## Abstract

Starting from Maxwell’s equations and taking the nonlinearity of linear electro-optic effect as a perturbation, we derive general wave coupling equations of quasi-phase-matched (QPM) linear electro-optic effect. And then we use the equations to study the electro-optic effect in PPLN. The numerical results indicate that the QPM condition plays an important role in electro-optic coupling. In addition, the coupling is very sensitive to the temperature and incident light wavelength, but it has a large tolerance to the direction of incident light.

©2006 Optical Society of America

## 1. Introduction

The “quasi-phase matching” proposed by Bloembergen *et al*. in 1962 provides an efficient method to match a nonlinear process in phase [1]. In order to achieve QPM, several useful and reliable techniques have been developed for modulating the nonlinear coefficient, including periodic or quasi-periodic electric-field poling of ferroelectric materials [2–8], and orientation patterning of semiconductors such as GaN [9]. The QPM technology has been widely used in nonlinear frequency conversions and receives more and more research interests [2–9]. Recently, the concept and technique of QPM were used for the linear electro-optic effect in the materials whose electro-optic coefficients are periodically modulated, and some interesting phenomena have been observed [10–13]. Lu *et al*. [10] used the coupled-mode theory developed by Yariv [14] to investigate the electro-optic coupling in a PPLN crystal. However, it is applicable only for some special cases (see the discussion below). On the other hand, a wave coupling theory, different from the refractive index ellipsoid and coupled-mode ones, of linear electro-optic effect was developed by She and Lee [15] and it has been generalized to the case of absorption medium [16]. However, these theories can only deal with the phenomenon of linear electro-optic effect in a single-domain bulk medium and no QPM analyses have been involved in them. In this paper, we generalize further the wave coupling theory of linear electro-optic effect to the case of QPM, then seek for the general solution of the resultant equations, and finally give some numerical analyses to the non-absorbing PPLN crystal, whose refractive indices and electro-optic coefficients are real.

## 2. Theory

Linear electro-optic effect can be regarded as a nonlinear interaction between a light wave and an external electric field [15]. The total electric field **E** participating in the process of linear electro-optic effect can be expressed as

where [**E**^{ω}exp(-*iωt*)/2+c.c.] is the light field with frequency *ω*, **E**(0) is the dc electric filed or slow varying electric field, and c.c. denotes the complex conjugate. In general, there exist two independent plane electromagnetic wave components for a monochromatic light wave with frequency ω propagating in a birefringent crystal, i.e.,

where **E**ω_{1} and **E**ω_{2} denote two perpendicular components when *k*
_{1}=*k*
_{2}, or two independent components experiencing different refractive indices when *k*
_{1}≠*k*
_{2}. Let

where **a, b** and c are three unit vectors and **a·b**=0, *A*
_{1}(*r*) and *A*
_{2}(*r*) are the normalized amplitudes of the two waves, *n*
_{1} and *n*
_{2} are the unperturbed refractive indices of the two waves with different polarizations. We assume that the crystal is in the clamped condition, i.e., the converse piezoelectric and photo-elastic effects are suppressed [17], and other second-order nonlinear effects are so weak because of phase mismatch that only linear electro-optic effect has to be considered. Similarly to Ref. [15], starting from Maxwell’s equations and taking the linear electro-optic effect as a perturbation, we have the following wave coupling equations under the slow varying amplitude approximation:

Here, *g*(*r*) is the structure function of the material, Δ*k*’=*k*
_{2}-*k*
_{2}, and

where *r _{effi}* (

*i*=1,2,3) is the effective electro-optic coefficient [15]. Note that the wave coupling Eqs. (4a) and (4b) are different from those of coupled-mode theory developed by Yariv [14]. In Yariv’s equations, only the first term on the right-hand side of either of Eqs. (4a) and (4b) is included. In fact, the second term will influence the phase of light and cannot be ignored.

If *g*(*r*) is a periodic function of *r* with period Λ due to periodically modulated electrooptic coefficient, it can be written as a Fourier series,

where the *m*th-harmonic grating wave vector α=2*π _{m}*/

*Λ*is very close to -Δ

*k*’. Substituting Eq. (6) into Eqs. (4a) and (4b), and neglecting those components that make little contributions to electro-optic effect because of phase mismatch, we have

where Δ*k*=Δ*k*’+*α _{m}*, and

Equations (7a) and (7b) are the resultant coupling equations of quasi-phase-matched linear electro-optic effect. Let the initial normalized amplitudes of the two waves be *A*
_{1}(0) and *A*
_{2}(0), respectively, and then the solution of Eqs. (7a) and (7b) can be obtained as follows:

where

Equations (9) to (16) can be used to describe the linear electro-optic effect in QPM materials with an external electric field applied along arbitrary direction. Some numerical analyses for the PPLN will be shown in section 3.

## 3. Numerical Analyses

Without loss of generality, we take a 2.5*cm* long PPLN, whose poling direction is *Z*-axis of the crystal, as the working material. The structure function *g*(*r*) of PPLN is

whose Fourier transform coefficient is found to be

where *D≡l*/Λ and *l* is the length of one positive section. We find that the optimum value of *D* is 0.5 for *m*=1, and 0.25 or 0.75 for *m*=2, since |*G _{m}*| is maximized at these values of

*D*. For the case of

*G*

_{0}=0 (it can only be achieved with

*D*=0.5), there is only one term left on the right hand of the coupling Eqs. (7a) and (7b), which are the same as those in the coupled mode theory [10, 14]. Specially, if all the reciprocal vectors cannot compensate the natural birefringence and

*v*=

_{1q}*v*=0, $\frac{{\mathit{dA}}_{1}\left(r\right)}{\mathit{dr}}\approx \frac{{\mathit{dA}}_{2}\left(r\right)}{\mathit{dr}}\approx 0$, meaning that the light can hardly be modulated by external field. Let the angle between wave vector

_{2q}**k**

_{1}(or

**k**

_{2}) and Z-axis be

*θ*. The projection of

**k**

_{1}(or

**k**

_{2}) in the

*XY*-plane is then making an angle

*φ*with the

*X*-axis. In the following studies, the external field direction is fixed to lie in the

*YZ*-plane and make an angle 45° with

*Z*-axis, i.e., $\mathbf{c}=(0,\sqrt{2}\u20442,\sqrt{2}\u20442)$. The Sellmeier equations for pure LiNbO

_{3}are given by [18]:

$${{n}_{e}}^{2}=4.5567+2.605\times {10}^{-7}{T}^{2}+\frac{0.970\times {10}^{5}+2.70\times {10}^{-2}{T}^{2}}{{\lambda}^{2}-{\left(2.01\times {10}^{2}+5.4\times {10}^{-5}{T}^{2}\right)}^{2}}-2.24\times {10}^{-8}{\lambda}^{2},$$

where *T* is the absolute temperature and *λ* is the wavelength of incident light in *nm*.

First we consider a simple structure that *D* is 0.5 and its first-order reciprocal vector satisfies the QPM condition when the light with wavelength *λ*=632.8*nm* propagates along the *X*-axis (i.e., *θ*=0.5*π*, *φ*=0) at temperature *T*=298K. Assuming the incident light is an extraordinary wave, the initial condition is given by *A*
_{1}(0)=0, *A*
_{2}(0)=1. The output intensities and phases of these two waves are shown in Figs. 1(a) and 1(b) when the QPM condition is satisfied. From the Fig. 1(a), one can find that the intensity of the extraordinary wave is totally converted to that of the ordinary wave at 0.9*kV/cm*. And the phases of the output waves are very regular due to the absence of *v _{1q}* and

*v*. Moreover, the output intensity of extraordinary (or ordinary) wave is modulated by the external electric field, which is applicable in optical modulation, optical switching, optical splitting and electric field detection, etc. If the light doesn’t propagate along the

_{2q}*X*-axis exactly, QPM condition doesn’t hold any more, because the effective period length becomes Λ′=Λ/(sin

*θ*cos

*φ*) and the refractive index of extraordinary wave has changed. The numerical results, for

*θ*=0.495

*π*and

*φ*=0.001π, are shown in Figs. 1(c) and 1(d). One can see that the output intensities and the phases of the two waves become somewhat complicated. As a comparison, when a light propagates along the

*X*-axis of a single-domain bulk LiNbO

_{3}whose structure function

*g*(

*r*)=1, the coupling between the two waves is very weak because of phase mismatch and the two waves behave experiencing different refractive indices independently.

As mentioned above, the polar angle *θ* and azimuth angle *φ* have their influence on the QPM condition. In other words, we can tune *θ* and *φ*(by altering incident direction of light) to meet QPM condition for the case of small deviation from QPM condition. Fixing the external electric field *E*
_{0} at 0.9*kV/cm* and varying *θ* from 0.49π to 0.51π and *φ* from -0.01π to 0.01π, we get the function relation between output intensity of ordinary wave and the angles (*θ* and *φ*), shown in Fig. 2. We can see that there exists a platform near the *X*-axis, which means that the coupling has a large tolerance to the change of angles *θ* and *φ* near the *X*-axis.

Besides *θ* and *φ*, temperature and the wavelength of light can also influence the QPM condition. The temperature influences the QPM condition primarily by changing the refractive indices [see Eq. (19)]. Still assuming an extraordinary light propagates along *X*-axis under a 0.9*kV/cm* applied field, we investigate the effect of the temperature *T* and the wavelength *λ* on the output intensities of the extraordinary and ordinary waves. The results are shown in Fig. 3. One can see that a 0.3K variation in temperature can lead to 50% decrease in the conversion efficiency. Therefore, a constant temperature system is necessary to achieve the stable coupling. The electro-optic coupling is also very sensitive to the incident light wavelength. The conversion efficiency drops to about 50% when the wavelength shifts by 0.1*nm*. This property makes it suitable to be used as a narrowband filter, whose passing frequency is governed by the QPM condition.

If Λ is doubled and *D*=0.75, the second-order reciprocal vector will satisfy the QPM condition. In this case, *v _{1q}* and

*v*, which don’t appear in coupled-mode theory [10, 14], but now should be considered since

_{2q}*G*

_{0}≠0. For comparison, we study both the cases with and without consideration of

*v*and

_{1q}*v*. The energy exchange between the extraordinary wave and the ordinary one and the phases of the two waves are shown in Fig. 4. Due to the presence of

_{2q}*v*and

_{1q}*v*, one cannot get a 100% conversion from the extraordinary wave to the ordinary one, and the phases of the two waves are irregular, which is very different from the case neglecting

_{2q}*v*and

_{1q}*v*[Figs. 4(c) and 4(d)]. So, one can draw a conclusion that

_{2q}*v*and

_{1q}*v*also play their roles in electro-optic effect.

_{2q}## 4. Conclusion

In conclusion, we have presented a wave coupling theory of quasi-phase-matched linear electro-optic effect and found the analytical solutions for the resultant equations. The numerical investigations indicate that besides *κ _{q}, κ_{q}** [see Eqs. (7a) and (7b)],

*v*and

_{1q}*v*, the QPM condition plays an important role in the electro-optic coupling process. And the QPM condition is very sensitive to the temperature and light wavelength.

_{2q}## Acknowledgments

The authors acknowledge the financial support from the National Natural Science Foundation of China (Grant Nos. 10374121 and 10574167).

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