Abstract

It has been demonstrated previously that a thin phase hologram, recorded in a weak reference condition, is capable of inverting a complex field. Using a computer simulation of the properties of a thin phase hologram, we find the operating conditions and dynamic range for wave-front inversion. The conclusions of the simulation are used for designing an experiment to invert a circulant matrix and the results of the experiment well support the analysis.

© 1984 Optical Society of America

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References

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  1. J. W. Goodman, “Coherent Optical Image Deblurring,” in Coherent Optical Engineering, F. T. Arecchi, V. Degiorgio, Eds. (North-Holland, Amsterdam, 1977), pp. 263–280.
  2. G. W. Stroke, R. G. Zech, “A Posteriori Image-Correcting ‘Deconvolution’ by Holographic Fourier-Transform Division,” Phys. Lett. A 25, 89 (1967).
    [CrossRef]
  3. A. Lohmann, H. W. Werlich, “Holographic Production of Spatial Filters for Code Translation and Image Restoration,” Phys. Lett. A 25, 570 (1967).
    [CrossRef]
  4. C. Zetzsche, “Simplified Realization of the Holographic Inverse Filter: A New Method,” Appl. Opt. 21, 1077 (1982).
    [CrossRef] [PubMed]
  5. S. I. Ragnarsson, “A New Holographic Method of Generating a High Efficiency, Extended-Range Spatial Filter with Application to Restoration of Defocused Images,” Phys. Scr. 2, 145 (1970).
    [CrossRef]
  6. D. Tichenor, “Extended Range Image Deblurring Filters,” Ph.D. Thesis, Dept. of Electrical Engineering, Stanford U. (1974).
  7. J. W. Goodman, “Architectural Development of Optical Data Procession Systems,” Proc. Inst. Radio Electron. Eng. Aust. 2, 139 (1982).
  8. Q. Cao, J. W. Goodman, “Coherent Optical Techniques for Diagonalization and Inversion of Circulant Matrices and Circulant Approximations to Toeplitz Matrices,” Appl. Opt. 23, 803 (1984).
    [CrossRef] [PubMed]
  9. C. E. K. Mees, The Theory of the Photographic Process (Macmillan, New York, 1954).
  10. “Kodak Plates and Films for Science and Industry,” Kodak Data Book (Eastman Kodak Co., Rochester, N.Y., 1962).
  11. R. L. van Renesee, F. A. J. Bouts, “Efficiency of Bleaching Agents for Holography,” Optik 38, 156 (1973).
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).
  13. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 55.
  14. A. Kozma, “Photographic Recording of Modulated Light,” J. Opt. Soc. Am. 56, 428 (1966).
    [CrossRef]
  15. Q. Cao, “Coherent Optical Techniques for Computing Eigenvalues and Inverse of Circulant Matrices,” Ph.D Thesis, Dept. of Electrical Engineering, Stanford U. (1984).

1984 (1)

1982 (2)

C. Zetzsche, “Simplified Realization of the Holographic Inverse Filter: A New Method,” Appl. Opt. 21, 1077 (1982).
[CrossRef] [PubMed]

J. W. Goodman, “Architectural Development of Optical Data Procession Systems,” Proc. Inst. Radio Electron. Eng. Aust. 2, 139 (1982).

1973 (1)

R. L. van Renesee, F. A. J. Bouts, “Efficiency of Bleaching Agents for Holography,” Optik 38, 156 (1973).

1970 (1)

S. I. Ragnarsson, “A New Holographic Method of Generating a High Efficiency, Extended-Range Spatial Filter with Application to Restoration of Defocused Images,” Phys. Scr. 2, 145 (1970).
[CrossRef]

1967 (2)

G. W. Stroke, R. G. Zech, “A Posteriori Image-Correcting ‘Deconvolution’ by Holographic Fourier-Transform Division,” Phys. Lett. A 25, 89 (1967).
[CrossRef]

A. Lohmann, H. W. Werlich, “Holographic Production of Spatial Filters for Code Translation and Image Restoration,” Phys. Lett. A 25, 570 (1967).
[CrossRef]

1966 (1)

Bouts, F. A. J.

R. L. van Renesee, F. A. J. Bouts, “Efficiency of Bleaching Agents for Holography,” Optik 38, 156 (1973).

Burckhardt, C. B.

J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 55.

Cao, Q.

Q. Cao, J. W. Goodman, “Coherent Optical Techniques for Diagonalization and Inversion of Circulant Matrices and Circulant Approximations to Toeplitz Matrices,” Appl. Opt. 23, 803 (1984).
[CrossRef] [PubMed]

Q. Cao, “Coherent Optical Techniques for Computing Eigenvalues and Inverse of Circulant Matrices,” Ph.D Thesis, Dept. of Electrical Engineering, Stanford U. (1984).

Collier, J.

J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 55.

Goodman, J. W.

Q. Cao, J. W. Goodman, “Coherent Optical Techniques for Diagonalization and Inversion of Circulant Matrices and Circulant Approximations to Toeplitz Matrices,” Appl. Opt. 23, 803 (1984).
[CrossRef] [PubMed]

J. W. Goodman, “Architectural Development of Optical Data Procession Systems,” Proc. Inst. Radio Electron. Eng. Aust. 2, 139 (1982).

J. W. Goodman, “Coherent Optical Image Deblurring,” in Coherent Optical Engineering, F. T. Arecchi, V. Degiorgio, Eds. (North-Holland, Amsterdam, 1977), pp. 263–280.

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

Kozma, A.

Lin, L. H.

J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 55.

Lohmann, A.

A. Lohmann, H. W. Werlich, “Holographic Production of Spatial Filters for Code Translation and Image Restoration,” Phys. Lett. A 25, 570 (1967).
[CrossRef]

Mees, C. E. K.

C. E. K. Mees, The Theory of the Photographic Process (Macmillan, New York, 1954).

Ragnarsson, S. I.

S. I. Ragnarsson, “A New Holographic Method of Generating a High Efficiency, Extended-Range Spatial Filter with Application to Restoration of Defocused Images,” Phys. Scr. 2, 145 (1970).
[CrossRef]

Stroke, G. W.

G. W. Stroke, R. G. Zech, “A Posteriori Image-Correcting ‘Deconvolution’ by Holographic Fourier-Transform Division,” Phys. Lett. A 25, 89 (1967).
[CrossRef]

Tichenor, D.

D. Tichenor, “Extended Range Image Deblurring Filters,” Ph.D. Thesis, Dept. of Electrical Engineering, Stanford U. (1974).

van Renesee, R. L.

R. L. van Renesee, F. A. J. Bouts, “Efficiency of Bleaching Agents for Holography,” Optik 38, 156 (1973).

Werlich, H. W.

A. Lohmann, H. W. Werlich, “Holographic Production of Spatial Filters for Code Translation and Image Restoration,” Phys. Lett. A 25, 570 (1967).
[CrossRef]

Zech, R. G.

G. W. Stroke, R. G. Zech, “A Posteriori Image-Correcting ‘Deconvolution’ by Holographic Fourier-Transform Division,” Phys. Lett. A 25, 89 (1967).
[CrossRef]

Zetzsche, C.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Optik (1)

R. L. van Renesee, F. A. J. Bouts, “Efficiency of Bleaching Agents for Holography,” Optik 38, 156 (1973).

Phys. Lett. A (2)

G. W. Stroke, R. G. Zech, “A Posteriori Image-Correcting ‘Deconvolution’ by Holographic Fourier-Transform Division,” Phys. Lett. A 25, 89 (1967).
[CrossRef]

A. Lohmann, H. W. Werlich, “Holographic Production of Spatial Filters for Code Translation and Image Restoration,” Phys. Lett. A 25, 570 (1967).
[CrossRef]

Phys. Scr. (1)

S. I. Ragnarsson, “A New Holographic Method of Generating a High Efficiency, Extended-Range Spatial Filter with Application to Restoration of Defocused Images,” Phys. Scr. 2, 145 (1970).
[CrossRef]

Proc. Inst. Radio Electron. Eng. Aust. (1)

J. W. Goodman, “Architectural Development of Optical Data Procession Systems,” Proc. Inst. Radio Electron. Eng. Aust. 2, 139 (1982).

Other (7)

C. E. K. Mees, The Theory of the Photographic Process (Macmillan, New York, 1954).

“Kodak Plates and Films for Science and Industry,” Kodak Data Book (Eastman Kodak Co., Rochester, N.Y., 1962).

J. W. Goodman, “Coherent Optical Image Deblurring,” in Coherent Optical Engineering, F. T. Arecchi, V. Degiorgio, Eds. (North-Holland, Amsterdam, 1977), pp. 263–280.

D. Tichenor, “Extended Range Image Deblurring Filters,” Ph.D. Thesis, Dept. of Electrical Engineering, Stanford U. (1974).

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

J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 55.

Q. Cao, “Coherent Optical Techniques for Computing Eigenvalues and Inverse of Circulant Matrices,” Ph.D Thesis, Dept. of Electrical Engineering, Stanford U. (1984).

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Figures (12)

Fig. 1
Fig. 1

Typical H-D curve.

Fig. 2
Fig. 2

Typical modulation function of film.

Fig. 3
Fig. 3

Hologram formed by two plane waves.

Fig. 4
Fig. 4

eff1 vs E0 for Er > Eo: (a) (Er = 500 ergs/cm2, C = 1 rad/den); (b) (Er = 600 ergs/cm2, C = 1 rad/den); (c) (Er = 700 ergs/cm2, C = 1 rad/den).

Fig. 5
Fig. 5

eff1 vs Eo for ErEo: (a) (Er = 1 erg/cm2, C = 1 rad/den); (b) (Er = 10 ergs/cm2, C = 1 rad/den); (c) (Er = 100 ergs/cm2, C = 1 rad/den); (d) (Er = 1.3 ergs/cm2, C = 1 rad/den).

Fig. 6
Fig. 6

eff1 vs Eo for ErEo and smaller C: (a) (Er = 1 erg/cm2, C = 0.5 rad/den); (b) (Er = 1 erg/cm2, C = 0.1 rad/den); (c) (Er = 2 erg/cm2, C = 0.1 rad/den).

Fig. 7
Fig. 7

Input matrix pattern.

Fig. 8
Fig. 8

Generalized optical Fourier transform geometry: G ( x , y ) = - d 1 λ d 2 Δ · exp { j K 2 [ 1 d 2 ( 1 - l d 1 Δ d 2 ) ] ( x 3 2 + y 3 2 ) } · [ g ( x 1 , y 1 ) ] v x = d 1 / λ d 2 Δ x 3 , v y = d 1 / λ d 2 Δ y 3 .

Fig. 9
Fig. 9

Optical system for recording Λ−1.

Fig. 10
Fig. 10

Results of Eigenvalues: (a) eigenvalue pattern with a sinc window; (b) measurement of eigenvalues.

Fig. 11
Fig. 11

Results of inverse eigenvalues: (a) inverse eigenvalue pattern; (b) measurement of inverted eigenvalues.

Fig. 12
Fig. 12

Results of the inverse matrix: (a) inverse matrix pattern; (b) profiles of six elements of the inverse matrix pattern.

Tables (7)

Tables Icon

Table I Program Flow Chart

Tables Icon

Table II Output Data from Simulation: (a) Er > Eo, C = 1 rad/den, Er = 500 ergs/cm2; (b) Er > Eo, C = 1 rad/den, Er = 600 ergs/cm2; (c) Er > Eo, C = 1 rad/den, Er = 700 ergs/cm2

Tables Icon

Table III Output Data from Simulation: (a) ErEo, C = 1 rad/den, Er = 1 ergs/cm2; (b) ErEo, C = 1 rad/den, Er = 10 ergs/cm2; ErEo, C = 1 rad/den, Er = 100 ergs/cm2; (d) ErE0, C = 1 rad/den, Er = 1.3 ergs/cm2

Tables Icon

Table IV Output Data from Simulation: (a) ErEo, C = 0.5 rad/den, Er = 1 erg/cm2; (b) ErEo, C = 0.1 rad/den, Er = 1 erg/cm2; (c) ErEo, C = 0.1 rad/den, Er = 2 ergs/cm2

Tables Icon

Table V Efficiency of Two Types of Inversion

Tables Icon

Table VI Effect of Beam Ratios on Types of Inversion

Tables Icon

Table VII Holographic Process

Equations (24)

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C = [ 1 2 3 4 4 1 2 3 3 4 1 2 2 3 4 1 ] .
D = log ( 1 τ ) ,
E = I T ,
D = D 0 + γ log E ,
Δ ϕ Δ D = C ,
E = I · T = { O 2 + R 2 + O R * exp [ j ( ϕ o = ϕ r ) ] + O * R exp [ - j ( ϕ o - ϕ r ) ] } · T ,
E = E o r [ 1 + v cos ( ϕ o - ϕ r ) ] ,
ν = 2 E o E r E o + E r
Δ ϕ = C γ log E .
Δ ϕ = C γ log { E o r [ 1 + v cos ( ϕ o - ϕ r ) ] } .
t = exp ( j Δ ϕ ) = exp ( j C γ log E o r · exp { j C γ log [ 1 + v cos ( ϕ o - ϕ r ) ] } .
v ~ 2 E r E o 1 , log [ 1 + v cos ( ϕ o - ϕ r ) ] ~ v cos ( ϕ o - ϕ r ) .
t exp ( j C γ log E o r ) · exp { [ j C γ v cos ( ϕ o - ϕ r ) ] } exp ( j C γ log E o r ) · k = - ( j ) k J k ( C γ v ) exp [ j k ( ϕ o - ϕ r ) ] ,
exp ( j M cos α ) = k = - J k ( M ) exp ( j k α ) ( j ) k
J 1 ( C γ v ) ~ C γ v ,
t - 1 exp j C γ log E o r · C γ v exp ( - j ϕ o ) exp ( j ϕ d ) O R O 2 + R 2 exp ( - j ϕ o ) ,
t - 1 { exp ( j ϕ d ) 1 O exp ( - j ϕ o ) O 0 , 0 O = 0.
log E = log E o r + log ( 1 + v cos ω x ) ,
ϕ = ϕ ( E o r ) + Δ ϕ ( v , ω x ) ,
t ~ exp [ j ϕ ( E o r ) ] · exp [ j Δ ϕ ( v , ω x ) ] ,
C = [ 1 0 1 1 1 0 0 1 1 ] .
Λ = [ 2 0 0 0 1 + j 3 2 0 , 0 0 1 - j 3 2 ] C - 1 = 1 2 1 1 - 1 - 1 1 1 1 - 1 1 ,
s 2 λ Δ 1 ,
G ( x , y ) = - d 1 λ d 2 Δ · exp { j K 2 [ 1 d 2 ( 1 - l d 1 Δ d 2 ) ] ( x 3 2 + y 3 2 ) } · [ g ( x 1 , y 1 ) ] v x = d 1 / λ d 2 Δ x 3 , v y = d 1 / λ d 2 Δ y 3 .

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