## 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

## 1. Introduction

In the coming all-optical communication systems, the optical frequency converter, the ultrafast all-optical signal processing and switching will become the primary elements of optical devices [1]. They can be realized via the quadratic nonlinear processes, e.g., sum- and difference-frequency mixing (SFM and DFM), of three interacting waves in nonlinear mediums [1, 2]. Periodically poled ferroelectrics crystals have been used extensively in nonlinear optical devices because of their large effective nonlinear coefficients *d*_{eff}
that are accessible through QPM and the possibility that they can be engineered to suit a noncritical interaction configuration [3]. In the uniform QPM grating, however, the conversion bandwidth and the detune of pump bandwidth is less than 90 nm and ~0.1 nm under the condition of *L*>25 mm, respectively [2, 3]. A quantitative description of the quadratic nonlinear effects for QPM normally employs a group of coupled-mode equations (CME) [2, 4, 5]. To overcome problems for the uncertainty of the propagation constant, errors in the fabrication process, and the fluctuation of laser wavelength or of temperature in the practical QPM devices, etc., the chirped [6] or segmented grating [2], and the method of phase shift [7], multiple phase-matching wavelength [8] or detuning the pump [3] were proposed and/or experimentally realized to broaden the QPM bandwidth. Almost all of their theoretical analyses, however, were directly obtained from the differential equation groups of CME of QPM-TWM. In this paper, we obtain expressions of QPM-TWM, under the undepleted-pump approximation (i.e., small-signal approximation), which can remarkably reduce operations by avoiding the iterative calculation of CME and can optimize arbitrary segments with arbitrary length and phase shift for broadening the QPM bandwidth. The simulative and emulational results are given.

## 2. Optimal design

In the Periodically poled crystal waveguide (e.g. LiNbO_{3}), the loss, group velocity mismatch (GVM) and higher-order dispersion of the material can normally be ignored at the length of ~30mm [3, 8, 9]^{*}.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, 4]. Fig.1 shows an aperiodic grating with arbitrary phase-shifted segments for this calculation. The sign of the nonlinear coefficient is periodically inverted, and its structure is divided into *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

and for QPM-SFM as

where *N*_{l,j}
(j=1,2,3,4) is the element of the operator *N*_{l}
, ${N}_{l}^{D}$
_{,1}={cosh(*Q*_{l}*L*_{l}
)+[*i*Δ*k*/2*Q*_{l}
]sinh(*Q*_{l}*L*_{l}
)}*e*
_{1}, ${N}_{l}^{D}$
_{,2}=-*i*(*M*
_{1}/*Q*_{l}
)sinh(*Q*_{l}*L*_{l}
)*e*
_{2}, ${N}_{l}^{D}$
_{,3}=-*i*(*M*
_{2}/*Q*_{l}
)sinh(*Q*_{l}*L*_{l}
)*e*
_{2}, ${N}_{l}^{S}$
_{,1}={cos(*Q*_{l}*L*_{l}
)+[*i*Δ*k*/(2*Q*_{l}
)]sinh(*Q*_{l}*L*_{l}
)}*e*
_{1}, ${N}_{l}^{S}$
_{,2}=-*i*(${M}_{1}^{*}$/*Q*_{l}
)sin(*Q*_{l}*L*_{l}
)*e*
_{2}, ${N}_{l}^{S}$
_{,3}=-*i*(*M*
_{2}/*Q*_{l}
)sin(*Q*_{l}*L*_{l}
)${e}_{2}^{*}$, *Q*_{l}
=[*M*
_{1}
${M}_{2}^{*}$±(Δ*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

where

Obviously, using the matrix operation of Eq. (2) can avoid the iterative calculation of differential equation groups of CME in the QPM chirp grating. As results, the operational process is far simplified and the optimization of QPM chirp grating becomes available.

The wavelength conversion employing DFM can accommodate greater than terahertz modulation bandwidth [2], and SFM can be used to implement a low power all-optical gate (AOG) [10]. To enhance the stability and bandwidth of the pump wave and extend the conversion bandwidth of the signal wave in QPM-TWM, we can optimize the QPM structure with arbitrary segments and phase shifts from Eq.(2). Table 1 and 2 show the optimal bandwidth Δ*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

data [2]: the pump power *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.

Fig.2(a)-(c) show optimizations of the signal and pump bandwidth in DFM and of the AOG bandwidth in SFM, respectively. These results come from the simulation of *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

In the PPLN waveguides, the propagation loss coefficient *α* is ~0.35 dB/cm at 1550 nm and ~0.7dB/cm at 775 nm [10]. To more accurately describe QPM-TWM, the loss should be taken into account in CME. Fig.3 shows relationships of the conversion efficiency *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

We obtain expressions of QPM-TWM with arbitrary grating structure and phase shift, for the first time, under the small-signal approximation. Employing the expressions, we can optimize the aperiodic grating to enhance the bandwidth and stability of the signal and pump wave. Three optimal examples for the signal and pump bandwidth of the wavelength conversion in DFM and for the AOG bandwidth in SFM are given. The optimal results in Table 1–2 coming from Eq.(2) is compared with Fig.2 calculating from a set of differential equations of CME. Although the conversion efficiency *η* decreases by comparing loss free with loss, the conversion bandwidth Δ*λ* changes a little when taking into account the propagation loss.

## Acknowledgement:

The authors thank the supports by the National Natural Science Foundation of China (69990540) and the 863 High Technology Research Development Program.

## Footnotes

^{*} | In Ref.[3], a 40-mm-long nonuniform QPM grating was theoretically obtained and experimentally fabricated; a 42-mm-long wavelength conversion section is demonstrated in Ref.[8]; and a 30-mm-long uniform device based on DFM in PPLN is theoretically and experimentally studied in Ref.[9]. All of experimental results reported were approximately consistent with the theoretical analysis that the loss, GVM and high-order dispersion of the material were neglected. |

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