Lab VII, Problem 1 : Interference Due to a Double SlitEmily YoungApril 24, 2015

Physics 1202W, Professor: Thomas Walsh, TA: Sergey Monin

AbstractA diode laser was used to illuminate a pair of slits separated by an unknown

distance, resulting in an interference pattern. The distance between the slits was examined by predicting the distance between interference minima using Huygens’

principle, and was confirmed to within experimental error.

Introduction

As a part of an investigation of virus properties, the size of small viruses was investigated. To represent these viruses, a series of slides with double slits were used to represent the viruses. The distance between the slits, representing virus size, was determined by illuminating the slits with a diode laser and examining the resulting diffraction patterns on a screen a known distance from the slits. To accurately determine the size of viruses, it is necessary to predict the distance between interference minima. This distance was achieved using Huygens’ principle, creating an equation expressing the distance between minima as a function of the wavelength, distance to the screen, and distance between slits.

Prediction

The distance between interference minima due to patterns produced when a pair of slits are illuminated with coherent light can be predicted using Huygen’s principle. At a point P on a screen distance D from the object, the distance y from the midpoint to point P can be expressed in terms of the angle the vector to point P makes with the horizontal θ, as shown in Figure 1.

Figure 1Using the pythagorean theorem, y can be expressed as

y= Dtanθ (1)

Figure 2 depicts the relevant quantities used in this experiment:

Figure 2The pathway from one end of the object to point P x1 and the other end of the object to point P x2

differ by ∆x. x2-x1= ∆x (2)

The path difference ∆x and the distance between the two slits d are related such that

sinθ= ∆x/d (3)The wavelength of the light source λ is equal to the path difference ∆x (Huygen’s principle). Relating this to the definition of given by equation 3, this gives

dsinθ =mλ (4)Where m is the order- the resulting interference pattern shows points where constructive and destructive interference occur. At points where waves from both ends of the object are exactly in phase, constructive interference occurs and this produces a maximum (m= 0, ±1, ±2, ±3,...)- resulting in a bright spot on the screen. When the waves are exactly out of phase, destructive interference occurs, producing a minimum (m= ±½, ±3/2, ±5/2,...)- resulting in a dark spot on the screen. The minimum can also be expressed as n, where m in equation 4 is replaced with (n + ½) and (n= 0, 1, 2,....).

The distance between the slits is d, which is given by equation 4. Plugging in equation 1 to give the angle gives

d= λ/sin(tan-1(y/D)) (5)Where y is the distance between maxima (or minima).

Because the distance to the screen D is very large compared to the distance between the slits d, the approximation sinθ≈tanθ can be made, and simplifying equation 6 gives

d= λD/y (6)

Rearranging for the distance between minima y givesy= λD/d (7)

Equation 7 predicts the distance between minima y of an interference pattern resulting from the illumination of a pair of slits distanced by d with a coherent light of wavelength λ distance D from the screen.

Given this equation, an increase in the size of the object would result in a decrease in the distance between maxima (or minima). This equation also predicts that with increasing distance between the object and the screen, the distance between maxima (or minima) increases.

The predicted interference pattern is such that where the path length difference ∆x is a whole number integer multiple of λ, a bright spot appears, and where the path length difference ∆x is a multiple of ½λ, a dark spot appears, with the brightest spot appearing in the middle and diminishing on both sides, as shown in the following diagram-

Figure 3: The Predicted Interference Pattern

Procedure

Figure 4: The mechanism used in this experiment

Figure 4 depicts the mechanism used to create a magnetic field in this experiment. A diode laser of known frequency (and by association, known wavelength λ) was used to illuminate a pair of slits separated by a distance d. The resulting waves hit a screen a distance D away- the interference between the waves from each slit produced an interference pattern on the screen giving information about the distance between the slits. At points where the waves from each slit interfered constructively, bright spots were produced. At points where the waves interfered destructively, dark spots were produced.

Data

Three sets of experiments with varying combinations of slit widths w and slit separations d (all with the same wavelength λ) were carried out, each with two trials using different distances between the slits and screen D.

λ=650nm w= .04mm d=.25mm

Trial D (m) m0 (mm)

n0 (mm)

m1 (mm)

n1 (mm)

m2 (mm)

n2 (mm)

yavg exp. (mm)

yavg theor. (mm)

1 .5 0 1.2 1.9 2.5 3.1 3.8 1.3 1.3

2 .25 0 .85 1.3 1.65 2.2 2.45 0.8 0.65

Uncertainty of yavg (Exp.): ±0.1mmUncertainty of yavg (Theor.): ±0.01mmExpected Range of yavg: ±0.11mmStandard Deviation of yavg: ±0.075mmAverage Percentage Error of yavg: ±9.37%

λ=650nm w= .04mm d=.50mm

Trial D (m) m0 (mm)

n0 (mm)

m1 (mm)

n1 (mm)

m2 (mm)

n2 (mm)

yavg exp. (mm)

yavg theor. (mm)

3 .5 0 1.25 1.9 2.5 3.2 3.7 1.225 0.65

4 .25 0 .85 1.3 1.65 2.2 2.45 0.85 0.325

Uncertainty of yavg (Exp.): ±0.1mmUncertainty of yavg (Theor.): ±0.01mmExpected Range of yavg: ±0.11mmStandard Deviation of yavg: ±0.525mmAverage Percentage Error of yavg: ±53.9%

λ=650nm w= .08mm d=.50mm

Trial D (m) m0 (mm)

n0 (mm)

m1 (mm)

n1 (mm)

m2 (mm)

n2 (mm)

yavg exp. (mm)

yavg theor. (mm)

5 .5 0 0.6 0.9 1.3 1.6 1.8 0.6 0.65

6 .25 0 .35 0.55 0.8 1.0 1.2 0.425 0.325

Uncertainty of yavg (Exp.): ±0.1mmUncertainty of yavg (Theor.): ±0.01mmExpected Range of yavg: ±0.11mmStandard Deviation of yavg: ±0.075mmAverage Percentage Error of yavg: ±15.6%

Graphs 1~6

Graphs 1-6: the experimental vs. theoretical distance between maxima/minimaGraphs 1-6 show the distance between the maxima/minima of the interference pattern produced by illuminating a pair of slits with a wavelength λ. The green line depicts the experimental results. The purple line shows the results expected, which were produced using equal increments of the expected y value from equation 7.

Analysis

The theoretical and experimental uncertainties for the distance between maxima (or minima) yavg were determined by calculating the maximum possible variation in either direction by accounting for any uncertainty in the measurements used to calculate the figures. The expected range of error the value of yavg in each trial was calculated by combining the uncertainty parameters for the theoretical and experimental values to calculate a “worst case scenario” or maximum possible expected error.

In each trial, yavg represents the average distance between points of maximum interference of minimum interference. With the exception of trials 3 and 4, the standard deviation fell within the expected parameters of ±0.11mm- ±0.075mm and ±0.075mm (for trials 1-2 and 5-6, respectively). The similarity between the experimental, measured results and the predicted results in trials 1-2 and 5-6 is reflected in the graphs. The graphs also reflect the exceptions- trials 3 and 4. With a standard deviation of ±0.525mm in the value of yavg, this fell outside the expected parameters. This can be explained by the fact that, in the equation used to predicted the y value (equation 7), the width of the slits was ignored. In trials 5 and 6, the slit width was increased- as expected, just as an increased distance between the slit width decreased the distance between minima, so did increasing the width of the slits themselves (this is due to interference between the rays of a single slit).

A more thorough and accurate experiment would involve a larger number of trials, as well as an equation predicting the distance between minima involving the slit width.

Conclusion

In order to accurately determine the size of small viruses as a part of an investigation of virus properties, a series of slits representing the viruses were illuminated with a diode laser to produce an interference pattern. Using Huygen’s principle and simple mathematics, an equation expressing the distance between minima of an interference pattern resulting from the illumination of a pair of slits was predicted as a function of the wavelength of the diode laser beam, distance to the screen, and distance between the slits. With the exception of two trials having an error resulting from the lack of slit width being represented in the equation, the results fell within the predicted range of ±0.11mm at ±0.075mm, confirming the expectations for this experiment.