Abstract

The transformation of generalized radiance by space-invariant linear systems, based on Walther’s second definition is analyzed. The transfer function for a generalized radiance function is introduced. Its form in the case of quasihomogeneous sources is discussed on the basis of two important examples: free space and an isoplanatic optical system.

© 1980 Optical Society of America

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References

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  1. See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1970), Chap. 4.8.
  2. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  3. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 63, 1622–1623 (1973).
    [CrossRef]
  4. A. Walther, “Propagation of the generalized radiance through lenses,” J. Opt. Soc. Am. 68, 1606–1611 (1978).
    [CrossRef]
  5. E. W. Marchand and E. Wolf, “Radiometry with sources of any state of coherence,” J. Opt. Soc. Am. 64, 1219–1226 (1974).
    [CrossRef]
  6. E. W. Marchand and E. Wolf, “Walther’s definition of generalized radiance,” J. Opt. Soc. Am. 64, 1273–1274 (1974).
    [CrossRef]
  7. W. H. Carter and E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975).
    [CrossRef]
  8. W. H. Carter and E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  9. B. Steinle and H. P. Baltes, “Radiant intensity and spatial coherence for finite planar sources,” J. Opt. Soc. Am. 67, 241–247 (1977).
    [CrossRef]
  10. E. Wolf, “The radiant intensity from planar sources of any state of coherence,” J. Opt. Soc. Am. 68, 1597–1605 (1978).
    [CrossRef]
  11. H. P. Baltes, J. Geist, and A. Walther, “Radiometry and coherence,” in Topics in Current Physics, Vol. 9, edited by H. P. Baltes (Springer, Berlin, 1978).
    [CrossRef]
  12. A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary states of coherence,” J. Opt. Soc. Am. 69, 192–199 (1979).
    [CrossRef]
  13. A detailed discussion of physical reality of different definitions of generalized radiance, see Ref. 12.
  14. See, for example, Ref. 1, Chap. 9.5.
  15. We use f′ here because the symbol f we reserve for the spatial frequency vector of the generalized radiance; however, both f′ and f define the spatial frequency vectors. Bearing in mind the different limitations of Fourier spectra of amplitude and generalized radiance, respectively, such a distinction seems to be justified.
  16. See Ref. 8, Appendix A.
  17. See, for example, Ref. 8.
  18. See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6.
  19. J. R. Shewell and E. Wolf, “Inverse diffraction and a new reciprocity theorem,” J. Opt. Soc. Am. 58, 1596–1603 (1968);E. Lalor “Inverse wave propagator,” J. Math. Phys. 9, 2001–2006 (1968).
    [CrossRef]
  20. See Ref. 18, Chap. 3.

1979 (1)

1978 (2)

1977 (2)

1975 (1)

1974 (2)

1973 (1)

1968 (2)

Baltes, H. P.

B. Steinle and H. P. Baltes, “Radiant intensity and spatial coherence for finite planar sources,” J. Opt. Soc. Am. 67, 241–247 (1977).
[CrossRef]

H. P. Baltes, J. Geist, and A. Walther, “Radiometry and coherence,” in Topics in Current Physics, Vol. 9, edited by H. P. Baltes (Springer, Berlin, 1978).
[CrossRef]

Born, M.

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1970), Chap. 4.8.

Carter, W. H.

Friberg, A. T.

Geist, J.

H. P. Baltes, J. Geist, and A. Walther, “Radiometry and coherence,” in Topics in Current Physics, Vol. 9, edited by H. P. Baltes (Springer, Berlin, 1978).
[CrossRef]

Goodman, J. W.

See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6.

Marchand, E. W.

Shewell, J. R.

Steinle, B.

Walther, A.

Wolf, E.

J. Opt. Soc. Am. (11)

A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
[CrossRef]

J. R. Shewell and E. Wolf, “Inverse diffraction and a new reciprocity theorem,” J. Opt. Soc. Am. 58, 1596–1603 (1968);E. Lalor “Inverse wave propagator,” J. Math. Phys. 9, 2001–2006 (1968).
[CrossRef]

E. W. Marchand and E. Wolf, “Radiometry with sources of any state of coherence,” J. Opt. Soc. Am. 64, 1219–1226 (1974).
[CrossRef]

W. H. Carter and E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975).
[CrossRef]

B. Steinle and H. P. Baltes, “Radiant intensity and spatial coherence for finite planar sources,” J. Opt. Soc. Am. 67, 241–247 (1977).
[CrossRef]

W. H. Carter and E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
[CrossRef]

E. Wolf, “The radiant intensity from planar sources of any state of coherence,” J. Opt. Soc. Am. 68, 1597–1605 (1978).
[CrossRef]

A. Walther, “Propagation of the generalized radiance through lenses,” J. Opt. Soc. Am. 68, 1606–1611 (1978).
[CrossRef]

A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary states of coherence,” J. Opt. Soc. Am. 69, 192–199 (1979).
[CrossRef]

E. W. Marchand and E. Wolf, “Walther’s definition of generalized radiance,” J. Opt. Soc. Am. 64, 1273–1274 (1974).
[CrossRef]

A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 63, 1622–1623 (1973).
[CrossRef]

Other (9)

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1970), Chap. 4.8.

A detailed discussion of physical reality of different definitions of generalized radiance, see Ref. 12.

See, for example, Ref. 1, Chap. 9.5.

We use f′ here because the symbol f we reserve for the spatial frequency vector of the generalized radiance; however, both f′ and f define the spatial frequency vectors. Bearing in mind the different limitations of Fourier spectra of amplitude and generalized radiance, respectively, such a distinction seems to be justified.

See Ref. 8, Appendix A.

See, for example, Ref. 8.

See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6.

H. P. Baltes, J. Geist, and A. Walther, “Radiometry and coherence,” in Topics in Current Physics, Vol. 9, edited by H. P. Baltes (Springer, Berlin, 1978).
[CrossRef]

See Ref. 18, Chap. 3.

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Figures (3)

FIG. 1
FIG. 1

Illustration of the characteristic ranges of transferred frequencies for the case when the relation (30) is equality: p/λ + fl = f0. fl is the radius of the region where the Fourier spectrum of the intensity distribution in the plane of the quasihomogeneous source is different from zero, f0 is the cutoff frequency of the CTF, and p is the projection of the unit vector of observation s onto the (x,y) plane. In the double-hatched region the Fourier spectrum of generalized radiance (or, the source intensity) is ideally transferred by well-corrected optical systems with circular apertures. Outside the hatched areas the Fourier spectrum of generalized radiance is not transferred at all by any isoplanatic optical system with a circular pupil.

FIG. 2
FIG. 2

Geometry of the diffraction system. The direction of observation is determined by the unit vector s situated at angle θ with respect to z axis. The vectors r0 and r are the position vectors in the planes (x0,y0) and (x,y), respectively. A typical sinusoidal source intensity [see Eq. (43)] is represented by the spatial frequency vectors ±f1; Λ = 1/f1 is the “grating constant” of this intensity distribution. The angle α is measured between the vector f1 and the projection p of the unit vector s onto the (x,y) plane.

FIG. 3
FIG. 3

Illustration of the relation (46) in polar coordinates (W,α) where the dimensionless parameter W = 2 z λ f 1 2 and α is the angle between the vector f1 and p. The relation W(α) is illustrated in the range 0°–90°, for five values of observation angle θ: 0°, 15°, 30°, 45°, and 60°.

Equations (48)

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Ũ ( f ) = F ̂ [ U ( r ) ] = U ( r ) e i 2 π f · r d 2 r ,
U ( r ) = F ̂ ( 1 ) [ Ũ ( f 1 ) ] = Ũ ( f ) e i 2 π f · r d 2 f ,
B ( r , p ) = 2 ω k s z Re [ U ( r ) Ũ * ( p / λ ) e i k p · r ] ,
B ( + ) ( r , p ) = ω k s z U ( r ) Ũ * ( p / λ ) e i k p · r .
B ( r , p ) = B ( + ) ( r , p ) + B ( ) ( r , p )
B ( ) ( r , p ) = [ B ( + ) ( r , p ) ] * .
B ( + ) ( f , p ) = ω k s z Ũ ( p / λ + f ) Ũ * ( p / λ )
B ( ) ( f , p ) = ω k s z Ũ * ( p / λ f ) Ũ ( p / λ ) .
U ( r ) = h ( r r 0 ) U 0 ( r 0 ) d 2 r 0 ,
Ũ ( f ) = H ( f ) Ũ 0 ( f ) ,
B ( ± ) ( r , p ) = g ( ± ) ( r r 0 , p ) B 0 ( ± ) ( r 0 , p ) d 2 r 0 ,
g ( + ) ( r , p ) = H * ( p / λ ) h ( r ) exp ( i k p · r )
B ( ± ) ( f , p ) = G ( ± ) ( f , p ) B 0 ( ± ) ( f , p ) ,
G ( + ) ( f , p ) = F ̂ [ g ( + ) ( r , p ) ] = H * ( p / λ ) H ( p / λ + f )
G ( ) ( f , p ) = H ( p / λ ) H * ( p / λ f ) .
B 0 ( r , p ) = 2 ω k s z I 0 ( r ) μ ( p / λ ) ,
B 0 ( + ) ( r , p ) = B 0 ( ) ( r , p ) = ( 1 / 2 ) B 0 ( r , p ) .
λ f I 1 / 1 ,
B ( r , p ) = g ( r r 0 , p ) B 0 ( r 0 , p ) d 2 r 0 ,
g ( r , p ) = ( 1 / 2 ) [ g ( + ) ( r , p ) + g ( ) ( r , p ) ]
B ( r , p ) = 2 ω k s z μ ( p / λ ) g ( r r 0 , p ) I 0 ( r 0 ) d 2 r 0 .
B ( f , p ) = G ( f , p ) B 0 ( f , p ) ,
G ( f , p ) = ( 1 / 2 ) ] G ( + ) ( f , p ) + G ( ) ( f , p ) ]
B ( f , p ) = F ̂ [ B ( r , p ) ] .
G ( f , p ) = ( 1 / 2 ) [ H * ( p / λ ) H ( f + p / λ ) + H ( p / λ ) H * ( p / λ f ) ] .
G * ( f , p ) = G ( f , p )
G ( 0 , p ) = | H ( p / λ ) | 2 .
H ( f ) = P ( λ d i f ) ,
P ( r p ) = P 0 ( r p ) exp [ i Φ ( r p ) ] ,
P 0 ( r p ) = circ ( r p / ρ ) ,
G ( f , p ) = 1 2 circ ( p p 0 ) [ circ ( | p / λ + f | f 0 ) + circ ( | p / λ + f | f 0 ) ] ,
p / λ + f I f 0 .
g ( r , p ) = δ ( r )
f I > f 0 2 ( p / λ ) 2 .
H ( f ) = { exp ( ikz 1 ( λ f ) 2 ) for f 1 / λ exp ( k z ( λ f ) 2 1 ) for f > 1 / λ ,
H ( p / λ ± f ) = { exp ( i 2 π z ( 1 / λ ) 2 ( p / λ ± f ) 2 ) for | p / λ ± f | 1 / λ exp ( 2 π z ( p / λ ± f ) 2 ( 1 / λ ) 2 ) for | p / λ ± f | > 1 / λ .
G ( f , p ) = 1 2 exp [ ( ikz s z ) ( 1 1 2 λ p · f s z 2 ( λ f s z ) 2 ) ] × circ ( | λ f + p | ) + 1 2 exp [ ( ikz s z ) × ( 1 1 + 2 λ p · f s z 2 ( λ f s z ) 2 ) ] circ ( | λ f p | ) .
2 λ f I cos 2 θ .
λ f I > cos θ .
G ( f , p ) = exp [ i ( 2 π z / s z ) p · f ] ,
g ( r r 0 , p ) = δ ( r r 0 ( z / s z ) p )
B ( r , p ) = B 0 ( r ( z / s z ) p , p ) .
( R R 0 ) × s = 0 .
G ( f , p ) = exp ( i 2 π z s z p · f ) cos { ( π z λ s z ) [ f 2 + ( p · f s z ) 2 ] } .
I ( r 0 ) = A [ 1 + a cos ( 2 π f 1 · r 0 + β ) ] ,
B ( r , p ) = A ω k s z μ ( p / λ ) { 1 + a M ( f 1 , p ) × cos [ 2 π f 1 ( r z s z p ) + β ] } ,
M ( f 1 , p ) = M ( f 1 , α , θ ) = cos { ( π z λ f 1 2 cos 3 θ ) [ 1 ( sin α sin θ ) 2 ] } ,
cos 3 θ 1 ( sin α sin θ ) 2 = 2 z λ f 1 2 = W .