Abstract

The characteristics of the one-way image transmission system presented in Part 1 are investigated in detail [ Appl. Opt. 22, 2192 ( 1983)]. First, a general expression of the expectation of the transmitted image is derived for turbulence that may be typical in image transmission in the horizontal direction. Then, with the help of numerical examples, the image quality is discussed in terms of the point spread function for both thin layer and uniformly distributed turbulence. It is shown that the image transmission system is effective especially where turbulence exists relatively close to the transmission plane.

© 1983 Optical Society of America

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  1. O. Ikeda, T. Suzuki, T. Sato, Appl. Opt., 22, 2192 (1983).
    [CrossRef] [PubMed]
  2. B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, Appl. Phys. Lett. 41, 141 (1982).
    [CrossRef]
  3. J. W. Goodman, D. W. Jackson, M. Lehmann, J. Knotts, Appl. Opt. 8, 1581 (1969).
    [CrossRef] [PubMed]
  4. V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971), p. 36.

1983

1982

B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, Appl. Phys. Lett. 41, 141 (1982).
[CrossRef]

1969

Cronin-Golomb, M.

B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, Appl. Phys. Lett. 41, 141 (1982).
[CrossRef]

Fischer, B.

B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, Appl. Phys. Lett. 41, 141 (1982).
[CrossRef]

Goodman, J. W.

Ikeda, O.

Jackson, D. W.

Knotts, J.

Lehmann, M.

Sato, T.

Suzuki, T.

Tatarski, V. I.

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971), p. 36.

White, J. O.

B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, Appl. Phys. Lett. 41, 141 (1982).
[CrossRef]

Yariv, A.

B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, Appl. Phys. Lett. 41, 141 (1982).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, Appl. Phys. Lett. 41, 141 (1982).
[CrossRef]

Other

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971), p. 36.

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Figures (11)

Fig. 1
Fig. 1

Schematic of the image transmission system using a pointlike reflector and four-wave mixing (Ref.1): BS, beam splitter; M, mirror; BSO, Bi12SiO20 crystal as medium for four-wave mixing.

Fig. 2
Fig. 2

Coordinates of the image transmission system: F(α), original image on the input image plane α = (α,β); f(x), Fourier-transformed input image on the transmission plane x = (x,y) which represents the four-wave mixing region; g(r), transmitted image on the observation plane r = (ξ,η), where rR = (ξRR) is the position of the reflector.

Fig. 3
Fig. 3

Integration paths of the ten auto- and cross-correlation terms in a(x, x′, r, rR) in Eq. (3′): (a), (b), … correspond to the first, second, … terms, respectively, in the exponential function in Eq. (3′).

Fig. 4
Fig. 4

Examples of PSFs for thin layer turbulence. The parameters used are zf = 200 mm, z0 = 1000 mm, D = 10 mm, λ = 0.5 μm (k = 2π/λ), l0 = 0.5 mm, L = 2.5 mm, zt = 125 mm, σc1 = π [Eq. (18)]. The top shows the profile of the peak intensity I[−(z0/zf)α0], α0= (α0,0), and the bottom shows the intensity distribution patterns I(r), where normalization in image intensity is made for each case of the position α0. The minimum level displayed is about one-tenth of the maximum value of each pattern.

Fig. 5
Fig. 5

Transition of the peak intensity I[− (z0/zf)α0] with a change of position of the turbulent layer zt/z0. The parameters used are the same as in Fig. 4 except that α0 is fixed at (0.075,0) mm and zt/z0 is varied from zero to unity.

Fig. 6
Fig. 6

Transition of the intensity distribution pattern with a change of position of the turbulent layer zt/z0, where the parameters used are the same as in Fig. 5. The patterns from (a) through (e) are normalized in intensity but not for the 3-D representations for the cases zt/z0 = 0.4 and 0.9.

Fig. 7
Fig. 7

Profiles of a [xn,−(z0/zf)α0] in Eq. (20) with respect to xn = (0,yn) for various values of zt/z0, where α0 = (0.075,0) mm and the other parameters are the same as in Fig. 5.

Fig. 8
Fig. 8

Profiles of a [xn,−(z0/zf)α0] in Eq. (20) with respect to xn = (xn,0) for various values of zt/z0, where α0 = (0.075,0) mm and the other parameters are the same as in Fig. 5.

Fig. 9
Fig. 9

Example of transmitted image for a letter input image X. The displayed area is 1 mm ×1 mm and the reflector is positioned 0.8 mm down from the center of the transmitted letter image X. The parameters used are the same as in Fig. 5 except for σc1 = 3π. (Degradation of the transmitted image is not significant within the displayed area for σc1 = π).

Fig. 10
Fig. 10

Example of PSFs for uniformly distributed turbulence. The top shows the profile of the peak intensity I [−(z0/zf)α0], where α0 = (α0,0), and the bottom shows the intensity distribution patterns I(r), where image-intensity normalization is made for each case of the position α0. The parameters used are z0 = 1000 mm, zf = 200 mm, D = 10 mm, λ = 0.5 μm, σc2 = π [Eq. (24)], L → ∞.

Fig. 11
Fig. 11

Integration paths between transmission and observation planes for the seventh term in the exponential function in Eq. (3′).

Equations (44)

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g ( r ) = α R I F ( α ) d α 2 x R P C × d x 2 exp { j [ ( k / z f ) α + ( k / z 0 ) ( r r R ) ] x } × exp [ j x r ψ ( p 12 ) d p 12 j x r R ψ ( p 11 ) d p 11 ] + f 0 x R P C d x 2 exp [ j ( k / z 0 ) ( r r R ) x ] × exp [ j x r ψ ( p 12 ) d p 12 j x r R ψ ( p 11 ) d p 11 ] ,
E { | g ( r ) | 2 } = α , α R I d α 2 d α 2 F ( α ) F * ( α ) × x , x R P C d x 2 d x 2 a ( x , x , r , r R ) × exp [ j ( k / z f ) ( α x α x ) + j ( k / z 0 ) ( r r R ) ( x x ) ] + | f 0 | 2 x , x R P C d x 2 d x 2 a ( x , x , x , x R ) × exp [ j ( k / z 0 ) ( r r R ) ( x x ) ] + 2 Re { f 0 * α R I d α 2 R ( α ) × x , x R P C d x 2 d x 2 a ( x , x , r , r R ) × exp [ j ( k / z f ) α x + j ( k / z 0 ) ( r r R ) ( x x ) ] } ,
a ( x , x , r , r R ) = E { exp [ j x r ψ ( p 12 ) d p 12 j x r R ψ ( p 11 ) d p 11 j x r ψ ( p 22 ) d p 22 + j x r R ψ ( p 21 ) d p 21 ] }
= exp ( 1 2 E { [ x r ψ ( p 12 ) d p 12 x r R ψ ( p 11 ) d p 11 x r ψ ( p 22 ) d p 22 + x r R ψ ( p 21 ) d p 21 ] 2 } ) = exp [ 1 2 E { ψ ( p 12 ) ψ ( p 12 ) } d p 12 d p 12 1 2 E { ψ ( p 11 ) ψ ( p 11 ) } d p 11 d p 11 1 2 E { ψ ( p 22 ) ψ ( p 22 ) } d p 22 d p 22 1 2 E { ψ ( p 21 ) ψ ( p 21 ) } d p 21 d p 21 + E { ψ ( p 12 ) ψ ( p 11 ) } d p 12 d p 11 + E { ψ ( p 12 ) ψ ( p 22 ) } d p 12 d p 22 E { ψ ( p 12 ) ψ ( p 21 ) } d p 12 d p 21 E { ψ ( p 11 ) ψ ( p 22 ) } d p 11 d p 22 + E { ψ ( p 11 ) ψ ( p 21 ) } d p 11 d p 21 + E { ψ ( p 22 ) ψ ( p 21 ) } d p 22 d p 21 ] .
E { ψ ( p ) ψ ( p ) } = σ 2 ( p + p 2 ) χ ( p p ) ,
σ 2 ( p + p 2 ) = σ 0 2 exp [ ( z + z 2 z t ) 2 / 2 L 2 ] ,
χ ( p p ) = exp ( | p p | 2 / 2 l 0 2 ) ,
Γ ( x , x , r , r R ) x r x r R E { ψ ( p 12 ) } ψ ( p 21 ) } d p 12 d p 21 = ( 2 π ) 1 / 2 l 0 σ 0 2 exp [ | x x | 2 2 l 0 2 ] × 0 z 0 exp [ ( z z t ) 2 2 L 2 | r r R | 2 + | x x | 2 2 ( r r R ) ( x x ) 2 l 0 2 z 0 2 z 2 ( r r R ) ( x x ) | x x | 2 l 0 2 z 0 z ] d z .
a ( x , x , r , r R ) = exp { 1 2 [ Γ ( x , x , r , r ) + Γ ( x , x , r R , r R ) + Γ ( x , x , r , r ) + Γ ( x , x , r R , r R ) ] + Γ ( x , x , r , r R ) + Γ ( x , x , r , r ) Γ ( x , x , r , r R ) Γ ( x , x , r R , r ) + Γ ( x , x , r R , r R ) + Γ ( x , x , r , r R ) } = exp [ 2 ( 2 π ) 1 / 2 l 0 σ 0 2 0 z 0 exp [ ( z z t ) 2 2 L 2 ] × ( 1 exp [ 1 2 | r r R l 0 | 2 ( z z 0 ) 2 ] exp [ 1 2 | x x l 0 | 2 ( 1 z z 0 ) 2 ] + 1 2 + , exp { 1 2 [ ( z z 0 ) ( r r R l 0 ) ± ( 1 z z 0 ) ( x x l 0 ) ] 2 } ) d z ] ] .
x p = x , x n = x x , y p = y , y n = y y , ξ = ξ ξ R , η = η η R ,
a ( x x , r r R ) = a ( x n , y n , ξ , η ) ,
E { | g ( ξ , η ) | 2 } ( 3 ) = 2 Re { [ f 0 * D 4 ( 2 π ) 2 ] ( α , β ) R I d α d β F ( α , β ) A ( ω ξ , ω η , ξ , η ) sinc ( D ω ξ 2 ) sinc [ ( D 2 ) ( ω ξ + α k z f ) ] × sinc ( D ω η 2 ) sinc [ ( D 2 ) ( ω η + β k z f ) ] } ,
A ( ω ξ , ω η , ξ , η ) = a ( x x , y n , ξ , η ) × exp ( j ω ξ x n + j ω η y n ) d x n d y n .
F ( α , β ) = 0 for ( α 2 + β 2 ) 1 / 2 λ z f / D ,
x r ψ ( p 12 ) d p 12 x r R ψ ( p 11 ) d p 11 for | r r R | λ z 0 / D .
F ( α ) = δ ( α α 0 )
x p = ( x + x ) / 2 , y p = ( y + y ) / 2 , x n = x x , y n = y y ,
I ( r ) = D D D D d x n d y n ( D | x n | ) ( D | y n | ) a ( x n , r r R ) × exp { [ j ( k / z f ) α 0 + j ( k / z 0 ) ( r r R ) ] x n } ,
σ c 1 2 = 2 π l 0 L σ 0 2
a ( x n , r r R ) = exp [ 2 σ c 1 2 ( 1 exp [ 1 2 | r r R l 0 | 2 ( z t z 0 ) 2 ] exp [ 1 2 | x n l 0 | 2 ( 1 z t z 0 ) 2 ] + + , 1 2 exp { 1 2 [ ( z t z 0 ) ( r r R l 0 ) ± ( 1 z t z 0 ) ( x n l 0 ) ] 2 } ) ] ,
a [ x n , ( z 0 / z f ) α 0 ] = exp [ 2 σ c 1 2 ( 1 exp [ 1 2 | α 0 l 0 | 2 ( z t z f ) 2 ] exp [ 1 2 | x n l 0 | 2 ( 1 z t z f ) 2 ] + + , 1 2 exp { 1 2 [ ( z t z f ) ( α 0 l 0 ) ± ( 1 z t z 0 ) ( x n l 0 ) ] 2 } ) ] .
a [ x n , ( z 0 / z f ) α 0 ] exp [ σ c 1 2 4 l 0 4 ( z t z f ) 2 ( 1 z t z 0 ) 2 × ( | x n | 2 | α 0 | 2 + | x n α 0 | 2 ) ] ,
a [ x n , ( z 0 / z f ) α 0 ] = exp ( 2 σ c 1 2 { 1 exp [ 1 2 | α 0 l 0 | 2 ( z t z f ) 2 ] } × { 1 exp [ 1 2 | x n l 0 | 2 ( 1 z t z 0 ) 2 ] } ) .
a [ x n , ( z 0 / z f ) α 0 ] = exp [ 2 σ c 2 2 z 0 0 z 0 d z ( 1 exp [ 1 2 ( z z f ) 2 | α 0 l 0 | 2 ] exp [ 1 2 ( 1 z z 0 ) 2 | x n l 0 | 2 ] + + , 1 2 exp { 1 2 [ ( z z f ) ( α 0 l 0 ) ± ( 1 z z 0 ) ( x n l 0 ) ] 2 } ) ] ,
σ c 2 2 = ( 2 π ) 1 / 2 l 0 z 0 σ 0 2 .
a [ x n , ( z 0 / z f ) α 0 ] exp [ ( σ c 2 2 / 60 l 0 4 ) ( z 0 / z f ) 2 ( | α 0 | 2 | x n | 2 + 2 | α 0 x n | 2 ) ] .
Γ ( x , x , r , r R ) x r d p 12 x r R d p 12 E { ψ ( p 12 ) ψ ( p 21 ) } = σ 0 2 exp [ ( z 2 + z 1 2 z t ) 2 2 L 2 | p 21 p 12 | 2 2 l 0 2 ] d p 12 d p 21 ,
p 21 = ( x 1 , y 1 , z 1 ) , p 12 = ( x 2 , y 2 , z 2 ) .
x 1 = a 1 z 1 + b 1 , x 2 = a 2 z 2 + b 2 , y 1 = c 1 z 1 + d 1 , y 2 = c 2 z 2 + d 2 ,
x = ( x , y ) = ( b 2 , d 2 ) , x = ( x , y ) = ( b 1 , d 1 ) , r R = ( ξ R , η R ) = ( a 1 z 0 + b 1 , c 1 z 0 + d 1 ) , r = ( ξ , η ) = ( a 2 z 0 + b 2 , c 2 z 0 + d 2 ) .
a 1 = ( ξ R x ) / z 0 , a 2 = ( ξ x ) / z 0 , b 1 = x , b 2 = x , c 1 = ( η R y ) / z 0 , c 2 = ( η y ) / z 0 , d 1 = y , d 2 = y .
Γ ( x , x , r , r R ) = σ 0 2 0 z 0 0 z 0 exp [ ( z 2 + z 1 2 z t ) 2 2 L 2 ( z 2 z 1 ) 2 + ( a 2 z 2 + b 2 a 1 z 1 b 1 ) 2 + ( c 2 z 2 + d 2 c 1 z 1 d 1 ) 2 2 l 0 2 ] d z 2 d z 1 cos θ 1 cos θ 2 ,
| a 1 | , | a 2 | , | c 1 | , | c 2 | 1
z p = ( z 2 + z 1 ) / 2 , z n = z 2 z 1 ,
Γ ( x , x , r , r R ) = σ 0 2 exp { [ ( a 2 + a 1 ) ( d 2 d 1 ) ( c 2 + c 1 ) ( b 2 b 1 ) ] 2 + 4 [ ( b 2 b 1 ) 2 + ( d 2 d 1 ) 2 ] 8 l 0 2 × 0 z 0 d z p exp ( ( z p z t ) 2 2 L 2 [ ( a 2 a 1 ) 2 + ( c 2 c 1 ) 2 + ( a 2 c 1 a 1 c 2 ) 2 ] z p 2 2 l 0 2 { 2 ( a 2 a 1 ) ( b 2 b 1 ) + 2 ( c 2 c 1 ) ( d 2 d 1 ) ( a 2 c 1 a 1 c 2 ) [ ( a 2 + a 1 ) ( d 2 d 1 ) ( b 2 b 1 ) ( c 2 + c 1 ) ] } z p 2 l 0 2 ) × w ( z p ) w ( z p ) d z n exp ( { z n + [ ( a 2 + a 1 ) ( b 2 b 1 ) + ( c 2 + c 1 ) ( d 2 d 1 ) + ( a 2 2 a 1 2 + c 2 2 c 1 2 ) z p ] / 2 } 2 2 l 0 2 ) ,
w ( z p ) = z 0 | 2 z p z 0 | , 0 < z p < z 0 = 0 , otherwise .
z 0 l 0
w ( z p ) w ( z p ) d z n exp { } = ( 2 π ) 1 / 2 l 0 .
θ S = x r ψ ( p ) d p .
var { θ s } = E { ψ ( p ) ψ ( p ) } d p d p .
var { θ s } = ( 2 π ) 1 / 2 l 0 σ 0 2 0 z 0 exp [ ( z z t ) 2 / 2 L 2 ] d z .
( 1 ) var { θ s } = ( 2 π ) 1 / 2 l 0 σ 0 2 z 0 for L ,
( 2 ) var { θ s } = 2 π l 0 L σ 0 2 for L z 0 and L z t z 0 L ,
( 3 ) var { θ s } = π l 0 L σ 0 2 for L z 0 and z t 0 or L .

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