Chapter 25 Optical Instruments. Analysis generally involves the laws of reflection and refraction Analysis uses the procedures of geometric optics To.

Chapter 25

Optical Instruments

Optical Instruments Analysis generally involves the

laws of reflection and refraction Analysis uses the procedures of

geometric optics To explain certain phenomena, the

wave nature of light must be used

The Camera The single-lens photographic camera is

an optical instrument Components

Opaque, light-tight box Converging lens

Produces a real image Film behind the lens

Receives the image

Digital Camera Image is formed on

an electric device CCD – Charge-coupled

device CMOS –

Complementary metal-oxide semiconductor

Both convert the image into digital form

The image can be stored in the camera’s memory

Camera Operation Proper focusing leads to sharp images

The lens-to-film distance will depend on the object distance and on the focal length of the lens

The shutter is a mechanical device that is opened for selected time intervals

Most cameras have an aperture of adjustable diameter to further control the intensity of the light reaching the film

With a small-diameter aperture, only light from the central portion reaches the film, and spherical aberration is minimized

Camera Operation, Intensity Light intensity is a measure of the rate

at which energy is received by the film per unit area of the image The intensity of the light reaching the film is

proportional to the area of the lens The brightness of the image formed on

the film depends on the light intensity Depends on both the focal length and the

diameter of the lens

Camera, f-numbers The ƒ-number of a camera is the

ratio of the focal length of the lens to its diameter ƒ-number = f/D The ƒ-number is often given as a

description of the lens “speed” A lens with a low f-number is a “fast” lens

Camera, f-numbers, cont Increasing the setting from one ƒ-number

to the next higher value decreases the area of the aperture by a factor of 2

The lowest ƒ-number setting on a camera corresponds to the aperture wide open and the maximum possible lens area in use

Simple cameras usually have a fixed focal length and a fixed aperture size, with an ƒ-number of about 11 Most cameras with variable ƒ-numbers

adjust them automatically

The Eye

The normal eye focuses light and produces a sharp image

Essential parts of the eye

Cornea – light passes through this transparent structure

Aqueous Humor – clear liquid behind the cornea

The Eye – Parts, cont The pupil

A variable aperture An opening in the iris

The crystalline lens Most of the refraction takes place

at the outer surface of the eye Where the cornea is covered with a

film of tears

The Eyes – Parts, final The iris is the colored portion of the eye

It is a muscular diaphragm that controls pupil size

The iris regulates the amount of light entering the eye by dilating the pupil in low light conditions and contracting the pupil in high-light conditions

The f-number of the eye is from about 2.8 to 16

The Eye – Operation The cornea-lens system focuses light

onto the back surface of the eye This back surface is called the retina The retina contains receptors called rods

and cones These structures send impulses via the optic

nerve to the brain The brain converts these impulses into our

conscious view of the world

The Eye – Operation, cont Rods and Cones

Chemically adjust their sensitivity according to the prevailing light conditions

The adjustment takes about 15 minutes This phenomena is “getting used to the dark”

Accommodation The eye focuses on an object by varying the shape of

the crystalline lens through this process An important component is the ciliary muscle which

is situated in a circle around the rim of the lens Thin filaments, called zonules, run from this muscle

to the edge of the lens

The Eye – Focusing The eye can focus on a distant object

The ciliary muscle is relaxed The zonules tighten This causes the lens to flatten, increasing its

focal length For an object at infinity, the focal length of

the eye is equal to the fixed distance between lens and retina

This is about 1.7 cm

The Eye – Focusing, cont The eye can focus on near objects

The ciliary muscles tense This relaxes the zonules The lens bulges a bit and the focal

length decreases The image is focused on the retina

The Eye – Near and Far Points The near point is the closest distance for which

the lens can accommodate to focus light on the retina

Typically at age 10, this is about 18 cm Average is about 25 cm It increases with age, to 500 cm or more at age 60

The far point of the eye represents the largest distance for which the lens of the relaxed eye can focus light on the retina

Normal vision has a far point of infinity

Conditions of the Eye Eyes may suffer a mismatch between

the focusing power of the lens-cornea system and the length of the eye

Eyes may be Farsighted

Light rays reach the retina before they converge to form an image

Nearsighted Person can focus on nearby objects but not those

far away

Farsightedness

Also called hyperopia The image focuses behind the retina Can usually see far away objects

clearly, but not nearby objects

Correcting Farsightedness

A converging lens placed in front of the eye can correct the condition

The lens refracts the incoming rays more toward the principle axis before entering the eye

This allows the rays to converge and focus on the retina

Nearsightedness

Also called myopia In axial myopia the nearsightedness is caused

by the lens being too far from the retina In refractive myopia, the lens-cornea system is

too powerful for the normal length of the eye

Correcting Nearsightedness

A diverging lens can be used to correct the condition

The lens refracts the rays away from the principle axis before they enter the eye

This allows the rays to focus on the retina

Presbyopia and Astigmatism Presbyopia is due to a reduction in

accommodation ability The cornea and lens do not have sufficient

focusing power to bring nearby objects into focus on the retina

Condition can be corrected with converging lenses

In astigmatism, the light from a point source produces a line image on the retina Produced when either the cornea or the lens or

both are not perfectly symmetric

Diopters Optometrists and ophthalmologists

usually prescribe lenses measured in diopters The power of a lens in diopters equals

the inverse of the focal length in meters

1ƒ

Simple Magnifier A simple magnifier consists of a

single converging lens This device is used to increase the

apparent size of an object The size of an image formed on

the retina depends on the angle subtended by the eye

The Size of a Magnified Image

When an object is placed at the near point, the angle subtended is a maximum

The near point is about 25 cm

When the object is placed near the focal point of a converging lens, the lens forms a virtual, upright, and enlarged image

Angular Magnification Angular magnification is defined as

The angular magnification is at a maximum when the image formed by the lens is at the near point of the eye q = - 25 cm Calculated by

o

angle with lensm

angle without lens

max

251

cmm

q

Magnification by a Lens With a single lens, it is possible to

achieve angular magnification up to about 4 without serious aberrations

With multiple lenses, magnifications of up to about 20 can be achieved The multiple lenses can correct for

aberrations

Compound Microscope

A compound microscope consists of two lenses

Gives greater magnification than a single lens

The objective lens has a short focal length, ƒo<1 cm

The ocular lens (eyepiece) has a focal length, ƒe, of a few cm

Compound Microscope, cont

The lenses are separated by a distance L L is much greater than either focal length

The approach to analysis is the same as for any two lenses in a row The image formed by the first lens becomes

the object for the second lens The image seen by the eye, I2, is virtual,

inverted and very much enlarged

Magnifications of the Compound Microscope

The lateral magnification of the microscope is

The angular magnification of the eyepiece of the microscope is

The overall magnification of the microscope is the product of the individual magnifications

ƒl

ll o

q LM

p

25ƒe

e

cmm

25ƒ ƒl e

o e

L cmm M m

Other Considerations with a Microscope The ability of an optical

microscope to view an object depends on the size of the object relative to the wavelength of the light used to observe it For example, you could not observe

an atom (d 0.1 nm) with visible light (λ 500 nm)

Telescopes Two fundamental types of telescopes

Refracting telescope uses a combination of lenses to form an image

Reflecting telescope uses a curved mirror and a lens to form an image

Telescopes can be analyzed by considering them to be two optical elements in a row The image of the first element becomes the

object of the second element

Refracting Telescope The two lenses are arranged

so that the objective forms a real, inverted image of a distant object

The image is near the focal point of the eyepiece

The two lenses are separated by the distance ƒo + ƒe which corresponds to the length of the tube

The eyepiece forms an enlarged, inverted image of the first image

Angular Magnification of a Telescope The angular magnification depends on

the focal lengths of the objective and eyepiece

Angular magnification is particularly important for observing nearby objects Very distant objects still appear as a small

point of light

ƒƒ

o

o e

m

Disadvantages of Refracting Telescopes Large diameters are needed to

study distant objects Large lenses are difficult and

expensive to manufacture The weight of large lenses leads to

sagging which produces aberrations

Reflecting Telescope Helps overcome some of the

disadvantages of refracting telescopes Replaces the objective lens with a mirror The mirror is often parabolic to overcome

spherical aberrations In addition, the light never passes

through glass Except the eyepiece Reduced chromatic aberrations

Reflecting Telescope, Newtonian Focus The incoming rays

are reflected from the mirror and converge toward point A

At A, a photographic plate or other detector could be placed

A small flat mirror, M, reflects the light toward an opening in the side and passes into an eyepiece

Examples of Telescopes Reflecting Telescopes

Largest in the world are 10 m diameter Keck telescopes on Mauna Kea in Hawaii

Largest single mirror in US is 5 m diameter on Mount Palomar in California

Refracting Telescopes Largest in the world is Yerkes Observatory

in Wisconsin Has a 1 m diameter

Resolution The ability of an optical system to

distinguish between closely spaced objects is limited due to the wave nature of light

If two sources of light are close together, they can be treated as non-coherent sources

Because of diffraction, the images consist of bright central regions flanked by weaker bright and dark rings

Rayleigh’s Criterion If the two sources are separated so that

their central maxima do not overlap, their images are said to be resolved

The limiting condition for resolution is Rayleigh’s Criterion When the central maximum of one image

falls on the first minimum of another image, they images are said to be just resolved

The images are just resolved when their angular separation satisfies Rayleigh’s criterion

Just Resolved If viewed through a slit of

width a, and applying Rayleigh’s criterion, the limiting angle of resolution is

For the images to be resolved, the angle subtended by the two sources at the slit must be greater than θmin

min a

Barely Resolved (Left) and Not Resolved (Right)

Resolution with Circular Apertures The diffraction pattern of a circular

aperture consists of a central, circular bright region surrounded by progressively fainter rings

The limiting angle of resolution depends on the diameter, D, of the aperture

min 1.22D

Resolving Power of a Diffraction Grating If λ1 and λ2 are nearly equal

wavelengths between which the grating spectrometer can just barely distinguish, the resolving power, R, of the grating is

All the wavelengths are nearly the same

2 1

R

Resolving Power of a Diffraction Grating, cont A grating with a high resolving power

can distinguish small differences in wavelength

The resolving power increases with order number R = Nm

N is the number of lines illuminated m is the order number

All wavelengths are indistinguishable for the zeroth-order maximum

m = 0 so R = 0

Michelson Interferometer The Michelson Interferometer is an

optical instrument that has great scientific importance

It splits a beam of light into two parts and then recombines them to form an interference pattern It is used to make accurate length

measurements

Michelson Interferometer, schematic

A beam of light provided by a monochromatic source is split into two rays by a partially silvered mirror M

One ray is reflected to M1 and the other transmitted to M2

After reflecting, the rays combine to form an interference pattern

The glass plate ensures both rays travel the same distance through glass

Measurements with a Michelson Interferometer

The interference pattern for the two rays is determined by the difference in their path lengths

When M1 is moved a distance of λ/4, successive light and dark fringes are formed This change in a fringe from light to dark is called

fringe shift The wavelength can be measured by counting the

number of fringe shifts for a measured displacement of M

If the wavelength is accurately known, the mirror displacement can be determined to within a fraction of the wavelength