Abstract

We study the spatial distribution of quantum noise in singly resonant second-harmonic generation. Calculations are performed below threshold for spatial modulational instability. For parameters for which the intracavity fields are modulationally stable the spatial spectrum shows maximum squeezing at k=0, whereas under conditions of modulational instability we find maximum squeezing at finite wave number k=kc, where kc corresponds to the classical critical wave number.

© 2002 Optical Society of America

Full Article  |  PDF Article

Corrections

P. Lodahl and M. Saffman, "Spatial quantum noise in singly resonant second-harmonic generation: errata," Opt. Lett. 27, 551-551 (2002)
https://www.osapublishing.org/ol/abstract.cfm?uri=ol-27-7-551

References

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  1. H. J. Kimble, Phys. Rep. 219, 227 (1992).
    [CrossRef]
  2. G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994).
    [CrossRef] [PubMed]
  3. M. Vaupel, A. Maître, and C. Fabre, Phys. Rev. Lett. 83, 5278 (1999).
    [CrossRef]
  4. R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
    [CrossRef]
  5. L. A. Lugiato and F. Castelli, Phys. Rev. Lett. 68, 3284 (1992).
    [CrossRef] [PubMed]
  6. L. A. Lugiato and A. Gatti, Phys. Rev. Lett. 70, 3868 (1993).
    [CrossRef] [PubMed]
  7. G. Grynberg and L. A. Lugiato, Opt. Commun. 101, 69 (1993).
    [CrossRef]
  8. E. M. Nagasako, R. W. Boyd, and G. S. Agarwal, Phys. Rev. A 55, 1412 (1997).
    [CrossRef]
  9. M. I. Kolobov, Rev. Mod. Phys. 71, 1539 (1999).
    [CrossRef]
  10. A. Gatti, E. Brambilla, L. A. Lugiato, and M. I. Kolobov, Phys. Rev. Lett. 83, 1763 (1999).
    [CrossRef]
  11. S.-K. Choi, M. Vasilyev, and P. Kumar, Phys. Rev. Lett. 83, 1938 (1999).
    [CrossRef]
  12. S.-K. Choi, M. Vasilyev, and P. Kumar, Phys. Rev. Lett. 84, 1361 (2000), erratum of Ref. 11.
    [CrossRef]
  13. C. Fabre, J. B. Fouet, and A. Maître, Opt. Lett. 25, 76 (2000).
    [CrossRef]
  14. P. Lodahl and M. Saffman, Phys. Rev. A 60, 3251 (1999).
    [CrossRef]
  15. A. Gatti and L. Lugiato, Phys. Rev. A 52, 1675 (1995).
    [CrossRef] [PubMed]
  16. M. J. Collett and R. B. Levien, Phys. Rev. A 43, 5068 (1991).
    [CrossRef] [PubMed]
  17. R. Paschotta, M. Collett, P. Kürz, K. Fiedler, H.-A. Bachor, and J. Mlynek, Phys. Rev. Lett. 72, 3807 (1994).
    [CrossRef] [PubMed]
  18. C. Cabrillo, J. L. Roldán, and P. García-Fernández, J. Opt. Soc. Am. B 17, 440 (2000).
    [CrossRef]
  19. Note that the model studied in Ref. 17 did not include the effect of phase mismatch on the second harmonic, which is described by a factor of g in Eq. (1b).
  20. C. W. Gardiner and M. J. Collett, Phys. Rev. A 31, 3761 (1985).
    [CrossRef] [PubMed]
  21. S. Reynaud and A. Heidmann, Opt. Commun. 71, 209 (1989).
    [CrossRef]
  22. The abrupt transition between k=0 and k finite as a result of the delta functions is an artifact of the assumption of an infinitely extended plane-wave pump. Allowing for a pump beam of finite extent would ease the transition.
  23. J. Jeffers and G.-L. Oppo, Phys. Rev. A 60, 2393 (1999).
    [CrossRef]
  24. L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
    [CrossRef] [PubMed]

2000 (3)

1999 (6)

J. Jeffers and G.-L. Oppo, Phys. Rev. A 60, 2393 (1999).
[CrossRef]

P. Lodahl and M. Saffman, Phys. Rev. A 60, 3251 (1999).
[CrossRef]

M. Vaupel, A. Maître, and C. Fabre, Phys. Rev. Lett. 83, 5278 (1999).
[CrossRef]

M. I. Kolobov, Rev. Mod. Phys. 71, 1539 (1999).
[CrossRef]

A. Gatti, E. Brambilla, L. A. Lugiato, and M. I. Kolobov, Phys. Rev. Lett. 83, 1763 (1999).
[CrossRef]

S.-K. Choi, M. Vasilyev, and P. Kumar, Phys. Rev. Lett. 83, 1938 (1999).
[CrossRef]

1997 (2)

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

E. M. Nagasako, R. W. Boyd, and G. S. Agarwal, Phys. Rev. A 55, 1412 (1997).
[CrossRef]

1995 (1)

A. Gatti and L. Lugiato, Phys. Rev. A 52, 1675 (1995).
[CrossRef] [PubMed]

1994 (2)

R. Paschotta, M. Collett, P. Kürz, K. Fiedler, H.-A. Bachor, and J. Mlynek, Phys. Rev. Lett. 72, 3807 (1994).
[CrossRef] [PubMed]

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994).
[CrossRef] [PubMed]

1993 (2)

L. A. Lugiato and A. Gatti, Phys. Rev. Lett. 70, 3868 (1993).
[CrossRef] [PubMed]

G. Grynberg and L. A. Lugiato, Opt. Commun. 101, 69 (1993).
[CrossRef]

1992 (2)

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

L. A. Lugiato and F. Castelli, Phys. Rev. Lett. 68, 3284 (1992).
[CrossRef] [PubMed]

1991 (1)

M. J. Collett and R. B. Levien, Phys. Rev. A 43, 5068 (1991).
[CrossRef] [PubMed]

1989 (1)

S. Reynaud and A. Heidmann, Opt. Commun. 71, 209 (1989).
[CrossRef]

1987 (1)

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef] [PubMed]

1985 (1)

C. W. Gardiner and M. J. Collett, Phys. Rev. A 31, 3761 (1985).
[CrossRef] [PubMed]

Agarwal, G. S.

E. M. Nagasako, R. W. Boyd, and G. S. Agarwal, Phys. Rev. A 55, 1412 (1997).
[CrossRef]

Baboiu, D.-M.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

Bachor, H.-A.

R. Paschotta, M. Collett, P. Kürz, K. Fiedler, H.-A. Bachor, and J. Mlynek, Phys. Rev. Lett. 72, 3807 (1994).
[CrossRef] [PubMed]

Boyd, R. W.

E. M. Nagasako, R. W. Boyd, and G. S. Agarwal, Phys. Rev. A 55, 1412 (1997).
[CrossRef]

Brambilla, E.

A. Gatti, E. Brambilla, L. A. Lugiato, and M. I. Kolobov, Phys. Rev. Lett. 83, 1763 (1999).
[CrossRef]

Brambilla, M.

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994).
[CrossRef] [PubMed]

Cabrillo, C.

Castelli, F.

L. A. Lugiato and F. Castelli, Phys. Rev. Lett. 68, 3284 (1992).
[CrossRef] [PubMed]

Choi, S.-K.

S.-K. Choi, M. Vasilyev, and P. Kumar, Phys. Rev. Lett. 84, 1361 (2000), erratum of Ref. 11.
[CrossRef]

S.-K. Choi, M. Vasilyev, and P. Kumar, Phys. Rev. Lett. 83, 1938 (1999).
[CrossRef]

Collett, M.

R. Paschotta, M. Collett, P. Kürz, K. Fiedler, H.-A. Bachor, and J. Mlynek, Phys. Rev. Lett. 72, 3807 (1994).
[CrossRef] [PubMed]

Collett, M. J.

M. J. Collett and R. B. Levien, Phys. Rev. A 43, 5068 (1991).
[CrossRef] [PubMed]

C. W. Gardiner and M. J. Collett, Phys. Rev. A 31, 3761 (1985).
[CrossRef] [PubMed]

Fabre, C.

C. Fabre, J. B. Fouet, and A. Maître, Opt. Lett. 25, 76 (2000).
[CrossRef]

M. Vaupel, A. Maître, and C. Fabre, Phys. Rev. Lett. 83, 5278 (1999).
[CrossRef]

Fiedler, K.

R. Paschotta, M. Collett, P. Kürz, K. Fiedler, H.-A. Bachor, and J. Mlynek, Phys. Rev. Lett. 72, 3807 (1994).
[CrossRef] [PubMed]

Fouet, J. B.

Fuerst, R. A.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

García-Fernández, P.

Gardiner, C. W.

C. W. Gardiner and M. J. Collett, Phys. Rev. A 31, 3761 (1985).
[CrossRef] [PubMed]

Gatti, A.

A. Gatti, E. Brambilla, L. A. Lugiato, and M. I. Kolobov, Phys. Rev. Lett. 83, 1763 (1999).
[CrossRef]

A. Gatti and L. Lugiato, Phys. Rev. A 52, 1675 (1995).
[CrossRef] [PubMed]

L. A. Lugiato and A. Gatti, Phys. Rev. Lett. 70, 3868 (1993).
[CrossRef] [PubMed]

Grynberg, G.

G. Grynberg and L. A. Lugiato, Opt. Commun. 101, 69 (1993).
[CrossRef]

Heidmann, A.

S. Reynaud and A. Heidmann, Opt. Commun. 71, 209 (1989).
[CrossRef]

Jeffers, J.

J. Jeffers and G.-L. Oppo, Phys. Rev. A 60, 2393 (1999).
[CrossRef]

Kimble, H. J.

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

Kolobov, M. I.

M. I. Kolobov, Rev. Mod. Phys. 71, 1539 (1999).
[CrossRef]

A. Gatti, E. Brambilla, L. A. Lugiato, and M. I. Kolobov, Phys. Rev. Lett. 83, 1763 (1999).
[CrossRef]

Kumar, P.

S.-K. Choi, M. Vasilyev, and P. Kumar, Phys. Rev. Lett. 84, 1361 (2000), erratum of Ref. 11.
[CrossRef]

S.-K. Choi, M. Vasilyev, and P. Kumar, Phys. Rev. Lett. 83, 1938 (1999).
[CrossRef]

Kürz, P.

R. Paschotta, M. Collett, P. Kürz, K. Fiedler, H.-A. Bachor, and J. Mlynek, Phys. Rev. Lett. 72, 3807 (1994).
[CrossRef] [PubMed]

Lawrence, B.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

Lefever, R.

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef] [PubMed]

Levien, R. B.

M. J. Collett and R. B. Levien, Phys. Rev. A 43, 5068 (1991).
[CrossRef] [PubMed]

Lodahl, P.

P. Lodahl and M. Saffman, Phys. Rev. A 60, 3251 (1999).
[CrossRef]

Lugiato, L.

A. Gatti and L. Lugiato, Phys. Rev. A 52, 1675 (1995).
[CrossRef] [PubMed]

Lugiato, L. A.

A. Gatti, E. Brambilla, L. A. Lugiato, and M. I. Kolobov, Phys. Rev. Lett. 83, 1763 (1999).
[CrossRef]

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994).
[CrossRef] [PubMed]

L. A. Lugiato and A. Gatti, Phys. Rev. Lett. 70, 3868 (1993).
[CrossRef] [PubMed]

G. Grynberg and L. A. Lugiato, Opt. Commun. 101, 69 (1993).
[CrossRef]

L. A. Lugiato and F. Castelli, Phys. Rev. Lett. 68, 3284 (1992).
[CrossRef] [PubMed]

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef] [PubMed]

Maître, A.

C. Fabre, J. B. Fouet, and A. Maître, Opt. Lett. 25, 76 (2000).
[CrossRef]

M. Vaupel, A. Maître, and C. Fabre, Phys. Rev. Lett. 83, 5278 (1999).
[CrossRef]

Mlynek, J.

R. Paschotta, M. Collett, P. Kürz, K. Fiedler, H.-A. Bachor, and J. Mlynek, Phys. Rev. Lett. 72, 3807 (1994).
[CrossRef] [PubMed]

Nagasako, E. M.

E. M. Nagasako, R. W. Boyd, and G. S. Agarwal, Phys. Rev. A 55, 1412 (1997).
[CrossRef]

Oppo, G.-L.

J. Jeffers and G.-L. Oppo, Phys. Rev. A 60, 2393 (1999).
[CrossRef]

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994).
[CrossRef] [PubMed]

Paschotta, R.

R. Paschotta, M. Collett, P. Kürz, K. Fiedler, H.-A. Bachor, and J. Mlynek, Phys. Rev. Lett. 72, 3807 (1994).
[CrossRef] [PubMed]

Reynaud, S.

S. Reynaud and A. Heidmann, Opt. Commun. 71, 209 (1989).
[CrossRef]

Roldán, J. L.

Saffman, M.

P. Lodahl and M. Saffman, Phys. Rev. A 60, 3251 (1999).
[CrossRef]

Stegeman, G. I.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

Torruellas, W. E.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

Trillo, S.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

Vasilyev, M.

S.-K. Choi, M. Vasilyev, and P. Kumar, Phys. Rev. Lett. 84, 1361 (2000), erratum of Ref. 11.
[CrossRef]

S.-K. Choi, M. Vasilyev, and P. Kumar, Phys. Rev. Lett. 83, 1938 (1999).
[CrossRef]

Vaupel, M.

M. Vaupel, A. Maître, and C. Fabre, Phys. Rev. Lett. 83, 5278 (1999).
[CrossRef]

Wabnitz, S.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

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

Opt. Commun. (2)

G. Grynberg and L. A. Lugiato, Opt. Commun. 101, 69 (1993).
[CrossRef]

S. Reynaud and A. Heidmann, Opt. Commun. 71, 209 (1989).
[CrossRef]

Opt. Lett. (1)

Phys. Rep. (1)

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

Phys. Rev. A (7)

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994).
[CrossRef] [PubMed]

E. M. Nagasako, R. W. Boyd, and G. S. Agarwal, Phys. Rev. A 55, 1412 (1997).
[CrossRef]

J. Jeffers and G.-L. Oppo, Phys. Rev. A 60, 2393 (1999).
[CrossRef]

P. Lodahl and M. Saffman, Phys. Rev. A 60, 3251 (1999).
[CrossRef]

A. Gatti and L. Lugiato, Phys. Rev. A 52, 1675 (1995).
[CrossRef] [PubMed]

M. J. Collett and R. B. Levien, Phys. Rev. A 43, 5068 (1991).
[CrossRef] [PubMed]

C. W. Gardiner and M. J. Collett, Phys. Rev. A 31, 3761 (1985).
[CrossRef] [PubMed]

Phys. Rev. Lett. (9)

R. Paschotta, M. Collett, P. Kürz, K. Fiedler, H.-A. Bachor, and J. Mlynek, Phys. Rev. Lett. 72, 3807 (1994).
[CrossRef] [PubMed]

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, L. A. Lugiato, and M. I. Kolobov, Phys. Rev. Lett. 83, 1763 (1999).
[CrossRef]

S.-K. Choi, M. Vasilyev, and P. Kumar, Phys. Rev. Lett. 83, 1938 (1999).
[CrossRef]

S.-K. Choi, M. Vasilyev, and P. Kumar, Phys. Rev. Lett. 84, 1361 (2000), erratum of Ref. 11.
[CrossRef]

M. Vaupel, A. Maître, and C. Fabre, Phys. Rev. Lett. 83, 5278 (1999).
[CrossRef]

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

L. A. Lugiato and F. Castelli, Phys. Rev. Lett. 68, 3284 (1992).
[CrossRef] [PubMed]

L. A. Lugiato and A. Gatti, Phys. Rev. Lett. 70, 3868 (1993).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

M. I. Kolobov, Rev. Mod. Phys. 71, 1539 (1999).
[CrossRef]

Other (2)

The abrupt transition between k=0 and k finite as a result of the delta functions is an artifact of the assumption of an infinitely extended plane-wave pump. Allowing for a pump beam of finite extent would ease the transition.

Note that the model studied in Ref. 17 did not include the effect of phase mismatch on the second harmonic, which is described by a factor of g in Eq. (1b).

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Figures (2)

Fig. 1
Fig. 1

Squeezing spectra for the amplitude (solid curves) and the phase (dashed curves) quadratures as a function of transverse wave number for Δ=ξ=ω=0 and A10=1/6. The thinner and thicker curves are the fundamental and the second harmonic, respectively.

Fig. 2
Fig. 2

Squeezing spectra for the S1+θ=0 quadrature for Δ=0, ξ=5, ω=0, and A102=0.6 (thinner curve), A102=1.3, (dashed curve), and A102=2.5 (bolder solid curve). The threshold for spatial instability sets in at A102=3.56.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

tAˆ1= -1+iΔAˆ1+fξAˆ1Aˆ12+i2Aˆ1-g*ξAˆ1+Aˆ2,inAˆ,1,in,
Aˆ2,out= gξAˆ12+Aˆ2,in.
tb1= -1+iΔb1+2fξA102b1+fξA102b1*+i2b1-g*ξA10*b2,in+b1,in,
b1,out= 2b1-b1,in,
b2,out= 2gξA10b1+b2,in.
Pj,outθ,k,ω= βj,outk,ωexp-iθ+βj,out*k,-ωexpiθ,
Sj+θ,k,ω=Pj,out+θ,k,ω2Pj,in+θ,k,ω2=Pj,outθ,k,ω2+Pj,outθ,k,ωPj,out*θ,-k,ωPj,inθ,k,ω2,

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