Abstract

In this paper, optical parametric amplification based on the degenerate four-wave mixing principle in a one-dimensional photonic bandgap (PBG) structure has been numerically studied. First, the multiple scale method was introduced to derive a complete set of nonlinear coupled-mode equations for a finite structure with different inhomogeneous nonlinear coefficients than those used in previous works. This finite structure is composed of 680 dielectric layers, which are alternating half-wave/eight-wave films. The wavelengths of the pump, signal, and idler pulses have been determined from the transmission spectrum, which was illustrated by using the transfer matrix method. The parametric interaction of the pump, signal, and idler pulses inside PBG structure has been numerically simulated by using the split-step Fourier transform method. The results of the simulation have shown that the intensities of the signal and idler have exponential growth with respect to the number of layers in the medium. Meanwhile, pump wavevector detuning directly affects the intensities of both pulses due to a band-edge phase-matching condition that might be achieved from only one optimal detuning parameter. Moreover, both the amplification gain and the conversion efficiency of the idler pulse have been shown to be dependent on the bandwidth of the pump pulse spectrum. A very narrow pulse, with a bandwidth much less than the relevant transmission peak, enables the highest amplification and conversion efficiency in this medium because the most efficient phase-matched condition occurs in this situation. Finally, the conversion efficiency grows exponentially with input pump intensity for several input signal intensities. Furthermore, the maximum conversion efficiencies directly vary with input signal intensity.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  7. B. Y. Soon, J. W. Haus, M. Scalora, and C. Sibilia, “One-dimensional photonic crystal optical limiter,” Opt. Express 11, 2007–2018 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2012 (1)

2008 (1)

2004 (1)

J. W. Haus, P. Powers, P. Bojja, M. Torres-Cisneros, M. Scalora, M. J. Bloemer, N. Akozbek, and M. A. Meneses-Nava, “Enhanced tunable terahertz generation in photonic band-gap structures,” Laser Phys. 14, 1–8 (2004).

2003 (1)

2002 (1)

2001 (3)

A. V. Tarasishin, S. A. Magnitskii, and A. M. Zheltikov, “Matching phase and group velocities in second-harmonic generation in finite one-dimensional photonic band-gap structures,” Laser Phys. 11, 31–38 (2001).

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, “Enhancement of sum frequency generation near the photonic band gap edge under quasi phase matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Efficient nonlinear infrared parametric generation on one-dimensional photonic band gap structures,” Opt. Commun. 189, 135–142 (2001).
[CrossRef]

1999 (1)

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

1996 (1)

1995 (1)

1994 (4)

M. Scalora and M. E. Crenshaw, “A beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

1976 (3)

J. P. van der Ziel and M. Ilegems, “Optical second harmonic generation in periodic multilayer GaAs-Al0.3Ga0.7As structures,” Appl. Phys. Lett. 28, 437–439 (1976).
[CrossRef]

J. P. van der Ziel and M. Ilegems, “Second harmonic generation in a thin AlAs-GaAs multilayer structure with wave propagation in the plane of the layers,” Appl. Phys. Lett. 29, 200–202 (1976).
[CrossRef]

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

1970 (1)

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

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).

Akozbek, N.

J. W. Haus, P. Powers, P. Bojja, M. Torres-Cisneros, M. Scalora, M. J. Bloemer, N. Akozbek, and M. A. Meneses-Nava, “Enhanced tunable terahertz generation in photonic band-gap structures,” Laser Phys. 14, 1–8 (2004).

Balakin, A. V.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, “Enhancement of sum frequency generation near the photonic band gap edge under quasi phase matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

Bertolotti, M.

M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Efficient nonlinear infrared parametric generation on one-dimensional photonic band gap structures,” Opt. Commun. 189, 135–142 (2001).
[CrossRef]

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

J. W. Haus, P. Powers, P. Bojja, M. Torres-Cisneros, M. Scalora, M. J. Bloemer, N. Akozbek, and M. A. Meneses-Nava, “Enhanced tunable terahertz generation in photonic band-gap structures,” Laser Phys. 14, 1–8 (2004).

M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Efficient nonlinear infrared parametric generation on one-dimensional photonic band gap structures,” Opt. Commun. 189, 135–142 (2001).
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and L. 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, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Bojja, P.

J. W. Haus, P. Powers, P. Bojja, M. Torres-Cisneros, M. Scalora, M. J. Bloemer, N. Akozbek, and M. A. Meneses-Nava, “Enhanced tunable terahertz generation in photonic band-gap structures,” Laser Phys. 14, 1–8 (2004).

Bowden, C. M.

M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Efficient nonlinear infrared parametric generation on one-dimensional photonic band gap structures,” Opt. Commun. 189, 135–142 (2001).
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and L. 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, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Bushuev, V. A.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, “Enhancement of sum frequency generation near the photonic band gap edge under quasi phase matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

Centini, M.

M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Efficient nonlinear infrared parametric generation on one-dimensional photonic band gap structures,” Opt. Commun. 189, 135–142 (2001).
[CrossRef]

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

Crenshaw, M. E.

M. Scalora and M. E. Crenshaw, “A beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
[CrossRef]

D’Aguanno, G.

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

de Sterke, C. M.

Dowling, J. P.

M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Foy, P. W.

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

Haus, J. W.

J. W. Haus, P. Powers, P. Bojja, M. Torres-Cisneros, M. Scalora, M. J. Bloemer, N. Akozbek, and M. A. Meneses-Nava, “Enhanced tunable terahertz generation in photonic band-gap structures,” Laser Phys. 14, 1–8 (2004).

B. Y. Soon, J. W. Haus, M. Scalora, and C. Sibilia, “One-dimensional photonic crystal optical limiter,” Opt. Express 11, 2007–2018 (2003).
[CrossRef]

J. W. Haus, B. Y. Soon, M. Scalora, C. Sibilia, and I. V. Mel’nikov, “Coupled-mode equations for Kerr media with periodically modulated linear and nonlinear coefficients,” J. Opt. Soc. Am. B 19, 2282–2291 (2002).
[CrossRef]

M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Efficient nonlinear infrared parametric generation on one-dimensional photonic band gap structures,” Opt. Commun. 189, 135–142 (2001).
[CrossRef]

J. W. Haus, “Photonic band gap structure,” in The Handbook of Nanotechnology: Nanometer Structures—Theory, Modeling, and Simulation, A. Lakhtakia, ed. (Prentice-Hall, 2004), pp. 45–108.

Horowitz, M.

Huang, N.

Ilegems, M.

J. P. van der Ziel and M. Ilegems, “Optical second harmonic generation in periodic multilayer GaAs-Al0.3Ga0.7As structures,” Appl. Phys. Lett. 28, 437–439 (1976).
[CrossRef]

J. P. van der Ziel and M. Ilegems, “Second harmonic generation in a thin AlAs-GaAs multilayer structure with wave propagation in the plane of the layers,” Appl. Phys. Lett. 29, 200–202 (1976).
[CrossRef]

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

Liu, H.

Magnitskii, S. A.

A. V. Tarasishin, S. A. Magnitskii, and A. M. Zheltikov, “Matching phase and group velocities in second-harmonic generation in finite one-dimensional photonic band-gap structures,” Laser Phys. 11, 31–38 (2001).

Mantsyzov, B. I.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, “Enhancement of sum frequency generation near the photonic band gap edge under quasi phase matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

Marhic, M. E.

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators, and Related Devices (Cambridge University, 2008).

Mel’nikov, I. V.

Meneses-Nava, M. A.

J. W. Haus, P. Powers, P. Bojja, M. Torres-Cisneros, M. Scalora, M. J. Bloemer, N. Akozbek, and M. A. Meneses-Nava, “Enhanced tunable terahertz generation in photonic band-gap structures,” Laser Phys. 14, 1–8 (2004).

Mikulyak, R. M.

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

Nefedov, L.

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and L. 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.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, “Enhancement of sum frequency generation near the photonic band gap edge under quasi phase matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

Petrov, E. V.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, “Enhancement of sum frequency generation near the photonic band gap edge under quasi phase matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

Powers, P.

J. W. Haus, P. Powers, P. Bojja, M. Torres-Cisneros, M. Scalora, M. J. Bloemer, N. Akozbek, and M. A. Meneses-Nava, “Enhanced tunable terahertz generation in photonic band-gap structures,” Laser Phys. 14, 1–8 (2004).

Scalora, M.

J. W. Haus, P. Powers, P. Bojja, M. Torres-Cisneros, M. Scalora, M. J. Bloemer, N. Akozbek, and M. A. Meneses-Nava, “Enhanced tunable terahertz generation in photonic band-gap structures,” Laser Phys. 14, 1–8 (2004).

B. Y. Soon, J. W. Haus, M. Scalora, and C. Sibilia, “One-dimensional photonic crystal optical limiter,” Opt. Express 11, 2007–2018 (2003).
[CrossRef]

J. W. Haus, B. Y. Soon, M. Scalora, C. Sibilia, and I. V. Mel’nikov, “Coupled-mode equations for Kerr media with periodically modulated linear and nonlinear coefficients,” J. Opt. Soc. Am. B 19, 2282–2291 (2002).
[CrossRef]

M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Efficient nonlinear infrared parametric generation on one-dimensional photonic band gap structures,” Opt. Commun. 189, 135–142 (2001).
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and L. 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 and M. E. Crenshaw, “A beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Shkurinov, A. P.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, “Enhancement of sum frequency generation near the photonic band gap edge under quasi phase matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

Sibilia, C.

B. Y. Soon, J. W. Haus, M. Scalora, and C. Sibilia, “One-dimensional photonic crystal optical limiter,” Opt. Express 11, 2007–2018 (2003).
[CrossRef]

J. W. Haus, B. Y. Soon, M. Scalora, C. Sibilia, and I. V. Mel’nikov, “Coupled-mode equations for Kerr media with periodically modulated linear and nonlinear coefficients,” J. Opt. Soc. Am. B 19, 2282–2291 (2002).
[CrossRef]

M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Efficient nonlinear infrared parametric generation on one-dimensional photonic band gap structures,” Opt. Commun. 189, 135–142 (2001).
[CrossRef]

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

Soon, B. Y.

Steel, M. J.

Sun, Q.

Tarasishin, A. V.

A. V. Tarasishin, S. A. Magnitskii, and A. M. Zheltikov, “Matching phase and group velocities in second-harmonic generation in finite one-dimensional photonic band-gap structures,” Laser Phys. 11, 31–38 (2001).

Toroker, Z.

Torres-Cisneros, M.

J. W. Haus, P. Powers, P. Bojja, M. Torres-Cisneros, M. Scalora, M. J. Bloemer, N. Akozbek, and M. A. Meneses-Nava, “Enhanced tunable terahertz generation in photonic band-gap structures,” Laser Phys. 14, 1–8 (2004).

van der Ziel, J. P.

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

J. P. van der Ziel and M. Ilegems, “Optical second harmonic generation in periodic multilayer GaAs-Al0.3Ga0.7As structures,” Appl. Phys. Lett. 28, 437–439 (1976).
[CrossRef]

J. P. van der Ziel and M. Ilegems, “Second harmonic generation in a thin AlAs-GaAs multilayer structure with wave propagation in the plane of the layers,” Appl. Phys. Lett. 29, 200–202 (1976).
[CrossRef]

Wang, Z.

Weiner, M.

M. Weiner, Ultrafast Optics (Wiley, 2009).

Wen, J.

Zheltikov, A. M.

A. V. Tarasishin, S. A. Magnitskii, and A. M. Zheltikov, “Matching phase and group velocities in second-harmonic generation in finite one-dimensional photonic band-gap structures,” Laser Phys. 11, 31–38 (2001).

Appl. Phys. Lett. (4)

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

J. P. van der Ziel and M. Ilegems, “Optical second harmonic generation in periodic multilayer GaAs-Al0.3Ga0.7As structures,” Appl. Phys. Lett. 28, 437–439 (1976).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Schematic of degenerate four-wave mixing with pulsed pump and signal inputs.

Fig. 2.
Fig. 2.

(a) Transmission spectrum of a PBG stack composed of 680 dielectric layers. (b) Electric field distributions (of λ1=1,550nm) inside the PBG stack when band-edge resonance is satisfied.

Fig. 3.
Fig. 3.

Snapshot of the OPA process in a nonlinear PBG stack (black bar) where the pump, signal, and idler pulses are represented by the blue (top plot in each set), red (middle), and green (bottom) lines, respectively. In this simulation, we determined that the normalized input pump and signal intensities are 4.500 and 0.011GW/cm2, while stack length L is about 154 μm. Figures 3(a)–3(c) are immediately captured when total time are 0, 200, and 400, respectively.

Fig. 4.
Fig. 4.

Amplified signal pulse (blue line) and the generated idler pulse intensities (red line) with respect to structure length.

Fig. 5.
Fig. 5.

Output intensities of the (a) signal and (b) idler pulses, which both depend on pump wave-vector detuning and total structure length (L).

Fig. 6.
Fig. 6.

(a) Amplification gain of the signal pulse when the pulse widths are 0.08 (red line), 0.16 (blue line), and 0.32 (black line) μm, with respect to the number of layer pairs. (b) Conversion efficiency of idler pulse generation when the pulse widths are 0.08 (red line), 0.16 (blue line), and 0.32 (black line) μm with respect to the number of layer pairs.

Fig. 7.
Fig. 7.

Conversion efficiency of idler pulse generation for a PBG stack with respect to input pump intensity when input signal intensity is 11.25 (red), 45.00 (blue), and 180.00 (black) MW/cm2 with medium length of 307 μm.

Equations (38)

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2E(z,t)z2ε^(z,ω)c22E(z,t)t2=4πc22PNL(z,t)t2.
E=j=13Aj±exp[±i(kjzωjt)]+c.c.,
PNL(z,t)=μχ(3)(z)E3(z,t).
2z2=2z02+2μz0z1+2μ2z0z2+,
2t2=2t02+2μt0t1+2μ2t0t2+.
E=E0+μE1+μ2E2+,
f(z)=l=Flexp(2πilz/d),
dπΔk+2δ(1)=δ(2)+δ(3),
z=πdz1andAj±=N0(j)aj±eiπdδ(j)z1
a1+z+dπvg(1)a1+tiδ(1)a1+iκ1(1)a1=iN0(1)[(|a1+|2+2|a1|2+2|a2+|2+2|a2+|2+2|a3+|2+2|a3|2)a1++2a2+a3+a1+*]+iN1(1)[(2|a1+|2+|a1|2+2|a2+|2+2|a2|2+2|a3+|2+2|a3|2)a1+(2a2a2+*+2a3a3+*)a1+]+iN1(1)[a1+a1*+2a2+a2*+2a3+a3*]a1+iN2(1)[a12a1+*],
a1z+dπvg(1)a1tiδ(1)a1iκ1(1)a1+=iN0(1)[(2|a1+|2+|a1|2+2|a2+|2+2|a2|2+2|a3+|2+2|a3|2)a1+2a2a3a1*]+iN1(1)[(|a1+|2+2|a1|2+2|a2+|2+2|a2|2+2|a3+|2+2|a3|2)a1++(2a2+a2*+2a3+a3*)a1]+iN1(1)[a1a1+*+2a2a2+*+2a3a3+*]a1+iN2(1)[a1+2a1*],
a2+z+dπvg(2)a2+tiδ(2)a2+iκ1(2)a2=iN0(2)[(2|a1+|2+2|a1|2+|a2+|2+2|a2|2+2|a3+|2+2|a3|2)a2++2a1+2a3+*]+iN1(2)[(2|a1+|2+2|a1|2+2|a2+|2+|a2|2+2|a3+|2+2|a3|2)a2+(2a1a1+*+2a3a3+*)a2+]+iN1(2)[2a1+a1*+a2+a2*+2a3+a3*]a2++iN2(2)[a22a2+*],
a2z+dπvg(2)a2tiδ(2)a2iκ1(2)a2+=iN0(2)[(2|a1+|2+2|a1|2+2|a2+|2+|a2|2+2|a3+|2+2|a3|2)a2+2a12a3*]+iN1(2)[(2|a1+|2+2|a1|2+|a2+|2+2|a2|2+2|a3+|2+2|a3|2)a2++(2a1+a1*+2a3+a3*)a2]+iN1(2)[2a1a1+*+a2a2+*+2a3a3+*]a2+iN2(2)[a2+2a2*],
a3+z+dπvg(3)a3+tiδ(3)a3+iκ1(3)a3=iN0(3)[(2|a1+|2+2|a1|2+2|a2+|2+2|a2|2+|a3+|2+2|a3|2)a3++2a1+2a2+*]+iN1(3)[(2|a1+|2+2|a1|2+2|a2+|2+2|a2|2+2|a3+|2+|a3|2)a3+(2a1a1+*+2a2a2+*)a3+]+iN1(3)[2a1+a1*+2a2+a2*+a3+a3*]a3++iN2(3)[a32a3+*],
a3z+dπvg(3)a3tiδ(3)a3iκ1(3)a3+=iN0(3)[(2|a1+|2+2|a1|2+2|a2+|2+2|a2|2+2|a3+|2+|a3|2)a3+2a12a2*]+iN1(3)[(2|a1+|2+2|a1|2+2|a2+|2+2|a2|2+|a3+|2+2|a3|2)a3++(2a1+a1*+2a2+a2*)a3]+iN1(3)[2a1a1+*+2a2a2+*+a3a3+*]a3+iN2(3)[a3+2a3*],
Efpump(0,t)=a1+(0,t),Efsignal(0,t)=a2+(0,t),Eb(L,t)=0.
1vgAt=(D+K+N)A,
A(t+Δt)=exp(vgΔt2D)exp(vgΔt2K)×exp(vgΔtN)exp(vgΔt2K)exp(vgΔt2D)A(t).
exp(vgΔtN)exp(vgΔtNd)+vgΔtNo.
a1+(z,0)=a10exp[(zLc)2/σ12],a2+(z,0)=a20exp[(zLc)2/σ22].
Gs=P2+(L)P2+(0)=|a2+(L)|2|a2+(0)|2.
η=P3+(L)P1+(0)=|a3+(L)|2|a1+(0)|2.
A=[a1+a1a2+a2a3+a3]T.
D=[z+iδ(1)000000z+iδ(1)000000z+iδ(2)000000z+iδ(2)000000z+iδ(3)000000z+iδ(3)],
K=[0iκ+1(1)0000iκ1(1)00000000iκ+1(2)0000iκ1(2)00000000iκ+1(3)0000iκ1(3)0].
N=[N11N120000N21N22000000N33N340000N43N44000000N55N560000N65N66],
N11=iΓ0(1)[|a1+|2+2|a1|2+2|a2+|2+2|a2|2+2|a3+|2+2|a3|2+2a2a3+a1+*a1+]+iΓ1(1)[2a2a2+*+2a3a3+*]+iΓ1(1)[a1+a1*+2a2+a2*+2a3+a3*],
N22=iΓ0(1)[2|a1+|2+|a1|2+2|a2+|2+2|a2|2+2|a3+|2+2|a3|2+2a2a3a1*a1]+iΓ1(1)[2a2+a2*+2a3+a3*]+iΓ1(1)[a1a1+*+2a2a2+*+2a3a3+*],
N33=iΓ0(2)[2|a1+|2+2|a1|2+|a2+|2+2|a2|2+2|a3+|2+2|a3|2+2a1+2a3+*a3+]+iΓ1(2)[2a1a1+*+2a3a3+*]+iΓ1(2)[2a1+a1*+a2+a2*+2a3+a3*],
N44=iΓ0(2)[2|a1+|2+2|a1|2+2|a2+|2+|a2|2+2|a3+|2+2|a3|2+2a12a3*a3]+iΓ1(2)[2a1+a1*+2a3+a3*]+iΓ1(2)[2a1a1+*+a2a2+*+2a3a3+*],
N55=iΓ0(3)[2|a1+|2+2|a1|2+2|a2+|2+2|a2|2+|a3+|2+2|a3|2+2a1+2a2+*a2+]+iΓ1(3)[2a1a1+*+2a2a2+*]+iΓ1(3)[2a1+a1*+2a2+a2*+a3+a3*],
N66=iΓ0(3)[2|a1+|2+2|a1|2+2|a2+|2+2|a2|2+2|a3+|2+|a3|2+2a12a2*a2]+iΓ1(3)[2a1+a1*+2a2+a2*]+iΓ1(3)[2a1a1+*+2a2a2+*+a3a3+*].
N12=iΓ1(1)[2|a1+|2+|a1|2+2|a2+|2+2|a2|2+2|a3+|2+2|a3|2]+iΓ2(1)[a1a1+*],
N21=iΓ1(1)[|a1+|2+2|a1|2+2|a2+|2+2|a2|2+2|a3+|2+2|a3|2]+iΓ2(1)[a1+a1*],
N34=iΓ1(2)[2|a1+|2+2|a1|2+2|a2+|2+|a2|2+2|a3+|2+2|a3|2]+iΓ2(2)[a2a2+*],
N43=iΓ1(2)[2|a1+|2+2|a1|2+|a2+|2+2|a2|2+2|a3+|2+2|a3|2]+iΓ2(2)[a2+a2*],
N56=iΓ1(3)[2|a1+|2+2|a1|2+2|a2+|2+2|a2|2+2|a3+|2+|a3|2]+iΓ2(3)[a3a3+*],
N65=iΓ1(3)[2|a1+|2+2|a1|2+2|a2+|2+2|a2|2+|a3+|2+2|a3|2]+iΓ2(3)[a3+a3*].

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