Fraunhofer Diffraction

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.

OA Intensity profiles θΜΙΝ Plane waves Aperture Shadow Fresnel Diffract ion Fraunhofer Diffract ion Z X Intensity Z ~ a2/λ

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.

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.

Z axis

Figure 3.2. Geometry for the K irchoff-Fresnel diffraction integral. Incident waves Aperture Observation plane y’ x’ dA’

r

R

z

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:

( )

0

, i kz t

INC

E z t =E e −ω

The field in the aperture i.e., where z=0, is then E_INC

( )

(

)

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:

r = −RG Gr′

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

( )

(

) (

)

0

Source strength Huygens spherical wave

i t ikr E e e dE P dA r ω λ −  _′   =_{  }× _     = ×

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

( )

∫∫

The detector measures the light intensity at P, rather than the electric field strength. Intensity is given by the magnitude of the time averaged Poynting vector,

2

0 0 ˆ

2 2

SG = ×EG GB µ = E Z z. Therefore, the detector measures:

( )

( )

2

I P = E P Z

Where Z0 = µ ε0 0 =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:

( )

(

)

This prediction is subject to the condition that the observation point is far enough away so that 2

za λ. From a practical perspective, 2

10

z a 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 2

10

z a 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.

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.

He-Ne laser Beam expander Lx2 Lx1 diffracting slit lead screw (1mm pitch) Calibrated translation stage a analyzing slit 9V R Photodiode I(x,z) _DMM

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: 1, 2, 3, n z x n n a λ   = _{ } = ± ± ±   "

Graph the quantity ax_n/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:

(

)

(

)

  on log-log paper. We expect a slope of -2. • The intensity at the central maximum is predicted to be:

(

)

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

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:

(

) (

)

(

)

Assume that the beam has a Gaussian intensity distribution:

( )

4. Geometric problems in setting up the experiment.

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.

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