Abstract

We investigate the properties of layered media in the framework of frequency conversion. We detail guidelines to ensure phase matching among wavelength pairs within a broad wavelength range, and we show which conversion performance can be theoretically found.

© 2002 Optical Society of America

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References

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  1. N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
    [CrossRef]
  2. S. Somekh and A. Yariv, “Phase-matchable nonlinear optical interactions in periodic thin films,” Appl. Phys. Lett. 21, 140–141 (1972).
    [CrossRef]
  3. C. L. Tang and P. B. Bey, “Phase matching in second-harmonic generation using artificial periodic structures,” IEEE J. Quantum Electron. 9, 9–17 (1973).
    [CrossRef]
  4. J. P. van der Ziel, M. Ilegemes, P. W. Foy, and R. M. Mikulyak, “Phase-matched second harmonic generation in a periodic AlGaAs waveguide,” Appl. Phys. Lett. 29, 775–777 (1976).
    [CrossRef]
  5. M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
    [CrossRef]
  6. A. V. Balakin, V. A. Bushuev, N. I. Koroteev, B. I. Mantsyzov, I. A. Ozheredov, A. P. Shkurinov, D. Boucher, and P. Masselin, “Enhancement of second-harmonic generation with femtosecond laser pulses near the photonic band edge for different polarizations of incident light,” Opt. Lett. 24, 793–795 (1999).
    [CrossRef]
  7. M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
    [CrossRef]
  8. G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
    [CrossRef]
  9. P. Yeh, A. Yariv, and C. S. Hong, “Electromagnetic propagation in stratified media,” J. Opt. Soc. Am. 67, 423–448 (1977).
    [CrossRef]
  10. A. Yariv, Optical Electronics in Modern Communications (Oxford U., New York, 1997), Chap. 8, pp. 273–325.
  11. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chaps. 8 and 9, pp. 109–140.
  12. An initial choice that does not satisfy Eqs. (3) physically represents a backreflected wave that is sustained by a field that, at the medium’s output, constitutes the superposition of two counterpropagating waves with suitable complex amplitudes. This is a physically meaningful and acceptable solution, but it is not the solution of the problem that we are dealing with.
  13. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN 77 (Cambridge U. Press, Cambridge, 1999), Chap. 9, pp. 340–386.
  14. A. Taflove, Computational Electrodynamics. The Finite Difference Time Domain Method (Artech House, Norwood, Mass., 1995).
  15. J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with application to photonic band structures,” Phys. Rev. A 46, 612–622 (1992).
    [CrossRef] [PubMed]
  16. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
    [CrossRef]
  17. J. P. Dowling, “Dipole emission in finite photonic bandgap structures: an exactly solvable one-dimensional model,” J. Lightwave Technol. 17, 2142–2151 (1999).
    [CrossRef]
  18. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, 1999), Chap. 1; see also Ref. 19.
  19. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Chap. 5.
  20. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995), Chap. 4.

1999 (4)

1997 (1)

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

1996 (1)

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

1992 (1)

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with application to photonic band structures,” Phys. Rev. A 46, 612–622 (1992).
[CrossRef] [PubMed]

1977 (1)

1976 (1)

J. P. van der Ziel, M. Ilegemes, P. W. Foy, and R. M. Mikulyak, “Phase-matched second harmonic generation in a periodic AlGaAs waveguide,” Appl. Phys. Lett. 29, 775–777 (1976).
[CrossRef]

1973 (1)

C. L. Tang and P. B. Bey, “Phase matching in second-harmonic generation using artificial periodic structures,” IEEE J. Quantum Electron. 9, 9–17 (1973).
[CrossRef]

1972 (1)

S. Somekh and A. Yariv, “Phase-matchable nonlinear optical interactions in periodic thin films,” Appl. Phys. Lett. 21, 140–141 (1972).
[CrossRef]

1970 (1)

N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
[CrossRef]

Balakin, A. V.

Bendickson, J. M.

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Bertolotti, M.

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[CrossRef]

Bey, P. B.

C. L. Tang and P. B. Bey, “Phase matching in second-harmonic generation using artificial periodic structures,” IEEE J. Quantum Electron. 9, 9–17 (1973).
[CrossRef]

Bloembergen, N.

N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
[CrossRef]

Bloemer, M. J.

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Boucher, D.

Bowden, C. M.

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with application to photonic band structures,” Phys. Rev. A 46, 612–622 (1992).
[CrossRef] [PubMed]

Bushuev, V. A.

Centini, M.

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[CrossRef]

D’Aguanno, G.

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

Dowling, J. P.

J. P. Dowling, “Dipole emission in finite photonic bandgap structures: an exactly solvable one-dimensional model,” J. Lightwave Technol. 17, 2142–2151 (1999).
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with application to photonic band structures,” Phys. Rev. A 46, 612–622 (1992).
[CrossRef] [PubMed]

Foy, P. W.

J. P. van der Ziel, M. Ilegemes, P. W. Foy, and R. M. Mikulyak, “Phase-matched second harmonic generation in a periodic AlGaAs waveguide,” Appl. Phys. Lett. 29, 775–777 (1976).
[CrossRef]

Haus, J. W.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Hong, C. S.

Ilegemes, M.

J. P. van der Ziel, M. Ilegemes, P. W. Foy, and R. M. Mikulyak, “Phase-matched second harmonic generation in a periodic AlGaAs waveguide,” Appl. Phys. Lett. 29, 775–777 (1976).
[CrossRef]

Koroteev, N. I.

Manka, A. S.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Mantsyzov, B. I.

Masselin, P.

Mikulyak, R. M.

J. P. van der Ziel, M. Ilegemes, P. W. Foy, and R. M. Mikulyak, “Phase-matched second harmonic generation in a periodic AlGaAs waveguide,” Appl. Phys. Lett. 29, 775–777 (1976).
[CrossRef]

Nefedov, I.

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

Ozheredov, I. A.

Scalora, M.

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Shkurinov, A. P.

Sibilia, C.

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999).
[CrossRef]

Sievers, A. J.

N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
[CrossRef]

Somekh, S.

S. Somekh and A. Yariv, “Phase-matchable nonlinear optical interactions in periodic thin films,” Appl. Phys. Lett. 21, 140–141 (1972).
[CrossRef]

Tang, C. L.

C. L. Tang and P. B. Bey, “Phase matching in second-harmonic generation using artificial periodic structures,” IEEE J. Quantum Electron. 9, 9–17 (1973).
[CrossRef]

van der Ziel, J. P.

J. P. van der Ziel, M. Ilegemes, P. W. Foy, and R. M. Mikulyak, “Phase-matched second harmonic generation in a periodic AlGaAs waveguide,” Appl. Phys. Lett. 29, 775–777 (1976).
[CrossRef]

Viswanathan, R.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Yariv, A.

P. Yeh, A. Yariv, and C. S. Hong, “Electromagnetic propagation in stratified media,” J. Opt. Soc. Am. 67, 423–448 (1977).
[CrossRef]

S. Somekh and A. Yariv, “Phase-matchable nonlinear optical interactions in periodic thin films,” Appl. Phys. Lett. 21, 140–141 (1972).
[CrossRef]

Yeh, P.

Appl. Phys. Lett. (3)

N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
[CrossRef]

S. Somekh and A. Yariv, “Phase-matchable nonlinear optical interactions in periodic thin films,” Appl. Phys. Lett. 21, 140–141 (1972).
[CrossRef]

J. P. van der Ziel, M. Ilegemes, P. W. Foy, and R. M. Mikulyak, “Phase-matched second harmonic generation in a periodic AlGaAs waveguide,” Appl. Phys. Lett. 29, 775–777 (1976).
[CrossRef]

IEEE J. Quantum Electron. (1)

C. L. Tang and P. B. Bey, “Phase matching in second-harmonic generation using artificial periodic structures,” IEEE J. Quantum Electron. 9, 9–17 (1973).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

Opt. Lett. (2)

Phys. Rev. A (2)

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with application to photonic band structures,” Phys. Rev. A 46, 612–622 (1992).
[CrossRef] [PubMed]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Phys. Rev. E (2)

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

Other (8)

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, 1999), Chap. 1; see also Ref. 19.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), Chap. 5.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995), Chap. 4.

A. Yariv, Optical Electronics in Modern Communications (Oxford U., New York, 1997), Chap. 8, pp. 273–325.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chaps. 8 and 9, pp. 109–140.

An initial choice that does not satisfy Eqs. (3) physically represents a backreflected wave that is sustained by a field that, at the medium’s output, constitutes the superposition of two counterpropagating waves with suitable complex amplitudes. This is a physically meaningful and acceptable solution, but it is not the solution of the problem that we are dealing with.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN 77 (Cambridge U. Press, Cambridge, 1999), Chap. 9, pp. 340–386.

A. Taflove, Computational Electrodynamics. The Finite Difference Time Domain Method (Artech House, Norwood, Mass., 1995).

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of the incident, reflected, and transmitted waves that are part of the shooting routine.

Fig. 2
Fig. 2

Solid curve, transmission of a nine-cell stack. The elementary cell is displayed in the inset: its white zones are air layers, whereas, the shaded zones represent layers of a physical material assumed to be nondispersive, with refractive index n=3.3. Dotted curve, density of modes for the stack.

Fig. 3
Fig. 3

Field distribution in the nine-layer stack whose transmission curve is shown in Fig. 2. (a) λ=644.6 nm (high-frequency edge of the sixth bandgap). (b) λ=725.6 nm (low-frequency edge of the sixth bandgap).

Fig. 4
Fig. 4

(a) Cross section of a ridge waveguide suitable for frequency conversion. (b) Schematic diagram of the frequency converter.

Fig. 5
Fig. 5

Transmission of the 99-cell stack used for frequency conversion near λ0=1550 nm.

Fig. 6
Fig. 6

Filled circles, conversion efficiency for a 99-layer stack with the elementary cell described in text; open triangles, conversion efficiency for a 100-layer stack; open squares, conversion efficiency for a 297-layer stack.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

d2ESdz2=-ωSnS(z)c02ES-ωSc02dNL(z)EPEI*,
d2EI*dz2=-ωInI(z)c02EI*-ωIc02dNL(z)EP*ES,
d2EPdz2=-ωPnP(z)c02ES-ωPc02dNL(z)EIES,
d2 Re[E(m)(z)]dz2=-2πλ0n(z)2 Re[E(m)(z)]-[β0n(z)]2 Re[E(m)(z)],
d2 Im[E(m)(z)]dz2=-2πλ0n(z)2 Im[E(m)(z)]-[β0n(z)]2 Im[E(m)(z)],
Re[E(m)(0)]=Re(A0)+Re(B0),
d Re[E(m)(z)]dzz=0=β0[Im(A0)-Im(B0)],
Im[E(m)(0)]=Im(A0)+Im(B0),
d Im[E(m)(z)]dzz=0=β0[-Re(A0)+Re(B0)],
E(z=L+)=Aout,dE(z)dzz=L+=-iβ0Aout.
Re[E(m)(L-)]=Re[Aout],
Im[E(m)(L-)]=Im(Aout),
d Re[E(m)(z)]dzz=L-=β0 Im(Aout),
d Im[E(m)(z)]dzz=L-=-β0 Re(Aout),
β0 Re[E(m)(L-)]+d Im[E(m)(z)]dzz=L-=0,
β0 Im[E(m)(L-)]-d Re[E(m)(z)]dzz=L-=0.
Φ(ωNl+m)=(Nl+m)π.
Φ(ω0)=Nl+N2πN evenNl+N±12πN odd.
Φ(ωP)=Nmπ±π,
Φ(ω-1)=Φ(ω0)-π,Φ(ω+1)=Φ(ω0)+π,
Φ(ω-1)+Φ(ω+1)-Φ(ωP)=2Φ(ω0)-Φ(ωP)=0.

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