Abstract

The concept of a quasihomogeneous source is introduced. Unlike a source that is strictly homogeneous in its statistical properties, a quasihomogeneous source may be finite. Many physical sources, both primary and secondary ones, are adequately approximated by this model. Coherence and radiometric properties of light generated by such sources (assumed, for simplicity, to be planar) are discussed and an important reciprocity relation is shown to exist between light in the far zone and in the source plane. This relation implies that the degree of coherence in the far zone is given by the classic form of the van Cittert-Zernike theorem, even though the source may have a high degree of spatial coherence over arbitrarily large areas. The reciprocity relation also provides a generalization of a recently derived result that expresses the angular dependence of the radiant intensity in terms of the degree of spatial coherence of light in the source plane. The dependence of all the basic radiometric quantities on the distribution of the optical intensity across the source and on the degree of spatial coherence of the light emerging from the source is discussed and is illustrated, for some typical sources, by computed curves.

© 1977 Optical Society of America

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  1. E. Wolf and W. H. Carter, "Angular Distribution of Radiant Intensity from Sources of Different Degrees of Spatial Coherence," Opt. Commun. 13, 205–209 (1975).
  2. The converse problem of determining the degree of coherence in the source plane from the knowledge of the angular distribution of the radiant intensity was studied by W. H. Carter and E. Wolf in, "Coherence Properties of Lambertian and Non-Lambertian Sources," J. Opt. Soc. Am. 65, 1067–1071 (1975).
  3. Some special cases were also treated by H. P. Baltes, B. Steinle, and G. Antes in, "Spectral Coherence Area and the Radiant Intensity from Statistically Homogeneous and Isotropic Planar Sources," Opt. Commun. 18, 242–246 (1976).
  4. The properties of this correlation coefficient (also called the complex degree of spectral coherence or the spectral correlation coefficient) were recently studied by L. Mandel and E. Wolf in, "Spectral coherence and the concept of cross-spectral purity, " J. Opt. Soc. Am. 66, 529–535 (1976). This correlation coefficient must be distinguished from the more familiar complex degree of coherence (cf. Ref. 6, Sec. 10.3.1), often denoted by γ(r1, r2, τ) [or γ12(τ), even though γ(r1, r2, 0) is also a measure of spatial coherence.
  5. Such a theorem was alluded to by A. Walther in his important paper, "Radiometry and Coherence," J. Opt. Soc. Am. 58, 1256–1259 (1968). However Walther did not formulate the theorem in precise mathematical terms, nor did he state to what class of sources it applies.
  6. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Sec. 10.4.2.
  7. In physical optics the quantity I(r) defined by Eq. (2. 1), is usually referred to simply as intensity (at frequency ω). We use here the adjective "optical" to distinguish it clearly from the radiometric concept of radiant intensity, that we will also encounter in the present paper.
  8. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961) (reprinted by Dover, New York, 1967), Sec. 1. 3.
  9. A. C. Schell, "The Multiple Plate Antenna," Doctoral dissertation (Massachusetts Institute of Technology, 1961) (unpublished), Sec. 7.5.
  10. See also R. A. Shore, "Partially Coherent Diffraction by a Circular Aperture," in Electromagnetic Theory and Antennas, edited by E. C. Jordan (Pergamon, London, 1963), Part 2, pp. 787–795; and A. K. Jaiswal, G. P. Agrawal, and C. L. Mehta, "Coherence Functions in the Far-field Diffraction Plane," Nuovo Cimento B 15, 295–307 (1973).
  11. Strictly speaking this condition only ensures that |µ(r1, r2)| has appreciable values when the separation |r1 - r2| is small compared with the linear dimensions of the source. This condition does not ensure that µ(r1, r2) is a function of r1 - r2 only over the whole domain occupied by the secondary source. The size of the domain where µ(r1, r2) is only a function of r1 - r2 depends on the imaging properties of the optical system and must be separately examined in each particular case. In any case µ(r1, r2) cannot be expected to be a function of r1 - r2 only when either of the two points is close to the boundary of the source, whether the source is a secondary or a primary one. This fact is, however, of no great practical consequence if the source is sufficiently large.
  12. Our concept of a quasihomogeneity bears some similarity to that of local homogeneity, well known in the statistical theory of turbulence (cf. Ref. 8, Secs. 5 and 7), but the two concepts are not equivalent. A factorization (of the mutual intensity rather than of the cross-spectral density function) of the form (2. 9) was assumed in a recent paper by S. Wadaka and T. Sato, "Merits of symmetric scanning for detection of coherence function in incoherent imaging system," J. Opt. Soc. Am. 66, 145–147 (1976). However in this work no motivation for this factorization is given, nor are any restrictions stated on the behavior of the two factors.
  13. Equation (2. 9) is also analogous to a basic relation in the theory of locally stationary random processes, established by R. A. Silverman, in "Locally Stationary Random Processes," IRE Trans. Information Theory 3, 182–187 (1957), Eq. (11).
  14. The situation here is somewhat similar to that encountered in connection with the Kirchhoff approximation in elementary diffraction theory (cf. Ref. 6, Sec. 8.3.2).
  15. See, for example R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, Now York, 1961), pp. 160–161.
  16. E. W. Marchand and E. Wolf, "Angular Correlation and the Far-Zone Behavior of Partially Coherent Fields," J. Opt. Soc. Am. 62, 379–385 (1972), Eq. (34).
  17. E. W. Marchand and E. Wolf, "Radiometry with Sources of Any State of Coherence," J. Opt. Soc. Am. 64, 1219–1226 (1974), Eq. (42).
  18. Equation (16) of Ref. 1. There is a misprint in that equation: Jω(f) should be replaced by Jω(S) on the left-hand side of that equation.
  19. The expression (5. 1) for the radiance was proposed by A. Walther in the paper quoted in Ref. 5. Essentially the same expression for the radiance was proposed in a somewhat different context by L. S. Dolin, in Izv. VUZov: Radiophys. 7, 559–563 (1964), whose title, in English translation, is "Description in Weakly Inhomogeneous Wave Fields."
  20. A function g(r) is said to be non-negative definite if for any non-negative integer N, any set of N vectors ri, and any set of N (real or complex) numbers ai, [equation] That this condition is obeyed by our degree of spatial coherence follows at once from the non-negative definiteness of the cross-spectral density function W(r1, r2), established in the Appendix of Ref. 4, and from Eq. (2.7) above if we take into account the fact that I(r)≥0.
  21. For an account of Bochner's theorem see, for example, R. R. Goldberg, Fourier Transforms (Cambridge U. P., Cambridge, England, 1965), Chap. 5.
  22. A. Walther, "Radiometry and Coherence," J. Opt. Soc. Am. 63, 1622–1623 (1973)..
  23. E. W. Marchand and E. Wolf, "Walther's Definition of Generalized Radiance," J. Opt. Soc. Am. 64, 1273–1274 (1974).
  24. A. Walther, "Reply to Marchand and Wolf," J. Opt. Soc. Am. 64, 1275 (1974).
  25. It has been shown in Ref. 17 that, in general, the radiant emittance of a partially coherent source acquires negative values.
  26. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Natl. Bur. Stds. (U.S. GPO, Washington, D. C., 1964), p. 319; also (Dover, New York, 1965); W. L. Miller and A. R. Gordon, J. Phys. Chem. 35, 2785 (1931).
  27. The integral in Eq. (7.3) extends only formally over the complete plane z=0, since I(0)(r) vanishes at point r outside the domain occupied by the source.
  28. The limit kσg→∞ must be interpreted with some caution, since for a quasihomogeneous source we must have σg«l, where l represents a typical linear dimension of the source. Hence in proceeding to the limit kσg/l we must allow kl→∞ in such a way that σg/l tends to some number that is much smaller than unity.

1976 (2)

1975 (2)

1974 (3)

1973 (1)

1972 (1)

1968 (1)

1957 (1)

Equation (2. 9) is also analogous to a basic relation in the theory of locally stationary random processes, established by R. A. Silverman, in "Locally Stationary Random Processes," IRE Trans. Information Theory 3, 182–187 (1957), Eq. (11).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Natl. Bur. Stds. (U.S. GPO, Washington, D. C., 1964), p. 319; also (Dover, New York, 1965); W. L. Miller and A. R. Gordon, J. Phys. Chem. 35, 2785 (1931).

Antes, G.

Some special cases were also treated by H. P. Baltes, B. Steinle, and G. Antes in, "Spectral Coherence Area and the Radiant Intensity from Statistically Homogeneous and Isotropic Planar Sources," Opt. Commun. 18, 242–246 (1976).

Baltes, H. P.

Some special cases were also treated by H. P. Baltes, B. Steinle, and G. Antes in, "Spectral Coherence Area and the Radiant Intensity from Statistically Homogeneous and Isotropic Planar Sources," Opt. Commun. 18, 242–246 (1976).

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Sec. 10.4.2.

Bracewell, R.

See, for example R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, Now York, 1961), pp. 160–161.

Carter, W. H.

Goldberg, R. R.

For an account of Bochner's theorem see, for example, R. R. Goldberg, Fourier Transforms (Cambridge U. P., Cambridge, England, 1965), Chap. 5.

Mandel, L.

Marchand, E. W.

Schell, A. C.

A. C. Schell, "The Multiple Plate Antenna," Doctoral dissertation (Massachusetts Institute of Technology, 1961) (unpublished), Sec. 7.5.

Shore, R. A.

See also R. A. Shore, "Partially Coherent Diffraction by a Circular Aperture," in Electromagnetic Theory and Antennas, edited by E. C. Jordan (Pergamon, London, 1963), Part 2, pp. 787–795; and A. K. Jaiswal, G. P. Agrawal, and C. L. Mehta, "Coherence Functions in the Far-field Diffraction Plane," Nuovo Cimento B 15, 295–307 (1973).

Silverman, R. A.

Equation (2. 9) is also analogous to a basic relation in the theory of locally stationary random processes, established by R. A. Silverman, in "Locally Stationary Random Processes," IRE Trans. Information Theory 3, 182–187 (1957), Eq. (11).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Natl. Bur. Stds. (U.S. GPO, Washington, D. C., 1964), p. 319; also (Dover, New York, 1965); W. L. Miller and A. R. Gordon, J. Phys. Chem. 35, 2785 (1931).

Steinle, B.

Some special cases were also treated by H. P. Baltes, B. Steinle, and G. Antes in, "Spectral Coherence Area and the Radiant Intensity from Statistically Homogeneous and Isotropic Planar Sources," Opt. Commun. 18, 242–246 (1976).

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961) (reprinted by Dover, New York, 1967), Sec. 1. 3.

Walther, A.

Wolf, E.

IRE Trans. Information Theory (1)

Equation (2. 9) is also analogous to a basic relation in the theory of locally stationary random processes, established by R. A. Silverman, in "Locally Stationary Random Processes," IRE Trans. Information Theory 3, 182–187 (1957), Eq. (11).

J. Opt. Soc. Am. (8)

The converse problem of determining the degree of coherence in the source plane from the knowledge of the angular distribution of the radiant intensity was studied by W. H. Carter and E. Wolf in, "Coherence Properties of Lambertian and Non-Lambertian Sources," J. Opt. Soc. Am. 65, 1067–1071 (1975).

E. W. Marchand and E. Wolf, "Angular Correlation and the Far-Zone Behavior of Partially Coherent Fields," J. Opt. Soc. Am. 62, 379–385 (1972), Eq. (34).

E. W. Marchand and E. Wolf, "Radiometry with Sources of Any State of Coherence," J. Opt. Soc. Am. 64, 1219–1226 (1974), Eq. (42).

The properties of this correlation coefficient (also called the complex degree of spectral coherence or the spectral correlation coefficient) were recently studied by L. Mandel and E. Wolf in, "Spectral coherence and the concept of cross-spectral purity, " J. Opt. Soc. Am. 66, 529–535 (1976). This correlation coefficient must be distinguished from the more familiar complex degree of coherence (cf. Ref. 6, Sec. 10.3.1), often denoted by γ(r1, r2, τ) [or γ12(τ), even though γ(r1, r2, 0) is also a measure of spatial coherence.

Such a theorem was alluded to by A. Walther in his important paper, "Radiometry and Coherence," J. Opt. Soc. Am. 58, 1256–1259 (1968). However Walther did not formulate the theorem in precise mathematical terms, nor did he state to what class of sources it applies.

A. Walther, "Radiometry and Coherence," J. Opt. Soc. Am. 63, 1622–1623 (1973)..

E. W. Marchand and E. Wolf, "Walther's Definition of Generalized Radiance," J. Opt. Soc. Am. 64, 1273–1274 (1974).

A. Walther, "Reply to Marchand and Wolf," J. Opt. Soc. Am. 64, 1275 (1974).

Opt. Commun. (2)

E. Wolf and W. H. Carter, "Angular Distribution of Radiant Intensity from Sources of Different Degrees of Spatial Coherence," Opt. Commun. 13, 205–209 (1975).

Some special cases were also treated by H. P. Baltes, B. Steinle, and G. Antes in, "Spectral Coherence Area and the Radiant Intensity from Statistically Homogeneous and Isotropic Planar Sources," Opt. Commun. 18, 242–246 (1976).

Other (17)

The situation here is somewhat similar to that encountered in connection with the Kirchhoff approximation in elementary diffraction theory (cf. Ref. 6, Sec. 8.3.2).

See, for example R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, Now York, 1961), pp. 160–161.

Equation (16) of Ref. 1. There is a misprint in that equation: Jω(f) should be replaced by Jω(S) on the left-hand side of that equation.

The expression (5. 1) for the radiance was proposed by A. Walther in the paper quoted in Ref. 5. Essentially the same expression for the radiance was proposed in a somewhat different context by L. S. Dolin, in Izv. VUZov: Radiophys. 7, 559–563 (1964), whose title, in English translation, is "Description in Weakly Inhomogeneous Wave Fields."

A function g(r) is said to be non-negative definite if for any non-negative integer N, any set of N vectors ri, and any set of N (real or complex) numbers ai, [equation] That this condition is obeyed by our degree of spatial coherence follows at once from the non-negative definiteness of the cross-spectral density function W(r1, r2), established in the Appendix of Ref. 4, and from Eq. (2.7) above if we take into account the fact that I(r)≥0.

For an account of Bochner's theorem see, for example, R. R. Goldberg, Fourier Transforms (Cambridge U. P., Cambridge, England, 1965), Chap. 5.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Sec. 10.4.2.

In physical optics the quantity I(r) defined by Eq. (2. 1), is usually referred to simply as intensity (at frequency ω). We use here the adjective "optical" to distinguish it clearly from the radiometric concept of radiant intensity, that we will also encounter in the present paper.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961) (reprinted by Dover, New York, 1967), Sec. 1. 3.

A. C. Schell, "The Multiple Plate Antenna," Doctoral dissertation (Massachusetts Institute of Technology, 1961) (unpublished), Sec. 7.5.

See also R. A. Shore, "Partially Coherent Diffraction by a Circular Aperture," in Electromagnetic Theory and Antennas, edited by E. C. Jordan (Pergamon, London, 1963), Part 2, pp. 787–795; and A. K. Jaiswal, G. P. Agrawal, and C. L. Mehta, "Coherence Functions in the Far-field Diffraction Plane," Nuovo Cimento B 15, 295–307 (1973).

Strictly speaking this condition only ensures that |µ(r1, r2)| has appreciable values when the separation |r1 - r2| is small compared with the linear dimensions of the source. This condition does not ensure that µ(r1, r2) is a function of r1 - r2 only over the whole domain occupied by the secondary source. The size of the domain where µ(r1, r2) is only a function of r1 - r2 depends on the imaging properties of the optical system and must be separately examined in each particular case. In any case µ(r1, r2) cannot be expected to be a function of r1 - r2 only when either of the two points is close to the boundary of the source, whether the source is a secondary or a primary one. This fact is, however, of no great practical consequence if the source is sufficiently large.

Our concept of a quasihomogeneity bears some similarity to that of local homogeneity, well known in the statistical theory of turbulence (cf. Ref. 8, Secs. 5 and 7), but the two concepts are not equivalent. A factorization (of the mutual intensity rather than of the cross-spectral density function) of the form (2. 9) was assumed in a recent paper by S. Wadaka and T. Sato, "Merits of symmetric scanning for detection of coherence function in incoherent imaging system," J. Opt. Soc. Am. 66, 145–147 (1976). However in this work no motivation for this factorization is given, nor are any restrictions stated on the behavior of the two factors.

It has been shown in Ref. 17 that, in general, the radiant emittance of a partially coherent source acquires negative values.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Natl. Bur. Stds. (U.S. GPO, Washington, D. C., 1964), p. 319; also (Dover, New York, 1965); W. L. Miller and A. R. Gordon, J. Phys. Chem. 35, 2785 (1931).

The integral in Eq. (7.3) extends only formally over the complete plane z=0, since I(0)(r) vanishes at point r outside the domain occupied by the source.

The limit kσg→∞ must be interpreted with some caution, since for a quasihomogeneous source we must have σg«l, where l represents a typical linear dimension of the source. Hence in proceeding to the limit kσg/l we must allow kl→∞ in such a way that σg/l tends to some number that is much smaller than unity.

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