Abstract

A new type of self-imaging effect connected with Fourier imaging of the radiance function in physical radiometry is analyzed.

© 1983 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 quasi-homogeneous 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, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Vol. 9.
    [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. T. Jannson, “Radiance transfer function,” J. Opt. Soc. Am. 70, 1544–1549 (1980).
    [CrossRef]
  14. See, for example, J. T. Winthrop and C. R. Worthington, “Theory of Fresnel images I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. 55, 373–380 (1965).
    [CrossRef]
  15. T. Jannson and J. Jannson, “Temporal self-imaging effect in single-mode fibers,” J. Opt. Soc. Am. 71, 1373–1376 (1981).
  16. See, for example, Ref. 13, Eq. (3).
  17. See Ref. 13, Eq. (21).
  18. See Ref. 13, Eq. (42).
  19. See, for example, J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
    [CrossRef]

1981 (1)

1980 (1)

1979 (2)

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]

See, for example, J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

1978 (2)

1977 (2)

1975 (1)

1974 (2)

1973 (1)

1968 (1)

1965 (1)

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, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Vol. 9.
[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, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Vol. 9.
[CrossRef]

Jahns, J.

See, for example, J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Jannson, J.

Jannson, T.

Lohmann, A. W.

See, for example, J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Marchand, E. W.

Steinle, B.

Walther, A.

Winthrop, J. T.

Wolf, E.

Worthington, C. R.

J. Opt. Soc. Am. (13)

See, for example, J. T. Winthrop and C. R. Worthington, “Theory of Fresnel images I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. 55, 373–380 (1965).
[CrossRef]

A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (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 quasi-homogeneous 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]

T. Jannson, “Radiance transfer function,” J. Opt. Soc. Am. 70, 1544–1549 (1980).
[CrossRef]

T. Jannson and J. Jannson, “Temporal self-imaging effect in single-mode fibers,” J. Opt. Soc. Am. 71, 1373–1376 (1981).

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]

Opt. Commun. (1)

See, for example, J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Other (5)

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

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

See, for example, Ref. 13, Eq. (3).

See Ref. 13, Eq. (21).

See Ref. 13, Eq. (42).

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Figures (2)

Fig. 1
Fig. 1

Geometry of the diffraction system for one-dimensional periodic spatial distribution of source intensity. The direction of observation is determined by unit vector s situated at angle θ with respect to the 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 is represented by the grating vectors ±f0; Λ = 1/f0 is the spatial period (or the grating constant) of this intensity distribution. By analogy, for a more-general source distribution with nonsinusoidal, periodic intensity, we have, according to Eq. (13), Kn = nK0, where K0 = 2πf0 and n = 0, 1, 2, …. The angle α0 is between the vector f0 and the projection vector p.

Fig. 2
Fig. 2

Illustration of the self-imaging effect in physical radiometry. At distance zq the periodic radiance distribution, B1(x1, p), is disturbed, whereas at distance z = q the radiance distribution, B2(x2, p), is reconstructed. The direction of observation, determined by unit vector s (or p), is fixed.

Equations (17)

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B ( r , p ) = s z Re { U ( r ) Ũ * ( p / λ ) exp ( i k p · r ) } ,
Ũ ( f ) = F ̂ { U ( r ) } = U ( r ) exp ( i 2 π f · r ) d 2 r
B 0 ( r 0 , p ) = s z I 0 ( r 0 ) μ ( p / λ ) ,
B ( f , p ) = G ( f , p ) B 0 ( f , p ) ,
B ( f , p ) = F ̂ { B ( r , p ) }
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 ] } .
G ( f , p ) = exp ( i 2 π z s z p · f ) M ( f , θ , α ) ,
M ( f , θ , α ) = cos [ π z λ f 2 W ( α , θ ) ]
W ( α , θ ) = cos 3 θ 1 ( sin α sin θ ) 2 .
0 W 1 .
( R R 0 ) × s = 0 ,
lim λ 0 M = 1
I 0 ( r 0 ) = n = 0 An cos ( K n · r 0 + Φ n ) ,
G n ( f , p ) = exp ( i 2 π z s z p · f ) M n ( f , θ , α ) ,
M n ( f , θ , α ) = { M ( f n , θ , α 0 ) for f = f n 0 for f f n ,
z m = m q ,
q = 2 Λ 2 W ( α 0 θ ) λ