Summer Proj II

Project II:

Slits and Fiber Diffraction Study

(Jeremy) Yu Gong

28 June 2014

Section I: Single Slit Diffraction

1. Theory

The Slit diffraction happens when the size of the obstacle is relatively close to the size of

the wave length of the incidental rays, consequently, all the parallel rays would be treated

as many elemental sources around the slit. As any slit(s)-based diffractions occur due to

the integer numbers of the phase difference along the path of the light rays, it allows us to

approach its analytical result by calculating the physical path difference of two typical

light rays from the slit.

Two approximations are therefore, to be introduced; for a far distance in between the

screen and the slit, as the following figure shows:

If we name the major reference angle as ,

and the minor angle as , there is a physical relation which satisfies the maximum super

position (the bright pattern), such that:

Meanwhile, the geometric relation must be satisfied with the vertical length difference as

the width of the slit, such that:

By taking the ratio of the above equations, we shall have the exact form of the bright

pattern condition, where:

(For complete proof and analysis, see: "Theory and Diffraction Analysis", Page 2-3)

The physical condition during the actual experiment is that, the value of the major

reference angle is usually much greater than the minor angle, consequently, the tangent

value of the half minor angle would have result its production with the cosine function

relatively much smaller than the sine value of the major reference angle, therefore, we

could have the famous diffraction relation as follows:

In the meantime, The uncertainty caused by the above approximation could be concluded

by the ratio of the approximated and exact expressions, such that: for any desired a%

uncertainty, there is:

As a mathematical consequence, for any less than a% uncertainty, there is:

If we name the ith bright-to-center distance as , for any less than a% uncertainty, the

requirements for such distance is that:

Where, we have defined the uncertainty factor:

(for complete proof and analysis, see: "Theory and Diffraction Analysis", Page 4-6)

For a near-field setting up, however, the top and bottom beams could be treated as parallel

rays, as the following figure indicates:

Assuming that the electric field is evenly spared along the slit's length; we could thus

define the linear field density so that:

Thus, at any vertical point y along the slit, for a small distance dy, there is:

Where, is the phase difference caused by the path difference, such that:

Therefore, for the overall effects on the pattern screen, there is:

Where, we define:

As the intensity is generally proportional to the square value of the electric field, we shall

hence have:


One may also notice that, the value of such intensity expression is mathematically

undefined when the value of is zero, we could, however, approach such issue by

applying the L'Hôpital's Rule for a close looking at the limit expression at such point:

Therefore, it implies that:

Which indicates that, the intensity value by the very center of the pattern is equal to the

intensity of the incidental beam.

2. Experimental Settings and Measurements

The actual design of the experiment contains several parts; as the intensity of any single

slit pattern is decaying much faster than those of the double slits', it is quite hard for

counting the number of the bright patterns while measuring the actual bright-to-center

distance by naked human eyes; therefore, we have set up a high resolution digital camera

along the optical reference axis, in order to capture relatively accurate but enlarged

pictures for all different diffraction patterns before we could have any numerical analysis

done. The distance from the laser source to the transparent screen is much shorter than

those of the double slits', however, by zooming into the screen where its frame is at the

same edge of the photographs, we shall have a easier time to measure the ratio of the

actual size of the diffraction patterns.

As the following figure indicates:

As soon as an appreciated clear pattern was on the screen, the camera would have its

picture taken, meanwhile, we should have measured the separation in between the screen

and the slit.

We have taken four pictures for four different slits including: lens 12 (smaller) slits 1A

and 2A; lens 12 (regular) slits (3), and (4). The indexation is shown on the following

figure:

After the photograph-takings and the data measurements, we have the following

calculations and conclusions for the varies of the single slits as the following table shows:

(For further details, see: "Data Analysis and Measurements Manual", Page 3, 9, and 12)

Lens 12 (smaller) 1A:

Photographing details: 1/25 s; 7.1 f; 3008x2000 picture resolution

Separation: (cm) 1st: (cm) 2nd: (cm) 3rd: (cm) 4th: (cm) Uncertainty: (cm)

30.0 0.69 0.98 1.43 1.93 0.07

Converted length: (cm) 0.35 0.50 0.73 0.99 0.04

Sine Value 0.0118 0.0168 0.0245 0.0330 Converted width:

(mm)

Uncertainty 0.00133 0.00133 0.00134 0.00134 0.0708

Width: (mm) 0.0536 0.0755 0.0776 0.0767

Converted Uncertainty:

(mm) Uncertainty:

(mm) 0.00605 0.00601 0.00424 0.00311 0.00485

The width of the slit is equal to:

Lens 12 (smaller) 2A:



30.0 0.71 0.99 1.27 1.83 0.07



(mm)

Uncertainty 0.00133 0.00133 0.00133 0.00134 0.0738

Width: (mm) 0.0521 0.0747 0.0874 0.0809


(mm)

Uncertainty: (mm) 0.00572 0.00589 0.00537 0.00345 0.00511


Lens 12 (regular) (3):



30.0 0.34 0.56 0.82 1.06 0.07



(mm)

Uncertainty 0.00133 0.00133 0.00133 0.00133 0.129

Width: (mm) 0.1088 0.1321 0.1353 0.1396


(mm)

Uncertainty: (mm) 0.02493 0.01838 0.01286 0.01027 0.0166





30.0 0.58 0.79 1.05 1.24 0.07



(mm)

Uncertainty 0.00133 0.00133 0.00133 0.00133 0.0956

Width: (mm) 0.0638 0.0936 0.1057 0.1193


(mm)

Uncertainty: (mm) 0.00857 0.00924 0.00785 0.00751 0.00829


The following picture contains the looks of a typical single slit diffraction pattern, which

is created by lens 12 (regular) (3), the figure blew, shows the approximated function of its

intensity:

The above picture with respect to the exact slit width should provide a general idea of how

fast the intensity is decaying for single slit diffractions; more importantly, it also implies

that, the smaller the width gets, the slower the intensity will decay, however, smaller slit

width requires much more intensive incidental beam, which creates a harsh condition for

the actual experimental performances.

Section II: Double Slits Diffraction

1. Theory

Not too much more different from the single slit diffraction approximation patterns, we

could generally apply the same near-field, where the actual setting up is almost the same

as the single slits' but an additional separation in between of the width length, rightfully as

the following figure shows:

We should as well, assume the same evenly spared linear electric field density so that:

Again, for any small vertical distance at position y, there is:

Where, is the phase difference caused by the path difference:

Along the vertical axis, the overall field effects on the pattern screen should be:

Where, we define:

As the intensity is generally proportional to the square value of the electric field, we shall

hence have:

(for complete proof and analysis, see: "Theory and Diffraction Analysis", Page 8-10)

Since we have already proven the value of the intensity for a single slit is 1, whereby the

origin, the value of the cosine part is also 1, whenever is zero. Hence, the value for the

intensity function still is zero when is equal to zero.

2. Experimental Settings and Measurements

The design of the experiment aims to enlarge the actual size of each bright patterns. As the

width of the double slits diffraction is usually small, which in turn, creates a much slower

decay rate than the single ones; however, within certain distance, the size of bright

patterns are too small to be measured; we have thus, separated the double slits filter and

the pattern screen by relatively longer distance than the single ones. The following figure

indicates the general set up for the experiment and measurements:

We have set up roughly about eleven different separations for the double slits filter and the

pattern screen; in a same manner, we have later on measured the bright-to-center distance

for every 2nd, 4th, 6th, and 8th bright patterns.

We have measured the separation in between the width of each double slits, thus obtained

several proper results for the lens 12 (small) slits 1B, 2B; lens 12 (regular) slits (2), (3),

and (4). The results are showing as follows:

(For further details, see: "Data Analysis and Measurements Manual", Page 3, 7-8, 13-14)

Lens 12 (small) 1B:

Table of measurements and results, see Appendix, Page1-3

The separation in between two widths is:

Lens 12 (small) 2B:




Table of measurements and results, see Appendix, Page 5



Table of measurements and results, see Appendix, Page 6-7





3. Intensity Analysis

As the result goes, we have acknowledged that the actual intensity function serves as a

combination of two periodic functions, which consequently have two effective "amplitude

functions". The first amplitude determines the decay of the overall periodic functions

despite what they are, and the second one is from the single slit intensity function, which

implies no matter what the rest of the periodic functions are, the rest periodic functions

never affects the value of the intensity they could reach (which also satisfies the

conservation of energy):

For a double slits intensity function, the slits terms are simply a cosine function with a

linear factor times the , however, it is consist a lot batter for the further analysis of the N-

slits intensity functions later on. The following figure shows a double slits diffraction

pattern for the lens 12 (regular) slits (3), screen-to-slits distance 135.0 cm:

The figure shows the converted intensity function for the above patten:

One must have noticed that, in between the bright patterns on the photograph, there occur

several "missing bright ones", where, they were seemingly to be existing; however, by a

close looking of the double slits intensity function for such pattern, there are several "local

maximums" between 0.2 and 0.4, where, the actual "height" or the intensity amplitude are

restrained by the "overall" single slit function. Therefore, those "missing ones" must be

bright patterns in theory, yet for an experimental observation, their intensities were

relatively too low to be detected.

Here is another typical double slits diffraction pattern from the lens 12 (small) slits 1B,

105.0 cm for screen-to-slits distance:

As the width of the slits is quite bigger than the previous pattern, by taking a close look at

its intensity functioning, where:

Apparently, there weren't that many of bright patterns inside of the single slit intensity

amplitude, however, it is seemingly that only one "actual bright" was hidden by the

restrain overall single slit intensity.

To explain such phenomena, we could approach its reason by the ratio of the separation

and the width, as for any double slit intensity functions, there is:

Assumingly that, for any time period that of an integer times of the production of w and

(w+d), there always occurs w of local maxima for the single-slit pattern, and (w+d) of

local maxima for the double slits pattern, mathematically, the number of the "bright

patterns" that contained within one single period of the overall single slit pattern must be:

More importantly, the above result will bring us significant advantage for the N-slits

analysis later on.

Section III: N-Slits Diffraction and its Approximation

1. Theory

Likewise the approach for the double slits, we could build an N-slits model basically by

adding up more equal separations and widths along certain distance, while taking exactly

the same linear electric field density, so that:

Where:

As the following figure shows:

Consequently, the overall integral for the selected length should be alternated by:

The eventual results after the integral should be:

If we define certain operator, such that, for any function of (Nx), there is:

Therefore, we could have easily concluded the above results for the intensity of N-Slits’

diffraction:

Where, as we defined:


It is somehow, hard to experimentally approach the physical meaning hidden behind such

charming phenomena; however, by the graphical characteristics of such function, we may

confidently confine certain important physical behaviors of N-Slits diffraction intensity

patterns. For instance, the following figure indicates a 10-slits diffraction intensity

pattern versus a double slits’:

The black curve represents a simple single-slit intensity function, meanwhile, the blue

curve stands for double-slits intensity, and the red curve is the actual function for the 10-

slits intensity. From the above figure, one may easily notice that, for any N-slits

diffraction pattern that has its width (w) and its separation (d) would share exactly the

same ith bright-to-center distance as a double-slits diffraction pattern that has the same

width (w) and separation (d). The only difference is that the width for the dark patterns on

the screen is seemingly longer for the N-slits than those for the double-slits.

Therefore, we are able to measure the tiny length of the separations for any N-slits based

on the same method we did for the double slits.

2. Experimental Approach

According to our theory, the bright pattern caused by the N-slits generally shares the same

“bright-to-center” distance as those for a double-slits that has the same width and

separation. Therefore, the following expression still applies:

We hence repeated the experiment for N-slits based on the same pattern that we did for the

double slits, by simply replacing the double silts filter with the N-slits; as the following

figure shows:

There were roughly eleven different positions on the optical axis we have set up each N-

slits filter, shortly after, we have measured the bright-to-center distance for every 1st, 2

nd,

3rd

, and 4th bright patterns for each positions of each filters.

We have hence obtained several proper results for the lens 12 (regular) slits (2), (3), (4),

and (5), which are shown as follows:

(For further details, see: "Data Analysis and Measurements Manual", Page 4, 10-11, 15-16)


Table of measurements and results, see Appendix, Page10











The following picture is a typical N-slits diffraction pattern with Lens 12 (regular) (2),

135.0 cm from filter to screen:

One must notice that the long distance between the bright patterns are caused by the

relatively large numbers of the slits meanwhile, the width in between each pairs of double

slits must be small; moreover, it is obvious that the length of separation between each slits

must not be that bigger comparing to the width, consequently, it means the ratio of the

separation and the width is not too big to create too many bright patterns within one

pattern length of the first decay period of the overall signal slit amplitude. The following

figure shows a intensity fitting function for seven-slits diffraction pattern for the same

width and separations:

In the above figure, the black curve is the overall single slit amplitude, the blue curve

indicates the intensity for a double slit of this set up, while the red one is the actual

intensity for seven-slits with the same slit width and separation; the following figure is a

"amplitude-free" actual curve of the seven slits diffraction:

The following picture shows another typical N-slits diffraction pattern, which was taken

for the Lens 12 (regular) (5), 150.0 cm filter to screen distance; which has a relatively

large separation/width ratio, however less numbers of the slits within unit slit-length:

One may notice that, due to the relatively bigger separation/width ratio, the number of the

bright pattern within one decay period of the single slit amplitude is much more than the

previous sample, while the less numbers of the slits within each unit slit-length has

decreased the separation for all the dark patterns, which in turn, makes the such N-slits

diffraction pattern look similar as a regular double slits pattern.

The following figure shows a intensity fitting function for five-slits within the unit slit-

length:

In the above figure, the black curve is the overall single slit amplitude, the blue curve

indicates the intensity for a double slit of this set up, while the red one is the actual

intensity for seven-slits with the same slit width and separation. As the numbers of the

slits within each unit length is not that many, which turns out to be a similar intensity

function like the double slits ones, however, there always are those "little bright" patterns

for the N-slits in between two bright patterns for the double slits', while their intensity is

relatively too small for human eyes to spot; in general, the 'amplitude-free" intensity

function looks still likewise a double pattern, as the following figure shows:

Section IV: 500/mm, 1000/mm slits fiber, and hallow cylinder fiber

1. 500/mm, 1000/mm Slits Fibers

During this experiment section, we have measured the curve of the diffraction pattern for

500/mm, and 1000/mm slits fibers, the setting for the experiment is as the following

picture shows:

As the above figure indicates, we were to capture the picture of the diffraction patterns for

each angles of rotations, by fixing the fiber-to-screen distance along the optical axis.

meanwhile, as we have found during the previous sections. The width of each silts will

decrease the decay period of the overall intensity; meanwhile the separation/width ratio for

each pairs of the double silts determine the numbers of the bright patterns within one

single decay period. More importantly, the numbers of the pairs of the slits within each

unit length will increase the length of the dark patterns, which relevantly would result

more noticeable bright patterns on the screen.

We have captured varies of the looks of the diffraction patterns for different values of the

angular rotations, for instance, the following set of figures contains the bright patterns

distributions for a 500/mm, and 1000/mm slits' fibers:

500/mm, 0 degree, 15.0 cm fiber to screen separation:





The reason for such phenomenon is that, for any requited angle of the bright pattern, the

light always choose to hit the screen at the spot where the shortest distance along the

optical axis is; therefore, when the rotation happens, it has somehow, decreased the filter

to screen distance for each bright patterns despite the center one, as a result, the diffraction

patterns no longer distribute as a straight line, but a curve.

2. Hallow Cylinder Fiber

During this experiment we have fixed the distance from the hallow cylinder fiber to the

pattern screen, change only the vertical angle of it, while capturing varies pictures with

respect to its fixed fiber-to-screen distance, as the following figure shows:

We have observed that, the scattered pattern was separated all 360 degrees, while by

tilting it with respect the vertical axis, it yielded different looks of the patterns as the

following pictures indicate:

256.5 cm separated, 90 degrees:




The assumption for such phenomenon is that, the pattern was not due to any slit-based

diffractions, it was most probably caused by the path difference of the scattered light rays,

as the following picture shows:

For any randomly scattered light rays within a small size would be relatively treated as

parallel rays, the above figure indicates that along certain tangent plane of the hallow

cylinder fiber, the two rays may concentrate along the same direction, while having an

integer numbers of the wave length difference; more importantly, such path difference

would have generated a bright pattern, in the meantime, it is possible as well for an integer

numbers of the half wave length to be differed from each of the parallel rays, which in

turn, would have created a dark pattern. More importantly, we could assume that the

scattering was partly symmetric, which for a cylinder screen plane, they shall have the

same nature of diffraction patterns, as the following figure shows:

As a result, by tilting the vertical angle of the hallow cylinder fiber, the actual looks of the

pattern on the screen would vary with respect to the angles; geometrically, one may treat

the curve on the physical screen as the curve which is generated by a plane cutting into a

cone surface.

Section V: Conclusions

Our experiment had taken places in two weeks, while many important futures of slits-

based diffractions were concluded. As a result, we have acknowledged that:

1. For any single slit diffractions, the shorter the width is, the longer its decay period for the

intensity will be, mathematically, the intensity function for a single slit pattern is that:

Where:

2. For any double slits diffractions, the greater the separation/width ratio (d/w) is, the more

bright pattern would have occurred within one decay period of its single slit amplitude,

mathematically, the intensity function for a double slit of width w, and separation d is that:

Where:

3. For any N-slits diffractions, the more pairs of double-slits for the unit length of the filter,

the longer the dark patterns on the screen there would have been; which also keeps the

total number of the bright patterns as the double slits should have generated, however, it

may seemingly as so, yet there still are several local maxima occurred in between of each

bright patterns for a double slits diffractions. Mathematically, the intensity function for a

N-slit which has its width of w, separation of d, and N pairs of double-slits for each unit

filter length is that:

Where:

4. For a large numbers of N slits per unit length, the pattern starts to have visible curvature

on its pattern screen, which is due to the shortest optical axis distance for generating its ith

bright patterns, as a result, its overall looking would have a parabolic curve along its

pattern axis.

5. The diffraction pattern generated by a hallow cylinder fiber is not of any slit-based

diffraction patterns. the bright and dark patterns are created because of an integer numbers

of whole wave length or half wave length difference along the path of parallel scattered

rays, respectively.

Summer Proj II

Documents

diffraction analysis

single slit diffraction

intensity value

sine value

tangent value

square value

major reference angle

physical path difference