Abstract

The transformation of a noisy classical optical signal in a nonlinear-optical converter is discussed with pump depletion taken into account. By means of the Hamiltonian formalism, the possibility of noise transfer between the interacting waves without a change in the total noise in the system is demonstrated. The conditions for noise reduction in both quadratures of the optical signal are found.

© 1993 Optical Society of America

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References

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  1. K. Wiesenfeld, B. McNamara, Phys. Rev. Lett. 55, 13 (1985).
    [CrossRef] [PubMed]
  2. P. Bryant, K. Wiesenfeld, B. McNamara, J. Appl. Phys. 62, 2898 (1987).
    [CrossRef]
  3. M. F. Bocko, J. Battiato, Phys. Rev. Lett. 60, 1763 (1988).
    [CrossRef] [PubMed]
  4. H. Takahasi, in Advances in Communication Systems, A. V. Balakrishnan, ed. (Academic, New York, 1965).
  5. B. Yurke, J. S. Denker, Phys. Rev. A 29, 1419 (1984).
    [CrossRef]
  6. New Physical Principles of Optical Processing of Information, S. A. Akhmanov, M. A. Vorontsov, eds. (Nauka, Moscow, 1990).
  7. H. J. Kimble, D. F. Walls, eds., feature issue on squeezed states of the electromagnetic field, J. Opt. Soc. Am. B 4, 1450–1741 (1987).
    [CrossRef]
  8. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).
  9. Yu. G. Pavlenko, Hamiltonian Method in Electrodynamics and Quantum Mechanics (Moscow University, Moscow, 1985).
  10. V. I. Arnold, Mathematical Methods of Classical Mechanics (Nauka, Moscow, 1989).
    [CrossRef]

1988

M. F. Bocko, J. Battiato, Phys. Rev. Lett. 60, 1763 (1988).
[CrossRef] [PubMed]

1987

1985

K. Wiesenfeld, B. McNamara, Phys. Rev. Lett. 55, 13 (1985).
[CrossRef] [PubMed]

1984

B. Yurke, J. S. Denker, Phys. Rev. A 29, 1419 (1984).
[CrossRef]

Arnold, V. I.

V. I. Arnold, Mathematical Methods of Classical Mechanics (Nauka, Moscow, 1989).
[CrossRef]

Battiato, J.

M. F. Bocko, J. Battiato, Phys. Rev. Lett. 60, 1763 (1988).
[CrossRef] [PubMed]

Bloembergen, N.

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

Bocko, M. F.

M. F. Bocko, J. Battiato, Phys. Rev. Lett. 60, 1763 (1988).
[CrossRef] [PubMed]

Bryant, P.

P. Bryant, K. Wiesenfeld, B. McNamara, J. Appl. Phys. 62, 2898 (1987).
[CrossRef]

Denker, J. S.

B. Yurke, J. S. Denker, Phys. Rev. A 29, 1419 (1984).
[CrossRef]

McNamara, B.

P. Bryant, K. Wiesenfeld, B. McNamara, J. Appl. Phys. 62, 2898 (1987).
[CrossRef]

K. Wiesenfeld, B. McNamara, Phys. Rev. Lett. 55, 13 (1985).
[CrossRef] [PubMed]

Pavlenko, Yu. G.

Yu. G. Pavlenko, Hamiltonian Method in Electrodynamics and Quantum Mechanics (Moscow University, Moscow, 1985).

Takahasi, H.

H. Takahasi, in Advances in Communication Systems, A. V. Balakrishnan, ed. (Academic, New York, 1965).

Wiesenfeld, K.

P. Bryant, K. Wiesenfeld, B. McNamara, J. Appl. Phys. 62, 2898 (1987).
[CrossRef]

K. Wiesenfeld, B. McNamara, Phys. Rev. Lett. 55, 13 (1985).
[CrossRef] [PubMed]

Yurke, B.

B. Yurke, J. S. Denker, Phys. Rev. A 29, 1419 (1984).
[CrossRef]

J. Appl. Phys.

P. Bryant, K. Wiesenfeld, B. McNamara, J. Appl. Phys. 62, 2898 (1987).
[CrossRef]

J. Opt. Soc. Am. B

Phys. Rev. A

B. Yurke, J. S. Denker, Phys. Rev. A 29, 1419 (1984).
[CrossRef]

Phys. Rev. Lett.

K. Wiesenfeld, B. McNamara, Phys. Rev. Lett. 55, 13 (1985).
[CrossRef] [PubMed]

M. F. Bocko, J. Battiato, Phys. Rev. Lett. 60, 1763 (1988).
[CrossRef] [PubMed]

Other

H. Takahasi, in Advances in Communication Systems, A. V. Balakrishnan, ed. (Academic, New York, 1965).

New Physical Principles of Optical Processing of Information, S. A. Akhmanov, M. A. Vorontsov, eds. (Nauka, Moscow, 1990).

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

Yu. G. Pavlenko, Hamiltonian Method in Electrodynamics and Quantum Mechanics (Moscow University, Moscow, 1985).

V. I. Arnold, Mathematical Methods of Classical Mechanics (Nauka, Moscow, 1989).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Transformation of the uncertainty-space projection. Initial conditions: |A10R| = 1, |A20R| = 0.5, φ10R = 0°, φ20R = −90°. (a) The first wave is initially coherent, and (b) the second wave is initially noisy.

Fig. 2
Fig. 2

Transformation of the uncertainty-space projection. Initial conditions: |A10R| = 1, |A20R| = 0.5, φ10R = −135°, φ20R = 0°. (a) The first wave is initially noisy, and (b) the second wave is initially coherent.

Fig. 3
Fig. 3

Variance Δk versus interaction length z (initial values for the quantities |Ak0R|, φk0R are the same as in Fig. 1 for a1–a3 and the same as in Fig. 2 for b1–b3). Initial values of Δk: Δ20 = 0.25 (a1–a3), Δ10 = 0.00 (a1), Δ10 = 0.10 (a2), Δ10 = 0.25 (a3); Δ10 = 0.25 (b1–b3), Δ20 = 0.00 (b1), Δ20 = 0.10 (b2), Δ20 = 0.25 (b3).

Fig. 4
Fig. 4

Transformation of the first wave’s phase noise. Initial conditions: |A10| = 1, |A20| = 1, φ20 = 0°.

Fig. 5
Fig. 5

Transformation of a noisy pulse in a nonlinear converter (initial values for the quantities |A10R|, φ10R in the center of the pulse are the same as in Fig. 2). a, The amplitude |A10R| and the phase φ10R of a pulse at the input of the frequency converter, b, The amplitude |A1R| and the phase φ1R of the pulse at the interaction length z = 1.3.

Equations (20)

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d A 1 d z = i σ A 1 * A 2 , d A 2 d z = i σ A 1 2 ,
H = 1 2 ( a 1 * 2 a 2 + a 1 2 a 2 * ) , a 1 = 1 2 A 1 | A 10 | , a 2 = 1 2 A 2 | A 10 | z = 2 σ | A 10 | z .
a k = X k + i Y k 2 ,
a k = I k exp ( i φ k ) .
[ a n , i a k * ] = [ X n , Y k ] = [ φ n , I k ] = δ n k , n , k = 1 , 2 ; d X k d z = [ X k , H Q ] , d Y k d z = [ Y k , H Q ] , H Q = 1 2 [ ( X 1 2 Y 1 2 ) X 2 + 2 X 1 Y 1 Y 2 ] ; d I k d z = [ I k , H I ] , d φ k d z = [ φ k , H I ] , H I = 2 I 1 I 2 cos ( φ 2 2 φ 1 ) .
V = d X 1 d Y 1 d X 2 d Y 2 = invariant ,
S 1 + S 2 = d X 1 d Y 1 + d X 2 d Y 2 = invariant .
X 10 = X 10 ( u ) , Y 10 = Y 10 ( u ) , X 20 = X 20 ( υ ) , Y 20 = Y 20 ( υ ) .
S k = d X k d Y k = 1 2 ( X k d Y k Y k d X k ) = 1 2 ( X k Y k Q i 0 Y k X k Q i 0 ) Q i 0 u d u + 1 2 ( X k Y k Q i 0 Y k X k Q i 0 ) Q i 0 υ d υ ,
Q 10 = X 10 , Q 20 = Y 10 , Q 30 = X 20 , Q 40 = Y 20 ,
S 1 = J 11 + J 12 , S 2 = J 22 + J 21 .
X 10 = X 10 R + r 1 cos ( u ) , Y 10 = Y 10 R + r 1 sin ( u ) , X 20 = X 20 R + r 2 cos ( υ ) , Y 20 = Y 20 R + r 2 sin ( υ ) , 0 u , υ 2 π .
( X k 0 R ) 2 + ( Y k 0 R ) 2 r k 2 ,
S 1 ( z ) = [ 1 D 1 ( z ) ] S 10 + D 2 ( z ) S 20 , S 2 ( z ) = [ 1 D 2 ( z ) ] S 20 + D 1 ( z ) S 10 , S 1 + S 2 = S 10 + S 20 ,
D = D 1 = D 2 = 2 I 10 R z 2 2 2 I 10 R I 20 R sin ( φ 20 R 2 φ 10 R ) z 3 , D = 2 σ | A 10 R | 2 z 2 [ 1 σ | A 20 R | sin ( φ 20 R 2 φ 10 R ) z ] ,
S 1 ( z ) = S 10 + D ( z ) ( S 20 S 10 ) , S 2 ( z ) = S 20 D ( z ) ( S 20 S 10 ) .
S 1 = S k 0 ( X 1 , Y 1 ) ( X k 0 , Y k 0 ) d X k 0 d Y k 0 , S 2 = S k 0 ( X 2 , Y 2 ) ( X k 0 , Y k 0 ) d X k 0 d Y k 0 .
S 1 + S 2 = invariant .
Δ k = [ ( X k X k ) 2 + ( Y k Y k ) 2 ] 1 / 2 ,
X 1 ( t ) = exp ( t 2 ) [ X 1 R + ξ 1 ( t ) ] , X 1 R = const ., ξ 1 ( t ) = 0 , ξ 1 ( t ) 2 = σ 2 , Y 1 ( t ) = exp ( t 2 ) [ Y 1 R + η 1 ( t ) ] , Y 1 R = const ., η 1 ( t ) = 0 , η 1 ( t ) 2 = σ 2 ,

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