Abstract

A pulsed optical parametric oscillator (OPO) is pumped at 355 nm and injection seeded at two distinct idler wavelengths at 855 nm; these are separated by a frequency interval Δ falling within the 350-GHz free-running OPO bandwidth. Above-threshold operation of this dual-frequency seeded OPO causes sidebands to be observed at multiples of Δ in the spectrum of the 607-nm signal radiation. The sideband spacing varies smoothly as Δ is tuned (with the injection-seeded OPO cavity misaligned to reduce its effective finesse), and corresponding sidebands are observed on the transmitted 355-nm pump radiation. This is interpreted as direct evidence of backconversion of signal and idler waves in a pulsed OPO, consistent with previous temporal observations and other aspects of OPO performance. Signal-wave sidebands are observed well beyond the regular free-running OPO gain profile, which is understood in terms of the different phase-matching conditions for the various OPO pump and idler (seed) frequencies that are involved.

© 1997 Optical Society of America

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References

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  1. S. E. Harris, “Tunable optical parametric oscillators,” Proc. IEEE 57, 2096–2113 (1969).
    [CrossRef]
  2. J. E. Bjorkholm, “Some effects of spatially nonuniform pumping in pulsed optical parametricoscillators,” IEEE J. Quantum Electron. QE-7, 109–118 (1971).
    [CrossRef]
  3. R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
    [CrossRef]
  4. Y. X. Fan, R. C. Eckardt, R. L. Byer, J. Nolting, and R. Wallenstein, “Visible BaB2O4 optical parametric oscillatorpumped by a single-axial-mode pulsed source,” Appl. Phys. Lett. 53, 2014–2016 (1988).
    [CrossRef]
  5. M. J.Johnson, J. G. Haub, and B. J. Orr, “Numerical simulations and realizationof a tunable, injection-seeded optical parametric oscillator,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSATechnical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CWE2; M. J. Johnson, “Development of pulsed, tunable, optical parametric oscillators for spectroscopic applications,” Ph.D.dissertation (Macquarie University, Sydney, Australia, 1995).
  6. A. V. Smith, W. J. Alford, T. D. Raymond, and M. S. Bowers, “Comparison of a numerical model with measured performance of a seeded, nanosecond KTP optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2253–2267 (1995).
    [CrossRef]
  7. D. Hamoir, L. Duffault, J.-L. Ayral, F. Simon, and M. J. Damzen, “Multimodeoptical parametric oscillation with cross-interactions between signal-idlerpairs,” Opt. Commun. (to be published).
  8. J. G. Haub, M. J. Johnson, A. J. Powell, and B. J. Orr, “Bandwidth characteristics of a pulsed optical parametric oscillator:application to degenerate four-wave mixing spectroscopy,” Opt. Lett. 20, 1637–1639 (1995).
    [CrossRef] [PubMed]
  9. J. G. Haub, M. J. Johnson, B. J. Orr, and R. Wallenstein, “A continuously tunable, injection-seeded β-barium borate opticalparametric oscillator: spectroscopic applications,” Appl. Phys. Lett. 58, 1718–1720 (1991); A. Fix, T. Schröder, R. Wallenstein, J. G. Haub, M. J. Johnson, and B. J. Orr, “A tunable β-barium borate optical parametric oscillator: operatingcharacteristics with and without injection seeding,” J. Opt. Soc. Am. B 10, 1744–1750 (1993); J. G. Haub, M. J. Johnson, and B. J. Orr, “Spectroscopic and nonlinear-optical applications of a tunable β-bariumborate optical parametric oscillator,” J. Opt. Soc. Am. B JOBPDE 10, 1765–1777 (1993).
    [CrossRef]
  10. B. J. Orr, M. J. Johnson, and J. G. Haub, “Spectroscopicapplications of tunable optical parametric oscillators”, in Tunable Laser Applications, F. J. Duarte, ed. (Marcel Dekker, New York, 1995), Chap. 2, pp. 11–83.
  11. M. J. Johnson, J. G. Haub, and B. J. Orr, “Continuously tunable, narrowband operation of an injection-seeded ring-cavityoptical parametric oscillator: spectroscopic applications,” Opt. Lett. 20, 1277–1279 (1995); J. G. Haub, R. M. Hentschel, M. J. Johnson, and B. J. Orr, “Controlling the performance of a pulsed optical parametric oscillator:a survey of techniques and spectroscopic applications,” J. Opt. Soc. Am. B 12, 2128–2141 (1995).
    [CrossRef] [PubMed]
  12. G. W. Baxter, M. J. Johnson, J. G. Haub, and B. J. Orr, “OPO CARS: coherent anti-Stokes Raman spectroscopy using tunable opticalparametric oscillators injection-seeded by external-cavity diode lasers,” Chem. Phys. Lett. 251, 211–218 (1996).
    [CrossRef]
  13. W. A. Neuman and S. P. Velsko, “Spectral propertiesof optical parametric oscillators,” in Proceedings ofInternational Conference on Lasers '95, V. J. Corcoran and T. A. Goldman, eds. (STS, McLean, Va., 1996), pp. 718–725.
  14. V. G. Dmitriev, G. G. Gurzadyn, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).
  15. K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, “Hot spots in parametric fluorescence with a pump beam of finite crosssection,” IEEE J. Quantum Electron. 31, 769–781 (1995).
    [CrossRef]

1996 (1)

G. W. Baxter, M. J. Johnson, J. G. Haub, and B. J. Orr, “OPO CARS: coherent anti-Stokes Raman spectroscopy using tunable opticalparametric oscillators injection-seeded by external-cavity diode lasers,” Chem. Phys. Lett. 251, 211–218 (1996).
[CrossRef]

1995 (3)

1988 (1)

Y. X. Fan, R. C. Eckardt, R. L. Byer, J. Nolting, and R. Wallenstein, “Visible BaB2O4 optical parametric oscillatorpumped by a single-axial-mode pulsed source,” Appl. Phys. Lett. 53, 2014–2016 (1988).
[CrossRef]

1979 (1)

R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
[CrossRef]

1971 (1)

J. E. Bjorkholm, “Some effects of spatially nonuniform pumping in pulsed optical parametricoscillators,” IEEE J. Quantum Electron. QE-7, 109–118 (1971).
[CrossRef]

1969 (1)

S. E. Harris, “Tunable optical parametric oscillators,” Proc. IEEE 57, 2096–2113 (1969).
[CrossRef]

Alford, W. J.

Baumgartner, R. A.

R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
[CrossRef]

Baxter, G. W.

G. W. Baxter, M. J. Johnson, J. G. Haub, and B. J. Orr, “OPO CARS: coherent anti-Stokes Raman spectroscopy using tunable opticalparametric oscillators injection-seeded by external-cavity diode lasers,” Chem. Phys. Lett. 251, 211–218 (1996).
[CrossRef]

Bjorkholm, J. E.

J. E. Bjorkholm, “Some effects of spatially nonuniform pumping in pulsed optical parametricoscillators,” IEEE J. Quantum Electron. QE-7, 109–118 (1971).
[CrossRef]

Bowers, M. S.

Byer, R. L.

Y. X. Fan, R. C. Eckardt, R. L. Byer, J. Nolting, and R. Wallenstein, “Visible BaB2O4 optical parametric oscillatorpumped by a single-axial-mode pulsed source,” Appl. Phys. Lett. 53, 2014–2016 (1988).
[CrossRef]

R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
[CrossRef]

Chakmakjian, S. H.

K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, “Hot spots in parametric fluorescence with a pump beam of finite crosssection,” IEEE J. Quantum Electron. 31, 769–781 (1995).
[CrossRef]

Cheung, E. C.

K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, “Hot spots in parametric fluorescence with a pump beam of finite crosssection,” IEEE J. Quantum Electron. 31, 769–781 (1995).
[CrossRef]

Eckardt, R. C.

Y. X. Fan, R. C. Eckardt, R. L. Byer, J. Nolting, and R. Wallenstein, “Visible BaB2O4 optical parametric oscillatorpumped by a single-axial-mode pulsed source,” Appl. Phys. Lett. 53, 2014–2016 (1988).
[CrossRef]

Fan, Y. X.

Y. X. Fan, R. C. Eckardt, R. L. Byer, J. Nolting, and R. Wallenstein, “Visible BaB2O4 optical parametric oscillatorpumped by a single-axial-mode pulsed source,” Appl. Phys. Lett. 53, 2014–2016 (1988).
[CrossRef]

Harris, S. E.

S. E. Harris, “Tunable optical parametric oscillators,” Proc. IEEE 57, 2096–2113 (1969).
[CrossRef]

Haub, J. G.

G. W. Baxter, M. J. Johnson, J. G. Haub, and B. J. Orr, “OPO CARS: coherent anti-Stokes Raman spectroscopy using tunable opticalparametric oscillators injection-seeded by external-cavity diode lasers,” Chem. Phys. Lett. 251, 211–218 (1996).
[CrossRef]

J. G. Haub, M. J. Johnson, A. J. Powell, and B. J. Orr, “Bandwidth characteristics of a pulsed optical parametric oscillator:application to degenerate four-wave mixing spectroscopy,” Opt. Lett. 20, 1637–1639 (1995).
[CrossRef] [PubMed]

Johnson, M. J.

G. W. Baxter, M. J. Johnson, J. G. Haub, and B. J. Orr, “OPO CARS: coherent anti-Stokes Raman spectroscopy using tunable opticalparametric oscillators injection-seeded by external-cavity diode lasers,” Chem. Phys. Lett. 251, 211–218 (1996).
[CrossRef]

J. G. Haub, M. J. Johnson, A. J. Powell, and B. J. Orr, “Bandwidth characteristics of a pulsed optical parametric oscillator:application to degenerate four-wave mixing spectroscopy,” Opt. Lett. 20, 1637–1639 (1995).
[CrossRef] [PubMed]

Koch, K.

K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, “Hot spots in parametric fluorescence with a pump beam of finite crosssection,” IEEE J. Quantum Electron. 31, 769–781 (1995).
[CrossRef]

Liu, J. M.

K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, “Hot spots in parametric fluorescence with a pump beam of finite crosssection,” IEEE J. Quantum Electron. 31, 769–781 (1995).
[CrossRef]

Moore, G. T.

K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, “Hot spots in parametric fluorescence with a pump beam of finite crosssection,” IEEE J. Quantum Electron. 31, 769–781 (1995).
[CrossRef]

Nolting, J.

Y. X. Fan, R. C. Eckardt, R. L. Byer, J. Nolting, and R. Wallenstein, “Visible BaB2O4 optical parametric oscillatorpumped by a single-axial-mode pulsed source,” Appl. Phys. Lett. 53, 2014–2016 (1988).
[CrossRef]

Orr, B. J.

G. W. Baxter, M. J. Johnson, J. G. Haub, and B. J. Orr, “OPO CARS: coherent anti-Stokes Raman spectroscopy using tunable opticalparametric oscillators injection-seeded by external-cavity diode lasers,” Chem. Phys. Lett. 251, 211–218 (1996).
[CrossRef]

J. G. Haub, M. J. Johnson, A. J. Powell, and B. J. Orr, “Bandwidth characteristics of a pulsed optical parametric oscillator:application to degenerate four-wave mixing spectroscopy,” Opt. Lett. 20, 1637–1639 (1995).
[CrossRef] [PubMed]

Powell, A. J.

Raymond, T. D.

Smith, A. V.

Wallenstein, R.

Y. X. Fan, R. C. Eckardt, R. L. Byer, J. Nolting, and R. Wallenstein, “Visible BaB2O4 optical parametric oscillatorpumped by a single-axial-mode pulsed source,” Appl. Phys. Lett. 53, 2014–2016 (1988).
[CrossRef]

Appl. Phys. Lett. (1)

Y. X. Fan, R. C. Eckardt, R. L. Byer, J. Nolting, and R. Wallenstein, “Visible BaB2O4 optical parametric oscillatorpumped by a single-axial-mode pulsed source,” Appl. Phys. Lett. 53, 2014–2016 (1988).
[CrossRef]

Chem. Phys. Lett. (1)

G. W. Baxter, M. J. Johnson, J. G. Haub, and B. J. Orr, “OPO CARS: coherent anti-Stokes Raman spectroscopy using tunable opticalparametric oscillators injection-seeded by external-cavity diode lasers,” Chem. Phys. Lett. 251, 211–218 (1996).
[CrossRef]

IEEE J. Quantum Electron. (3)

J. E. Bjorkholm, “Some effects of spatially nonuniform pumping in pulsed optical parametricoscillators,” IEEE J. Quantum Electron. QE-7, 109–118 (1971).
[CrossRef]

R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
[CrossRef]

K. Koch, E. C. Cheung, G. T. Moore, S. H. Chakmakjian, and J. M. Liu, “Hot spots in parametric fluorescence with a pump beam of finite crosssection,” IEEE J. Quantum Electron. 31, 769–781 (1995).
[CrossRef]

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

Opt. Lett. (1)

Proc. IEEE (1)

S. E. Harris, “Tunable optical parametric oscillators,” Proc. IEEE 57, 2096–2113 (1969).
[CrossRef]

Other (7)

D. Hamoir, L. Duffault, J.-L. Ayral, F. Simon, and M. J. Damzen, “Multimodeoptical parametric oscillation with cross-interactions between signal-idlerpairs,” Opt. Commun. (to be published).

W. A. Neuman and S. P. Velsko, “Spectral propertiesof optical parametric oscillators,” in Proceedings ofInternational Conference on Lasers '95, V. J. Corcoran and T. A. Goldman, eds. (STS, McLean, Va., 1996), pp. 718–725.

V. G. Dmitriev, G. G. Gurzadyn, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

M. J.Johnson, J. G. Haub, and B. J. Orr, “Numerical simulations and realizationof a tunable, injection-seeded optical parametric oscillator,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSATechnical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CWE2; M. J. Johnson, “Development of pulsed, tunable, optical parametric oscillators for spectroscopic applications,” Ph.D.dissertation (Macquarie University, Sydney, Australia, 1995).

J. G. Haub, M. J. Johnson, B. J. Orr, and R. Wallenstein, “A continuously tunable, injection-seeded β-barium borate opticalparametric oscillator: spectroscopic applications,” Appl. Phys. Lett. 58, 1718–1720 (1991); A. Fix, T. Schröder, R. Wallenstein, J. G. Haub, M. J. Johnson, and B. J. Orr, “A tunable β-barium borate optical parametric oscillator: operatingcharacteristics with and without injection seeding,” J. Opt. Soc. Am. B 10, 1744–1750 (1993); J. G. Haub, M. J. Johnson, and B. J. Orr, “Spectroscopic and nonlinear-optical applications of a tunable β-bariumborate optical parametric oscillator,” J. Opt. Soc. Am. B JOBPDE 10, 1765–1777 (1993).
[CrossRef]

B. J. Orr, M. J. Johnson, and J. G. Haub, “Spectroscopicapplications of tunable optical parametric oscillators”, in Tunable Laser Applications, F. J. Duarte, ed. (Marcel Dekker, New York, 1995), Chap. 2, pp. 11–83.

M. J. Johnson, J. G. Haub, and B. J. Orr, “Continuously tunable, narrowband operation of an injection-seeded ring-cavityoptical parametric oscillator: spectroscopic applications,” Opt. Lett. 20, 1277–1279 (1995); J. G. Haub, R. M. Hentschel, M. J. Johnson, and B. J. Orr, “Controlling the performance of a pulsed optical parametric oscillator:a survey of techniques and spectroscopic applications,” J. Opt. Soc. Am. B 12, 2128–2141 (1995).
[CrossRef] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

Nonlinear-optical processes in a pulsed OPO, satisfying the resonance condition of Eq. (1). Vertical bars indicate spectral features of pump and output radiation; each set of triangles joined by a dashed line represents a coherent three-wave process, with downward-pointing arrows designating generated waves. (a) Signal–idler pairs (frequencies ωS, ωI) in the output spectrum of a broadband, free-running OPO, generated by a monochromatic pump wave (frequency ωP). (b) Signal-wave output obtained by narrow-band injection seeding in the idler wavelength range. (c) Coherent sum-frequency mixing of the narrow-band signal and idler waves generating a backconverted pump wave at ωP.

Fig. 2
Fig. 2

Single-shot spectra of the 607-nm signal-wave output of a ring-cavity BBO OPO, pumped at 355 nm by a single-mode Nd:YAG laser at two times above threshold. Trace (a) shows the broadband output of a free-running BBO OPO (with no injection seeding). The traces (b) were obtained at two different detected light levels by injection-seeding the OPO at its 855-nm idler wavelength with a pair of single-mode diode lasers, separated in output frequency by Δ=150 GHz. Sidebands of order q=1, 2, and 3 are marked by the corresponding number of asterisks, on either side of stronger (unmarked) signal peaks that correspond to the two distinctly seeded idler frequencies. Single-frequency seeding with each of the diode lasers in turn yields sideband-free signal-wave spectra, as in traces (c) and (d).

Fig. 3
Fig. 3

Dual-frequency seeded signal-wave spectra recorded as in lower trace (b) of Fig. 2 except that the injection-seeding frequency interval Δ is stepped through 150, 120, 85, 50, and 0 GHz as indicated. The first-order sidebands follow this variation smoothly, according to Eq. (3).

Fig. 4
Fig. 4

Single-shot spectra of transmitted OPO pump radiation at 355 nm. Traces (a) and (b) are recorded under conditions that correspond to (c) and (d) of Fig. 2, with single-frequency injection seeding. Traces (c) and (d) show that first-order pump-wave sidebands appear if dual-frequency injection seeding with Δ =150 GHz is used under conditions that give rise to signal-wave sidebands, as in (b) of Fig. 2. The sidebands are displaced by ±Δ from the pump fundamental. The upper of traces (d) is recorded with a tenfold increase of instrumental gain, relative to the lower. The only difference between trace (c) and traces (d) is that the latter are recorded with a lower (aperture-determined) detected intensity to bring the fundamental peak on scale and show the relative magnitude of the pump-wave sidebands.

Fig. 5
Fig. 5

Comparison of OPO pump-wave and signal-wave spectra, showing the dependence of dual-frequency injection-seeded sideband intensities on variations of the pump-laser intensity. The three traces are recorded with Δ=150 GHz as in (b) of Fig. 2 and (c) of Fig. 4 (for 607-nm signal and 355-nm pump waves, respectively), except that the pump intensity is stepped through 2.0, 1.9, and 1.7 times above OPO threshold as indicated.

Fig. 6
Fig. 6

Schematic relationship (in the style of Fig. 1) of pump, signal, and idler frequencies in an OPO operated with dual-frequency, idler-wave injection seeding. (a), Two OPO signal output frequencies ωS±(0) are connected to the single-pump frequency ωP(0), and the two injection-seeded idler frequencies ωI(0) separated by an interval Δ, in a manner consistent with Eq. (2). (b) Original pump frequency ωP(0) is regenerated by backconversion of either signal–idler pair (ωS±(0), ωI(0)), as in a monochromatically seeded OPO. By contrast, (c) illustrates the generation of first-order pump-wave sidebands (ωP±(1)) through cross-interacting backconversion of two different pairs of dual-frequency seeded signal and idler waves (ωS±(0), ωI±(0)), as in Eq. (4).

Fig. 7
Fig. 7

Relationship of first-order pump, signal, and idler sideband frequencies to the corresponding zero-order frequencies of a dual-frequency seeded OPO, following the style of Figs. 1 and 6. (a) Composite of Fig. 6. (b) and (c) Corresponding generation of first-order signal- and idler-wave sidebands (ωS±(1) and ωI±(1), respectively).

Fig. 8
Fig. 8

Multiplex OPO CARS spectra12 [broadband in trace (a) and dual frequency in trace (b)] for N2 in furnace air at 1100 K. These are compared in the inverted trace (c) with the spectrum at 607 nm of corresponding dual-frequency OPO signal output (including accompanying sidebands) that serves as Stokes radiation in the CARS process to which trace (b) refers. A two-dimensional 8-bit intensified CCD video camera is used to record these results, rather than the linear diode array employed in Figs. 25. Each trace is the average of 50 OPO shots and is spatially integrated over the vertical profile of the CARS or OPO signal beams. The weak shoulders on the low-frequency side of each peak in trace (b) are considered to be spurious. The absence of CARS peaks that correspond to OPO sidebands indicates that (in this case, at least) the presence of those sidebands will not necessarily give rise to spectroscopic ambiguities.

Tables (1)

Tables Icon

Table 1 Values of Phase Mismatch (Δk/2π) Calculated in Units of Δ for Various Processes in a Type I Collinearly Phase-Matched BBO OPO, Oriented at θ32.5°, Pumped at λP(0)=354.7  nm, and Injection-Seeded with a Frequency Interval Δ=5 cm-1 (150 GHz)

Equations (27)

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ωP=ωS+ωI.
ωS±(0)=ωP(0)-ωI(0),
ωS±(q)=ωS±(0)±qΔ,
ωP±(1)=ωS±(0)+ωI±(0)=ωP(0)±Δ,
ωS±(q)=ωP±(r)-ωI(q-r)=ωS±(0)±qΔ,
Δk=kP-kI-kS,
nαe(θ)=nαo[(1+tan2 θ)/(1+(nαo/nαe)2 tan2 θ)]1/2,
Δk(ωP, ωS, ωI)/2π=[nP(ωP)]ωP-[nS(ωS)]ωS-[nI(ωI)]ωI.
ω¯P=ωP(0),ω¯S=(ωS+(0)+ωS-(0))/2,
ω¯I=(ωI+(0)+ωI-(0))/2.
Δk(ω¯P, ω¯S, ω¯I)/2π=n¯Pω¯P-n¯Sω¯S-n¯Iω¯I,
Δk(ω¯P±Δ, ω¯S±1/2Δ, ω¯I±1/2Δ)/2π
=[nP(ω¯P±Δ)](ω¯P±Δ)-[nS(ω¯S±1/2Δ)]×(ω¯S±1/2Δ)-[nI(ω¯I±1/2Δ)](ω¯I±1/2Δ),
Δk(ω¯P±Δ, ω¯S±3/2Δ, ω¯I±1/2Δ)/2π
=[nP(ω¯P±Δ)](ω¯P±Δ)-[nS(ω¯S±3/2Δ)]×(ω¯S±3/2Δ)-[nI(ω¯I1/2Δ)](ω¯I1/2Δ).
Δk(ω¯P, ω¯S±3/2Δ, ω¯I3/2Δ)/2π
=n¯Pω¯P-[nS(ω¯S±3/2Δ)](ω¯S±3/2Δ)
-[nI(ω¯I3/2Δ)](ω¯I3/2Δ).
nα(ωα)=n¯α+nα(ωα-ω¯α)+,
Δk(ω¯P±Δ, ω¯S±1/2Δ, ω¯I±1/2Δ)/2π
=±Δ{[(n¯P+nPω¯P)-(n¯S+nSω¯S)]-1/2[(nI+nIω¯I)-(n¯S+nSω¯S)]}+,
Δk(ω¯P±Δ, ω¯S±3/2Δ, ω¯I1/2 Δ)/2π
=±Δ{[(n¯P+nPω¯P)-(n¯S+nSω¯S)]+1/2[(n¯I+nIω¯I)-(n¯S+nSω¯S)]}+,
Δk(ω¯P, ω¯S±3/2Δ, ω¯I3/2Δ)/2π
=±3/2Δ[(n¯I+nIω¯I)-(n¯S+nSω¯S)]+,
Δk(ω¯P±rΔ, ω¯S±(q+1/2)Δ, ω¯I(q-r+1/2)Δ)2π
=±Δ{r[(n¯P+nPω¯P)-(n¯S+nSω¯S)]-(q-r+1/2)[(n¯I+nIω¯I)-(n¯S+nSω¯S)]}+.

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