Abstract

The characteristics of extended-range focal-line images generated in partially coherent light by annular-aperture logarithmic axicons are analyzed in terms of a certain generalized radiometric model. This radiometric description, which makes use of radiance propagation along geometric rays, is valid in the asymptotic short-wavelength limit, and it yields both the spectral density and the spatial coherence distributions of the images. The radiometrically obtained values on the image axis and in transverse planes are assessed against direct partially coherent wave-theoretical calculations. It is shown that the radiometric technique produces remarkably accurate results for the intensity and the spatial coherence profiles. Compared with diffraction calculations, the radiometric method is much faster, especially in conditions of relatively incoherent illumination.

© 1996 Optical Society of America

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References

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  1. J. Sochacki, A. Kolodziejczyk, Z. Jaroszewics, S. Bara, “Nonparaxial design of generalized axicons,” Appl. Opt. 31, 5326–5330 (1992).
    [CrossRef] [PubMed]
  2. Z. Jaroszewicz, J. Sochacki, A. Kolodziejczyk, L. R. Staronski, “Apodized annular-aperture logarithmic axicon: smoothness and uniformity of intensity distributions,” Opt. Lett. 18, 1893–1895 (1993).
    [CrossRef] [PubMed]
  3. A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
    [CrossRef] [PubMed]
  4. A. T. Friberg, S. Y. Popov, “Partially coherently illuminated uniform-intensity holographic axicons,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 224–227.
  5. S. Y. Popov, A. T. Friberg, “Linear axicons in partially coherent light,” Opt. Eng. 34, 2567–2573 (1995).
    [CrossRef]
  6. Z. Jaroszewicz, J. F. Roman Dopazo, “Polychromatic illumination of logarithmic annular-aperture diffractive axicon,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 228–231.
  7. J. T. Foley, E. Wolf, “Radiometry with quasihomogeneous sources,” J. Mod. Opt. 42, 787–798 (1995).
    [CrossRef]
  8. E. Wolf, “Radiometric model for propagation of coherence,” Opt. Lett. 19, 2024–2026 (1994).
    [CrossRef] [PubMed]
  9. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
    [CrossRef]
  10. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1258 (1968).
    [CrossRef]
  11. K. Kim, E. Wolf, “Propagation law for Walther’s first generalized radiance function and its short-wavelength limit with quasihomogeneous sources,” J. Opt. Soc. Am. A 4, 1233–1236 (1987).
    [CrossRef]
  12. W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  13. R. C. Fairchild, J. R. Fienup, “Computer-originated aspheric holographic optical elements,” Opt. Eng. 21, 133–140 (1982).
  14. A. Walther, “Propagation of the generalized radiance through lenses,” J. Opt. Soc. Am. 68, 1606–1610 (1978).
    [CrossRef]
  15. A. T. Friberg, “Propagation of a generalized radiance in paraxial optical systems,” Appl. Opt. 30, 2443–2446 (1991).
    [CrossRef] [PubMed]
  16. Although analogous results pertaining to arbitrary optical systems have been suggested,14 to our knowledge the asymptotic invariance of generalized radiance functions along ray paths has been proven beyond the paraxial regime only as regards propagation in free space [see, for example, A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992)]. In the present case axicon images are analyzed in paraxial conditions, in which Eqs. (9) lead effectively to a position-dependent ray matrix for the hologram, and so the radiance invariance in the asymptotic short-wavelength limit can be justified much as before.14,15
    [CrossRef]
  17. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965), Secs. 9.6 and 9.7.
  18. M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), Chap. 1.

1995 (2)

S. Y. Popov, A. T. Friberg, “Linear axicons in partially coherent light,” Opt. Eng. 34, 2567–2573 (1995).
[CrossRef]

J. T. Foley, E. Wolf, “Radiometry with quasihomogeneous sources,” J. Mod. Opt. 42, 787–798 (1995).
[CrossRef]

1994 (1)

1993 (1)

1992 (2)

1991 (1)

1989 (1)

1987 (1)

1982 (1)

R. C. Fairchild, J. R. Fienup, “Computer-originated aspheric holographic optical elements,” Opt. Eng. 21, 133–140 (1982).

1978 (1)

1977 (1)

1976 (1)

1968 (1)

Agarwal, G. S.

Bara, S.

Beran, M. J.

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), Chap. 1.

Carter, W. H.

Fairchild, R. C.

R. C. Fairchild, J. R. Fienup, “Computer-originated aspheric holographic optical elements,” Opt. Eng. 21, 133–140 (1982).

Fienup, J. R.

R. C. Fairchild, J. R. Fienup, “Computer-originated aspheric holographic optical elements,” Opt. Eng. 21, 133–140 (1982).

Foley, J. T.

Friberg, A. T.

Jaroszewics, Z.

Jaroszewicz, Z.

Z. Jaroszewicz, J. Sochacki, A. Kolodziejczyk, L. R. Staronski, “Apodized annular-aperture logarithmic axicon: smoothness and uniformity of intensity distributions,” Opt. Lett. 18, 1893–1895 (1993).
[CrossRef] [PubMed]

Z. Jaroszewicz, J. F. Roman Dopazo, “Polychromatic illumination of logarithmic annular-aperture diffractive axicon,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 228–231.

Kim, K.

Kolodziejczyk, A.

Mandel, L.

Parrent, G. B.

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), Chap. 1.

Popov, S. Y.

S. Y. Popov, A. T. Friberg, “Linear axicons in partially coherent light,” Opt. Eng. 34, 2567–2573 (1995).
[CrossRef]

A. T. Friberg, S. Y. Popov, “Partially coherently illuminated uniform-intensity holographic axicons,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 224–227.

Roman Dopazo, J. F.

Z. Jaroszewicz, J. F. Roman Dopazo, “Polychromatic illumination of logarithmic annular-aperture diffractive axicon,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 228–231.

Sochacki, J.

Staronski, L. R.

Turunen, J.

Vasara, A.

Walther, A.

Wolf, E.

Appl. Opt. (2)

J. Mod. Opt. (1)

J. T. Foley, E. Wolf, “Radiometry with quasihomogeneous sources,” J. Mod. Opt. 42, 787–798 (1995).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Opt. Eng. (2)

R. C. Fairchild, J. R. Fienup, “Computer-originated aspheric holographic optical elements,” Opt. Eng. 21, 133–140 (1982).

S. Y. Popov, A. T. Friberg, “Linear axicons in partially coherent light,” Opt. Eng. 34, 2567–2573 (1995).
[CrossRef]

Opt. Lett. (2)

Other (4)

A. T. Friberg, S. Y. Popov, “Partially coherently illuminated uniform-intensity holographic axicons,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 224–227.

Z. Jaroszewicz, J. F. Roman Dopazo, “Polychromatic illumination of logarithmic annular-aperture diffractive axicon,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 228–231.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965), Secs. 9.6 and 9.7.

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964), Chap. 1.

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Figures (9)

Fig. 1
Fig. 1

Illustration of the geometry associated with axial line-image formation by an annular-aperture holographic axicon. The hologram is characterized by a phase function φ(ρ) such that (nonparaxially) ∂φ/∂ρ = −sin θ.

Fig. 2
Fig. 2

Apodization function t(ρ) given by Eq. (3), with R 1 = 2.5 mm, R 2 = 5.0 mm, and Δ = 20 mm−1. The value of the square |t(ρ)|2, corresponding to intensity reduction, at the aperture’s center radius ρ = 3.75 mm is 0.95.

Fig. 3
Fig. 3

Notations used in the radiometric calculation of the cross-spectral and spectral densities on the axis of the axicon line image.

Fig. 4
Fig. 4

On-axis spectral density distributions predicted by the radiometric model (solid curve) and by the wave-theoretical model (dashed curve) for an apodized, annular-aperture logarithmic axicon of R 1 = 2.5 mm and R 2 = 5.0 mm that, in coherent light, produces a nearly uniform axial intensity between d 1 = 100 mm and d 2 = 200 mm. The curves are calculated from Eqs. (12)(14) and (15), respectively, with S 0 = 1, λ = 0.633 μm, and Δ = 20 mm−1. The transverse correlation distance of the axicon illumination is (a) σ g = 1.0 mm, (b) σ g = 0.25 mm.

Fig. 5
Fig. 5

On-axis distributions of the magnitude of the complex degree of spatial coherence when one of the field points is located at z = 150 mm. Both the radiometric (solid curve) and the wave-theoretical (dashed curve) results are shown for σ g = 0.5 mm. The axicon parameters are as in Fig. 4.

Fig. 6
Fig. 6

Notations related to the radiometric evaluation of the cross-spectral and the spectral densities at off-axis points in the image region. The generalized radiance remains invariant along the (skewed) geometric rays.

Fig. 7
Fig. 7

Transverse distributions of the spectral density in the center of the image line at z = 150 mm based on the radiometric model (solid curve) and on the wave-theoretical model (dashed curve). The curves are calculated from Eqs. (17), (18) and (19), (20), respectively. The system parameters are as in Fig. 4; the transverse coherence width of the irradiance is (a) σ g = 1.0 mm, (b) σ g = 0.25 mm.

Fig. 8
Fig. 8

Transverse distributions of the magnitude of the complex degree of spatial coherence at z = 150 mm, where one point is on the axis and the other is varying along a radial direction. Both the radiometric result (solid curve) and the wave-theoretical result (open circles) are shown; the dotted curve corresponds to the relative intensity profile. The transverse coherence width of the illumination is σ g = 0.5 mm, and the other parameters are as in Fig. 4.

Fig. 9
Fig. 9

Three-dimensional spectral density profiles of the axicon line images calculated with the radiometric model of Eqs. (17) and (18) with the incident generalized radiance given by the right-hand side of Eq. (14). The transverse coherence width of the illumination is (a) σ g = 0.25 mm, (b) σ g = 0.1 mm. Other parameters are as in Fig. 4.

Equations (20)

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h ( ρ ) = t ( ρ ) exp [ i k φ ( ρ ) ] ,
φ ( ρ ) = - 1 2 a log [ d 1 + a ( ρ 2 - R 1 2 ) ] ,
t ( ρ ) = { 0.5 + arctan [ Δ ( R 2 - ρ ) ] / π } × { 0.5 + arctan [ Δ ( ρ - R 1 ) ] / π } ,
W 0 ( ρ 1 , ρ 2 ) = S 0 exp [ - ( ρ 1 - ρ 2 ) 2 / 2 σ g 2 ] .
W ( r 1 , r 2 ) = ( k / 2 π ) 2 exp [ - i k ( z 1 - z 2 ) ] / z 1 z 2 × A t ( ρ 1 ) t ( ρ 2 ) W 0 ( ρ 1 , ρ 2 ) × exp { - i k [ φ ( ρ 1 ) - φ ( ρ 2 ) ] } exp { - i k [ ( ρ 1 - ρ 1 ) 2 / 2 z 1 - ( ρ 2 - ρ 2 ) 2 / 2 z 2 ] } d 2 ρ 1 d 2 ρ 2 ,
μ ( r 1 , r 2 ) = W ( r 1 , r 2 ) / [ S ( r 1 ) S ( r 2 ) ] 1 / 2 ,
W ( r 1 , r 2 ) = B ( r 1 + r 2 2 , s ) exp [ i k s · ( r 2 - r 1 ) ] d Ω ,
B ( r 1 + r 2 2 , s ) = B ( 0 ) [ ρ - ( s z ) / s z , s ] ,
s x ( x , y ) = s 0 x ( x , y ) - φ ( x , y ) x , s y ( x , y ) = s 0 y ( x , y ) - φ ( x , y ) y ,
B 0 ( ρ , s 0 ) = S 0 ( k σ g ) 2 2 π s 0 z exp [ - 1 2 ( k σ g ) 2 s 0 2 ] ,
d Ω = ρ d ρ d ϕ cos θ ( ρ 2 + z 2 ) ,
W ( z 1 , z 2 ) = 2 π R B ( 0 ) ( ρ , θ ) × exp [ i k cos θ ( z 2 - z 1 ) ] ρ cos θ ( ρ 2 + z 2 ) d ρ ,
sin θ 0 = sin θ - ρ d 1 + a ( ρ 2 - R 1 2 ) ,
B ( 0 ) ( ρ , θ ) = t ( ρ ) 2 S 0 ( k σ g ) 2 2 π cos θ 0 × exp [ - 1 2 ( k σ g ) 2 sin 2 θ 0 ] ,
W ( z 1 , z 2 ) = S 0 k 2 exp [ - i k ( z 1 - z 2 ) ] / z 1 z 2 × R t ( ρ 1 ) t ( ρ 2 ) exp [ - ( ρ 1 2 + ρ 2 2 ) / 2 σ g 2 ] × I 0 ( ρ 1 ρ 2 / σ g 2 ) exp { - i k [ φ ( ρ 1 ) - φ ( ρ 2 ) ] } × exp [ - i k ( ρ 1 2 / 2 z 1 - ρ 2 2 / 2 z 2 ) ] × ρ 1 ρ 2 d ρ 1 d ρ 2 ,
d Ω = ρ d ρ d ϕ cos θ ( ρ 2 + z 2 + l 2 - 2 l ρ cos ϕ ) ,
W ( l 1 , l 2 , z ) = A B ( 0 ) ( ρ , ϕ , s x , s y ) exp [ i k s x ( l 2 - l 1 ) ] × ρ s z ( ρ 2 + z 2 + l 2 - 2 l ρ cos ϕ ) d ρ d ϕ ,
s 0 x = s x - cos ϕ ρ d 1 + a ( ρ 2 - R 1 2 ) , s 0 y = s y - sin ϕ ρ d 1 + a ( ρ 2 - R 1 2 ) .
W ( l 1 , l 2 , z ) = S 0 ( k / 2 π z ) 2 exp [ - i k ( l 1 2 - l 2 2 ) / 2 z ] × R t ( ρ 1 ) t ( ρ 2 ) exp [ - ρ 1 2 + ρ 2 2 ) / 2 σ g 2 ] × C ( ρ 1 , ρ 2 ; l 1 , l 2 , z ) exp { - i k [ φ ( ρ 1 ) - φ ( ρ 2 ) ] } × exp [ - i k ( ρ 1 2 - ρ 2 2 ) / 2 z ) ] ρ 1 ρ 2 d ρ 1 d ρ 2 ,
C ( ρ 1 , ρ 2 ; l 1 , l 2 , z ) = 0 2 π exp [ ρ 1 ρ 2 cos ( ϕ 1 - ϕ 2 ) / σ g 2 ] × exp [ i k ( l 1 ρ 1 cos ϕ 1 - l 2 ρ 2 cos ϕ 2 ) / z ] × d ϕ 1 d ϕ 2 .

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