Abstract

We propose two designs with which one can obtain a circular light disk with both uniform irradiance and phase, namely, a homodisk on a plane target from a TEM0,0 Gaussian beam. The first design is pure-phase filtering masks based on wave theory and computer simulation. The second design, an aspherical lens, satisfies the two requirements successively. We used geometrical optics to obtain uniform irradiance with the aspherical lens. The theoretical results of both designs are reasonably good.

© 1998 Optical Society of America

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References

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  1. V. V. Kotlyar, I. V. Nikolsky, V. A. Soifer, “Adaptive iterative algorithm for focusators synthesis,” Optik (Stuttgart) 88, 17–19 (1991).
  2. J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, S. Bara, “Nonparaxial design of generalized axicons,” Appl. Opt. 31, 5326–5330 (1992).
    [CrossRef] [PubMed]
  3. J. Sochacki, Z. Jaroszewicz, L. R. Staronski, A. Kolodziejczyk, “Annular-aperture logarithmic axicon,” J. Opt. Soc. Am. A 10, 1765–1768 (1993).
    [CrossRef]
  4. J. Rosen, A. Yariv, “Synthesis of an arbitrary axial field profile by computer-generated holograms,” Opt. Lett. 19, 843–845 (1994).
    [CrossRef] [PubMed]
  5. G. Zhang, B. Dong, G. Yang, B. Gu, “Design of diffractive phase axicon illuminated by a Gaussian profile beam,” Acta Phys. Sin. (Overseas Ed.) 5, 354–364 (1996).
    [CrossRef]
  6. G. Zhang, B. Dong, G. Yang, “A new design of diffractive phase elements for generating multi-focal annuli,” Acta Phys. Sin. 45, 1647–1654 (1996).
  7. Y. Kato, K. Kima, “Random phase-shifting of laser beam for absorption profile smoothing and instability suppression in laser produced plasmas,” Appl. Phys. B 29, 186–187 (1982); Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
    [CrossRef]
  8. R. H. Lehmberg, S. P. Obenschain, “Use of induced spatial incoherence for uniform illumination of laser fusion targets,” Opt. Commun. 46, 27–31 (1983).
    [CrossRef]
  9. X. Deng, X. Liang, Z. Chen, W. Yu, R. Ma, “Uniform illumination of large targets using a lens array,” Appl. Opt. 25, 377–381 (1986).
    [CrossRef] [PubMed]
  10. B. Dong, B. Gu, G. Yang, “Effective algorithm for the reconstruction of real images from Hartley-transform modulus only: simulation calculations,” Optik (Stuttgart) 90, 107–116 (1992); G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitude-phase retrieval in any linear transform system and its applications,” Int. J. Mod. Phys. B 7, 3151–3224 (1993).
    [CrossRef]
  11. See, for example, P. W. Milonni, J. H. Eberly , Lasers (Wiley, New York, 1991).
  12. See, for example, R. A. Monzingo, T. W. Miller , Introduction to Adaptive Arrays (Wiley, New York, 1980).
  13. See, for example, H. F. Harmuth , Sequence Theory, Foundations and Applications (Academic, Boston, 1997).
  14. M. De, L. N. Hazra, “Real-time image restoration through Walsh filtering,” Opt. Acta 24, 211–220 (1977).
    [CrossRef]
  15. W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), pp. 410–415.
  16. See, for example, D. G. Feitelson , Optical Computing (MIT, Cambridge, Mass., 1988).
  17. L. Cheng, Optics, Principles and Development (Science Press, Beijing, China, 1990), p. 115.

1996 (2)

G. Zhang, B. Dong, G. Yang, B. Gu, “Design of diffractive phase axicon illuminated by a Gaussian profile beam,” Acta Phys. Sin. (Overseas Ed.) 5, 354–364 (1996).
[CrossRef]

G. Zhang, B. Dong, G. Yang, “A new design of diffractive phase elements for generating multi-focal annuli,” Acta Phys. Sin. 45, 1647–1654 (1996).

1994 (1)

1993 (1)

1992 (2)

J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, S. Bara, “Nonparaxial design of generalized axicons,” Appl. Opt. 31, 5326–5330 (1992).
[CrossRef] [PubMed]

B. Dong, B. Gu, G. Yang, “Effective algorithm for the reconstruction of real images from Hartley-transform modulus only: simulation calculations,” Optik (Stuttgart) 90, 107–116 (1992); G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitude-phase retrieval in any linear transform system and its applications,” Int. J. Mod. Phys. B 7, 3151–3224 (1993).
[CrossRef]

1991 (1)

V. V. Kotlyar, I. V. Nikolsky, V. A. Soifer, “Adaptive iterative algorithm for focusators synthesis,” Optik (Stuttgart) 88, 17–19 (1991).

1986 (1)

1983 (1)

R. H. Lehmberg, S. P. Obenschain, “Use of induced spatial incoherence for uniform illumination of laser fusion targets,” Opt. Commun. 46, 27–31 (1983).
[CrossRef]

1982 (1)

Y. Kato, K. Kima, “Random phase-shifting of laser beam for absorption profile smoothing and instability suppression in laser produced plasmas,” Appl. Phys. B 29, 186–187 (1982); Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

1977 (1)

M. De, L. N. Hazra, “Real-time image restoration through Walsh filtering,” Opt. Acta 24, 211–220 (1977).
[CrossRef]

Bara, S.

Chen, Z.

Cheng, L.

L. Cheng, Optics, Principles and Development (Science Press, Beijing, China, 1990), p. 115.

De, M.

M. De, L. N. Hazra, “Real-time image restoration through Walsh filtering,” Opt. Acta 24, 211–220 (1977).
[CrossRef]

Deng, X.

Dong, B.

G. Zhang, B. Dong, G. Yang, “A new design of diffractive phase elements for generating multi-focal annuli,” Acta Phys. Sin. 45, 1647–1654 (1996).

G. Zhang, B. Dong, G. Yang, B. Gu, “Design of diffractive phase axicon illuminated by a Gaussian profile beam,” Acta Phys. Sin. (Overseas Ed.) 5, 354–364 (1996).
[CrossRef]

B. Dong, B. Gu, G. Yang, “Effective algorithm for the reconstruction of real images from Hartley-transform modulus only: simulation calculations,” Optik (Stuttgart) 90, 107–116 (1992); G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitude-phase retrieval in any linear transform system and its applications,” Int. J. Mod. Phys. B 7, 3151–3224 (1993).
[CrossRef]

Eberly, J. H.

See, for example, P. W. Milonni, J. H. Eberly , Lasers (Wiley, New York, 1991).

Feitelson, D. G.

See, for example, D. G. Feitelson , Optical Computing (MIT, Cambridge, Mass., 1988).

Gu, B.

G. Zhang, B. Dong, G. Yang, B. Gu, “Design of diffractive phase axicon illuminated by a Gaussian profile beam,” Acta Phys. Sin. (Overseas Ed.) 5, 354–364 (1996).
[CrossRef]

B. Dong, B. Gu, G. Yang, “Effective algorithm for the reconstruction of real images from Hartley-transform modulus only: simulation calculations,” Optik (Stuttgart) 90, 107–116 (1992); G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitude-phase retrieval in any linear transform system and its applications,” Int. J. Mod. Phys. B 7, 3151–3224 (1993).
[CrossRef]

Harmuth, H. F.

See, for example, H. F. Harmuth , Sequence Theory, Foundations and Applications (Academic, Boston, 1997).

Hazra, L. N.

M. De, L. N. Hazra, “Real-time image restoration through Walsh filtering,” Opt. Acta 24, 211–220 (1977).
[CrossRef]

Jaroszewicz, Z.

Kato, Y.

Y. Kato, K. Kima, “Random phase-shifting of laser beam for absorption profile smoothing and instability suppression in laser produced plasmas,” Appl. Phys. B 29, 186–187 (1982); Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Kima, K.

Y. Kato, K. Kima, “Random phase-shifting of laser beam for absorption profile smoothing and instability suppression in laser produced plasmas,” Appl. Phys. B 29, 186–187 (1982); Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Kolodziejczyk, A.

Kotlyar, V. V.

V. V. Kotlyar, I. V. Nikolsky, V. A. Soifer, “Adaptive iterative algorithm for focusators synthesis,” Optik (Stuttgart) 88, 17–19 (1991).

Lehmberg, R. H.

R. H. Lehmberg, S. P. Obenschain, “Use of induced spatial incoherence for uniform illumination of laser fusion targets,” Opt. Commun. 46, 27–31 (1983).
[CrossRef]

Liang, X.

Ma, R.

Miller, T. W.

See, for example, R. A. Monzingo, T. W. Miller , Introduction to Adaptive Arrays (Wiley, New York, 1980).

Milonni, P. W.

See, for example, P. W. Milonni, J. H. Eberly , Lasers (Wiley, New York, 1991).

Monzingo, R. A.

See, for example, R. A. Monzingo, T. W. Miller , Introduction to Adaptive Arrays (Wiley, New York, 1980).

Nikolsky, I. V.

V. V. Kotlyar, I. V. Nikolsky, V. A. Soifer, “Adaptive iterative algorithm for focusators synthesis,” Optik (Stuttgart) 88, 17–19 (1991).

Obenschain, S. P.

R. H. Lehmberg, S. P. Obenschain, “Use of induced spatial incoherence for uniform illumination of laser fusion targets,” Opt. Commun. 46, 27–31 (1983).
[CrossRef]

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), pp. 410–415.

Rosen, J.

Sochacki, J.

Soifer, V. A.

V. V. Kotlyar, I. V. Nikolsky, V. A. Soifer, “Adaptive iterative algorithm for focusators synthesis,” Optik (Stuttgart) 88, 17–19 (1991).

Staronski, L. R.

Yang, G.

G. Zhang, B. Dong, G. Yang, B. Gu, “Design of diffractive phase axicon illuminated by a Gaussian profile beam,” Acta Phys. Sin. (Overseas Ed.) 5, 354–364 (1996).
[CrossRef]

G. Zhang, B. Dong, G. Yang, “A new design of diffractive phase elements for generating multi-focal annuli,” Acta Phys. Sin. 45, 1647–1654 (1996).

B. Dong, B. Gu, G. Yang, “Effective algorithm for the reconstruction of real images from Hartley-transform modulus only: simulation calculations,” Optik (Stuttgart) 90, 107–116 (1992); G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitude-phase retrieval in any linear transform system and its applications,” Int. J. Mod. Phys. B 7, 3151–3224 (1993).
[CrossRef]

Yariv, A.

Yu, W.

Zhang, G.

G. Zhang, B. Dong, G. Yang, B. Gu, “Design of diffractive phase axicon illuminated by a Gaussian profile beam,” Acta Phys. Sin. (Overseas Ed.) 5, 354–364 (1996).
[CrossRef]

G. Zhang, B. Dong, G. Yang, “A new design of diffractive phase elements for generating multi-focal annuli,” Acta Phys. Sin. 45, 1647–1654 (1996).

Acta Phys. Sin. (1)

G. Zhang, B. Dong, G. Yang, “A new design of diffractive phase elements for generating multi-focal annuli,” Acta Phys. Sin. 45, 1647–1654 (1996).

Acta Phys. Sin. (Overseas Ed.) (1)

G. Zhang, B. Dong, G. Yang, B. Gu, “Design of diffractive phase axicon illuminated by a Gaussian profile beam,” Acta Phys. Sin. (Overseas Ed.) 5, 354–364 (1996).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

Y. Kato, K. Kima, “Random phase-shifting of laser beam for absorption profile smoothing and instability suppression in laser produced plasmas,” Appl. Phys. B 29, 186–187 (1982); Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

M. De, L. N. Hazra, “Real-time image restoration through Walsh filtering,” Opt. Acta 24, 211–220 (1977).
[CrossRef]

Opt. Commun. (1)

R. H. Lehmberg, S. P. Obenschain, “Use of induced spatial incoherence for uniform illumination of laser fusion targets,” Opt. Commun. 46, 27–31 (1983).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (2)

V. V. Kotlyar, I. V. Nikolsky, V. A. Soifer, “Adaptive iterative algorithm for focusators synthesis,” Optik (Stuttgart) 88, 17–19 (1991).

B. Dong, B. Gu, G. Yang, “Effective algorithm for the reconstruction of real images from Hartley-transform modulus only: simulation calculations,” Optik (Stuttgart) 90, 107–116 (1992); G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitude-phase retrieval in any linear transform system and its applications,” Int. J. Mod. Phys. B 7, 3151–3224 (1993).
[CrossRef]

Other (6)

See, for example, P. W. Milonni, J. H. Eberly , Lasers (Wiley, New York, 1991).

See, for example, R. A. Monzingo, T. W. Miller , Introduction to Adaptive Arrays (Wiley, New York, 1980).

See, for example, H. F. Harmuth , Sequence Theory, Foundations and Applications (Academic, Boston, 1997).

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), pp. 410–415.

See, for example, D. G. Feitelson , Optical Computing (MIT, Cambridge, Mass., 1988).

L. Cheng, Optics, Principles and Development (Science Press, Beijing, China, 1990), p. 115.

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Figures (5)

Fig. 1
Fig. 1

Schematic illustration of a filter H that converts a Gaussian profile beam into a homodisk D of radius ρ d .

Fig. 2
Fig. 2

Achieved (a) intensity and (b) phase for λ = 0.6328 μm, w 0 = 1 mm, L = 100 mm, and ρ d = 10 mm for an N = 512 phase filtering mask.

Fig. 3
Fig. 3

Ray-tracing geometry for the design of the aspherical lens (AL).

Fig. 4
Fig. 4

Illustration of the AL for λ = 1.06 μm, w 0 = 5 mm, L = 200 mm, and ℑ = 25I 0. K = 21 and r 1, r 2, … , r 21 are coordinates of the positions of coating rings.

Fig. 5
Fig. 5

Illustration of the AL for λ = 1.06 μm, w 0 = 2 mm, L = 100 mm, and ℑ = I 0 when K = 30.

Tables (3)

Tables Icon

Table 1 Values of φ in the Phase Transparency in Units of 2π/32

Tables Icon

Table 2 Standard Deviations of Intensity and Phase Distributions

Tables Icon

Table 3 Sequential Values of r for the Phase Correction Rings

Equations (25)

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E r = E 0 exp [ - r / w 0 2 ] ,
t n = exp i φ n , n = 1 ,   2 , ,   N .
n - 1 N   R ,   n N   R .
E 1 ( r ) = E ( r ) t n as n - 1 N   R < r < n N   R .
E 2 Q m = E 2 , m = 0 R   r d r   exp [ - ( r / w 0 ) 2 ] exp ( i φ ) 2   0 π d θ × exp   i 2 π λ ( r   cos   θ - ρ m ) 2 + ( r   sin   θ ) 2 + L 2 ,
E 2 , m E 2 , m * = I m for m = 1 ,   2 , ,   M ,
I m = 1 0 for for m = 1 ,   2 , ,   M m = M + 1 ,   M + 2 , ,   M .
m = 1 M ( ϕ m - ϕ ) 2 = 0 ,
ϕ m = arg E 2 , m ,       ϕ = 1 M m = 1 M   ϕ m .
Ψ ( φ 1 ,   φ 2 , ,   φ N ) = m = 1 M ( E 2 , m E 2 , m * - I m ) 2 + m = 1 M ( ϕ m - ϕ ) 2 .
Φ = 𝒲 · P
Ψ ( p 1 ,   p 2 , ,   p N ) = m = 1 M ( E 2 , m E 2 , m * - I m ) 2 + m = 1 M ( ϕ m - ϕ ) 2 ,
F = I ( r ) 2 π r d r ,
I ( r ) = I 0 exp ( - 2 r 2 / w 0 2 ) .
ρ d ρ = C   exp ( - 2 r 2 / w 0 2 ) r d r ,
ρ = C [ 1 - exp ( - 2 r 2 / w 0 2 ) ] 1 / 2 ,
d z / d r = tan   β .
n   sin   β = sin α + β = sin   α   cos   β + cos   α   sin   β ,
sin   α = sin   β [ n   cos   β ± ( 1 - n 2 sin 2   β ) 1 / 2 ] ,
L - z tan   α = ρ - r ,
tan   α = 1 L - z { C [ 1 - exp ( - 2 r 2 / w 0 2 ) - r ] 1 / 2 } .
tan   α 1 + tan 2   α 1 / 2 = sin   β [ n   cos   β - ( 1 - n 2 sin 2   β ) 1 / 2 ] ,
z ( r ) = 0 r tan   β ( r ) d r ,
p ( r ) = p ( ABE ) = n [ d 0 + z ( r ) ] + [ L - z ( r ) ] cos [ α ( r ) ] ,
d d r   p ( r ) Δ r = λ 16 .

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