Abstract

We propose a novel method of designing diffractive axicons for use in spatially partially coherent illumination. The design procedure is based on the results obtained by the stationary-phase method. The technique leads to a coherence-dependent differential equation with appropriate boundary conditions for the axicon phase function. We demonstrate the method with annular-aperture axicons generating extended focal line segments of uniform on-axis intensity.

© 1998 Optical Society of America

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References

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  1. Z. Jaroszewicz, Axicons: Design and Propagation Properties, Vol. 5 of Research & Development Treatises (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1997).
  2. V. P. Koronkevich, I. A. Mikhaltsova, E. G. Churin, and Yu. I. Yurlov, Appl. Opt. 34, 5761 (1995).
    [CrossRef] [PubMed]
  3. N. Davidson, A. A. Friesem, and E. Hasman, Opt. Commun. 88, 326 (1992).
    [CrossRef]
  4. J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, and S. Bara, Appl. Opt. 31, 5326 (1992).
    [CrossRef] [PubMed]
  5. J. Sochacki, Z. Jaroszewicz, L. R. Staronski, and A. Kolodziejczyk, J. Opt. Soc. Am. A 10, 1765 (1993).
    [CrossRef]
  6. Z. Jaroszewicz, J. Sochacki, A. Kolodziejczyk, and L. R. Staronski, Opt. Lett. 18, 1893 (1993).
    [CrossRef] [PubMed]
  7. S. Yu. Popov and A. T. Friberg, Pure Appl. Opt. 7, 537 (1998).
    [CrossRef]
  8. A. T. Friberg, J. Opt. Soc. Am. A 13, 743 (1996).
    [CrossRef]
  9. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Secs. 4.3 and 4.4.
    [CrossRef]
  10. S. Yu. Popov and A. T. Friberg, Opt. Eng. 34, 2567 (1995).
    [CrossRef]
  11. A. T. Friberg and S. Yu. Popov, Appl. Opt. 35, 3039 (1996).
    [CrossRef] [PubMed]
  12. Z. Jaroszewicz, J. F. Roman Dopato, and C. Gomez-Reino, Appl. Opt. 35, 1025 (1996).
    [CrossRef] [PubMed]
  13. A. T. Friberg and S. Yu. Popov, in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 224.
  14. J. Turunen and F. Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Akademie, Berlin, 1997).
  15. One might argue that C1 could be determined from energy conservation. This is not so simple, however, since the integral of I0,z over the image segment along the z axis has no obvious physical meaning [cf. Ref. 4 and L. R. Staronski, J. Sochacki, Z. Jaroszewicz, and A. Kolodziejczyk, J. Opt. Soc. Am. A 9, 2091 (1992). Considerations of energy conservation must be based on the energy-flux vector.
    [CrossRef]

1998 (1)

S. Yu. Popov and A. T. Friberg, Pure Appl. Opt. 7, 537 (1998).
[CrossRef]

1996 (3)

1995 (2)

1993 (2)

1992 (3)

Bara, S.

Churin, E. G.

Davidson, N.

N. Davidson, A. A. Friesem, and E. Hasman, Opt. Commun. 88, 326 (1992).
[CrossRef]

Friberg, A. T.

S. Yu. Popov and A. T. Friberg, Pure Appl. Opt. 7, 537 (1998).
[CrossRef]

A. T. Friberg, J. Opt. Soc. Am. A 13, 743 (1996).
[CrossRef]

A. T. Friberg and S. Yu. Popov, Appl. Opt. 35, 3039 (1996).
[CrossRef] [PubMed]

S. Yu. Popov and A. T. Friberg, Opt. Eng. 34, 2567 (1995).
[CrossRef]

A. T. Friberg and S. Yu. Popov, in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 224.

Friesem, A. A.

N. Davidson, A. A. Friesem, and E. Hasman, Opt. Commun. 88, 326 (1992).
[CrossRef]

Gomez-Reino, C.

Hasman, E.

N. Davidson, A. A. Friesem, and E. Hasman, Opt. Commun. 88, 326 (1992).
[CrossRef]

Jaroszewicz, Z.

Kolodziejczyk, A.

Koronkevich, V. P.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Secs. 4.3 and 4.4.
[CrossRef]

Mikhaltsova, I. A.

Popov, S. Yu.

S. Yu. Popov and A. T. Friberg, Pure Appl. Opt. 7, 537 (1998).
[CrossRef]

A. T. Friberg and S. Yu. Popov, Appl. Opt. 35, 3039 (1996).
[CrossRef] [PubMed]

S. Yu. Popov and A. T. Friberg, Opt. Eng. 34, 2567 (1995).
[CrossRef]

A. T. Friberg and S. Yu. Popov, in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 224.

Roman Dopato, J. F.

Sochacki, J.

Staronski, L. R.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Secs. 4.3 and 4.4.
[CrossRef]

Yurlov, Yu. I.

Appl. Opt. (4)

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

N. Davidson, A. A. Friesem, and E. Hasman, Opt. Commun. 88, 326 (1992).
[CrossRef]

Opt. Eng. (1)

S. Yu. Popov and A. T. Friberg, Opt. Eng. 34, 2567 (1995).
[CrossRef]

Opt. Lett. (1)

Pure Appl. Opt. (1)

S. Yu. Popov and A. T. Friberg, Pure Appl. Opt. 7, 537 (1998).
[CrossRef]

Other (4)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Secs. 4.3 and 4.4.
[CrossRef]

Z. Jaroszewicz, Axicons: Design and Propagation Properties, Vol. 5 of Research & Development Treatises (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1997).

A. T. Friberg and S. Yu. Popov, in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 224.

J. Turunen and F. Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Akademie, Berlin, 1997).

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Figures (4)

Fig. 1
Fig. 1

Axial intensity distributions produced by a logarithmic axicon [Eq. (4)] for unit-amplitude partially coherent illumination with σg of (a) , (b) 10, (c) 5, (d) 3, and (e) 1.5 mm. The curves were numerically calculated from the usual diffraction integral with a slight apodization. The parameters are r=2.5 mm, R=5.0 mm, d1=100 mm, d2=200 mm, and λ=632.8 nm.

Fig. 2
Fig. 2

Differences of the axicon phase retardations kφρ designed to generate an image segment of axially uniform intensity from the corresponding logarithmic phase kφcohρ for coherence widths σg of (b) 10, (c) 5, (d) 3, and (e) 1.5 mm. All other parameters are as in Fig. 1.

Fig. 3
Fig. 3

Axial intensity distributions produced by the diffractive axicons optimized for σg of (a) , (b) 10, (c) 5, (d) 3, and (e) 1.5 mm. The method of intensity calculation and the system parameters are the same as in Fig. 1.

Fig. 4
Fig. 4

Derivatives φ1ρ [Eq. (3)] of the optimized axicon phase functions for coherence widths σg of (a) , (b) 10, (c) 5, (d) 3, and (e) 1.5 mm and in the limit of σg0 (dashed line). The parameters are as in Fig. 1.

Equations (4)

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Isp0,zA2πk/z2TρcI0ρc2/σg2×exp-ρc2/σg2ρc2ψ2ρc-1.
φ2ρ-ρ-1φ1ρ-C1I0ρ2/σg2×exp-ρ2/σg2φ1ρ2=0,
φ1ρ=-2ρ/C1ρ2 exp-ρ2/σg2I0ρ2/σg2+I1ρ2/σg2+C2,
φρcoh=-2a-1 log1+aρ2-r2/d1,

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