Abstract

We present a design method for diffractive axicons in spatially partially coherent Gaussian Schell-model illumination. The method of stationary phase applied to the Fresnel diffraction integral for on-axis intensity leads, on requiring a uniform axial image profile, to a second-order differential equation for the optimal axicon phase function. The first integral can be formally performed, and the phase function is subsequently obtained numerically. The correctness of the synthesized phase profiles is confirmed by numerical simulations using partially coherent Fresnel diffraction theory. The effects of input-beam spot size and coherence width are assessed, and influences of different forms of apodization, including asymmetric functions for narrow incident beams, in annular-aperture diffractive axicons are examined.

© 2002 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
  5. J. Pu, H. Zhang, S. Nemoto, W. Zhang, W. Zhang, “Annular-aperture diffractive axicons illuminated by Gaussian beams,” J. Opt. A 1, 730–734 (1999).
    [CrossRef]
  6. S. Yu. Popov, A. T. Friberg, “Design of diffractive axicons for partially coherent light,” Opt. Lett. 23, 1639–1641 (1998).
    [CrossRef]
  7. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Secs. 4.3.2 and 5.6.4.
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965), Eq. (9.6.16).
  14. A. Erdelyi, Asymptotic Expansions (Dover, New York, 1956), Sec. 2.9, Eq. (2).
  15. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 7, Eq. (3.10).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  18. S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
    [CrossRef]
  19. A. T. Friberg, “Stationary-phase analysis of generalised axicons,” J. Opt. Soc. Am. A 13, 743–750 (1996).
    [CrossRef]
  20. N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Opt. A 1, 552–559 (1999).
    [CrossRef]
  21. M. Hazewinkel, ed., Encyclopaedia of Mathematics (Reidel, Dordrecht, The Netherlands, 1998), Vol. 1 (A–B), p. 359.
  22. M. Braun, Differential Equations and Their Applications, 4th ed. (Springer, Berlin, 1993), Chap. 12, Eq. (11).
  23. J. Turunen, F. Wyrowski, eds., Diffractive Optics for Industrial and Commercial Application (Akademie Verlag, Berlin, 1997).
  24. S. Yu. Popov, A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
    [CrossRef]
  25. M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
    [CrossRef]
  26. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. Shirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
    [CrossRef]

1999

J. Pu, H. Zhang, S. Nemoto, W. Zhang, W. Zhang, “Annular-aperture diffractive axicons illuminated by Gaussian beams,” J. Opt. A 1, 730–734 (1999).
[CrossRef]

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Opt. A 1, 552–559 (1999).
[CrossRef]

A. T. Friberg, S. Yu. Popov, “Effects of partial spatial coherence with uniform-intensity diffractive axicons,” J. Opt. Soc. Am. A 16, 1049–1058 (1999).
[CrossRef]

1998

S. Yu. Popov, A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[CrossRef]

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

S. Yu. Popov, A. T. Friberg, “Design of diffractive axicons for partially coherent light,” Opt. Lett. 23, 1639–1641 (1998).
[CrossRef]

1996

1993

1992

1991

R. Gase, “The multimode laser radiation as a Gaussian Schell-model beam,” J. Mod. Opt. 38, 1107–1115 (1991).
[CrossRef]

1988

1980

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

1954

Ambrosini, D.

Bagini, V.

Bara, S.

Borghi, R.

Braun, M.

M. Braun, Differential Equations and Their Applications, 4th ed. (Springer, Berlin, 1993), Chap. 12, Eq. (11).

Erdelyi, A.

A. Erdelyi, Asymptotic Expansions (Dover, New York, 1956), Sec. 2.9, Eq. (2).

Friberg, A. T.

A. T. Friberg, S. Yu. Popov, “Effects of partial spatial coherence with uniform-intensity diffractive axicons,” J. Opt. Soc. Am. A 16, 1049–1058 (1999).
[CrossRef]

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Opt. A 1, 552–559 (1999).
[CrossRef]

S. Yu. Popov, A. T. Friberg, “Design of diffractive axicons for partially coherent light,” Opt. Lett. 23, 1639–1641 (1998).
[CrossRef]

S. Yu. Popov, A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[CrossRef]

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

A. T. Friberg, “Stationary-phase analysis of generalised axicons,” J. Opt. Soc. Am. A 13, 743–750 (1996).
[CrossRef]

Gase, R.

R. Gase, “The multimode laser radiation as a Gaussian Schell-model beam,” J. Mod. Opt. 38, 1107–1115 (1991).
[CrossRef]

Gori, F.

Herman, R. M.

Honkanen, M.

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

Jaroszewicz, Z.

Karen, E.

Kettunen, V.

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Opt. A 1, 552–559 (1999).
[CrossRef]

Kolodziejczyk, A.

Kuittinen, M.

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Opt. A 1, 552–559 (1999).
[CrossRef]

Lautanen, J.

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

Lavi, S.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Secs. 4.3.2 and 5.6.4.

McLeod, J. H.

Nemoto, S.

J. Pu, H. Zhang, S. Nemoto, W. Zhang, W. Zhang, “Annular-aperture diffractive axicons illuminated by Gaussian beams,” J. Opt. A 1, 730–734 (1999).
[CrossRef]

Pacileo, A. M.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 7, Eq. (3.10).

Perrone, M. R.

M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

Piegari, A.

M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

Popov, S. Yu.

A. T. Friberg, S. Yu. Popov, “Effects of partial spatial coherence with uniform-intensity diffractive axicons,” J. Opt. Soc. Am. A 16, 1049–1058 (1999).
[CrossRef]

S. Yu. Popov, A. T. Friberg, “Design of diffractive axicons for partially coherent light,” Opt. Lett. 23, 1639–1641 (1998).
[CrossRef]

S. Yu. Popov, A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[CrossRef]

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

Prochaska, R.

Pu, J.

J. Pu, H. Zhang, S. Nemoto, W. Zhang, W. Zhang, “Annular-aperture diffractive axicons illuminated by Gaussian beams,” J. Opt. A 1, 730–734 (1999).
[CrossRef]

Santarsiero, M.

Scaglione, S.

M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

Schnabel, B.

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

Sergienko, N.

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Opt. A 1, 552–559 (1999).
[CrossRef]

Sochacki, J.

Soroko, L. M.

L. M. Soroko, Meso-Optics—Foundations and Applications (World Scientific, Singapore, 1996), Chap. 2.

Spagnolo, G. Shirripa

Stamnes, J. J.

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Opt. A 1, 552–559 (1999).
[CrossRef]

Staronski, L. R.

Turunen, J.

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Opt. A 1, 552–559 (1999).
[CrossRef]

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

Vahimaa, P.

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Opt. A 1, 552–559 (1999).
[CrossRef]

Wiggins, T. A.

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Secs. 4.3.2 and 5.6.4.

Zhang, H.

J. Pu, H. Zhang, S. Nemoto, W. Zhang, W. Zhang, “Annular-aperture diffractive axicons illuminated by Gaussian beams,” J. Opt. A 1, 730–734 (1999).
[CrossRef]

Zhang, W.

J. Pu, H. Zhang, S. Nemoto, W. Zhang, W. Zhang, “Annular-aperture diffractive axicons illuminated by Gaussian beams,” J. Opt. A 1, 730–734 (1999).
[CrossRef]

J. Pu, H. Zhang, S. Nemoto, W. Zhang, W. Zhang, “Annular-aperture diffractive axicons illuminated by Gaussian beams,” J. Opt. A 1, 730–734 (1999).
[CrossRef]

Appl. Opt.

IEEE J. Quantum Electron.

M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

J. Mod. Opt.

R. Gase, “The multimode laser radiation as a Gaussian Schell-model beam,” J. Mod. Opt. 38, 1107–1115 (1991).
[CrossRef]

J. Opt. A

J. Pu, H. Zhang, S. Nemoto, W. Zhang, W. Zhang, “Annular-aperture diffractive axicons illuminated by Gaussian beams,” J. Opt. A 1, 730–734 (1999).
[CrossRef]

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Opt. A 1, 552–559 (1999).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

Opt. Lett.

Pure Appl. Opt.

S. Yu. Popov, A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[CrossRef]

Other

M. Hazewinkel, ed., Encyclopaedia of Mathematics (Reidel, Dordrecht, The Netherlands, 1998), Vol. 1 (A–B), p. 359.

M. Braun, Differential Equations and Their Applications, 4th ed. (Springer, Berlin, 1993), Chap. 12, Eq. (11).

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

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Secs. 4.3.2 and 5.6.4.

L. M. Soroko, Meso-Optics—Foundations and Applications (World Scientific, Singapore, 1996), Chap. 2.

Z. Jaroszewicz, Axicons: Design and Propagation Properties, Vol. 5 of Research & Development Treatises (SPIE Polish Chapter, Warsaw, 1997).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965), Eq. (9.6.16).

A. Erdelyi, Asymptotic Expansions (Dover, New York, 1956), Sec. 2.9, Eq. (2).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 7, Eq. (3.10).

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Figures (9)

Fig. 1
Fig. 1

Geometry of the annular-aperture axicon and illustration of the notation. The inset shows the on-axis profile of image intensity.

Fig. 2
Fig. 2

Axial intensity distributions for a diffractive axicon optimized for uniform illumination with the spatial coherence width σμ given by (a) ∞, (b) 10, (c) 5, (d) 3, and (e) 1.5 mm. The apodization profile is the arctangent function given by Eq. (23).

Fig. 3
Fig. 3

Axial intensity distributions of an axicon in uniform, fully coherent illumination with the use of arctangent apodization (solid curve), super-Gaussian apodization [Eq. (24)] with n=16 and w=1.09 mm (dotted–dashed curve), and super-Gaussian apodization with n=10 and w=1.04 mm (dashed curve).

Fig. 4
Fig. 4

On-axis intensity profiles of an axicon optimized for a GSM beam with σI=5 mm and σμ equal to (a) ∞, (b) 10, (c) 5, (d) 3, and (e) 1.5 mm. The apodization is super-Gaussian with n=10 and w=1.04 mm.

Fig. 5
Fig. 5

Same as Fig. 4 but for σI=3 mm.

Fig. 6
Fig. 6

Same as Fig. 4 but for σI=1.5 mm.

Fig. 7
Fig. 7

Derivatives of the optimized phase functions for intensity width σI=5 mm and spatial coherence widths σμ of (a) ∞, (b) 10, (c) 5, (d) 3, and (e) 1.5 mm.

Fig. 8
Fig. 8

Same as Fig. 7 but for σI=3 mm.

Fig. 9
Fig. 9

On-axis intensity distributions of an axicon optimized for σI=1.5 mm and σμ given by (a) ∞, (b) 10, (c) 5, (d) 3, and (e) 1.5 mm. The apodization is super-Gaussian and asymmetric with n=4 and w=0.80 mm in the outer part of the axicon and n=18 and w=1.08 mm in the inner part.

Equations (26)

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I(ρ)=A exp(-ρ2/2σI2)
μ(ρ1, ρ2)=exp[-(ρ1-ρ2)2/2σμ2],
Win(ρ1, ρ2)=[I(ρ1)I(ρ2)]1/2μ(ρ1, ρ2)=A exp[-(ρ12+ρ22)/4σI2]×exp[-(ρ1-ρ2)2/2σμ2].
I(0, z)=k2πz2Win(ρ1, ρ2)t(ρ1)t(ρ2)×exp{-ik[φ(ρ1)-φ(ρ2)]}×exp[-ik(ρ12-ρ22)/2z]d2ρ1d2ρ2.
02π02π exp[ρ1ρ2 cos(θ1-θ2)/σμ2]dθ1dθ2
=(2π)2I0(ρ1ρ2/σμ2),
I(0, z)=Akz2r1r2r1r2t(ρ1)t(ρ2)exp[-(ρ12+ρ22)/4σI2]×exp[-(ρ12+ρ22)/2σμ2]I0(ρ1ρ2/σμ2)ρ1ρ2×exp{-ik[φ(ρ1)-φ(ρ2)]}×exp[-ik(ρ12-ρ22)/2z]dρ1dρ2.
Isp(0, z)=A2πkz2T(ρc)I0ρc2σμ2×exp-ρc212σI2+1σμ2ρc21ψ(2)(ρc, z),
ψ(ρ, z)=ρ2/2z+φ(ρ)
φ(1)(ρc)=-ρc/z,
ψ(2)(ρc, z)=1/z+φ(2)(ρc).
φ(2)(ρc)-φ(1)(ρc)ρc-C1T(ρc)exp-ρc212σI2+1σμ2×I0ρc2σμ2[φ(1)(ρc)]2=0,
g(ρ)=-C1I0ρ2σμ2exp-ρ222σI2+1σμ2,
x(ρ)-1ρx(ρ)+g(ρ)x2(ρ)=0,
y(ρ)+1ρy(ρ)=g(ρ),
y(ρ)=exp[-F(ρ)]rρ exp[F(ρ)]g(ρ)dρ+C,
F(ρ)=rρ1ρdρ=log ρ-log r,
φ(1)(ρ)=-ρC1r1ρρI0ρ2σμ2exp-ρ212σI2+1σμ2dρ+C2-1,
φ(1)(r1)=-r1/d1,
φ(1)(r2)=-r2/d2.
φ(ρ)=-1aρ22+σI2 loga+b exp-ρ22σI2+const.,
a=[d2 exp(-r12/2σI2)-d1 exp(-r22/2σI2)]/[exp(-r12/2σI2)-exp(-r22/2σI2)],
b=-(d2-d1)/[exp(-r12/2σI2)-exp(-r22/2σI2)]
t(ρ)={0.5+arctan[Δ(r˜1-ρ)/π]}{0.5+arctan[Δ(ρ-r˜2)/π]}.
t(ρ)=exp-ρ-r¯wn,
dφ(ρ)dρ=-sin θ,

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