## Abstract

Expressions for the quasi-phase-matching (QPM) three-wave mixing (TWM) with arbitrary grating structure and phase shift are obtained in this paper, for the first time, under the small-signal approximation. The expressions can be extensively applied to the all-optical signal processing for TWM, in which the signal and pump bandwidth of the wavelength conversion in DFM and the all-optical gate (AOG) bandwidth in SFM are all optimized. The optimal results from our expressions are compared with the results from the coupled-mode equations of QPM-TWM. Compared with loss free, the propagation loss in waveguides can decrease the conversion efficiency, but only a little change for the bandwidth.

© 2001 Optical Society of America

Full Article | PDF Article**OSA Recommended Articles**

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### References

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- G.. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse, compression and solitions,” Opt. Quantum Electron. 28, 1691–1740 (1996).

[Crossref] -
X. -M. Liu, H. -Y. Zhang, and Y. -L. Guo, “Theoretical Analyses and Optimizations for Wavelength Conversion by Quasi-Phase-Matching Difference-Frequency Generation,” J. Lightwave Technol. 19, 1785–1792 (2001).

[Crossref] - M. H. Chou, I. Brener, K. R. Parameswaran, and M. M. Fejer, “Stability and bandwidth enhancement of difference frequency generation (DFM)-based wavelength conversion by pump detuning,” Electron. Lett. 35, 978–980 (1999).

[Crossref] - A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: an analytical study,” Phy. Rev. A 54, 3455–3471 (1996).

[Crossref] - X. -M. Liu and M. -D. Zhang, “Theoretical Studies for the Special States of the Cascaded Quadratic Nonlinear Effects”, J. Opt. Soc.Am. B18, (2001), (to be published in November).

[Crossref] - T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).

[Crossref] -
K. Mizuuchi and K. Yamamoto, “Waveguide second-harmonic generation device with broadened flat quasi-phase-matching response by use of a grating structure with located phase shifts”, Opt. Lett. 23, 1880–1882 (1998).

[Crossref] -
M. H. Chou, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO3 waveguides,” Opt. Lett. 24, 1157–1159 (1999).

[Crossref] -
M. H. Chou, J. Hauden, M. A. Arbore, I. Brener, and M. M. Fejer, “1.5-um-band wavelength conversion based on difference-frequency generation in LiNbO3 waveguides with integrated coupling structures,” Opt. Lett. 23, 1004–1006 (1998).

[Crossref] - K. R. Parameswaran, M. Fujimura, M. H. Chou, and M. M. Fejer, “low power all-optical gate based on sum frequency mixing in APE wave guides in PPLN,” IEEE Photon. Technol. Lett. 12, 654–657 (2000).

[Crossref]

#### 2001 (1)

X. -M. Liu, H. -Y. Zhang, and Y. -L. Guo, “Theoretical Analyses and Optimizations for Wavelength Conversion by Quasi-Phase-Matching Difference-Frequency Generation,” J. Lightwave Technol. 19, 1785–1792 (2001).

[Crossref]

#### 2000 (1)

K. R. Parameswaran, M. Fujimura, M. H. Chou, and M. M. Fejer, “low power all-optical gate based on sum frequency mixing in APE wave guides in PPLN,” IEEE Photon. Technol. Lett. 12, 654–657 (2000).

[Crossref]

#### 1999 (2)

M. H. Chou, I. Brener, K. R. Parameswaran, and M. M. Fejer, “Stability and bandwidth enhancement of difference frequency generation (DFM)-based wavelength conversion by pump detuning,” Electron. Lett. 35, 978–980 (1999).

[Crossref]

M. H. Chou, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO3 waveguides,” Opt. Lett. 24, 1157–1159 (1999).

[Crossref]

#### 1998 (2)

M. H. Chou, J. Hauden, M. A. Arbore, I. Brener, and M. M. Fejer, “1.5-um-band wavelength conversion based on difference-frequency generation in LiNbO3 waveguides with integrated coupling structures,” Opt. Lett. 23, 1004–1006 (1998).

[Crossref]

K. Mizuuchi and K. Yamamoto, “Waveguide second-harmonic generation device with broadened flat quasi-phase-matching response by use of a grating structure with located phase shifts”, Opt. Lett. 23, 1880–1882 (1998).

[Crossref]

#### 1996 (2)

A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: an analytical study,” Phy. Rev. A 54, 3455–3471 (1996).

[Crossref]

G.. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse, compression and solitions,” Opt. Quantum Electron. 28, 1691–1740 (1996).

[Crossref]

#### 1990 (1)

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).

[Crossref]

#### Arbore, M. A.

M. H. Chou, J. Hauden, M. A. Arbore, I. Brener, and M. M. Fejer, “1.5-um-band wavelength conversion based on difference-frequency generation in LiNbO3 waveguides with integrated coupling structures,” Opt. Lett. 23, 1004–1006 (1998).

[Crossref]

#### Brener, I.

M. H. Chou, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO3 waveguides,” Opt. Lett. 24, 1157–1159 (1999).

[Crossref]

M. H. Chou, I. Brener, K. R. Parameswaran, and M. M. Fejer, “Stability and bandwidth enhancement of difference frequency generation (DFM)-based wavelength conversion by pump detuning,” Electron. Lett. 35, 978–980 (1999).

[Crossref]

M. H. Chou, J. Hauden, M. A. Arbore, I. Brener, and M. M. Fejer, “1.5-um-band wavelength conversion based on difference-frequency generation in LiNbO3 waveguides with integrated coupling structures,” Opt. Lett. 23, 1004–1006 (1998).

[Crossref]

#### Chou, M. H.

K. R. Parameswaran, M. Fujimura, M. H. Chou, and M. M. Fejer, “low power all-optical gate based on sum frequency mixing in APE wave guides in PPLN,” IEEE Photon. Technol. Lett. 12, 654–657 (2000).

[Crossref]

M. H. Chou, I. Brener, K. R. Parameswaran, and M. M. Fejer, “Stability and bandwidth enhancement of difference frequency generation (DFM)-based wavelength conversion by pump detuning,” Electron. Lett. 35, 978–980 (1999).

[Crossref]

M. H. Chou, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO3 waveguides,” Opt. Lett. 24, 1157–1159 (1999).

[Crossref]

[Crossref]

#### Fejer, M. M.

K. R. Parameswaran, M. Fujimura, M. H. Chou, and M. M. Fejer, “low power all-optical gate based on sum frequency mixing in APE wave guides in PPLN,” IEEE Photon. Technol. Lett. 12, 654–657 (2000).

[Crossref]

[Crossref]

[Crossref]

[Crossref]

#### Fujimura, M.

[Crossref]

#### Guo, Y. -L.

X. -M. Liu, H. -Y. Zhang, and Y. -L. Guo, “Theoretical Analyses and Optimizations for Wavelength Conversion by Quasi-Phase-Matching Difference-Frequency Generation,” J. Lightwave Technol. 19, 1785–1792 (2001).

[Crossref]

#### Hagan, D. J.

G.. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse, compression and solitions,” Opt. Quantum Electron. 28, 1691–1740 (1996).

[Crossref]

#### Hauden, J.

[Crossref]

#### Kobyakov, A.

A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: an analytical study,” Phy. Rev. A 54, 3455–3471 (1996).

[Crossref]

#### Lederer, F.

A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: an analytical study,” Phy. Rev. A 54, 3455–3471 (1996).

[Crossref]

#### Liu, X. -M.

X. -M. Liu, H. -Y. Zhang, and Y. -L. Guo, “Theoretical Analyses and Optimizations for Wavelength Conversion by Quasi-Phase-Matching Difference-Frequency Generation,” J. Lightwave Technol. 19, 1785–1792 (2001).

[Crossref]

X. -M. Liu and M. -D. Zhang, “Theoretical Studies for the Special States of the Cascaded Quadratic Nonlinear Effects”, J. Opt. Soc.Am. B18, (2001), (to be published in November).

[Crossref]

#### Mizuuchi, K.

K. Mizuuchi and K. Yamamoto, “Waveguide second-harmonic generation device with broadened flat quasi-phase-matching response by use of a grating structure with located phase shifts”, Opt. Lett. 23, 1880–1882 (1998).

[Crossref]

#### Nishihara, H.

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).

[Crossref]

#### Parameswaran, K. R.

[Crossref]

[Crossref]

[Crossref]

#### Stegeman, G.. I.

G.. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse, compression and solitions,” Opt. Quantum Electron. 28, 1691–1740 (1996).

[Crossref]

#### Suhara, T.

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).

[Crossref]

#### Torner, L.

[Crossref]

#### Yamamoto, K.

K. Mizuuchi and K. Yamamoto, “Waveguide second-harmonic generation device with broadened flat quasi-phase-matching response by use of a grating structure with located phase shifts”, Opt. Lett. 23, 1880–1882 (1998).

[Crossref]

#### Zhang, H. -Y.

[Crossref]

#### Zhang, M. -D.

X. -M. Liu and M. -D. Zhang, “Theoretical Studies for the Special States of the Cascaded Quadratic Nonlinear Effects”, J. Opt. Soc.Am. B18, (2001), (to be published in November).

[Crossref]

#### Electron. Lett. (1)

[Crossref]

#### IEEE J. Quantum Electron. (1)

[Crossref]

#### IEEE Photon. Technol. Lett. (1)

[Crossref]

#### J. Lightwave Technol. (1)

[Crossref]

#### Opt. Lett. (3)

[Crossref]

[Crossref]

[Crossref]

#### Opt. Quantum Electron. (1)

[Crossref]

#### Phy. Rev. A (1)

[Crossref]

#### Other (1)

X. -M. Liu and M. -D. Zhang, “Theoretical Studies for the Special States of the Cascaded Quadratic Nonlinear Effects”, J. Opt. Soc.Am. B18, (2001), (to be published in November).

[Crossref]

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### Figures (3)

**Fig.1.**

Model of the QPM grating structure with the phase shift. Directions of arrows are those of the nonlinear coefficient.

**Fig.2.**

ptimal results for the 3-segment without the phase shift, (a) for the signal bandwidth, (b) for the pump bandwidth in DFM, and (c) for the pump bandwidth for the AOG in SFM. The subscripts of in and out express the input and output waves, and Δ_{1, 2, 3} express the bandwidth, which contents the fluctuation of <2 dB, of the signal, converted and pump waves, respectively. All of parameters see Table 1–2 and their usages.

**Fig.3**

Curves of the conversion efficiency _{1} for QPM-DFM without phase-shifted segments, where the dashed-dot and solid lines correspond to take into account the loss or not. The lines with the same color have the same segment number. The simulation results are from CME, which are very consistent with results from Eq.(2a) (see Table 1) in the ideal condition

### Tables (2)

**Table 1.** Optimized Bandwidth Δ^{a}

**Table 2.** Optimized Bandwidth Δ^{a}

### Equations (6)

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