Abstract

It was recently shown that coherence theory provides a simple and elegant means of interpreting and extending the Lau effect. The details of this analysis are presented along with experimental results which show good qualitative agreement with the predicted intensity distributions. In addition an unexpected intensity modulation is observed which is readily explained in terms of the present analysis.

© 1981 Optical Society of America

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References

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  1. E. Lau, Ann. Phys. 6, 417 (1948).
    [CrossRef]
  2. J. Jahns, A. W. Lohmann, Opt. Commun. 28, 263 (1979).
    [CrossRef]
  3. F. Gori, Opt. Commun. 31, 4 (1979).
    [CrossRef]
  4. R. Sudol, B. J. Thompson, Opt. Commun. 31, 105 (1979).
    [CrossRef]
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 10.
  6. The periodic nature of the complex degree of spatial coherence produced by a periodic spatially incoherent source has been previously noted. See, for example, M. Françon, Optical Interferometry (Academic, New York, 1966), p. 178; G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977), p. 104; B. J. Thompson, C. Roychoudhuri, Opt. Acta 26, 21 (1979). These last authors have also considered periodic partially coherent sources.
    [CrossRef]
  7. L. Mandel, E. Wolf, J. Opt. Soc. Am. 66, 529 (1976).
    [CrossRef]
  8. For those more familiar with the mutual intensity function we have included, for the sake of completeness, in the Appendix the relationship among the mutual intensity function, the cross-spectral density function, and their normalized forms under the condition of quasi-monochromatic light.
  9. With the grating periodicity occurring in only one direction, it is sufficient that we present a 1-D analysis. The constants which have been omitted in front of the various integral expressions are those normally associated with a 2-D analysis. We have chosen not to carry the functional form of the second dimension since it adds no new information and unnecessarily complicates the expressions.
  10. E. Wolf, J. Opt. Soc. Am. 68, 6 (1978).
    [CrossRef]
  11. For a general review of the propagation of partially coherent light, see, for example, Ref. 5. It will be noted that several equations contained therein are similar to ours with the mutual intensity J being replaced by the cross-spectral density W and the mean frequency ω¯ by the specific frequency ω. We have, however, presented these equations as they apply to the Lau experiment and also to clarify the framework of our analysis.
  12. F. D. Feiock, J. Opt. Soc. Am. 68, 485 (1978).
    [CrossRef]
  13. In writing Eq. (8) we have considered the grating G1 to be a strictly periodic function. In this case we are assuming that the slowly varying function iω(x) ultimately limits the actual extent of this grating.
  14. Referring to Ref. 13, the assumption of constant iω implies grating G1 to be of infinite extent. Actually, Gori3 has pointed out (heuristically) that finite grating size has little effect as long as a sufficient number of slits are illuminated. In our notation the condition turns out to be N ≫ αp/β(2α), where N is the number of slits, p is grating period, and 2a equals the slit width.
  15. P. M. Woodward, Probablility and Information Theory with Applications to Radar (Pergamon, New York, 1953).
  16. Comments similar to those in Ref. 13 also apply here.
  17. We have assumed a grating consisting of an odd number of slits centered on the optical axis of the system. This in no way limits the generality of the analysis and is used only for simplicity.
  18. Strictly speaking, the complex degree of spatial coherence cannot have this functional form since the δ function is not a properly normalizable function. This form is a result of our infinite grating assumption, which is a mathematical convenience for the purpose of the analysis. Including the finite extent of a physical grating will eliminate this mathematical difficulty. What we have done, then, in going from Eq. (26) to (32) is to normalize the area of the Dirac δ function corresponding to zero separation, i.e., σ = 0, to unity.
  19. R. A. Shore, B. J. Thompson, R. E. Whitney, J. Opt. Soc. Am. 56, 733 (1966).
    [CrossRef]
  20. R. Sudol, “Fresnel Images, Coherence Theory, and the Lau Effect,” to be presented at the SPIE 24th International Technical Symposium and Instrument Display (July 1980).

1979 (3)

J. Jahns, A. W. Lohmann, Opt. Commun. 28, 263 (1979).
[CrossRef]

F. Gori, Opt. Commun. 31, 4 (1979).
[CrossRef]

R. Sudol, B. J. Thompson, Opt. Commun. 31, 105 (1979).
[CrossRef]

1978 (2)

1976 (1)

1966 (1)

1948 (1)

E. Lau, Ann. Phys. 6, 417 (1948).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 10.

Feiock, F. D.

Françon, M.

The periodic nature of the complex degree of spatial coherence produced by a periodic spatially incoherent source has been previously noted. See, for example, M. Françon, Optical Interferometry (Academic, New York, 1966), p. 178; G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977), p. 104; B. J. Thompson, C. Roychoudhuri, Opt. Acta 26, 21 (1979). These last authors have also considered periodic partially coherent sources.
[CrossRef]

Gori, F.

F. Gori, Opt. Commun. 31, 4 (1979).
[CrossRef]

Jahns, J.

J. Jahns, A. W. Lohmann, Opt. Commun. 28, 263 (1979).
[CrossRef]

Lau, E.

E. Lau, Ann. Phys. 6, 417 (1948).
[CrossRef]

Lohmann, A. W.

J. Jahns, A. W. Lohmann, Opt. Commun. 28, 263 (1979).
[CrossRef]

Mandel, L.

Shore, R. A.

Sudol, R.

R. Sudol, B. J. Thompson, Opt. Commun. 31, 105 (1979).
[CrossRef]

R. Sudol, “Fresnel Images, Coherence Theory, and the Lau Effect,” to be presented at the SPIE 24th International Technical Symposium and Instrument Display (July 1980).

Thompson, B. J.

Whitney, R. E.

Wolf, E.

E. Wolf, J. Opt. Soc. Am. 68, 6 (1978).
[CrossRef]

L. Mandel, E. Wolf, J. Opt. Soc. Am. 66, 529 (1976).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 10.

Woodward, P. M.

P. M. Woodward, Probablility and Information Theory with Applications to Radar (Pergamon, New York, 1953).

Ann. Phys. (1)

E. Lau, Ann. Phys. 6, 417 (1948).
[CrossRef]

J. Opt. Soc. Am. (4)

Opt. Commun. (3)

J. Jahns, A. W. Lohmann, Opt. Commun. 28, 263 (1979).
[CrossRef]

F. Gori, Opt. Commun. 31, 4 (1979).
[CrossRef]

R. Sudol, B. J. Thompson, Opt. Commun. 31, 105 (1979).
[CrossRef]

Other (12)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 10.

The periodic nature of the complex degree of spatial coherence produced by a periodic spatially incoherent source has been previously noted. See, for example, M. Françon, Optical Interferometry (Academic, New York, 1966), p. 178; G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977), p. 104; B. J. Thompson, C. Roychoudhuri, Opt. Acta 26, 21 (1979). These last authors have also considered periodic partially coherent sources.
[CrossRef]

For a general review of the propagation of partially coherent light, see, for example, Ref. 5. It will be noted that several equations contained therein are similar to ours with the mutual intensity J being replaced by the cross-spectral density W and the mean frequency ω¯ by the specific frequency ω. We have, however, presented these equations as they apply to the Lau experiment and also to clarify the framework of our analysis.

For those more familiar with the mutual intensity function we have included, for the sake of completeness, in the Appendix the relationship among the mutual intensity function, the cross-spectral density function, and their normalized forms under the condition of quasi-monochromatic light.

With the grating periodicity occurring in only one direction, it is sufficient that we present a 1-D analysis. The constants which have been omitted in front of the various integral expressions are those normally associated with a 2-D analysis. We have chosen not to carry the functional form of the second dimension since it adds no new information and unnecessarily complicates the expressions.

R. Sudol, “Fresnel Images, Coherence Theory, and the Lau Effect,” to be presented at the SPIE 24th International Technical Symposium and Instrument Display (July 1980).

In writing Eq. (8) we have considered the grating G1 to be a strictly periodic function. In this case we are assuming that the slowly varying function iω(x) ultimately limits the actual extent of this grating.

Referring to Ref. 13, the assumption of constant iω implies grating G1 to be of infinite extent. Actually, Gori3 has pointed out (heuristically) that finite grating size has little effect as long as a sufficient number of slits are illuminated. In our notation the condition turns out to be N ≫ αp/β(2α), where N is the number of slits, p is grating period, and 2a equals the slit width.

P. M. Woodward, Probablility and Information Theory with Applications to Radar (Pergamon, New York, 1953).

Comments similar to those in Ref. 13 also apply here.

We have assumed a grating consisting of an odd number of slits centered on the optical axis of the system. This in no way limits the generality of the analysis and is used only for simplicity.

Strictly speaking, the complex degree of spatial coherence cannot have this functional form since the δ function is not a properly normalizable function. This form is a result of our infinite grating assumption, which is a mathematical convenience for the purpose of the analysis. Including the finite extent of a physical grating will eliminate this mathematical difficulty. What we have done, then, in going from Eq. (26) to (32) is to normalize the area of the Dirac δ function corresponding to zero separation, i.e., σ = 0, to unity.

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Figures (12)

Fig. 1
Fig. 1

System for producing and observing the Lau effect. An incoherent source is imaged incoherently by lens L1 onto the grating G1. Light propagating from G1 illuminates grating G2, and the Fourier transform of the field propagating from G2 is produced by lens L2 in the plane F.

Fig. 2
Fig. 2

Lau fringes produced by an extended white-light source. (This figure can be related to later discussions by noting that z 0 = 3 p 2 / 7 λ ¯, where λ ¯ is the mean wavelength.)

Fig. 3
Fig. 3

Transmittance t(u) of the grating G2.

Fig. 4
Fig. 4

Intensity distribution in the fringe pattern produced by the Lau effect.

Fig. 5
Fig. 5

Case 1; β = 1, α = even integer, z0 = αp2/λ: (a) locations of the centers of the apertures in the grating G2; (b) amplitude and phase of the complex degree of spatial coherence μ G 2 ( ) ( ) of the illumination falling on the grating G2; α = 2; (c) amplitude and phase of the complex degree of spatial coherence μ G 2 ( ) ( ) of the illumination falling on the grating G2; α = 4.

Fig. 6
Fig. 6

Case 2; β = 1; α = odd integer, z0 = αp2/λ: (a) locations of the centers of the apertures in the grating G2; (b) amplitude and phase of the complex degree of spatial coherence μ G 2 ( ) ( ) of the illumination falling on the grating G2; α = 1; (c) amplitude and phase of the complex degree of spatial coherence μ G 2 ( ) ( ) of the illumination falling on the grating G2; α = 3.

Fig. 7
Fig. 7

Case 3; α = 1, β = even integer, z0 = p2/βλ: (a) locations of the centers of the apertures in the grating G2; (b) amplitude and phase of the complex degree of spatial coherence μ G 2 ( ) ( ) of the illumination falling on the grating G2; β = 2; (c) amplitude and phase of the complex degree of spatial coherence μ G 2 ( ) ( ) of the illumination falling on the grating G2; β = 4.

Fig. 8
Fig. 8

Case 4, α = 1, β = odd integer, z0 = p2/βλ: (a) locations of the centers of the apertures in the grating G2; (b) amplitude and phase of the complex degree of spatial coherence μ G 2 ( ) ( ) of the illumination falling on the grating G2; β = 3; (c) amplitude and phase of the complex degree of spatial coherence μ G 2 ( ) ( ) of the illumination falling of the grating G2; β = 5.

Fig. 9
Fig. 9

Case 5, αβ = even integer; α ≠ 1 ≠ β; z0 = αp2/βλ: (a) locations of the centers of the apertures in the grating G2; (b) amplitude and phase of the complex degree of spatial coherence μ G 2 ( ) ( ) of the illumination falling on the grating G2; α = 2, β = 3; (c) amplitude and phase of the complex degree of spatial coherence μ G 2 ( ) ( ) of the illumination falling on the grating G2, α = 4, β = 3.

Fig. 10
Fig. 10

Case 6, αβ = odd integer; α ≠ 1 ≠ β; z0 = αp2/βλ: (a) locations of the centers of the apertures in the grating G2; amplitude and phase of the complex degree of spatial coherence μ G 2 ( ) ( ) of the illumination falling on the grating G2, α = 3, β = 5; (c) amplitude and phase of the complex degree of spatial coherence μ G 2 ( ) ( ) of the illumination falling on the grating G2; α = 7, β = 5.

Fig. 11
Fig. 11

Illustration of the intensity distribution in the Lau fringes. R is a proportionality constant related to the total number of slits in both gratings and the intensity iω. (a) α = 1, β = 1, z1 ≃ 45.8 cm; (b) α = 2, β = 1, z0 ≃ 91.5 cm; (c) α = 1, β = 2, z0 ≃ 22.9 cm; (d) α = 1, β = 3, z0 ≃ 15.3 cm; (e) α = 2, β = 3, z0 ≃ 30.5 cm; (f) α = 3, β = 5, z0 ≃ 27.5 cm.

Fig. 12
Fig. 12

Cross-spectral density function for quasi-monochromatic light.

Equations (50)

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W G 1 ( + ) ( x 1 , x 2 , ω ) = g ( x 1 ) g * ( x 2 ) W G 1 ( ) ( x 1 , x 2 , ω ) ,
W G 2 ( ) ( u 1 , u 2 , ω ) = g ( x 1 ) g * ( x 2 ) W G 1 ( ) ( x 1 , x 2 , ω ) × exp { i k [ ( x 1 u 1 ) 2 ( x 2 u 2 ) 2 ] / 2 z 0 } d x 1 d x 2 .
W F ( y 1 , y 2 , ω ) = exp ( i ϕ ) t ( u 1 ) t * ( u 2 ) W G 1 ( ) ( u 1 , u 2 , ω ) × exp [ i k ( u 1 y 1 u 2 y 2 ) f ] d u 1 d u 2 ,
I F ( y , ω ) = W ( y , y , ω ) ,
I F ( y , ω ) = t ( u 1 ) t * ( u 2 ) W G 2 ( ) ( u 1 , u 2 , ω ) × exp [ i k ( u 1 u 2 ) y / f ] d u 1 d u 2 ,
W G 1 ( + ) ( x 1 , x 2 , ω ) = i ω ( x 1 ) δ ( x 1 x 2 ) .
W G 2 ( ) ( u 1 , u 2 , ω ) = exp [ i k ( u 1 2 u 2 2 ) / 2 z 0 ] i ω ( x 1 ) | g ( x 1 ) | 2 × exp [ i k ( u 1 u 2 ) x 1 / z 0 ] d x 1 .
| g ( x ) | 2 = | t ( x ) | 2 = t ( x ) = t ( x + p ) .
t ( x ) = n = C n exp ( i 2 π n x / p ) ,
C n = 1 p p / 2 p / 2 t ( x ) exp ( i 2 π n x / p ) d x .
W G 2 ( ) ( u 1 , u 2 , ω ) = exp [ i k ( u 1 2 u 2 2 ) / 2 z 0 ] × n = C n i ω ( x 1 ) × exp { i x 1 [ ( k ( u 1 u 2 ) / z 0 ) 2 π n / p ] } d x 1 .
W G 2 ( ) ( u 1 , u 2 , ω ) = i ω n = C n × exp [ i π n ( u 1 + u 2 ) / p ] δ ( u 1 u 2 λ z 0 n / p )
σ = ( u 1 u 2 ) , u = u 2
u 1 = σ + u , u 2 = u .
W G 2 ( ) ( σ , u , ω ) = i ω n = C n exp ( i 2 π n u / p ) × exp [ i π λ z 0 ( n / p ) 2 ] δ ( σ λ z 0 n / p ) ,
I F ( y , ω ) = i ω n = C n B n exp [ i π λ z 0 ( n / p ) 2 ] × exp [ i 2 π n y / p ( f / z 0 ) ] ,
B n = t ( u ) t ( u + λ z 0 n / p ) exp ( i 2 π n u / p ) d u .
P ( u ) = t ( u ) t ( u + λ z 0 n / p ) ,
P ( u ) = t ( u ) for { λ z 0 / p = α p ; α = integer for all n or λ z 0 / p = α p / β ; α / β = rational number integer ; for all n = γ β , γ = integer ; a < p / 2 β .
0 for { λ z 0 / p = α p / β ; α / β = rational number integer ; n γ β , γ = integer ; a < p / 2 β .
t ( u ) = m = ( N 1 ) 2 ( N 1 ) 2 t p ( u m p ) ,
t p ( u ) = { 1 , | u | a 0 , | u | > a
B n = m = ( N 1 ) 2 ( N 1 ) 2 exp ( i 2 π m n ) p / 2 p / 2 t p ( u ) exp ( i 2 π n u / p ) d u .
B n L p / 2 p / 2 t p ( u ) exp ( i 2 π n u / p ) d u ; for conditions in Eq . ( 18 a ) .
B n p L C * n ; conditioned on Eq . ( 18 a ) ,
z 0 = ( α p 2 ) / ( β λ ) .
exp [ i π λ z 0 ( n / p ) 2 ] = exp [ i π ( α β ) n 2 ] .
W G 2 ( ) ( σ , u , ω ) = i ω n = C n × exp [ i π ( α β ) n 2 ] exp ( i 2 π n u / p ) δ ( σ α β n p ) ;
I F ( y , ω ) = { i ω p L n = C n 2 ( 1 ) n × exp [ i 2 π n y / p β ( f / z 0 ) ] ; α β odd , i ω p L n = C n 2 × exp [ i 2 π n y / p β ( f / z 0 ) ] ; α β even ;
C n = 1 p p / 2 p / 2 t ( u ) exp [ i 2 π n u / ( p / β ) d u ] ,
p = p β · f z 0 = λ f α p ,
W = ( 2 a ) · 2 f z 0 = ( 2 a ) · 2 f λ p 2 · β α .
μ ( x 1 , x 2 , ω ) = W ( x 1 , x 2 , ω ) I ( x 1 , ω ) I ( x 2 , ω ) ,
μ G 2 ( ) ( σ , u , ω ) = n = C n exp ( i 2 π n u / p ) exp ( i π α β n 2 ) δ ( σ α p n / β ) ,
W ( r 1 , r 2 , ω ) = 1 2 π Γ ( r 1 , r 2 , τ ) exp ( i ω τ ) d τ ,
Γ ( r 1 , r 2 , τ ) = 0 W ( r 1 , r 2 , ω ) exp ( i ω τ d ω )
Δ ω ω ¯ ,
Δ ω τ 2 π .
ω = ω ω ¯ ,
ω = ω + ω ¯ , d ω = d ω ,
Γ ( r 1 , r 2 , τ ) = ω ¯ W ( r 1 , r 2 , ω + ω ¯ ) exp ( i ( ω + ω ¯ ) τ ) d ω .
Γ ( r 1 , r 2 , τ ) exp ( i ω ¯ τ ) Δ ω / 2 Δ ω / 2 W ( r 1 , r 2 , ω + ω ¯ ) exp ( i ω τ ) d ω .
Γ ( r 1 , r 2 , τ ) exp ( i ω ¯ τ ) Δ ω / 2 Δ ω / 2 W ( r 1 , r 2 , ω + ω ¯ ) d ω ,
Γ ( r 1 , r 2 , τ ) = exp ( i ω ¯ τ ) 0 W ( r 1 , r 2 , ω ) d ω ,
Γ ( r 1 , r 2 , τ ) = Γ ( r 1 , r 2 , 0 ) exp ( i ω ¯ τ ) ,
Γ ( r 1 , r 2 , 0 ) = 0 ( r 1 , r 2 , ω ) d ω .
Γ ( r 1 , r 2 , 0 ) ( Δ ω ) W ( r 1 , r 2 , ω ¯ ) .
γ ( r 1 , r 2 , 0 ) = Γ ( r 1 , r 2 , 0 ) Γ ( r 1 , r 1 , 0 ) Γ ( r 2 , r 2 , 0 ) .
γ ( r 1 , r 2 , 0 ) W ( r 1 , r 2 , ω ¯ ) W ( r 1 , r 1 , ω ¯ ) W ( r 2 , r 2 , ω ¯ ) .
γ ( r 1 , r 2 , 0 ) μ ( r 1 , r 2 , ω ¯ ) .

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