Abstract

We analyze a new scheme for generating squeezed states in a short semiconductor AlxGa1−xAs waveguide with χ(3) nonlinearity at below half the band-gap energy. We find that for a Gaussian pulse the amount of squeezing achievable is limited by the squeezed-state detection phase mismatch caused by the pump self-phase modulation and is also degraded slightly by the pump–probe phase mismatch that is due to the different nonlinear refractive indices experienced by the pump and the probe beams. We show theoretically that the amount of squeezing observed can be increased by use of either a short pulse or a pulse with matched phase variation as the local oscillator. In a centimeter-long AlxGa1−xAs waveguide more than 85% (8.2 dB) of squeezing potentially can be obtained, limited mainly by two-photon absorption.

© 1995 Optical Society of America

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References

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  1. C. M. Caves, Phys. Rev. D 23, 1693 (1981).
    [CrossRef]
  2. H. P. Yuen and J. H. Shapiro, IEEE Trans. Inform. Theory IT-24, 657 (1978).
    [CrossRef]
  3. R. E. Slusher and B. Yurke, J. Lightwave Technol. 8, 466 (1990).
    [CrossRef]
  4. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
    [CrossRef] [PubMed]
  5. R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
    [CrossRef] [PubMed]
  6. Ling-An Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
    [CrossRef] [PubMed]
  7. M. W. Maeda, P. Kumar, and J. H. Shapiro, Opt. Lett. 3, 161 (1986).
  8. S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987).
    [CrossRef] [PubMed]
  9. A. Heidmann, R. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
    [CrossRef] [PubMed]
  10. S. T. Ho, N. C. Wong, and J. H. Shapiro, Opt. Lett. 16, 840 (1991).
    [CrossRef] [PubMed]
  11. H. J. Kimble, Phys. Rep. 219, 227 (1992).
    [CrossRef]
  12. S. T. Ho, P. Kumar, and J. H. Shapiro, J. Opt. Soc. Am. B 8, 37 (1991).
    [CrossRef]
  13. M. D. Levenson, R. M. Shelby, and S. H. Perlmutter, Opt. Lett. 10, 514 (1985).
    [CrossRef] [PubMed]
  14. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), Chaps. 4 and 7.
  15. R. M. Shelby, M. D. Levenson, and P. W. Bayer, Phys. Rev. B 31, 5244 (1985).
    [CrossRef]
  16. K. Bergman, H. A. Haus, and M. Shirasaki, Appl. Phys. B 55, 242 (1992).
    [CrossRef]
  17. R. M. Shelby, P. D. Drummond, and S. J. Carter, Phys. Rev. A 42, 2966 (1990).
    [CrossRef] [PubMed]
  18. B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
    [CrossRef] [PubMed]
  19. R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, Phys. Rev. Lett. 59, 2566 (1987).
    [CrossRef] [PubMed]
  20. K. Bergman, H. A. Haus, E. P. Ippen, and M. Shirasaki, Opt. Lett. 19, 290 (1994).
    [CrossRef] [PubMed]
  21. K. Bergman and H. A. Haus, Opt. Lett. 16, 663 (1991).
    [CrossRef] [PubMed]
  22. K. Bergman, H. A. Haus, E. P. Ippen, and M. Shirasaki, in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 142–143, paper WII1.
  23. K. Bergman, C. R. Doerr, H. A. Haus, and M. Shirasaki, Opt. Lett. 18, 643 (1993).
    [CrossRef] [PubMed]
  24. M. Shirasaki and H. A. Haus, Opt. Lett. 17, 1225 (1992).
    [CrossRef] [PubMed]
  25. M. Shirasaki and H. A. Haus, J. Opt. Soc. Am. B 7, 30 (1990). Also see Eq. (24) in this paper.
    [CrossRef]
  26. M. Rosenbluh and R. M. Shelby, Phys. Rev. Lett. 66, 153 (1991).
    [CrossRef] [PubMed]
  27. S. T. Ho, C. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slusher, Appl. Phys. Lett. 59, 2558 (1991).
    [CrossRef]
  28. Our initial experiments show that GAWBS noise is negligible in the semiconductor waveguide.
  29. 29. See Eq. (14), where Xs = (½)[ks + i(γ/Ip)], and ks = (4π/λ)n(2).
  30. In this case we can use the instantaneous model of the four-wave mixing process. See, e.g., L. G. Joneckis and J. H. Shapiro, J. Opt. Soc. Am. B 10, 1102 (1993).
    [CrossRef]
  31. In the case of a square pump pulse, the shape of the LO pulse does not affect the amount of squeezing detectable as long as LO pulse has a uniform phase and its profile is enclosed in that of the square pump pulse.
  32. O. Aytur and P. Kumar, Opt. Lett. 15, 390 (1990).
    [CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), Chaps. 4 and 7.

Aytur, O.

O. Aytur and P. Kumar, Opt. Lett. 15, 390 (1990).
[CrossRef]

Bayer, P. W.

R. M. Shelby, M. D. Levenson, and P. W. Bayer, Phys. Rev. B 31, 5244 (1985).
[CrossRef]

Bergman, K.

K. Bergman, H. A. Haus, and M. Shirasaki, Appl. Phys. B 55, 242 (1992).
[CrossRef]

K. Bergman, H. A. Haus, E. P. Ippen, and M. Shirasaki, Opt. Lett. 19, 290 (1994).
[CrossRef] [PubMed]

K. Bergman and H. A. Haus, Opt. Lett. 16, 663 (1991).
[CrossRef] [PubMed]

K. Bergman, H. A. Haus, E. P. Ippen, and M. Shirasaki, in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 142–143, paper WII1.

K. Bergman, C. R. Doerr, H. A. Haus, and M. Shirasaki, Opt. Lett. 18, 643 (1993).
[CrossRef] [PubMed]

Camy, G.

A. Heidmann, R. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Carter, S. J.

R. M. Shelby, P. D. Drummond, and S. J. Carter, Phys. Rev. A 42, 2966 (1990).
[CrossRef] [PubMed]

Caves, C. M.

C. M. Caves, Phys. Rev. D 23, 1693 (1981).
[CrossRef]

DeVoe, R. G.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

Doerr, C. R.

K. Bergman, C. R. Doerr, H. A. Haus, and M. Shirasaki, Opt. Lett. 18, 643 (1993).
[CrossRef] [PubMed]

Drummond, P. D.

R. M. Shelby, P. D. Drummond, and S. J. Carter, Phys. Rev. A 42, 2966 (1990).
[CrossRef] [PubMed]

Fabre, C.

A. Heidmann, R. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Giacobino, E.

A. Heidmann, R. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Grangier, P.

R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, Phys. Rev. Lett. 59, 2566 (1987).
[CrossRef] [PubMed]

B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[CrossRef] [PubMed]

Hall, J. L.

Ling-An Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Haus, H. A.

K. Bergman, H. A. Haus, and M. Shirasaki, Appl. Phys. B 55, 242 (1992).
[CrossRef]

K. Bergman, H. A. Haus, E. P. Ippen, and M. Shirasaki, Opt. Lett. 19, 290 (1994).
[CrossRef] [PubMed]

K. Bergman and H. A. Haus, Opt. Lett. 16, 663 (1991).
[CrossRef] [PubMed]

K. Bergman, H. A. Haus, E. P. Ippen, and M. Shirasaki, in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 142–143, paper WII1.

K. Bergman, C. R. Doerr, H. A. Haus, and M. Shirasaki, Opt. Lett. 18, 643 (1993).
[CrossRef] [PubMed]

M. Shirasaki and H. A. Haus, Opt. Lett. 17, 1225 (1992).
[CrossRef] [PubMed]

M. Shirasaki and H. A. Haus, J. Opt. Soc. Am. B 7, 30 (1990). Also see Eq. (24) in this paper.
[CrossRef]

Heidmann, A.

A. Heidmann, R. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Ho, S. T.

S. T. Ho, N. C. Wong, and J. H. Shapiro, Opt. Lett. 16, 840 (1991).
[CrossRef] [PubMed]

S. T. Ho, P. Kumar, and J. H. Shapiro, J. Opt. Soc. Am. B 8, 37 (1991).
[CrossRef]

S. T. Ho, C. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slusher, Appl. Phys. Lett. 59, 2558 (1991).
[CrossRef]

Hobson, W. S.

S. T. Ho, C. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slusher, Appl. Phys. Lett. 59, 2558 (1991).
[CrossRef]

Hollberg, L. W.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

Horowicz, R.

A. Heidmann, R. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Ippen, E. P.

K. Bergman, H. A. Haus, E. P. Ippen, and M. Shirasaki, in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 142–143, paper WII1.

K. Bergman, H. A. Haus, E. P. Ippen, and M. Shirasaki, Opt. Lett. 19, 290 (1994).
[CrossRef] [PubMed]

Islam, M. N.

S. T. Ho, C. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slusher, Appl. Phys. Lett. 59, 2558 (1991).
[CrossRef]

Itaya, Y.

S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987).
[CrossRef] [PubMed]

Joneckis, L. G.

In this case we can use the instantaneous model of the four-wave mixing process. See, e.g., L. G. Joneckis and J. H. Shapiro, J. Opt. Soc. Am. B 10, 1102 (1993).
[CrossRef]

Kimble, H. J.

Ling-An Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

H. J. Kimble, Phys. Rep. 219, 227 (1992).
[CrossRef]

Kumar, P.

M. W. Maeda, P. Kumar, and J. H. Shapiro, Opt. Lett. 3, 161 (1986).

S. T. Ho, P. Kumar, and J. H. Shapiro, J. Opt. Soc. Am. B 8, 37 (1991).
[CrossRef]

O. Aytur and P. Kumar, Opt. Lett. 15, 390 (1990).
[CrossRef]

LaPorta, A.

R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, Phys. Rev. Lett. 59, 2566 (1987).
[CrossRef] [PubMed]

Levenson, M. D.

M. D. Levenson, R. M. Shelby, and S. H. Perlmutter, Opt. Lett. 10, 514 (1985).
[CrossRef] [PubMed]

R. M. Shelby, M. D. Levenson, and P. W. Bayer, Phys. Rev. B 31, 5244 (1985).
[CrossRef]

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

Levi, A. F. J.

S. T. Ho, C. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slusher, Appl. Phys. Lett. 59, 2558 (1991).
[CrossRef]

Machida, S.

S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987).
[CrossRef] [PubMed]

Maeda, M. W.

M. W. Maeda, P. Kumar, and J. H. Shapiro, Opt. Lett. 3, 161 (1986).

Mertz, J. C.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

Perlmutter, S. H.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

M. D. Levenson, R. M. Shelby, and S. H. Perlmutter, Opt. Lett. 10, 514 (1985).
[CrossRef] [PubMed]

Potasek, M. J.

B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[CrossRef] [PubMed]

R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, Phys. Rev. Lett. 59, 2566 (1987).
[CrossRef] [PubMed]

Reynaud, S.

A. Heidmann, R. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

Rosenbluh, M.

M. Rosenbluh and R. M. Shelby, Phys. Rev. Lett. 66, 153 (1991).
[CrossRef] [PubMed]

Shapiro, J. H.

In this case we can use the instantaneous model of the four-wave mixing process. See, e.g., L. G. Joneckis and J. H. Shapiro, J. Opt. Soc. Am. B 10, 1102 (1993).
[CrossRef]

M. W. Maeda, P. Kumar, and J. H. Shapiro, Opt. Lett. 3, 161 (1986).

S. T. Ho, P. Kumar, and J. H. Shapiro, J. Opt. Soc. Am. B 8, 37 (1991).
[CrossRef]

S. T. Ho, N. C. Wong, and J. H. Shapiro, Opt. Lett. 16, 840 (1991).
[CrossRef] [PubMed]

H. P. Yuen and J. H. Shapiro, IEEE Trans. Inform. Theory IT-24, 657 (1978).
[CrossRef]

Shelby, R. M.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

M. D. Levenson, R. M. Shelby, and S. H. Perlmutter, Opt. Lett. 10, 514 (1985).
[CrossRef] [PubMed]

R. M. Shelby, P. D. Drummond, and S. J. Carter, Phys. Rev. A 42, 2966 (1990).
[CrossRef] [PubMed]

R. M. Shelby, M. D. Levenson, and P. W. Bayer, Phys. Rev. B 31, 5244 (1985).
[CrossRef]

M. Rosenbluh and R. M. Shelby, Phys. Rev. Lett. 66, 153 (1991).
[CrossRef] [PubMed]

Shirasaki, M.

M. Shirasaki and H. A. Haus, J. Opt. Soc. Am. B 7, 30 (1990). Also see Eq. (24) in this paper.
[CrossRef]

K. Bergman, C. R. Doerr, H. A. Haus, and M. Shirasaki, Opt. Lett. 18, 643 (1993).
[CrossRef] [PubMed]

M. Shirasaki and H. A. Haus, Opt. Lett. 17, 1225 (1992).
[CrossRef] [PubMed]

K. Bergman, H. A. Haus, and M. Shirasaki, Appl. Phys. B 55, 242 (1992).
[CrossRef]

K. Bergman, H. A. Haus, E. P. Ippen, and M. Shirasaki, in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 142–143, paper WII1.

K. Bergman, H. A. Haus, E. P. Ippen, and M. Shirasaki, Opt. Lett. 19, 290 (1994).
[CrossRef] [PubMed]

Slusher, R. E.

R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, Phys. Rev. Lett. 59, 2566 (1987).
[CrossRef] [PubMed]

B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[CrossRef] [PubMed]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

R. E. Slusher and B. Yurke, J. Lightwave Technol. 8, 466 (1990).
[CrossRef]

S. T. Ho, C. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slusher, Appl. Phys. Lett. 59, 2558 (1991).
[CrossRef]

Soccolich, C. E.

S. T. Ho, C. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slusher, Appl. Phys. Lett. 59, 2558 (1991).
[CrossRef]

Valley, J. F.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

Walls, D. F.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

Wong, N. C.

S. T. Ho, N. C. Wong, and J. H. Shapiro, Opt. Lett. 16, 840 (1991).
[CrossRef] [PubMed]

Wu, H.

Ling-An Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Wu, Ling-An

Ling-An Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Yamamoto, Y.

S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987).
[CrossRef] [PubMed]

Yuen, H. P.

H. P. Yuen and J. H. Shapiro, IEEE Trans. Inform. Theory IT-24, 657 (1978).
[CrossRef]

Yurke, B.

R. E. Slusher and B. Yurke, J. Lightwave Technol. 8, 466 (1990).
[CrossRef]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[CrossRef] [PubMed]

R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, Phys. Rev. Lett. 59, 2566 (1987).
[CrossRef] [PubMed]

Other (32)

C. M. Caves, Phys. Rev. D 23, 1693 (1981).
[CrossRef]

H. P. Yuen and J. H. Shapiro, IEEE Trans. Inform. Theory IT-24, 657 (1978).
[CrossRef]

R. E. Slusher and B. Yurke, J. Lightwave Technol. 8, 466 (1990).
[CrossRef]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

Ling-An Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

M. W. Maeda, P. Kumar, and J. H. Shapiro, Opt. Lett. 3, 161 (1986).

S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987).
[CrossRef] [PubMed]

A. Heidmann, R. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, Phys. Rev. Lett. 59, 2555 (1987).
[CrossRef] [PubMed]

S. T. Ho, N. C. Wong, and J. H. Shapiro, Opt. Lett. 16, 840 (1991).
[CrossRef] [PubMed]

H. J. Kimble, Phys. Rep. 219, 227 (1992).
[CrossRef]

S. T. Ho, P. Kumar, and J. H. Shapiro, J. Opt. Soc. Am. B 8, 37 (1991).
[CrossRef]

M. D. Levenson, R. M. Shelby, and S. H. Perlmutter, Opt. Lett. 10, 514 (1985).
[CrossRef] [PubMed]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), Chaps. 4 and 7.

R. M. Shelby, M. D. Levenson, and P. W. Bayer, Phys. Rev. B 31, 5244 (1985).
[CrossRef]

K. Bergman, H. A. Haus, and M. Shirasaki, Appl. Phys. B 55, 242 (1992).
[CrossRef]

R. M. Shelby, P. D. Drummond, and S. J. Carter, Phys. Rev. A 42, 2966 (1990).
[CrossRef] [PubMed]

B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[CrossRef] [PubMed]

R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, Phys. Rev. Lett. 59, 2566 (1987).
[CrossRef] [PubMed]

K. Bergman, H. A. Haus, E. P. Ippen, and M. Shirasaki, Opt. Lett. 19, 290 (1994).
[CrossRef] [PubMed]

K. Bergman and H. A. Haus, Opt. Lett. 16, 663 (1991).
[CrossRef] [PubMed]

K. Bergman, H. A. Haus, E. P. Ippen, and M. Shirasaki, in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 142–143, paper WII1.

K. Bergman, C. R. Doerr, H. A. Haus, and M. Shirasaki, Opt. Lett. 18, 643 (1993).
[CrossRef] [PubMed]

M. Shirasaki and H. A. Haus, Opt. Lett. 17, 1225 (1992).
[CrossRef] [PubMed]

M. Shirasaki and H. A. Haus, J. Opt. Soc. Am. B 7, 30 (1990). Also see Eq. (24) in this paper.
[CrossRef]

M. Rosenbluh and R. M. Shelby, Phys. Rev. Lett. 66, 153 (1991).
[CrossRef] [PubMed]

S. T. Ho, C. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slusher, Appl. Phys. Lett. 59, 2558 (1991).
[CrossRef]

Our initial experiments show that GAWBS noise is negligible in the semiconductor waveguide.

29. See Eq. (14), where Xs = (½)[ks + i(γ/Ip)], and ks = (4π/λ)n(2).

In this case we can use the instantaneous model of the four-wave mixing process. See, e.g., L. G. Joneckis and J. H. Shapiro, J. Opt. Soc. Am. B 10, 1102 (1993).
[CrossRef]

In the case of a square pump pulse, the shape of the LO pulse does not affect the amount of squeezing detectable as long as LO pulse has a uniform phase and its profile is enclosed in that of the square pump pulse.

O. Aytur and P. Kumar, Opt. Lett. 15, 390 (1990).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Scheme to generate squeezing in an AlxGa1−xAs waveguide. Ep is a strong classical pump, and âs and âs are its vacuum sidebands (quantized probes).

Fig. 2
Fig. 2

Effect of nonlinear pump–probe phase mismatch Δκ on the amount of squeezing as function of waveguide length for a square pump pulse with 4.5-GW/cm2 pulse peak intensity, uniform-phase LO pulse, and no nonlinear absorption: (a) Δκ ≠ 0 calculated with Δκ = 2(κsκp)Ip and the parameters given in the text; (b) Δκ = 0.

Fig. 3
Fig. 3

Effect of SSD phase mismatch on squeezing for a Gaussian pump pulse with uniform phase under its pulse profile and without nonlinear absorption: (a) the LO pulse has the same width as the Gaussian pump pulse; (b), (c), (d) narrow uniform-phase LO pulses, where the LO pulse width is taken to be one half, one fourth, and one eighth, respectively, of that of the pump pulse. (e) The square pump pulse case for comparison.

Fig. 4
Fig. 4

Variation of −Φab across the Gaussian pump pulse as a function of waveguide length.

Fig. 5
Fig. 5

Waveguide length dependence of the nonlinear phase terms Φab (dashed curve) and −2κpIpl (dotted curve) at the pump pulse center. The solid curve represents their difference −2κpIpl + Φab. The dotted–dashed line corresponds to phase π.

Fig. 6
Fig. 6

Improved detection of squeezing with two kinds of matched LO pulse: (a) output pump pulse from the same waveguide, (b) optimal LO pulse from another waveguide. (c) For comparison, we also show the square pump pulse.

Fig. 7
Fig. 7

Squeezing with nonlinear absorption: (a) Gaussian pump pulse with a nearly matched LO pulse from another waveguide, (b) Gaussian pump pulse with a reused pump pulse, (c) Gaussian pump pulse with a narrower uniform-phase LO pulse, (d) Gaussian pump pulse with a uniform-phase LO pulse having the same pulse width as the pump, (e) square pump pulse. (f) Relative pump amplitude Ep(l)/Ep(0).

Fig. 8
Fig. 8

Limit of the amount of squeezing S for different ratios R = XrIp/γ when γl ≫ 1.

Fig. 9
Fig. 9

Amount of squeezing achievable in a long waveguide for different ratios R. The pump pulse is a Gaussian pulse, and the LO pulse is a matched one optimized in another waveguide. Solid curve, R = 16.5; dashed curve, R = 33; dotted curve, R = 0.01. All other parameters are the same as those that we use for the practical waveguide discussed through this paper and in Ref. 27.

Equations (66)

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E p z = i κ p I p E p ,
a ^ s z = - γ a ^ s + i κ s I p a ^ s + i X s I p exp ( i 2 κ p I p z ) a ^ s + G ^ s ( z ) ,
a ^ s z = - γ a ^ s - i κ s I p a ^ s - i X s * I p exp ( - i 2 κ p I p z ) a ^ s + G ^ s ( z ) .
a ^ s z = - 2 [ α ( 2 ) I p F s ( 2 ) + α ( 3 ) I p 2 F s ( 3 ) ] × a ^ s + i κ s I p F s ( 2 ) a ^ s + i X s I p F s ( 2 ) × exp ( i 2 κ p F p ( 2 ) I p z ) a ^ s + + G ^ s ( z ) ,
F s ( 2 ) = u s ( x , y ) u p 2 ( x , y ) u s ( x , y ) d x d y u s 2 ( x , y ) d x d y .
a ˜ s z = - γ a ˜ s + i X s I p exp [ i 2 ( κ p - κ s ) I p z ] × a ˜ s + G ^ s ( z ) exp ( - i κ s I p z ) ,
a ˜ s z = - γ a ˜ s - i X s * I p exp [ - i 2 ( κ p - κ s ) I p z ] × a ˜ s + G ^ s ( z ) exp ( i κ s I p z ) ,
b ^ s z = ( - γ + i Δ κ / 2 ) b ^ s + i X s I p b ^ s + G ^ s exp ( - i κ p I p z ) ,
b ^ s z = ( - γ - i Δ κ / 2 ) b ^ s + i X s * I p b ^ s + G ^ s exp ( i κ p I p z ) .
a ^ s ( l ) = exp ( - γ l ) exp ( i κ p I p l ) [ μ ( l ) a ^ s ( 0 ) + ν ( l ) a ^ s ( 0 ) ] + Γ ^ 1 ( l ) exp ( i κ p I p l ) ,
a ^ s ( l ) = exp ( - γ l ) exp ( - i κ p I p l ) [ μ * ( l ) a ^ s ( 0 ) + ν * ( l ) a ^ s ( 0 ) ] + Γ ^ 2 ( l ) exp ( i κ p I p l ) ,
μ ( l ) = cosh ( Ω l ) + ( i Δ κ / 2 Ω ) sinh ( Ω l ) ,
ν ( l ) = ( i X s I p / Ω ) sinh ( Ω l ) ,
Ω = ( X s I p 2 - Δ κ 2 / 4 ) 1 / 2 ,
Γ ^ 1 ( l ) = 0 l d z exp [ - γ ( l - z ) ] [ exp ( - i κ p I p z ) × μ ( l - z ) G ^ s ( z ) + exp ( i κ p I p z ) × ν ( l - z ) G ^ s ( z ) ] ,
Γ ^ 2 ( l ) = 0 l d z exp [ - γ ( l - z ) ] [ exp ( i κ p I p z ) × μ * ( l - z ) G ^ s ( z ) + exp ( - i κ p I p z ) × ν * ( l - z ) G ^ ( z ) ] .
X ^ ( φ ) = [ ( a ^ s + a ^ s ) exp ( - i φ ) + ( a ^ s + a ^ s ) exp ( i φ ) ] / ( 2 2 ) ,
[ Δ X ^ ( φ ) ] 2 = ¼ + ½ [ c + a 2 + b 2 × sin ( 2 φ - 2 κ p I p l + Φ a b ) ] ,
a = exp ( - 2 γ l ) Re ( μ ν ) + Re ( Γ ^ 1 Γ ^ 2 ) ,
b = exp ( - 2 γ l ) Im ( μ ν ) + Im ( Γ ^ 1 Γ ^ 2 ) ,
c = exp ( - 2 γ l ) ν 2 + Γ ^ 1 + Γ ^ 1 ,
Φ a b = arctan ( a / b ) .
S = [ 1 - 4 ( Δ X ( φ ) ] 2 ) × 100 % .
2 φ - 2 κ p I p l + Φ a b = 2 n π - π / 2.
I p ( z c ) = I p ( 0 ) exp ( - z c 2 ) ,
S tot = j S ( z c j ) I LO ( z c j ) j I LO ( z c j ) ,
φ ( z c ) = φ c + κ p I p ( z c ) l ,
φ ( z c ) = φ c + κ p I p ( z c ) l = φ c + κ p I p ( z c ) l + κ p I p ( z c ) Δ l ,
Re ( μ ν ) = - X i I p 2 Ω sinh ( 2 Ω l ) - X r I p Δ κ 2 Ω 2 sinh 2 ( Ω l ) ,
Im ( μ ν ) = - X r I p 2 Ω sinh ( 2 Ω l ) - Δ κ X i I p 2 Ω 2 sinh 2 ( Ω l ) ,
Re ( Γ 1 Γ 2 ) = - 2 γ ( 2 n ¯ + 1 ) ( X i I p 8 Ω { [ exp [ 2 ( Ω - γ ) l ] - 1 Ω - γ + exp [ - 2 ( Ω - γ ) l ] - 1 Ω + γ } + X r I p Δ κ 8 Ω 2 × { [ exp [ 2 ( Ω - γ ) l ] - 1 Ω - γ - [ exp [ 2 ( Ω - γ ) l ] - 1 Ω + γ + exp ( - 2 γ l ) - 1 γ } ) ,
Im ( Γ 1 Γ 2 ) = 2 γ ( 2 n ¯ + 1 ) ( X r I p 8 Ω { [ exp [ 2 ( Ω - γ ) l ] - 1 Ω - γ + exp [ - 2 ( Ω - γ ) l ] - 1 Ω + γ } + X i I p Δ κ 8 Ω 2 × { [ exp [ 2 ( Ω - γ ) l ] - 1 Ω - γ - exp [ - 2 ( Ω - γ ) l ] - 1 Ω + γ + 2 [ exp ( - 2 γ l ) - 1 ] γ } ) ,
ν 2 = X s I p 2 Ω 2 sinh 2 ( Ω z ) ,
Γ 1 Γ 1 ) = 1 4 γ n ¯ { exp [ 2 ( Ω - γ ) l ] - 1 Ω - γ - exp [ - 2 ( Ω - γ ) l ] - 1 Ω + γ - 2 [ exp ( - 2 γ l ) - 1 ] γ } ) + [ 2 γ n ¯ ( Δ κ 2 Ω ) 2 + 2 γ ( n ¯ + 1 ) ( X s I p Ω ) 2 ] × { exp [ 2 ( Ω - γ ) l ] - 1 Ω - γ - exp [ - 2 ( Ω - γ ) l ] - 1 Ω + γ + 2 [ exp ( - 2 γ l ) - 1 ] γ } ,
X r = κ p = 2 π λ n ( 2 ) ,             X i = γ 2 I p .
Δ κ = 2 ( κ s - κ p ) I p = 2 X r I p .
Re ( μ ν ) - ( X r I p l ) 2 - ½ ( γ l ) ,
Im ( μ ν ) - ( X r I p l ) 2 - ½ ( X r I p l ) ( γ l ) ,
Re Γ 1 Γ 2 - ( X r I p l ) 2 ( γ l ) ,
Im Γ 1 Γ 2 ( X r I p l ) ( γ l ) ,
Γ 1 Γ 1 ( X r I p l ) 2 ( γ l ) ,
ν 2 = ( X r I p l ) 2 .
a - ( X r I p l ) 2 + [ / ( X r I p l ) 2 - ½ ] ( γ l ) ,
b ( X r I p l ) - ³ / ( X r I p l ) ( γ l ) ,
c ( X r I p l ) 2 [ 1 - / ( γ l ) ] .
a 2 + b 2 = ( X r I p l ) { 1 + [ ( - ½ γ l ) + ( X r I p l ) 2 ( 1 - / γ l ) ] } 1 / 2 ( X r I p l ) - ¼ ( X r I p l ) ( γ l ) .
[ Δ X ( φ ) ] 2 = ¼ + ½ ( c - a 2 + b 2 ) , ¼ - ½ ( X r I p l )
S = { 1 - 4 [ Δ X ( φ ) ] 2 } × 100 % = 2 ( X r I p l ) × 100 % , = 2 κ p I p l × 100 % .
a 2 + b 2 = ( X r I p l ) 2 { 1 + [ - / ( γ l ) + 1 ( X r I p l ) 2 ( 1 - ½ γ l ) ] } 1 / 2 ( X r I p l ) 2 [ 1 - / ( γ l ) ] .
[ Δ X ( φ ) ] 2 ( γ l ) ,
S = [ 1 - ½ ( γ l ) ] × 100 % .
S = [ 1 - ( γ p l ) ] × 100 % ,
Re Γ 1 Γ 2 - 1 3 + 2 3 ( X r I p γ ) 2 ,
Im Γ 1 Γ 2 X r I p 3 γ ,
exp ( - 2 γ l ) Re ( μ ν ) 0 ,
exp ( - 2 γ l ) Im ( μ ν ) 0 ,
exp ( - 2 γ l ) ν 2 0 ,
Γ 1 Γ 1 ( X r I p γ ) 2 + 1 6 ,
a = exp ( - 2 γ l ) Re ( μ ν ) + Re Γ 1 Γ 1 - - R 2 ,
b = exp ( - 2 γ l ) Im ( μ ν ) + Im Γ 1 Γ 1 R ,
c = exp ( - 2 γ l ) ν 2 + Γ 1 Γ 1 R 2 + .
[ Δ X ( φ ) ] 2 = ¼ + ½ ( c - a 2 + b 2 ) = ¼ + ½ { ( R 2 + ) - [ ( - R 2 - ) 2 + R 2 9 ] 1 / 2 } .
[ Δ X ( φ ) ] 2 = ¼ + ½ { ( R 2 + ) - [ 1 + ( 4 R 4 + 5 R 2 ) ] 1 / 2 } ¼ - ¹ / ₁₂ ( 1 + R 2 ) ,
S = ( 1 + R 2 ) × 100 % .
[ Δ X ( φ ) ] 2 = ¼ + ½ { ( R 2 + ) - ( + R 2 ) × [ 1 + R 2 / 9 ( R 2 + ) 2 ] 1 / 2 } .
R = X r I p γ = ( 2 π / λ ) n ( 2 ) I p 2 [ α ( 2 ) + α ( 3 ) I p ] I p π λ n ( 2 ) α ( 2 ) .

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