Fraunhofer Diffraction - University of Colorado Boulder

Physics 3340 Spring 2001

Fraunhofer Diffraction

Purpose

The experiment will test the theory of Fraunhofer diffraction at a single slit by comparing
a careful measurement of the angular dependence of intensity of diffracted laser light with
predictions of the theory. A qualitative overview of interference and diffraction phenomena will
round off the experiment.

Light waves are known to be transversely polarized waves. Unfortunately, the majority of
diffraction problems of practical interest can be solved only for the simpler case of waves in a
scalar field. The lack of a sound fundamental basis for diffraction theory emphasizes the need for
experimental tests of its ability to correctly predict optical phenomena.

Introduction

"Diffraction" refers to the spreading of waves and appearance of fringes that occur when
a wave front is constricted by an aperture in a screen that is otherwise opaque. The light pattern
changes as you move away from the aperture, being characterized by three regions.

Plane waves Intensity profiles

X

θΜΙΝ Z
OA

Aperture In ten s ity Z ~ a2/λ
Fraunhofer
Shadow Fres n el Diffract ion
Diffract ion

Figure 3.1 Diffraction of plane waves at an aperture.

1. In the shadow region, close to the aperture, the boundary of the transmitted light is sharp
and resembles the aperture in shape.

2. As you move away into the Fresnel region, the beam width remains comparable to that in
the aperture, but narrow fringes appear at the edges.

3. Far away, in the Fraunhofer region, the beam spreads to a width much greater than that of
the aperture and is flanked by many weaker fringes.

Fraunhofer Diffraction 3.1 Spring 2001

The Fraunhofer region is chosen for the experiment because the broader fringes are easier

to measure with an optical detector of finite aperture, and the calculations are more

straightforward than in region 2.

y’ P(x,y)
x’

R xy
r

Z axis

dA’ z Observation
Aperture plane

Incident
waves

Figure 3.2. Geometry for the Kirchoff-Fresnel diffraction integral.

The Huygens-Fresnel principle governs diffraction phenomena: "Every unobstructed
element of a wavefront acts as a source of spherical waves with the same frequency as the
primary wave. The amplitude of the optical field beyond is a superposition of all these wavelets
taking account of their amplitudes and phases."

The Kirchoff-Fresnel diffraction integral gives quantitative expression to these ideas.
Consider plane waves incident on an aperture from the left, as shown in Figure 3.2. The incident
field is described via:

( )EINC z, t = E0ei(kz−ωt)
The field in the aperture i.e., where z=0, is then EINC (0, t ) = E0e−iωt . A typical element of
the wave front of area dA' and at position rG′ = (x′, y′, 0) then acts as a source of Huygens

wavelets. Our light detector sits at a point P in the observation screen, at a vector distance, R,

from the origin of the aperture. Note that the distance of the detector, P, from the element dA', is

given by: G rG′
R
r = −

The field at P due to the element dA' is then equal to:

dE (P) =  E0 e − iω t dA′  ×  eikr 
 λ   r 
  

= (Source strength )× (Huygens spherical wave)

The field at P due to the entire aperture is then a superposition of the wavelets from all
the elemental areas:

Fraunhofer Diffraction 3.2 Spring 2001

∫∫( )EP=  E0 e − iω t dA′  ×  eikr 
 λ   r 
Aperture   
Area

The detector measures the light intensity at P, rather than the electric field strength.

IGntensGityGis given by the magnitude of the time averaged Poynting vector,
S = E × B 2µ0 = E 2 2Z0 zˆ . Therefore, the detector measures:

I (P) = E (P) 2 2Z0

Where Z0 = µ0 ε0 = 377Ω is the characteristic impedance (the ratio of the electric

field magnitude to the magnetic field magnitude) of free space.

In the present experiment, the aperture is a slit of width, a, while the detector is a

photodiode at position, P. Evaluation of the Kirchoff-Fresnel integral for the slit gives the

following prediction for the diffracted intensity:

( )I  a2  sin 2 π xa 
P (x, z) = I INC  λz  π λz 
 
xa 2
λ z 

This prediction is subject to the condition that the observation point is far enough away so
that z a2 λ . From a practical perspective, z 10 a2 λ is sufficiently far away for the theory

to be quite accurate.

Notice that the formula applies to a situation in which plane waves of uniform intensity

are incident normal to a long narrow slit of uniform width. The rays from different parts of the

slit to a given observing point are effectively parallel (the Fraunhofer condition). The design of

the experiment must mimic these conditions as closely as is possible. The condition on
parallelism of rays is adequately satisfied at a distance of z 10 a2 λ or more, since the error

depends on the square of this quantity.
The general form of the intensity equation as a function of detector position, x, at a fixed

distance, z, from the slit is sketched below on linear and logarithmic scales. For the range of
intensities that you are likely to measure, only the log scale can represent the weaker high order

fringes. You should plot the data on 5 cycle log-linear graph paper. If you use a plotting program
to display the data, again, be sure to use a logarithmic intensity scale.

Fraunhofer Diffraction 3.3 Spring 2001

Apparatus

A beam expander increases the diameter of the laser beam so that the light is uniform

over the width of the diffracting slit. The intensity of the diffracted light is measured with a

photodiode. The photocurrent is determined from the voltage developed across a load resistor,

using a digital multimeter. An analyzing slit of width, w, restricts the effective aperture of the

photodiode so that the detailed shape of the diffraction pattern can be discerned. The photodiode

and analyzing slit can be moved horizontally (the x-direction) by a calibrated translation stage

driven by tuning a lead screw of pitch 1 mm. The revolution counter indicates the integral

number of mm. Each small division on the calibrated knob represents 0.01 mm. Be sure to

disengage the worm drive by tuning the lifting screw before use.

diffracting analyzing R 9V
slit slit

Lx1 Lx2 I(x,z) DMM

He-Ne
laser

Photodiode lead screw
(1mm pitch)
Beam a
expander

Calibrated
translation
stage

Figure 3.3 Experimental arrangement for Fraunhofer Diffraction.

Fraunhofer Diffraction 3.4 Spring 2001

Outline of the Experiment

1. Set up the apparatus
• Make up the diffraction and analyzing slits from razor blades.
• Measure the width of each with the measuring microscope to 2% if possible.
• On the optic bench, set up the laser, beam expander, diffracting slit, and calibrated
translator. Mount the photodiode with analyzing slit on the translator.
• With the laser on, check that all the light passing through the analyzing slit also enters the
detector for all positions of the translator. Draw the appearance of the fringes.
• Connect the photodiode circuit. Check that the output is not saturated when the detector is
at the center of the central maximum.

2. Measurements
• Test the symmetry. Measure the intensities and positions for all the measurable maxima
and minima on both sides of the center. The theory predicts that the pattern should be
symmetric about the center. If it is not symmetric, then either the experiment has
problems or the theory is wrong (or both!).
• If there is an asymmetry, assume that it is the experiment. Fix the problem before you go
ahead. It is most likely a combination of non-uniform slit width, tilted slit, and detector at
the wrong height.
• For the main data run, measure intensity versus detector position with about 5 points per
peak. Make a step-wise scan from the x=0 position, to the maximum positive x; then
return step-wise through the x=0 position and out to the greatest negative x. Then return
to zero. Why is this procedure necessary? Do you need to improve on the procedure?
Graph your results on 5-cycle log-linear paper.
• Compare the intensity of the full laser beam with the intensity at the center of the central
maximum.

3. Compare you data with the following predictions of the Kirchoff-Fresnel single-slit formula.

• For the zeros of intensity, the position of the n-th zero should satisfy:

xn = n  λz  n = ±1, ±2, ±3,"
 a 

Graph the quantity axn/z versus n for the various zeros. This plot should give a line with a
slope of λ. Calculate the value of λ and compare it to the result you found from the

Ronchi grating in the Projection Microscope laboratory.

• For maxima of intensity, the relative intensity of the n-th subsidiary maximum is

predicted to be approximately:

Fraunhofer Diffraction 3.5 Spring 2001

I In 0) =  1 1 2 n = ±1, ±2, ±3,"
 n+ 2 
(x = π 2

Graph In versus  n + 1  on log-log paper. We expect a slope of -2.
 2 
I (x = 0)

• The intensity at the central maximum is predicted to be:

I (x = 0) = a2

IINC λ z

4. Overview of interference and diffraction.

In your experimental kit, you will find two transparency slides with several frames of
circular apertures. Slide A has four rows and three columns with single circular and
rectangular apertures (rows 1-4 in column A) and pairs of apertures with different
separations (rows 1-4 in columns b and c). Slide B has single apertures and several arrays
of apertures. You can use the inspection microscope to examine the slides. Examine the
diffraction patterns produced with the transparencies and interpret what you see.

Illuminate each frame separately, using an aperture to restrict the size of the laser beam.
Change the frame by moving the transparency over the aperture. Stick some magnetic
tape on the slide and use a steel aperture plate so that the magnetic force holds the slide in
position. Shown below are some pictures of several of the patterns you should be able to
find.

Slide A 1a Slide A 2a

Slide B 1b Slide B 2a Slide B 3b
Fraunhofer Diffraction 3.6 Spring 2001

Problems

1. Verify that the equations of part 3) in the experiment do indeed follow from the Kirchoff-
Fresnel formula for single slit diffraction

2. Slit widths

• What is the maximum width, a, of the diffracting slit if it is to satisfy the
condition for Fraunhofer diffraction when the detector is 1 meter away from the slit,
assuming that the wavelength is 0.6 microns?

• Calculate the width, w, of the scanning slit such that it samples only one tenth of
the width of a subsidiary fringe. This width is needed so that you can properly map
the shape of each fringe.

3. Laser beam.

What is the minimum diameter needed for the expanded laser beam if the intensity is to

be uniform to within 5% over the width of the diffracting slit? In other words, what is the

minimum diameter needed so that: I

I (x = 0) − I (x = a 2) = 0.05 5%
I (x = 0) a

Assume that the beam has a Gaussian intensity distribution: I0/e

( )Ir = I e− r2
R2

0

4. Geometric problems in setting up the experiment. Dr

a. If the diffracting slit is tapered instead of being uniform in width, what would the
diffraction pattern look like? Sketch the fringes. What effect would it have on the
intensity distribution observed with the photodiode?

b. If the slit is uniform, but slightly off the vertical position, how would the
diffraction pattern be changed? Suppose the detector is correctly positioned for
the central fringe, due to the non-vertical slit and is constrained to move
horizontally, how would the apparent intensities of the fringes be changed? Sketch
roughly the logarithmic graphs for the correct and incorrect geometries.

Fraunhofer Diffraction 3.7 Spring 2001