Abstract

It is shown that, in general, the size of the image of a point source generated by time-reversed wave-front devices is governed by considerations similar to those determining the resolving power of lenses. If the device subtends an angle α at the image, then the image has a lateral dimension ~λ/sin α. However, in some cases, when α is large and the imaging device is thin or the laser beams have special polarizations, images several times larger might be obtained.

© 1981 Optical Society of America

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References

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  1. D. M. Bloom and G. C. Bjorklund, “Conjugate wave-front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592–594 (1977).
    [Crossref]
  2. A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978); also author’s reply to comments by Ivakin and et al., IEEE J. Quantum Electron. QE-15, 524 (1979), and references therein.
    [Crossref]
  3. M. D. Levenson, “High resolution imaging by wave-front conjugation,” Opt. Lett. 5, 182–184 (1980).
    [Crossref]
  4. R. W. Hellwarth, “Generation of time-reversed wave fronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977).
    [Crossref]
  5. J. H. Marburger and J. F. Lam, “Effect of nonlinear index changes in degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
    [Crossref]
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

1980 (1)

1979 (1)

J. H. Marburger and J. F. Lam, “Effect of nonlinear index changes in degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
[Crossref]

1978 (1)

A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978); also author’s reply to comments by Ivakin and et al., IEEE J. Quantum Electron. QE-15, 524 (1979), and references therein.
[Crossref]

1977 (2)

D. M. Bloom and G. C. Bjorklund, “Conjugate wave-front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592–594 (1977).
[Crossref]

R. W. Hellwarth, “Generation of time-reversed wave fronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977).
[Crossref]

Bjorklund, G. C.

D. M. Bloom and G. C. Bjorklund, “Conjugate wave-front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592–594 (1977).
[Crossref]

Bloom, D. M.

D. M. Bloom and G. C. Bjorklund, “Conjugate wave-front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592–594 (1977).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hellwarth, R. W.

Lam, J. F.

J. H. Marburger and J. F. Lam, “Effect of nonlinear index changes in degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
[Crossref]

Levenson, M. D.

Marburger, J. H.

J. H. Marburger and J. F. Lam, “Effect of nonlinear index changes in degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
[Crossref]

Yariv, A.

A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978); also author’s reply to comments by Ivakin and et al., IEEE J. Quantum Electron. QE-15, 524 (1979), and references therein.
[Crossref]

Appl. Phys. Lett. (2)

D. M. Bloom and G. C. Bjorklund, “Conjugate wave-front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592–594 (1977).
[Crossref]

J. H. Marburger and J. F. Lam, “Effect of nonlinear index changes in degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
[Crossref]

IEEE J. Quantum Electron. (1)

A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978); also author’s reply to comments by Ivakin and et al., IEEE J. Quantum Electron. QE-15, 524 (1979), and references therein.
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Other (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (7)

Fig. 1
Fig. 1

Schematic of a time-reversed wave-front imaging device.

Fig. 2
Fig. 2

Illustration of the definitions of r, r′ and x (see text).

Fig. 3
Fig. 3

Illustration of the definitions of θ, l(θ,ϕ) (see text).

Fig. 4
Fig. 4

Cylindrical device subtending a small angle (see text).

Fig. 5
Fig. 5

Cylindrical device subtending a large angle (see text).

Fig. 6
Fig. 6

A thin cylindrical device subtending a large angle (see text).

Fig. 7
Fig. 7

Sketch of L(z) for the device of Fig. 6 (see text).

Equations (35)

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j ( r , t ) = j ( r , Ω ) e i Ω t d Ω ,
j ( r , Ω ) = j ( 1 ) ( r , Ω ) + j ( 2 ) ( r , Ω ) + j ( 3 ) ( r , Ω ) + ,
j α ( 1 ) ( r , Ω ) = σ α β ( Ω ) E β ( r , Ω ) ,
j α ( 2 ) ( r , Ω ) = δ ( Ω - Ω 1 - Ω 2 ) σ α μ ν ( 2 ) ( Ω 1 , Ω 2 ) × E μ ( r , Ω 1 ) E ν ( r , Ω 2 ) d Ω 1 d Ω 2 ,
j α ( 3 ) ( r , ) = δ ( Ω - Ω 1 - Ω 2 - Ω 3 ) σ α μ ν τ ( 3 ) ( Ω 1 , Ω 2 , Ω 3 ) × E μ ( r , Ω 1 ) E ν ( r , Ω 2 ) E τ ( r , Ω 3 ) d Ω 1 d Ω 2 d Ω 3 ,
σ α β ( Ω ) = σ ( Ω ) δ α β ,
σ α μ ν ( 2 ) = 0 ,
σ α μ ν τ ( 3 ) ( Ω 1 , Ω 2 , Ω 3 ) = σ 3 ( Ω 1 , Ω 2 , Ω 3 ) K α μ ν τ ,
K α μ ν τ = δ α μ δ ν τ + δ α ν δ μ τ + δ α τ δ μ ν .
E = E ( s ) + E ( 1 ) + E ( 2 ) ,
E ( 1 ) ( r , Ω ) = 1 e i p · r δ ( Ω + ω ) + 1 * e - i p · r δ ( Ω - ω ) ,
E ( 2 ) ( r , Ω ) = 2 e - i p · r δ ( Ω + ω ) + 2 * e i p · r δ ( Ω - ω ) ,
E ( 1 ) ( r , Ω 1 ) E ( 2 ) ( r , Ω 2 ) = 1 2 δ ( Ω + ω ) δ ( Ω + ω ) + 1 * 2 * δ ( Ω - ω ) δ ( Ω - ω ) + two other terms ,
E ( s ) ( r , Ω ) = E ( r ) δ ( Ω + ω ) + E * ( r ) δ ( Ω - ω ) .
j ( r , Ω ) = L ¯ · E ( r ) δ ( Ω - ω ) + L ¯ * · E * ( r ) δ ( Ω + ω ) ,
E ( s ) ( r , t ) = 1 r [ E e - i ( k r - ω t ) + E * e i ( k r - ω t ) ] ( I - r ˆ r ˆ ) · e ,
E ( r ) = 1 r E e - i k r ( I - r ˆ r ˆ ) · e .
1 r e i k r ( I - r ˆ r ˆ ) · L ¯ · E ( r ) δ ( Ω - ω ) + 1 r e - i k r ( I - r ˆ r ˆ ) · L ¯ * · E * ( r ) δ ( Ω + ω ) ,
E ( x , Ω ) = V d r 1 r ( I - r ˆ r ˆ ) · [ L ¯ · E ( r ) e i k r δ ( Ω - ω ) + L * · E * ( r ) e - i k r δ ( Ω + ω ) ] ,
E ( x , Ω ) = V d r 1 r r ( I - r ˆ r ˆ ) · [ E e i k ( r - r ) δ ( Ω - ω ) L ¯ + E * e - i k ( r - r ) δ ( Ω + ω ) L ¯ * ] · ( I - r ˆ r ˆ ) · e .
F ( x ) = V d r 1 r r e i k ( r - r )
F ( x ) = V d r r r - x exp [ i k ( r - r - x ) ] .
r - x = r - 1 r x · r + 0 ( x 2 / r 2 ) ,
F ( x ) = V d r 1 r 2 exp [ - i k x · r / r ) .
F ( x ) = V d r sin θ d θ d θ exp ( - i k x cos θ ) ,
F ( x ) = sin θ d θ d ϕ l ( θ , ϕ ) exp ( - i k x cos θ ) .
l ( θ ) = d ϕ l ( θ , ϕ ) ,
F ( x ) = sin θ d θ l ( θ ) exp ( - i k x cos θ ) = - 1 1 d z L ( z ) exp ( - i k x z ) ,
l ( θ ) = L ( cos θ ) .
F ( x ) = - sin α sin α d z L ( z ) exp ( - i k x z ) = L - sin α sin α d z exp ( i k x z ) = L α sin ( k x sin α ) / k x sin α ,
k x sin α = π / 2 ,
x / λ 1 / sin α ,
F ( x ) = cos α 1 d z L ( z ) exp ( - i k x z ) = L α cos α 1 d z exp ( - i k x z ) = L α exp [ 1 / 2 i k x ( 1 + cos α ) ] × sin [ 1 / 2 k x ( 1 - cos α ) ] / [ 1 / 2 k x ( 1 - cos α ) ] ,
x / λ 1 / sin 2 α .
( I - r ˆ r ˆ ) · L ¯ · ( I - r ˆ r ˆ ) · e ( I - r ˆ r ˆ ) · L ¯ · ( I - r ˆ r ˆ ) · e