Abstract

Quantum noise in a model of singly resonant frequency doubling, including phase mismatch and driving in the harmonic mode, is analyzed. The use of a nonlinear normalization allows us to disentangle in the spectra the squeezing induced by the system dynamics from the deleterious effect of the noise coming from the various inputs. The physical insight gained permits the elaboration of general criteria to optimize noise-suppression performance. The subsequent application to the specific system here addressed reveals excellent squeezing behavior. In particular, unlimited degrees of squeezing in the harmonic mode are possible by means of an adequate phase mismatch or driving in the harmonic mode. This is in contrast with the standard phase-matched second-harmonic generation in which the squeezing is limited to 1/9. The applicability of the model, as well as possible experimental implementations, is extensively discussed.

© 2000 Optical Society of America

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  32. K. Kasai, G. Jiangrui, and C. Fabre, “Observation of squeezing using cascaded nonlinearity,” Europhys. Lett. 40, 25–30 (1997).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  36. H. Schmidt, K. L. Campman, and A. Imamoglu, “Tunneling induced transparency: Fano interference in intersubband transitions,” Appl. Phys. Lett. 70, 3455–3457 (1997).
    [CrossRef]
  37. J. Faist, F. Capasso, and L. N. Pfeiffer, “Controlling the sign of quantum interference by tunnelling from quantum wells,” Nature 390, 589–591 (1997).
    [CrossRef]
  38. H. Schmidt and A. Imamoglu, “Nonlinear optical devices based on a transparency in semiconductors intersubband transitions,” Opt. Commun. 131, 333–338 (1996).
    [CrossRef]
  39. S. Schiller, G. Breitenbach, and J. Mlynek, “Subharmonic-pumped continuous-wave parametric oscillator,” Appl. Phys. Lett. 68, 3374–3376 (1996).
    [CrossRef]
  40. S. Schiller, R. Bruckmeier, and A. G. White, “Classical and quantum properties of the subharmonic pumped parametric oscillator,” Opt. Commun. 138, 158–171 (1997).
    [CrossRef]
  41. J. L. Sorensen and E. S. Polzik, “Internally pumped subthreshold OPO,” Appl. Phys. B 66, 711–718 (1998).
    [CrossRef]
  42. A. G. White, P. K. Lam, and H.-A. Bachor, “Classical and quantum signatures of competing χ(2) nonlinearities,” Phys. Rev. A 55, 4511–4515 (1997).
    [CrossRef]

1998 (2)

K. V. Kheruntsyan, G. Yu. Kryuchkyan, and K. G. Petrosyan, “Controlling instability and squeezing from a cascaded frequency doubler,” Phys. Rev. A 57, 535–547 (1998).
[CrossRef]

J. L. Sorensen and E. S. Polzik, “Internally pumped subthreshold OPO,” Appl. Phys. B 66, 711–718 (1998).
[CrossRef]

1997 (8)

A. G. White, P. K. Lam, and H.-A. Bachor, “Classical and quantum signatures of competing χ(2) nonlinearities,” Phys. Rev. A 55, 4511–4515 (1997).
[CrossRef]

S. Schiller, R. Bruckmeier, and A. G. White, “Classical and quantum properties of the subharmonic pumped parametric oscillator,” Opt. Commun. 138, 158–171 (1997).
[CrossRef]

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471–475 (1997).
[CrossRef]

C. Cabrillo, J. L. Roldán, and P. García-Fernández, “Squeezing enhancement by competing nonlinearities: almost perfect squeezing without instabilities,” Phys. Rev. A 56, 5131–5134 (1997).
[CrossRef]

K. V. Kheruntsyan, D. S. Krahmer, and K. G. Petrossian, “Wigner function for a generalized model of parametric oscillator: phase-space tristability, competition and nonclassical effects,” Opt. Commun. 139, 157–164 (1997).
[CrossRef]

K. Kasai, G. Jiangrui, and C. Fabre, “Observation of squeezing using cascaded nonlinearity,” Europhys. Lett. 40, 25–30 (1997).
[CrossRef]

H. Schmidt, K. L. Campman, and A. Imamoglu, “Tunneling induced transparency: Fano interference in intersubband transitions,” Appl. Phys. Lett. 70, 3455–3457 (1997).
[CrossRef]

J. Faist, F. Capasso, and L. N. Pfeiffer, “Controlling the sign of quantum interference by tunnelling from quantum wells,” Nature 390, 589–591 (1997).
[CrossRef]

1996 (8)

H. Schmidt and A. Imamoglu, “Nonlinear optical devices based on a transparency in semiconductors intersubband transitions,” Opt. Commun. 131, 333–338 (1996).
[CrossRef]

S. Schiller, G. Breitenbach, and J. Mlynek, “Subharmonic-pumped continuous-wave parametric oscillator,” Appl. Phys. Lett. 68, 3374–3376 (1996).
[CrossRef]

A. G. White, M. S. Taubman, and H.-A. Bachor, “Experimental test of modular noise propagation theory for quantum optics,” Phys. Rev. A 54, 3400–3404 (1996).
[CrossRef] [PubMed]

G. Yu. Kryuchkyan and K. V. Kheruntsyan, “Exact quantum theory of a parametrically driven dissipative anharmonic oscillator,” Opt. Commun. 127, 230–236 (1996).
[CrossRef]

A. G. White, J. Mlynek, and S. Schiller, “Cascaded second-order nonlinearity in an optical cavity,” Europhys. Lett. 35, 425–430 (1996).
[CrossRef]

K. Schneider, R. Bruckmeier, H. Hansen, S. Schiller, and J. Mlynek, “Bright squeezed-light generation by a continuous-wave semimonolithic parametric amplifier,” Opt. Lett. 21, 1396–1398 (1996).
[CrossRef] [PubMed]

S. Youn, S.-K. Choi, P. Kumar, and R.-D. Li, “Observation of sub-Poissonian light in traveling-wave second-harmonic generation,” Opt. Lett. 21, 1597–1599 (1996).
[CrossRef] [PubMed]

J. Faist, C. Sirtori, F. Capasso, S.-N. G. Chu, L. N. Pfeiffer, and K. W. West, “Tunable Fano interference in intersubband absorption,” Opt. Lett. 21, 985–987 (1996).
[CrossRef] [PubMed]

1995 (7)

1994 (1)

R. Paschotta, M. Collett, and P. Kurz, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

1993 (2)

P. Kurz, R. Paschotta, K. Fiedler, and J. Mlynek, “Bright squeezed light by second-harmonic generation in a monolithic resonator,” Europhys. Lett. 24, 449–454 (1993).
[CrossRef]

C. Cabrillo and F. J. Bermejo, “Large quadrature squeezing at high intensities,” Phys. Rev. A 48, 2433–2436 (1993).
[CrossRef] [PubMed]

1992 (4)

C. Cabrillo, F. J. Bermejo, P. García-Fernández, R. Toral, P. Colet, and M. San Miguel, “Transient behavior of a parametric amplifier with an added fourth-order interaction,” Phys. Rev. A 45, 3216–3223 (1992).
[CrossRef] [PubMed]

C. Cabrillo and F. J. Bermejo, “Control of the squeezing spectrum by means of a fourth-order interaction,” Phys. Lett. A 170, 300–304 (1992).
[CrossRef]

C. Sirtori, F. Capasso, and D. L. Sivco, “Resonant Stark tuning of second-order susceptibility in coupled quantum wells,” Appl. Phys. Lett. 60, 151–153 (1992).
[CrossRef]

E. S. Polzik, J. Carri, and H. J. Kimble, “Atomic spectroscopy with squeezed light for sensitivity beyond the vacuum state limit,” Appl. Phys. B 55, 279–290 (1992).
[CrossRef]

1991 (2)

P. García-Fernández, P. Colet, R. Toral, M. San Miguel, and F. J. Bermejo, “Squeezing resulting from a fourth-order interaction in a degenerate parametric amplifier with absorption losses,” Phys. Rev. A 43, 4923–4929 (1991).
[CrossRef]

M. J. Collett and R. B. Levien, “Two-photon-loss model of intracavity second-harmonic generation,” Phys. Rev. A 43, 5068–5072 (1991).
[CrossRef] [PubMed]

1990 (2)

A. Sizmann, R. J. Horowicz, and G. Wagner, “Observation of amplitude squeezing of the up-converted mode in second harmonic generation,” Opt. Commun. 80, 138–142 (1990).
[CrossRef]

C. Fabre, E. Giacobino, and A. Heidmann, “Squeezing in detuned degenerate optical parametric oscillators,” Quantum Opt. 2, 159–187 (1990).
[CrossRef]

1989 (1)

S. Reynaud, C. Fabre, and E. Giacobino, “Photon noise reduction by passive optical bistable systems,” Phys. Rev. A 40, 1440–1446 (1989).
[CrossRef] [PubMed]

1988 (1)

S. F. Pereira, X. Min, H. J. Kimble, and J. L. Hall, “Generation of squeezed light by intracavity frequency doubling,” Phys. Rev. A 38, 4931–4934 (1988).
[CrossRef] [PubMed]

1984 (1)

M. J. Collet and C. W. Gardiner, “Squeezing of intracavity and travelling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[CrossRef]

Bachor, H.-A.

A. G. White, P. K. Lam, and H.-A. Bachor, “Classical and quantum signatures of competing χ(2) nonlinearities,” Phys. Rev. A 55, 4511–4515 (1997).
[CrossRef]

A. G. White, M. S. Taubman, and H.-A. Bachor, “Experimental test of modular noise propagation theory for quantum optics,” Phys. Rev. A 54, 3400–3404 (1996).
[CrossRef] [PubMed]

T. C. Ralph, M. S. Taubman, A. G. White, D. E. McClelland, and H.-A. Bachor, “Squeezed light from second-harmonic generation: experiment versus theory,” Opt. Lett. 20, 1316–1318 (1995).
[CrossRef] [PubMed]

Bermejo, F. J.

C. Cabrillo and F. J. Bermejo, “Large quadrature squeezing at high intensities,” Phys. Rev. A 48, 2433–2436 (1993).
[CrossRef] [PubMed]

C. Cabrillo and F. J. Bermejo, “Control of the squeezing spectrum by means of a fourth-order interaction,” Phys. Lett. A 170, 300–304 (1992).
[CrossRef]

C. Cabrillo, F. J. Bermejo, P. García-Fernández, R. Toral, P. Colet, and M. San Miguel, “Transient behavior of a parametric amplifier with an added fourth-order interaction,” Phys. Rev. A 45, 3216–3223 (1992).
[CrossRef] [PubMed]

P. García-Fernández, P. Colet, R. Toral, M. San Miguel, and F. J. Bermejo, “Squeezing resulting from a fourth-order interaction in a degenerate parametric amplifier with absorption losses,” Phys. Rev. A 43, 4923–4929 (1991).
[CrossRef]

Breitenbach, G.

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471–475 (1997).
[CrossRef]

S. Schiller, G. Breitenbach, and J. Mlynek, “Subharmonic-pumped continuous-wave parametric oscillator,” Appl. Phys. Lett. 68, 3374–3376 (1996).
[CrossRef]

G. Breitenbach, T. Müller, S. F. Pereira, J.-Ph. Poizat, S. Schiller, and J. Mlynek, “Squeezed vacuum from a monolithic optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2304–2309 (1995).
[CrossRef]

Bruckmeier, R.

S. Schiller, R. Bruckmeier, and A. G. White, “Classical and quantum properties of the subharmonic pumped parametric oscillator,” Opt. Commun. 138, 158–171 (1997).
[CrossRef]

K. Schneider, R. Bruckmeier, H. Hansen, S. Schiller, and J. Mlynek, “Bright squeezed-light generation by a continuous-wave semimonolithic parametric amplifier,” Opt. Lett. 21, 1396–1398 (1996).
[CrossRef] [PubMed]

Cabrillo, C.

C. Cabrillo, J. L. Roldán, and P. García-Fernández, “Squeezing enhancement by competing nonlinearities: almost perfect squeezing without instabilities,” Phys. Rev. A 56, 5131–5134 (1997).
[CrossRef]

C. Cabrillo and F. J. Bermejo, “Large quadrature squeezing at high intensities,” Phys. Rev. A 48, 2433–2436 (1993).
[CrossRef] [PubMed]

C. Cabrillo, F. J. Bermejo, P. García-Fernández, R. Toral, P. Colet, and M. San Miguel, “Transient behavior of a parametric amplifier with an added fourth-order interaction,” Phys. Rev. A 45, 3216–3223 (1992).
[CrossRef] [PubMed]

C. Cabrillo and F. J. Bermejo, “Control of the squeezing spectrum by means of a fourth-order interaction,” Phys. Lett. A 170, 300–304 (1992).
[CrossRef]

Campman, K. L.

H. Schmidt, K. L. Campman, and A. Imamoglu, “Tunneling induced transparency: Fano interference in intersubband transitions,” Appl. Phys. Lett. 70, 3455–3457 (1997).
[CrossRef]

Capasso, F.

J. Faist, F. Capasso, and L. N. Pfeiffer, “Controlling the sign of quantum interference by tunnelling from quantum wells,” Nature 390, 589–591 (1997).
[CrossRef]

J. Faist, C. Sirtori, F. Capasso, S.-N. G. Chu, L. N. Pfeiffer, and K. W. West, “Tunable Fano interference in intersubband absorption,” Opt. Lett. 21, 985–987 (1996).
[CrossRef] [PubMed]

C. Sirtori, F. Capasso, and D. L. Sivco, “Resonant Stark tuning of second-order susceptibility in coupled quantum wells,” Appl. Phys. Lett. 60, 151–153 (1992).
[CrossRef]

Carri, J.

E. S. Polzik, J. Carri, and H. J. Kimble, “Atomic spectroscopy with squeezed light for sensitivity beyond the vacuum state limit,” Appl. Phys. B 55, 279–290 (1992).
[CrossRef]

Choi, S.-K.

Chu, S.-N. G.

Colet, P.

C. Cabrillo, F. J. Bermejo, P. García-Fernández, R. Toral, P. Colet, and M. San Miguel, “Transient behavior of a parametric amplifier with an added fourth-order interaction,” Phys. Rev. A 45, 3216–3223 (1992).
[CrossRef] [PubMed]

P. García-Fernández, P. Colet, R. Toral, M. San Miguel, and F. J. Bermejo, “Squeezing resulting from a fourth-order interaction in a degenerate parametric amplifier with absorption losses,” Phys. Rev. A 43, 4923–4929 (1991).
[CrossRef]

Collet, M. J.

M. J. Collet and C. W. Gardiner, “Squeezing of intracavity and travelling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[CrossRef]

Collett, M.

R. Paschotta, M. Collett, and P. Kurz, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

Collett, M. J.

M. J. Collett and R. B. Levien, “Two-photon-loss model of intracavity second-harmonic generation,” Phys. Rev. A 43, 5068–5072 (1991).
[CrossRef] [PubMed]

Fabre, C.

K. Kasai, G. Jiangrui, and C. Fabre, “Observation of squeezing using cascaded nonlinearity,” Europhys. Lett. 40, 25–30 (1997).
[CrossRef]

C. Fabre, E. Giacobino, and A. Heidmann, “Squeezing in detuned degenerate optical parametric oscillators,” Quantum Opt. 2, 159–187 (1990).
[CrossRef]

S. Reynaud, C. Fabre, and E. Giacobino, “Photon noise reduction by passive optical bistable systems,” Phys. Rev. A 40, 1440–1446 (1989).
[CrossRef] [PubMed]

Faist, J.

J. Faist, F. Capasso, and L. N. Pfeiffer, “Controlling the sign of quantum interference by tunnelling from quantum wells,” Nature 390, 589–591 (1997).
[CrossRef]

J. Faist, C. Sirtori, F. Capasso, S.-N. G. Chu, L. N. Pfeiffer, and K. W. West, “Tunable Fano interference in intersubband absorption,” Opt. Lett. 21, 985–987 (1996).
[CrossRef] [PubMed]

Fiedler, K.

P. Kurz, R. Paschotta, K. Fiedler, and J. Mlynek, “Bright squeezed light by second-harmonic generation in a monolithic resonator,” Europhys. Lett. 24, 449–454 (1993).
[CrossRef]

García-Fernández, P.

C. Cabrillo, J. L. Roldán, and P. García-Fernández, “Squeezing enhancement by competing nonlinearities: almost perfect squeezing without instabilities,” Phys. Rev. A 56, 5131–5134 (1997).
[CrossRef]

C. Cabrillo, F. J. Bermejo, P. García-Fernández, R. Toral, P. Colet, and M. San Miguel, “Transient behavior of a parametric amplifier with an added fourth-order interaction,” Phys. Rev. A 45, 3216–3223 (1992).
[CrossRef] [PubMed]

P. García-Fernández, P. Colet, R. Toral, M. San Miguel, and F. J. Bermejo, “Squeezing resulting from a fourth-order interaction in a degenerate parametric amplifier with absorption losses,” Phys. Rev. A 43, 4923–4929 (1991).
[CrossRef]

Gardiner, C. W.

M. J. Collet and C. W. Gardiner, “Squeezing of intracavity and travelling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[CrossRef]

Giacobino, E.

C. Fabre, E. Giacobino, and A. Heidmann, “Squeezing in detuned degenerate optical parametric oscillators,” Quantum Opt. 2, 159–187 (1990).
[CrossRef]

S. Reynaud, C. Fabre, and E. Giacobino, “Photon noise reduction by passive optical bistable systems,” Phys. Rev. A 40, 1440–1446 (1989).
[CrossRef] [PubMed]

Hall, J. L.

S. F. Pereira, X. Min, H. J. Kimble, and J. L. Hall, “Generation of squeezed light by intracavity frequency doubling,” Phys. Rev. A 38, 4931–4934 (1988).
[CrossRef] [PubMed]

Hansen, H.

Heidmann, A.

C. Fabre, E. Giacobino, and A. Heidmann, “Squeezing in detuned degenerate optical parametric oscillators,” Quantum Opt. 2, 159–187 (1990).
[CrossRef]

Horowicz, R. J.

A. Sizmann, R. J. Horowicz, and G. Wagner, “Observation of amplitude squeezing of the up-converted mode in second harmonic generation,” Opt. Commun. 80, 138–142 (1990).
[CrossRef]

Imamoglu, A.

H. Schmidt, K. L. Campman, and A. Imamoglu, “Tunneling induced transparency: Fano interference in intersubband transitions,” Appl. Phys. Lett. 70, 3455–3457 (1997).
[CrossRef]

H. Schmidt and A. Imamoglu, “Nonlinear optical devices based on a transparency in semiconductors intersubband transitions,” Opt. Commun. 131, 333–338 (1996).
[CrossRef]

Jiangrui, G.

K. Kasai, G. Jiangrui, and C. Fabre, “Observation of squeezing using cascaded nonlinearity,” Europhys. Lett. 40, 25–30 (1997).
[CrossRef]

Kasai, K.

K. Kasai, G. Jiangrui, and C. Fabre, “Observation of squeezing using cascaded nonlinearity,” Europhys. Lett. 40, 25–30 (1997).
[CrossRef]

Kheruntsyan, K. V.

K. V. Kheruntsyan, G. Yu. Kryuchkyan, and K. G. Petrosyan, “Controlling instability and squeezing from a cascaded frequency doubler,” Phys. Rev. A 57, 535–547 (1998).
[CrossRef]

K. V. Kheruntsyan, D. S. Krahmer, and K. G. Petrossian, “Wigner function for a generalized model of parametric oscillator: phase-space tristability, competition and nonclassical effects,” Opt. Commun. 139, 157–164 (1997).
[CrossRef]

G. Yu. Kryuchkyan and K. V. Kheruntsyan, “Exact quantum theory of a parametrically driven dissipative anharmonic oscillator,” Opt. Commun. 127, 230–236 (1996).
[CrossRef]

Kimble, H. J.

E. S. Polzik, J. Carri, and H. J. Kimble, “Atomic spectroscopy with squeezed light for sensitivity beyond the vacuum state limit,” Appl. Phys. B 55, 279–290 (1992).
[CrossRef]

S. F. Pereira, X. Min, H. J. Kimble, and J. L. Hall, “Generation of squeezed light by intracavity frequency doubling,” Phys. Rev. A 38, 4931–4934 (1988).
[CrossRef] [PubMed]

Kohler, S.

S. Schiller, S. Kohler, R. Paschotta, and J. Mlynek, “Squeezing and quantum nondemolition measurements with an optical parametric amplifier,” Appl. Phys. B 60, S77–S88 (1995).

Krahmer, D. S.

K. V. Kheruntsyan, D. S. Krahmer, and K. G. Petrossian, “Wigner function for a generalized model of parametric oscillator: phase-space tristability, competition and nonclassical effects,” Opt. Commun. 139, 157–164 (1997).
[CrossRef]

Kryuchkyan, G. Yu.

K. V. Kheruntsyan, G. Yu. Kryuchkyan, and K. G. Petrosyan, “Controlling instability and squeezing from a cascaded frequency doubler,” Phys. Rev. A 57, 535–547 (1998).
[CrossRef]

G. Yu. Kryuchkyan and K. V. Kheruntsyan, “Exact quantum theory of a parametrically driven dissipative anharmonic oscillator,” Opt. Commun. 127, 230–236 (1996).
[CrossRef]

Kumar, P.

Kurz, P.

R. Paschotta, M. Collett, and P. Kurz, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

P. Kurz, R. Paschotta, K. Fiedler, and J. Mlynek, “Bright squeezed light by second-harmonic generation in a monolithic resonator,” Europhys. Lett. 24, 449–454 (1993).
[CrossRef]

Lam, P. K.

A. G. White, P. K. Lam, and H.-A. Bachor, “Classical and quantum signatures of competing χ(2) nonlinearities,” Phys. Rev. A 55, 4511–4515 (1997).
[CrossRef]

Levien, R. B.

M. J. Collett and R. B. Levien, “Two-photon-loss model of intracavity second-harmonic generation,” Phys. Rev. A 43, 5068–5072 (1991).
[CrossRef] [PubMed]

Li, R.-D.

Marte, M. A. M.

M. A. M. Marte, “Sub-poissonian twin beams via competing nonlinearities,” Phys. Rev. Lett. 76, 4815–4818 (1995).
[CrossRef]

M. A. M. Marte, “Nonlinear dynamics and quantum noise for competing χ(2) nonlinearities,” J. Opt. Soc. Am. B 12, 2296–2303 (1995).
[CrossRef]

McClelland, D. E.

Min, X.

S. F. Pereira, X. Min, H. J. Kimble, and J. L. Hall, “Generation of squeezed light by intracavity frequency doubling,” Phys. Rev. A 38, 4931–4934 (1988).
[CrossRef] [PubMed]

Mlynek, J.

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471–475 (1997).
[CrossRef]

K. Schneider, R. Bruckmeier, H. Hansen, S. Schiller, and J. Mlynek, “Bright squeezed-light generation by a continuous-wave semimonolithic parametric amplifier,” Opt. Lett. 21, 1396–1398 (1996).
[CrossRef] [PubMed]

A. G. White, J. Mlynek, and S. Schiller, “Cascaded second-order nonlinearity in an optical cavity,” Europhys. Lett. 35, 425–430 (1996).
[CrossRef]

S. Schiller, G. Breitenbach, and J. Mlynek, “Subharmonic-pumped continuous-wave parametric oscillator,” Appl. Phys. Lett. 68, 3374–3376 (1996).
[CrossRef]

S. Schiller, S. Kohler, R. Paschotta, and J. Mlynek, “Squeezing and quantum nondemolition measurements with an optical parametric amplifier,” Appl. Phys. B 60, S77–S88 (1995).

G. Breitenbach, T. Müller, S. F. Pereira, J.-Ph. Poizat, S. Schiller, and J. Mlynek, “Squeezed vacuum from a monolithic optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2304–2309 (1995).
[CrossRef]

P. Kurz, R. Paschotta, K. Fiedler, and J. Mlynek, “Bright squeezed light by second-harmonic generation in a monolithic resonator,” Europhys. Lett. 24, 449–454 (1993).
[CrossRef]

Müller, T.

Paschotta, R.

S. Schiller, S. Kohler, R. Paschotta, and J. Mlynek, “Squeezing and quantum nondemolition measurements with an optical parametric amplifier,” Appl. Phys. B 60, S77–S88 (1995).

R. Paschotta, M. Collett, and P. Kurz, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

P. Kurz, R. Paschotta, K. Fiedler, and J. Mlynek, “Bright squeezed light by second-harmonic generation in a monolithic resonator,” Europhys. Lett. 24, 449–454 (1993).
[CrossRef]

Pereira, S. F.

G. Breitenbach, T. Müller, S. F. Pereira, J.-Ph. Poizat, S. Schiller, and J. Mlynek, “Squeezed vacuum from a monolithic optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2304–2309 (1995).
[CrossRef]

S. F. Pereira, X. Min, H. J. Kimble, and J. L. Hall, “Generation of squeezed light by intracavity frequency doubling,” Phys. Rev. A 38, 4931–4934 (1988).
[CrossRef] [PubMed]

Petrossian, K. G.

K. V. Kheruntsyan, D. S. Krahmer, and K. G. Petrossian, “Wigner function for a generalized model of parametric oscillator: phase-space tristability, competition and nonclassical effects,” Opt. Commun. 139, 157–164 (1997).
[CrossRef]

Petrosyan, K. G.

K. V. Kheruntsyan, G. Yu. Kryuchkyan, and K. G. Petrosyan, “Controlling instability and squeezing from a cascaded frequency doubler,” Phys. Rev. A 57, 535–547 (1998).
[CrossRef]

Pfeiffer, L. N.

J. Faist, F. Capasso, and L. N. Pfeiffer, “Controlling the sign of quantum interference by tunnelling from quantum wells,” Nature 390, 589–591 (1997).
[CrossRef]

J. Faist, C. Sirtori, F. Capasso, S.-N. G. Chu, L. N. Pfeiffer, and K. W. West, “Tunable Fano interference in intersubband absorption,” Opt. Lett. 21, 985–987 (1996).
[CrossRef] [PubMed]

Poizat, J.-Ph.

Polzik, E. S.

J. L. Sorensen and E. S. Polzik, “Internally pumped subthreshold OPO,” Appl. Phys. B 66, 711–718 (1998).
[CrossRef]

E. S. Polzik, J. Carri, and H. J. Kimble, “Atomic spectroscopy with squeezed light for sensitivity beyond the vacuum state limit,” Appl. Phys. B 55, 279–290 (1992).
[CrossRef]

Ralph, T. C.

Reynaud, S.

S. Reynaud, C. Fabre, and E. Giacobino, “Photon noise reduction by passive optical bistable systems,” Phys. Rev. A 40, 1440–1446 (1989).
[CrossRef] [PubMed]

Roldán, J. L.

C. Cabrillo, J. L. Roldán, and P. García-Fernández, “Squeezing enhancement by competing nonlinearities: almost perfect squeezing without instabilities,” Phys. Rev. A 56, 5131–5134 (1997).
[CrossRef]

San Miguel, M.

C. Cabrillo, F. J. Bermejo, P. García-Fernández, R. Toral, P. Colet, and M. San Miguel, “Transient behavior of a parametric amplifier with an added fourth-order interaction,” Phys. Rev. A 45, 3216–3223 (1992).
[CrossRef] [PubMed]

P. García-Fernández, P. Colet, R. Toral, M. San Miguel, and F. J. Bermejo, “Squeezing resulting from a fourth-order interaction in a degenerate parametric amplifier with absorption losses,” Phys. Rev. A 43, 4923–4929 (1991).
[CrossRef]

Schiller, S.

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471–475 (1997).
[CrossRef]

S. Schiller, R. Bruckmeier, and A. G. White, “Classical and quantum properties of the subharmonic pumped parametric oscillator,” Opt. Commun. 138, 158–171 (1997).
[CrossRef]

S. Schiller, G. Breitenbach, and J. Mlynek, “Subharmonic-pumped continuous-wave parametric oscillator,” Appl. Phys. Lett. 68, 3374–3376 (1996).
[CrossRef]

A. G. White, J. Mlynek, and S. Schiller, “Cascaded second-order nonlinearity in an optical cavity,” Europhys. Lett. 35, 425–430 (1996).
[CrossRef]

K. Schneider, R. Bruckmeier, H. Hansen, S. Schiller, and J. Mlynek, “Bright squeezed-light generation by a continuous-wave semimonolithic parametric amplifier,” Opt. Lett. 21, 1396–1398 (1996).
[CrossRef] [PubMed]

G. Breitenbach, T. Müller, S. F. Pereira, J.-Ph. Poizat, S. Schiller, and J. Mlynek, “Squeezed vacuum from a monolithic optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2304–2309 (1995).
[CrossRef]

S. Schiller, S. Kohler, R. Paschotta, and J. Mlynek, “Squeezing and quantum nondemolition measurements with an optical parametric amplifier,” Appl. Phys. B 60, S77–S88 (1995).

Schmidt, H.

H. Schmidt, K. L. Campman, and A. Imamoglu, “Tunneling induced transparency: Fano interference in intersubband transitions,” Appl. Phys. Lett. 70, 3455–3457 (1997).
[CrossRef]

H. Schmidt and A. Imamoglu, “Nonlinear optical devices based on a transparency in semiconductors intersubband transitions,” Opt. Commun. 131, 333–338 (1996).
[CrossRef]

Schneider, K.

Sirtori, C.

J. Faist, C. Sirtori, F. Capasso, S.-N. G. Chu, L. N. Pfeiffer, and K. W. West, “Tunable Fano interference in intersubband absorption,” Opt. Lett. 21, 985–987 (1996).
[CrossRef] [PubMed]

C. Sirtori, F. Capasso, and D. L. Sivco, “Resonant Stark tuning of second-order susceptibility in coupled quantum wells,” Appl. Phys. Lett. 60, 151–153 (1992).
[CrossRef]

Sivco, D. L.

C. Sirtori, F. Capasso, and D. L. Sivco, “Resonant Stark tuning of second-order susceptibility in coupled quantum wells,” Appl. Phys. Lett. 60, 151–153 (1992).
[CrossRef]

Sizmann, A.

A. Sizmann, R. J. Horowicz, and G. Wagner, “Observation of amplitude squeezing of the up-converted mode in second harmonic generation,” Opt. Commun. 80, 138–142 (1990).
[CrossRef]

Sorensen, J. L.

J. L. Sorensen and E. S. Polzik, “Internally pumped subthreshold OPO,” Appl. Phys. B 66, 711–718 (1998).
[CrossRef]

Sundar, K.

K. Sundar, “Highly amplitude squeezed states of the radiation field,” Phys. Rev. Lett. 75, 2116–2119 (1995).
[CrossRef] [PubMed]

Taubman, M. S.

A. G. White, M. S. Taubman, and H.-A. Bachor, “Experimental test of modular noise propagation theory for quantum optics,” Phys. Rev. A 54, 3400–3404 (1996).
[CrossRef] [PubMed]

T. C. Ralph, M. S. Taubman, A. G. White, D. E. McClelland, and H.-A. Bachor, “Squeezed light from second-harmonic generation: experiment versus theory,” Opt. Lett. 20, 1316–1318 (1995).
[CrossRef] [PubMed]

Toral, R.

C. Cabrillo, F. J. Bermejo, P. García-Fernández, R. Toral, P. Colet, and M. San Miguel, “Transient behavior of a parametric amplifier with an added fourth-order interaction,” Phys. Rev. A 45, 3216–3223 (1992).
[CrossRef] [PubMed]

P. García-Fernández, P. Colet, R. Toral, M. San Miguel, and F. J. Bermejo, “Squeezing resulting from a fourth-order interaction in a degenerate parametric amplifier with absorption losses,” Phys. Rev. A 43, 4923–4929 (1991).
[CrossRef]

Tsuchida, H.

Wagner, G.

A. Sizmann, R. J. Horowicz, and G. Wagner, “Observation of amplitude squeezing of the up-converted mode in second harmonic generation,” Opt. Commun. 80, 138–142 (1990).
[CrossRef]

West, K. W.

White, A. G.

A. G. White, P. K. Lam, and H.-A. Bachor, “Classical and quantum signatures of competing χ(2) nonlinearities,” Phys. Rev. A 55, 4511–4515 (1997).
[CrossRef]

S. Schiller, R. Bruckmeier, and A. G. White, “Classical and quantum properties of the subharmonic pumped parametric oscillator,” Opt. Commun. 138, 158–171 (1997).
[CrossRef]

A. G. White, M. S. Taubman, and H.-A. Bachor, “Experimental test of modular noise propagation theory for quantum optics,” Phys. Rev. A 54, 3400–3404 (1996).
[CrossRef] [PubMed]

A. G. White, J. Mlynek, and S. Schiller, “Cascaded second-order nonlinearity in an optical cavity,” Europhys. Lett. 35, 425–430 (1996).
[CrossRef]

T. C. Ralph, M. S. Taubman, A. G. White, D. E. McClelland, and H.-A. Bachor, “Squeezed light from second-harmonic generation: experiment versus theory,” Opt. Lett. 20, 1316–1318 (1995).
[CrossRef] [PubMed]

Youn, S.

Appl. Phys. B (3)

E. S. Polzik, J. Carri, and H. J. Kimble, “Atomic spectroscopy with squeezed light for sensitivity beyond the vacuum state limit,” Appl. Phys. B 55, 279–290 (1992).
[CrossRef]

S. Schiller, S. Kohler, R. Paschotta, and J. Mlynek, “Squeezing and quantum nondemolition measurements with an optical parametric amplifier,” Appl. Phys. B 60, S77–S88 (1995).

J. L. Sorensen and E. S. Polzik, “Internally pumped subthreshold OPO,” Appl. Phys. B 66, 711–718 (1998).
[CrossRef]

Appl. Phys. Lett. (3)

S. Schiller, G. Breitenbach, and J. Mlynek, “Subharmonic-pumped continuous-wave parametric oscillator,” Appl. Phys. Lett. 68, 3374–3376 (1996).
[CrossRef]

C. Sirtori, F. Capasso, and D. L. Sivco, “Resonant Stark tuning of second-order susceptibility in coupled quantum wells,” Appl. Phys. Lett. 60, 151–153 (1992).
[CrossRef]

H. Schmidt, K. L. Campman, and A. Imamoglu, “Tunneling induced transparency: Fano interference in intersubband transitions,” Appl. Phys. Lett. 70, 3455–3457 (1997).
[CrossRef]

Europhys. Lett. (3)

K. Kasai, G. Jiangrui, and C. Fabre, “Observation of squeezing using cascaded nonlinearity,” Europhys. Lett. 40, 25–30 (1997).
[CrossRef]

A. G. White, J. Mlynek, and S. Schiller, “Cascaded second-order nonlinearity in an optical cavity,” Europhys. Lett. 35, 425–430 (1996).
[CrossRef]

P. Kurz, R. Paschotta, K. Fiedler, and J. Mlynek, “Bright squeezed light by second-harmonic generation in a monolithic resonator,” Europhys. Lett. 24, 449–454 (1993).
[CrossRef]

J. Opt. Soc. Am. B (2)

Nature (2)

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471–475 (1997).
[CrossRef]

J. Faist, F. Capasso, and L. N. Pfeiffer, “Controlling the sign of quantum interference by tunnelling from quantum wells,” Nature 390, 589–591 (1997).
[CrossRef]

Opt. Commun. (5)

H. Schmidt and A. Imamoglu, “Nonlinear optical devices based on a transparency in semiconductors intersubband transitions,” Opt. Commun. 131, 333–338 (1996).
[CrossRef]

G. Yu. Kryuchkyan and K. V. Kheruntsyan, “Exact quantum theory of a parametrically driven dissipative anharmonic oscillator,” Opt. Commun. 127, 230–236 (1996).
[CrossRef]

K. V. Kheruntsyan, D. S. Krahmer, and K. G. Petrossian, “Wigner function for a generalized model of parametric oscillator: phase-space tristability, competition and nonclassical effects,” Opt. Commun. 139, 157–164 (1997).
[CrossRef]

A. Sizmann, R. J. Horowicz, and G. Wagner, “Observation of amplitude squeezing of the up-converted mode in second harmonic generation,” Opt. Commun. 80, 138–142 (1990).
[CrossRef]

S. Schiller, R. Bruckmeier, and A. G. White, “Classical and quantum properties of the subharmonic pumped parametric oscillator,” Opt. Commun. 138, 158–171 (1997).
[CrossRef]

Opt. Lett. (5)

Phys. Lett. A (1)

C. Cabrillo and F. J. Bermejo, “Control of the squeezing spectrum by means of a fourth-order interaction,” Phys. Lett. A 170, 300–304 (1992).
[CrossRef]

Phys. Rev. A (11)

C. Cabrillo and F. J. Bermejo, “Large quadrature squeezing at high intensities,” Phys. Rev. A 48, 2433–2436 (1993).
[CrossRef] [PubMed]

M. J. Collett and R. B. Levien, “Two-photon-loss model of intracavity second-harmonic generation,” Phys. Rev. A 43, 5068–5072 (1991).
[CrossRef] [PubMed]

M. J. Collet and C. W. Gardiner, “Squeezing of intracavity and travelling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[CrossRef]

A. G. White, M. S. Taubman, and H.-A. Bachor, “Experimental test of modular noise propagation theory for quantum optics,” Phys. Rev. A 54, 3400–3404 (1996).
[CrossRef] [PubMed]

S. Reynaud, C. Fabre, and E. Giacobino, “Photon noise reduction by passive optical bistable systems,” Phys. Rev. A 40, 1440–1446 (1989).
[CrossRef] [PubMed]

S. F. Pereira, X. Min, H. J. Kimble, and J. L. Hall, “Generation of squeezed light by intracavity frequency doubling,” Phys. Rev. A 38, 4931–4934 (1988).
[CrossRef] [PubMed]

P. García-Fernández, P. Colet, R. Toral, M. San Miguel, and F. J. Bermejo, “Squeezing resulting from a fourth-order interaction in a degenerate parametric amplifier with absorption losses,” Phys. Rev. A 43, 4923–4929 (1991).
[CrossRef]

C. Cabrillo, F. J. Bermejo, P. García-Fernández, R. Toral, P. Colet, and M. San Miguel, “Transient behavior of a parametric amplifier with an added fourth-order interaction,” Phys. Rev. A 45, 3216–3223 (1992).
[CrossRef] [PubMed]

C. Cabrillo, J. L. Roldán, and P. García-Fernández, “Squeezing enhancement by competing nonlinearities: almost perfect squeezing without instabilities,” Phys. Rev. A 56, 5131–5134 (1997).
[CrossRef]

K. V. Kheruntsyan, G. Yu. Kryuchkyan, and K. G. Petrosyan, “Controlling instability and squeezing from a cascaded frequency doubler,” Phys. Rev. A 57, 535–547 (1998).
[CrossRef]

A. G. White, P. K. Lam, and H.-A. Bachor, “Classical and quantum signatures of competing χ(2) nonlinearities,” Phys. Rev. A 55, 4511–4515 (1997).
[CrossRef]

Phys. Rev. Lett. (3)

K. Sundar, “Highly amplitude squeezed states of the radiation field,” Phys. Rev. Lett. 75, 2116–2119 (1995).
[CrossRef] [PubMed]

M. A. M. Marte, “Sub-poissonian twin beams via competing nonlinearities,” Phys. Rev. Lett. 76, 4815–4818 (1995).
[CrossRef]

R. Paschotta, M. Collett, and P. Kurz, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

Quantum Opt. (1)

C. Fabre, E. Giacobino, and A. Heidmann, “Squeezing in detuned degenerate optical parametric oscillators,” Quantum Opt. 2, 159–187 (1990).
[CrossRef]

Other (3)

C. W. Gardiner, Quantum Noise (Springer-Verlag, Berlin, 1991), Chap. 5.3.

P. Tombesi and H. P. Yuen, “Enhanced squeezing in an optically bistable 2-photon medium,” in Coherence and Quantum Optics V, L. Mandel and E. Wolf, eds. (Plenum, New York, 1984).

P. Tombesi, “Oversqueezing via 4-order interaction,” in Quantum Optics IV, J. D. Harvey and D. F. Walls, eds. (Springer, New York, 1986).

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Figures (6)

Fig. 1
Fig. 1

Dependence of Ki and Kr with respect to the phase mismatch.

Fig. 2
Fig. 2

Noise spectra at zero frequency of the fundamental mode following an optimum path for three escape efficiencies of the cavity, including the ideal case γˆc=1. The curves above the divergences are not physical.

Fig. 3
Fig. 3

Noise spectra at zero frequency of the harmonic mode following a nearly optimum path with respect to the phase mismatch for the SHG-like case.

Fig. 4
Fig. 4

Squeezing in the harmonic mode along an optimum path with respect to the normalized intracavity photon number (m) for the SHG-like case, compared with the phase-matched SHG case and the minimum static limit.

Fig. 5
Fig. 5

Noise spectra at zero frequency (harmonic mode) following an optimum path with respect to the normalized input harmonic amplitude (ηin). The curves are not physical above the divergences.

Fig. 6
Fig. 6

Noise spectra at zero frequency (harmonic mode) for Δk=δˆ=0 as function of the normalized intracavity photon number at various distances from the dynamic instability.

Equations (76)

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

dα/dt=-[γ+iδ+νK(Δk)|α|2]α+2γcαin.
K(Δk)=20Lm0zu*(Δk, z)u(Δk, z)dzdz/Lm2,
dα/dt=-[γ+μ|α|2+i(δ+Γ|α|2)]α+2γcαin.
μνKr(Δk)=νsinc ΔkLm22,
ΓνKi(Δk)=2νΔkLmsinc ΔkLm2cos ΔkLm2-1,
dadt=-[γ+iδ+(μ+iΓ)aa]a+2μabin+2γcain+2γswin,
aout=2γca-ain,
bout=μa2-bin,
δa=a-α,
δain,out=ain,out-αin,out,
δbin,out=bin,out-βin,out,
dδadt=-[γ+iδ+2(μ+iΓ)|α|2]δa+[2μβin-(μ+iΓ)α2]δa+2μα*δbin+2γcδain+2γswin,
δaout=2γcδa-δain,
δbout=2αμδa-δbin,
dαdt=-[γ+iδ+(μ+iΓ)|α|2]α+2μα*βin+2γcαin.
α=2γc{[γ+μn-i(δ+Γn)]αin+2μβinαin*}(γ+μn)2+(δ+Γn)2-4µ|βin|2,
|α|exp[i(θ-φin/2)][(γ+μn)2+(δ+Γn)2-4µ|βin|2]
=|αin|2γc{[γ+μn-i(δ+Γn)]×exp[i(θin-φin/2)]+2μ|βin|
×exp[-i(θin-φin/2)]}.
0=n[(γ+μn)2+(δ+Γn)2-4µ|βin|2]2-2γc|αin|2{(γ+μn)2+(δ+Γn)2+4µ|βin|2+4μ|βin|[(γ+μn)cos(2θin-φin)+(δ+Γn)sin(2θin-φin)]}.
cos(θ-φin/2)=|αin||α|2γc (γ+μn+2μ|βin|)cos(θin-φin/2)+(δ+Γn)sin(θin-φin/2)(γ+μn)2+(δ+Γn)2-4µ|βin|2,
sin(θ-φin/2)=|αin||α|2γc (γ+μn-2μ|βin|)sin(θin-φin/2)-(δ+Γn)cos(θin-φin/2)(γ+μn)2+(δ+Γn)2-4µ|βin|2.
2γc|αin|2=n[(γ+μn)2+(δ+Γn)2-4µ|βin|2]2|γ+μn+2μ|βin|exp[i(2θin-φin)]|2+4μ|βin|(δ+Γn)sin(2θin-φin).
λ±=-(γ+2µn)±[|(μ+iΓ)α2-2μβin|2-(δ+2Γn)2]1/2.
S(ω)=C-δXϕout(t)δXϕout(t+τ)exp(-iωτ)dτCδXϕout(ω)δXϕout(-ω),
S(ω)=C[δaout(ω)δaout(-ω)+Re{exp(-i2ϕ)δaout(ω)δaout(-ω)}],
S(ω)=C[δaout(ω)δaout(-ω)-|δaout(ω)δaout(-ω)|],
ϕopt=(ω)-π2,
S-,+(ω)=1+:δaout(ω)δaout(-ω):|:δaout(ω)δaout(-ω):|,
S-,+a(ω)=1+4γc|B| N-,+D,
S-,+b(ω)=1+8µn|B| N-,+D,
N-,+=2|B|(γ+2µn)[(γ+2µn)2-(δ+2Γn)2+|B|2+ω2]2+4(γ+2µn)2(δ+2Γn)2,
D=[(γ+2µn)2+(δ+2Γn)2-|B|2-ω2]2+4(γ+2µn)2ω2.
δaout(ω)δaout(-ω)
=4γcB[ω2+|B|2+(γ+2µn)2-(δ+2Γn)2
+i2(γ+2µn)(δ+2Γn)]/D,
δbout(ω)δbout(-ω)
=8µα2B[ω2+|B|2+(γ+2µn)2-(δ+2Γn)2
+i2(γ+2µn)(δ+2Γn)]/D.
dδcdτ=-(1+iΔ)δc+B˜δc+2γ˜nlδrin+2γ˜cδcin+2γ˜ssin,
ca exp(-iθ),
cin,outain,outγtexp(-iθ),
rin,outbin,outγtexp(-i2θ),
sinwinγtexp(-iθ).
δcout=2γ˜cδc-δcin,
δrout=2γ˜nlδc-δrin.
γ˜nl+γ˜c+γ˜s=1.
dδcdτ=-(1+iΔ)δc+B˜δc+n=1N2γ˜nδcinn,
n=1Nγ˜n=1.
δcoutn=2γ˜nδc-δcinn.
S-,+n(ω˜)=1+: S-,+n(ω˜):=1+2γ˜n : S-,+(ω˜):,
dδcdτ=-[1+iΔ]δc+B˜δc+2δcinref,
δcoutref=2δc-δcinref.
S-,+n(ω˜)=1+γ˜n : S-,+ref(ω˜):.
SMn=1-γ˜n,
: S-,+ref(ω˜):
=4|B˜| 2|B˜|±(1+ω˜2+|B˜|2-Δ2)2+4Δ2(1-ω˜2-|B˜|2+Δ2)2+4ω˜2.
1+Δ2=|B˜|2.
: SIref(ω˜):=4|B˜| 2|B˜|-4|B˜|2+ω˜2(ω˜2+4)ω˜2(ω˜2+4).
: S-,+opt:=4|B˜|(1±|B˜|)2.
S-,+ref(ω˜)=[2|B˜|±(ω˜2+|B˜|2+1-Δ2)2+4Δ2]2(1-ω˜2-|B˜|2+Δ2)2+4ω˜2.
dδcdτ=-[1+2mKr+i(δˆ+2mKi)]δc+[Krηin-(Kr+iKi)m]δc+2mKrδrin+2γˆcδcin+2γˆssin,
ηin2νγβin exp(-i2θ),
S-,+a(ω˜)=1+γ˜c: S-,+ref(ω˜):.
SMa=1-γ˜c=γs+2µnγs+γc+2µn=γˆs+2Kr(Δk)m1+2Kr(Δk)m.
: S-,+opt:=4mπ(π±m)2.
S-,+a(0)=1γˆc 4mπ(π±m)2.
π2-arctanΓnγc-γs.
S-,+b(ω˜)=1+γ˜nl : S-,+ref(ω˜) :,
SMb=1-γ˜nl=γγ+2µn=11+2Kr(Δk)m.
S-a(0)+S-b(0)=2-γ˜c-γ˜nl=2-γcγ+2µn-2µnγ+2µn=1,
δˆ±=-2mKi(Δk)±m2[Ki(Δk)2-3Kr(Δk)2]-4Kr(Δk)m-1.
|B˜|=121+Ki(Δk)Kr(Δk)21/2.
1+2m=|ηin-m|,
S-,+b(0)=1+2m±|ηin-m|1+2m±|ηin-m|2.
bout(ηin-2m)exp[i(2θ+π)].

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