Abstract

We derive the properties of a pump-resonant singly resonant optical parametric oscillator for which the pump and one of the parametrically generated waves share a common cavity. Wave-vector mismatch and focusing effects are taken into account. We calculate the oscillation threshold and its dependence on emission wavelengths as the result of mode shape changes. The conversion efficiencies for signal and idler waves are calculated. It is shown that one can maximize the conversion efficiencies by optimizing mirror transmissivities. The interference effects that occur in a standing-wave geometry and the mode content of the nonresonant wave are also analyzed.

© 1999 Optical Society of America

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References

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  1. P. E. Powers, T. J. Kulp, and S. E. Bisson, “Continuous tuning of a continuous-wave periodically poled lithium niobate optical parametric oscillator by use of a fan-out grating design,” Opt. Lett. 23, 159–161 (1998).
    [CrossRef]
  2. G. M. Gibson, M. H. Dunn, and M. J. Padgett, “Application of a continuously tunable, cw optical parametric oscillator for high-resolution spectroscopy,” Opt. Lett. 23, 40–42 (1998).
    [CrossRef]
  3. R. Al-Tahtamouni, K. Bencheikh, R. Storz, K. Schneider, M. Lang, J. Mlynek, and S. Schiller, “Long-term stable operation and absolute frequency stabilization of continuous-wave doubly-resonant optical parametric oscillators,” Appl. Phys. B 66, 733–740 (1998).
    [CrossRef]
  4. F. Kühnemann, K. Schneider, A. Hecker, A. A. E. Martis, W. Urban, S. Schiller, and J. Mlynek, “Photoacoustic trace gas detection with a single-frequency continuous-wave optical parametric oscillator,” Appl. Phys. B 66, 741–746 (1998).
    [CrossRef]
  5. K. Schneider, P. Kramper, O. Mor, S. Schiller, and J. Mlynek, “Continuous-wave, single-frequency parametric oscillator for the 1.45–4.0 μm range,” in Advanced Solid State Lasers, W. R. Bosenberg and M. M. Fejer, eds., Vol. 19 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 256–258.
  6. K. Schneider, P. Kramper, S. Schiller, and J. Mlynek, “Toward an optical synthesizer: a single-frequency parametric oscillator using periodically poled LiNbO3,” Opt. Lett. 22, 1293–1295 (1997).
    [CrossRef]
  7. K. Schneider and S. Schiller, “Narrow-linewidth, pump-enhanced singly-resonant parametric oscillator pumped at 532 nm,” Appl. Phys. B 65, 775–777 (1997).
    [CrossRef]
  8. R. L. Byer, “Parametric fluorescence and optical parametric oscillation,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1968).
  9. G. Robertson, M. J. Padgett, and M. H. Dunn, “Continuous-wave singly resonant pump-enhanced type II LiB3O5 optical parametric oscillator,” Opt. Lett. 21, 1735–1737 (1994).
    [CrossRef]
  10. S. T. Yang, R. C. Eckardt, and R. L. Byer, “Continuous-wave singly resonant optical parametric oscillator pumped by a single-frequency resonantly doubled Nd:YAG laser,” Opt. Lett. 18, 971–973 (1993).
    [CrossRef] [PubMed]
  11. Further work on the PR SRO includes that of D. Chen, D. Hinkley, J. Pyo, J. Swenson, and R. Fields, “Single-frequency low-threshold continuous-wave 3-μm periodically poled lithium niobate optical parametric oscillator,” J. Opt. Soc. Am. B 15, 1693–1697 (1998); M. Scheidt, M. E. Klein, and K. J. Boller, “Spiking in pump enhanced idler resonant optical parametric oscillators,” Opt. Commun. 149, 108–112 (1998).
    [CrossRef]
  12. S. Guha, F. Wu, and J. Falk, “The effect of focusing on parametric oscillation,” IEEE J. Quantum Electron. QE-18, 907–912 (1982).
    [CrossRef]
  13. S. T. Yang, R. C. Eckardt, and R. L. Byer, “Power and spectral characteristics of continuous-wave parametric oscillators: the doubly to singly resonant transition,” J. Opt. Soc. Am. B 10, 1684–1695 (1993).
    [CrossRef]
  14. 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]
  15. S. Guha, “Focusing dependence of the efficiency of a singly resonant optical parametric oscillator,” Appl. Phys. B 66, 663–676 (1998).
    [CrossRef]
  16. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  17. I. S. Gradstein and I. M. Rhyzik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 837, Eq. (7.373.2).
  18. C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B 66, 685–700 (1998).
    [CrossRef]
  19. The maximization may be implemented with a wedged quasi-phase-matched crystal. See G. Imeshev, M. Proctor, and M. M. Fejer, “Phase correction in double-pass quasi-phase-matched second-harmonic generation with a wedged crystal,” Opt. Lett. 23, 165–167 (1998).
    [CrossRef]

1998

R. Al-Tahtamouni, K. Bencheikh, R. Storz, K. Schneider, M. Lang, J. Mlynek, and S. Schiller, “Long-term stable operation and absolute frequency stabilization of continuous-wave doubly-resonant optical parametric oscillators,” Appl. Phys. B 66, 733–740 (1998).
[CrossRef]

F. Kühnemann, K. Schneider, A. Hecker, A. A. E. Martis, W. Urban, S. Schiller, and J. Mlynek, “Photoacoustic trace gas detection with a single-frequency continuous-wave optical parametric oscillator,” Appl. Phys. B 66, 741–746 (1998).
[CrossRef]

S. Guha, “Focusing dependence of the efficiency of a singly resonant optical parametric oscillator,” Appl. Phys. B 66, 663–676 (1998).
[CrossRef]

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B 66, 685–700 (1998).
[CrossRef]

G. M. Gibson, M. H. Dunn, and M. J. Padgett, “Application of a continuously tunable, cw optical parametric oscillator for high-resolution spectroscopy,” Opt. Lett. 23, 40–42 (1998).
[CrossRef]

P. E. Powers, T. J. Kulp, and S. E. Bisson, “Continuous tuning of a continuous-wave periodically poled lithium niobate optical parametric oscillator by use of a fan-out grating design,” Opt. Lett. 23, 159–161 (1998).
[CrossRef]

The maximization may be implemented with a wedged quasi-phase-matched crystal. See G. Imeshev, M. Proctor, and M. M. Fejer, “Phase correction in double-pass quasi-phase-matched second-harmonic generation with a wedged crystal,” Opt. Lett. 23, 165–167 (1998).
[CrossRef]

1997

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, P. Kramper, S. Schiller, and J. Mlynek, “Toward an optical synthesizer: a single-frequency parametric oscillator using periodically poled LiNbO3,” Opt. Lett. 22, 1293–1295 (1997).
[CrossRef]

K. Schneider and S. Schiller, “Narrow-linewidth, pump-enhanced singly-resonant parametric oscillator pumped at 532 nm,” Appl. Phys. B 65, 775–777 (1997).
[CrossRef]

1994

1993

1982

S. Guha, F. Wu, and J. Falk, “The effect of focusing on parametric oscillation,” IEEE J. Quantum Electron. QE-18, 907–912 (1982).
[CrossRef]

Al-Tahtamouni, R.

R. Al-Tahtamouni, K. Bencheikh, R. Storz, K. Schneider, M. Lang, J. Mlynek, and S. Schiller, “Long-term stable operation and absolute frequency stabilization of continuous-wave doubly-resonant optical parametric oscillators,” Appl. Phys. B 66, 733–740 (1998).
[CrossRef]

Bencheikh, K.

R. Al-Tahtamouni, K. Bencheikh, R. Storz, K. Schneider, M. Lang, J. Mlynek, and S. Schiller, “Long-term stable operation and absolute frequency stabilization of continuous-wave doubly-resonant optical parametric oscillators,” Appl. Phys. B 66, 733–740 (1998).
[CrossRef]

Bisson, S. E.

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]

Byer, R. L.

Cohadon, P. F.

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B 66, 685–700 (1998).
[CrossRef]

Dunn, M. H.

Eckardt, R. C.

Fabre, C.

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B 66, 685–700 (1998).
[CrossRef]

Falk, J.

S. Guha, F. Wu, and J. Falk, “The effect of focusing on parametric oscillation,” IEEE J. Quantum Electron. QE-18, 907–912 (1982).
[CrossRef]

Fejer, M. M.

Gatti, A.

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B 66, 685–700 (1998).
[CrossRef]

Gibson, G. M.

Guha, S.

S. Guha, “Focusing dependence of the efficiency of a singly resonant optical parametric oscillator,” Appl. Phys. B 66, 663–676 (1998).
[CrossRef]

S. Guha, F. Wu, and J. Falk, “The effect of focusing on parametric oscillation,” IEEE J. Quantum Electron. QE-18, 907–912 (1982).
[CrossRef]

Hecker, A.

F. Kühnemann, K. Schneider, A. Hecker, A. A. E. Martis, W. Urban, S. Schiller, and J. Mlynek, “Photoacoustic trace gas detection with a single-frequency continuous-wave optical parametric oscillator,” Appl. Phys. B 66, 741–746 (1998).
[CrossRef]

Imeshev, G.

Kramper, P.

Kühnemann, F.

F. Kühnemann, K. Schneider, A. Hecker, A. A. E. Martis, W. Urban, S. Schiller, and J. Mlynek, “Photoacoustic trace gas detection with a single-frequency continuous-wave optical parametric oscillator,” Appl. Phys. B 66, 741–746 (1998).
[CrossRef]

Kulp, T. J.

Lang, M.

R. Al-Tahtamouni, K. Bencheikh, R. Storz, K. Schneider, M. Lang, J. Mlynek, and S. Schiller, “Long-term stable operation and absolute frequency stabilization of continuous-wave doubly-resonant optical parametric oscillators,” Appl. Phys. B 66, 733–740 (1998).
[CrossRef]

Lugiato, L.

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B 66, 685–700 (1998).
[CrossRef]

Marte, M. A. M.

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B 66, 685–700 (1998).
[CrossRef]

Martis, A. A. E.

F. Kühnemann, K. Schneider, A. Hecker, A. A. E. Martis, W. Urban, S. Schiller, and J. Mlynek, “Photoacoustic trace gas detection with a single-frequency continuous-wave optical parametric oscillator,” Appl. Phys. B 66, 741–746 (1998).
[CrossRef]

Mlynek, J.

F. Kühnemann, K. Schneider, A. Hecker, A. A. E. Martis, W. Urban, S. Schiller, and J. Mlynek, “Photoacoustic trace gas detection with a single-frequency continuous-wave optical parametric oscillator,” Appl. Phys. B 66, 741–746 (1998).
[CrossRef]

R. Al-Tahtamouni, K. Bencheikh, R. Storz, K. Schneider, M. Lang, J. Mlynek, and S. Schiller, “Long-term stable operation and absolute frequency stabilization of continuous-wave doubly-resonant optical parametric oscillators,” Appl. Phys. B 66, 733–740 (1998).
[CrossRef]

K. Schneider, P. Kramper, S. Schiller, and J. Mlynek, “Toward an optical synthesizer: a single-frequency parametric oscillator using periodically poled LiNbO3,” Opt. Lett. 22, 1293–1295 (1997).
[CrossRef]

Padgett, M. J.

Powers, P. E.

Proctor, M.

Ritsch, H.

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B 66, 685–700 (1998).
[CrossRef]

Robertson, G.

Schiller, S.

R. Al-Tahtamouni, K. Bencheikh, R. Storz, K. Schneider, M. Lang, J. Mlynek, and S. Schiller, “Long-term stable operation and absolute frequency stabilization of continuous-wave doubly-resonant optical parametric oscillators,” Appl. Phys. B 66, 733–740 (1998).
[CrossRef]

F. Kühnemann, K. Schneider, A. Hecker, A. A. E. Martis, W. Urban, S. Schiller, and J. Mlynek, “Photoacoustic trace gas detection with a single-frequency continuous-wave optical parametric oscillator,” Appl. Phys. B 66, 741–746 (1998).
[CrossRef]

K. Schneider, P. Kramper, S. Schiller, and J. Mlynek, “Toward an optical synthesizer: a single-frequency parametric oscillator using periodically poled LiNbO3,” Opt. Lett. 22, 1293–1295 (1997).
[CrossRef]

K. Schneider and S. Schiller, “Narrow-linewidth, pump-enhanced singly-resonant parametric oscillator pumped at 532 nm,” Appl. Phys. B 65, 775–777 (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]

Schneider, K.

F. Kühnemann, K. Schneider, A. Hecker, A. A. E. Martis, W. Urban, S. Schiller, and J. Mlynek, “Photoacoustic trace gas detection with a single-frequency continuous-wave optical parametric oscillator,” Appl. Phys. B 66, 741–746 (1998).
[CrossRef]

R. Al-Tahtamouni, K. Bencheikh, R. Storz, K. Schneider, M. Lang, J. Mlynek, and S. Schiller, “Long-term stable operation and absolute frequency stabilization of continuous-wave doubly-resonant optical parametric oscillators,” Appl. Phys. B 66, 733–740 (1998).
[CrossRef]

K. Schneider, P. Kramper, S. Schiller, and J. Mlynek, “Toward an optical synthesizer: a single-frequency parametric oscillator using periodically poled LiNbO3,” Opt. Lett. 22, 1293–1295 (1997).
[CrossRef]

K. Schneider and S. Schiller, “Narrow-linewidth, pump-enhanced singly-resonant parametric oscillator pumped at 532 nm,” Appl. Phys. B 65, 775–777 (1997).
[CrossRef]

Schwob, C.

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B 66, 685–700 (1998).
[CrossRef]

Storz, R.

R. Al-Tahtamouni, K. Bencheikh, R. Storz, K. Schneider, M. Lang, J. Mlynek, and S. Schiller, “Long-term stable operation and absolute frequency stabilization of continuous-wave doubly-resonant optical parametric oscillators,” Appl. Phys. B 66, 733–740 (1998).
[CrossRef]

Urban, W.

F. Kühnemann, K. Schneider, A. Hecker, A. A. E. Martis, W. Urban, S. Schiller, and J. Mlynek, “Photoacoustic trace gas detection with a single-frequency continuous-wave optical parametric oscillator,” Appl. Phys. B 66, 741–746 (1998).
[CrossRef]

White, A. G.

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]

Wu, F.

S. Guha, F. Wu, and J. Falk, “The effect of focusing on parametric oscillation,” IEEE J. Quantum Electron. QE-18, 907–912 (1982).
[CrossRef]

Yang, S. T.

Appl. Phys. B

R. Al-Tahtamouni, K. Bencheikh, R. Storz, K. Schneider, M. Lang, J. Mlynek, and S. Schiller, “Long-term stable operation and absolute frequency stabilization of continuous-wave doubly-resonant optical parametric oscillators,” Appl. Phys. B 66, 733–740 (1998).
[CrossRef]

F. Kühnemann, K. Schneider, A. Hecker, A. A. E. Martis, W. Urban, S. Schiller, and J. Mlynek, “Photoacoustic trace gas detection with a single-frequency continuous-wave optical parametric oscillator,” Appl. Phys. B 66, 741–746 (1998).
[CrossRef]

K. Schneider and S. Schiller, “Narrow-linewidth, pump-enhanced singly-resonant parametric oscillator pumped at 532 nm,” Appl. Phys. B 65, 775–777 (1997).
[CrossRef]

S. Guha, “Focusing dependence of the efficiency of a singly resonant optical parametric oscillator,” Appl. Phys. B 66, 663–676 (1998).
[CrossRef]

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B 66, 685–700 (1998).
[CrossRef]

IEEE J. Quantum Electron.

S. Guha, F. Wu, and J. Falk, “The effect of focusing on parametric oscillation,” IEEE J. Quantum Electron. QE-18, 907–912 (1982).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

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.

Other

Further work on the PR SRO includes that of D. Chen, D. Hinkley, J. Pyo, J. Swenson, and R. Fields, “Single-frequency low-threshold continuous-wave 3-μm periodically poled lithium niobate optical parametric oscillator,” J. Opt. Soc. Am. B 15, 1693–1697 (1998); M. Scheidt, M. E. Klein, and K. J. Boller, “Spiking in pump enhanced idler resonant optical parametric oscillators,” Opt. Commun. 149, 108–112 (1998).
[CrossRef]

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

I. S. Gradstein and I. M. Rhyzik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 837, Eq. (7.373.2).

R. L. Byer, “Parametric fluorescence and optical parametric oscillation,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1968).

K. Schneider, P. Kramper, O. Mor, S. Schiller, and J. Mlynek, “Continuous-wave, single-frequency parametric oscillator for the 1.45–4.0 μm range,” in Advanced Solid State Lasers, W. R. Bosenberg and M. M. Fejer, eds., Vol. 19 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 256–258.

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Figures (7)

Fig. 1
Fig. 1

Schematic of a ring PR SRO. The pump wave is coupled in through the left-hand mirror (power reflectivity Rp). The resonant signal wave is coupled out at some other mirror with power transmission Ts=1-Rs. The idler wave is fully coupled out at a particular mirror and is not resonated. Signal and pump wave power losses, typically distributed over the various optical elements and the crystal, are lumped together as Vs and Vp.

Fig. 2
Fig. 2

Schematic of a standing-wave PR SRO. The pump wave is coupled in through the left-hand mirror (power reflectivity Rp). The resonant signal wave is coupled out at the two end mirrors with power transmissions 1-|rs1|2 and 1-|rs2|2. The idler wave is partially coupled out at the right-hand mirror (power transmission 1-|ri|2) and fully coupled out at the front mirror. Pump wave losses now include the partial transmission through the back mirror.

Fig. 3
Fig. 3

Optimization of a PR SRO for maximum external output power (a) on the resonated wave and (b) on the nonresonant wave versus pump power. The resonated wave is taken to be the signal. Solid curves, quantum conversion efficiencies. The short-dashed curve in the top figure is the conversion efficiency for the nonresonant wave. In the bottom figure this efficiency is zero. Dashed–dotted curves, the corresponding optimum normalized output coupler transmission of the resonated wave (Ts/Vs). Long-dashed curves, the corresponding optimum pump input coupler transmission Tp/Vp.

Fig. 4
Fig. 4

Normalized nonlinearity as a function of phase mismatch. The focus is at the back crystal face; the focusing strength is zR/L=0.4; oscillation frequencies are ζ=0.7.

Fig. 5
Fig. 5

Maximum of the normalized nonlinearity with respect to phase mismatch as a function of focusing parameter of the resonator: (a) focus at the end of the crystal, (b) focus at the center of the crystal. The curves are for various values ζ=nrωr/npωp, where r denotes the resonant parametrically generated wave. In both plots, from the top, the values are 0.9, 0.7, 0.5, and 0.2. The nonlinearity for a doubly resonant OPO is given by the ζ=0 curves. Note that the conventional focusing parameter ξ=L/2zR.

Fig. 6
Fig. 6

Normalized nonlinearity for a cavity geometry in which the idler wave double-passes the nonlinear crystal, leading to interference effects that depend on the relative phase ϕ. Complete reflection after the first pass is assumed (|ri|=1). For each value of ϕ,hsp is maximized with respect to the wave-vector mismatch. The focus position is the back crystal face; zR/L=0.4; ζ=0.7.

Fig. 7
Fig. 7

Power fraction of the nonresonated parametrically generated wave that is in a TEM00 mode as a function of its frequency ωnr=cknr/nnr (solid curve). The phase mismatch at each ωnr is chosen to maximize E. Also shown are the corresponding waist position z0,nr (short-dashed curve) and waist wnr (long-dashed curve). The focus position of the pump and the resonant waves is chosen to lie at the end of the crystal with focusing strength zR/L=0.4.

Equations (113)

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E˜(x, y, z)=ωn m,n=0Amn(z)ψmn(x, y, z),
dAmn(z)dz=-i μ0c2ωn - -
×P˜NL(x, y, z)ψmn*(x, y, z)dxdy.
P˜NL,p(x, y, z)=20dE˜sE˜i exp(iΔkz),
P˜NL,s(x, y, z)=20dE˜pE˜i* exp(-iΔkz),
P˜NL,i(x, y, z)=20dE˜pE˜s* exp(-iΔkz),
E˜p(x, y, z)=ωp/np Ap(z)ψ00, p(x, y, z),
E˜s(x, y, z)=ωs/ns As(z)ψ00,s(x, y, z),
dAp(z)dz=-iξAs(z)mnAmn,i(z)Omn(z)exp(iΔkz),
dAs(z)dz=-iξAp(z)mnAmn,i*(z)Omn*(z)exp(-iΔkz),
dAmn,i(z)dz=-iξAp(z)As*(z)Omn*(z)exp(-iΔkz),
Omn(z)=--ψ00,p*(x, y, z)×ψ00,s(x, y, z)ψmn,i(x, y, z)dx dy.
Amn,i(z)=-iξzImn*(z)Ap(0)As*(0),
Imn(z)=1z0zOmn(z)exp(iΔkz)dz.
As(z)=As(0)+½(ξz)2D(z)As(0)|Ap(0)|2,
Ap(z)=Ap(0)-½(ξz)2D*(z)Ap(0)|As(0)|2.
D(z)=n,m=0Dmn(z),
Dmn(z)=2z20zzImn(z)Omn*(z)exp(-iΔkz)dz.
ωs|As(0)|2+ωp|Ap(0)|2=ωs|As(z)|2+ωp|Ap(z)|2+ωimn|Amn,i(z)|2.
Re D(z)=mn|Imn(z)|2.
Es(x, y, z=0)=Es(x, y, z=0)|rt.
Es(x, y, z=0)|rt=[Rs(1-Vs)]1/2P [Es(x, y, z=L), Lrt-L],
Es(x, y, z=L)
=exp(-iksL+iωst)ωs/nsAs(L)ψ00(x, y, L).
P [Es(x, y, z=L), Lrt-L]
=exp(2iψs)exp(-iksLrt+iωst)ωs/nsAs(L)
×ψ00(x, y, 0).
ψs=q arctan τ (zq),
As(0)=[Rs(1-Vs)]1/2 exp(2iψs)exp(-iksLrt)As(L).
arg[exp(2iψs)exp(-iksLrt)As(L)/As(0)]=0.
Ep(x, y, 0)|rt=Tpωp/np Ap,inψ00(x, y, 0)×exp(iωpt)+[Rp(1-Vp)]1/2×P [Ep(x, y, L),Lrt-L].
Ap(0)=Tp Ap,in+[Rp(1-Vp)]1/2×exp(2iψp)exp(-ikpLrt)Ap(L)
=Ap,in Tp1-rp exp(2iψp)exp(-ikpLrt)Ap(L)/Ap(0),
Ap,r
=-RpAp,in
+[Tp(1-Vp)]1/2 exp(2iψp)exp(-ikpLrt)Ap(L)
=Ap,inRp
×Tp1-rp exp(2iψp)exp(-ikpLrt)Ap(L)/Ap(0)-1.
arg[exp(2iψp)exp(-ikpLrt)Ap(L)/Ap(0)]=0.
Ap,r=Ap,inRpTp1-rp|Ap(L)/Ap(0)|-1,
Ap(0)=Ap,in Tp1-rp|Ap(L)/Ap(0)|.
|As(L)|2/|As(0)|2=1Rs(1-Vs),
Ap(0)2=η(ξL)2 Re D,
TprpAp,inAp(0)=1rp-1+12(ξL)2 Re D|As(0)|2.
Ppth=η/E=Ts+Vs(1-Ts)(1-Vs)ETs+VsE,
E=2μ0d2ωiωscninsnpL2 Re D.
Pinth=(1-rp)2TpPpth=1(1-Ts)(1-Vs)Tp+VpTp (Tp+Vp)(Ts+Vs)4E.
|As(0)|2=2η1-rprp|Apth(0)|2Ap,inAinth-1,
Ps(0)=(1-Ts)(1-Vs)(1-Tp/2)(1-Vp/2)ωsωp4Ts+VsTpTp+Vp×PinthPinPinth-1.
Ps,out=4(1-Tp/2)(1-Vp/2)ωsωpTsTs+VsTpTp+Vp×PinthPinPinth-1.
Pi,out=η ωiωsPs(0)=ωiωsEPpthPs(0)=4(1-Tp/2)(1-Vp/2)ωiωpTpTp+Vp×PinthPinPinth-1.
PrPin=1Rp2TpTp+VpPinthPin-12.
Pr,totPin,tot=(1-β)+β PrPin,
Amn,i(z)=-iξzImn*(z)Ap(0)As*(0),
As(L)=As(0)[1+1/2(ξL)2D(L)|Ap(0)|2],
Ap(L)=Ap(0)[1-1/2(ξL)2D*(L)|As(0)|2].
Amn,i(z)=Amn,i(0)-iξzImn*(z)Ap(0)As*(0),
As(L)=As(0)[1+½(ξL)2D(L)|Ap(0)|2]+(ξL)2Y˜*(L)Ap*(0)Ap(0)As(0),
Ap(L)=Ap(0)[1-½(ξL)2D*(L)|As(0)|2]-(ξL)2Y˜(L)As*(0)As(0)Ap(0),
Y(L)=mnr˜mn,iImn*(L)Imn(L).
As(0)=r˜s2 As(L),Ap(0)=r˜p2 Ap(L).
As(L)rs2 As(0)=1+(ξL)2[½D(L)+½D(L)+(r˜s2)-1r˜p2Y*(L)]|Ap(0)|2,
Ap(L)rp2 Ap(0)=1-(ξL)2[½D*(L)+½D*(L)
+r˜s2(r˜p2)-1Y(L)]| As(0)|2.
As(0)=r˜s1(1-Vs)As(L),
Ap(0)=Ap,in Tp1-rp|Ap(L)/Ap(0)|.
Apth(0)2
=η2(ξL)2 Re[½D(L)+½D(L)+(r˜s2*)-1r˜p2*Y(L)],
Tprp|rp2|Ap,inAp(0)=1rp|rp2|-1+(ξL)2|As(0)|2 Re[½D(L)+½D(L)+r˜s2(r˜p2)-1Y(L)].
r˜s1=rs1 exp(-iksL)exp[2i arctan τ(L)],
r˜s2=rs2 exp(-iksL)exp[2i arctan τ (L)],
r˜p2=rp2 exp(-ikpL)exp[2i arctan τ (L)],
r˜mn,i=ri exp(-ikiL)exp{i(m+n+1)×[arctan τi(L)-arctan τi(0)]}.
D(L)=D*(L),Imn(L)=exp(iΔkL)Imn*(L),
Ppth=Ts+Vs2Esp,
Pinth=Tp+VpTp (Tp+Vp)(Ts+Vs)8Esp,
Esp=2μ0d2ωiωscninsnpL2mn|Imn(L)|2(1+|ri|cos[2ΔkL-2 arg Imn(L)+2(m+n+1)×arctan τ (L)+ϕ)].
Pminth=VpVsE,
Pminth=VpVs2Esp.
TsVs=PinPminth-1,TpVp=PinPminth,
Psmax=ωsωp(Pin-Pminth)2,
Pin1+PminthPin2=4Pinth.
Pi,out=ωiωpPin(Pin-Pminth).
Ts=0,TpVp=PinPminth,
Pimax=ωiωp(Pin-Pminth),Pin1+PminthPin2=4Pinth.
Lopt=Vf,sVf, pαsαp1/2.
Pminth=(αsVf, p+αpVf,s)2Eh.
Pminth|ringPminth|st.w.=hsphVf,s|ringVf,s|st.w..
Imn(L)=1L0L--ψ00, p*(x, y, z)×ψ00,s(x, y, z)ψmn,i(x, y, z)dxdy×exp(iΔkz)dz.
ψ00, p*(x, y, z)=2π 1wp(1+iτ)exp-x2+y2wp2(1+iτ),
ψ00,s(x, y, z)=2π 1ws(1-iτ)exp-x2+y2ws2(1-iτ),
zR=npωpwp22c=nsωsws22c,
ψmn,i(x, y, z)
=22m2nπm!n!exp[+i(m+n)arctan τi]wi(1-iτi)
×exp-x2+y2wi2(1-iτi)Hm2xwi(z)Hn2ywi(z).
(1+τ2)-1[wi-2+ws-2+wp-2-iτ
×(wp-2-ws-2-wi-2)].
wi-2+ws-2=wp-21-(kp-ks-ki) λp2π,
-- exp-2 x2+y2wp2(z)Hm2xwi(z)Hn2ywi(z)dxdy.
π2wp2(z) m!n!(m/2)!(n/2)!wp2(z)wi2(z)-1(m+n)/2.
Imn*(L)=2π2m2n m!n!(m/2)!(n/2)!wpwswiζ(m+n)/2×exp(-iΔkz0) zRLτ(0)τ(L)×exp[-iΔkzRτ-i(m+n)arctan τ]1+iτdτ,
E=4μ0d2ωs2ωi2πc2np2ωp×Lm,neven m!n!2n+m[(m/2)!(n/2)!]2ζm+nhm+n,
h2N=zR2Lτ (0)τ (L) exp(-iΔkzRτ-2iN arctan τ)1+iτdτ2.
E=4μ0d2ωs2ωi2πc2np2ωpLh,h=N=0,1,ζ2Nh2N.
E=4μ0d2ωs2ωi2πc2np2ωpLzR2Lτ(0)τ(L)τ(0)τ(L) ×exp[+iΔkzR(τ-τ)](1-iτ)(1+iτ)-ζ2(1+iτ)(1-iτ)dτ dτ.
h=L2zRsinc2(ΔkL/2)1-ζ2,
Esp(|ri|=1)
=4μ0d2ωs2ωi2πc2np2ωpLhsp,
hsp=(h+hc cos ϕ-hs sin ϕ)=2ζ2Nh2N cos2[ΔkL-arg Imn(L)+ϕ/2].
hspMax(h)=sinc2(ΔkL/2)[1+cos(ΔkL+ϕ)],
I00*(L)=2π wpwswiz¯R,i exp(-iΔkz0) zRLI˜,
I=τ (0)τ (L) 2 exp(-iΔkzRτ)(1+ζ)(iτ+z¯R,i+iΔz¯0)+(1-ζ)(iz¯R,iτ+1-Δz¯0τ)dτ,
f00=|I00(L)|2m,neven|Imn(L)|2=z¯R,izR2hL|I˜|2.

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