Fraunhofer Diffraction with a Laser Source

Second Year Laboratory

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Health and Safety Instructions.

There is no significant risk to this experiment when performed under controlled laboratory conditions. However:

• This experiment involves the use of a class 2 diode laser. This has a low power, visible, red beam. Eye damage can occur if the laser beam is shone directly into the eye. Thus, the following safety precautions MUST be taken:-

• Never look directly into the beam. The laser key, which is used to switch the laser on and off, preventing any accidents, must be obtained from and returned to the lab technician before and after each lab session. • Do not remove the laser from its mounting on the optical bench and do not attempt to use the laser in any other circumstances, except for the recording of the diffraction patterna. • Be aware that the laser beam may be reflected off items such as jewellery etc

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26-09-08

Department of Physics and Astronomy, University of Sheffield Second Year Laboratory

1. Aims The aim of this experiment is to study the phenomenon of diffraction using monochromatic light from a laser source. The diffraction patterns from single and multiple slits will be measured and compared to the patterns predicted by Fourier methods. Finally, Babinet’s principle will be applied to determine the width of a human hair from the diffraction pattern it produces.

2. Apparatus • Diode Laser – see Laser Safety Guidelines before use, • Photodiode, • Optics Bench, • Traverse, • Box of apertures, • PicoScope Analogue to Digital Converter, • Voltmeter, • Personal Computer.

3. Background (a) General The phenomenon of diffraction occurs when waves pass through an aperture whose width is comparable to their wavelength. It will be investigated in this experiment using light from a diode laser. This is a highly monochromatic source whose long coherence length makes if particularly useful in diffraction experiments, as it produces bright and well-defined diffraction patterns. A photodiode is used to detect these patterns and an analogue to digital converter will enable the data to be stored on a computer. Fourier analysis will allow these patterns to be compared to the theoretical predictions.

(b) Diffraction at a Single Slit Consider parallel, monochromatic light incident on a single slit of width b, as shown in Figure 1 below. After passing through the slit, light does not travel only in its original direction. Instead we should consider each portion of the slit as a source of light waves (Huygen’s Principle).

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Figure 1 Diffraction at a slit As semi-circular wavefronts propagate out from each portion, they will interfere. The resultant intensity, as viewed on a distant screen, depends on the angle θ. A full description of the diffraction pattern is given below using Fourier theory, but first the positions of the minima will be established by simpler reasoning.

(i) Simple Theory

Consider Figure 2. • Dividing the slit in half shows that the path difference between waves from the centre and the edge of the slit is (b/2)sinθ. θ • There will be complete destructive interference b/2 between waves from the two halves of the slit when this path difference is equal to half a wavelength, i.e. (b/2) sin θ = λ/2 , b sinθ = λ . (b/2) sin θ • Similarly, dividing the slit into 4 or 6 equal parts, Figure 2 Diffraction shows that destructive interference occurs for geometry (b/4) sinθ = λ/2 , (b/6) sinθ = λ/2 , etc.

Thus, minima occur in the diffraction pattern for sinθ = mλ / b, m = ±1, ±2, ±3, etc. (1) Note: sinθ = 0 is not a minimum but the principal maximum – see Fig. 4.

(ii) Fourier Theory Fourier theory can be used to predict the form of the diffraction pattern from a given aperture. In optics, the amplitude of the diffraction pattern is given by the Fourier transform of the aperture function, which is a mathematical method of representing the transmission of light from an aperture. The aperture function f(x) for the single slit is given by the so called “top-hat” function and reflects the transmission of light through the aperture. f(x) = 0 = opaque f(x) = 1 = transparent

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For the single-slit the aperture function is given by :- f(x) = 1 for –b/2 < x < +b/2 f(x) = 0 for elsewhere which is shown in Figure 3.

f(x)

−b/2 +b/2 x

Figure 3 The single slit and its transmission function f(x)

The amplitude ()kF of the diffracted light is given by the Fourier transform of the aperture function:-

+∞ = ∫ ()ikx dx.exfF(k) (2) ∞− where 2π k = sinθ . (3) λ k is known as the propagation constant. Note that it has dimensions of inverse length. θ is the angular distance from the centre of the diffraction pattern (see Fig. 2) and λ is the wavelength of the laser. This leads to an intensity variation given by:

2 I β )( ⎡sin β ⎤ = ⎢ ⎥ (4) I )0( ⎣ β ⎦ where πb β = sinθ (5) λ as plotted in Figure 4. Equation 4 has the value 1.0 at θ = 0 so the graph shows the intensity at any point relative to this central maximum of 1.0 at β = 0 .

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Figure 4 The single slit diffraction pattern. For the full derivation of this result please refer to any recommended textbook - e.g. “Optics” by Hecht. The final result for the diffraction pattern is obtained by substituting for β in Eq 4, giving:

2 ⎡ ⎛ bsinθπ ⎞⎤ sin⎜ ⎟ I θ )( ⎢ λ ⎥ = ⎢ ⎝ ⎠⎥ . (6) I )0( ⎢ bsinθπ ⎥ ⎢ ⎥ ⎣ λ ⎦

(c) Diffraction at two slits (i) Simple Theory Consider two slits, a distance a apart. In the classic Young’s slits experiment the width of the slits is assumed to be negligible. So to derive the interference pattern, two point sources of light waves are considered.

The path difference between waves from the two slits is Δ = a sin θ. θ Hence there are maxima in the interference pattern when the interference is constructive: a θ Δ = a sin θ = mλ, m = 0, ±1, ±2, etc., (6a) and minima when the interference is destructive: Figure 5 Double slits Δ = a sin θ = (m+ ½)λ, m = 0, ±1, ±2, etc.. (6b) asinϑ

More generally, at any point, the electric field due to light from the top slit is

E1 = E0 sin ωt ,

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and the field due to light from the bottom slit is

E2 = E0 sin (ωt + φ), where φ / 2π = Δ / λ so φ = 2π a sinθ / λ. So at any point, the total electric field is

E = E0 sin ωt + E0 sin (ωt + φ) = 2 E0 cos(φ/2) sin(ωt + φ/2). The light intensity is proportional to the square of the magnitude of the electric field, 2 2 2 I ∝ 4 E0 cos (φ/2) sin (ωt + φ/2). When, as in this experiment, the time-averaged intensity is measured, the diffraction pattern is a cos2 distribution (as shown in Figure 7b): 2 2 I(θ) = Imax cos (φ/2) = Imax cos (πa sinθ /λ) (7)

(ii) Fourier Theory The same result can be obtained using Fourier theory. In Fourier theory, an ideal (infinitely narrow) aperture is represented by a delta function. So the diffraction pattern for Young’s slits is obtained by taking the Fourier Transform of two delta functions a distance a apart – see Figure 6.

δ(−a/2) δ(+a/2)

β x Figure 6a: aperture function for two ideal slits. Figure 6b: interference pattern for two ideal slits.

(iii) Two Slits of Finite Width In practice the slits have finite width, so the diffraction pattern observed for double slits is a convolution of the double slit and single slit patterns, as shown in Figure 7. Such convolutions can be derived neatly using Fourier theory.

Slit width = delta function β

Figure 7: Development of the diffraction pattern for two slits Finite slit width

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(d) Diffraction from Multiple Slits The diffraction pattern for N identical apertures of finite width can be derived as follows: • The diffraction pattern for one aperture is given by the Fourier Transform of the aperture function. • The diffraction pattern for N ideal apertures is given by the Fourier Transform of the appropriate array of delta functions. • The Fourier Transform for N finite apertures is simply the product of the two above functions. So for N slits, Amplitude of Fourier Transform of Fourier Transform Diffraction Pattern of = Top Hat Function × of N Delta Multiple Aperture Functions For N slits of separation a and width b, the intensity distribution is given by

2 2 I θ )( ⎡ β ⎤ ⎡sinsin1 Nγ ⎤ = 2 ⎢ ⎥ ⎢ ⎥ , (8) )0( NI ⎣ β ⎦ ⎣ sinγ ⎦ πb πa where β = sinθ as before, and γ = sinθ . Note that the first (β ) term tends to 1.0 λ λ when β =0, but the second (γ ) term tends to N2 when γ =0. The final version of equation 8 is then

2 2 ⎡ ⎛ πbsinθ ⎞⎤ ⎡ ⎛ aN sinθπ ⎞⎤ sin⎜ ⎟ sin⎜ ⎟ I θ )( 1 ⎢ λ ⎥ ⎢ λ ⎥ = ⎢ ⎝ ⎠⎥ ⎢ ⎝ ⎠⎥ . (9) I )0( N 2 ⎢ πbsinθ ⎥ ⎢ ⎛ πasinθ ⎞ ⎥ ⎢ ⎥ ⎢ sin⎜ ⎟ ⎥ ⎣ λ ⎦ ⎣ ⎝ λ ⎠ ⎦ Once again, refer to any of the recommended textbooks for a derivation of this result. For a double slit, equation (8) gives: -

2 ⎡ ⎛ πbsinθ ⎞⎤ ⎢sin⎜ ⎟⎥ I θ )( ⎝ λ ⎠ 2 ⎛ πasinθ ⎞ = ⎢ ⎥ cos ⎜ ⎟ . (10) I )0( ⎢ πbsinθ ⎥ ⎝ λ ⎠ ⎢ ⎥ ⎣ λ ⎦ At the other limit, where N is large, we are dealing with a diffraction grating. In this limit the πa diffraction pattern has principal maxima whenever sin = nπθ , where n is an integer. This λ occurs when sinθ = na λ , (11) which is the standard formula for the diffraction angles from a diffraction grating.

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4 Experimental procedure: (a) Single slit diffraction 1. Place slide number 46993 from the aperture box into the aperture holder. Turn on the laser and position the slide so that the laser is centred on the single slit (A). Adjust the slit until the diffraction pattern is horizontal. This ensures that the photodiode is correctly aligned with the diffraction pattern. The height of the photodiode may need adjustment so that it detects the pattern effectively.

2. Start the PicoScope software on the PC. As soon as PicoScope is opened it will begin logging data. To begin taking the measurements, select channel 1 as trace A with initially a gain of x1. The timebase should be set to 1s/div. Conduct a dry run by moving the photodiodee across the diffraction pattern by pressing the appropriate direction button on the controller. Adjust the gain (on screen) to ensure that you can clearly see the pattern on the screen. Note the gain that you use (although this is not applied to the data saved to disk).

3. Once you are satisfied that the data can be clearly seen, you should click on the STOP button and click on CLEAR. Move the traverse back to the beginning of its travel and close the black box to avoid unwanted background light being picked up by the photodiode. Now click “GO” and start the photodiode moving back across the pattern.

4. When you have collected the data click “STOP”. Then select “Copy as Text” on the “Edit” menu. You should now be able to paste the data series into Excel. 5. TURN OFF THE LASER. 6. It should now be possible to produce a graph of your results as explained in the Section 4.3 below. Note:- The photodiode may begin to pick up the signal only when it gets close to the central maximum. There are two ways to deal with this. One is just to plot the central part of the diffraction pattern, by changing the scale on the x-axis, when you have plotted your data in the way explained in Section 4.3 below. The other possibility, which also saves time is to begin taking readings further towards the centre of the diffraction pattern, instead of at the edge, taking care to properly renormalise the x axis of your graph. 7. In Excel copy the first page which contains your raw data and then work on the new page, so that your basic raw data is always preserved. 8. For this first and all subsequent parts of the experiment, the x-axis of your graph needs to be converted to angle of diffraction θ. The software outputs data points with the time as the x axis in ms. 9. The next step is to convert the x axis from time to the transverse position of the photodiode. You can do this by first measuring a value of the velocity of the complete traverse of the detector. Note that you will have to make a sensible choice of the centre of the pattern to fix the point with x = 0 correctly, so that x has positive and negative values about the central maximum. 10. Finally, measure the distance from the slit to the photodiode as accurately as possible and use trigonometry to convert the x axis to θ. This will allow you to plot the diffraction

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pattern against θ . If you normalise the detector output (y axis) to unity at the central maximum this will allow you to compare your results directly with the predictions of Eq. (5). Note: Try to do each of these steps in a separate column of your Excel worksheet and make a record of each calculation performed in your Laboratory Diary.

-3-2-10123 sin theta

Diffraction order (m) Figure 8. Determination of the slit width b.

11. Once you have calibrated the x axis, you will be able to determine the values of sinθ that correspond to minima in the diffraction pattern. Plot a graph of these values of sinθ against diffraction order number m, as shown in Figure 8. From Eq. (1) we see that the slope of this graph is equal to λ / b. Hence we can determine the value of b, given that the wavelength of the laser light is 633 nm. Measure the slit width with the travelling microscope and check that the value that you obtain agrees with the one you have deduced from your diffraction pattern. 12. Having calibrated the x-axis and determined the slit width, you should now be able to compare your experimental diffraction pattern with theory. In order to compare your findings with theoretical predictions, superimpose a graph of equation (6) onto the experimental graph of relative intensity against θ. Note that Excel will not plot equation (6) correctly for the central point at θ = 0 . This is because the programme is trying to divide by zero. If L’Hôpital’s rule is applied, it can be shown that equation (6) tends to 1 as θ tends to zero. This data point should be inserted manually. 13. The experimental intensity pattern which has been converted to relative intensity, should be compared with the one predicted by theory and any differences between the two should be explained. 14. Use equation (6) to calculate the expected intensity of the secondary maxima relative to the central maximum. Compare this value with the value from the experimental data. (Hint: Use your graph from part 15 to find the values of θ and I for the two maxima.)

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(b) Multiple slit diffraction 15. Insert slide number 46993 into the aperture holder with the double slit (B) aligned with the laser. Using the same procedure as for the single slit, measure the diffraction pattern from the double slit. Make sure that the sampling rate is the same as before. 16. Now superimpose your results from the single slit onto the double slit graph. Explain how the shape for the single slit fits onto the double slit pattern. (Hint – think about convolution theory). 17. Now use equation (10) to superimpose theoretical predictions for the double slit onto your experimental data, using the same procedure as for the single slit, remembering that the point at θ = 0 , I(θ ) =1. Compare your experimental graph with the theoretical one, and explain any differences. You should be able to determine the slit separation and the width of the slits by again determining the values of sinθ at which I()θ is a minimum and using equations (1) and (6b). 18. Repeat the procedure for the case of 4 slits (C). Determine the width and separation of the slits.

(c) Diffraction from a Human Hair There are many instances in physics where measurements of size cannot be made directly. The last part of this experiment explores this problem and makes use of Babinet’s principle for diffraction from complementary apertures.

Two screens are said to be complimentary if S2 is the screen that is formed when the transparent parts of S1 are made opaque, and the opaque parts are made transparent. Consider a point P, which is a point at which, in the absence of any diffracting screen, the disturbance is zero. Babinet’s theorem states that the intensity at P caused by the introduction of S1 is the same as that caused by the introduction of S2. In other words: apart from the direct light, complementary screens produce identical diffraction patterns. With regards to Fourier theory, a complementary screen to the single slit used in Part 4 would look as below:- Single Slit Complementary Aperture f(x) f(x)

−b/2 +b/2 x −b/2 +b/2 x

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In this case, the aperture function is replaced by one where the light is transmitted in all the regions where the original was opaque, and vice versa. Babinet’s principle states that this complementary screen will give a diffraction pattern that is exactly the same as the single slit, apart from in the straight-through direction. It is possible to make a screen that behaves in a very similar way to a single slit by placing a human hair between two glass slides.

19. Take a strand of hair and place it between two glass slides so that it lies as straight as possible. Now repeat the procedure undertaken for the single slit but with this new aperture in place. 20. Compare the intensity variation for the hair with the single slit pattern to determine if Babinet’s principle has been proved. Explain any differences between the two. Remember that one important difference is that the slit will interrupt most of the incident beam of the laser, but this will not be so for the human hair. 21. Use your intensity variation for the hair, and knowledge about the position of the minima and maxima, to estimate the diameter of the hair. Compare this with a value measured using a travelling microscope.