Abstract

A design method is presented for computing the phase functions of an energy efficient system using two holographic elements for converting a Gaussian beam into a uniform beam with rectangular support in the far field of the source. The method is based on a modification of the Gerchberg-Saxton algorithm which includes an x-y separability constraint on the phase of one of the holographic elements. A beamforming system was fabricated using this method, and experimental results were obtained which support the design approach.

© 1989 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Ogland, “Mirror System for Uniform Beam Transformation in High-Power Annular Lasers,” Appl. Opt. 17, 2917–2923 (1978).
    [CrossRef] [PubMed]
  2. P. W. Rhodes, D. L. Shealy, “Refractive Optical Systems for Irradiance Redistribution of Collimated Radiation: Their Design and Analysis,” Appl. Opt. 19, 3545–3553 (1980).
    [CrossRef] [PubMed]
  3. D. Shafer, “Gaussian to Flat-Top Intensity Distributing Lens,” Opt. Laser Technol. 3, 159–160 (June1982).
    [CrossRef]
  4. J. Cederquist, A. M. Tai, “Computer-Generated Holograms for Geometric Transformation,” Appl. Opt. 23, 3099–3104 (1984).
    [CrossRef] [PubMed]
  5. C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic Conversion of a Gaussian Beam to a Uniform Beam,” Proc. Soc. Photo-Opt. Instrum. Eng. 883, 220–229 (1988).
  6. W. B. Veldkamp, “Laser Beam Profile Shaping with Interlaced Binary Diffraction Gratings,” Appl. Opt. 21, 3209–3212 (1982).
    [CrossRef] [PubMed]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 57.
  8. O. Bryngdahl, “Optical Map Transformations,” Opt. Commun. 10, 164–168 (1974).
    [CrossRef]
  9. O. Bryngdahl, “Geometrical Transformations in Optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
    [CrossRef]
  10. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 749.
  11. C.-Y. Han, Y. Ishii, K. Murata, “Reshaping Collimated Laser Beams with Gaussian Profile to Uniform Profiles,” Appl. Opt. 22, 3644–3647 (1983).
    [CrossRef] [PubMed]
  12. J. R. Fienup, “Reconstruction and Synthesis Applications of an Iterative Algorithm,” Proc. Soc. Photo. Opt. Instrum. Eng. 373, 147–160 (1981).
  13. R. W. Gerchberg, W. O. Saxton, “A Practical Algorithm for Determination of Phase from Image and Diffraction Plane Pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  14. W.-H. Lee, “Method for Converting a Gaussian Laser Beam into a Uniform Beam,” Opt. Commun. 36, 469–471 (1981).
    [CrossRef]
  15. H. O. Bartelt, “Applications of the Tandem Component: an Element with Optimum Light Efficiency,” Appl. Opt. 24, 3811–3816 (1985).
    [CrossRef] [PubMed]
  16. N. C. Gallagher, B. Liu, “Method for Computing Kinoforms that Reduces Image Reconstruction Error,” Appl. Opt. 12, 2328–2335 (1973).
    [CrossRef] [PubMed]
  17. J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297–305 (1980).
    [CrossRef]
  18. F. Wyrowski, O. Bryngdahl, “Iterative Fourier-Transform Algorithm Applied to Computer Holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
    [CrossRef]
  19. R. C. Fairchild, J. R. Fienup, “Computer Originated Aspheric Holographic Optical Elements,” Opt. Eng. 21, 133–140 (1982).
    [CrossRef]
  20. D. K. Angell, “Improved Diffraction Efficiency of Silver Halide (Sensitized) Gelatin,” Appl. Opt. 26, 4692–4702 (1987).
    [CrossRef] [PubMed]
  21. B. J. Chang, C. D. Leonard, “Dichromated Gelatin for the Fabrication of Holographic Optical Elements,” Appl. Opt. 18, 2407–2417 (1979).
    [CrossRef] [PubMed]

1988

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic Conversion of a Gaussian Beam to a Uniform Beam,” Proc. Soc. Photo-Opt. Instrum. Eng. 883, 220–229 (1988).

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-Transform Algorithm Applied to Computer Holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
[CrossRef]

1987

1985

1984

1983

1982

R. C. Fairchild, J. R. Fienup, “Computer Originated Aspheric Holographic Optical Elements,” Opt. Eng. 21, 133–140 (1982).
[CrossRef]

D. Shafer, “Gaussian to Flat-Top Intensity Distributing Lens,” Opt. Laser Technol. 3, 159–160 (June1982).
[CrossRef]

W. B. Veldkamp, “Laser Beam Profile Shaping with Interlaced Binary Diffraction Gratings,” Appl. Opt. 21, 3209–3212 (1982).
[CrossRef] [PubMed]

1981

W.-H. Lee, “Method for Converting a Gaussian Laser Beam into a Uniform Beam,” Opt. Commun. 36, 469–471 (1981).
[CrossRef]

J. R. Fienup, “Reconstruction and Synthesis Applications of an Iterative Algorithm,” Proc. Soc. Photo. Opt. Instrum. Eng. 373, 147–160 (1981).

1980

J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

P. W. Rhodes, D. L. Shealy, “Refractive Optical Systems for Irradiance Redistribution of Collimated Radiation: Their Design and Analysis,” Appl. Opt. 19, 3545–3553 (1980).
[CrossRef] [PubMed]

1979

1978

1974

O. Bryngdahl, “Optical Map Transformations,” Opt. Commun. 10, 164–168 (1974).
[CrossRef]

O. Bryngdahl, “Geometrical Transformations in Optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
[CrossRef]

1973

1972

R. W. Gerchberg, W. O. Saxton, “A Practical Algorithm for Determination of Phase from Image and Diffraction Plane Pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Aleksoff, C. C.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic Conversion of a Gaussian Beam to a Uniform Beam,” Proc. Soc. Photo-Opt. Instrum. Eng. 883, 220–229 (1988).

Angell, D. K.

Bartelt, H. O.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 749.

Bryngdahl, O.

Cederquist, J.

Chang, B. J.

Ellis, K. K.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic Conversion of a Gaussian Beam to a Uniform Beam,” Proc. Soc. Photo-Opt. Instrum. Eng. 883, 220–229 (1988).

Fairchild, R. C.

R. C. Fairchild, J. R. Fienup, “Computer Originated Aspheric Holographic Optical Elements,” Opt. Eng. 21, 133–140 (1982).
[CrossRef]

Fienup, J. R.

R. C. Fairchild, J. R. Fienup, “Computer Originated Aspheric Holographic Optical Elements,” Opt. Eng. 21, 133–140 (1982).
[CrossRef]

J. R. Fienup, “Reconstruction and Synthesis Applications of an Iterative Algorithm,” Proc. Soc. Photo. Opt. Instrum. Eng. 373, 147–160 (1981).

J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

Gallagher, N. C.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A Practical Algorithm for Determination of Phase from Image and Diffraction Plane Pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 57.

Han, C.-Y.

Ishii, Y.

Lee, W.-H.

W.-H. Lee, “Method for Converting a Gaussian Laser Beam into a Uniform Beam,” Opt. Commun. 36, 469–471 (1981).
[CrossRef]

Leonard, C. D.

Liu, B.

Murata, K.

Neagle, B. D.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic Conversion of a Gaussian Beam to a Uniform Beam,” Proc. Soc. Photo-Opt. Instrum. Eng. 883, 220–229 (1988).

Ogland, J. W.

Rhodes, P. W.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A Practical Algorithm for Determination of Phase from Image and Diffraction Plane Pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Shafer, D.

D. Shafer, “Gaussian to Flat-Top Intensity Distributing Lens,” Opt. Laser Technol. 3, 159–160 (June1982).
[CrossRef]

Shealy, D. L.

Tai, A. M.

Veldkamp, W. B.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 749.

Wyrowski, F.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

O. Bryngdahl, “Optical Map Transformations,” Opt. Commun. 10, 164–168 (1974).
[CrossRef]

W.-H. Lee, “Method for Converting a Gaussian Laser Beam into a Uniform Beam,” Opt. Commun. 36, 469–471 (1981).
[CrossRef]

Opt. Eng.

R. C. Fairchild, J. R. Fienup, “Computer Originated Aspheric Holographic Optical Elements,” Opt. Eng. 21, 133–140 (1982).
[CrossRef]

J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

Opt. Laser Technol.

D. Shafer, “Gaussian to Flat-Top Intensity Distributing Lens,” Opt. Laser Technol. 3, 159–160 (June1982).
[CrossRef]

Optik (Stuttgart)

R. W. Gerchberg, W. O. Saxton, “A Practical Algorithm for Determination of Phase from Image and Diffraction Plane Pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Proc. Soc. Photo-Opt. Instrum. Eng.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic Conversion of a Gaussian Beam to a Uniform Beam,” Proc. Soc. Photo-Opt. Instrum. Eng. 883, 220–229 (1988).

Proc. Soc. Photo. Opt. Instrum. Eng.

J. R. Fienup, “Reconstruction and Synthesis Applications of an Iterative Algorithm,” Proc. Soc. Photo. Opt. Instrum. Eng. 373, 147–160 (1981).

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 749.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 57.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1
Fig. 1

Holographic beamformer. Holographic element H1 redistributes the energy of the incident beam into the desired near field intensity distribution. Holographic element H2 places the correct phase on the beam. On propagation to the far field, a uniform intensity and phase results.

Fig. 2
Fig. 2

Beam profiles: (a) truncated Gaussian profile; (b) desired near field modulus; (c) desired far field modulus.

Fig. 3
Fig. 3

Equivalent systems for holographic redistribution of energy. In (a), the complex amplitude distributions are related by a Fourier transform. If the holographic element is placed in contact with the lens, as in (b), the intensity distribution at the output plane is unchanged. In (c), the quadratic phase of the lens is added to the holographic element.

Fig. 4
Fig. 4

Determination of the mapping function through conservation of energy. The energy bounded by 0 ≤ x′ ≤ x in (a) must equal that bounded by 0 ≤ u′ ≤ u in (b).

Fig. 5
Fig. 5

Simulated results for the beamformer designed by the stationary phase method: (a) near field modulus; (b) far field modulus.

Fig. 6
Fig. 6

Gerchberg-Saxton algorithm.

Fig. 7
Fig. 7

Near field modulus obtained using the Gerchberg-Saxton algorithm with a random initial phase estimate.

Fig. 8
Fig. 8

Synthesis results using the Gerchberg-Saxton algorithm with the initial phase estimate from the stationary phase method: (a) near field modulus; (b) far field modulus.

Fig. 9
Fig. 9

Hybrid iterative algorithm. The separability constraint is applied after each set of Gerchberg-Saxton iterations.

Fig. 10
Fig. 10

Synthesis results for the beamformer designed using the hybrid algorithm: (a) near field modulus; (b) far field modulus.

Fig. 11
Fig. 11

Experimental results for CGH computed by the stationary phase method: (a) intensity distribution at a plane 375 mm from the CGH; (b) line scan through (a).

Fig. 12
Fig. 12

Beamformer experimental results: (a) intensity distribution at a plane 375 mm from the CGH; (b) same result from the COHOE; (c) near field intensity distribution directly after H2; (d) far field intensity distribution; (e)–(h) line scans through the distributions in (a)–(d), respectively.

Fig. 13
Fig. 13

COHOE fabrication. The desired first diffraction order of the CGH is imaged onto the COHOE plane and interfered with a reference beam.

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

A N ( x , y ) = sinc ( 6 x / x 0 , 6 y / y 0 ) ( 1 | x | / x 0 ) ( 1 | y | / y 0 ) ,
a 2 ( u , υ ) = i λ f a 1 ( x , y ) exp [ i ϕ ( x , y ) i 2 π λ f ( x u + y υ ) ] d x d y ,
a 2 ( u , υ ) = exp [ i π λ f ( u 2 + υ 2 ) ] a 2 ( u , υ ) .
ϕ ( x , y ) = ϕ ( x , y ) π λ f ( x 2 + y 2 ) .
( x y ) ( u υ ) = [ ( λ f / 2 π ) ϕ ( x , y ) / x ( λ f / 2 π ) ϕ ( x , y ) / y ] .
( x y ) ( u υ ) = [ g ( x , y ) h ( x , y ) ] ,
ϕ x = 2 π λ f g ( x , y ) , ϕ y = 2 π λ f h ( x , y ) .
ϕ ( x , y ) = 2 π λ f [ 0 x g ( x ) d x + 0 y h ( y ) d y ] π λ f ( x 2 + y 2 ) .
I g ( x ) = { I 0 exp ( x 2 / 2 α 2 ) | x | x 0 , 0 | x | > x 0 ,
I N ( u ) = { I N 0 sinc 2 ( 6 u / u 0 ) ( 1 | u | / u 0 ) 2 | u | u 0 , 0 | u | > u 0 .
I 0 0 x exp ( x 2 / 2 α 2 ) d x = I N 0 0 u sinc 2 ( 6 u / u 0 ) × ( 1 | u | / u 0 ) 2 d u ,
I g ( x ) d x = I N ( u ) d u .
I N 0 = I 0 x 0 x 0 exp ( x 2 / 2 α 2 ) d x u 0 u 0 sinc 2 ( 6 u / u 0 ) ( 1 | u | / u 0 ) 2 d u ,
I G ( x ) = { I 0 exp ( x 2 / 2 α 2 ) | x | x 0 , 0 | x | > x 0 ,
I N ( x ) = { I N 0 sinc 2 ( N x / x 0 ) ( 1 | x | / x 0 ) 2 | x | x 0 , 0 | x | > x 0 ,
I N 0 = I 0 .
I G ( x ) I N ( x ) for | x | x 0 .
I 0 exp [ ( N 1 / 2 N ) 2 x 0 2 2 α 2 ] = I 0 sinc 2 [ ( N 1 / 2 ) ] × [ 1 ( N 1 / 2 N ) ] 2 .
α = { ( N 1 / 2 N ) 2 x 0 2 2 ln [ 4 N 2 ( N 1 / 2 ) 2 π 2 ] } 1 / 2 .
η = x 0 x 0 I 0 sinc 2 ( N x / x 0 ) ( 1 | x | / x 0 ) 2 d x x 0 x 0 I 0 exp ( x 2 / 2 α 2 ) d x ,

Metrics