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Optics Learning by Computing with Examples using MATLAB_Dieter Moller_2007

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188	4. COHERENCE

C1COH2S

Intensity of Two Sources Separated by s

Superposition of two double slit patterns. The slits have width d and separation a; one pattern is untilted with ψ 0, the other tilted by ψ x/Z, and distance from sources to slit is Z. Distance from slit to screen is X, coordinate on screen is Y , Y/X θ. By enlarging ψ, starting from 0, one ﬁnds the ﬁrst fringe disappearance. If ψ is further enlarged, the fringes reappear, but now the minima are not zero. Another point of view: fringes may disappear for constant s and changing a.

θ ≡ −.006, −.00599.. .006

d ≡ .05

a : 1

Z ≡ 9000

λ : .0005

I 1(θ) :

sin (π) · dλ · sin(θ)

· cos π ·

π · dλ · (sin(θ)) 2

· (sin(θ))

s1 ≡ 0

I I 1(θ) :

sin

π · dλ · (sin(θ)) 2

· cos π ·

· (sin(θ) + sin(ψ1))

ψ1 ≡

π · λ · (sin(θ))

I I 2(θ) :

sin

π · dλ · (sin(θ)) 2

· cos π ·

· (sin(θ) + sin(ψ2))

s2 ≡ 1.5

(sin(θ))

ψ2 :

π ·

sin

π · dλ · (sin(θ)) 2

· (sin(θ) + sin(ψ3))

s3 ≡ 2.25

I I 3(θ) :

(sin(θ))

· cos π ·

ψ3 :

π ·

4.1. SPATIAL COHERENCE

189

sin

π · dλ · (sin(θ)) 2

· (sin(θ) + sin(ψ4))

s4 ≡ 2.6

I I 4(θ) :

(sin(θ))

· cos π ·

ψ4 :

π ·

Application 4.1.

1.Change d to 0.1 and determine the s value for disappearance of fringes. Save the changed values for future comparisons.

2.Change λ to 0.0006 and determine the s value for disappearance of fringes. Save the changed values for future comparisons.

3.Change a to 1.2 and determine the s values for disappearance of fringes. Save the changed values for future comparisons.

4.1.3 Coherence Condition

We have seen that the two source points produce superimposed intensity patterns for certain distances s between them. In our case we found fringes for distance s 0, and for a larger distance s 2.25mm, the fringes disappeared. For a small separation s of the two source points, one would observe fringes similar to the pattern produced by one source point. The coherence condition tells us how large one can make the separation s and still have a fringe pattern similar to the one produced by one source point only. For the discussion, we assume that the openings of the two slits are very small and therefore we may omit diffraction.

190	4. COHERENCE

ψ	ψ λ

ψ ω

ω	ω λ
ω

FIGURE 4.2 The condition a sin ψ a(s/Z) λ/2 is equivalent to sa/Z s sin ω λ/2 and tells us that the product of the distance between the source times the opening angle must be similar to that of a pointlike source.

We then have for the pattern generated by sources S1 and S2,
I1(θ, 0)	I0	{cos[(πa/λ) sin θ]}2	(4.3)
I2(θ, ψ)	I0	{cos[(πa/λ)(sin θ − sin ψ)]}2.	(4.4)

The center of the pattern of I1 is at (a sin θ) 0 and the ﬁrst minimum is at (a sin θ) λ/2. We want to ﬁnd the magnitude of the angle ψ for which we will see only one fringe pattern. If the center of the pattern of I2 is at the minimum of I1 (i.e., for (a sin ψ) λ/2), the angle ψ is too large and we are at the position where fringes disappear for the ﬁrst time. Therefore, in order to observe just one fringe pattern of the superposition of I1 and I2 we have to require that

(a sin ψ) λ/2.

(4.5)

Equation (4.5) may also be written in good approximation as s · a/Z, where a/Z is the opening angle ω from the source points (Figure 4.2) and one has

s · a/Z s · ω λ/2.

(4.6)

The opening angle ω seen from the middle of the two source points is given as a/Z and the product of this angle times the separation s of the source points must be small compared to half of the wavelength. We state this as

(source size) times (opening angle ω) must be λ/2.

(4.7)

This is called the coherence condition.

4.1.4 Extended Source

The discussion of the double star was the ﬁrst step in investigating the question of how far away a source has to be to qualify as a point source, even if it has a

4.1. SPATIAL COHERENCE

191

ﬁnite diameter. To discuss an extended source we look at a line source made of a sequence of point sources. The distance between the source points is considered inﬁnitesimally small, but there is no ﬁxed phase relation between the light of one source point with respect to any other. We apply to the line source what we have done for two source points, and now have to integrate I (θ, ψ) over the angle ψ from 0 to ψs s/Z

		s/Z
I0	0	{sin[(πd/λ)(sin θ − sin ψ)]/[(πd/λ)(sin θ − sin ψ)]}2
	· {cos[πa/λ)(sin θ − sin ψ)]}2dψ.		(4.8)

The integration may be interpreted as the superposition of the intensity fringe pattern, produced by all source points between S1 and S2. The upper integration limit, that is, the angle s/Z, is taken as such a numerical value that we can compare the angles with calculations of the fringes for the two point sources. The ﬁrst graph in FileFig 4.2 shows the intensity interference pattern for a line source of length s 1 mm. The second graph for the length of s 1.5 mm again shows a fringe pattern. The third graph shows the disappearance of the intensity pattern at a length of s 4.5, and the fourth graph shows the reappearance at s 5. We see that the integrated intensity pattern is produced by an extended source of length s 4.5 mm. All fringes cancel for s 4.5 mm. This value is twice as large as s 2.25 mm, the value we found for the two source points.

We may associate the length of a line source with the diameter of a circular source. The disappearance of the interference pattern occurs when the diameter of the circular source is twice the distance of the two source points. In Figure 4.3 we show an experimental setup to observe a fringe pattern of an extended source.

The coherence condition (Eq. (4.7)) may be applied to the extended source as well. One has that the area s times the solid angle a/Z of the source must

FIGURE 4.3 Laboratory setup for the observation of fringes from an extended source. Experimental values: B 20 m, X 1 m, width of S1 and S2 .4 mm, b 6 mm, λ .00057 mm. At 2a 2 mm, the fringes disappear for the ﬁrst time, and the upper limit of the experiment is 2a 4 mm. (Adapted from Einf¨uhrung in die Optic, R.W. Pohl, Springer-Verlag, Heidelberg,

1948.)

192	4. COHERENCE

be small compared to (1/2)λ. This condition must be obeyed when setting up Young’s experiment. The source point is the illuminated hole of the ﬁrst screen and must be so small and far away that the wave at the next two holes has a ﬁxed phase relation. The size of the hole must be determined by the opening angle a/Z of the experimental setup. On the other hand, a star may be very large, but if the star is so far away that the product s times the opening angle ω (see Figure 4.2) is small compared to λ/2, the coherence condition is met. As the result, the light from the star is like parallel light when entering the double slit. Fringes may be observed and the light of the “large star" is called spatially coherent.

FileFig 4.2 (C2COHEX)

Graphs are shown for the superposition of the integrated intensities I (θ, ψ0) depending on the upper limit of the integration values ψ0 of the angle ψ0 s/Z. Four values of ψ0 are used, corresponding to s 1 mm, s 1.5 mm, s 4.5 mm, and s 5 mm. Parameters used are the separation of the two openings a 1 mm, opening of the slits d 0.05 mm, wavelength λ 0.0005 mm, distance from source to the double slit Z 9000 mm, and distance from aperture to observation screen X 4000 mm. By choosing the same values for a and λ as used in FileFig 4.1, we ﬁnd fringes disappear for the ﬁrst time for the size of the source of s 4.5 mm, that is, twice as large as found for the distance between the two point sources. Fringe patterns are observed for separations s smaller than 4.5 mm. For s 4.5 mm we have disappearance of fringes for the ﬁrst time. For s larger than 4.5 mm the fringe pattern reappears.

C2COHEX

Intensity of an Extended Source

Width is s and interference diffraction is on a double slit. Slit openings are d and separation a, distance from source to slit Z, from slit to screen X, coodinare on screen is Y , and small angle approximation Y/X θ.

a : 1

d ≡ .05

0 ≤ ψ ≤ 2

k ≡ 0. .200 θk ≡ .01 − k · .0001

λ : .0005

Z 9000

s1 ≡ 1

ψ1 ≡

I 1k :

ψ1 sin

π · dλ · (sin(θk ) + sin(ψ))

· cos π ·

· (sin(θk ) + sin(ψ))

dψ.

π ·

· (sin(θk ) + sin(ψ))

4.1. SPATIAL COHERENCE		193

s2 ≡ 1.5

ψ2 ≡

I 2k :

ψ2

sin

π · dλ · (sin(θk ) + sin(ψ))

π ·

· (sin(θk ) + sin(ψ))

π ·

(sin(θk ) + sin(ψ))

· cos

dψ.

s3 ≡ 4.5

ψ3 ≡

I 3k :

ψ3

sin

π · dλ · (sin(θk ) + sin(ψ))

π ·

· (sin(θk ) + sin(ψ))

π ·

(sin(θk ) + sin(ψ))

· cos

dψ.

194	4. COHERENCE

s4 ≡ 5

ψ4 ≡

I Ik :

ψ1

sin

π · dλ · (sin(θk ) + sin(ψ))

π ·

· (sin(θk ) + sin(ψ))

π ·

· (sin(θk ) + sin(ψ))

· cos

dψ.

Application 4.2.

1.Change d to 0.1 and determine the s value for disappearance of fringes. Compare to FileFig 4.1.

2.Change λ to 0.0006 and determine the s value for disappearance of fringes. Compare to FileFig 4.1.

3.Change a to 1.2 and determine the s values for disappearance of fringes. Compare to FileFig 4.1.

4.1.5 Visibility

4.1.5.1 Visibility for Two Point Sources

We have discussed appearance and disappearance of fringe patterns for two point sources at variable distances, and for extended sources of variable diameters. To

4.1. SPATIAL COHERENCE

195

measure how well the fringes may be seen, Michelson has deﬁned the visibility of fringes as the absolute value of

V	Itot,max − Itot,min	.	(4.9)

	Itot,max + Itot,min

The intensities are taken in the nominator and denominator of one fringe. For the pattern shown in the ﬁrst graph of FileFig 4.2, one has Imax 1 and Imin 0 and obtains for the visibility, V 1. When considering the superposition of two intensity patterns, one has to determine Imax and Imin depending on the angle ψ. We saw that the angle ψ is small because Z is large and applied the small angle approximation (i.e., sin θ Y/X and sin ψ Y /X). From Eqs. (4.3) and (4.4), disregarding the diffraction factors, we have for the total intensity Itot (i.e., the sum of Eqs. (4.3) and (4.4)),

Itot (Y ) {cos[(πa/λX)(Y )]}2 + {cos[(πa/λX)(Y − Y )]}2.	(4.10)
Using 2 cos2 α/2 1 + cos α, we get
Itot (Y ) 1 + 1/2{cos[(2πa/λX)(Y )]} + 1/2{cos[(2πa/λX)(Y − Y )]},	(4.11)
and with cos α + cos β 2 cos{(α + β)/2} cos{(α − β)/2} one has
Itot (Y ) 1 + {cos[(2πa/λX)(Y /2)]}{cos[(2πa/λX)(Y − Y /2)]}.	(4.12)

The minimum and maximum values of Itot depend only on Y , since Y is ﬁxed by the angle ψ for which we want to determine the visibility. The maximum value of {cos(2πa/λX)(Y − Y /2)} is 1 because that is the maximum of the cos-function and similarly the minimum value is −1. We have therefore

Itot,max 1	+ {cos[(2πa/λX)(Y /2)]}
Itot,min 1	− {cos[(2πa/λX)(Y /2)]}	(4.13)
and using the deﬁnition of Eq. (4.9) one obtains for the visibility
V \| cos(πa/λX)(Y )\|.		(4.14)

For small values of ψ Y /X s/Z we have a maximum. The ﬁrst zero of the visibility is for πaY /λX π/2; that is, Y /X s/Z λ/2a.

In FileFig 4.3 we have plotted Eq. (4.14), using similar values to those in FileFig 4.1. We observe that the visibility is ﬁrst zero for s 2.25 mm, the value we determined in FileFig 4.1 for the ﬁrst disappearance of the fringes. For larger values of s we found reappearance of fringes (see the fourth graph of FileFig 4.1).

The interval from s 0 to the value of s for which the visibility is ﬁrst zero is called the coherence interval.

196	4. COHERENCE
	FileFig 4.3	(C3VIS2S)

Graph of the visibility for two point sources.

C3VIS2S is only on the CD.

Application 4.3.

1.Change λ to 0.0006 and determine the s value for V 0. Compare to FileFig 4.1.

2.Change a to 1.2 and determine the s value for V 0. Compare to FileFig 4.1.

4.1.5.2 Visibility for an Extended Source (Line Source)

The visibility for an extended source is obtained by determination of Itot,max and Itot,min of the total integrated intensity. Again, disregarding the diffraction factor, one has

		s/Z
Itot (Y ) I0	0	{cos[(πa/λ)(sin θ − sin ψ)]}2dψ.	(4.15)
We calculate in small angle approximation the integral
		Y /X
Itot (Y ) I0	0	{cos[(πa/λ)(Y/X − Y /X)]}2d(Y /X).	(4.16)

First one has to go through the steps of Eqs. (4.10) to (4.12) and determine the maxima and minima. The second step is the integration of Eq. (4.14) and the result is

V \|{sin(πa/λX)(Y )}/(πa/λX)(Y )\|,	(4.17)
where Y /X s/Z. The ﬁrst minimum	of the visibility is obtained at

(πa/λX)Y π, (that is for Y /X λ/a). This is twice the value we found for the two point sources at distance s. In FileFig 4.4 we have plotted, for similar values to those used in FileFig 4.2, Eq. (4.17) as a function of the length s of the extended source. One ﬁnds the ﬁrst disappearance of the fringes at s 4.5, the same value determined in FileFig 4.2.

FileFig 4.4 (C4VISEX)

Graph of the visibility for an extended source.

C4VISEX is only on the CD.

4.1. SPATIAL COHERENCE

197

Application 4.4.

1.Change λ to 0.0006 and determine the s value for V 0. Compare to FileFig 4.2.

2.Change a to 1.2 and determine the s value for V 0. Compare to FileFig 4.2.

4.1.6 Michelson Stellar Interferometer

In the discussion of the intensity pattern produced byYoung’s experiment for the two source points and the extended source, we assumed a ﬁxed separation a of the two openings. We studied the change in the intensity pattern depending on the separation s of the source points and the diameter s of the extended source. Michelson was interested in the measurement of the diameter of a star. In the Young’s experiment we discussed above, the separation a of the two slits was ﬁxed and the distance between the two source points s was varied. Application of this experiment to the measurement of the diameter of a star is very difﬁcult because both parameters s and a are ﬁxed. Michelson wanted to modify Young’s experiment by making the separation a variable. However, since the diameter he wanted to measure was so small, the setup of Young’s experiment had to be modiﬁed.

We start with Young’s experiment for two source points. The fringe pattern of the superposition of the two patterns on the observation screen depends on the distance of the two source points. This is shown in Fig. 4.4a. The intensities of the two source points are for small angles

uI A2	{cos(πaθ/λ)}2	(4.18)
uI I A2	{cos(πa(θ − φ)/λ)}2.	(4.19)

The angle φ is what we wish to determine. In the setup of Figure 4.4a the displacement of the intensity pattern of source I with respect to source II is limited by the size of the distance a between the two slits. The modiﬁcation is shown in Figure 4.4b. A new parameter h of the large distance between the ﬁrst two mirrors is introduced. Changing the length h produces a change of the fringe pattern. From the variation of the fringes for two values of h one can determine the angle φ. In Figure 4.4b we show the modiﬁed setup and the two intensity patterns are now given as

uI A2	{cos(πaθ/λ)}2	(4.20)
uII A2	{cos[π(aθ − hφ)/λ]}2.	(4.21)

The superposition uI + uI I will show maxima and minima depending on h for ﬁxed φ and a. We may calculate the angle φ from the observed h values for a resulting maximum and the following minimum. For a resulting maximum we

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