q8 : You will find that the analyzer must rotate through twicethe angle of the initial polarizer. Is this really true? What is thestate of polarization in this case after P1, QWP1, and QWP2?

Page 71: Lab

66 CHAPTER 6. POLARIZATION OF LIGHT

Ang. Disp. (θ) Intensity (Volts) I/I0 cos2 θ

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

75

80

85

90

Table 6.1: Law of Malus

Page 72: Lab

Chapter 7

Resolving Power of a Grating

and Limit of Resolution

7.1 Theory

For a discussion of diffraction gratings, refer to Chapter 17 of Jenkins andWhite or Chapter 10 of Hecht and Zajac.

The resolving power of a grating or a prism is given by

R ≡λ

δλ, (7.1)

where λ is the wavelength resolved from the wavelength λ + δλ. Consideran assembly of N slits with a separation of d between each slit (d is referredto as the grating constant). If the mth maximum in intensity occurs at anangle θ, then the extreme optical path difference between waves leaving slit1 and the Nth slit is mNλ. The first minimum occurs at θ + δθ when theextreme path difference is mNλ + δλ since waves from the Nth slit will beλ/2 wavelengths out of phase with the wave from the N/2 slit.

The Rayleigh criterion for resolution requires that if two sources are justresolved, the central maximum of the diffraction pattern from one source fallsupon the minimum of the other. Then if we are to resolve a second compo-nent differing in wavelength by δλ, it must produce a maximum at the firstminimum described above. For the ray of wavelength λ + δλ, mN (λ + δλ)must be the extreme path length difference for a maximum so that

mNλ + λ = mN (λ + δλ) . (7.2)

67

Page 73: Lab

68CHAPTER 7. RESOLVING POWER OF A GRATING AND LIMIT OF RESOLUTION

Qty. Description

1 transmission grating (2000 lines/inch or 300 lines/mm)

1 student spectrometer focused for parallel light

1 vernier calipers for adjusting effective size of grating

1 mercury vapor source

Table 7.1: Apparatus for the Resolving Power of a Diffraction Grating ex-

periment.

This equation then leads to the following expression for the resolving powerof the diffraction grating

R =λ

δλ= mN . (7.3)

7.2 Experimental Procedure

Make at least three measurements of each quantity and then deter-mine the average and the standard deviation of that quantity.

7.2.1 Resolving Power of a Grating

Mount the transmission grating so that the incident light is normal and thecaliper lies between the collimator and the grating. The caliper jaws shouldbe parallel to the lines of the grating and as close to the grating as possible.The jaws of the caliper provide a variable aperture which determines thenumber of lines on the grating which are used. With the calipers fully openthe spectral orders are scanned. You should be able to see several orderspresent on either side of the central image. Set the scope on the yellow linesof the first orders and vary the calipers until the lines are resolved. Recordthe order and the width of the slit. Repeat, going to higher orders, until thelast observable order is resolved for both sides of the central image. Recordyour measurements in Table 7.3. Table 7.2 provides the wavelengths of themercury spectrum.

Page 74: Lab

7.3. LIMIT OF RESOLUTION 69

Color Wavelength (nm)

Red 623.4

Yellow 579.0

Yellow 577.0

Green 546.1

Blue 435.8

Violet 407.8

Violet 404.7

Table 7.2: The mercury spectrum.

7.2.2 Determination of the Grating Constant d

Measure the angular position of several known lines of a given light sourceand record your measurements in Table 7.4. Knowing the angular position,the order, and the wavelength, d may be determined from

mλ = d sin θ . (7.4)

7.3 Limit of Resolution

Set the the long vertical filament lamp securely on the table. Place a redfilter in front of the lamp followed by the resolution source object (RSO) slitplate as shown. The RSO consists of rows of slits. The long dimension ofthe resolution object plate should be vertical. The top row has slits withlargest separation. Slits in each successive row have half as much separationas the row above it. Make sure you do not place the filter too close to thelamp. Stand back about 3-4 m from the plate. Hold the slide with the slitfilmslide with the long dimension vertical such that the CAL monogram is in theupper right corner. Hold the film close to the eye (almost touching the eye)and look through the slide at the resolution source object. In the first columnon the slitfilm you will see single slits of different widths. Choose a slit withsome aperture size, say D, and focus your attention on one particular row ofobject slits on the RSO plate. Move back and forth until you are just ableto resolve two adjacent slits. Determine the distance ` where you are justable to resolve two adjacent slits. Compute the angle subtended at the eyeby two adjacent slits. Compare this to λ/D.

Page 75: Lab

70CHAPTER 7. RESOLVING POWER OF A GRATING AND LIMIT OF RESOLUTION

D

l

Lamp

Filter

Resolution SlitObject Plate

SlitfilmApertures

Figure 7.1: Limit of resolution set up

Looking through the same aperture replace the red filter by blue. Whatqualitative change do you see? Move back and forth and determine thedistance ` where you are just able to resolve two adjacent slits. Once againcompute the angle subtended at your eye by two adjacent slits. Comparethis to λ/D.

Change D and repeat the steps outlined above. Choose a different rowof slits and repeat the steps in the previous two paragraphs.

Q1 : For a fixed D and ` which color allows you better resolu-tion? How does resolution change with `? Your eye has a finiteaperture too (the pupil). Are you limited by the pupil or by theslitwidth in your experiment? How do you decide? What do youlearn from this experiment?

Remove the RSO. You can now use the slitfilm to observe dozens ofdiffraction patterns. Investigate at least two of these qualitatively and includethem in your report.

Page 76: Lab

7.4. EXPERIMENTAL DATA 71

7.4 Experimental Data

Caliper Width

order m trial 1 trial 2 trial 3 average std dev N

1

2

3

4

5

6

Table 7.3: Resolving Power of a Grating. Using yellow lines of of each spectral

order to determine R. Caliper width is recorded when lines are just resolved.

Effective number of slits N may be obtained from this measurement and

known number of slits per inch.

Q2 : Rederive the relationship for resolution using a diagramshowing all quantities. Compute the resolution necessary to re-solve the mercury 579.07 nm and 576.98 nm doublet. Comparethe observed and the computed resolutions.

Page 77: Lab

72CHAPTER 7. RESOLVING POWER OF A GRATING AND LIMIT OF RESOLUTION

Angular Position

λ order m trial 1 trial 2 trial 3 average std dev d

Table 7.4: Determination of the grating constant d. d may be determined

from mλ = d sin θ.

Page 78: Lab

Chapter 8

Image Formation and Spatial

Filtering

8.1 Theory

For discussions of the diffraction theory of image formation and/or spatialfiltering, see Sec. 28.12 of Jenkins and White or Sec. 14.1 of Hecht andZajac.

Every image is a synthesis of a diffraction pattern. The purpose of anoptical instrument is to synthesize a diffraction pattern to form an images ofan object. The image that you see in the eyepiece of an optical instrument isalways an accurate synthesis of some diffraction pattern — but not necessar-ily the diffraction pattern of the real object you are trying to see. The finalimage delivered by these instruments is critically dependent on how much ofthe real object’s diffraction pattern is utilized. Since optical elements havefinite apertures no instrument utilizes all of an object’s diffraction pattern.It follows that the images formed by optical instruments are to some extentfalse. This includes images formed by the naked eye.

In this experiment we study how the image of a real object may be falsifiedwhen only a part of its diffraction pattern is utilized. For our investigations,we will use a simple object that has a repetitive structure — a piece of screenmesh. The diffraction pattern of this object is simple and repetitive.

The form of the above pattern may be understood by considering thewire mesh as a pair of diffraction gratings with their lines crossed at 90.The first grating of the pair provides, say, the familiar horizontal row ofdots. The second grating disperses each of these dots into a vertical column

73

Page 79: Lab

74 CHAPTER 8. IMAGE FORMATION AND SPATIAL FILTERING

a1a2 a2a1

b2

b2

b1

b1

0

c1

c1

c2

c2

d2

d2

d1

d1

Screen Diffraction Pattern

Figure 8.1: Diffraction pattern of a wire mesh.

of dots, thus building up the pattern of Fig. 8.1.

Consider now the effect of suppressing certain parts of the “full” diffrac-tion pattern of Fig. 8.1, and using a lens to synthesize the remaining ordersinto an image. We first use a mask that accepts only the orders an of Fig. 8.1,as shown in Fig. 8.2. Since an is the diffraction pattern of a set of verticalwires, we will see in our eyepiece nothing but a field of vertical wires — thehorizontal wires will have wholly disappeared. Similarly, when we pick offand synthesize the vertical orders bn we will see only horizontal wires. Mostinteresting is the case in which we pick off only diagonal orders, say cn. Thistime we will see a very real-looking set of wires at 45 — wires which do not,of course, exist in the real object. Should we add the orders dn with a maskhaving an X-shaped cutout, the eyepiece would reveal a very convincing pieceof mesh, but rotated 45 with respect to the real mesh.

The above is a preview of what to expect, so you will know if you haveset up the apparatus correctly. Later in the course of this experiment youwill use masks which will admit the central maximum only, central maximumplus second-order only, first-order only, and various other combinations. Inthese cases you will be asked to anticipate what the image will look like,and you will have to explain what you see in the eyepiece, on the basis ofdiffraction theory.

The ideas of this experiment, although developed in a simple and qualita-tive way, serves as an introduction to the important topic of spatial filteringand Fourier Optics. The diffraction pattern of Fig. 8.1 is the two-dimensional

Page 80: Lab

8.1. THEORY 75

Diffraction Pattern Image

Figure 8.2: Spatial filtering.

Fourier transform of the object, the object being viewed as a mathematicalfunction. The image seen by the eyepiece is, in turn, the Fourier transformof the diffraction pattern. In this manner information is transmitted fromthe object to the eye. If all the information were transmitted, the imagewould be a perfect representation of the object. However, due to the fi-nite size of optical apertures, the image function is not strictly proportionalto the object function, but is modified in certain respects. The totality ofthese modifications is called the optical transfer function (OTF) of the sys-tem. Anything done deliberately to alter the OTF is called spatial filtering.Spatial filtering can often be used to “falsify” an image in useful ways, forexample, filling in the spaces between scan lines of a TV picture, or removingscratches from a transparency. The filtering may be done either optically,by suitable masks (as we shall do in this experiment), or analytically, byappropriate programming of a computer.

Page 81: Lab

76 CHAPTER 8. IMAGE FORMATION AND SPATIAL FILTERING

Qty. Description

1 optical bench with component carriers

1 eyepiece

1 microscope slide in holder for mounting on bench

1 candle and matches for soot-mask microscope slide

1 lens of 15 to 25 cm focal length

1 piece of screen mesh, 30 to 100 wires per cm

1 flashlight and mount

1 white card or screen

1 variable slit on rotating mount

Table 8.1: Apparatus for the Spatial Filtering experiment.

8.2 Experimental Procedure

Record your observations in the format suggested in Sec. 8.3. Step 1.Set up the equipment approximately as shown in Fig. 8.3. For the moment,put a white card in the holder intended for the microscope slide (“mask”).Focus the diffraction pattern of the mesh at the plane of the mask and thendraw this pattern as indicated in Sec. 8.3.

Step 2.Remove the white card and – without disturbing the bulb-lens-mask spacing– focus an image of the mesh just in front of the eyepiece. Move the eyepiece

eye piece eyemasklenswire mashlamp iris

≈ 60cm ≈ 20cm ≈ 20cm 80cm≈

Figure 8.3: An outline of the experimental setup.

Page 82: Lab

8.2. EXPERIMENTAL PROCEDURE 77

back and forth until you can see a clear image of the mesh in it. We are nowready to have some fun with spatial filtering.Step 3.Mount an adjustable slit in the holder for the mask and move the carrier untila crisp diffraction pattern appears on the slit. Open the slit and adjust itsorientation so that only the orders an of Fig. 8.1 pass through the slit. Viewthe result in the eyepiece. If successful, you should see clear, distinct, andreal-looking vertical lines. If not, try adjusting the slit width and orientation.Light coming around the slide may be a problem, in which case you shoulduse a cardboard screen with a hole just big enough to admit the diffractionpattern, between the lens and the slide. Insert this “stop” in the opticalpath during the actual viewing through the eyepiece only, otherwise, it willinterfere with your work at other times.Step 4.Reposition and use the slit for bn and the cn and observe the effects on theimage of the mesh. Draw in Sec. 8.3 the cn mask and the pattern it produces.Make sure that you preserve the relative orientation of the mask and whatyou see in the eyepiece.Step 5. (Optional)Now you are ready to try some effects not previewed in Sec. 8.1. In every casebelow, make a drawing anticipating what you will see, before you actuallyperform the experiment. Afterwards, draw what you did see, together withthe mask that produced it, in their correct spatial relation. Use the adjustableslit or the appropriate slides for

(i) The central maximum, 0,(ii) 0 and a2,(iii) a1 only,(iv) 0 and c1,(v) 0, c1, and d1.

Compare the spacing of the wires in cases (ii) and (iii) with each otherand with the spacing seen when the mask is not present. Make as gooda quantitative estimate as possible. One way to do this is to look throughthe eyepiece and simultaneously view with the other eye an illuminated scale;or arrange a crude reticule at the focus of the eyepiece.Step 6.Make an oblong opening for a1 and 0. While observing the view in the eye-piece, move a paper clip slowly from left to right against the back (uncoatedside) of the slide. When the wire of the paper clip blocks only the central

Page 83: Lab

78 CHAPTER 8. IMAGE FORMATION AND SPATIAL FILTERING

maximum, a notable change should occur. Describe this with a diagram inSec. 8.3. Now enlarge the oblong until it includes all orders an and 0. Oncemore block 0 with the paper clip. What happens this time?Step 7.Install the variable slit in place of the glass mask and its holder. Focus thediffraction pattern carefully at the plane of the slit. Arrange the centering ofthe pattern so that the central maximum just falls into the slit, both whenthe slit is vertical and when it is horizontal. Now observe the mesh throughthe eyepiece while rotating the slit from horizontal to vertical or vice versa.You will certainly see 45 wires, and probably also wires inclined at otherangles due to higher-order spectra being captured. Next, view one set of lineswith the slit’s orientation fixed, and gradually widen and narrow the slit tosee the mechanism by which one set of wires is discriminated against.

Page 84: Lab

8.3. EXPERIMENTAL DATA 79

8.3 Experimental Data

Step 1. Draw the diffraction pattern of mesh as seen on white card in placeof mask.Step 4. Draw the cn mask and the resulting image pattern.Step 5.(i) Passing the central maximum only : Draw (a) the mask, (b) the predictedpattern, and (c) the observed image pattern.(ii) Passing the central maximum and a2 : Draw (a) the mask, (b) thepredicted pattern, and (c) the observed pattern.(iii) Passing a1 only : Draw (a) the mask, (b) the predicted pattern, and (c)the observed pattern.(iv) Passing the central maximum and c1 : Draw (a) the mask, (b) thepredicted pattern, and (c) the observed pattern.(v) Passing the central maximum, c1, and d1 : Draw (a) the mask, (b) thepredicted pattern, and (c) the observed pattern.How do the spacings of the horizontal wires in patterns observed in (ii) and(iii) compare?How do the spacings of the vertical wires in patterns observed in (ii) and (iii)compare?How does the spacing of the wires in pattern (ii) compare with the spacingin the absence of the mask?How does the spacing of the wires in pattern (iii) compare with the spacingin the absence of the mask?Step 6. Passing the central maximum and a1 : Draw (a) the observed imagepattern. Also draw (b) the pattern with central maximum blocked.Passing the central maximum and the an : Draw (a) the observed imagepattern. Also draw (b) the pattern with central maximum blocked.Step 7. Draw the observed image pattern and compare with the patternwithout mask.

Page 85: Lab

Chapter 9

Holography

9.1 Theory

Our goal in this lab is to understand the concepts involved in the making ofa hologram. For discussions of the theory of holography, see Chapter 31 ofJenkins and White or Sec. 14.3 of Hecht and Zajac. There are several goodarticles about holography in the American Journal of Physics, including oneon pages 954–957 in Volume 43 (No. 11).

An ordinary photograph is a record of the intensity of light |E|2 reflectedfrom an object. It has no phase information. There is no parallax stored inthe film. This means the information about the phase relationship betweenlight waves reflected from different parts of the objects is not recorded. It isonly a two dimensional record of the object. Also, on a given piece of filmonly one part of the image may be stored.

A hologram, on the other hand, not only records the intensity informationof the light reflected by the object but also the phase relationship betweenlight rays from different parts of the object. Displaying the hologram resultsin reconstructing both the intensity and phase pattern of the original lightwaves. It reproduces a set of light rays identical to the ones that are actuallyscattered by the object. Since all the phase and intensity information ispresent a true three dimensional image having the same parallax as the objectis formed. The image is in every way the same as the object itself. Of coursethe object is not physically present, but all the light rays that would comefrom it are present. The result is that you couldn’t test for the existenceof the object by vision alone. In every sense, the image would be visuallyreal. This is what a hologram does. Another interesting difference between

80

Page 86: Lab

9.2. EXPERIMENTAL PROCEDURE 81

a photograph and a hologram is that the entire image is stored on a piece ofany size.

How do we record the information about the phase? This is done byilluminating the object with a light with long coherence length. This is wherelasers come in. Although laser light is coherent its coherence length is finite.The finite length over which the beam will remain coherent is known as thecoherence length Lc of the laser. Since it is the coherence of the beam thatwe wish to utilize in constructing the interference pattern on the hologram itis clear that we cannot use an object whose dimensions exceed the coherencelength. Also beam path differences cannot exceed one coherence length.Therefore we strive to keep all paths of approximately the same length. Thistype of coherence is known as temporal coherence and tells us about thequality of the beam for a given length of time t = Lc/c. Another type ofcoherence, spatial coherence, is also important in making holograms. Spatialcoherence refers to the uniformity of phase across the beam at a given instantof time. For holography , we need a beam with uniform cross section knownas the TEM00 mode. Mathematically this mode corresponds to a Gaussian(classic bell curve) beam profile. To further enhance the uniformity of ourbeam, we shall use a spatial filter. We also need a method to diverge thebeam enough to cover the object. Any irregularities in the beam or the lenswill be greatly magnified in so doing. The purpose of the spatial filter is thustwofold: it expands the beam to the necessary size, and secondly it cleansup the beam, i.e. produces a beam of uniform cross section. Aligning thespatial filter and keeping beam irregularities can be a trying experience. Bepatient, the fruits of your care and hard work will be evident in the qualityof the hologram. So persevere!

9.2 Experimental Procedure

Safety Rules for the Laser LabDo not look directly into the laser beam or its reflectionDo not touch any optical surfacesUse a nonreflecting white card to locate the beamWatch for others when redirecting the beamDo not eat in the laser lab or the darkroomWear gloves while working in the darkroomWhen mixing acids and water, always add the acid to the waterWash your hands several times after working with darkroom chemicals

Page 87: Lab

82 CHAPTER 9. HOLOGRAPHY

Qty. Description

1 HeNe laser

1 pinhole or spatial filter for output of laser

1 shutter to control exposure times

1 front surface mirror mounted to cast beam through spatial filter

1 spatial filter to cast broad beam upon object

1 holographic film plate (4× 5 in.) holder

1 object whose image is to be recorded

1 shock table

Table 9.1: Optical apparatus for a reflection hologram.

Qty. Description

4 trays (developer, bath, bleach, fixer)

Developer and chemicals(do not expose to room light)

red-sensitive holographic plates (AGFA 8E75-HD NAH, 4× 5 in.)

1 timer

tongs or gloves for handling plates

1 empty plate box for drying plates

Table 9.2: Darkroom apparatus for the Holography experiment.

9.2.1 Transmission and Reflection Holograms

We describe here only how to produce a single-beam reflection hologram. Theimage of this type of hologram may be viewed using an overhead projectoras light source.

As shown in Fig. 9.1, the expanded laser beam illuminates both the plateand the object behind the plate. The name “reflection” hologram refers tothe fact the light reflected from the object interferes with the light incidenton the plate, creating the holographic image.

Depending on the size of shock table available, it may be necessary oradvantageous to use a front-surface mirror to reflect the beam from the laserthrough the spatial filter. This additional path length facilitates expansion ofthe beam and makes the task of aligning the components somewhat simpler.

Page 88: Lab

9.2. EXPERIMENTAL PROCEDURE 83

beam expander

plate holder

object

shutter

Laser

aperture

Figure 9.1: Single beam reflection hologram. It is true that transmission

holograms yield the best depth, parallax, and resolution, but they are not

viewable in ordinary light. Reflection holograms (also known as Bragg’s angle

holograms) can be viewed in sunlight or with a point source of light without

any need for special lighting situations.

The alignment of the spatial filter is somewhat difficult, but when successful,the resulting images are of quite good quality.

Use the following procedure to expose a holographic plate.The laser must be allowed to “warm up” for a time sufficient for it to

stabilize (at least an hour or so). Set the shutter for the exposure timedesired — this may range from 0.5 seconds to 4 or 5 seconds. Ensure thatthe beam fully illuminates the space described by the plate holder. Theobject to be imaged should be placed behind the plate holder, close enoughto it for good illumination, but not so close that it will be jarred duringthe process of inserting the plate into the holder. We have had good resultswith a variety of objects ranging from the ubiquitous die to an illustratedcoffee mug and a toy airplane made of light wood. Ivory figurines shouldalso work well. Bear in mind that the nature of the image will affect theexposure, developer, and bleach times. Several attempts with a given objectwill probably be necessary to produce a good quality hologram.

Have everyone stand away from the table and then turn the lights off.Open the light-tight box of plates and carefully remove one plate using yourfingertips only. These plates have an emulsion on one side only; the emulsionside should be placed toward the object (away from the incident laser beam).A moist finger will find the emulsion side to be considerably more “tacky”than the glass side. After determining which side has the emulsion, it isprobably wise to stick to some sort of convention when handling the plate.For instance, hold the emulsion side toward yourself when carrying it andthen — of course — keep the emulsion side up when placing the plate in the

Page 89: Lab

84 CHAPTER 9. HOLOGRAPHY

darkroom trays to avoid scratching the film.

Now slide the plate into the plate holder, being careful to avoid touchingthe surfaces of the plate as well as the object. Make sure that the plateslides fully down into the holder. Replace the cover of the light-tight platebox holding the unexposed plates.

Move over to the shutter control, and then wait at least thirty secondsfor any vibrations or air currents to settle down. Any dust in the path of thebeam or motion of the components will degrade the quality of the image.

Open the shutter for the desired length of time (we are presently gettinggood with results with exposure times on the order of one half of one second).Then — with gloves on — remove the plate from its holder. If you like,you may first remove the object to avoid knocking it into the emulsion ofthe plate. Holding the emulsion side of the plate toward yourself, proceedto the darkroom, where the chemicals should have been prepared per theinstructions in Subsection 9.2.2.

Close the door of the darkroom. Do not use a “safety” lamp. Do notturn on the lights in the laser lab, as the darkroom door is not light-tight.

Place the plate in the developer tray, emulsion side up. Agitate thedeveloper by rocking the tray gently back and forth. Do this for anywherefrom about 1.5 to 2.5 minutes. As with all of the timings given here, youwill have to make several trials to determine the best combination of timesto use.

Remove the plate using either a pair of tongs or your gloved hands. Placethe plate emulsion side up in a tray set in the bottom of the right-hand sink.The tray should be full of tap water, with enough flow from the tap that thebath is continually being replenished. There is no need to agitate the bathsince the water is flowing. Leave the plate in the water bath for about threeminutes. Rinse the developer from your gloves.

Remove the plate using either a pair of tongs or your gloved hands. Placethe plate emulsion side up in the bleach tray for about one minute, gentlyagitating the tray as before. Avoid dripping water in the other trays.

Remove the plate using either a pair of tongs or your gloved hands. Becareful to avoid dripping bleach from the plate into the other trays. Placethe plate emulsion side up in the water-bath tray for about three minutes.Rinse the bleach from your gloves.

Remove the plate using either a pair of tongs or your gloved hands. Placethe plate emulsion side up in the fixer tray for about two minutes, gentlyagitating the tray as before. Avoid dripping water in the other trays.

At this point, you may turn on a “safety” light if you like. Finally, placethe developed plate into a rack from an empty light-tight plate box for drying.

Page 90: Lab

9.2. EXPERIMENTAL PROCEDURE 85

It takes at least ten minutes for a plate to dry fully — do not rush this step.When leaving the darkroom, be sure to close the door to preserve the

quality of the developer. You may now turn the laser lab lights on.After the plate has dried, it may be viewed by standing so that the light

from the sun or an overhead projector is cast over your shoulder. Holdingthe plate in front of you and to the side from which the light is coming, youshould be able to view the image from many different angles. Any swirly graypatterns seen on the film are not the interference pattern you are trying torecord. They are a result of optical imperfections and dust in the beam path.The pattern you are recording is microscopic and it would be a good idea toexamine the film under a microscope to see the complex three dimensionalinterference pattern you have recorded on the film.

9.2.2 Darkroom Chemicals Preparation

The holographic chemicals are all stocked in the darkroom and are mixedon an as-needed basis. The reason for this is that the shelf life of the mixeddeveloper is only a day or so!

Discuss with your instructor and follow the instruction on the chemicalpacket for preparing chemicals for developing.

Figure 9.2 shows the additional complexity involved in producing a split-beam reflection hologram.

Figures 9.3 and 9.4 illustrate the setups required to produce transmissionholograms.

Page 91: Lab

86 CHAPTER 9. HOLOGRAPHY

aperture

beam splitter

50/50beam splitter

reference beam

plate

object

reference mirror

reference lens

card

object mirror

object lenscard object mirror

object lens

laser

shutter

Figure 9.2: Split beam reflection hologram. Note that the arrangement shown

will yield a dynamic hologram when viewed with a point source of light.

9.2.3 Viewing a Transmission Hologram

In order to view a transmission hologram, you need to place the plate in thesame position relative to the laser beam as when the exposure was made.Therefore, if the optical table is still set up from the exposure , just place theplate into the mount and view. It is not difficult to align if the equipmenthas been moved, just view the image until it is clearly visible. Be careful inlooking at the plate with the laser beam, do not look directly down the beamand beware of regions of specular reflection where the beam will be comingstraight through the film.

Notice that one can observe the image from various angles and actually seedifferent views of the image by just moving your eyes. The object appears toactually be there just behind the plate (virtual image). Now flip the hologramover and place back in the mount. The image will appear to be in front of the

Page 92: Lab

9.2. EXPERIMENTAL PROCEDURE 87

beam expandershutter

Laser

aperture

plate holder

object

Figure 9.3: Single beam transmission hologram.

aperture

beam splitter

50/50beam splitter

reference beam

plate

object

reference mirror

reference lens

object mirror

object lens

cardobject mirror

object lens

card

laser

Figure 9.4: Split beam transmission hologram. This is the basic setup. The

most sensible approach is to maintain a symmetry in the placement of the

components. This also gives the most satisfactory results.

Page 93: Lab

88 CHAPTER 9. HOLOGRAPHY

plate when viewed from a short distance. This image is called pseudoscopic:it is actually inside-out. It is possible to place an object in the path of theimage for projection when viewed this way (real image, pseudoscopic). Nowhit the plate with just the undispersed dot of the laser beam. Notice thatwith just this dot , it is possible to view the entire object. As you movethe viewing spot about, you will see the image from different points of view.The effect is the same as looking through a tiny hole to see the view out ofa window. In order to see the entire image, you will have to move your headabout to different positions, but it is all there! Thus the entire hologram is anassembled of images, one from each point of view, yet each viewpoint containsthe entire image. This is just an inkling of the power and information storedin a hologram!

Page 94: Lab

Chapter 10

Photoelectric Effect

In all experiments that we have performed so far, light can be described inthe language of rays and waves. We now look at situations where the wavepicture of light is inadequate to explain experimental observations. Thephotoelectric effect, together with the Compton effect and the Raman effect,requires light to be treated as a stream of particles. The experiment on thephotoelectric effect introduces you to the quantum aspect of light.

For a discussions of the photoelectric effect see Chapter 13 of Hecht orChapter 33 of Jenkins and White.

10.1 Theory

The photoelectric effect is the emission of electrons from a metal surface whenthe surface is illuminated by light. It was first noted by Heinrick Hertz in1887. Attempts to explain the experimental observations defied theoreticalphysics until 1905 when Einstein applied the then new Quantum Theory tothe effect and obtained complete agreement with experimental observations.The photoelectric effect thus provided one of the key tests of the new theoryand partly for his explanation of it Einstein received the Nobel prize.

Einstein’s basic assumption was that light consists of particles, or quanta,of energy. Each quantum has a well defined energy given by

E = hν (10.1)

Based on this assumption Einstein proposed the following relation for the

89

Page 95: Lab

90 CHAPTER 10. PHOTOELECTRIC EFFECT

Qty. Description

1 Mercury vapor lamp

1 Intensity filters (transmissivity 80%, 60%, 40%, and 20%)

2 UV blocking filters (green and yellow filters)

1 Photocell with a stand

1 Digital multimeter

1 patch cords

1 goniometer to hold the source and the photoelectric cell

1 Sliding mount with a grating and lens

Table 10.1: Apparatus for the photoelectric effect.

kinetic energy K of the emitted electrons :

K = hν −W0 (10.2)

where K is the kinetic energy of the electron, h is the Planck’s constant,ν is the frequency of the incident light and W0 is the work function of themetal being used. Work function is the minimum amount of work requiredto free an electron from the metal surface. An interesting consequence ofEq. (3) is that for a given metal the kinetic energy of the emitted electronsis independent of light intensity. It depends only on the frequency of theincident light. Would you have guessed that the kinetic energy wouldbe independent of light intensity? A wave model of light would havepredicted that the kinetic energy depends on light intensity.

The simple relation of Eq. (2) completely explains the photoelectric effectand suggests a way to determine Plancks constant h and the work functionW0. To experimentally investigate the Einstein relation we note that onemust determine the kinetic energy K of the electrons as a function of lightfrequency ν. Consider the setup shown in Figure 1. Suppose initially the ap-plied potential V is zero. Then when light strikes the metal surface, electronsare emitted with kinetic energy K. They proceed through the vacuum untilthey strike the other metal plate and are registered as an electrical currentby the ammeter. Now consider what happens as the potential V is increased.The electrons now experience a retarding force and when V is high enoughall the electrons emitted will be just brought to rest and no current will be

Page 96: Lab

10.2. EXPERIMENTAL PROCEDURE 91

ammeter

voltmeter

e

V

A+

light

metal

Figure 10.1: Setup to observe the photoelectric effect.

registered. When this happens we know the kinetic energy of the electrons.

eV =1

2mv2 = K = hν −W0. (10.3)

Thus we have a means of determining one of the parameters in the Einsteinrelation. Light frequency ν is selected by using an appropriate color filterwhich allows only the light of a certain frequency to pass through. Forexample, a red filter transmits light quanta of frequencies corresponding tothe red color and absorbs quanta of all other frequencies. Another approach isto use a grating to separate the incident light into its component frequencies.Different frequencies then can be selected from the spectrum with the helpof a slit that allows only a certain part of the spectrum to pass through.The rest of the spectrum is blocked. Thus by using the light of differentfrequencies and determining the stopping voltage we can determine both hand W0 in the Einstein relation.

10.2 Experimental Procedure

An outline of the apparatus is shown in Figure 2. Since the electric currentproduced by a light source of any reasonable intensity is very small (it variesfrom 10−12 − 10−9 A), it is difficult to measure the null point in current

Page 97: Lab

92 CHAPTER 10. PHOTOELECTRIC EFFECT

∗mercury lightsource

filtergrating

S C AMP

Voltmeter

Figure 10.2: An outline of the experimental setup.

with currents of this size. For this reason, in the experiment we measure thestopping voltage directly with the help of an high impedance amplifier.

The photoelectrons emitted from the photocathode move toward the an-ode producing a small photo-current. This photocurrent is used to charge asmall capacitance. When the potential on this capacitance reaches the stop-ping potential for the photoelectrons, the current decreases to zero and theanode-to-cathode potential stabilizes. This final voltage between the anodeand the cathode is therefore the stopping potential of the photoelectrons.This voltage is measured with the help of a high input impedance ( > 1012

Ω), unity gain amplifier [Fig. 2].

The procedure for recording the data is as follows. Turn the mercuryvapor lamp on and allow about 5 minutes for it to stabilize. Attach thegrating and lens sliding mount to the mercury lamp housing. Place a cardin front of the grating. You should see sharp mercury line spectra. Mountthe photoelectric cell on the support rod on the support assembly. Placethe support assembly over the pin at the end of the coupling bar assembly[Fig. 3]. Set the photocell directly in front of the light source and adjust thegrating/lens assembly so that a sharp image of the source aperture is seen onthe mask on the photocell. Rotate support assembly until the light in one ofspectral lines in the first order enters the photo cell. If you select the green

Page 98: Lab

10.2. EXPERIMENTAL PROCEDURE 93

Hg Source

Grating

Photocell

Mask andFilter Holder

Coupling BarAssembly

Support BaseAssembly

Figure 10.3: Details of the experimental setup.

or yellow spectral line, place the corresponding colored filter over the WhiteReflective Mask on the h/e apparatus. Place the Variable Transmission Filterso that the light passes through the section marked 100% and reaches thephotodiode.

Depress momentarily the zero switch S on the photocell to discharge anyaccumulated charge on the storage capacitor. Read the output voltage onthe voltmeter and approximately how much time is required to recharge theinstrument to the maximum voltage. The maximum voltage is the stoppingvoltage. Move the Variable Transmission Filter so that next section is directlyin front of the incoming light. Record the stopping voltage and approximatecharge time for different transmission fractions. Repeat the procedure forall available lines in the spectrum and record these observations in yournotebook. Repeat the procedure for the second order spectral lines.

Plot a graph of intensity versus stopping potential, and intensity versuscharging time. Explain the effect of intensity on the stopping potential andcharging time.

Plot a graph of the stopping potential V vs light frequency ν (ν = c/λ).The stopping potential gives us the kinetic energy of the electrons via K =eV . Using this in Eq. (2) and rearranging it we obtain

V = (h/e) ν −W0/e . (10.4)

Thus the slope of the V vs. ν graph determines the ratio h/e and the interceptdetermines the ratio W0/e. If V0 is the intercept in volts then W0 = −eV0.We will automatically get the work function W0 in electron volts by droppinge: W0 = -V0 eV (electron-volts). Assign experimental error to your measured

Page 99: Lab

94 CHAPTER 10. PHOTOELECTRIC EFFECT

values of h and W0. The following constants are given :

e = 1.602× 10−19 C (10.5)

c = 2.998× 108 m/s (10.6)

Page 100: Lab

Appendix A

Error Analysis

Scientific knowledge is based on the results of measurements of physical quan-tities. Therefore we must understand how to express the results of mea-surements and how to analyze and draw meaningful conclusions from them.Since all measurements are subject to errors or uncertainties, the result ofany physical measurement must consists of two essential components: (1) anumerical value (in a specified system of units) giving the best possible esti-mate of the quantity measured, and (2) the degree of uncertainty associatedwith this value. For example, a measurement of the width of a table wouldgive a result such as 85.3 ± 0.1 cm. We can minimize the error by usingmore sophisticated apparatus or techniques but we cannot eliminate it.

It is important to keep in mind that error is not the difference betweenthe measurement and some accepted “exact” value. Accepted values are not“right” answers. They are just measurements made by other people whichhave errors associated with them as well. For example, the “true” value ofthe ratio of the proton mass to that of the electron is 1836.153± .001.

Error does not mean mean “blunder.” Reading a scale backwards, orreading a centimeter scale as inches are blunders which can be caught andshould be avoided. One should never refer to ”human error” as an excuse fordeviation of your measured results from an “accepted” value.

Error refers to the uncertainty in measurements. It is an intrinsic part of

95

Page 101: Lab

96 APPENDIX A. ERROR ANALYSIS

the measurement. Although we cannot eliminate error, it can be character-ized. For instance, the repeated measurements may cluster tightly togetheror they may spread widely. This pattern can be analyzed systematically.

It is also important to distinguish between the accuarcy and precision ofa result. The accuracy of an experiment is a measure of how close the resultof the experiment comes to the true value. The precision of an experiment,on the other hand, refers to how exactly is the result determined withoutreference to what the result means. For example, if in an experiment de-termining the length of a table, the table is found to be 1.752 m long, themeasurement indicates a high precision on the order of 1 mm. The correc-tions to this result due to thermal expansion will improve the accuracy butnot the precision.

The way in which the result of an experiment is written should reflect theprecision. To indicate the precision we write a number with as many digitsas are significant.

A.1 Significant Figures

The significant figures of a measured or calculated quantity are the meaning-ful digits in it - digits which can be trusted, even if some are zeros. There areconventions for expressing the results of measurements to properly indicatetheir significant figures.

• Any digit that is not zero is significant. Thus 639 has three significantfigures and 1.492 has four significant figures.

• Zeros between non zero digits are significant. Thus 1023 has four sig-nificant figures.

• Zeros to the left of the first non zero digit are not significant. Thus0.000069 has only two significant figures. This is more easily seen ifscientific notation is used to write it as 6.9×10−5.

• For numbers with decimal points, zeros to the right of a non zero digitare significant. Thus 3.00 has three significant figures and 0.040 has twosignificant figures. For this reason it is important to keep the trailingzeros to indicate the actual number of significant figures.

Page 102: Lab

A.1. SIGNIFICANT FIGURES 97

• For numbers without decimal points, trailing zeros may or may not besignificant. Thus, 400 indicates only one significant figure. To indicatethat the trailing zeros are significant a decimal point must be added oruse scientifc notaton to write it. For example, 400. or 4.00×102 hasthree significant figures, and 4× 102 has one significant figure.

• Exact numbers have an infinite number of significant digits. For exam-ple, if there are two oranges on a table, then the number of oranges is2.000... . Defined numbers are also like this. For example, the num-ber of centimeters per inch (2.54) has an infinite number of significantdigits, as does the speed of light (299792458 m/s).

There are also specific rules for consistently expressing the uncertaintyassociated with a number. In general, the last significant figure in any resultshould be of the same order of magnitude (i.e.. in the same decimal position)as the uncertainty. Also, the uncertainty should be rounded to one or twosignificant figures. Always work out the uncertainty after finding the numberof significant figures for the actual measurement.

Thus the correct format for reporting the focal length of a lens is:14.82 ± 0.02 cm OR 15.0±1.5 cm OR 15 ± 1 cm

By the same token14.82 ± 0.02385 cm is wrong but 14.82 ± 0.02 is acceptable10.0 ± 2 cm is wrong but 10.0 ± 2.0 cm is acceptable4 ± 0.5 is wrong but 4.0 ± 0.5 is acceptable

In practice, when doing mathematical calculations, it is a good idea tokeep an extra digit than is significant to reduce rounding errors. But in theend, the answer must be expressed with only the proper number of significantfigures. After addition or subtraction, the result is significant only to theplace determined by the largest last significant place in the original numbers.For example,

89.332 + 1.1 = 90.432

should be rounded to get 90.4 as the tenths place is the last significant place inthe second number. After multiplication or division, the number of significantfigures in the result is determined by the original number with the smallestnumber of significant figures. For example,

(2.80)× (4.5039) = 12.61092

Page 103: Lab

98 APPENDIX A. ERROR ANALYSIS

should be rounded off to 12.6 since 2.80 has three significant figures.

Read any good introductory physics textbook for an explanation for work-ing out significant figures.

A.2 Classification of Errors

Errors of measurement can be classified into two categories.

Systematic errors are errors which tend to shift all measurements in asystematic way so their mean value is displaced. This may be due to incorrectcalibration of equipment, consistently improper use of equipment or failureto properly account for some effect. In a sense, a systematic error is ratherlike a blunder. Large systematic errors can and must be eliminated in a goodexperiment. Small systematic errors will always be present. For instance, noinstrument can ever be calibrated perfectly.

Other sources of systematic errors are external effects which can changethe results of the experiment, but for which the corrections are not wellknown. In science, the reasons why several independent confirmations ofexperimental results are often required (especially using different techniques)is because different apparatus at different places may be affected by differentsystematic effects.

Random errors are the fluctuations in observations which yield resultsthat differ from experiment to experiment. These errors are beyond thecontrol of the observer. If we are measuring the decay rate of a radioactivesample we get different results from one time interval to another. This is dueto the inherently random nature of the process being investigated.

A.3 Normal Distribution

When a quantity x is measured a large number of times a pattern will beginto emerge from the data. A picture of the distribution of measurementsmight look like the histogram in Fig. 1. To obtain such a plot we divide

Page 104: Lab

A.3. NORMAL DISTRIBUTION 99

P(x)

µ-3σ µ-2σ µ-σ µ µ- σ µ-2σ µ-3σ

x

Figure A.1: A typical histogram of measured values of x and the normal

(Gaussian) distribution.

the range of observed values of x into many intervals of equal size, computethe fraction of observations that lie within each interval, and then plot thisfraction against the average of the observations in each respective interval.Figure A.1 shows one such plot.

If make an infinite number of measurements, we can determine the “true”distribution (called the parent distribution) that governs the probability ofgetting any particular result in one measurement. Making an infinite numberof measurements is costly and time consuming. So we must decide at somestage that the measurement is complete. It should be kept in mind thatrepeating the experiment is the only way to gain confidence in the accuracyof our result.

In the absence of any specific information or theoretical basis we assumethe parent distribution to be a Gaussian . This assumption will tend to bevalid for random errors because of the law of large numbers. The Gaussiandistribution, also called the Normal distribution, has the form :

P (x) =1√2π σ

e−(x−µ)2/2σ2

, (A.1)

Page 105: Lab

100 APPENDIX A. ERROR ANALYSIS

where µ is the center (the most probable value), and σ is the width (stan-dard deviation) of the distribution. The normal distribution is shown by thecontinuous curve in Fig. A.1. In terms of this distribution P (x)dx is theprobability that a measurement of x will yield a result between x and x+dx.For a Gaussian distribution, the measured values are distributed symmetri-cally around the true value µ. Approximately 68% of the measurements willfall within ±σ of µ, while 95% will be within ±2σ and only 5% will be morethan 2σ away from µ.

A.4 Sample Mean and Variance

In an actual experiment we measure a physical quantity a finite number oftimes, say N , with the results x1, x2, x3 · · · , xN . These measurements repre-sent a finite sample from the parent population. All of our calculations mustbe made from this sample. Then the average or mean of the our measure-ments

x =1

N

N∑

k=1

xk . (A.2)

is our best estimate of the true value µ of the parent distribution. 1

This is a reasonable estimate of the mean because random errors woulddiffer in sign and magnitude from measurement to measurement. So in thecalculation of the average or mean value of our sample, some of the randomvariations could be expected to cancel out the others. This is the best thatcan be done to deal with random errors: repeat the measurement manytimes, varying as many parameters as possible and use the sample averageas the best estimate of the true value of x. For example, in the case ofthe measurement of the focal length of a lens by using a distant object, useseveral different distant objects -large and small, different helpers to read thescale, etc.

We are also interested in an estimate of the standard deviation σ. Interms of our measurements the best estimate for σ would be the sample

1This estimate for finite N will differ from the true mean value; though, of course, for

larger N it will be closer to the parent mean.

Page 106: Lab

A.5. PROPAGATION OF ERRORS 101

standard deviation s given by

s2 ≈1

N − 1

N∑

k=1

(xk − µ)2

. (A.3)

Note that this formula has N−1 in the denominator. This is because we needat least one observation to determine x. That leaves only N −1 independentobservations to determine σ. What these numbers x and s mean is that ifwe made one more measurement of x, there is 68% probability that it willlie within x± s (and therefore 32% probability that it will disagree with theaccepted value x by more than s).

In addition to the uncertainty of one measurement, we are also interestedin the error of the mean, that is, how close is our estimate x to the true valueµ. This uncertainty in estimating µ is given by

σx =s√N

, (A.4)

where N is the number of independent measurements [ See Sec. 1.7 for aproof]. The uncertainty in the determination of the mean is equal to thestandard deviation of an individual measurement divided by the square rootof the number of independent measurements. This means that the accuracyof our estimate improves in proportion to the square root of the number ofmeasurements in the sample. This is a fundamental principle of statistics.Similarly, we could estimate the uncertainty in σµ itself. This however, isusually not required.

A.5 Propagation of Errors

Often the quantity of interest V is calculated from two measured quantitiesx and y by means of a theoretical formula V = f(x, y). Then V is uncertainas a result of the uncertainties in the measured quantities x and y. How dowe combine the errors in x and y to estimate the error in V ?

In general the errors in x and y are correlated. These correlations can becharacterized by the correlation coefficient ρxy defined by

ρxy = limN→∞

1

N σxσy

N∑

k=1

(xk − µx)(yk − µy) , (A.5)

Page 107: Lab

102 APPENDIX A. ERROR ANALYSIS

where µx, µy, σx and σy are given by equations similar to (2) and (4). ForN pairs of measurements of x and y the best estimate of ρxy is

ρxy =1

(N − 1) σxσy

N∑

k=1

(δxk δyk) ,

=1

(N − 1) σxσy

[N∑

k=1

xk yk −Nx y

](A.6)

For independent errors ρxy is equal to zero. For completely correlated errors,ρxy is either +1 or -1. We can now estimate errors in V .

From N measurements of x and y we can compute x and y and N valuesof V = f(x, y): V1 = f(x1, y1), V2 = f(x2, y2), . . . , VN = f(xN , yN). We shallmake the fundamental assumption that the most probable value of V is

V = f(x, y). (A.7)

Then the deviation of Vk can be expressed in terms of the deviations of xk

and yk by

δVk ≡ Vk − V ≈∂V

∂xδxk +

∂V

∂yδyk , (A.8)

where δxk = xk − x and δyk = yk − y, respectively, are the deviations of xk

and yk. It is easy to check that the sum of the deviations∑N

k=1 δVk vanishes.The average of the squares of deviations of V will be

1

N

N∑

k=1

(δVk)2 =

1

N

N∑

k=1

[(∂V

∂x

)2

(δxk)2 +

(∂V

∂y

)2

(δyk)2 + 2

(∂V

∂x

)(∂V

∂y

)δxk δyk

].

(A.9)

Taking the square root of both sides in the limit N → ∞, we obtain thestandard deviation of V

σV =

√(∂V

∂x

)2

σ2x +

(∂V

∂y

)2

σ2y + 2ρxy

(∂V

∂x

)(∂V

∂y

)σxσy . (A.10)

Equation (A.10) may be generalized for the case when V is a function of

Page 108: Lab

A.6. STANDARD DEVIATION OF AN AVERAGE 103

many measured quantities. For example, if V = V (x, y, z), then

σV =

[(∂V

∂x

)2

σ2x +

(∂V

∂y

)2

σ2y +

(∂V

∂z

)2

σ2z

+ 2ρxy

(∂V

∂x

)(∂V

∂y

)σxσy + 2ρxz

(∂V

∂x

)(∂V

∂z

)σxσz

+ 2ρyz

(∂V

∂y

)(∂V

∂z

)σyσz

] 12

. (A.11)

With a finite number of measurements N , we obtain the best estimate ofσV by substituting in Eq. (A.10) the best estimates sx and sy of σx and σy,and the best estimates of ρxy from Eq. (A.6). If measurements are known apriori to be independent, ρxy is set equal to zero.

A.6 Standard Deviation of an Average

We are now in a position to derive Eq. (A.4) for the standard deviationσx of the average. We make use of the fact that the average x is a quantitycomputed from the measured quantities x1, x2, . . . , xN by means of Eq. (A.2).Then

∂x

∂x1=

∂x

∂x2= · · · =

∂x

∂xN=

1

N, (A.12)

and since we are considering only the random errors in the measurementsxk, they may be considered independent, so that we may take their mutualcorrelation equal to zero. Eq. (A.10) modified for this case then gives theresult stated in Eq. (A.4)

σx =

√1

N

(s2

x1+ s2

x2+ s2

x3+ · · · + s2

xN

)=

s√N

, (A.13)

where we have used the fact that sxk = sx ≡ s for each value of k.

We can also estimate the standard deviation σV of the average of V . Weexpect the averages x and y to have the same degree of correlation as theindividual pairs of measurements. In fact, it is shown in advance treatisesthat ρxy = ρxy. Therefore we expect the standard deviations of averages

Page 109: Lab

104 APPENDIX A. ERROR ANALYSIS

to combine in the same way as the standard deviations of the individualmeasurements:

σV =

√(∂V

∂x

)2

σ2x +

(∂V

∂y

)2

σ2y + 2ρxy

(∂V

∂x

)(∂V

∂y

)σxσy . (A.14)

A.7 Special Rules for Combining Errors

The rules expressed by Eqs. (A.10) and (A.14) are valid regardless of themathematical form of V = V (x, y). In the following special cases a and b aresome constants, x and y are independent (ρxy = 0) variables and σx and σy

are the corresponding uncertainties.

A.7.1 Sum and difference: V = ax± by

∂V

∂x= a ,

∂V

∂y= ±b , (A.15)

σV =√

a2σ2x + b2σ2

y . (A.16)

For example, for V = x + y with x = 100± 3 and y = 6 ± 4,

V = 106±√

32 + 42 = 106± 5

A.7.2 Product of various powers: V = xp yq

∂V

∂x= pxp−1 yq ,

∂V

∂y= qxp yq−1 , (A.17)

σV =√

p2x2(p−1)y2qσ2x + q2x2py2(q−1)σ2

y . (A.18)

Page 110: Lab

A.7. SPECIAL RULES FOR COMBINING ERRORS 105

For fractional standard deviation we can write

σV

V=

p2(σx

x

)2

+ q2

(σy

y

)2

. (A.19)

For the special cases V = xy or V = x/y corresponding to p = 1 and q = ±1,we obtain

σV

V=

√(σx

x

)2

+ q2

(σy

y

)2

. (A.20)

For example, for V = xy and x = 100± 3 and y = 6± 4, we find

V = (100.0± 0.3)(6.0± 0.4) = 600± 600

√(0.3

100.0

)2

+

(0.4

6.0

)2

= 600± 40.

On the other hand, for V = x2 with x = 10± 1, we find

V = (10±1)2 = 102

[1± 2

1

10

]= 100±20 and not V = 102

[1±

√2

1

10

]= 100±14

A.7.3 Logarithm of a quantity: V = A ln x

In this case dV /dx = A/x. Then from Eq. (19) we obtain

σV = Aσx

x. (A.21)

A.7.4 Exponential of a quantity: V = Ae±bx

Using dV /dx = ±Abe±bx = ±bV in Eq.(19) we obtain

σV

V= B σx . (A.22)

Reference:

John R. Taylor, An Introduction to Error Analysis : The Study of Un-certainties in PhysicalMeasurements ( University Science Books, 1982).