Abstract

Within the framework of inverse diffractive optics, we present a design for diffractive axicons in twisted, spatially partially coherent fields, in particular twisted Gaussian Schell-model (TGSM) fields. The design is based on the method of stationary phase. A general modification is introduced to the inverse diffractive optics approach for improving the synthesized optical element to produce the desired intensity distribution. Both the design and modification are demonstrated with annular-aperture axicons generating uniform-intensity axial line segments in partially coherent TGSM illumination.

© 2013 Optical Society of America

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References

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  1. J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Akademie-Verlag, 1997).
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    [CrossRef]
  3. Z. Jaroszewicz, J. Sochacki, A. Kolodziejczyk, and L. R. Staronski, “Apodized annular-aperture logarithmic axicon smoothness and uniformity of intensity distributions,” Opt. Lett. 18, 1893–1895 (1993).
    [CrossRef]
  4. J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, and S. Bara, “Nonparaxial design of generalized axicons,” Appl. Opt. 31, 5326–5330 (1992).
    [CrossRef]
  5. J. Pu, H. Zhang, S. Nemoto, W. Zhang, and W. Zhang, “Annular-aperture diffractive axicons illuminated by Gaussian beams,” J. Opt. A 1, 730–734 (1999).
    [CrossRef]
  6. S. Y. Popov and A. T. Friberg, “Design of diffractive axicons for partially coherent light,” Opt. Lett. 23, 1639–1641 (1998).
    [CrossRef]
  7. A. Thaning, A. T. Friberg, S. Y. Popov, and Z. Jaroszewicz, “Design of diffractive axicons producing uniform line images in Gaussian Schell-model illumination,” J. Opt. Soc. Am. A 19, 491–496 (2002).
    [CrossRef]
  8. S. Y. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
    [CrossRef]
  9. Z. Jaroszewicz, J. Sochacki, A. Kolodziejczyk, and L. R. Staronski, “Apodized annular-aperture logarithmic axicon: smoothness and uniformity of the intensity distributions,” Opt. Lett. 18, 1893–1895 (1993).
    [CrossRef]
  10. G. Adamkiewicz, Z. Jaroszewicz, A. Koodziejczyk, and T. Osuch, “Apodized diffractive elements obtained with the help of HEBS glasses,” in EOS Topical Meeting Diffractive Optics (EOS, 2005), pp. 151–152.
  11. S. Y. Popov and A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
    [CrossRef]
  12. A. J. Cox and J. D’Anna, “Constant-axial-intensity nondiffracting beam,” Opt. Lett. 17, 232–234 (1992).
    [CrossRef]
  13. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [CrossRef]
  14. R. Simon, K. Sundar, and N. Mukunda, “Twisted Gaussian Schell-model beams. I. symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
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  15. K. Saundar, R. Simon, and N. Mukunda, “Twisted Gaussian Schell-model beams. II. spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A 10, 2017–2023 (1993).
    [CrossRef]
  16. A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
    [CrossRef]
  17. A. A. Alkelly, M. A. Shukri, and Y. S. Alarify, “Influences of twist phenomenon of partially coherent field with uniform-intensity diffractive axicons,” J. Opt. Soc. Am. A 29, 417–425(2012).
    [CrossRef]
  18. M. Shukri, A. A. Alkelly, and Y. S. Alarify, “Spatial correlation properties of twisted partially coherent light focused by diffractive axicons,” J. Opt. Soc. Am. A 29, 2019–2027(2012).
    [CrossRef]
  19. W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems (Von Hoffmann, 2001).
  20. M. R. Perrone, A. Piegari, and S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1487 (1993).
    [CrossRef]
  21. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
    [CrossRef]
  22. Y. Cai and Q. Lin, “Focusing properties of partially coherent twisted anisotropic Gaussian Schell-model beams,” Opt. Commun. 215, 239–245 (2003).
    [CrossRef]

2012

2006

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

2003

Y. Cai and Q. Lin, “Focusing properties of partially coherent twisted anisotropic Gaussian Schell-model beams,” Opt. Commun. 215, 239–245 (2003).
[CrossRef]

2002

1999

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

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

1998

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

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

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

1994

1993

1992

Adamkiewicz, G.

G. Adamkiewicz, Z. Jaroszewicz, A. Koodziejczyk, and T. Osuch, “Apodized diffractive elements obtained with the help of HEBS glasses,” in EOS Topical Meeting Diffractive Optics (EOS, 2005), pp. 151–152.

Alarify, Y. S.

Alkelly, A. A.

Bara, S.

Boyce, W. E.

W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems (Von Hoffmann, 2001).

Cai, Y.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

Y. Cai and Q. Lin, “Focusing properties of partially coherent twisted anisotropic Gaussian Schell-model beams,” Opt. Commun. 215, 239–245 (2003).
[CrossRef]

Cox, A. J.

D’Anna, J.

DiPrima, R. C.

W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems (Von Hoffmann, 2001).

Friberg, A. T.

He, S.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

Honkanen, M.

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

Jaroszewicz, Z.

Kolodziejczyk, A.

Koodziejczyk, A.

G. Adamkiewicz, Z. Jaroszewicz, A. Koodziejczyk, and T. Osuch, “Apodized diffractive elements obtained with the help of HEBS glasses,” in EOS Topical Meeting Diffractive Optics (EOS, 2005), pp. 151–152.

Lautanen, J.

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

Lin, Q.

Y. Cai and Q. Lin, “Focusing properties of partially coherent twisted anisotropic Gaussian Schell-model beams,” Opt. Commun. 215, 239–245 (2003).
[CrossRef]

Mukunda, N.

Nemoto, S.

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

Osuch, T.

G. Adamkiewicz, Z. Jaroszewicz, A. Koodziejczyk, and T. Osuch, “Apodized diffractive elements obtained with the help of HEBS glasses,” in EOS Topical Meeting Diffractive Optics (EOS, 2005), pp. 151–152.

Perrone, M. R.

M. R. Perrone, A. Piegari, and S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1487 (1993).
[CrossRef]

Piegari, A.

M. R. Perrone, A. Piegari, and S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1487 (1993).
[CrossRef]

Popov, S. Y.

A. Thaning, A. T. Friberg, S. Y. Popov, and Z. Jaroszewicz, “Design of diffractive axicons producing uniform line images in Gaussian Schell-model illumination,” J. Opt. Soc. Am. A 19, 491–496 (2002).
[CrossRef]

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

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

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

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

Pu, J.

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

Saundar, K.

Scaglione, S.

M. R. Perrone, A. Piegari, and S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1487 (1993).
[CrossRef]

Schnabel, B.

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

Shukri, M.

Shukri, M. A.

Simon, R.

Sochacki, J.

Staronski, L. R.

Sundar, K.

Tervonen, E.

Thaning, A.

Turunen, J.

S. Y. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and 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, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

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

Wyrowski, F.

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

Zhang, H.

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

Appl. Opt.

Appl. Phys. Lett.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

IEEE J. Quantum Electron.

M. R. Perrone, A. Piegari, and S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1487 (1993).
[CrossRef]

J. Opt. A

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

J. Opt. Soc. Am. A

Opt. Commun.

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

Y. Cai and Q. Lin, “Focusing properties of partially coherent twisted anisotropic Gaussian Schell-model beams,” Opt. Commun. 215, 239–245 (2003).
[CrossRef]

Opt. Lett.

Pure Appl. Opt.

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

Other

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

W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems (Von Hoffmann, 2001).

G. Adamkiewicz, Z. Jaroszewicz, A. Koodziejczyk, and T. Osuch, “Apodized diffractive elements obtained with the help of HEBS glasses,” in EOS Topical Meeting Diffractive Optics (EOS, 2005), pp. 151–152.

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Figures (10)

Fig. 1.
Fig. 1.

Geometry of an annular-aperture diffractive axicon with inner and outer radii R1 and R2, respectively, performing an axial focal line between d1 and d2, and notation of the corresponding on-axis profile of image intensity.

Fig. 2.
Fig. 2.

On-axis intensity distribution S(0,z) produced by the nonapodized axicon given by Eq. (11) for σI=1.5mm, in GSM beams. The system parameters are S0=1 R1=2.5mm, R2=5mm, d1=100mm, d2=200mm, and the wavelength λ=0.6328μm. The apodization is super-Gaussian with n=10 and ϖ=1.04mm.

Fig. 3.
Fig. 3.

Derivatives of the apodized (solid curves) and nonapodized (dashed curves) phase functions for σI=1.5mm. The order of the curves from top to bottom takes the same order as the associated legend.

Fig. 4.
Fig. 4.

Same as in Fig. 2, but by using the apodized design in Eq. (9) for η0.

Fig. 5.
Fig. 5.

On-axis intensity distribution S(0,z) produced by an axicon optimized for a GSM beam of σI=5mm, σμ=2mm, in TGSM beams with various values of twist parameter η. The system parameters are S0=1, R1=2.5mm, R2=5mm, d1=100mm, d2=200mm, and the wavelength λ=0.6328μm. The apodization is super-Gaussian with n=10 and ϖ=1.2mm.

Fig. 6.
Fig. 6.

Same as in Fig. 5 and employing the optimized phase function described in Eq. (9).

Fig. 7.
Fig. 7.

Same as Fig. 6 but for σI=10mm, σμ=3mm.

Fig. 8.
Fig. 8.

Three-dimensional spectral density profiles of the axicon line images calculated with the optimized phase function of Eq. (9) for σI=5mm, σμ=2mm, and twist parameter η: (a) η=0.0 and (b) η=0.6. The other parameters are the same as in Fig. 6.

Fig. 9.
Fig. 9.

Derivatives of the optimized phase functions optimized for beam width σI=5mm, spatial coherence width σμ=2mm, and various values of the twist parameter. The system parameters are the same as in Fig. 5.

Fig. 10.
Fig. 10.

Same as in Fig. 8 for σI=10mm, σμ=3mm.

Equations (11)

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W(ρ1,ρ2)=S0exp[(ρ12+ρ22)/4σI2]exp(|ρ1ρ2|2/2σμ2)exp[iηρ1ρ2sin(ϕ1ϕ2)/σμ2],
S(0,z)=S0k2z2at(ρ1)t(ρ2)exp{ik[ψ(ρ1,z)ψ(ρ2,z)]}exp[ρ12+ρ224σI2ρ12+ρ222σμ2]I0(ρ1ρ21η2σμ2)ρ1ρ2dρ1dρ2,
ψ(ρ,z)=φ(ρ)+ρ2/2z,
Ssp(0,z)S0(2πk/z2)ρc2T(ρc)f(ρc)[ψ(2)(ρc,z)]1,
f(ρc)=exp(ρc22σI2ρc2σμ2)I0(1η2σμ2ρc2),
φ(2)(ρ)ρ1φ(1)(ρ)C1f(ρ)T(ρ)[φ(1)(ρ)]2=0,
y(1)(ρ)+C1f(ρ)T(ρ)ρ=0.
y(ρ)=(C1rρf(s)T(s)sds+C2),
φ(1)(ρ)=ρ[C1R1ρexp(s22σI2s2σμ2)I0(1η2σμ2s2)T(s)sds+C2]1.
t(ρ)=exp{[(ρr¯)/ϖ]n},
φ(1)(ρ)=ρ[C1R1ρexp(s22σI2s2σμ2)I0(s2σμ2)sds+C2]1.

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