Estimation of Intensity Profile of Single Slit Diffraction Patterns

Joonsuk Huh (20125098)

(General Physics Experiment 2: Section 8)

Abstract: We estimated intensity profiles of single slit diffraction patterns via digital

image processing. We used slit width 0.02mm, 0.04mm, 0.08mm and 0.16mm. Then we

compared estimated intensity profiles with theoretical calculations based on phasor

diagram. We show that estimated intensity profiles well agree with theoretical calculations.

Theory & Introduction

Figure 1. Schematic of diffraction by slit with width a. D is the distance between the slit and the screen. y is the distance from the diffraction center.

Diffraction is a wave phenomenon such that

when a group of waves passes an obstacle or a hole,

its direction and intensity changes so that it looks

like spreading out.[1] It essentially comes from the

superposition of infinitely many electromagnetic

waves with slightly different phases. Let be

angular phase difference of wave at point 1 and

point 2 at Figure 1. Then the intensity of electro-

magnetic wave due to infinite superposition of

electromagnetic waves between point 1 and point 2

is given by[2]

= !!"#!!!!

!

(1)

where I is the intensity of the light of given and I0

is the maximum intensity of the light (when =0).

Figure 2. Phasor diagram for the diffracted light with net angular pahse difference .

This formula can be derived geometrically from

phasor diagram[2,3]. If we think a single slit as a

composition of infinitesimal width slits, then

diffracted light can be thought as an infinite

superposition of lights emitted from every point

between point 1 and 2 due to Huygens principle.

Therefore its phasor diagram is like Figure 2. For

each , the amplitude of superposed wave E is same

as the length of the line segment bc. From the

isosceles triangle abc, the length of bc is given by

= = 2sin !!

(2)

where R is the radius of the half-circle C. Because

the length of the arc bc is same as original

magnitude of wave E0, R can be written as

Figure 3. The path difference shown near the slit.

= !!!

(3)

If we combine (2), (3) and use the fact that the

intensity I is proportional to E2, then we get (1).

In the real experiment, is not the direct control

variable. Therefore we need to express as a

function of slit width a and other variables. From

Figure 3, one can see that path difference between

the wave starting from the point 1 and the wave

from the point 2 is given by

= sin (4)

Because of proportionality between and the

wavelength is and 2,

= !!!!= !!" !"#!

! (5)

Finally, from figure 1, for small , sin tan =

y/D. Therefore the final expression for I is

= !!"#!"#!"!"#!"

!

(6)

Where y is the vertical distance from the center of

the diffraction pattern and D is the distance between

the slit from the screen. Especially, from (6) one can

see minima occurs when

!"# =!"#!

= 1,2,3 (7)

Experimental Procedure

We used 650nm wavelength laser and slits of

width a = 0.02, 0.04, 0.08, 0.16mm respectively. Slit

to screen distance D = 1m. We took digital pictures

of diffraction patterns in dark room. We used Mathe

Figure 4. Estimated intensity profile of diffraction patterns from slit with width a=0.02, 0.04, 0.08, 0.16mm along with theoretical calculations.

Table 1. Distances ymin with theoretical calculations by Eq. 1. Estimated slit width a is presented with standard deviation (SD). All values are presented in mm unit.

matica packages LineProfile function to extract

intensity information of digital images along a line

segment. We normalized extracted intensity data by

dividing them by the maximum intensity value.

Then we interpolated normalized intensity data

points.

Result & Discussion

Figure 4 shows estimated intensity profile along

with theoretical calculations. Froms figure 4, we see

that as slit width a increases, the width of intensity

curve decreases. This can be shown from (7) and

also intuitively expected. From figure 4, one can see

that theory and experiments agree well.

Table 1 shows estimated distances ymin from the

center of the diffraction pattern to the minimum

intensity positions along with theoretical calcul-

ations by (7). We also estimated slit width a from

measured ymins using (7). Mean estimated value of a

and standard deviation (SD) is presented in Table 1.

We can see that theory and experimental values only

differ up to few millimeters. Specifically, their

mean % error between measurement and theory was

about 7.8%.

Conclusion

We observed single slit diffraction patterns of

650nm wavelength laser on the screen apart from the

slit by 1m. We used slits with width with width a =

0.02, 0.04, 0.08, 0.16mm. We estimated normalized

intensity profiles of diffraction patterns via digital

image processing using Mathematica software and

interpolated and plotted them as a function of

distance y from the center of the pattern. We saw

that as slit width a increases, the width of intensity

curve decreases as expected. We compared esti-

mated intensity profile with theoretical formula (6)

and found that theory and experiment agree well.

We also measured distance ymins from the center of

the diffraction pattern to the points of minimum

intensity. We compared these with approximation

formula equation (7) and confirmed that this formula

and experiment also agree well.

References [1] D. Halliday, R. Resnick, J. Walker, Fundamentals of Physcis (Wiley, 2010, 9th ed.), p.963. [2] D. Halliday, R. Resnick, J. Walker, Fundamentals of Physcis (Wiley, 2010, 9th ed.), p.997. [3] D. Halliday, R. Resnick, J. Walker, Fundamentals of Physcis (Wiley, 2010, 9th ed.), p.998.