Abstract

The design of diffractive phase elements (DPE’s) for generating pseudo-nondiffracting beams (PNDB’s) is carried out by employing the conjugate-gradient method. By selecting a trapezoid profile as the preset axial intensity distribution to guide the design and by using an error function with a weighting factor to evaluate the performance of the DPE, we develop a model design and obtain a satisfactory diffractive pattern of a single-segment PNDB. We demonstrate the effectiveness of the conjugate-gradient method in the design of DPE’s that produce multiple-segment PNDB’s in which two consecutive segments are separated by a dark region along the optical axis. We also evaluate the degree of nonuniformity of the axial intensity over a given axial region. The design results match the requirements quite well.

© 1998 Optical Society of America

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References

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  5. S. N. Khonina, V. V. Kotlyar, and 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]
  6. R. Piestun and J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. 19, 771–773 (1994).
    [CrossRef] [PubMed]
  7. R. Piestun, B. Spektor, and J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. Mod. Opt. 43, 1495–1507 (1996).
    [CrossRef]
  8. B. Z. Dong, G. Z. Yang, B. Y. Gu, and O. K. Ersoy, “Iterative optimization approach for designing an axicon with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A 13, 97–103 (1996).
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  11. B. Salik, J. Rosen, and A. Yariv, “One-dimensional beam shaping,” J. Opt. Soc. Am. A 12, 1702–1706 (1995).
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  12. J. Rosen, B. Salik, A. Yariv, and H. K. Liu, “Pseudonondiffracting slitlike beam and its analogy to the pseudonondispersing pulse,” Opt. Lett. 20, 423–425 (1995).
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  13. M. Avriel, Nonlinear Programming: Analysis and Methods (Englewood Cliffs, N.J., 1976), pp. 299–307.
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), pp. 12, 13, 59, 63.
  15. R. Piestun, B. Spektor, and J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A 13, 1837–1848 (1996).
    [CrossRef]

1996 (4)

1995 (3)

1994 (3)

1993 (1)

S. N. Khonina, V. V. Kotlyar, and 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]

1992 (1)

1991 (1)

Bará, S.

Davidson, N.

Dong, B. Z.

Ersoy, O. K.

Friberg, A. T.

Friesem, A. A.

Gu, B. Y.

Hasman, E.

Jaroszewicz, J.

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, and 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.

S. N. Khonina, V. V. Kotlyar, and 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]

Liu, H. K.

Piestun, R.

Rosen, J.

Salik, B.

Shamir, J.

Sochacki, J.

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, and 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]

Spektor, B.

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

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

Yang, G. Z.

Yariv, A.

J. Mod. Opt. (2)

S. N. Khonina, V. V. Kotlyar, and 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]

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

J. Opt. Soc. Am. A (5)

Opt. Lett. (6)

Other (2)

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), pp. 12, 13, 59, 63.

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of a diffractive optical system for producing PNDB’s.

Fig. 2
Fig. 2

Axial intensity distribution of a single-segment PNDB. The incident light is a uniform plane wave. The dots and the solid curve correspond to the calculated and the desired axial intensities (trapezoid profile), respectively. The relevant parameters are λ=0.6328 µm, R1m=4.0 mm, N1=256, and the weighting factor takes the form of Eq. (14).

Fig. 3
Fig. 3

Three-dimensional plot of a single-segment PNDB.

Fig. 4
Fig. 4

Phase distribution of the designed DPE for generating a single-segment PNDB.

Fig. 5
Fig. 5

Axial intensity distribution of a single-segment PNDB with a rectangular profile as the expected axial intensity distribution. The solid and the dashed curves correspond to the calculated and the desired axial intensities, respectively. (a) The weighting factor is chosen according to Eq. (14). (b) The weighting factor is chosen as w(α)=1/Nz. Other parameters are the same as in Fig. 2.

Fig. 6
Fig. 6

Axial intensity distribution of a single-segment PNDB. The incident light is a Gaussian-profile beam. The dots and the solid curve correspond to the calculated and the expected axial intensities, respectively. The relevant parameters are the same as in Fig. 2.

Fig. 7
Fig. 7

Axial intensity distribution of a double-segment PNDB. The dots and the solid curve correspond to the calculated and the desired axial intensities, respectively. The relevant parameters are the same as in Fig. 2.

Fig. 8
Fig. 8

Three-dimensional plot of a double-segment PNDB generated by the designed DPE.

Fig. 9
Fig. 9

(a) Axial intensity distribution of a triple-segment PNDB. (b) Axial intensity distribution of another triple-segment PNDB in which the middle segment is the shortest. The dots and the solid curve correspond to the calculated and the desired axial intensities, respectively. The relevant parameters are the same as in Fig. 2. (c) Phase distribution of the designed DPE for generating the triple-segment PNDB as shown in (a).

Fig. 10
Fig. 10

Three-dimensional plot of the triple-segment PNDB corresponding to Fig. 9(a).

Equations (20)

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U1(r1)=ρ1(r1)exp[iϕ1(r1)].
U2(r2α, zα)=ρ2(r2α, zα)exp[iϕ2(r2α, zα)].
U2(r2α, zα)=G(r2α, r1, zα)U1(r1)dr1,
G(r2α, r1, zα)=2πiλzαexp(i2πzα/λ)×exp[iπ(r2α2+r12)/λzα]×J02πr2αr1λzαr1,
U2(r2α, zα)=Gˆ(zα)U1(r1),
U2mα=n=1N1Gmn(zα)U1n,
U1n=ρ1n exp(iϕ1n),U2mα=ρ2mα exp(iϕ2mα),
n=1,2,3,,N1,m=1,2,3,,N2α,
α=1,2,3,,Nz.
E=α=1NzW(α)m=1N2αρ˜2mα-n=1N1Gmn(zα)ρ1n exp(iϕ1n)2,
Φ1(k+1)=Φ1(k)+τ(k)d(k),fork=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=α=1NzW(α)m=1N2α2(|Smα|-ρ˜2mα)|Smα|ϕ1n=α=1NzW(α)m=1N2α1-ρ˜2mα|Smα|Smα* Smαϕ1n+c.c.=-2 Imρ1n exp(iϕ1n)α=1NzW(α)m=1N2α1-ρ˜2mα|Smα|×Gmn(zα)Smα*,
Smα=k=1N1Gmk(zα)ρ1k exp(iϕ1k).
G(r2α=0, r1, zα)=2πiλzαexp(i2πzα/λ)exp(iπr12/λzα)r1.
w(α)=0.98Np inthesignalregion0.02Nz-Npinotherregions,
nonuniformity=(I2-I2)1/2I,
dz2πz2kR1m2-2πz,

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