# Chapter7 Experiment5: DiﬀractionofLightWaves

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Chapter 7

Experiment 5: Diﬀraction of Light Waves

WARNING

This experiment will employ Class III(a) lasers as a coherent, monochromatic light source. The student must read and understand the laser safety instructions on page 90 before attending this week’s laboratory.

7.1 Introduction

In this lab the phenomenon of diﬀraction will be explored. Diﬀraction is interference of a wave with itself. According to Huygen’s Principle waves propagate such that each point reached by a wavefront acts as a new wave source. The sum of the secondary waves emitted from all points on the wavefront propagate the wave forward. Interference between secondary waves emitted from diﬀerent parts of the wave front can cause waves to bend around corners and cause intensity ﬂuctuations much like interference patterns from separate sources. Some of these eﬀects were touched in the previous lab on interference.

General Information

We will observe the diﬀraction of light waves; however, diﬀraction occurs when any wave propagates around obstacles. Diﬀraction and the wave nature of particles leads to Heisenberg’s Uncertainty Principle and to the very reason why atoms are the sizes that they are.

In this lab the intensity patterns generated by monochromatic (laser) light passing through a single thin slit, a circular aperture, and around an opaque circle will be predicted and experimentally veriﬁed. The intensity distributions of monochromatic light diﬀracted

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CHAPTER 7: EXPERIMENT 5

a y

x Plane Wave

m = 4

m = 3

m = 2

m = 1

m = 0

m = -1

m = -2 m = -3

Diffraction Pattern

m = -4

Figure 7.1: A sketch of a plane wave incident upon a single slit being diﬀracted and the resulting intensity distribution. The intensity is on a logarithmic scale; since our eyes have logarithmic response, this much more closely matches the apparent brightness of the respectve dots. Keep in mind that this week we are concerned with the dark intensity MINIMA and that m = 0 is NOT a minimum.

from the described objects are based on:

1. the Superposition Principle, 2. the wave nature of light Disturbance:

Intensity vs. Position 1

(a) Amplitude: A = A0 sin(ωt + ϕ)

0.8

(b) Intensity: I ∝ ( A)2,

0.6

3. Huygen’s Principle - Light propagates in

such a way that each point reached by

0.4

the wave acts as a point source of a new

light wave. The superposition of all these

0.2

waves represents the propagation of the

light wave.

0

-7

-5

-3

-1

1

3

5

7

y (cm)

All calculations are based on the assumption that the distance, x, between the slit and the viewing screen is much larger than the slit width a:, i.e. x >> a. These results also apply to plane waves incident on the obstruction. This

Figure 7.2: A graphical plot of an example diﬀraction intensity distribution for a single slit.

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CHAPTER 7: EXPERIMENT 5

particular case is called Fraunhöfer scattering. Plane waves result from a laser in this case, but a point source very far away or at the focus of a lens also give plane waves. The calculations of this type of scattering are much simpler than the Fresnel scattering where the distant point source constraint is removed. Fresnel scattering assumes spherical wave-fronts.

Checkpoint

What is the diﬀerence between Fraunhofer and Fresnel scattering?

Checkpoint

Are the eyes sensitive to the amplitude, to the phase, or to the intensity of light? Are the eyes’ response linear, logarithmic, or something else?

7.2 The Experiment

We will observe the diﬀraction patterns from slits having several widths and apertures having diﬀerent diameters. Each will be illuminated with a laser’s coherent light and the resulting pattern will be characterized.

7.2.1 Diﬀraction From a Single Slit

A narrow slit of inﬁnite length and width a is

illuminated by a plane wave (laser beam) as shown

in Figure 7.1. The intensity distribution observed (on

a screen) at an angle θ with respect to the incident

direction is given by Equation (7.1). This relation is

derived in detail in the appendix and every student

Figure 7.3: A photograph of Pasco’s must make an eﬀort to go through its derivation. The

OS-8523 optics single slit set. We are mathematics used to calculate this relation are very

interested in the single slits at the top simple. The contributions from the ﬁeld at each small

right and the circular apertures at the area of the slit to the ﬁeld at a point on the screen are

top left.

added together by integration. Squaring this result

and disregarding sinusoidal ﬂuctuations in time gives

the intensity. The main diﬃculty in the calculation

is determining the relative phase of each small contribution. Figure 7.2 shows the expected

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CHAPTER 7: EXPERIMENT 5

shape of this distribution; mathematics can describe this distribution,

I(θ) sin α 2

πa

=

with α = sin(θ),

I (0)

α

λ

(7.1)

where a is the width of the slit, λ is the wavelength of the light, θ is the angle between the

optical axis and the propagation direction of the scattered light, I(θ) is the intensity of light

scattered to angle θ, and I(0) is the transmitted light intensity on the optical axis. When

θ = 0, α = 0, and the relative intensity is undeﬁned (0/0); however, when θ is very small and

yet not zero, the relative intensity is very close to 1. In fact, as θ gets closer to 0, this ratio

of intensities gets closer to 1. Using calculus we describe these situations using the limit as

α approaches 0,

sin α

lim

= 1.

α→0 α

Although 0/0 might be anything, in this particular case we might think that the ratio is

eﬀectively 1; this is the basis for the approximation sin θ ≈ θ when θ << 1 radian. Certainly,

the intensity of the light on the axis is deﬁned, is measurable, and is quite close to the

intensity near the optical axis. The numerator of this ratio can also be zero when the

denominator is nonzero. In these cases the intensity is predicted and observed to vanish.

These intensity minima occur when 0 = sin α for each time

πm = α and mλ = a sin θ for m = ±1, ±2, . . .

(7.2)

Figure 7.4: A sketch of our apparatus showing the view screen, the sample slide, the laser, and the laser adjustments. Our beam needs to be horizontal and at the level of the disk’s center.

We might note the similarity between this relation and the formula for interference maxima; we must avoid confusing the points that for diﬀraction these directions are intensity minima and for diﬀraction the optical axis (m = 0) is an exceptional maximum instead of an expected minimum.

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CHAPTER 7: EXPERIMENT 5

This relation is satisﬁed for integer values of m. Increasing values of m give minima at correspondingly larger angles. The ﬁrst minimum will be found for m = 1, the second for m = 2 and so forth. If is less than one for all values of θ, there will be no minima, i.e. when the size of the aperture is smaller than a wavelength (a < λ). This indicates that diﬀraction is most strongly caused by protuberances with sizes that are about the same dimension as a wavelength.

Four single slits (along with some double slits) are on a disk shown in Figure 7.3 photograph of Pasco’s OS-8523 optics single slit set. We are interested in the single slits at the top right and the circular apertures at the top left.. This situation is similar to the one diagrammed in Figure 7.1. To observe diﬀraction from a single slit, align the laser beam parallel to the table, at the height of the center of the disk, as shown in Figure 7.4. When the slit is perpendicular to the beam, the reﬂected light will re-enter the laser. The diﬀraction pattern you are expected to observe is shown in Figure 7.5.

Figure 7.5: Observed Diﬀraction Pattern. The pattern observed by one’s eyes does not die oﬀ as quickly in intensity as one expects when comparing the observed pattern with the calculated intensity proﬁle given by Equation (7.1) and shown in Figure 7.2. This is because the bright laser light saturates the eye. Thus the center and nearby fringes seem to vary slightly in size but all appear to be the same brightness.

Observe on the screen the diﬀerent patterns generated by all of the single slits on this mask. Note the characteristics of the pattern and the slit width for each in your Data. Is it possible from our cursory observations that mλ = a sin θ? If a quick and cheap observation can contradict this equation, then we need not spend more money and time.

Calculate the width of one of the four single slits. This quantity can be calculated from Equation (7.2) using measurements of the locations of the intensity minima. The wavelength of the HeNe laser is 6328 Å, (1 Å= 10−10 m). The quantity to be determined experimentally is sin θ. This can be done using trigonometry as shown in Figure 7.6.

Measure the slit width using several intensity minima of the diﬀraction pattern. Place a long strip of fresh tape on the screen and carefully mark all of the dark spots in the fringes. Be sure to record the manufacturer’s speciﬁed slit width, a. Avoid disturbing the laser, the slit, and the screen until all of the minima are marked and the distance, x, between the slit and the screen is measured. Is your screen perpendicular to your optical axis? Remove the slit and circle the undeﬂected laser spot. Ideally, this spot is exactly the center of the diﬀraction pattern. Make a table in your Data and record the order numbers, m, and the distances, ym, between the optical axis and the dark spots. It is more accurate to measure

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CHAPTER 7: EXPERIMENT 5

the distance 2 ym between the two mth minima on opposite sides of the axis; you might wish to avoid this additional complication. Note the units in the column headings and the errors in the table body.

Enter your data table into Vernier Software’s Ga3 graphical analysis program. A suitable setup ﬁle for Ga3 can be downloaded from the lab’s website at

http://groups.physics.northwestern.edu/lab/diﬀraction.html

Otherwise, change the column names and units to represent your data. Add a calculated column to hold your measured slit widths. Enter the column name and appropriate units and type

“m” * 632.8 * x / (c * “ym”)

into the Equation edit control. Use the “Column” button to obtain the quoted values; choose your column names instead of mine. Instead of “x” type the number of cm, m, etc. between your screen and slit; use the same units for x as for ym so that they cancel in the above ratio. What units remain to be the units of a? If you like, you can use the value “c” to convert these units to nm (c = 1), µm (c = 1000), mm (c = 1000000), or m (c = 109). Be sure to use the correct units in the column heading to represent your numbers. Plot your slit widths on the y-axis and the order number on the x-axis. Click the smallest number on the y-axis and change the range to include 0. Does it look like all of your data are the same within your error? If so, draw a box around your data points and Analyze/Statistics. Move the parameters box oﬀ of your data points.

Review Section 2.6 and specify your best estimate of slit width as a = (a¯ ± sa¯) U.

R y

x

Figure 7.6: A schematic diagram of our experiment showing how we can determine the scattering angle, θ, by measuring the distance from the optical axis, y, and the distance between the slits and the screen, x.

Checkpoint

The relation of sin2 α/α2 for α = 0 is an undeﬁned expression of 0/0. What is the limit of this relation for the limit α → 0?

96

CHAPTER 7: EXPERIMENT 5

Checkpoint

Which relation gives the position of the diﬀraction minima for a slit of width a, illuminated with plane wave light of wavelength λ?

Checkpoint

If you want to sharpen up a beam spot by inserting a narrow vertical slit into the beam, will the beam spot get more and more narrow as you close the slit? Explain.

7.2.2 Diﬀraction by a Circular Aperture

For a circular hole of diameter, a, the diﬀraction pattern of light with wavelength λ consists

of concentric rings, which are analogous to the bands which we obtained for the single slits.

The pattern for this intensity distribution can be calculated in the same way as for the single

slit (see Appendix), but because the aperture here is circular, it is more convenient to use

cylindrical coordinates (z, ρ, φ). The superposition principle requires us to integrate over a

disk, and the result is a Bessel function. The condition for observing a minimum of intensity

is found from the zeroes of the Bessel function, 0 = J1(πkm), and the minima are at angles

θ such that

λ sin θ = km a .

(7.3)

The ﬁrst several proportionality constants, km, are displayed in Table 7.1. These are slightly larger than the corresponding integers.

Table 7.1 m km

The disk provided with the apparatus has circular openings of two

1 1.220

diﬀerent diameters. Put the apertures in the laser beam one at a

2 2.233

time and observe how the diameter of the dark rings depends on the

3 3.238

aperture diameter. Record your observations in your Data and choose the

4 4.241

conﬁguration which gives the best diﬀraction pattern for detailed study.

5 5.243

Having the aperture perpendicular to the beam and in the center of the

6 6.244

beam gives the best results. Having the screen perpendicular to the beam

7 7.245

will avoid stretching the circles in the projection into ellipses.

8 8.245

Calculate the diameter of the aperture. To make this measurement

9 9.246

accurately, make the distance between the object and the screen large; you may choose to

use the wall as your screen. First, measure the distance x between the view screen and the

circular aperture. Without disturbing this distance, carefully mark the diameters of at least

5-6 dark rings in the pattern. A circular ring scale is provided for those who wish to use a

phone camera; align the bright central spot on the scale’s black center and take a photograph.

The ring diameters can then be read from the photograph; do not use a ruler because the

photograph changes the scale. Alternately, white paper can be marked on opposite sides of

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CHAPTER 7: EXPERIMENT 5

the dark rings and the ruler can be used to ﬁnd the diameters between the dots. Please do not mark on the scales. The angle in Equation (7.3) is between the optical axis (center of the central spot) and the dark ring. A little geometry and trigonometry should reveal that

sin θm ≈ tan θ = Rm = (2Rm)

x

2x

(7.4)

as shown in Figure 7.6. If the dark circle is irregular, use a typical value instead of the largest or smallest radius and consider its shape when estimating δRm. If you cannot get a good pattern, mark the slit set’s position with one of the elevator blocks and ask your TA for assistance; your aperture might need cleaned. Don’t forget to include your units and uncertainties.

Option 1: Statistics of Aperture Diameter

Measure the diameters of at least 5 dark rings and use each with the respective km from Table 7.1 to compute a measurement for the aperture diameter. Plot the aperture diameters on the y-axis and decide whether all predictions are equal or whether there is a dependence on m. Compute the statistics and report the diameter, a = a¯ ± sa¯.

Option 2: Measure the km

The roundoﬀ error in k1 is limited to ±0.0005 and you have estimated uncertainties for x and R1. The uncertainty in our wavelength is much smaller than these so estimate the error in aperture diameter using

1.22λx

a≈

and δa = a

R1

δR1 2 0.0005 2

+

.

R1

1.220

(7.5)

Measure

the

value

of

the

constant

km

in

front

of

the

ratio

λ a

in

Equation

(7.3)

for

the second order minimum, m = 2, and the third order minimum, m = 3. Do this by

using as aperture diameter a the value previously obtained above and by measuring sin θ

corresponding to the second minimum and then the third minimum,

km ≈ aRm and δkm = km λx

δa 2 +

a

δRm 2 .

Rm

In measuring the constant km you are determining the zeroes of “the ﬁrst Bessel function of the ﬁrst kind.” Bessel functions come in two kinds, Jm and Km, and they solve math problems with cylindrical symmetry.

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CHAPTER 7: EXPERIMENT 5

Checkpoint

Beams of particles act like waves with very short wavelengths when scattered by protuberances such as other particles. If a target of Pb and one of C are bombarded with a beam of protons, which target will show the sharpest diﬀraction pattern, the large size Pb target or the small C one? (Hint: The pattern due to an aperture is identical to the pattern caused by its photographic negative: a disk in empty space.)

Checkpoint

What size object will generate an observable diﬀraction pattern if placed in the path of light with wavelength λ?

Checkpoint

What is responsible for the factor 1.22 in Equation (7.3)?

7.2.3 The Poisson Spot

Figure 7.7: A sketch illustrating the optical beam line used to observe the Poisson spot in the center of a sphere’s shadow.

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CHAPTER 7: EXPERIMENT 5

Historical Aside

In 1818 Augustin-Jean Fresnel entered a competition sponsored by the French Academy to try to clear up some outstanding questions about light. His paper was on a wave theory of light in which he showed that the refraction and reﬂection of light and all of their characteristics could be explained if we only allowed light to be treated as a wave. His theory even explained the results of Young’s double-slit experiment. This was about 30 years after Sir Isaac Newton presented a corpuscle theory of light in which light particles moved from a source to a destination but could be bent by material surfaces.

Everyone knows to hide behind a rock or a tree if someone is throwing stones or shooting bullets at him since the projectiles will either be stopped by the obstacle or pass harmlessly by. One is not so safe from waves in a swimming pool, however, because waves will bend around obstacles having the size of their wavelengths or smaller.

Historical Aside

Many scientists at the meeting were having problems with Fresnel’s paper because everyone can see that objects cast shadows. Simeon Denis Poisson, one of the judges, was particularly disturbed by Fresnel’s paper and he used wave theory to show that if light is to be described as a wave phenomenon, then a bright spot would be visible at the center of the shadow of a circular opaque obstacle, a result which he felt proved the absurdity of the wave theory of light.

This surprising prediction, fashioned by Poisson as the death blow to wave theory, was almost immediately veriﬁed experimentally by Dominique Arago, another member of the judges committee. The spot actually exists! The spot is still called the Poisson spot (more recently “Poisson-Arago spot” has come into fashion) despite Argo’s having discovered it. I guess this just goes to show that some of our mistakes will never be lived down. . .

Historical Aside

Fresnel won the competition, but it would require James Clerk Maxwell (sixty years later) to locate the medium in which the waves of light propagate.

In less than 60 seconds you can now settle a controversy that has preoccupied the minds of the brightest philosophers and scientists for centuries. Is light described as a stream of particles or as a wave phenomenon? In other words, does the Poisson spot exist?

100

Experiment 5: Diﬀraction of Light Waves

WARNING

This experiment will employ Class III(a) lasers as a coherent, monochromatic light source. The student must read and understand the laser safety instructions on page 90 before attending this week’s laboratory.

7.1 Introduction

In this lab the phenomenon of diﬀraction will be explored. Diﬀraction is interference of a wave with itself. According to Huygen’s Principle waves propagate such that each point reached by a wavefront acts as a new wave source. The sum of the secondary waves emitted from all points on the wavefront propagate the wave forward. Interference between secondary waves emitted from diﬀerent parts of the wave front can cause waves to bend around corners and cause intensity ﬂuctuations much like interference patterns from separate sources. Some of these eﬀects were touched in the previous lab on interference.

General Information

We will observe the diﬀraction of light waves; however, diﬀraction occurs when any wave propagates around obstacles. Diﬀraction and the wave nature of particles leads to Heisenberg’s Uncertainty Principle and to the very reason why atoms are the sizes that they are.

In this lab the intensity patterns generated by monochromatic (laser) light passing through a single thin slit, a circular aperture, and around an opaque circle will be predicted and experimentally veriﬁed. The intensity distributions of monochromatic light diﬀracted

91

CHAPTER 7: EXPERIMENT 5

a y

x Plane Wave

m = 4

m = 3

m = 2

m = 1

m = 0

m = -1

m = -2 m = -3

Diffraction Pattern

m = -4

Figure 7.1: A sketch of a plane wave incident upon a single slit being diﬀracted and the resulting intensity distribution. The intensity is on a logarithmic scale; since our eyes have logarithmic response, this much more closely matches the apparent brightness of the respectve dots. Keep in mind that this week we are concerned with the dark intensity MINIMA and that m = 0 is NOT a minimum.

from the described objects are based on:

1. the Superposition Principle, 2. the wave nature of light Disturbance:

Intensity vs. Position 1

(a) Amplitude: A = A0 sin(ωt + ϕ)

0.8

(b) Intensity: I ∝ ( A)2,

0.6

3. Huygen’s Principle - Light propagates in

such a way that each point reached by

0.4

the wave acts as a point source of a new

light wave. The superposition of all these

0.2

waves represents the propagation of the

light wave.

0

-7

-5

-3

-1

1

3

5

7

y (cm)

All calculations are based on the assumption that the distance, x, between the slit and the viewing screen is much larger than the slit width a:, i.e. x >> a. These results also apply to plane waves incident on the obstruction. This

Figure 7.2: A graphical plot of an example diﬀraction intensity distribution for a single slit.

92

CHAPTER 7: EXPERIMENT 5

particular case is called Fraunhöfer scattering. Plane waves result from a laser in this case, but a point source very far away or at the focus of a lens also give plane waves. The calculations of this type of scattering are much simpler than the Fresnel scattering where the distant point source constraint is removed. Fresnel scattering assumes spherical wave-fronts.

Checkpoint

What is the diﬀerence between Fraunhofer and Fresnel scattering?

Checkpoint

Are the eyes sensitive to the amplitude, to the phase, or to the intensity of light? Are the eyes’ response linear, logarithmic, or something else?

7.2 The Experiment

We will observe the diﬀraction patterns from slits having several widths and apertures having diﬀerent diameters. Each will be illuminated with a laser’s coherent light and the resulting pattern will be characterized.

7.2.1 Diﬀraction From a Single Slit

A narrow slit of inﬁnite length and width a is

illuminated by a plane wave (laser beam) as shown

in Figure 7.1. The intensity distribution observed (on

a screen) at an angle θ with respect to the incident

direction is given by Equation (7.1). This relation is

derived in detail in the appendix and every student

Figure 7.3: A photograph of Pasco’s must make an eﬀort to go through its derivation. The

OS-8523 optics single slit set. We are mathematics used to calculate this relation are very

interested in the single slits at the top simple. The contributions from the ﬁeld at each small

right and the circular apertures at the area of the slit to the ﬁeld at a point on the screen are

top left.

added together by integration. Squaring this result

and disregarding sinusoidal ﬂuctuations in time gives

the intensity. The main diﬃculty in the calculation

is determining the relative phase of each small contribution. Figure 7.2 shows the expected

93

CHAPTER 7: EXPERIMENT 5

shape of this distribution; mathematics can describe this distribution,

I(θ) sin α 2

πa

=

with α = sin(θ),

I (0)

α

λ

(7.1)

where a is the width of the slit, λ is the wavelength of the light, θ is the angle between the

optical axis and the propagation direction of the scattered light, I(θ) is the intensity of light

scattered to angle θ, and I(0) is the transmitted light intensity on the optical axis. When

θ = 0, α = 0, and the relative intensity is undeﬁned (0/0); however, when θ is very small and

yet not zero, the relative intensity is very close to 1. In fact, as θ gets closer to 0, this ratio

of intensities gets closer to 1. Using calculus we describe these situations using the limit as

α approaches 0,

sin α

lim

= 1.

α→0 α

Although 0/0 might be anything, in this particular case we might think that the ratio is

eﬀectively 1; this is the basis for the approximation sin θ ≈ θ when θ << 1 radian. Certainly,

the intensity of the light on the axis is deﬁned, is measurable, and is quite close to the

intensity near the optical axis. The numerator of this ratio can also be zero when the

denominator is nonzero. In these cases the intensity is predicted and observed to vanish.

These intensity minima occur when 0 = sin α for each time

πm = α and mλ = a sin θ for m = ±1, ±2, . . .

(7.2)

Figure 7.4: A sketch of our apparatus showing the view screen, the sample slide, the laser, and the laser adjustments. Our beam needs to be horizontal and at the level of the disk’s center.

We might note the similarity between this relation and the formula for interference maxima; we must avoid confusing the points that for diﬀraction these directions are intensity minima and for diﬀraction the optical axis (m = 0) is an exceptional maximum instead of an expected minimum.

94

CHAPTER 7: EXPERIMENT 5

This relation is satisﬁed for integer values of m. Increasing values of m give minima at correspondingly larger angles. The ﬁrst minimum will be found for m = 1, the second for m = 2 and so forth. If is less than one for all values of θ, there will be no minima, i.e. when the size of the aperture is smaller than a wavelength (a < λ). This indicates that diﬀraction is most strongly caused by protuberances with sizes that are about the same dimension as a wavelength.

Four single slits (along with some double slits) are on a disk shown in Figure 7.3 photograph of Pasco’s OS-8523 optics single slit set. We are interested in the single slits at the top right and the circular apertures at the top left.. This situation is similar to the one diagrammed in Figure 7.1. To observe diﬀraction from a single slit, align the laser beam parallel to the table, at the height of the center of the disk, as shown in Figure 7.4. When the slit is perpendicular to the beam, the reﬂected light will re-enter the laser. The diﬀraction pattern you are expected to observe is shown in Figure 7.5.

Figure 7.5: Observed Diﬀraction Pattern. The pattern observed by one’s eyes does not die oﬀ as quickly in intensity as one expects when comparing the observed pattern with the calculated intensity proﬁle given by Equation (7.1) and shown in Figure 7.2. This is because the bright laser light saturates the eye. Thus the center and nearby fringes seem to vary slightly in size but all appear to be the same brightness.

Observe on the screen the diﬀerent patterns generated by all of the single slits on this mask. Note the characteristics of the pattern and the slit width for each in your Data. Is it possible from our cursory observations that mλ = a sin θ? If a quick and cheap observation can contradict this equation, then we need not spend more money and time.

Calculate the width of one of the four single slits. This quantity can be calculated from Equation (7.2) using measurements of the locations of the intensity minima. The wavelength of the HeNe laser is 6328 Å, (1 Å= 10−10 m). The quantity to be determined experimentally is sin θ. This can be done using trigonometry as shown in Figure 7.6.

Measure the slit width using several intensity minima of the diﬀraction pattern. Place a long strip of fresh tape on the screen and carefully mark all of the dark spots in the fringes. Be sure to record the manufacturer’s speciﬁed slit width, a. Avoid disturbing the laser, the slit, and the screen until all of the minima are marked and the distance, x, between the slit and the screen is measured. Is your screen perpendicular to your optical axis? Remove the slit and circle the undeﬂected laser spot. Ideally, this spot is exactly the center of the diﬀraction pattern. Make a table in your Data and record the order numbers, m, and the distances, ym, between the optical axis and the dark spots. It is more accurate to measure

95

CHAPTER 7: EXPERIMENT 5

the distance 2 ym between the two mth minima on opposite sides of the axis; you might wish to avoid this additional complication. Note the units in the column headings and the errors in the table body.

Enter your data table into Vernier Software’s Ga3 graphical analysis program. A suitable setup ﬁle for Ga3 can be downloaded from the lab’s website at

http://groups.physics.northwestern.edu/lab/diﬀraction.html

Otherwise, change the column names and units to represent your data. Add a calculated column to hold your measured slit widths. Enter the column name and appropriate units and type

“m” * 632.8 * x / (c * “ym”)

into the Equation edit control. Use the “Column” button to obtain the quoted values; choose your column names instead of mine. Instead of “x” type the number of cm, m, etc. between your screen and slit; use the same units for x as for ym so that they cancel in the above ratio. What units remain to be the units of a? If you like, you can use the value “c” to convert these units to nm (c = 1), µm (c = 1000), mm (c = 1000000), or m (c = 109). Be sure to use the correct units in the column heading to represent your numbers. Plot your slit widths on the y-axis and the order number on the x-axis. Click the smallest number on the y-axis and change the range to include 0. Does it look like all of your data are the same within your error? If so, draw a box around your data points and Analyze/Statistics. Move the parameters box oﬀ of your data points.

Review Section 2.6 and specify your best estimate of slit width as a = (a¯ ± sa¯) U.

R y

x

Figure 7.6: A schematic diagram of our experiment showing how we can determine the scattering angle, θ, by measuring the distance from the optical axis, y, and the distance between the slits and the screen, x.

Checkpoint

The relation of sin2 α/α2 for α = 0 is an undeﬁned expression of 0/0. What is the limit of this relation for the limit α → 0?

96

CHAPTER 7: EXPERIMENT 5

Checkpoint

Which relation gives the position of the diﬀraction minima for a slit of width a, illuminated with plane wave light of wavelength λ?

Checkpoint

If you want to sharpen up a beam spot by inserting a narrow vertical slit into the beam, will the beam spot get more and more narrow as you close the slit? Explain.

7.2.2 Diﬀraction by a Circular Aperture

For a circular hole of diameter, a, the diﬀraction pattern of light with wavelength λ consists

of concentric rings, which are analogous to the bands which we obtained for the single slits.

The pattern for this intensity distribution can be calculated in the same way as for the single

slit (see Appendix), but because the aperture here is circular, it is more convenient to use

cylindrical coordinates (z, ρ, φ). The superposition principle requires us to integrate over a

disk, and the result is a Bessel function. The condition for observing a minimum of intensity

is found from the zeroes of the Bessel function, 0 = J1(πkm), and the minima are at angles

θ such that

λ sin θ = km a .

(7.3)

The ﬁrst several proportionality constants, km, are displayed in Table 7.1. These are slightly larger than the corresponding integers.

Table 7.1 m km

The disk provided with the apparatus has circular openings of two

1 1.220

diﬀerent diameters. Put the apertures in the laser beam one at a

2 2.233

time and observe how the diameter of the dark rings depends on the

3 3.238

aperture diameter. Record your observations in your Data and choose the

4 4.241

conﬁguration which gives the best diﬀraction pattern for detailed study.

5 5.243

Having the aperture perpendicular to the beam and in the center of the

6 6.244

beam gives the best results. Having the screen perpendicular to the beam

7 7.245

will avoid stretching the circles in the projection into ellipses.

8 8.245

Calculate the diameter of the aperture. To make this measurement

9 9.246

accurately, make the distance between the object and the screen large; you may choose to

use the wall as your screen. First, measure the distance x between the view screen and the

circular aperture. Without disturbing this distance, carefully mark the diameters of at least

5-6 dark rings in the pattern. A circular ring scale is provided for those who wish to use a

phone camera; align the bright central spot on the scale’s black center and take a photograph.

The ring diameters can then be read from the photograph; do not use a ruler because the

photograph changes the scale. Alternately, white paper can be marked on opposite sides of

97

CHAPTER 7: EXPERIMENT 5

the dark rings and the ruler can be used to ﬁnd the diameters between the dots. Please do not mark on the scales. The angle in Equation (7.3) is between the optical axis (center of the central spot) and the dark ring. A little geometry and trigonometry should reveal that

sin θm ≈ tan θ = Rm = (2Rm)

x

2x

(7.4)

as shown in Figure 7.6. If the dark circle is irregular, use a typical value instead of the largest or smallest radius and consider its shape when estimating δRm. If you cannot get a good pattern, mark the slit set’s position with one of the elevator blocks and ask your TA for assistance; your aperture might need cleaned. Don’t forget to include your units and uncertainties.

Option 1: Statistics of Aperture Diameter

Measure the diameters of at least 5 dark rings and use each with the respective km from Table 7.1 to compute a measurement for the aperture diameter. Plot the aperture diameters on the y-axis and decide whether all predictions are equal or whether there is a dependence on m. Compute the statistics and report the diameter, a = a¯ ± sa¯.

Option 2: Measure the km

The roundoﬀ error in k1 is limited to ±0.0005 and you have estimated uncertainties for x and R1. The uncertainty in our wavelength is much smaller than these so estimate the error in aperture diameter using

1.22λx

a≈

and δa = a

R1

δR1 2 0.0005 2

+

.

R1

1.220

(7.5)

Measure

the

value

of

the

constant

km

in

front

of

the

ratio

λ a

in

Equation

(7.3)

for

the second order minimum, m = 2, and the third order minimum, m = 3. Do this by

using as aperture diameter a the value previously obtained above and by measuring sin θ

corresponding to the second minimum and then the third minimum,

km ≈ aRm and δkm = km λx

δa 2 +

a

δRm 2 .

Rm

In measuring the constant km you are determining the zeroes of “the ﬁrst Bessel function of the ﬁrst kind.” Bessel functions come in two kinds, Jm and Km, and they solve math problems with cylindrical symmetry.

98

CHAPTER 7: EXPERIMENT 5

Checkpoint

Beams of particles act like waves with very short wavelengths when scattered by protuberances such as other particles. If a target of Pb and one of C are bombarded with a beam of protons, which target will show the sharpest diﬀraction pattern, the large size Pb target or the small C one? (Hint: The pattern due to an aperture is identical to the pattern caused by its photographic negative: a disk in empty space.)

Checkpoint

What size object will generate an observable diﬀraction pattern if placed in the path of light with wavelength λ?

Checkpoint

What is responsible for the factor 1.22 in Equation (7.3)?

7.2.3 The Poisson Spot

Figure 7.7: A sketch illustrating the optical beam line used to observe the Poisson spot in the center of a sphere’s shadow.

99

CHAPTER 7: EXPERIMENT 5

Historical Aside

In 1818 Augustin-Jean Fresnel entered a competition sponsored by the French Academy to try to clear up some outstanding questions about light. His paper was on a wave theory of light in which he showed that the refraction and reﬂection of light and all of their characteristics could be explained if we only allowed light to be treated as a wave. His theory even explained the results of Young’s double-slit experiment. This was about 30 years after Sir Isaac Newton presented a corpuscle theory of light in which light particles moved from a source to a destination but could be bent by material surfaces.

Everyone knows to hide behind a rock or a tree if someone is throwing stones or shooting bullets at him since the projectiles will either be stopped by the obstacle or pass harmlessly by. One is not so safe from waves in a swimming pool, however, because waves will bend around obstacles having the size of their wavelengths or smaller.

Historical Aside

Many scientists at the meeting were having problems with Fresnel’s paper because everyone can see that objects cast shadows. Simeon Denis Poisson, one of the judges, was particularly disturbed by Fresnel’s paper and he used wave theory to show that if light is to be described as a wave phenomenon, then a bright spot would be visible at the center of the shadow of a circular opaque obstacle, a result which he felt proved the absurdity of the wave theory of light.

This surprising prediction, fashioned by Poisson as the death blow to wave theory, was almost immediately veriﬁed experimentally by Dominique Arago, another member of the judges committee. The spot actually exists! The spot is still called the Poisson spot (more recently “Poisson-Arago spot” has come into fashion) despite Argo’s having discovered it. I guess this just goes to show that some of our mistakes will never be lived down. . .

Historical Aside

Fresnel won the competition, but it would require James Clerk Maxwell (sixty years later) to locate the medium in which the waves of light propagate.

In less than 60 seconds you can now settle a controversy that has preoccupied the minds of the brightest philosophers and scientists for centuries. Is light described as a stream of particles or as a wave phenomenon? In other words, does the Poisson spot exist?

100

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