Abstract

We analyze the frequency-shifted conjugate beam generated by nearly degenerate four-wave mixing in an array of single-mode waveguides. We show that if the array sufficiently samples the signal beam, a high-quality conjugate will be generated when the pump beams are counterpropagating. The ability of a waveguide mode to couple to incoming beams over a range of incident directions relaxes the strict phase-matching requirement present in bulk-media four-wave mixing. A shift in the propagation direction of one of the pump beams changes the propagation direction of the generated beam while maintaining the spatial phase coherence.

© 1989 Optical Society of America

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References

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  1. See, e.g., H. Nakajima, R. Frey, Appl. Phys. Lett. 47, 769 (1985);R. Nietzke, P. Fenz, W. Elsasser, E. O. Gobel, Appl. Phys. Lett. 51, 1298 (1987);M. Lucente, J. G. Fujimoto, G. M. Carter, Appl. Phys. Lett. 53, 1897 (1988).
    [Crossref]
  2. G. P. Agrawal, J. Opt. Soc. Am. B 5, 147 (1988).
    [Crossref]
  3. For a review, see R. W. Hellwarth, in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983), pp. 127–133.
  4. The waveguide propagating beam-coupling analysis is derived from H. Kressel, J. K. Butler, Semiconductor Lasers and Heterojunction LEDs (Academic, New York, 1977), p. 192.
  5. See, e.g., R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).
  6. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), p. 401 and following.
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2.

1988 (1)

1985 (1)

See, e.g., H. Nakajima, R. Frey, Appl. Phys. Lett. 47, 769 (1985);R. Nietzke, P. Fenz, W. Elsasser, E. O. Gobel, Appl. Phys. Lett. 51, 1298 (1987);M. Lucente, J. G. Fujimoto, G. M. Carter, Appl. Phys. Lett. 53, 1897 (1988).
[Crossref]

Agrawal, G. P.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), p. 401 and following.

Butler, J. K.

The waveguide propagating beam-coupling analysis is derived from H. Kressel, J. K. Butler, Semiconductor Lasers and Heterojunction LEDs (Academic, New York, 1977), p. 192.

Frey, R.

See, e.g., H. Nakajima, R. Frey, Appl. Phys. Lett. 47, 769 (1985);R. Nietzke, P. Fenz, W. Elsasser, E. O. Gobel, Appl. Phys. Lett. 51, 1298 (1987);M. Lucente, J. G. Fujimoto, G. M. Carter, Appl. Phys. Lett. 53, 1897 (1988).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2.

Hellwarth, R. W.

For a review, see R. W. Hellwarth, in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983), pp. 127–133.

Kressel, H.

The waveguide propagating beam-coupling analysis is derived from H. Kressel, J. K. Butler, Semiconductor Lasers and Heterojunction LEDs (Academic, New York, 1977), p. 192.

Nakajima, H.

See, e.g., H. Nakajima, R. Frey, Appl. Phys. Lett. 47, 769 (1985);R. Nietzke, P. Fenz, W. Elsasser, E. O. Gobel, Appl. Phys. Lett. 51, 1298 (1987);M. Lucente, J. G. Fujimoto, G. M. Carter, Appl. Phys. Lett. 53, 1897 (1988).
[Crossref]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), p. 401 and following.

Appl. Phys. Lett. (1)

See, e.g., H. Nakajima, R. Frey, Appl. Phys. Lett. 47, 769 (1985);R. Nietzke, P. Fenz, W. Elsasser, E. O. Gobel, Appl. Phys. Lett. 51, 1298 (1987);M. Lucente, J. G. Fujimoto, G. M. Carter, Appl. Phys. Lett. 53, 1897 (1988).
[Crossref]

J. Opt. Soc. Am. B (1)

Other (5)

For a review, see R. W. Hellwarth, in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983), pp. 127–133.

The waveguide propagating beam-coupling analysis is derived from H. Kressel, J. K. Butler, Semiconductor Lasers and Heterojunction LEDs (Academic, New York, 1977), p. 192.

See, e.g., R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), p. 401 and following.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2.

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Figures (1)

Fig. 1
Fig. 1

Geometry for 4wm in a waveguide array with incident pump beams E1 and E2, signal beam Es, and generated beam Ec.

Equations (16)

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E j ( x , y , z ) = d k jx d k jy exp [ i k j r ] j ( k jx , k jy ) ,
| k j | 2 = k jx 2 + k jy 2 + k jz 2 = ω j 2 c 2
j k j = 0 ,
E s ( x , y , 0 ) = m , n E smn ( x , y , 0 ) = m , n A smn ( 0 ) ψ ( x m x 0 , y n y 0 ) ,
A smn ( 0 ) = d k sx d k sy sy ( k sx , k sy ) × d x d y ψ ( x m x 0 , y n y 0 ) × exp [ i ( k sx x + k sy y ) ]
= d k sx d k sy sy ( k sx , k xy ) × exp [ i ( m k sx x 0 + n k sy y 0 ) ] γ ( k sx , k xy ) ,
γ ( k xy , k sy ) = d x d y ψ ( x , y ) exp [ i ( k sx x + k sy y ) ] ,
E cmn ( x , y , 0 ) = A cmn ( 0 ) ψ ( x m x 0 , y n y 0 ) ,
A cmn ( 0 ) = F ( A 1 mn , A 2 mn , ω 0 , Ω , β , L , Γ ) A smn * ( 0 ) = F mn A smn * ( 0 ) .
Γ ( Ω ) = d x d y χ ( 3 ) ( x , y , ω 0 , Ω ) | ψ ( x , y ) | 4 .
E c ( x , y , z ) = d k cx d k cy × exp ( i k c r ) mn cmn ( k cx , k cy ) ,
cmn ( k cx , k cy ) = 1 4 π 2 E cmn ( x , y , 0 ) × exp [ i ( k cx x + k cy y ) ] d x d y .
E c ( x , y , z ) = 1 4 π 2 d k cx d k cy exp ( i k c r ) × d k sx d k sy γ * ( k sx , k sy ) γ ( k cx , k cy ) × s * ( k sx , k sy ) m , n F mn exp [ i ( m k sx x 0 + n k sy y 0 + m k cx x 0 + n k cy y 0 ) ] .
E c ( x , y , z ) = F d k sx d k sy × exp ( i k s r ) | γ ( k sx , k sy ) | 2 s * ( k sx , k sy ) ,
A c m m = A cmn exp [ i 2 ( m k 1 x x 0 + n k 1 y y 0 ) ] ,
A * s m n = A * s m n exp [ i 2 ( m k 1 x x 0 + n k 1 y y 0 ) ] ,

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