Abstract

We employ the conjugate-gradient method to design diffractive phase elements (DPE’s) that implement the predefined axial-intensity modulation over a given axial region. We introduce an error function for guiding the design of DPE’s and for evaluating their performance. Numerical simulations are carried out for several examples, for instance, the design of the DPE that generates four foci with equal or unequal spacing between the consecutive foci along the optical axis and the DPE that produces a sinusoidlike axial-intensity modulation. The obtained results show that the designed DPE’s can satisfactorily match practical requirements.

© 1998 Optical Society of America

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  1. F. Wyrowski, “Design theory of diffractive elements in the paraxial domain,” J. Opt. Soc. Am. A 10, 1553–1561 (1993).
    [CrossRef]
  2. J. N. Mait, “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A 12, 2145–2158 (1995).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  5. R. P. MacDonald, J. Chrostowski, S. A. Boothroyd, B. A. Syrett, “Holographic formation of a diode laser nondiffracting beam,” Appl. Opt. 32, 6470–6474 (1993).
    [CrossRef] [PubMed]
  6. R. Piestun, J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. 19, 771–773 (1994).
    [CrossRef] [PubMed]
  7. J. Rosen, “Synthesis of nondiffracting beams in free space,” Opt. Lett. 19, 369–371 (1994).
    [PubMed]
  8. V. V. Kotlyar, S. N. Khonina, V. A. Soifer, “Algorithm for the generation of non-diffracting Bessel modes,” J. Mod. Opt. 42, 1231–1239 (1995).
    [CrossRef]
  9. C. Paterson, R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).
    [CrossRef]
  10. C. Paterson, R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
    [CrossRef]
  11. L. Niggl, T. Lanzl, M. Maier, “Properties of Bessel beams generated by periodic gratings of circular symmetry,” J. Opt. Soc. Am. A 14, 27–33 (1997).
    [CrossRef]
  12. N. Davidson, A. A. Friesem, E. Hasman, “Holographic axilens: high resolution and long focal depth,” Opt. Lett. 16, 523–525 (1991).
    [CrossRef] [PubMed]
  13. J. Sochacki, S. Bará, Z. Jaroszewicz, A. Kolodziejczyk, “Phase retardation of the uniform-intensity axilens,” Opt. Lett. 17, 7–9 (1992).
    [CrossRef] [PubMed]
  14. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Mod. Opt. 40, 761–769 (1993).
    [CrossRef]
  15. B.-Z. Dong, G.-Z. Yang, B.-Y. Gu, O. K. Erosy, “Iterative optimization approach for designing an axicon with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A 13, 97–103 (1996).
    [CrossRef]
  16. M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39, 1245–1251 (1992).
    [CrossRef]
  17. J. Rosen, A. Yariv, “Synthesis of an arbitrary axial field profile by computer-generated holograms,” Opt. Lett. 19, 843–845 (1994).
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  18. B. Salik, J. Rosen, A. Yariv, “One-dimensional beam shaping,” J. Opt. Soc. Am. A 12, 1702–1706 (1995).
    [CrossRef]
  19. R. Piestun, B. Spektor, J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A 13, 1837–1848 (1996).
    [CrossRef]
  20. R. Piestun, B. Spektor, J. Shamir, “Unconventional light distribution in three-dimensional domains,” J. Mod. Opt. 43, 1495–1507 (1996).
    [CrossRef]
  21. V. V. Kotlyar, S. N. Khonina, V. A. Soifer, “Iterative calculation of diffractive optical elements focusing into a three-dimensional domain and onto the surface of the body of rotation,” J. Mod. Opt. 43, 1509–1524 (1996).
    [CrossRef]
  22. M. Avriel, Nonlinear Programming: Analysis and Methods (Prentice-Hall, Englewood Cliffs, N. J., 1976), pp. 299–307.
  23. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  24. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
    [CrossRef] [PubMed]
  25. J. R. Fienup, “Gradient-search phase retrieval algorithm for inverse synthetic aperture radar,” Opt. Eng. 13, 3237–3242 (1994).
    [CrossRef]
  26. G. Leone, R. Pierri, F. Soldovieri, “Reconstruction of complex signals from intensities of Fourier-transform pairs,” J. Opt. Soc. Am. A 13, 1546–1556 (1996).
    [CrossRef]
  27. S. T. Teiwes, B. Schillinger, T. Beth, F. Wyrowski, “Efficient design of paraxial diffractive phase elements with descent search methods,” in Diffractive and Holographic Optics II, Ivan Cindrich, ed., Proc. SPIE2404, 40–49 (1994).
    [CrossRef]
  28. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), pp. 13, 58, and 63.

1997

1996

C. Paterson, R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).
[CrossRef]

C. Paterson, R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

B.-Z. Dong, G.-Z. Yang, B.-Y. Gu, O. K. Erosy, “Iterative optimization approach for designing an axicon with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A 13, 97–103 (1996).
[CrossRef]

R. Piestun, B. Spektor, J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A 13, 1837–1848 (1996).
[CrossRef]

R. Piestun, B. Spektor, J. Shamir, “Unconventional light distribution in three-dimensional domains,” J. Mod. Opt. 43, 1495–1507 (1996).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, V. A. Soifer, “Iterative calculation of diffractive optical elements focusing into a three-dimensional domain and onto the surface of the body of rotation,” J. Mod. Opt. 43, 1509–1524 (1996).
[CrossRef]

G. Leone, R. Pierri, F. Soldovieri, “Reconstruction of complex signals from intensities of Fourier-transform pairs,” J. Opt. Soc. Am. A 13, 1546–1556 (1996).
[CrossRef]

1995

1994

1993

1992

J. Sochacki, S. Bará, Z. Jaroszewicz, A. Kolodziejczyk, “Phase retardation of the uniform-intensity axilens,” Opt. Lett. 17, 7–9 (1992).
[CrossRef] [PubMed]

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39, 1245–1251 (1992).
[CrossRef]

N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[CrossRef]

1991

1989

1982

Avriel, M.

M. Avriel, Nonlinear Programming: Analysis and Methods (Prentice-Hall, Englewood Cliffs, N. J., 1976), pp. 299–307.

Bará, S.

Beth, T.

S. T. Teiwes, B. Schillinger, T. Beth, F. Wyrowski, “Efficient design of paraxial diffractive phase elements with descent search methods,” in Diffractive and Holographic Optics II, Ivan Cindrich, ed., Proc. SPIE2404, 40–49 (1994).
[CrossRef]

Boothroyd, S. A.

Chrostowski, J.

Davidson, N.

N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[CrossRef]

N. Davidson, A. A. Friesem, E. Hasman, “Holographic axilens: high resolution and long focal depth,” Opt. Lett. 16, 523–525 (1991).
[CrossRef] [PubMed]

Dong, B.-Z.

Doskolovich, L. L.

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39, 1245–1251 (1992).
[CrossRef]

Erosy, O. K.

Fienup, J. R.

Friberg, A. T.

Friesem, A. A.

N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[CrossRef]

N. Davidson, A. A. Friesem, E. Hasman, “Holographic axilens: high resolution and long focal depth,” Opt. Lett. 16, 523–525 (1991).
[CrossRef] [PubMed]

Golub, M. A.

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39, 1245–1251 (1992).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), pp. 13, 58, and 63.

Gu, B.-Y.

Hasman, E.

N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[CrossRef]

N. Davidson, A. A. Friesem, E. Hasman, “Holographic axilens: high resolution and long focal depth,” Opt. Lett. 16, 523–525 (1991).
[CrossRef] [PubMed]

Jaroszewicz, Z.

Kazanskiy, N. L.

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39, 1245–1251 (1992).
[CrossRef]

Kharitonov, S. I.

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39, 1245–1251 (1992).
[CrossRef]

Khonina, S. N.

V. V. Kotlyar, S. N. Khonina, V. A. Soifer, “Iterative calculation of diffractive optical elements focusing into a three-dimensional domain and onto the surface of the body of rotation,” J. Mod. Opt. 43, 1509–1524 (1996).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, V. A. Soifer, “Algorithm for the generation of non-diffracting Bessel modes,” J. Mod. Opt. 42, 1231–1239 (1995).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Mod. Opt. 40, 761–769 (1993).
[CrossRef]

Kolodziejczyk, A.

Kotlyar, V. V.

V. V. Kotlyar, S. N. Khonina, V. A. Soifer, “Iterative calculation of diffractive optical elements focusing into a three-dimensional domain and onto the surface of the body of rotation,” J. Mod. Opt. 43, 1509–1524 (1996).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, V. A. Soifer, “Algorithm for the generation of non-diffracting Bessel modes,” J. Mod. Opt. 42, 1231–1239 (1995).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Mod. Opt. 40, 761–769 (1993).
[CrossRef]

Lanzl, T.

Leone, G.

MacDonald, R. P.

Maier, M.

Mait, J. N.

Niggl, L.

Paterson, C.

C. Paterson, R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

C. Paterson, R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).
[CrossRef]

Pierri, R.

Piestun, R.

Rosen, J.

Salik, B.

Schillinger, B.

S. T. Teiwes, B. Schillinger, T. Beth, F. Wyrowski, “Efficient design of paraxial diffractive phase elements with descent search methods,” in Diffractive and Holographic Optics II, Ivan Cindrich, ed., Proc. SPIE2404, 40–49 (1994).
[CrossRef]

Shamir, J.

Smith, R.

C. Paterson, R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).
[CrossRef]

C. Paterson, R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

Sochacki, J.

Soifer, V. A.

V. V. Kotlyar, S. N. Khonina, V. A. Soifer, “Iterative calculation of diffractive optical elements focusing into a three-dimensional domain and onto the surface of the body of rotation,” J. Mod. Opt. 43, 1509–1524 (1996).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, V. A. Soifer, “Algorithm for the generation of non-diffracting Bessel modes,” J. Mod. Opt. 42, 1231–1239 (1995).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Mod. Opt. 40, 761–769 (1993).
[CrossRef]

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39, 1245–1251 (1992).
[CrossRef]

Soldovieri, F.

Spektor, B.

R. Piestun, B. Spektor, J. Shamir, “Unconventional light distribution in three-dimensional domains,” J. Mod. Opt. 43, 1495–1507 (1996).
[CrossRef]

R. Piestun, B. Spektor, J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A 13, 1837–1848 (1996).
[CrossRef]

Syrett, B. A.

Teiwes, S. T.

S. T. Teiwes, B. Schillinger, T. Beth, F. Wyrowski, “Efficient design of paraxial diffractive phase elements with descent search methods,” in Diffractive and Holographic Optics II, Ivan Cindrich, ed., Proc. SPIE2404, 40–49 (1994).
[CrossRef]

Turunen, J.

Vasara, A.

Wyrowski, F.

F. Wyrowski, “Design theory of diffractive elements in the paraxial domain,” J. Opt. Soc. Am. A 10, 1553–1561 (1993).
[CrossRef]

S. T. Teiwes, B. Schillinger, T. Beth, F. Wyrowski, “Efficient design of paraxial diffractive phase elements with descent search methods,” in Diffractive and Holographic Optics II, Ivan Cindrich, ed., Proc. SPIE2404, 40–49 (1994).
[CrossRef]

Yang, G.-Z.

Yariv, A.

Appl. Opt.

J. Mod. Opt.

R. Piestun, B. Spektor, J. Shamir, “Unconventional light distribution in three-dimensional domains,” J. Mod. Opt. 43, 1495–1507 (1996).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, V. A. Soifer, “Iterative calculation of diffractive optical elements focusing into a three-dimensional domain and onto the surface of the body of rotation,” J. Mod. Opt. 43, 1509–1524 (1996).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, V. A. Soifer, “Algorithm for the generation of non-diffracting Bessel modes,” J. Mod. Opt. 42, 1231–1239 (1995).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Mod. Opt. 40, 761–769 (1993).
[CrossRef]

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39, 1245–1251 (1992).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[CrossRef]

C. Paterson, R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).
[CrossRef]

C. Paterson, R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

Opt. Eng.

J. R. Fienup, “Gradient-search phase retrieval algorithm for inverse synthetic aperture radar,” Opt. Eng. 13, 3237–3242 (1994).
[CrossRef]

Opt. Lett.

Other

M. Avriel, Nonlinear Programming: Analysis and Methods (Prentice-Hall, Englewood Cliffs, N. J., 1976), pp. 299–307.

S. T. Teiwes, B. Schillinger, T. Beth, F. Wyrowski, “Efficient design of paraxial diffractive phase elements with descent search methods,” in Diffractive and Holographic Optics II, Ivan Cindrich, ed., Proc. SPIE2404, 40–49 (1994).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), pp. 13, 58, and 63.

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Figures (4)

Fig. 1
Fig. 1

Sketch of diffractive optical system for producing axial-intensity modulation.

Fig. 2
Fig. 2

(a) Axial-intensity distribution of diffraction field generated by the designed DPE for producing four focal spots with equal spacing along the z axis. (b) Transverse-intensity distribution on the four focal planes at z=30.0, 31.0, 32.0, and 33.0 mm. The transverse coordinates are measured in units of the diffraction-limited spot size A1 (4.758 µm) of a lens with a focal length of f=30.0 mm. The relevant parameters are λ=780 nm, R1m=3.0 mm, and N1=1024.

Fig. 3
Fig. 3

(a) Axial-intensity distribution of the diffraction pattern generated by the designed DPE for producing four focal spots with unequal spacing along the z axis. (b) Transverse-intensity distribution on the four focal planes at z=30.0, 31.7, 33.0, and 34.0 mm. The transverse coordinates are measured in units of A1, which is the same as that in Fig. 2(b). Other relevant parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

(a) Phase distribution of the designed DPE for producing eight periods of nearly sinusoidlike modulation over a given range of [100.0, 120.0] mm. (b) Calculated axial-intensity distribution. The period of the sinusoidlike modulation is approximately 2.4 mm. (c) Axial-intensity distribution evaluated over a larger region than that in the design. The relevant parameters are λ=780 nm, R1m=3.0 mm, and N1=1024.

Equations (23)

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U1(X1)=ρ1(X1)exp[iϕ1(X1)],
U2(X2α, zα)=ρ2(X2α, zα)exp[iϕ2(X2α, zα)].
U2(X2α, zα)=G(X2α, X1, zα)U1(X1)dX1,
G(X2α, X1, zα)=1iλzαexp(ikzα)×expi k2zα(X2α-X1)2.
U2(X2α, zα)=Gˆ(zα)U1(X1),
U1n=ρ1n exp(iϕ1n),
n=1, 2, 3, , N1,
U2mα=ρ2mα exp(iϕ2mα),
m=1, 2, 3, , N2α,
α=1, 2, 3, , Nz,
U2mα=n=1N1Gmn(zα)U1n.
E=α=1Nzm=1N2αρ˜2mα-n=1N1Gmn(zα)ρ1n exp(iϕ1n)2,
Φ1(k+1)=Φ1(k)+τ(k)d(k),k=0, 1, 2, 3, ,
d(k)=-E(Φ1(k))+β(k-1)d(k-1),k=1, 2, 3, , 
β(k-1)=|E(Φ1(k))|2|E(Φ1(k-1))|2,k=1, 2, 3, .
E(Φ1(k+1))<E(Φ1(k))
E(Φ1)ϕ1n=-2 Imρ1n exp(iϕ1n)×α=1Nzm=1N2α1-ρ˜2mα|Smα|Gmn(zα)Smα*,
Smα=k=1N1Gmk(zα)ρ1k exp(iϕ1k),
m=1, 2, 3, , N2α.
G(r2α, r1, zα)=2πiλzαexp(i2πzα/λ)×exp[iπ(r2α2+r12)/λzα]×J02πr2αr1λzαr1,
U2mα=n=1N1Gmn(zα)U1n=2πiλzαexp(i2πzα/λ)exp(iπr2mα2/λzα)×n=1N1ρ1n exp(iϕ1n) 1Δr1nr1nr1n+Δr1n×exp(iπξ2/λzα)J02πr2mαξλzαξdξ,
ρ2mα2=α=18A rectz-z0-(α-1)zT0.5Wz+B,
rect(x)=1,|x|10,otherwise,

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