Abstract

In this paper we extend the calculation of the propagation of the mutual power spectrum, Γˆ(x1,x2,ν), for nonlinear dielectrics to the case of optically lossy media. In particular, in the quasi-monochromatic limit, the dependence of the mutual coherence function of the second harmonic wave on the full fourth-order coherence of the incoming radiant flux is discussed. An experiment utilizing this phenomenon is suggested.

© 1967 Optical Society of America

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References

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  1. N. Bloembergen, Nonlinear Optics (W. A. Benjamin, Inc., New York, N. Y., 1965).
  2. M. J. Beran and J. B. DeVelis, J. Opt. Soc. Am. 57, 186 (1967).
    [CrossRef]
  3. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, N. Y., 1964).
  4. M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs. New Jersey, 1964).
  5. W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964).
    [CrossRef]
  6. J. Ward (private communication).
  7. M. Hercher, Appl. Phys. Letters 7, 39 (1965).
    [CrossRef]

1967 (1)

1965 (1)

M. Hercher, Appl. Phys. Letters 7, 39 (1965).
[CrossRef]

1964 (1)

W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964).
[CrossRef]

Beran, M. J.

M. J. Beran and J. B. DeVelis, J. Opt. Soc. Am. 57, 186 (1967).
[CrossRef]

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs. New Jersey, 1964).

Bloembergen, N.

N. Bloembergen, Nonlinear Optics (W. A. Benjamin, Inc., New York, N. Y., 1965).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, N. Y., 1964).

DeVelis, J. B.

Hercher, M.

M. Hercher, Appl. Phys. Letters 7, 39 (1965).
[CrossRef]

Martienssen, W.

W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964).
[CrossRef]

Parrent, G. B.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs. New Jersey, 1964).

Spiller, E.

W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964).
[CrossRef]

Ward, J.

J. Ward (private communication).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, N. Y., 1964).

Am. J. Phys. (1)

W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964).
[CrossRef]

Appl. Phys. Letters (1)

M. Hercher, Appl. Phys. Letters 7, 39 (1965).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (4)

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, N. Y., 1964).

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs. New Jersey, 1964).

J. Ward (private communication).

N. Bloembergen, Nonlinear Optics (W. A. Benjamin, Inc., New York, N. Y., 1965).

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Figures (1)

Fig. 1
Fig. 1

Experiment to measure the second-harmonic intensity as a function of the temporal coherence of the incoming radiation. L is a single-mode high power laser; A is a pinhole aperture; G is the rotating ground glass; L1 is a collimating lens; S is a beam splitter; L2 and L3 are focusing lenses; C is the nonlinear crystal; F is a filter to pass only 2ν; D1 and D2 are photodetectors; and M1 and M2 are oscilloscope monitors.

Equations (40)

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2 V ˆ r ( x , ν ) + k 2 V ˆ r ( x , ν ) = - 4 π c ( μ / ) 1 2 k i σ V ˆ r ( x , ν ) ,
V ˆ r ( x , v ) = - V ( x , t ) e 2 π i ν t d t ,
J ˆ ( x , ν ) = σ L ( ν ) V ˆ r ( x , ν ) + ν 1 + ν 2 = ν σ N L ( ν 1 , ν 2 , ν 1 + ν 2 = ν ) × V ˆ r ( x , ν 1 ) V ˆ r ( x , ν 2 ) ,
2 V ˆ r ( x , ν ) + [ k 2 + 4 π k c ( μ / ) 1 2 i σ L ( ν ) ] V ˆ r ( x , ν ) = 4 π k c ( μ / ) 1 2 i ν 1 + ν 2 = ν σ N L ( ν 1 , ν 2 ) V ˆ r ( x , ν 2 ) V ˆ r ( x , ν 2 ) .
V ( x , t ) = V r ( x , t ) + i [ V r ( x , t ) ] ,
V ( x , t ) = 0 V ˆ ( x , ν ) e - 2 π i ν t d ν
V ˆ ( x , ν ) = - V ( x , t ) e 2 π i ν t d t ν > 0 = 0 ν < 0.
V ˆ ( x , ν ) = 2 V r ( x , ν ) ν > 0 = 0 ν < 0 ,
V ˆ * ( x , ν ) = 0 ν > 0 = 2 V ˆ r ( x , ν ) ν < 0 ;
V ˆ ( x , - ν ) = V ˆ * ( x , ν ) ,
2 V ˆ ( x , ν ) + K D 2 ( ν ) V ˆ ( x , ν ) = - 4 π k c ( μ / ) 1 2 i ( 1 / 2 ) × { ν 1 + ν 2 = ν ν 1 , ν 2 > 0 σ N L ( ν 1 , ν 2 ) V ˆ ( x , ν 1 ) V ˆ ( x , ν 2 ) + ν 1 + ν 2 = ν ν 1 > 0 ν 2 < 0 σ N L ( ν 1 , ν 2 ) V ˆ ( x , ν 1 ) V ˆ * ( x , ν 2 ) + ν 1 + ν 2 = ν ν 1 < 0 ν 2 > 0 σ N L ( ν 1 , ν 2 ) V ˆ * ( x , ν 1 ) V ˆ ( x , ν 2 ) + ν 1 + ν 2 = ν ν 1 < 0 ν 2 < 0 σ N L ( ν 1 , ν 2 ) V ˆ * ( x , ν 1 ) V ˆ * ( x , ν 2 ) } ,
K D 2 ( ν ) = k 2 + ( 4 π k / c ) ( μ / ) 1 2 i σ L ( ν ) .
V T r ( x , t ) = V r ( x , t ) t T = 0 t > T ,
[ 1 2 + K D 2 ( ν ) ] [ 2 2 + K D 2 * ( ν ) ] Γ ˆ ( x 1 , x 2 , ν ) = ( 4 π 2 k 2 / c 2 ) μ / ν 1 + ν 2 = ν ν 3 + ν 4 = - ν σ N L ( ν 1 , ν 2 ) σ N L * ( ν 1 , ν 2 ) 2 Re × i = 1 7 L ˆ i ( x 1 , x 1 , x 2 , x 2 , ν 1 , ν 2 , ν 3 , ν 4 ) ,
Γ ˆ ( x 1 , x 2 , ν ) = lim T 1 2 T V ˆ T ( x 1 , ν ) V ˆ T * ( x 2 , ν ) ,
Γ ˆ ( x 1 , x 2 , ν ) = Γ ˆ 0 ( x 1 , x 2 , ν ) + Γ ˆ 1 ( x 1 , x 2 , ν ) ,
j = 1 4 [ j 2 + K D 2 ( ν ) ] L ˆ i ( x 1 , x 2 , x 3 , x 4 , ν 1 , ν 2 , ν 3 , ν 4 ) = 0
[ 1 2 + K D 2 ( ν ) ] [ 2 2 + K D * 2 ( ν ) ] Γ ˆ 1 ( x 1 , x 2 , ν ) = ( 8 π 2 k 2 / c 2 ) μ / ν 1 + ν 2 = ν ν 3 + ν 4 = - ν σ N L ( ν 1 , ν 2 ) σ N L * ( ν 3 , ν 4 ) × Re i = 1 7 L ˆ i ( x 1 , x 1 , x 2 , x 2 , ν 1 , ν 2 , ν 3 , ν 4 ) .
[ 1 2 + K D 2 ( 2 ν ) ] [ 2 2 + K D * 2 ( 2 ν ) ] Γ ˆ 1 ( x 1 , x 2 , 2 ν ) = ( 8 π 2 k 2 / c 2 ) μ / ν 1 ν 3 σ N L ( ν , ν ) 2 × Re L ˆ 3 ( x 1 , x 1 , x 2 , x 2 , ν 1 , ν 2 , ν 3 , ν 4 ) ,
L ˆ 3 = lim T 1 2 T V ˆ T ( x 1 , ν 1 ) V ˆ T ( x 2 , ν 3 ) V ˆ T * ( x 3 , ν 3 ) V ˆ T * ( x 4 , ν 4 )
L ˆ 3 ( x 1 , x 2 , x 3 , x 4 , ν 1 , ν 2 , ν 3 , ν 4 ) = G ( ν 1 , ν 2 , ν 3 , ν 4 ) exp [ i K D ( ν 1 ) z 1 + i K D ( ν 2 ) z 2 - i K D * ( ν 3 ) z 3 - i K D * ( ν 4 ) z 4 ] .
Γ ˆ p 1 ( x 1 , x 3 , 2 ν ) = ν 1 ν 3 Re B ( ν 1 , ν 2 , ν 3 , ν 4 ) exp { i [ K D ( ν 1 ) + K D ( ν 2 ) ] z 1 - i [ K D * ( ν 3 ) + K D * ( ν 4 ) ] z 3 } ,
B ( ν 1 , ν 2 , ν 3 , ν 4 ) = ( 8 π 2 k 2 / c 2 ) μ / G ( ν 1 , ν 2 , ν 3 , ν 4 ) σ N L ( ν ¯ , ν ¯ , 2 ν ¯ ) 2 { K D 2 ( 2 ν ) - [ K D ( ν 1 ) + K D ( ν 2 ) ] 2 } { K D * 2 ( 2 ν ) - [ K D * ( ν 3 ) + K D * ( ν 4 ) ] 2 } ,
Γ ˆ h 1 ( x 1 , x 3 , 2 ν ) = H ( 2 ν ) exp [ i K D ( 2 ν ) z 1 - i K D * ( 2 ν ) z 3 ] ,
H ( 2 ν ) = - ν 1 ν 3 Re B ( ν i ) ,
Γ ˆ inside 1 ( z , z , 2 ν ) = ν 1 ν 3 B ( ν i ) [ exp { - 4 z Im [ K D ( ν ¯ ) ] } + exp { - 2 z Im [ K D ( 2 ν ¯ ) ] } - 2 cos ( z { Re [ K D ( 2 ν ¯ ) ] - 2 Re [ K D ( ν ¯ ) ] } ) × exp ( - z { Im [ K D ( 2 ν ¯ ) ] - 2 Im [ K D ( ν ¯ ) ] } ) ] ν 1 ν 3 B ( ν i ) F ( z , ν ¯ , 2 ν ¯ ) ,
Γ ˆ inside 1 ( z 1 , z 1 , 2 ν ) = ν 1 ν 3 4 B ( ν i ) × sin 2 { z [ K D ( 2 ν ¯ ) - 2 K D ( ν ¯ ) ] / 2 } ,
Γ ˆ outside 1 ( x 1 , x 3 , 2 ν ) = Γ ˆ inside ( l , l , 2 ν ) exp [ i k ( z 1 - z 3 ) ] ,
z 1 = z 1 - l ,             z 3 = z 3 - l ,
Γ ˆ outside 1 ( x 1 , x 3 , 2 ν ¯ ) = ( 16 π 2 k 2 / c 2 ) ( μ / ) σ N L 2 F ( l , ν ¯ , 2 ν ¯ ) K D 2 ( 2 ν ¯ ) - 4 K D 2 ( ν ¯ ) 2 × ν 1 ν 3 ν 1 = ν ¯ + Δ ν ν 2 = - ν ¯ + Δ ν G ( ν 1 , ν ¯ + Δ ν - ν 1 , ν 3 , - ν ¯ + Δ ν - ν 3 ) × exp [ i k ( z 1 - z 3 ) ] .
Γ outside 1 ( x 1 , x 3 , τ ) = ( 16 π 2 k 2 μ / c 2 ) σ N L 2 F ( l , ν ¯ , 2 ν ¯ ) K D 2 ( 2 ν ¯ ) - 4 K D 2 ( ν ¯ ) 2 × exp { i [ k ( z 1 - z 3 ) - 2 π ν ¯ τ ] } × 0 0 0 Δ ν max d ν 1 d ν 3 d Δ ν exp ( - i 2 π Δ ν τ ) × G ( ν 1 , ν ¯ + Δ ν - ν 1 , ν 3 , - ν ¯ + Δ ν , - ν 3 ) × δ ( ν 1 - ν ¯ + Δ ν ) δ ( ν 3 + ν ¯ - Δ ν ) ,
Γ outside 1 ( x 1 , x 3 , τ ) = ( 16 π 2 k 2 μ / c 2 ) σ N L 2 F ( l , ν ¯ , 2 ν ¯ ) K D 2 ( 2 ν ¯ ) - 4 K D 2 ( ν ¯ ) 2 × exp [ i k ( z 1 - z 3 ) - 2 π i ν ¯ τ ] × 0 Δ ν max G ( ν ¯ + Δ ν , 0 , - ν ¯ + Δ ν , 0 ) × exp ( - 2 π i Δ ν τ ) d Δ ν .
G ( ν ¯ + Δ ν , 0 , - ν ¯ + Δ ν , 0 ) = I 2 ( ν ¯ ) { 2 ( Δ ν ) 1 2 / [ ( Δ ν ) 2 + ( Δ ν ) 1 2 2 ] } 2 ,
I ( 2 ν ¯ ) = I 2 ( ν ¯ ) ( 4 / 9 2 ν ¯ 2 ) σ N L 2 1 + ( 4 / 9 2 ν ¯ 2 ) [ 4 σ ( ν ¯ ) - σ ( 2 ν ¯ ) ] 2 F ( l , ν ¯ , 2 ν ¯ ) × 2 τ c [ ( Δ ν ) max τ c ( Δ ν ) max 2 τ c 2 + 1 + tan - 1 τ c ( Δ ν ) max ] .
Γ ˆ reflected 1 ( x 1 , x 3 , 2 ν ¯ ) = [ H ( 2 ν ¯ ) + ν 1 ν 3 Re B ( ν i ) ] × exp [ - i k ( z 1 - z 3 ) ] .
Γ 1 ( x 1 , x 3 ) = [ H + ( 8 π 2 k 2 / c 2 ) μ / σ N L 2 f ( τ c ) K D 2 ( 2 ν ¯ ) - 4 K D 2 ( ν ¯ ) 2 ] × exp [ - i k ( z 1 - z 3 ) ] ,
f ( τ c ) = 2 I 2 ( ν ¯ ) τ c { ( Δ ν ) max τ c / [ ( Δ ν ) max 2 τ c 2 + 1 ] + tan - 1 τ c ( Δ ν ) max } .
I ( 2 ν ¯ ) = ( 2 / 9 2 ν ¯ 2 ) σ N L 2 1 + ( 4 / 9 2 ν ¯ 2 ) [ 4 σ ( ν ¯ ) - σ ( 2 ν ¯ ) ] 2 f ( τ c ) + H .
Δ ν = ω r / a ,
( Δ ν ) laser Δ ν .