Abstract

The explicit form of the cross-spectral density function characterizing a large homogeneous 1-D Lambertian source is derived, and the associated coherent-mode representation in the space-frequency domain is developed. The buildup of the total radiant intensity from the various coherent-mode contributions is analyzed with proper account taken of the mode strengths. The results are compared with analogous results pertaining to a 1-D spatially strictly incoherent source. It is found that differences in the strengths of certain modes lead to a clear physical explanation of the characteristically different radiation properties of these two types of source.

© 1984 Optical Society of America

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References

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  1. R. C. Bourret, “Coherence Properties of Blackbody Radiation,” Nuovo Cimento 18, 347 (1960).
    [Crossref]
  2. C. L. Mehta, E. Wolf, “Coherence Properties of Blackbody Radiation. III. Cross-spectral Tensors,” Phys. Rev. 161, 1328 (1967).
    [Crossref]
  3. M. Beran, G. Parrent, “The Mutual Coherence of Incoherent Radiation,” Nuovo Cimento 27, 1049 (1963).
    [Crossref]
  4. A. Walther, “Radiometry and Coherence,” J. Opt. Soc. Am. 58, 1256 (1968).
    [Crossref]
  5. E. W. Marchand, E. Wolf, “Radiometry with Sources of any State of Coherence,” J. Opt. Soc. Am. 64, 1219 (1974).
    [Crossref]
  6. E. Wolf, W. H. Carter, “Angular Distribution of Radiant Intensity from Sources of Different States of Spatial Coherence,” Opt. Commun. 13, 205 (1975).
    [Crossref]
  7. E. Wolf, “Coherence and Radiometry,” J. Opt. Soc. Am. 68, 6 (1978).
    [Crossref]
  8. W. H. Carter, E. Wolf, “Coherence Properties of Lambertian and Non-Lambertian Sources,” J. Opt. Soc. Am. 65, 1067 (1975).
    [Crossref]
  9. E. Wolf, “New Spectral Representation of Random Sources and of the Partially Coherent Fields that they Generate,” Opt. Commun. 38, 3 (1981).
    [Crossref]
  10. E. Wolf, “New Theory of Partial Coherence in the Space-Frequency Domain. Part I: Spectra and Cross-spectra of Steady-State Sources,” J. Opt. Soc. Am. 72, 343 (1982).
    [Crossref]
  11. A. Starikov, E. Wolf, “Coherent-Mode Representation of Gaussian Schell-Model Sources and of Their Radiation Fields,” J. Opt. Soc. Am. 72, 923 (1982).
    [Crossref]
  12. E. W. Marchand, E. Wolf, “Angular Correlation and the Far-Zone Behavior of Partially Coherent Fields,” J. Opt. Soc. Am. 62, 379 (1972).
    [Crossref]
  13. Strictly speaking, the quantity J(θ,ω) defined here represents the radiant intensity produced by the source per unit length perpendicular to the (x,z) plane. This fact is related to the geometrical shape of any practical realization of a 1-D (or line) source. For the optical field to obey the 2-D Helmholtz equation (∂2/∂x2 + ∂2/∂z2 + k2)v(x,z;ω) = 0, the source clearly has to be infinite and uniform in the y direction.
  14. E. Wolf, “Radiant Intensity from Planar Sources of any State of Coherence,” J. Opt. Soc. Am. 68, 1597 (1978).
    [Crossref]
  15. W. H. Carter, “Radiant Intensity from Inhomogeneous Sources and the Concept of Averaged Cross-spectral Density,” Opt. Commun. 26, 1 (1978).
    [Crossref]
  16. E. Collett, E. Wolf, “New Equivalence Theorems for Planar Sources that Generate the Same Distributions of Radiant Intensity,” J. Opt. Soc. Am. 69, 942 (1979).
    [Crossref]
  17. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (U.S. GPO, Washington, D.C., 1964).
  18. E. Wolf, W. H. Carter, “On the Radiation Efficiency of Quasi-homogeneous Sources of Different Degrees of Spatial Coherence,” in Coherence and Quantum Optics, Vol. 4, L. Mandel, E. Wolf, Eds. (Plenum, New York, 1978), pp. 415–429.
  19. A. T. Friberg, “Radiation from Partially Coherent Sources,” Opt. Eng. 21, 362 (1982).
    [Crossref]
  20. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).
  21. A. T. Friberg, “Phase-space Methods for Partially Coherent Wavefields,” in Optics in Four Dimensions—1980, M. A. Machado, L. M. Narducci, Eds., Conf. Proc.65 (American Institute of Physics, New York, 1981), pp. 313–331.
  22. W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).
  23. B. Saleh, Photoelectron Statistics (Springer, Berlin, 1978).
  24. We point out that this choice differs slightly from that of Refs. 9–11, where each eigenvalue λn is considered as an integral part of the cross-spectral density of the corresponding mode. In our terminology all large homogeneous line sources possess the same mode cross-spectral density functions that depend on the source length L as shown by Eqs. (31) and (32). The various sources of this type then differ according to the relative strengths with which these mode contributions are superposed to obtain the total cross-spectral density function of the source. We note also that the present situation is somewhat analogous to the usual Karhunen-Loeve expansion of a stationary signal in the limit as the expansion interval approaches infinity (see Ref. 22, Sec. 6-4).
  25. For simplicity, we omit here any reference to the low and high spatial frequency parts of the optical field v(x,z;ω) across the line source [see Eq. (8) and (28)].
  26. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

1982 (3)

1981 (1)

E. Wolf, “New Spectral Representation of Random Sources and of the Partially Coherent Fields that they Generate,” Opt. Commun. 38, 3 (1981).
[Crossref]

1979 (1)

1978 (3)

E. Wolf, “Radiant Intensity from Planar Sources of any State of Coherence,” J. Opt. Soc. Am. 68, 1597 (1978).
[Crossref]

W. H. Carter, “Radiant Intensity from Inhomogeneous Sources and the Concept of Averaged Cross-spectral Density,” Opt. Commun. 26, 1 (1978).
[Crossref]

E. Wolf, “Coherence and Radiometry,” J. Opt. Soc. Am. 68, 6 (1978).
[Crossref]

1975 (2)

W. H. Carter, E. Wolf, “Coherence Properties of Lambertian and Non-Lambertian Sources,” J. Opt. Soc. Am. 65, 1067 (1975).
[Crossref]

E. Wolf, W. H. Carter, “Angular Distribution of Radiant Intensity from Sources of Different States of Spatial Coherence,” Opt. Commun. 13, 205 (1975).
[Crossref]

1974 (1)

1972 (1)

1968 (1)

1967 (1)

C. L. Mehta, E. Wolf, “Coherence Properties of Blackbody Radiation. III. Cross-spectral Tensors,” Phys. Rev. 161, 1328 (1967).
[Crossref]

1963 (1)

M. Beran, G. Parrent, “The Mutual Coherence of Incoherent Radiation,” Nuovo Cimento 27, 1049 (1963).
[Crossref]

1960 (1)

R. C. Bourret, “Coherence Properties of Blackbody Radiation,” Nuovo Cimento 18, 347 (1960).
[Crossref]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (U.S. GPO, Washington, D.C., 1964).

Beran, M.

M. Beran, G. Parrent, “The Mutual Coherence of Incoherent Radiation,” Nuovo Cimento 27, 1049 (1963).
[Crossref]

Bourret, R. C.

R. C. Bourret, “Coherence Properties of Blackbody Radiation,” Nuovo Cimento 18, 347 (1960).
[Crossref]

Carter, W. H.

W. H. Carter, “Radiant Intensity from Inhomogeneous Sources and the Concept of Averaged Cross-spectral Density,” Opt. Commun. 26, 1 (1978).
[Crossref]

W. H. Carter, E. Wolf, “Coherence Properties of Lambertian and Non-Lambertian Sources,” J. Opt. Soc. Am. 65, 1067 (1975).
[Crossref]

E. Wolf, W. H. Carter, “Angular Distribution of Radiant Intensity from Sources of Different States of Spatial Coherence,” Opt. Commun. 13, 205 (1975).
[Crossref]

E. Wolf, W. H. Carter, “On the Radiation Efficiency of Quasi-homogeneous Sources of Different Degrees of Spatial Coherence,” in Coherence and Quantum Optics, Vol. 4, L. Mandel, E. Wolf, Eds. (Plenum, New York, 1978), pp. 415–429.

Collett, E.

Davenport, W. B.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

Friberg, A. T.

A. T. Friberg, “Radiation from Partially Coherent Sources,” Opt. Eng. 21, 362 (1982).
[Crossref]

A. T. Friberg, “Phase-space Methods for Partially Coherent Wavefields,” in Optics in Four Dimensions—1980, M. A. Machado, L. M. Narducci, Eds., Conf. Proc.65 (American Institute of Physics, New York, 1981), pp. 313–331.

Goodman, J. W.

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

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

Marchand, E. W.

Mehta, C. L.

C. L. Mehta, E. Wolf, “Coherence Properties of Blackbody Radiation. III. Cross-spectral Tensors,” Phys. Rev. 161, 1328 (1967).
[Crossref]

Parrent, G.

M. Beran, G. Parrent, “The Mutual Coherence of Incoherent Radiation,” Nuovo Cimento 27, 1049 (1963).
[Crossref]

Root, W. L.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

Saleh, B.

B. Saleh, Photoelectron Statistics (Springer, Berlin, 1978).

Starikov, A.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (U.S. GPO, Washington, D.C., 1964).

Walther, A.

Wolf, E.

A. Starikov, E. Wolf, “Coherent-Mode Representation of Gaussian Schell-Model Sources and of Their Radiation Fields,” J. Opt. Soc. Am. 72, 923 (1982).
[Crossref]

E. Wolf, “New Theory of Partial Coherence in the Space-Frequency Domain. Part I: Spectra and Cross-spectra of Steady-State Sources,” J. Opt. Soc. Am. 72, 343 (1982).
[Crossref]

E. Wolf, “New Spectral Representation of Random Sources and of the Partially Coherent Fields that they Generate,” Opt. Commun. 38, 3 (1981).
[Crossref]

E. Collett, E. Wolf, “New Equivalence Theorems for Planar Sources that Generate the Same Distributions of Radiant Intensity,” J. Opt. Soc. Am. 69, 942 (1979).
[Crossref]

E. Wolf, “Coherence and Radiometry,” J. Opt. Soc. Am. 68, 6 (1978).
[Crossref]

E. Wolf, “Radiant Intensity from Planar Sources of any State of Coherence,” J. Opt. Soc. Am. 68, 1597 (1978).
[Crossref]

W. H. Carter, E. Wolf, “Coherence Properties of Lambertian and Non-Lambertian Sources,” J. Opt. Soc. Am. 65, 1067 (1975).
[Crossref]

E. Wolf, W. H. Carter, “Angular Distribution of Radiant Intensity from Sources of Different States of Spatial Coherence,” Opt. Commun. 13, 205 (1975).
[Crossref]

E. W. Marchand, E. Wolf, “Radiometry with Sources of any State of Coherence,” J. Opt. Soc. Am. 64, 1219 (1974).
[Crossref]

E. W. Marchand, E. Wolf, “Angular Correlation and the Far-Zone Behavior of Partially Coherent Fields,” J. Opt. Soc. Am. 62, 379 (1972).
[Crossref]

C. L. Mehta, E. Wolf, “Coherence Properties of Blackbody Radiation. III. Cross-spectral Tensors,” Phys. Rev. 161, 1328 (1967).
[Crossref]

E. Wolf, W. H. Carter, “On the Radiation Efficiency of Quasi-homogeneous Sources of Different Degrees of Spatial Coherence,” in Coherence and Quantum Optics, Vol. 4, L. Mandel, E. Wolf, Eds. (Plenum, New York, 1978), pp. 415–429.

J. Opt. Soc. Am. (9)

Nuovo Cimento (2)

M. Beran, G. Parrent, “The Mutual Coherence of Incoherent Radiation,” Nuovo Cimento 27, 1049 (1963).
[Crossref]

R. C. Bourret, “Coherence Properties of Blackbody Radiation,” Nuovo Cimento 18, 347 (1960).
[Crossref]

Opt. Commun. (3)

E. Wolf, W. H. Carter, “Angular Distribution of Radiant Intensity from Sources of Different States of Spatial Coherence,” Opt. Commun. 13, 205 (1975).
[Crossref]

W. H. Carter, “Radiant Intensity from Inhomogeneous Sources and the Concept of Averaged Cross-spectral Density,” Opt. Commun. 26, 1 (1978).
[Crossref]

E. Wolf, “New Spectral Representation of Random Sources and of the Partially Coherent Fields that they Generate,” Opt. Commun. 38, 3 (1981).
[Crossref]

Opt. Eng. (1)

A. T. Friberg, “Radiation from Partially Coherent Sources,” Opt. Eng. 21, 362 (1982).
[Crossref]

Phys. Rev. (1)

C. L. Mehta, E. Wolf, “Coherence Properties of Blackbody Radiation. III. Cross-spectral Tensors,” Phys. Rev. 161, 1328 (1967).
[Crossref]

Other (10)

Strictly speaking, the quantity J(θ,ω) defined here represents the radiant intensity produced by the source per unit length perpendicular to the (x,z) plane. This fact is related to the geometrical shape of any practical realization of a 1-D (or line) source. For the optical field to obey the 2-D Helmholtz equation (∂2/∂x2 + ∂2/∂z2 + k2)v(x,z;ω) = 0, the source clearly has to be infinite and uniform in the y direction.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

A. T. Friberg, “Phase-space Methods for Partially Coherent Wavefields,” in Optics in Four Dimensions—1980, M. A. Machado, L. M. Narducci, Eds., Conf. Proc.65 (American Institute of Physics, New York, 1981), pp. 313–331.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

B. Saleh, Photoelectron Statistics (Springer, Berlin, 1978).

We point out that this choice differs slightly from that of Refs. 9–11, where each eigenvalue λn is considered as an integral part of the cross-spectral density of the corresponding mode. In our terminology all large homogeneous line sources possess the same mode cross-spectral density functions that depend on the source length L as shown by Eqs. (31) and (32). The various sources of this type then differ according to the relative strengths with which these mode contributions are superposed to obtain the total cross-spectral density function of the source. We note also that the present situation is somewhat analogous to the usual Karhunen-Loeve expansion of a stationary signal in the limit as the expansion interval approaches infinity (see Ref. 22, Sec. 6-4).

For simplicity, we omit here any reference to the low and high spatial frequency parts of the optical field v(x,z;ω) across the line source [see Eq. (8) and (28)].

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

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (U.S. GPO, Washington, D.C., 1964).

E. Wolf, W. H. Carter, “On the Radiation Efficiency of Quasi-homogeneous Sources of Different Degrees of Spatial Coherence,” in Coherence and Quantum Optics, Vol. 4, L. Mandel, E. Wolf, Eds. (Plenum, New York, 1978), pp. 415–429.

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Figures (6)

Fig. 1
Fig. 1

Geometry and notation relating to the problem of radiation by a 1-D (line) source.

Fig. 2
Fig. 2

Normalized low-frequency parts of the cross-spectral density functions associated with the optical fields across a Lambertian and a spatially incoherent line source.

Fig. 3
Fig. 3

Polar diagrams of the radiant intensities generated by a Lambertian and a spatially incoherent line source.

Fig. 4
Fig. 4

Normalized spatial frequency spectra of a Lambertian and a spatially incoherent line source as a function of the spatial frequency variable f.

Fig. 5
Fig. 5

Relative eigenvalues λn(ω)/λ0(ω) associated with the coherent-mode representations of a Lambertian and a spatially incoherent line source. The dots have been calculated for each value of n from Eqs. (34) and (39) with L/λ = 40.

Fig. 6
Fig. 6

Polar diagrams of the normalized mode contributions Jn(θ,ω)/J0(0,ω) to the radiant intensity for n = 0, 1, and 2. The graphs have been calculated from Eq. (42) with kL = 40. The solid circle represents the total radiant intensity from a Lambertian line source, whereas the dashed curve represents the radiant intensity produced by a spatially incoherent line source.

Equations (52)

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T r W i j ( r 1 , r 2 ; ω ) = C T ( ω ) sin ( k r 1 - r 2 ) k r 1 - r 2 ,
C T ( ω ) = 4 ω 3 c 3 1 exp ( ω / k T ) - 1 ,
W L F ( ρ 1 , ρ 2 ; ω ) = C ( ω ) sin ( k ρ 1 - ρ 2 ) k ρ 1 - ρ 2 .
v ( ) ( R , θ ; ω ) = k 2 π cos θ v ˜ ( k sin θ , 0 ; ω ) exp [ i ( k R - π / 4 ) ] k R ,
v ˜ ( f , 0 ; ω ) = 1 2 π - v ( x , 0 ; ω ) exp ( - i f x ) d x
I ( ) ( R , θ ; ω ) = 2 π k cos 2 θ ( 1 R ) W ˜ 0 ( - k sin θ , k sin θ ; ω ) .
W ˜ 0 ( f 1 , f 2 ; ω ) = 1 ( 2 π ) 2 W 0 ( x 1 , x 2 ; ω ) × exp [ - i ( f 1 x 1 + f 2 x 2 ) ] d x 1 d x 2 .
W 0 ( x 1 , x 2 ; ω ) = v * ( x 1 , 0 ; ω ) v ( x 2 , 0 ; ω ) ,
J ( θ , ω ) = R I ( ) ( R , θ ; ω )
J ( θ , ω ) = 2 π k cos 2 θ W ˜ 0 ( - k sin θ , k sin θ ; ω ) .
W 0 ( x 1 , x 2 ; ω ) = F ( x 1 - x 2 , ω ) ,
W ˜ 0 ( - k sin θ , k sin θ ; ω ) = 1 ( 2 π ) 2 F ( x , ω ) × exp ( i k sin θ x ) d x d x ,
W ˜ 0 ( - k sin θ , k sin θ ; ω ) = L 2 π F ˜ ( - k sin θ , ω ) ,
J ( θ , ω ) = k L cos 2 θ F ˜ ( - k sin θ , ω ) ,
F L F ( x , ω ) = - k k F ˜ ( f , ω ) exp ( i f x ) d f .
F L F ( x , ω ) = 1 L - π / 2 π / 2 J ( θ , ω ) cos θ exp ( - i k sin θ x ) d θ .
J ( θ , ω ) = C ( ω ) cos θ ,
F L F ( x , ω ) = C ( ω ) L - π / 2 π / 2 exp ( - i k sin θ x ) d θ ,
F L F ( x , ω ) = C ( ω ) L 0 π exp ( i k cos θ x ) d θ .
J 0 ( x ) = 1 π 0 π exp ( i k cos θ x ) d θ
W 0 L F ( x 1 - x 2 , ω ) = π C ( ω ) L J 0 [ k ( x 1 - x 2 ) ] ,
δ L F ( x 1 - x 2 ) = 1 2 π - k k exp [ - i f ( x 1 - x 2 ) ] = ( k π ) sin k ( x 1 - x 2 ) k ( x 1 - x 2 ) .
W 0 ( x 1 - x 2 , ω ) = 2 π C ( ω ) L δ [ k ( x 1 - x 2 ) ] ,
W 0 L F ( x 1 - x 2 ; ω ) = 2 C ( ω ) L sin k ( x 1 - x 2 ) k ( x 1 - x 2 )
J ( θ , ω ) = C ( ω ) cos 2 θ .
F ˜ ( f , ω ) = C ( ω ) L 1 k 2 - f 2 ,
F ˜ ( f , ω ) = C ( ω ) k L .
W 0 L F ( x 1 , x 2 ; ω ) = n λ n ( ω ) W n ( x 1 , x 2 ; ω ) ,
W n ( x 1 , x 2 ; ω ) = ψ n * ( x 1 , ω ) ψ n ( x 2 , ω )
- L / 2 L / 2 π C ( ω ) L J 0 [ k ( x 1 - x 2 ) ] ψ n ( x 1 , ω ) d x 1 = λ n ( ω ) ψ n ( x 2 , ω ) .
ψ n ( x , ω ) = 1 L exp ( i κ n x ) ,
κ n = n · ( 2 π L ) ,             n = 0 , ± 1 , ± 2 , .
λ n ( ω ) = 2 π C ( ω ) L 1 k 2 - κ n 2 , when n k L 2 π ,
= 0 , when n > k L 2 π .
λ n ( ω ) λ 0 ( ω ) = 1 1 - n 2 ( λ / L ) 2 ,             ( n L λ ) ,
N 2 ( k L 2 π ) + 1 1.
W n ( x 1 , x 2 ; ω ) = 1 L exp [ - i κ n ( x 1 - x 2 ) ] ,
- L / 2 L / 2 2 C ( ω ) L sin k ( x 1 - x 2 ) k ( x 1 - x 2 ) ψ n ( x 1 , ω ) d x 1 = λ n ( ω ) ψ n ( x 2 , ω ) .
λ n ( ω ) = 2 π C ( ω ) k L , when n k L 2 π ,
= 0 , when n > k L 2 π ,
λ n ( ω ) λ 0 ( ω ) = 1 ,             ( n L λ ) ,
ψ n ( ) ( R , θ ; ω ) = k L 2 π cos θ exp [ i ( k R - π / 4 ) ] R × sin [ L ( κ n - k sin θ ) / 2 ] L ( κ n - k sin θ ) / 2 ,
I ( ) ( R , θ ; ω ) = k L 2 π cos 2 θ 1 R { sin [ L ( κ n - k sin θ ) / 2 ] L ( κ n - k sin θ ) / 2 } 2 ,
J ( θ , ω ) = k L 2 π cos 2 θ { sin [ L ( κ n - k sin θ ) / 2 ] L ( κ n - k sin θ ) / 2 } 2
J ( θ , ω ) = n λ n ( ω ) J n ( θ , ω ) ,
J n ( 0 , ω ) = k L 2 π , when n = 0 ,
= 0 , otherwise .
sin [ L ( κ n - k sin θ ) / 2 ] L ( κ n - k sin θ ) / 2 δ n , ( k L / 2 π ) sin θ ,
J ( θ , ω ) = 2 π C ( ω ) L 1 k 2 - k 2 sin 2 θ ( k L 2 π ) cos 2 θ = C ( ω ) cos θ .
J ( θ , ω ) = 2 π C ( ω ) k L ( k L 2 π ) cos 2 θ = C ( ω ) cos 2 θ ;
v ( x , 0 ; ω ) = n a n ( ω ) ψ n ( x , ω ) ,
a n * ( ω ) a m ( ω ) = λ n ( ω ) δ n , m .

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