Abstract

Incoherent broad-bandwidth cw-pumping singly resonant Ti:LiNbO3 integrated optical parametric oscillators, in the quasi-phase-matched nondegenerate type I {eee} configuration, may efficiently generate coherent signal output by the convection-induced phase-locking mechanism. The incoherence of the pump is absorbed by the idler wave, propagating at the same group velocity, while the finite temporal walk-off of the signal with respect to the pump and idler waves determines a signal coherence gain of more than 4 orders of magnitude.

© 2007 Optical Society of America

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2005 (1)

A. V. Smith, "Bandwidth and group-velocity effects in nanosecond optical parametric amplifiers and oscillators," J. Opt. Soc. Am. B 22, 1953-1965 (2005).
[CrossRef]

2004 (1)

C. Montes, A. Picozzi, and K. Gallo, "Ultra-coherent output from an incoherent cw-pumped singly resonant optical parametric oscillator," Opt. Commun. 237, 437-449 (2004).
[CrossRef]

2002 (1)

A. Picozzi, C. Montes, and M. Hælterman, "Coherence properties of the parametric three-wave interaction driven from an incoherent pump," Phys. Rev. E 66, 056605 (2002).
[CrossRef]

2001 (2)

A. Picozzi and M. Hælterman, "Parametric three-wave soliton generated from incoherent light," Phys. Rev. Lett. 86, 2010-2013 (2001).
[CrossRef] [PubMed]

G. Schreiber, D. Hofmann, W. Grundkötter, Y. L. Lee, H. Suche, V. Quiring, R. Ricken, and W. Sohler, "Nonlinear integrated optical frequency converters with periodically poled Ti:LiNbO3 waveguides," Proc. SPIE 4277, 144-160 (2001).
[CrossRef]

1999 (1)

1997 (2)

1996 (1)

1995 (1)

1993 (2)

1990 (1)

R. L. Berger, "Suppression of parametric instability by weakly incoherent laser beams," Phys. Rev. Lett. 65, 1207-1210 (1990).
[CrossRef] [PubMed]

1989 (1)

H. Suche and W. Sohler, "Integrated optical parametric oscillators," Optoelectron., Devices Technol. 41-20 (1989).

1988 (1)

A. Martins and J. T. Mendonça, "The non-linear three-wave interaction with a finite spectral width," Phys. Fluids 31, 3286-3294 (1988).
[CrossRef]

1987 (1)

G. P. Bava, I. Montrosset, W. Sohler, and H. Suche, "Numerical modeling of Ti:LiNbO3 integrated optical parametric oscillators," IEEE J. Quantum Electron. QE-23, 42-51 (1987).
[CrossRef]

1985 (1)

A. Martins and J. T. Mendonça, "Projection-operator method of the nonlinear three-wave interaction," Phys. Rev. A 31, 3898-3906 (1985).
[CrossRef] [PubMed]

1958 (1)

A. L. Shawlow and C. H. Townes, "Infrared and optical masers," Phys. Rev. 112, 1940-1949 (1958).
[CrossRef]

Arbore, M. A.

Bava, G. P.

G. P. Bava, I. Montrosset, W. Sohler, and H. Suche, "Numerical modeling of Ti:LiNbO3 integrated optical parametric oscillators," IEEE J. Quantum Electron. QE-23, 42-51 (1987).
[CrossRef]

Berger, R. L.

R. L. Berger, "Suppression of parametric instability by weakly incoherent laser beams," Phys. Rev. Lett. 65, 1207-1210 (1990).
[CrossRef] [PubMed]

Boller, K. J.

Bosenberg, W. R.

Byer, R. L.

Eckardt, R. C.

Falkovich, G.

V. E. Zakharov, V. S. L'vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I, Springer Series in Nonlinear Dynamics (Springer, 1992).

Fejer, M. M.

Gallo, K.

C. Montes, A. Picozzi, and K. Gallo, "Ultra-coherent output from an incoherent cw-pumped singly resonant optical parametric oscillator," Opt. Commun. 237, 437-449 (2004).
[CrossRef]

Grundkötter, W.

G. Schreiber, D. Hofmann, W. Grundkötter, Y. L. Lee, H. Suche, V. Quiring, R. Ricken, and W. Sohler, "Nonlinear integrated optical frequency converters with periodically poled Ti:LiNbO3 waveguides," Proc. SPIE 4277, 144-160 (2001).
[CrossRef]

D. Hofmann, G. Schreiber, W. Grundkötter, R. Ricken, and W. Sohler, "Mid-infrared continuous-wave singly resonant optical parametric oscillator with periodically poled Ti:LiNbO3 waveguide," in Conference on Lasers and Electro-Optics Europe (CLEO'00) (IEEE, 2000), paper CDM4.

D. Hofmann, H. Herrmann, G. Schreiber, W. Grundkötter, R. Ricken, and W. Sohler,"Continuous-wave mid-infrared optical parametric oscillators with periodically poled Ti:LiNbO3 waveguide," in 9th European Conference on Integrated Optics and Technical Exhibition ECIO'99 (EOS-European Optical Society, 1999), p. 21.

D. Hofmann, H. Herrmann, G. Schreiber, C. Haase, W. Grundkötter, R. Ricken, and W. Sohler, in Nonlinear Guided Waves and Their Applications, (Optical Society of America, 1999), p. 465.

Haase, C.

D. Hofmann, H. Herrmann, G. Schreiber, C. Haase, W. Grundkötter, R. Ricken, and W. Sohler, in Nonlinear Guided Waves and Their Applications, (Optical Society of America, 1999), p. 465.

Hælterman, M.

A. Picozzi, C. Montes, and M. Hælterman, "Coherence properties of the parametric three-wave interaction driven from an incoherent pump," Phys. Rev. E 66, 056605 (2002).
[CrossRef]

A. Picozzi and M. Hælterman, "Parametric three-wave soliton generated from incoherent light," Phys. Rev. Lett. 86, 2010-2013 (2001).
[CrossRef] [PubMed]

Herrmann, H.

D. Hofmann, H. Herrmann, G. Schreiber, W. Grundkötter, R. Ricken, and W. Sohler,"Continuous-wave mid-infrared optical parametric oscillators with periodically poled Ti:LiNbO3 waveguide," in 9th European Conference on Integrated Optics and Technical Exhibition ECIO'99 (EOS-European Optical Society, 1999), p. 21.

D. Hofmann, H. Herrmann, G. Schreiber, C. Haase, W. Grundkötter, R. Ricken, and W. Sohler, in Nonlinear Guided Waves and Their Applications, (Optical Society of America, 1999), p. 465.

Hofmann, D.

G. Schreiber, D. Hofmann, W. Grundkötter, Y. L. Lee, H. Suche, V. Quiring, R. Ricken, and W. Sohler, "Nonlinear integrated optical frequency converters with periodically poled Ti:LiNbO3 waveguides," Proc. SPIE 4277, 144-160 (2001).
[CrossRef]

D. Hofmann, G. Schreiber, W. Grundkötter, R. Ricken, and W. Sohler, "Mid-infrared continuous-wave singly resonant optical parametric oscillator with periodically poled Ti:LiNbO3 waveguide," in Conference on Lasers and Electro-Optics Europe (CLEO'00) (IEEE, 2000), paper CDM4.

D. Hofmann, H. Herrmann, G. Schreiber, W. Grundkötter, R. Ricken, and W. Sohler,"Continuous-wave mid-infrared optical parametric oscillators with periodically poled Ti:LiNbO3 waveguide," in 9th European Conference on Integrated Optics and Technical Exhibition ECIO'99 (EOS-European Optical Society, 1999), p. 21.

D. Hofmann, H. Herrmann, G. Schreiber, C. Haase, W. Grundkötter, R. Ricken, and W. Sohler, in Nonlinear Guided Waves and Their Applications, (Optical Society of America, 1999), p. 465.

Jundt, D.

Klein, M. E.

Lee, D. H.

Lee, Y. L.

G. Schreiber, D. Hofmann, W. Grundkötter, Y. L. Lee, H. Suche, V. Quiring, R. Ricken, and W. Sohler, "Nonlinear integrated optical frequency converters with periodically poled Ti:LiNbO3 waveguides," Proc. SPIE 4277, 144-160 (2001).
[CrossRef]

L'vov, V. S.

V. E. Zakharov, V. S. L'vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I, Springer Series in Nonlinear Dynamics (Springer, 1992).

Martins, A.

A. Martins and J. T. Mendonça, "The non-linear three-wave interaction with a finite spectral width," Phys. Fluids 31, 3286-3294 (1988).
[CrossRef]

A. Martins and J. T. Mendonça, "Projection-operator method of the nonlinear three-wave interaction," Phys. Rev. A 31, 3898-3906 (1985).
[CrossRef] [PubMed]

Mendonça, J. T.

A. Martins and J. T. Mendonça, "The non-linear three-wave interaction with a finite spectral width," Phys. Fluids 31, 3286-3294 (1988).
[CrossRef]

A. Martins and J. T. Mendonça, "Projection-operator method of the nonlinear three-wave interaction," Phys. Rev. A 31, 3898-3906 (1985).
[CrossRef] [PubMed]

Meyn, J. P.

Montes, C.

C. Montes, A. Picozzi, and K. Gallo, "Ultra-coherent output from an incoherent cw-pumped singly resonant optical parametric oscillator," Opt. Commun. 237, 437-449 (2004).
[CrossRef]

A. Picozzi, C. Montes, and M. Hælterman, "Coherence properties of the parametric three-wave interaction driven from an incoherent pump," Phys. Rev. E 66, 056605 (2002).
[CrossRef]

Montrosset, I.

G. P. Bava, I. Montrosset, W. Sohler, and H. Suche, "Numerical modeling of Ti:LiNbO3 integrated optical parametric oscillators," IEEE J. Quantum Electron. QE-23, 42-51 (1987).
[CrossRef]

Myers, L. E.

Picozzi, A.

C. Montes, A. Picozzi, and K. Gallo, "Ultra-coherent output from an incoherent cw-pumped singly resonant optical parametric oscillator," Opt. Commun. 237, 437-449 (2004).
[CrossRef]

A. Picozzi, C. Montes, and M. Hælterman, "Coherence properties of the parametric three-wave interaction driven from an incoherent pump," Phys. Rev. E 66, 056605 (2002).
[CrossRef]

A. Picozzi and M. Hælterman, "Parametric three-wave soliton generated from incoherent light," Phys. Rev. Lett. 86, 2010-2013 (2001).
[CrossRef] [PubMed]

Quiring, V.

G. Schreiber, D. Hofmann, W. Grundkötter, Y. L. Lee, H. Suche, V. Quiring, R. Ricken, and W. Sohler, "Nonlinear integrated optical frequency converters with periodically poled Ti:LiNbO3 waveguides," Proc. SPIE 4277, 144-160 (2001).
[CrossRef]

Ricken, R.

G. Schreiber, D. Hofmann, W. Grundkötter, Y. L. Lee, H. Suche, V. Quiring, R. Ricken, and W. Sohler, "Nonlinear integrated optical frequency converters with periodically poled Ti:LiNbO3 waveguides," Proc. SPIE 4277, 144-160 (2001).
[CrossRef]

D. Hofmann, H. Herrmann, G. Schreiber, C. Haase, W. Grundkötter, R. Ricken, and W. Sohler, in Nonlinear Guided Waves and Their Applications, (Optical Society of America, 1999), p. 465.

D. Hofmann, H. Herrmann, G. Schreiber, W. Grundkötter, R. Ricken, and W. Sohler,"Continuous-wave mid-infrared optical parametric oscillators with periodically poled Ti:LiNbO3 waveguide," in 9th European Conference on Integrated Optics and Technical Exhibition ECIO'99 (EOS-European Optical Society, 1999), p. 21.

D. Hofmann, G. Schreiber, W. Grundkötter, R. Ricken, and W. Sohler, "Mid-infrared continuous-wave singly resonant optical parametric oscillator with periodically poled Ti:LiNbO3 waveguide," in Conference on Lasers and Electro-Optics Europe (CLEO'00) (IEEE, 2000), paper CDM4.

Schreiber, G.

G. Schreiber, D. Hofmann, W. Grundkötter, Y. L. Lee, H. Suche, V. Quiring, R. Ricken, and W. Sohler, "Nonlinear integrated optical frequency converters with periodically poled Ti:LiNbO3 waveguides," Proc. SPIE 4277, 144-160 (2001).
[CrossRef]

D. Hofmann, G. Schreiber, W. Grundkötter, R. Ricken, and W. Sohler, "Mid-infrared continuous-wave singly resonant optical parametric oscillator with periodically poled Ti:LiNbO3 waveguide," in Conference on Lasers and Electro-Optics Europe (CLEO'00) (IEEE, 2000), paper CDM4.

D. Hofmann, H. Herrmann, G. Schreiber, W. Grundkötter, R. Ricken, and W. Sohler,"Continuous-wave mid-infrared optical parametric oscillators with periodically poled Ti:LiNbO3 waveguide," in 9th European Conference on Integrated Optics and Technical Exhibition ECIO'99 (EOS-European Optical Society, 1999), p. 21.

D. Hofmann, H. Herrmann, G. Schreiber, C. Haase, W. Grundkötter, R. Ricken, and W. Sohler, in Nonlinear Guided Waves and Their Applications, (Optical Society of America, 1999), p. 465.

Shawlow, A. L.

A. L. Shawlow and C. H. Townes, "Infrared and optical masers," Phys. Rev. 112, 1940-1949 (1958).
[CrossRef]

Small, D. L.

Smith, A. V.

A. V. Smith, "Bandwidth and group-velocity effects in nanosecond optical parametric amplifiers and oscillators," J. Opt. Soc. Am. B 22, 1953-1965 (2005).
[CrossRef]

Sohler, W.

G. Schreiber, D. Hofmann, W. Grundkötter, Y. L. Lee, H. Suche, V. Quiring, R. Ricken, and W. Sohler, "Nonlinear integrated optical frequency converters with periodically poled Ti:LiNbO3 waveguides," Proc. SPIE 4277, 144-160 (2001).
[CrossRef]

H. Suche and W. Sohler, "Integrated optical parametric oscillators," Optoelectron., Devices Technol. 41-20 (1989).

G. P. Bava, I. Montrosset, W. Sohler, and H. Suche, "Numerical modeling of Ti:LiNbO3 integrated optical parametric oscillators," IEEE J. Quantum Electron. QE-23, 42-51 (1987).
[CrossRef]

W. Sohler and H. Suche, in Digest of the Third International Conference on Integrated Optics and Optical Fiber Communication (Optical Society of America, 1981), p. 89.

D. Hofmann, G. Schreiber, W. Grundkötter, R. Ricken, and W. Sohler, "Mid-infrared continuous-wave singly resonant optical parametric oscillator with periodically poled Ti:LiNbO3 waveguide," in Conference on Lasers and Electro-Optics Europe (CLEO'00) (IEEE, 2000), paper CDM4.

D. Hofmann, H. Herrmann, G. Schreiber, W. Grundkötter, R. Ricken, and W. Sohler,"Continuous-wave mid-infrared optical parametric oscillators with periodically poled Ti:LiNbO3 waveguide," in 9th European Conference on Integrated Optics and Technical Exhibition ECIO'99 (EOS-European Optical Society, 1999), p. 21.

D. Hofmann, H. Herrmann, G. Schreiber, C. Haase, W. Grundkötter, R. Ricken, and W. Sohler, in Nonlinear Guided Waves and Their Applications, (Optical Society of America, 1999), p. 465.

Suche, H.

G. Schreiber, D. Hofmann, W. Grundkötter, Y. L. Lee, H. Suche, V. Quiring, R. Ricken, and W. Sohler, "Nonlinear integrated optical frequency converters with periodically poled Ti:LiNbO3 waveguides," Proc. SPIE 4277, 144-160 (2001).
[CrossRef]

H. Suche and W. Sohler, "Integrated optical parametric oscillators," Optoelectron., Devices Technol. 41-20 (1989).

G. P. Bava, I. Montrosset, W. Sohler, and H. Suche, "Numerical modeling of Ti:LiNbO3 integrated optical parametric oscillators," IEEE J. Quantum Electron. QE-23, 42-51 (1987).
[CrossRef]

W. Sohler and H. Suche, in Digest of the Third International Conference on Integrated Optics and Optical Fiber Communication (Optical Society of America, 1981), p. 89.

Townes, C. H.

A. L. Shawlow and C. H. Townes, "Infrared and optical masers," Phys. Rev. 112, 1940-1949 (1958).
[CrossRef]

Tsytovich, V. N.

V. N. Tsytovich, Nonlinear Effects in Plasma (Plenum, 1970).

Wallenstein, R.

Yang, S. T.

Zakharov, V. E.

V. E. Zakharov, V. S. L'vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I, Springer Series in Nonlinear Dynamics (Springer, 1992).

Zelmon, D. E.

IEEE J. Quantum Electron. (1)

G. P. Bava, I. Montrosset, W. Sohler, and H. Suche, "Numerical modeling of Ti:LiNbO3 integrated optical parametric oscillators," IEEE J. Quantum Electron. QE-23, 42-51 (1987).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Commun. (1)

C. Montes, A. Picozzi, and K. Gallo, "Ultra-coherent output from an incoherent cw-pumped singly resonant optical parametric oscillator," Opt. Commun. 237, 437-449 (2004).
[CrossRef]

Opt. Lett. (4)

Optoelectron., Devices Technol. (1)

H. Suche and W. Sohler, "Integrated optical parametric oscillators," Optoelectron., Devices Technol. 41-20 (1989).

Phys. Fluids (1)

A. Martins and J. T. Mendonça, "The non-linear three-wave interaction with a finite spectral width," Phys. Fluids 31, 3286-3294 (1988).
[CrossRef]

Phys. Rev. (1)

A. L. Shawlow and C. H. Townes, "Infrared and optical masers," Phys. Rev. 112, 1940-1949 (1958).
[CrossRef]

Phys. Rev. A (1)

A. Martins and J. T. Mendonça, "Projection-operator method of the nonlinear three-wave interaction," Phys. Rev. A 31, 3898-3906 (1985).
[CrossRef] [PubMed]

Phys. Rev. E (1)

A. Picozzi, C. Montes, and M. Hælterman, "Coherence properties of the parametric three-wave interaction driven from an incoherent pump," Phys. Rev. E 66, 056605 (2002).
[CrossRef]

Phys. Rev. Lett. (2)

R. L. Berger, "Suppression of parametric instability by weakly incoherent laser beams," Phys. Rev. Lett. 65, 1207-1210 (1990).
[CrossRef] [PubMed]

A. Picozzi and M. Hælterman, "Parametric three-wave soliton generated from incoherent light," Phys. Rev. Lett. 86, 2010-2013 (2001).
[CrossRef] [PubMed]

Proc. SPIE (1)

G. Schreiber, D. Hofmann, W. Grundkötter, Y. L. Lee, H. Suche, V. Quiring, R. Ricken, and W. Sohler, "Nonlinear integrated optical frequency converters with periodically poled Ti:LiNbO3 waveguides," Proc. SPIE 4277, 144-160 (2001).
[CrossRef]

Other (6)

D. Hofmann, G. Schreiber, W. Grundkötter, R. Ricken, and W. Sohler, "Mid-infrared continuous-wave singly resonant optical parametric oscillator with periodically poled Ti:LiNbO3 waveguide," in Conference on Lasers and Electro-Optics Europe (CLEO'00) (IEEE, 2000), paper CDM4.

W. Sohler and H. Suche, in Digest of the Third International Conference on Integrated Optics and Optical Fiber Communication (Optical Society of America, 1981), p. 89.

V. N. Tsytovich, Nonlinear Effects in Plasma (Plenum, 1970).

V. E. Zakharov, V. S. L'vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I, Springer Series in Nonlinear Dynamics (Springer, 1992).

D. Hofmann, H. Herrmann, G. Schreiber, W. Grundkötter, R. Ricken, and W. Sohler,"Continuous-wave mid-infrared optical parametric oscillators with periodically poled Ti:LiNbO3 waveguide," in 9th European Conference on Integrated Optics and Technical Exhibition ECIO'99 (EOS-European Optical Society, 1999), p. 21.

D. Hofmann, H. Herrmann, G. Schreiber, C. Haase, W. Grundkötter, R. Ricken, and W. Sohler, in Nonlinear Guided Waves and Their Applications, (Optical Society of America, 1999), p. 465.

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Figures (7)

Fig. 1
Fig. 1

Group-velocity dispersion for Ti : LiNbO 3 at 20 ° C : experimental waveguide dispersion relation. The three marked values correspond to the point of operation of the nondegenerate QPM three-wave interaction in the singly resonant type I { e e e } IOPO configuration (SR-IOPO).

Fig. 2
Fig. 2

SR-IOPO for a signal intensity reflectivity R = ρ s 2 = 98.01 % , a Fabry–Perot cavity length L = 8.00 cm and an intensity signal attenuation α s = 0.06 cm 1 : Signal and idler output power (in log units) as function of the input pump power of spectral width Δ ν p = 86.1 GHz , which coincide with the characteristics for a narrow coherent pump (plotted in the idler curve by the + symbols for the coherent pump, and by the * symbols for the incoherent pump).

Fig. 3
Fig. 3

SR-IOPO for a signal intensity reflectivity R = ρ s 2 = 98.01 % : Spatial amplitude evolution in the cavity of length L = 3.24 L n l = 8.00 cm at time t = 16384 t r = 18 μ s . The incoherent random phase modulated pump at the input ( z = 0 ) generates strong amplitude fluctuations during propagation to the output due to dispersion in the waveguide (upper graph). It transfers its incoherence to the growing (nonresonant) idler wave, moving at the same group velocity, whereas the resonant signal wave becomes highly coherent. The relative pump and idler fluctuation levels become almost the same as Fig. 4a, 4b.

Fig. 4
Fig. 4

SR-IOPO for a signal intensity reflectivity R = ρ s 2 = 98.01 % at time t = 16384 t r = 18 μ s : (a) Temporal pump evolution with a relative average fluctuating amplitude Δ A p ( L , t ) = 0.2349 around the average output amplitude A p ( L , t ) = 0.8297 . (b) Temporal idler evolution with almost the same relative average fluctuating amplitude Δ A i ( L , t ) = 0.2356 as the pump, around the average output amplitude A i ( L , t ) = 0.1825 . (c) Temporal signal evolution with a strongly reduced relative average fluctuating amplitude Δ A s ( L , t ) = 0.01708 around the average output amplitude A s ( L , t ) = 0.1198 .

Fig. 5
Fig. 5

SR-IOPO for a signal intensity reflectivity R = ρ s 2 = 98.01 % at time t = 16384 t r = 18 μ s : (a) Incoherent pump power spectrum showing the broad-bandwidth Δ ν p = 86.1 GHz stochasticity. (b) Idler power spectrum showing the same broad-bandwidth stochasticity absorbed from the incoherent pump. (c) Coherent signal power spectrum (single line) resulting from the convection-induced phase-locking mechanism. (d) Coherent signal power spectrum in logarithmic scale. [In Figs. 5a, 5b, 5c the power densities are normalized to the maximum of the pump power density].

Fig. 6
Fig. 6

SR-IOPO for a signal intensity reflectivity R = ρ s 2 = 98.01 % . Coherent signal power spectrum in log 10 scale showing the attractor tendency to coherence with temporal evolution at times: (a) t = 4096 t r = 4.5 μ s , (b) t = 8192 t r = 9 μ s , (c) t = 12288 t r = 13.5 μ s , and (d) t = 16384 t r = 18 μ s (where t r = 2 n s L c = 1.1 ns is the signal round-trip time). The signal wave attains its asymptotics and emerges from the cavity with a bandwidth Δ ν s < 2.0 × 10 5 Δ ν p .

Fig. 7
Fig. 7

SR-IOPO for a signal intensity reflectivity R = ρ s 2 = 98.01 % at time t = 16384 t r = 18 μ s [case (c) of Table 1]: (a) Incoherent pump power spectrum of bandwidth Δ ν p = 67.69 GHz . (b) Idler power spectrum showing the broad-bandwidth stochasticity absorbed from the incoherent pump. (c) Multiline signal power spectrum resulting from a nonefficient coherence transfer. (d) Signal power spectrum in logarithmic scale. [In Figs. 7a, 7b, 7c the power densities are normalized to the maximum of the pump power density].

Tables (1)

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Table 1 Signal Coherence Gain versus Incoherent Pump Bandwidth a

Equations (65)

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τ c e f f = ( v p Δ v ) τ c < τ 0 ,
v p v i < τ c v p γ i ,
λ p = 1.52121 μ m , n p = 2.14502 , v p c = 0.455796 ,
α p = 0.023 cm 1 ,
[ λ p * = 1.5350 μ m , n p * = 2.14458 , v p * c = 0.455871 ,
[ α p * = 0.023 cm 1 ] ,
λ i = 2.47731 μ m , n i = 2.11640 ,
v i c = v p c 2 × 10 7 , α i = 0.023 cm 1 ,
[ λ i * = 2.51410 μ m , n i * = 2.11524 ,
[ v i * c = v p * c 4 × 10 4 , α i * = 0.023 cm 1 ] ,
λ s = 3.94152 μ m , n s = 2.06027 , v s c = 0.445525 ,
α s = 0.060 cm 1 ,
Δ v v p = v s v p v p = 0.02253 1 44 .
ω p = ω s + ω i , k p k s k i = K ,
( t ± v p z + γ p + i β p t t ) A p ± = σ p A s ± A i ± ,
( t ± v s z + γ s + i β s t t ) A s ± = σ s A p ± A i ± * ,
( t ± v i z + γ i + i β i t t ) A i ± = σ i A p ± A s ± * ,
A p + ( 0 , t ) = E p ( 0 ) exp [ i Φ ( t ) ] + g n , p ,
A p ( L , t ) = g n , p ,
A s + ( 0 , t ) = ρ s A s ( 0 , t ) + g n , s ,
A s ( L , t ) = ρ s A s + ( L , t ) + g n , s ,
A i + ( 0 , t ) = g n , i ,
A i ( L , t ) = g n , i ,
σ j = 2 π d e f f v j f j , k , l λ j n j ,
σ p = 2866 m s V , σ i = 940 m s V , σ s = 262 m s V.
τ 0 = 2 E p ( 0 ) σ p = 0.181 ns , L n l = v p τ 0 = 2.47 cm.
τ c e f f τ 0 = v p Δ v τ c τ 0 = 0.998 < 1 .
I 1 τ c e f f τ 0 = v p Δ v τ c τ 0 = 0.905 < 1 ,
I 2 v i v p v p 1 τ c γ i = 2 × 10 4 < 1 .
Δ A j ( L , t ) [ A j ( L , t ) 2 A j ( L , t ) 2 ] 1 2 A j ( L , t ) , ( j = p , s , i ) ,
Δ A p ( L , t ) = 0.2349 , Δ A i ( L , t ) = 0.2356 ,
Δ A s ( L , t ) = 0.01708 .
Δ A p ( L , t ) = 0.2365 , Δ A i ( L , t ) = 0.2305 ,
Δ A s ( L , t ) = 0.02519 ,
Δ v v p > 1 π Δ ν p τ 0 .
E j ( r , t ) = F j ( x , y ) F j , max A j ( z , t ) exp ( i k j z ) ,
F j ( x , y ) = 2 π w j exp [ 2 ( x w j ) 2 ] 2 π d j y d j exp [ 1 2 ( y d j ) 2 ]
S j , e f f = 1 F j , max 2 = e 8 π w j d j ,
S p = 30.99 μ m 2 , S i = 58.78 μ m 2 , S s = 133.15 μ m 2
P j = I j S j , e f f = 1 2 n j ε 0 c A j 2 S j , e f f ,
σ j = 2 π d e f f v j f j , k , l λ j n j ,
f j , k , l = F j , max F k , max F l , max d x d y F j ( x , y ) F k ( x , y ) F l ( x , y ) .
d x d y F p ( x , y ) F i ( x , y ) F s ( x , y ) = 55265.1 m 1 ,
F p , max F i , max F s , max = 15.89 μ m ,
F i , max F p , max F s , max = 8.378 μ m ,
F s , max F p , max F i , max = 3.689 μ m.
f p = 0.8781 , f i = 0.4630 , f s = 0.2043 .
σ p = 2866 m s V , σ i = 940 m s V , σ s = 262 m s V.
( t + v p z + γ p + i β p t t ) A p = σ p A s A i ,
( t + v s z + γ s + i β s t t ) A s = σ s A p A i * ,
( t + v i z + γ i + i β i t t ) A i = σ i A p A s * .
τ 0 = 2 E p ( 0 ) σ p , L n l = v p τ 0 ,
τ = t τ 0 , ζ = z L n l , μ j = γ j τ 0 ,
a j = A j E p ( 0 ) .
( τ + ζ + μ p + i β ̃ p τ τ ) a p = 2 a s a i ,
( τ + ζ + μ s + i β ̃ s τ τ ) a s = κ s a p a i * ,
( τ + ζ + μ i + i β ̃ i τ τ ) a i = κ i a p a s * ,
κ s = 2 σ s σ p = 0.1827 , κ i = 2 σ i σ p = 0.6561 ,
τ 0 = 2 E p ( 0 ) σ p = 0.181 ns , L n l = v p τ 0 = 2.47 cm.
P p = 1250 mW , τ 0 = 0.185 ns ,
L n l = 2.52 cm , L = Λ n l L n l = 8.00 cm ,
Λ n l = L L n l = 3.17 , μ p = μ i = 0.028 , μ s = 0.074 ,
D A s = σ 2 0 t e γ i ( t t ) A p ( z ) A p * ( z ) A s ( x , t ) d t ,
A i ( x , t ) = σ 0 t e γ i ( t t ) A p ( z ) A s * ( x , t ) d t .
v i v p λ c γ i = τ c v p γ i ,

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