## Optimal design for the quasi-phase-matching three-wave mixing

Optics Express, Vol. 9, Issue 12, pp. 631-636 (2001)

http://dx.doi.org/10.1364/OE.9.000631

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### 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.

© Optical Society of America

## 1. Introduction

1. 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]

1. 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]

2. 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]

*d*

_{eff}that are accessible through QPM and the possibility that they can be engineered to suit a noncritical interaction configuration [3

3. 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]

*L*>25 mm, respectively [2

2. 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]

3. 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]

2. 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]

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

5. 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]

6. 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]

**19**, 1785–1792 (2001). [CrossRef]

7. 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]

8. 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 LiNbO_{3} waveguides,” Opt. Lett. **24**, 1157–1159 (1999). [CrossRef]

3. 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]

## 2. Optimal design

_{3}), the loss, group velocity mismatch (GVM) and higher-order dispersion of the material can normally be ignored at the length of ~30mm [3

**35**, 978–980 (1999). [CrossRef]

8. 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 LiNbO_{3} waveguides,” Opt. Lett. **24**, 1157–1159 (1999). [CrossRef]

9. 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 LiNbO_{3} waveguides with integrated coupling structures,” Opt. Lett. **23**, 1004–1006 (1998). [CrossRef]

^{*}.Under approximations of the slowly-varying-envelope, plane-wave and the first-order diffraction effect of the grating perturbation, the Maxwell equations for electric fields at three frequencies can be reduced to the CME with the Fourier components [2

**19**, 1785–1792 (2001). [CrossRef]

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

*m*segments. Each segment

*L*

_{j}may have the different grating period Λ

_{j}as the others, and a phase shift

*δ*

_{j}between neighbor segments is assumed (when no phase shift,

*δ*

_{j}=0). If the conversion efficiency is low or one fundamental wave

*E*

_{3}(named as the electric field of pump wave) is far more intensive than other one or two wave(s), the small-signal approximation is valid. Then the expressions of signal wave

*E*

_{1}and idler wave

*E*

_{2}for every segment can be solved for QPM-DFM as

*N*

_{l,j}(j=1,2,3,4) is the element of the operator

*N*

_{l},

_{,1}={cosh(

*Q*

_{l}

*L*

_{l})+[

*i*Δ

*k*/2

*Q*

_{l}]sinh(

*Q*

_{l}

*L*

_{l})}

*e*

_{1},

_{,2}=-

*i*(

*M*

_{1}/

*Q*

_{l})sinh(

*Q*

_{l}

*L*

_{l})

*e*

_{2},

_{,3}=-

*i*(

*M*

_{2}/

*Q*

_{l})sinh(

*Q*

_{l}

*L*

_{l})

*e*

_{2},

_{,1}={cos(

*Q*

_{l}

*L*

_{l})+[

*i*Δ

*k*/(2

*Q*

_{l})]sinh(

*Q*

_{l}

*L*

_{l})}

*e*

_{1},

_{,2}=-

*i*(

*Q*

_{l})sin(

*Q*

_{l}

*L*

_{l})

*e*

_{2},

_{,3}=-

*i*(

*M*

_{2}/

*Q*

_{l})sin(

*Q*

_{l}

*L*

_{l})

*Q*

_{l}=[

*M*

_{1}

*k*

_{l}/2)

^{2}]

^{1/2}(

*Q*

_{l}takes a plus for SFM and a minus for DFM),

*M*

_{j}=

*ω*

_{j}

*d*

_{eff}

*E*

_{3}(0)/(

*n*

_{j}

*c*) and

*ϕ*

_{i,j}=exp)(-

*in*

_{j}

*δ*

_{i,j}

*π*/

*λ*

_{j}) (

*j*=1,2),

*L*

_{l}=

*z*

_{l}-

*z*

_{l}

_{-1}where

*z*

_{l}

_{-1}and

*z*

_{l}are the input and output position of this segment respectively (see Fig.1),

*e*

_{1}=exp(-

*i*Δ

*k*

_{l}

*L*

_{l}/2) and

*e*

_{2}=exp[-

*i*Δ

*k*

_{l}(

*z*

_{l}

_{-1}+

*δ*

_{l}

_{-1}+

*L*

_{l}/2)] for the phase mismatching Δ

*k*

_{l}of the

*l*-th segment. Symbol of * expresses the conjugate operator.

*c, n, ω*and

*λ*are the speed of light in the vacuum, the index of refraction, the light frequency and wavelength, respectively. Under the small-signal approximation, therefore, solutions of CME of QPM-TWM for arbitrary chirp grating structure with arbitrary phase-shifted segment can be obtained from Eq. (1), namely

**19**, 1785–1792 (2001). [CrossRef]

10. 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]

*λ*of the signal and pump in DFM and of the pump for AOG in SFM with and without the phase-shifted segment

*δ*, respectively, where the length of each segment is assumed to be equal. If each segment is not equal, the optimization can also be obtained from Eq.(2). When

*L*

_{1}/

*L*

_{2}=1.71 and Λ

_{1,2}=16.215, 16.239 µm, for example, Δ

*λ*=109 nm for the wavelength conversion of the signal. This result is greater than the value of the equal 2-segment. Because of the limit of paper length, we cannot give the details of optimal results for unequal segments. In the simulation calculations, we employ the presently representative

**19**, 1785–1792 (2001). [CrossRef]

*P*

_{3}=100 mW for DFM and

*P*

_{3}=500 mWfor SFM, each input signal power

*P*

_{1}=1 mW, the effective channel waveguide cross section is 30µm

^{2},

*d*

_{eff}=15 pm/V,

*L*=30 mm, the pump wavelength

*λ*

_{3}=775nm for the calculation of the signal bandwidth and the signal wavelength

*λ*

_{1}=1550 nm for the calculation of the pump bandwidth in DFM, and

*λ*

_{3}=1550 nm for the AOG in SFM. In the simulation of Table 1, the conversion efficiency is assumed to be >-5 dB, and the fluctuation of the conversion bandwidth is <2dB (see Fig.2). In Table 2, the signal transmission is assumed to be <-10dB when P3=500 mW. From Table 1 and 2, it is easily seen that: ① the optimal bandwidth Δ

*λ*is broadened with the increase of the segment number

*m*, but the increment value decreases when

*m*>4, e.g., the signal Δ

*λ*=138 nm in the 5-segment against Δ

*λ*=132 nm in the 4-segment for DFM; ②Δ

*λ*is improved with the phase shift

*δ*against without

*δ*, e.g., the pump Δ

*λ*=0.51 nm with

*δ*against Δ

*λ*=0.41 nm without

*δ*for the 3-segment structure; ③ the stabilities of the pump in DFM and SFM are enhanced when having

*δ*and increasing the segment number

*m*; ④ Δ

*λ*for the aperiodic structure with

*δ*can increase over 2–5 times against for the uniform structure.

*VPItransmissionMake*

^{TM}by direct calculating the differential equation groups of CME of QPM-TWM, where all of parameters come from Table 1–2 and their usages. By comparison Table 1–2 coming from Eq.(2) with Fig.2a(a)-(c) calculating from CME, it is easily found that Eq.(2), expressed by the matrix, can not only successfully optimized arbitrary QPM grating structure with arbitrary phase shift but also greatly simplify the calculation. From Table 1–2 and Fig.2, we can see that the results are extremely consistent and the QPM bandwidth is multiply enhanced from the uniform to the aperiodic structure.

## 3. Discussions

*α*is ~0.35 dB/cm at 1550 nm and ~0.7dB/cm at 775 nm [10

10. 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]

*η*vs. the signal wavelength

*λ*

_{1}, where the dashed-dot lines include the propagation loss with the typical value in CME, but loss free for the solid lines. The numerical simulation comes from calculating the differential equation groups of CME with and without the propagation loss, and parameters all take from Table 1 and their usages. It can be seen, from Fig.3, that ①

*η*decreases ~1.2 dB but only ~1–2 nm change for the bandwidth Δ

*λ*when considering the loss, ② the results from CME is very consistent with those from Eq.(2a) (see Table 1) in the ideal condition. Eq.(2a) and (2b) are also very useful for optical parametric amplification (OPA) and optical parametric oscillator (OPO).

## 4. Conclusion

*η*decreases by comparing loss free with loss, the conversion bandwidth Δ

*λ*changes a little when taking into account the propagation loss.

## Acknowledgement:

## Footnotes

* | In Ref.[3
9. 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 LiNbO |

## References and links

1. | G.. I. Stegeman, D. J. Hagan, and L. Torner, “ |

2. | 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. |

3. | 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. |

4. | A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: an analytical study,” Phy. Rev. A |

5. | 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] |

6. | T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. |

7. | 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. |

8. | 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 LiNbO |

9. | 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 LiNbO |

10. | 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. |

**OCIS Codes**

(060.2630) Fiber optics and optical communications : Frequency modulation

(190.2620) Nonlinear optics : Harmonic generation and mixing

(230.1150) Optical devices : All-optical devices

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 14, 2001

Published: December 3, 2001

**Citation**

Xueming Liu, Hanyi Zhang, and Yanhe Li, "Optimal design for the quasi-phase- matching three-wave mixing," Opt. Express **9**, 631-636 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-12-631

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### References

- G. I. Stegeman, D. J. Hagan, L. Torner, "X(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, 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, 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. B 18, (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, 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, 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]

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