Interference and Diffraction Effects of Light

By Bobby Rohde

4-23-99

Abstract

This experiment was designed to study interference and diffraction effects in light.  The experiment had several sections dealing with passing a laser through single slits, double slits, multiple slits, wire meshes, and grazing a ruler.  The wavelength of the laser was found to be 676 +/- 6.9 nm (single slit), 595 +/-14 nm (double slit).

Introduction

The purpose of this experiment was to investigate interference and diffraction effects of light, with primary interest in methods of measuring the wavelength of light.  The first portion consisted of passing a laser through a single slit whose width was varied from 40 to 160 mm, and using the diffraction pattern to determine the wavelength which was found to be 676 +/- 6.9 nm.  The next section dealt with using two slits of varying width and separation and resulted in a value for the laser's wavelength of 595 +/- 14 nm.  The laser was then passed through both increased numbers of slits and wire meshes and the data was compared to the theory (see Results/Analysis Section 3, 4).  Finally the laser was shined at grazing incidence on the surface of a grooved metal engineering ruler and the resulting light pattern was analyzed to give a value of the laser's wavelength of, which however was flawed due to bad data.

Experiment

This experiment was naturally divided up into 5 parts consisting of: Single Slit, Double Slit, Multiple Slits, Wire Mesh and Ruler.  With the exception of the ruler whose technique is described below (section 5), the data for each of these sections was collected using a computer connected to a photodiode.  The photodiode was run along a track at a constant velocity measured to be 1.000 +/- 0.015 cm/sec as clocked by a stopwatch using multiple measurements and running distances and using the standard deviation as the error.  In this way it was possible to scan along the entire diffraction/interference pattern.  The data thus recorded was stored with the computer and analyzed later.  Where appropriate the width and separation of slits and similar items was measured using the travelling microscope.

Section 1 (Single Slit):

In this part of the lab a laser was set up so as to shine through a single slit and then onto the moving photodiode (figure 1).  Scans were recorded using three different width slits: 162, 80.7, and 43.3 mm.

Lab Setup (Fig 1)

Section 2 (Double Slit)

Same configuration except that the laser was shined through a double slit with the size and separation of the slit varied such that values of separation were found to be 97, 213, 470 mm and the slits had a width of 30/35, 37/39, and 38/35 mm.

Section 3 (N slits)

During this section the same setup was used to measure barriers with increased numbers of slits.  Sets of 2, 3, and 4 slits were measured.

Section 4 (Wire Mesh)

In this portion of the experiment the laser beam was shown through three separate fine wire meshes and each time the pattern created was recorded.  This was done to simulate the effect of passing the laser through a very large number of slits.

Section 5 (Ruler)

In this section a laser beam was shined on an engineer's ruler at grazing incidence so that the reflection from the surface and the interactions with the grooved measuring points on its surface were used to produce a diffraction pattern.  This was projected onto a screen position so as to be perpendicular to the unimpeded path of the laser beam.  On the screen was fastened a piece of paper upon which the pattern was displayed.  Using a pencil the location of the brightest spots of laser light were recorded, and later examined.

Ruler Configuration (fig 2)

Results/Analysis

Section 1:

In the first portion of this experiment with only slit the only effects to consider are diffraction effects and thus I(q)=I(0)(sin(a)/a)^2, where I is the intensity, q is the angle of the beam displaced from straight after passing through the slit and a = (p*a/l)*sin(q), with a equal to the width of the slit and l the wavelength of the laser.  From this we can find the expected maxima and minima of I as a function of l.  We then take the data and read the maxima and minima values directly from the graphs.  After converting between data point number to actual distance in meters by using the velocity (see Experiment, opening statements), and the sampling rate, we then used a linear fit with x vs a/(p*a), where x is the actual distance displaced.  By using a to be the appropriate maxima or minima values and looking at the slope of that line we arrived at values for D*l where D is the distance from the grid to the photodiode.  Thus we got the following results:

Measurements of l based on maxima values (Table 1)

Measurements of l based on minima values (Table 2)

Error in l : The error was arrived at from several places, including the uncertainty in measuring D and a, the uncertainty in the velocity of the photodiode, and the uncertainty in localizing the center of the graph and the uncertainty in locating the minima and maxima from the graph.  The biggest contributors tended to be error in D and uncertainty in finding the x position, however no terms for which a reasonable error could be described were neglected.  In a strict sense the error is not totally symmetric in angle because we are using a flat rather than a curved screen however since D x, this effect was negligible relative to other effects.

These measurements taken together give a net result of l = 676 +/- 6.91 nm.

As one can easily notice these values for l do not readily conform for the expected value of ~ 630 nm, however it is useful to notice that values as based on the maxima and minima do conform well to one another.  While I can't exactly say where this enters into the data I suspect the presence of a systematic influence in D, velocity, and/or sampling rate.  The most probable seems to me to be with the velocity.  The velocity measure was recorded at the end of the lab and the motor may have warmed up or for some other reason varied with respect to the time at which it was measured versus the velocity it gave when these measurements were taken at the beginning.

Single Slit with Width 162 mm (Fig 3)

Single Slit with Width 80.7 mm (Fig 4)

Single Slit with Width 43.4 mm (Fig 5)

Section 2:

In this portion of the lab, we used sets of double slits in order to determine the wavelength.  Using a similar method as above I read off the locations of the minima from the graph and used these together with knowledge that I(q)=4*I(0)(sin(a)/a)^2*cos^2(b) where everything is as in Section 1 with the addition that b = p*d/l*sin(q), from this we see that x_min = +/- (2*m-1)/2*l*D/d, where m is the integral number of the wave and d is the separation of the slits.  From this we can find the values of  by using a linear fit and reading the slope.  For this the data (see example data set, fig 6) from all three slits was used so that one could hold m constant.  From this we found the slopes by fitting 10 sets of 3 points each a line and then using a weighted average to find the best value, which was determined to be l = 595 +/- 14 nm.  This value is of course rather low except to say that the values across all three data sets were consistently low.

Section 3:

In this section we looked at the N-slit case for N =2, 3, and 4.  Qualitatively this is similar to the double slits discussed above, however the analysis is only crudely done so as to give an impression of what the data does rather than make a precise determination of the wavelength.

N=2 (fig 6)

N=3 (fig 7)

N=4 (fig 8)

As one can see the data at higher orders follows the same pattern as that for first order but with extra oscillations built-in.  The number of secondary maxima is equal to the number of slits - 2.

Section 4:

In this part of the experiment the wire meshes were used to examine the effect of passing light through a large number of slits.  The results were similar to N-Slit except that the increased number of slits made the secondary maxima impossible to see thus leaving only periodic spikes as shown in this example.

Grid Example

Section 5:

In this section a laser beam was reflected off a ruler and the diffraction pattern made by the marks in the ruler was studied.  We know that the diffraction pattern thus established is such that l = d*(x_p(x_p - x₀))/(2*p*D^2), where  is the wavelength, x₀ is the place where the unimpeded beam would hit, x_p is the location of the p^th maxima, d is the distance between increments on the ruler and D is the distance from the ruler to the screen.

First Data Analysis Attempt (Table 3)

d= 885 mm, D = 92 cm

From looking at the wavelength values it obvious that this is no good.  After a little consideration I realized that D was recorded not from the ruler but from a location where the slits used to be held and had not been remeasured.  Oops.  If one assumes that the wavelength is supposed to be 632 nm, and through some back calculation we get that D should have been 37.5 cm, which is reasonable.  In this case the results would have looked as follows:

Fudged Results
d= 885 +/- 2 mm, D = 37.5 +/- 0.5 cm (Error on D is as if measured)

The error on the wavelengths comes from the extrapolated error from the measurement limits on each of the terms in the equation for wavelength.

Thus we would have had a value of l = 632 +/- 1.9 nm

Conclusion

After looking at diffraction and interference in all these different ways I can honestly say that I am tired of doing these labs were I do things I have done many times before for no particular purpose except to demonstrate once again that I know what to do with a set of numbers and a bunch of formulae.  In order to report the same dull uninteresting types of results over and over with full technical analysis requires so much time as to make doing these measurements in this way both unfulfilling and a drain on ones resources for other classes and outside activities.  We have shown that exactly as predicted that the ruler gives a more accurate way of measuring these effects and that additionally trying to measure wavelength with a diffraction grating can give somewhat inconsistent answers, either because the equipment wasn't very consistent (possibly in the speed of the motor) or that I was nearly as good at measuring the quantities as I thought I was.  In conclusion it is fair to say that the data for this lab was very poor and the results are questionable.  I did what I could with the numbers I had.