Abstract

It is shown for a particular class of partially coherent fields that, when such a field is incident upon a circular aperture, the spectrum of the light at an observation point in the far zone of the aperture is different from the spectrum of the light in the aperture. An explicit expression for the spectrum at such observation points is obtained, and it is shown that the difference between the two spectra depends on the ratio of the radius of the aperture to the effective correlation length of the light in the aperture. Numerical examples are presented, and the relevant limiting cases are discussed.

© 1991 Optical Society of America

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References

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  1. E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
    [CrossRef] [PubMed]
  2. G. M. Morris, D. Faklis, “Effects of source correlation on the spectrum of light,” Opt. Commun. 62, 5–11 (1987).
    [CrossRef]
  3. E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
    [CrossRef]
  4. E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlations,” Opt. Commun. 62, 12–16 (1987).
    [CrossRef]
  5. E. Wolf, “Redshifts and blueshifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987). The sources discussed in Refs. 3–5 were three-dimensional primary sources. The corresponding results for planar secondary sources are essentially the same.
    [CrossRef] [PubMed]
  6. D. Faklis, M. Morris, “Spectral shifts produced by source correlations,” Opt. Lett. 13, 4–6 (1988).
    [CrossRef] [PubMed]
  7. F. Gori, G. Guattari, C. Palma, “Observation of optical redshifts and blueshifts produced by source correlations,” Opt. Commun. 67, 1–4 (1988).
    [CrossRef]
  8. H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–172 (1989).
    [CrossRef]
  9. J. T. Foley, “The effect of an aperture on the spectrum of partially coherent light,” Opt. Commun. 75, 347–352 (1990).
    [CrossRef]
  10. μis sometimes also referred to as the complex degree of spectral coherence.
  11. E. W. Marchand, E. Wolf, “Radiometry with sources of any state of coherence,”J. Opt. Soc. Am. 64, 1219–1226 (1974), Eqs. (2) and (12), with the factor cos2θ omitted because of the paraxial approximation.
    [CrossRef]
  12. These values of ω¯ and Γ were chosen so that s(0)(ω) corresponds, roughly, to the analogous spectrum observed in Ref. 2, their SIIconv(ω). This is a broadband spectrum.
  13. To within the resolution used to plot Figs. 4–6, there is no shift of peak of the spectrum. More detailed calculations show that small shifts do occur, even for fairly large values of a/ L¯. Such shifts are discussed, for on-axis observation points, in Ref. 9.
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 119–120.

1990

J. T. Foley, “The effect of an aperture on the spectrum of partially coherent light,” Opt. Commun. 75, 347–352 (1990).
[CrossRef]

1989

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–172 (1989).
[CrossRef]

1988

D. Faklis, M. Morris, “Spectral shifts produced by source correlations,” Opt. Lett. 13, 4–6 (1988).
[CrossRef] [PubMed]

F. Gori, G. Guattari, C. Palma, “Observation of optical redshifts and blueshifts produced by source correlations,” Opt. Commun. 67, 1–4 (1988).
[CrossRef]

1987

G. M. Morris, D. Faklis, “Effects of source correlation on the spectrum of light,” Opt. Commun. 62, 5–11 (1987).
[CrossRef]

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
[CrossRef]

E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlations,” Opt. Commun. 62, 12–16 (1987).
[CrossRef]

E. Wolf, “Redshifts and blueshifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987). The sources discussed in Refs. 3–5 were three-dimensional primary sources. The corresponding results for planar secondary sources are essentially the same.
[CrossRef] [PubMed]

1986

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef] [PubMed]

1974

Faklis, D.

D. Faklis, M. Morris, “Spectral shifts produced by source correlations,” Opt. Lett. 13, 4–6 (1988).
[CrossRef] [PubMed]

G. M. Morris, D. Faklis, “Effects of source correlation on the spectrum of light,” Opt. Commun. 62, 5–11 (1987).
[CrossRef]

Foley, J. T.

J. T. Foley, “The effect of an aperture on the spectrum of partially coherent light,” Opt. Commun. 75, 347–352 (1990).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 119–120.

Gori, F.

F. Gori, G. Guattari, C. Palma, “Observation of optical redshifts and blueshifts produced by source correlations,” Opt. Commun. 67, 1–4 (1988).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, C. Palma, “Observation of optical redshifts and blueshifts produced by source correlations,” Opt. Commun. 67, 1–4 (1988).
[CrossRef]

Joshi, K. C.

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–172 (1989).
[CrossRef]

Kandpal, H. C.

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–172 (1989).
[CrossRef]

Marchand, E. W.

Morris, G. M.

G. M. Morris, D. Faklis, “Effects of source correlation on the spectrum of light,” Opt. Commun. 62, 5–11 (1987).
[CrossRef]

Morris, M.

Palma, C.

F. Gori, G. Guattari, C. Palma, “Observation of optical redshifts and blueshifts produced by source correlations,” Opt. Commun. 67, 1–4 (1988).
[CrossRef]

Vaishya, J. S.

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–172 (1989).
[CrossRef]

Wolf, E.

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
[CrossRef]

E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlations,” Opt. Commun. 62, 12–16 (1987).
[CrossRef]

E. Wolf, “Redshifts and blueshifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987). The sources discussed in Refs. 3–5 were three-dimensional primary sources. The corresponding results for planar secondary sources are essentially the same.
[CrossRef] [PubMed]

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef] [PubMed]

E. W. Marchand, E. Wolf, “Radiometry with sources of any state of coherence,”J. Opt. Soc. Am. 64, 1219–1226 (1974), Eqs. (2) and (12), with the factor cos2θ omitted because of the paraxial approximation.
[CrossRef]

J. Opt. Soc. Am.

Nature (London)

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
[CrossRef]

Opt. Commun.

E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlations,” Opt. Commun. 62, 12–16 (1987).
[CrossRef]

G. M. Morris, D. Faklis, “Effects of source correlation on the spectrum of light,” Opt. Commun. 62, 5–11 (1987).
[CrossRef]

F. Gori, G. Guattari, C. Palma, “Observation of optical redshifts and blueshifts produced by source correlations,” Opt. Commun. 67, 1–4 (1988).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–172 (1989).
[CrossRef]

J. T. Foley, “The effect of an aperture on the spectrum of partially coherent light,” Opt. Commun. 75, 347–352 (1990).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef] [PubMed]

E. Wolf, “Redshifts and blueshifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987). The sources discussed in Refs. 3–5 were three-dimensional primary sources. The corresponding results for planar secondary sources are essentially the same.
[CrossRef] [PubMed]

Other

μis sometimes also referred to as the complex degree of spectral coherence.

These values of ω¯ and Γ were chosen so that s(0)(ω) corresponds, roughly, to the analogous spectrum observed in Ref. 2, their SIIconv(ω). This is a broadband spectrum.

To within the resolution used to plot Figs. 4–6, there is no shift of peak of the spectrum. More detailed calculations show that small shifts do occur, even for fairly large values of a/ L¯. Such shifts are discussed, for on-axis observation points, in Ref. 9.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 119–120.

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Figures (8)

Fig. 1
Fig. 1

Optical system.

Fig. 2
Fig. 2

Normalized far-zone spectrum s(v, ω) [Eq. (4.7)] with ω ¯ = 3.20 × 1015 s−1, Γ = 0.60 × 1015 s−1, and a/ L ¯ = 0.01. The units on ω are 1015 s−1. The arrow indicates the curve for which ω = ω ¯.

Fig. 3
Fig. 3

Normalized far-zone spectrum s(v, ω) with ω ¯ = 3.20 × 1015 s−1, Γ = 0.60 × 1015 s−1, and a/ L ¯ = 0.50. The units on ω are 1015 s−1. The arrow indicates the curve for which ω = ω ¯.

Fig. 4
Fig. 4

Normalized far-zone spectrum s(v, ω) with ω ¯= 3.20 × 1015 s−1, = 0.60 × 1015 s−1, and a/ L ¯ = 1.00. The units on ω are 1015 s−1. The arrow indicates the curve for which ω = ω ¯.

Fig. 5
Fig. 5

Normalized far-zone spectrum s(v, ω) with ω ¯ = 3.20 × 1015 s−1, Γ = 0.60 × 1015 s−1, and a/ L ¯ = 2.00. The units on ω are 1015 s−1. The arrow indicates the curve for which ω = ω ¯.

Fig. 6
Fig. 6

Normalized far-zone spectrum s(v, ω) with ω ¯ = 3.20 × 1015 s−1, Γ = 0.60 × 1015 s−1, and a/ L ¯ = 3.00. The units on ω are 1015 s−1. The arrow indicates the curve for which ω = ω ¯.

Fig. 7
Fig. 7

Normalized far-zone spectrum s(v, ω) with ω ¯ = 3.20 × 1015 s−1, Γ = 0.30 × 1015 s−1, and a/ L ¯ = 0.01. The units on ω are 1015 s−1. The arrow indicates the curve for which ω = ω ¯.

Fig. 8
Fig. 8

Normalized far-zone spectrum s(v, ω) with ω ¯ = 3.20 × 1015 s−1, Γ = 0.30 × 1015 s−1, and a/ L ¯ = 3.00. The units on ω are 1015 s−1. The arrow indicates the curve for which ω = ω ¯.

Equations (37)

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W ( 0 ) ( ρ 1 , ρ 2 , ω ) = S ( 0 ) ( ω ) μ ( 0 ) ( ρ 1 , ρ 2 , ω ) ,
S ( 0 ) ( ω ) = κ S ( s ) ( ω ) ,
μ ( 0 ) ( ρ 1 , ρ 2 , ω ) = Besinc ( k ρ 2 - ρ 1 a s / f ) .
Besinc ( u ) = 2 J 1 ( u ) u ,
k a s / f = 3.832 / L ( ω ) ,
S ( ρ , z , ω ) = ( k / 2 π z ) 2 A d 2 ρ 1 A d 2 ρ 2 W ( 0 ) ( ρ 1 , ρ 2 , ω ) × exp [ - i k ( ρ 2 - ρ 1 ) · ρ / z ] .
S ( ρ , z , ω ) = S ( 0 ) ( ω ) ( k / 2 π z ) 2 A d 2 ρ 1 A d 2 ρ 2 × Besinc ( k ρ 2 - ρ 1 a s / f ) exp [ - i k ( ρ 2 - ρ 1 ) · ρ / z ] .
S ( ρ , z , ω ) = S ( 0 ) ( ω ) M ( ρ , z , ω ) ,
M ( ρ , z , ω ) = 2 ( k a 2 z ) 2 0 1 C ( u ) Besinc ( 2 k a a s f u ) × J 0 ( 2 k a ρ z u ) u d u ,
C ( u ) = ( 2 / π ) [ cos - 1 ( u ) - u ( 1 - u 2 ) 1 / 2 ] .
M ( ρ , z , ω ) 2 ( k a 2 z ) 2 0 1 C ( u ) J 0 ( 2 k a ρ z u ) u d u = ( k a 2 2 z ) 2 Besinc 2 ( k a ρ z ) ,
S ( ρ , z , ω ) S ( 0 ) ( ω ) ( k a 2 2 z ) 2 Besinc 2 ( k a ρ z ) .
α = 2 k a ρ / z ,
β = 2 k a a s / f .
M ( ρ , z , ω ) = 2 ( k a 2 z ) 2 0 1 C ( u ) J 0 ( α u ) Besinc ( β u ) u d u = ( 2 k a 2 β z ) 2 β 0 1 C ( u ) J 0 ( α u ) J 1 ( β u ) d u = ( a f a s z ) 2 f ( α , β ) ,
f ( α , β ) = β 0 1 C ( u ) J 0 ( α u ) J 1 ( β u ) d u .
f ( α , β ) { 1 , α < β 0 , α > β .
S ( ρ , z , ω ) { ( a f / a s z ) 2 S ( 0 ) ( ω ) , ρ / z < a s / f 0 , ρ / z > a s / f .
S ( 0 ) ( ω ) = S 0 Γ 2 ( ω - ω ¯ ) 2 + Γ 2 ,
k ¯ a / z = 3.832 / a ¯ 1 ,
k ¯ a s / f = 3.832 / L ¯ .
S ( ρ , z , ω ) = 2 S ( 0 ) ( ω ) ( k a 2 z ) 2 0 1 C ( u ) J 0 [ 7.664 ( ρ a ¯ 1 ) ( ω ω ¯ ) u ] × Besinc [ 7.664 ( a L ¯ ) ( ω ω ¯ ) u ] u d u .
S coh ( 0 , z , ω ¯ ) = S 0 ( k ¯ a 2 / 2 z ) 2 .
v ρ / a ¯ 1 ,
s ( v , ω ) S ( ρ , z , ω ) S coh ( 0 , z , ω ¯ ) = 8 ( ω / ω ¯ ) 2 s ( 0 ) ( ω ) 0 1 C ( u ) J 0 [ 7.664 v ( ω ω ¯ ) u ] × Besinc [ 7.664 ( a L ¯ ) ( ω ω ¯ ) u ] u d u ,
s ( 0 ) ( ω ) = S ( 0 ) ( ω ) / S 0 = Γ 2 ( ω - ω ¯ ) 2 + Γ 2 .
P ( ρ ) = { 1 , ρ < a 0 , ρ > a ,
S ( ρ , z , ω ) = S ( 0 ) ( ω ) ( k 2 π z ) 2 d 2 ρ 1 d 2 ρ 2 P ( ρ 1 ) P ( ρ 2 ) × Besinc ( k ρ 2 - ρ 1 a s f ) exp [ - i k ( ρ 2 - ρ 1 ) · ρ / z ] ,
ρ = ρ 2 - ρ 1 ,
R = ( 1 / 2 ) ( ρ 1 + ρ 2 ) .
S ( ρ , z , ω ) = S ( 0 ) ( ω ) ( k 2 π z ) 2 P ( ρ ) Besinc ( k ρ a s f ) × exp ( - i k ρ · ρ / z ) d 2 ρ ,
P ( ρ ) = P [ R - ( 1 / 2 ) ρ ] P [ R + ( 1 / 2 ) ρ ] d 2 R .
P ( ρ ) = { π a 2 C ( ρ / 2 a ) , ρ < 2 a 0 , ρ > 2 a ,
C ( u ) = ( 2 / π ) [ cos - 1 ( u ) - u ( 1 - u 2 ) 1 / 2 ] .
S ( ρ , z , ω ) = S ( 0 ) ( ω ) π ( k a 2 π z ) 2 0 2 a 0 2 π C ( ρ 2 a ) Besinc ( k ρ a s f ) × exp ( - i k ρ · ρ / z ) ρ d ρ d θ = 1 2 S ( 0 ) ( ω ) ( k a z ) 2 0 2 a C ( ρ 2 a ) Besinc ( k ρ a s f ) × J 0 ( k ρ ρ z ) ρ d ρ .
S ( ρ , z , ω ) = S ( 0 ) ( ω ) M ( ρ , z , ω ) ,
M ( ρ , z , ω ) = 2 ( k a 2 z ) 2 0 1 C ( u ) Besinc ( 2 k a a s f u ) × J 0 ( 2 k a ρ z u ) u d u .

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