Abstract

We propose and analyze a complex-amplitude spatial filter consisting of a pair of facing phase conjugators with gains that have the special features of a large dynamic range and a high spatial resolution. We show analytically that the high resolution is maintained even if the imaging lenses inserted have large aberrations. The large-dynamic-range characteristics are evaluated numerically, taking into account the intensity-dependent nonlinear reflectivity of the phase conjugators, showing that the maximum intensity of the input wave allowed decreases with the increasing dynamic range required. Some consideration is also given to the design of the spatial filter.

© 1985 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Cederquist, S. H. Lee, “Coherent optical feedback for the analog solution of partial differential equations,” J. Opt. Soc. Am. 70, 944 (1980).
    [CrossRef]
  2. F. Laeri, T. Tschudi, J. Albers, “Coherent cw image amplifier and oscillator using two-wave interaction in a BaTiO3-crystal,” Opt. Commun. 47, 387 (1983).
    [CrossRef]
  3. R. P. Akins, S. H. Lee, “Two-stage injection-locked ring dye laser/amplifier for coherent image amplification,” J. Opt. Soc. Am. A 1, 533 (1984).
    [CrossRef]
  4. J. Feinberg, R. W. Hellwarth, “Phase-conjugating mirror with continuous-wave gain,” Opt. Lett. 5, 519 (1980).
    [CrossRef] [PubMed]
  5. B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, “Amplified reflection, transmission, and self-oscillation in real-time holography,” Opt. Lett. 6, 519 (1981).
    [CrossRef] [PubMed]
  6. O. Ikeda, T. Suzuki, T. Sato, “Image transmission through a turbulent medium using a point reflector and four-wave mixing in BSO crystal,” Appl. Opt. 22, 2192 (1983).
    [CrossRef] [PubMed]
  7. J. P. Huignard, J. P. Herriau, G. Rivet, “Phase-conjugation and spatial-frequency dependence of wave-front reflectivity in Bi12SiO20crystals,” Opt. Lett. 5, 102 (1980).
    [CrossRef]
  8. J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980).
    [CrossRef]
  9. A. Hardy, Y. Silberberg, “Saturation effects in phase-conjugate lasers,” J. Opt. Soc. Am. 73, 594 (1983).
    [CrossRef]
  10. O. Ikeda, T. Sato, H. Kojima, “Construction of Wiener filter using phase conjugate filter,” J. Opt. Soc. Am. A. (to be published).
  11. O. Ikeda, T. Suzuki, T. Sato, “High-accuracy surface profile measuring system using a BSO phase conjugating mirror,” Appl. Opt. 21, 4468 (1982).
    [CrossRef] [PubMed]

1984

1983

1982

1981

1980

Akins, R. P.

Albers, J.

F. Laeri, T. Tschudi, J. Albers, “Coherent cw image amplifier and oscillator using two-wave interaction in a BaTiO3-crystal,” Opt. Commun. 47, 387 (1983).
[CrossRef]

Cederquist, J.

Cronin-Golomb, M.

Feinberg, J.

J. Feinberg, R. W. Hellwarth, “Phase-conjugating mirror with continuous-wave gain,” Opt. Lett. 5, 519 (1980).
[CrossRef] [PubMed]

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980).
[CrossRef]

Fischer, B.

Hardy, A.

Heiman, D.

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980).
[CrossRef]

Hellwarth, R. W.

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980).
[CrossRef]

J. Feinberg, R. W. Hellwarth, “Phase-conjugating mirror with continuous-wave gain,” Opt. Lett. 5, 519 (1980).
[CrossRef] [PubMed]

Herriau, J. P.

Huignard, J. P.

Ikeda, O.

Kojima, H.

O. Ikeda, T. Sato, H. Kojima, “Construction of Wiener filter using phase conjugate filter,” J. Opt. Soc. Am. A. (to be published).

Laeri, F.

F. Laeri, T. Tschudi, J. Albers, “Coherent cw image amplifier and oscillator using two-wave interaction in a BaTiO3-crystal,” Opt. Commun. 47, 387 (1983).
[CrossRef]

Lee, S. H.

Rivet, G.

Sato, T.

Silberberg, Y.

Suzuki, T.

Tanguay, A. R.

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980).
[CrossRef]

Tschudi, T.

F. Laeri, T. Tschudi, J. Albers, “Coherent cw image amplifier and oscillator using two-wave interaction in a BaTiO3-crystal,” Opt. Commun. 47, 387 (1983).
[CrossRef]

White, J. O.

Yariv, A.

Appl. Opt.

J. Appl. Phys.

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

F. Laeri, T. Tschudi, J. Albers, “Coherent cw image amplifier and oscillator using two-wave interaction in a BaTiO3-crystal,” Opt. Commun. 47, 387 (1983).
[CrossRef]

Opt. Lett.

Other

O. Ikeda, T. Sato, H. Kojima, “Construction of Wiener filter using phase conjugate filter,” J. Opt. Soc. Am. A. (to be published).

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 (7)

Fig. 1
Fig. 1

Schematic of spatial filter using two facing phase conjugators with gains. a, Input wave; b, output wave; M, modulated readout wave. RO, plane readout wave; RF, plane reference wave; BS, beam splitter. Both RO and RF counterpropagate in PCl, and so do RF and M in PC2.

Fig. 2
Fig. 2

Block diagram of the spatial filter. f, Amplitude transmissivity of the attenuation filter; tL1, amplitude transmissivity of L1; tL2, amplitude transmissivity of L2 and BS; r10, amplitude reflectivity of PC1; r20, amplitude reflectivity of PC2 in the absence of modulation (m = 1); t1, amplitude transmissivity of PC1; tB, ratio of amplitude reflectivity to amplitude transmissivity of BS; (·)*, phase-conjugation operation; m, spatial modulation by readout wave in PC2; k, wave number of the optical field.

Fig. 3
Fig. 3

Linearity of the maximum filter gain |h|max to the input |a| for various reflectivities of R0.

Fig. 4
Fig. 4

Proportionality of the output |b|max = |h|max|a| to the input |a| for various reflectivities to R0.

Fig. 5
Fig. 5

Linearity of gain factor G = D/d to the input |a| for various reflectivities of R0.

Fig. 6
Fig. 6

Relation between gain factor G and the maximum allowed amplitude ac of input wave.

Fig. 7
Fig. 7

Four-wave mixing configuration using BaTiO3 crystal. A1, A2, pump waves (reference and readout waves, respectively); A3, input wave; A4, output phase-conjugated wave; E, electric field resulting from external application of voltage V; α, β, angles of A1, A3, respectively, relative to the crystal axis; l, interaction length.

Equations (33)

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

h ( X ) = C 0 f ( X ) 1 t L 2 2 r 10 r 20 m * ( X ) ,
C 0 = t L 1 t L2 t 1 t B exp [ j 4 k ( f 1 + f 2 ) ] .
| h | max = | C 0 | / ( 1 r 10 r 20 t L2 2 ) .
| h | min = | C 0 | / d ( 1 + r 10 r 20 t L2 2 ) .
D = d ( 1 + r 10 r 20 t L 2 2 ) 1 r 10 r 20 t L 2 2 .
a ( X 1 ) = a ( X 1 ) f ( X 1 ) .
b ( X 3 ) = r 10 r 20 t L 2 2 d 2 X 3 b ( X 3 ) m * ( X 3 ) P 1 ( X 3 , X 3 ) + t 1 t B t L 2 d 2 X 1 a ( X 1 ) P 2 ( X 1 , X 3 ) ,
P 1 ( X 3 , X 3 ) = d 2 X 2 h * ( X 3 , X 2 ) exp [ j ( k / 2 f 2 ) | X 2 | 2 jw ( X 2 ) ] d 2 X 1 h * ( X 2 , X 1 ) d 2 X 2 h ( X 1 , ( X 2 ) × exp [ j ( k / 2 f 2 ) | X 2 | 2 + jw ( X 2 ) ] h ( X 2 , X 3 ) ,
P 2 ( X 1 , X 3 ) = d 2 X 2 h ( X 1 , X 2 ) exp [ j ( k / 2 f 2 ) | X 2 | 2 + j w ( X 2 ) ] h X 2 , X 3 ) ,
P 1 ( X 3 , X 3 ) = d 2 X 2 h ( X 2 , X 3 ) h * ( X 2 , X 3 ) ,
b ( X 3 ) = r 10 r 20 t L 2 2 d 2 X 3 b ( X 3 ) m * ( X 3 ) P 1 ( X 3 X 3 ) + t 1 t B t L 2 a ( X 3 ) .
b ( X 3 ) = r 10 r 20 t L 2 2 b ( X 3 ) m * ( X 3 ) + t 1 t B t L 2 a ( X 3 )
b ( X 3 ) = t 1 t B t L 2 1 r 10 r 20 t L 2 2 m * ( X 3 ) a ( X 3 ) .
BW ( P 1 ) > BW ( b m * ) = BW [ a m * / ( 1 q 0 m * ) ] ,
r 2 = r 0 2 1 + ( I 3 / I P ) r 0 2 ,
R 0 = 1 t L2 2 G 1 G + 1 .
R 0 = | sinh ( γ l / 2 ) cosh [ ( γ l / 2 ) + ( ln q / 2 ) ] | 2 ,
γ = j ω exp ( j ψ ) 2 c cos [ ( α β ) / 2 ] r eff E P [ E 2 + E d 2 E 2 + ( E d + E P ) 2 ] 1 / 2 ,
ψ = tan 1 { [ E d ( E d + E P ) + E 2 ] / EE P } .
r eff = { n 0 4 r 13 cos α cos β + 2 n o 2 n e 2 r 42 cos 2 [ ( α + β ) / 2 ] + n e 4 r 33 sin α sin β } sin [ ( α + β ) / 2 ] / n e ,
h ( X , X ) h 0 exp [ j ( k / 4 f 2 ) | X X | 2 ] , h 0 j ( k / 4 π f 2 ) exp ( j 2 k f 2 ) ;
P 1 ( X 3 , X 3 ) = D P 2 | h 0 | 4 exp [ j ( k / 4 f 2 ) ( x 3 2 + y 3 2 x 3 2 y 3 2 ) ] × d x 2 d y 2 sinc [ ( k D P / 4 f 2 ) ( x 2 x 2 ) ] × sinc [ ( k D P / 4 f 2 ) ( y 2 y 2 ) ] exp [ jw ( x 2 , y 2 ) + j ( k / 2 f 2 ) ( x 3 x 2 + y 3 y 2 ) ] d x 2 d y 2 × exp [ jw ( x 2 , y 2 ) j ( k / 2 f 2 ) ( x 3 x 2 + y 3 y 2 ) ] .
F s ( f x , f y ) = ( x , y ) R L 2 d x d y sinc [ ( k D P / 4 f 2 ) x ] × sinc [ ( k D P / 4 f 2 ) y ] exp [ j 2 π ( f x x + f y y ) ] = { ( 4 π f 2 / k D P ) 2 for | f x | , | f y | < k D P / 8 π f 2 , 0 otherwise
F e ( f x , f y ) = ( x , y ) R L 2 d x d y exp [ jw ( x , y ) ] × exp [ j 2 π ( f x x + f y y ) ] , f x = f x + ( k / 4 π f 2 ) x 3 , f y = f y + ( k / 4 π f 2 ) y 3
X c = [ ( k / 4 π f 2 ) x 3 , ( k / 4 π f 2 ) y 3 ] ,
S w ; | f X c | < S w / 2 | F e ( f x , f y ) | 2 d f x d f y / A 0 = η , f = ( f x , f y ) , A ( k / 4 π 2 D
A 0 = | F e ( f x , f y ) | 2 d f x d f y = ( x , y ) R L 2 d x d y
( k / 4 π f 2 ) ( x 3 2 + y 3 2 ) 1 / 2 + S w / 2 < k D P / 8 π f 2
2 ( x 3 2 + y 3 2 ) 1 / 2 + S < D P ,
D I + S < D P .
P 1 ( X 3 , X 3 ) = | h 0 | 2 exp [ j ( k / 4 f 2 ) ( x 3 2 + y 3 2 x 3 2 y 3 2 ) ] × d x 2 d y 2 exp { j ( k / 2 f 2 ) [ ( x 3 x 3 ) x 2 + ( y 3 y 3 ) y 2 ] }
= ( x 2 , y 2 ) R L 2 d 2 X 2 h ( X 2 , X 3 ) h * ( X 2 , X 3 ) .
P 1 ( f ; X 3 ) = P 1 ( X 3 , X 3 X ) exp ( j 2 π f X ) d 2 X = | h 0 | exp ( j π / 2 ) ( x 2 , y 2 ) R L 2 d x 2 d y 2 × exp { j ( k / 4 f 2 ) [ ( x 2 x 3 2 λ f 2 f x ) 2 + ( y 2 y 3 2 λ f 2 f y ) 2 ] } .

Metrics