Abstract

We have developed a model for mode hopping in doubly resonant optical parametric oscillators. The pump produced two pairs of signal–idler modes. We obtain various steady states of interest, namely, when only the first pair oscillates with the other pair that has null amplitudes and vice versa. Stability analysis reveals that there can be an exchange of stability between the oscillating and the zero amplitude pairs. We derive conditions for the exchange of stability in terms of cavity parameters, which can change because of changes in the cavity length or because of fluctuations in the phase of the pump. We demonstrate the exchange of stability from numerical solutions of coupled nonlinear equations for five complex modes (i.e., the pump and the two signal idler pairs). For a specific choice of parameters we also demonstrate the excitation of both the pairs.

© 1997 Optical Society of America

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References

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  1. J. A. Giordmaine and R. C. Miller, Phys. Rev. Lett. 14, 973 (1965); Appl. Phys. Lett. 9, 298 (1966).
    [CrossRef]
  2. For the current state of the art, see, for example, the references in the feature on optical parametric oscillation and amplification, J. Opt. Soc. Am. B 10, 1654–1791, 2148–2243 (1993).
  3. R. L. Byer, in Treatise on Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, New York, 1973), pp. 588–702.
  4. R. J. Smith, in Advances in Lasers, A. K. Levine and A. J. DeMaria, eds. (Dekker, New York, 1976), Vol. 4.
  5. C. L. Tang, “Optical parametric processes and inorganic nonlinear optical crystals,” in Proceedings of the International School of Physics 〈Enrico Fermi〉 (North-Holland, Amsterdam, 1994), pp. 97–129.
  6. For a recent review, see, for example, C. L. Tang, Int. J. Nonlin. Opt. Phys. 3, 205 (1994).
    [CrossRef]
  7. F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973), pp. 153–176.
  8. S. E. Harris, Proc. IEEE 57, 2096 (1969).
    [CrossRef]
  9. J. Falk, IEEE J. Quantum Electron. QE-7, 230 (1971).
    [CrossRef]
  10. R. J. Smith, IEEE J. Quantum Electron. QE-9, 530 (1973).
    [CrossRef]
  11. G. S. Agarwal, Phys. Rev. A 18, 1490 (1978).
    [CrossRef]
  12. See, for example, S. J. Brosnan and R. L. Byer, IEEE J. Quantum Electron. QE-15, 415 (1979); L. Wu, Min Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987); C. D. Nabors, S. T. Yang, and R. L. Byer, J. Opt. Soc. Am. B 7, 815 (1990); H. J. Bakker, J. T. M. Kennis, H. J. Kop, and A. Lagendjik, Opt. Commun. 86, 58 (1991); W. S. Pelouch, P. E. Powers, and C. L. Tang, Opt. Lett. 17, 1070 (1992). Many recent studies of OPO’s focus on tunable ultrashort-pulse generation; approximately 10 appeared in Optics Letters alone in 1995–1996.
    [CrossRef] [PubMed]
  13. R. Graham and H. Haken, Z. Phys. 210, 276 (1968).
    [CrossRef]
  14. R. C. Eckardt, C. D. Nabors, W. J. Kozlovsky, and R. L. Byer, J. Opt. Soc. Am. B 8, 646 (1991).
    [CrossRef]
  15. Instabilities in OPO devices have been reported that are nearly singly resonant, with a small amount of feedback at the nonresonant frequency.9 Our model can also deal with such situations, and we can even predict how small this feedback will be to give stable operation. We do not discuss them here, as our main purpose is to treat instabilities in DRO’s.

1994 (1)

For a recent review, see, for example, C. L. Tang, Int. J. Nonlin. Opt. Phys. 3, 205 (1994).
[CrossRef]

1993 (1)

For the current state of the art, see, for example, the references in the feature on optical parametric oscillation and amplification, J. Opt. Soc. Am. B 10, 1654–1791, 2148–2243 (1993).

1991 (1)

1979 (1)

See, for example, S. J. Brosnan and R. L. Byer, IEEE J. Quantum Electron. QE-15, 415 (1979); L. Wu, Min Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987); C. D. Nabors, S. T. Yang, and R. L. Byer, J. Opt. Soc. Am. B 7, 815 (1990); H. J. Bakker, J. T. M. Kennis, H. J. Kop, and A. Lagendjik, Opt. Commun. 86, 58 (1991); W. S. Pelouch, P. E. Powers, and C. L. Tang, Opt. Lett. 17, 1070 (1992). Many recent studies of OPO’s focus on tunable ultrashort-pulse generation; approximately 10 appeared in Optics Letters alone in 1995–1996.
[CrossRef] [PubMed]

1978 (1)

G. S. Agarwal, Phys. Rev. A 18, 1490 (1978).
[CrossRef]

1973 (1)

R. J. Smith, IEEE J. Quantum Electron. QE-9, 530 (1973).
[CrossRef]

1971 (1)

J. Falk, IEEE J. Quantum Electron. QE-7, 230 (1971).
[CrossRef]

1969 (1)

S. E. Harris, Proc. IEEE 57, 2096 (1969).
[CrossRef]

1968 (1)

R. Graham and H. Haken, Z. Phys. 210, 276 (1968).
[CrossRef]

1965 (1)

J. A. Giordmaine and R. C. Miller, Phys. Rev. Lett. 14, 973 (1965); Appl. Phys. Lett. 9, 298 (1966).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, Phys. Rev. A 18, 1490 (1978).
[CrossRef]

Brosnan, S. J.

See, for example, S. J. Brosnan and R. L. Byer, IEEE J. Quantum Electron. QE-15, 415 (1979); L. Wu, Min Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987); C. D. Nabors, S. T. Yang, and R. L. Byer, J. Opt. Soc. Am. B 7, 815 (1990); H. J. Bakker, J. T. M. Kennis, H. J. Kop, and A. Lagendjik, Opt. Commun. 86, 58 (1991); W. S. Pelouch, P. E. Powers, and C. L. Tang, Opt. Lett. 17, 1070 (1992). Many recent studies of OPO’s focus on tunable ultrashort-pulse generation; approximately 10 appeared in Optics Letters alone in 1995–1996.
[CrossRef] [PubMed]

Byer, R. L.

R. C. Eckardt, C. D. Nabors, W. J. Kozlovsky, and R. L. Byer, J. Opt. Soc. Am. B 8, 646 (1991).
[CrossRef]

See, for example, S. J. Brosnan and R. L. Byer, IEEE J. Quantum Electron. QE-15, 415 (1979); L. Wu, Min Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987); C. D. Nabors, S. T. Yang, and R. L. Byer, J. Opt. Soc. Am. B 7, 815 (1990); H. J. Bakker, J. T. M. Kennis, H. J. Kop, and A. Lagendjik, Opt. Commun. 86, 58 (1991); W. S. Pelouch, P. E. Powers, and C. L. Tang, Opt. Lett. 17, 1070 (1992). Many recent studies of OPO’s focus on tunable ultrashort-pulse generation; approximately 10 appeared in Optics Letters alone in 1995–1996.
[CrossRef] [PubMed]

R. L. Byer, in Treatise on Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, New York, 1973), pp. 588–702.

Eckardt, R. C.

Falk, J.

J. Falk, IEEE J. Quantum Electron. QE-7, 230 (1971).
[CrossRef]

Giordmaine, J. A.

J. A. Giordmaine and R. C. Miller, Phys. Rev. Lett. 14, 973 (1965); Appl. Phys. Lett. 9, 298 (1966).
[CrossRef]

Graham, R.

R. Graham and H. Haken, Z. Phys. 210, 276 (1968).
[CrossRef]

Haken, H.

R. Graham and H. Haken, Z. Phys. 210, 276 (1968).
[CrossRef]

Harris, S. E.

S. E. Harris, Proc. IEEE 57, 2096 (1969).
[CrossRef]

Kozlovsky, W. J.

Midwinter, J. E.

F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973), pp. 153–176.

Miller, R. C.

J. A. Giordmaine and R. C. Miller, Phys. Rev. Lett. 14, 973 (1965); Appl. Phys. Lett. 9, 298 (1966).
[CrossRef]

Nabors, C. D.

Smith, R. J.

R. J. Smith, IEEE J. Quantum Electron. QE-9, 530 (1973).
[CrossRef]

R. J. Smith, in Advances in Lasers, A. K. Levine and A. J. DeMaria, eds. (Dekker, New York, 1976), Vol. 4.

Tang, C. L.

For a recent review, see, for example, C. L. Tang, Int. J. Nonlin. Opt. Phys. 3, 205 (1994).
[CrossRef]

C. L. Tang, “Optical parametric processes and inorganic nonlinear optical crystals,” in Proceedings of the International School of Physics 〈Enrico Fermi〉 (North-Holland, Amsterdam, 1994), pp. 97–129.

Zernike, F.

F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973), pp. 153–176.

IEEE J. Quantum Electron. (3)

J. Falk, IEEE J. Quantum Electron. QE-7, 230 (1971).
[CrossRef]

R. J. Smith, IEEE J. Quantum Electron. QE-9, 530 (1973).
[CrossRef]

See, for example, S. J. Brosnan and R. L. Byer, IEEE J. Quantum Electron. QE-15, 415 (1979); L. Wu, Min Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987); C. D. Nabors, S. T. Yang, and R. L. Byer, J. Opt. Soc. Am. B 7, 815 (1990); H. J. Bakker, J. T. M. Kennis, H. J. Kop, and A. Lagendjik, Opt. Commun. 86, 58 (1991); W. S. Pelouch, P. E. Powers, and C. L. Tang, Opt. Lett. 17, 1070 (1992). Many recent studies of OPO’s focus on tunable ultrashort-pulse generation; approximately 10 appeared in Optics Letters alone in 1995–1996.
[CrossRef] [PubMed]

Int. J. Nonlin. Opt. Phys. (1)

For a recent review, see, for example, C. L. Tang, Int. J. Nonlin. Opt. Phys. 3, 205 (1994).
[CrossRef]

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

For the current state of the art, see, for example, the references in the feature on optical parametric oscillation and amplification, J. Opt. Soc. Am. B 10, 1654–1791, 2148–2243 (1993).

R. C. Eckardt, C. D. Nabors, W. J. Kozlovsky, and R. L. Byer, J. Opt. Soc. Am. B 8, 646 (1991).
[CrossRef]

Phys. Rev. A (1)

G. S. Agarwal, Phys. Rev. A 18, 1490 (1978).
[CrossRef]

Phys. Rev. Lett. (1)

J. A. Giordmaine and R. C. Miller, Phys. Rev. Lett. 14, 973 (1965); Appl. Phys. Lett. 9, 298 (1966).
[CrossRef]

Proc. IEEE (1)

S. E. Harris, Proc. IEEE 57, 2096 (1969).
[CrossRef]

Z. Phys. (1)

R. Graham and H. Haken, Z. Phys. 210, 276 (1968).
[CrossRef]

Other (5)

Instabilities in OPO devices have been reported that are nearly singly resonant, with a small amount of feedback at the nonresonant frequency.9 Our model can also deal with such situations, and we can even predict how small this feedback will be to give stable operation. We do not discuss them here, as our main purpose is to treat instabilities in DRO’s.

F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973), pp. 153–176.

R. L. Byer, in Treatise on Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, New York, 1973), pp. 588–702.

R. J. Smith, in Advances in Lasers, A. K. Levine and A. J. DeMaria, eds. (Dekker, New York, 1976), Vol. 4.

C. L. Tang, “Optical parametric processes and inorganic nonlinear optical crystals,” in Proceedings of the International School of Physics 〈Enrico Fermi〉 (North-Holland, Amsterdam, 1994), pp. 97–129.

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Figures (4)

Fig. 1
Fig. 1

Temporal evolution of (a) mode amplitudes |a1|, |b1|, and |d| (curves 1, 2, and 3, respectively) and (b) logarithm of |a1|, |b1|, and |d| (curves 1, 2, 3, respectively) when |fa| is switched in accordance with Eq. (5) (see text) with γ=0.01, δ=0.01, |fb|=0.9, and t=500. Other parameters are Δωa=Δωb=0, κd=κa1=κa2=κb1=κb2=1, β=0.83×10-3, and |Fp|=4.8×103. (c) Shows |fa| and |fb| (curves 1 and 2) as functions of time.

Fig. 2
Fig. 2

Same as in Fig. 1, except that |fa| is switched in accordance with Eq. (6) (see text), with γ=0.01, δ=0.01, |fb|=0.9, and t=500.

Fig. 3
Fig. 3

Temporal evolution of |a1|, |b1|, and |d| for |fa|=|fb|=1.0. κa and κb are assumed to switch in accordance with Eqs. (8) and (9). (c) Shows κb/κa. Other parameters are as in Fig. 1.

Fig. 4
Fig. 4

(a), (b) Temporal evolution of |a1|, |b1|, and |d| for two different sets of initial conditions. The parameters are κd=κa1=κa2=κb1=κb2=1 and |fa|=|fb|=0.9 (dashed curves). The solid curves show the results when |fa| is switched at t=500 from value 0.9 to 0.91. Other parameters are as in Fig. 1.

Equations (43)

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a˙j=-κajaj+βfa exp(-iΔωat)da3-j*,
b˙j=-κbjbj+βfb exp(-iΔωbt)db3-j*,
d˙=-κdd-βfa* exp(iΔωat)a1a2-βfb* exp(iΔωbt)b1b2+Fp.
fs=sin(ΔksL/2)ΔksL/2exp(iΔksL/2),
κd=πc2Fd Lnd,
β=22deffVπ3/2(ωs1ωs2ωd)1/2.
|fa|=|fb|+δ tanh γ(t-t),
|fa|=|fb|-δ+δ[1+sign(t-t)]×{1-exp[-γ(t-t)]},
sign(x)=1x>0-1x<0.
κa=1+1/2[1+sign(t-t)]δ,
κb=1-δ+1/2[1+sign(t-t)]δ,
δ=δ2+3δ.
aj=Aj exp(-iΩajt),bj=Bj exp(-iΩbjt),
Ωa1+Ωa2=Δωa,Ωb1+Ωb2=Δωb.
(-κa1-iΩa1)A¯1+βfad¯A¯2*=0,
(-κa2-iΩa2)A¯2+βfad¯A¯1*=0,
(-κb1-iΩb1)B¯1+βfbd¯B¯2*=0,
(-κb2-iΩb2)B¯2+βfbd¯B¯1*=0,
-κdd¯-βfa*A¯1A¯2-βfb*B¯1B¯2+Fp=0.
|d¯|2=κa1κa2β2|fa|21+Δωa2κa12+κa22,
Ωa1=Δωaκa1κa1+κa2,Ωa2=Δωaκa2κa1+κa2.
κa1|A¯1|2=κa2|A¯2|2=Ca2.
A¯j=Caκajexp(-iφaj),B¯j=0,j=1, 2,
d¯=|d¯|exp(-iφ3),
φ3-φa1-φa2=ΔkaL2+tan-1 Δωaκa1+κa2,
Ca2=κa1κa2β|fa|×|Fp|2-κd2|d¯|2Δωa2Δωa2+(κa1+κa2)21/2-κd|d¯|(κa1+κa2)[Δωa2+(κa1+κa2)2]1/2,
sinφp-φ3+tan-1 Δωaκa1+κa2=κd|d¯||Fp|Δωa[Δωa2+(κa1+κa2)2]1/2.
b˙1=-κb1b1+βfbd¯b2*,
b˙2*=-κb2b2*+βfb*d¯*b1.
λ2+λ(κb1+κb2)+κb1κb2-β2|fb|2|d¯|2=0.
d¯=κa1κa2β|fa|.
λ1,2=-κb1+κb22±κb1+κb222+β2|fb|2|d¯|2-κb1κb21/2.
β|fb||d¯|>κb1κb2.
|fb||fa|>κb1κb2κa1κa21/2.
|d¯|2=κa1κa2β2|fa|21+Δωa2(κa1+κa2)2=κb1κb2β2|fb|21+Δωb2(κb1+κb2)2,
Ωsj=κsjΔωsκs1+κs2,j=1, 2,s=a, b.
aj=Caκajexp(-iφaj),bj=Cbκbjexp(-φbj),
j=1, 2,d=|d¯|exp(-iφ3)
φ3-φs1-φs2=ΔksL2+tan-1 Δωsκs1+κs2,
s=a,b.
-κd|d¯|-β|fa|κa1κa2Ca2 costan-1 -Δωaκa1+κa2-β|fb|κb1κb2Cb2 costan-1 -Δωbκb1+κb2=|Fp|cos(φp-φ3),
β|fa|κa1κa2Ca2 sintan-1 -Δωaκa1+κa2+β|fb|2κb1κb2Cb2 sintan-1 -Δωbκb1+κb2=|Fp|sin(φp-φ3).
-κd|d¯|-β|fa|κa1κa2Ca2-β|fb|κb1κb2Cb2=|Fp|.

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