Abstract

We present a new method of calculating the performance of nonlinear processes in a resonator. An optimization-based approach, conceptually similar to techniques used in nonlinear circuit analysis, is formulated and used to find the wave magnitudes that satisfy all of the boundary conditions and account for nonlinear optical effects. Unlike previous solution methods, this technique is applicable to any nonlinear process (second-order, third-order, etc.) and multiple coupled resonators, maintains the phase relations between the waves, and is exact. Examples are given for second-order nonlinear processes in a one-dimensional resonator.

© 2006 Optical Society of America

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References

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  1. J. A. Armstrong, N. Bloembergen, J. Ducuing and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
    [CrossRef]
  2. A. Ashkin, G. D. Boyd, and J. M. Dziedzic, "Resonant optical second harmonic generation and mixing," IEEE J. Quantum Electron. QE-2, 109-124 (1966).
    [CrossRef]
  3. V. Berger, "Second-harmonic generation in monolithic cavities," J. Opt. Soc. Am. B 14, 1351-1360 (1997).
    [CrossRef]
  4. C. Simonneau,  et al, "Second-harmonic generation in a doubly resonant semiconductor microcavity," Opt. Lett. 22, 1775-1777 (1997).
    [CrossRef]
  5. G. Klemens, C.-H. Chen, and Y. Fainman, "Design of optimized dispersive resonant cavities for nonlinear wave mixing," Opt. Express 13, 9388-9397 (2005).
    [CrossRef] [PubMed]
  6. D. S. Bethune, "Optical harmonic generation and mixing in multilayer media: analysis using optical transfer matrix techniques," J. Opt. Soc. Am. B 6, 910-916 (1989).
    [CrossRef]
  7. A. Calderone and J. Vigneron, "Computation of the electromagnetic harmonics generation by stratified systems containing nonlinear layers," Int. J. Quantum Chem. 70, 763-770 (1988).
    [CrossRef]
  8. M. G. Martemyanov, T. V. Dolgova and A. A. Fedyanin,"Optical third-harmonic generation in one-dimensional photonic crystals and microcavities," J. Exp. Theor. Phys. 98, 463-477 (2003).
    [CrossRef]
  9. S. El-Rabaie, V. F. Fusco, and C. Stewart,"Harmonic balance evaluation of nonlinear microwave circuits-a tutorial approach," IEEE Trans. on Education 31, 181-192 (1988).
    [CrossRef]
  10. I. Shoji, T. Kondo, A. Kitamoto, M. Shirane and R. Ito, "Absolute scale of second-order nonlinear-optical coeffients," J. Opt. Soc. Am. B 14, 2268-2294 (1997).
    [CrossRef]
  11. W. J. Tropf, M. E. Thomas, and T. J. Harris, "Properties of crystals and glasses," in Handbook of Optics: Volume II, (McGraw-Hill 1995).
  12. K. E. Atkinson, An Introduction to Numerical Analysis, (John Wiley 1989), Chap. 8.
  13. C. H. Chen, K. Tetz,W. Nakagawa, and Y. Fainman, "Wide-field-of-view GaAs/AlxOy one-dimensional photonic crystal filter," Appl. Opt. 44, 1503-1511 (2005).
    [CrossRef] [PubMed]
  14. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-Phase-Matched Second Harmonic Generation: Tuning and Tolerances," IEEE J. Quantum Electron. 28, 2631-2654 (1992).
    [CrossRef]

2005 (2)

2003 (1)

M. G. Martemyanov, T. V. Dolgova and A. A. Fedyanin,"Optical third-harmonic generation in one-dimensional photonic crystals and microcavities," J. Exp. Theor. Phys. 98, 463-477 (2003).
[CrossRef]

1997 (3)

1992 (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-Phase-Matched Second Harmonic Generation: Tuning and Tolerances," IEEE J. Quantum Electron. 28, 2631-2654 (1992).
[CrossRef]

1989 (1)

1988 (2)

A. Calderone and J. Vigneron, "Computation of the electromagnetic harmonics generation by stratified systems containing nonlinear layers," Int. J. Quantum Chem. 70, 763-770 (1988).
[CrossRef]

S. El-Rabaie, V. F. Fusco, and C. Stewart,"Harmonic balance evaluation of nonlinear microwave circuits-a tutorial approach," IEEE Trans. on Education 31, 181-192 (1988).
[CrossRef]

1966 (1)

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, "Resonant optical second harmonic generation and mixing," IEEE J. Quantum Electron. QE-2, 109-124 (1966).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Ashkin, A.

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, "Resonant optical second harmonic generation and mixing," IEEE J. Quantum Electron. QE-2, 109-124 (1966).
[CrossRef]

Berger, V.

Bethune, D. S.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Boyd, G. D.

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, "Resonant optical second harmonic generation and mixing," IEEE J. Quantum Electron. QE-2, 109-124 (1966).
[CrossRef]

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-Phase-Matched Second Harmonic Generation: Tuning and Tolerances," IEEE J. Quantum Electron. 28, 2631-2654 (1992).
[CrossRef]

Calderone, A.

A. Calderone and J. Vigneron, "Computation of the electromagnetic harmonics generation by stratified systems containing nonlinear layers," Int. J. Quantum Chem. 70, 763-770 (1988).
[CrossRef]

Chen, C. H.

Chen, C.-H.

Dolgova, T. V.

M. G. Martemyanov, T. V. Dolgova and A. A. Fedyanin,"Optical third-harmonic generation in one-dimensional photonic crystals and microcavities," J. Exp. Theor. Phys. 98, 463-477 (2003).
[CrossRef]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Dziedzic, J. M.

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, "Resonant optical second harmonic generation and mixing," IEEE J. Quantum Electron. QE-2, 109-124 (1966).
[CrossRef]

El-Rabaie, S.

S. El-Rabaie, V. F. Fusco, and C. Stewart,"Harmonic balance evaluation of nonlinear microwave circuits-a tutorial approach," IEEE Trans. on Education 31, 181-192 (1988).
[CrossRef]

Fainman, Y.

Fedyanin, A. A.

M. G. Martemyanov, T. V. Dolgova and A. A. Fedyanin,"Optical third-harmonic generation in one-dimensional photonic crystals and microcavities," J. Exp. Theor. Phys. 98, 463-477 (2003).
[CrossRef]

Fejer, M. M.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-Phase-Matched Second Harmonic Generation: Tuning and Tolerances," IEEE J. Quantum Electron. 28, 2631-2654 (1992).
[CrossRef]

Fusco, V. F.

S. El-Rabaie, V. F. Fusco, and C. Stewart,"Harmonic balance evaluation of nonlinear microwave circuits-a tutorial approach," IEEE Trans. on Education 31, 181-192 (1988).
[CrossRef]

Ito, R.

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-Phase-Matched Second Harmonic Generation: Tuning and Tolerances," IEEE J. Quantum Electron. 28, 2631-2654 (1992).
[CrossRef]

Kitamoto, A.

Klemens, G.

Kondo, T.

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-Phase-Matched Second Harmonic Generation: Tuning and Tolerances," IEEE J. Quantum Electron. 28, 2631-2654 (1992).
[CrossRef]

Martemyanov, M. G.

M. G. Martemyanov, T. V. Dolgova and A. A. Fedyanin,"Optical third-harmonic generation in one-dimensional photonic crystals and microcavities," J. Exp. Theor. Phys. 98, 463-477 (2003).
[CrossRef]

Nakagawa, W.

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Shirane, M.

Shoji, I.

Simonneau, C.

Stewart, C.

S. El-Rabaie, V. F. Fusco, and C. Stewart,"Harmonic balance evaluation of nonlinear microwave circuits-a tutorial approach," IEEE Trans. on Education 31, 181-192 (1988).
[CrossRef]

Tetz, K.

Vigneron, J.

A. Calderone and J. Vigneron, "Computation of the electromagnetic harmonics generation by stratified systems containing nonlinear layers," Int. J. Quantum Chem. 70, 763-770 (1988).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-Phase-Matched Second Harmonic Generation: Tuning and Tolerances," IEEE J. Quantum Electron. 28, 2631-2654 (1992).
[CrossRef]

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, "Resonant optical second harmonic generation and mixing," IEEE J. Quantum Electron. QE-2, 109-124 (1966).
[CrossRef]

IEEE Trans. on Education (1)

S. El-Rabaie, V. F. Fusco, and C. Stewart,"Harmonic balance evaluation of nonlinear microwave circuits-a tutorial approach," IEEE Trans. on Education 31, 181-192 (1988).
[CrossRef]

Int. J. Quantum Chem. (1)

A. Calderone and J. Vigneron, "Computation of the electromagnetic harmonics generation by stratified systems containing nonlinear layers," Int. J. Quantum Chem. 70, 763-770 (1988).
[CrossRef]

J. Exp. Theor. Phys. (1)

M. G. Martemyanov, T. V. Dolgova and A. A. Fedyanin,"Optical third-harmonic generation in one-dimensional photonic crystals and microcavities," J. Exp. Theor. Phys. 98, 463-477 (2003).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Other (2)

W. J. Tropf, M. E. Thomas, and T. J. Harris, "Properties of crystals and glasses," in Handbook of Optics: Volume II, (McGraw-Hill 1995).

K. E. Atkinson, An Introduction to Numerical Analysis, (John Wiley 1989), Chap. 8.

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Figures (5)

Fig. 1.
Fig. 1.

Field analysis of a single cavity. The operator A represents a phase-propagation matrix in the linear case, or coupled differential equations for a nonlinear dielectric. Analysis from left-to-right or right-to-left will produce self-consistent results for a correct set of vectors.

Fig. 2.
Fig. 2.

Calculated harmonic intensity for a singly-resonant 1.5 micrometer cavity of GaAs. The pump intensity is 10 MW/m2. Shown are results for different values of cavity finesse at the pump frequency.

Fig. 3.
Fig. 3.

Calculated harmonic intensity for a cavity of GaAs with phase-compensating mirrors. The pump intensity is 10 MW/m 2. Shown are results for different values of cavity finesse.

Fig. 4.
Fig. 4.

Calculated harmonic intensity for a cavity of GaAs with phase-compensating mirrors. Parameters are the same as in the previous case, but unlike the simple phase-compensation technique, the mirror phases at the fundamental and the harmonic frequencies are searched for the optimum values at each cavity length.

Fig. 5.
Fig. 5.

Calculated harmonic intensity for a cavity of GaAs with phase-compensating mirrors. The compensation is according to Equation 13. One curve with the same phase compensation of Fig. 3 is included for comparison. The pump intensity is 10 MW/m2. Shown are results for different values of cavity finesse.

Equations (13)

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E 1 = ( E 1 + E 1 ) , E 3 = ( E 3 + E 3 ) ,
M = 1 τ ( 1 ρ ρ 1 ) ,
A = ( exp ( ikd ) 0 0 exp ( ikd ) ) ,
( E 3 + E 3 ) = MAM 1 ( E 1 + E 1 ) ,
( E 1 + E 1 ) = ( 1 r ) , ( E 3 + E 3 ) = ( t 0 ) ,
d A 0 dz = j ω d c n 0 A 0 * A 1 exp ( j Δ k z ) α 0 A 0 ,
d A 1 dz = j 2 ω d c n 1 A 0 A 0 exp ( j Δ k z ) α 1 A 1 ,
d A p dz = j ω p d c n p A s A i exp ( j Δ k z ) α p A p ,
d A s dz = j ω s d c n s A p A s * exp ( j Δ k z ) α s A s ,
d A i dz = j ω i d c n i A p A i * exp ( j Δ k z ) α i A i ,
E 3 = MA ( M 1 E 1 ) , E 1 = M 1 A 1 ( ME 3 ) ,
R = E 3 MA ( M 1 E 1 ) + E 1 M 1 A 1 ( ME 3 ) ,
j e j k 1 d Δ k ( e j Δ k d 1 ) .

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