1. Introduction

Nonlinear optical (NLO) polymers based on second-order nonlinearity have been widely studied because of their potential applications in electro-optic (EO) devices [1, 2]. Various techniques have been reported for characterization of EO coefficients of poled polymer thin films such as the reflection ellipsometric method [3–7], Fabry-Perot (FP) [8, 9], attenuated total reflection (ATR) [10–12], Michelson interferometer [13, 14], and Mach-Zehnder interferometer (MZI) [14–19]. Compared to the reflection ellipsometric method first introduced by Teng and Man [3] as well as Shildkraut [4] (so called simple Teng-Man method), other methods such as FP, ATR, and MZI have the advantage of separate measurement of the EO coefficients $r_{13}$ and $r_{33}$. In addition, ATR is less susceptible to properties of transparent conducting oxide (TCO) because the TCO layer doesn’t affect the waveguide modes very much [12]. The reflection-type MZI (MZIR) method for characterization of the second-order nonlinearities in a poled polymer thin film was first proposed by Norwood et al. [14] in 1994, later analyzed by Shin et al. [18], and recently modified by Greenlee, et al. [19], who added the ability to independently determine the piezoelectric contribution by flipping the reflective sample. Singer et al. [16] and Qiu, et al. [17] demonstrated the method for the complex tensor components of EO coefficients with a MZI in a transmission scheme (MZIT). However, most analyses have used a simple analysis that ignores multiple reflections from a reflective layer, such as a TCO layer, and assumes a perfectly reflective metal electrode, in the MZIR case, or perfectly transmissive electrodes, in the MZIT case. Therefore, the simple analysis takes into account only optical properties of the EO materials, just as in the simple Teng-Man method, although the multilayer nature of the structures should be taken into account to obtain more reliable values for the EO coefficients [7].

In this paper, we derive analytic expressions for the rigorous MZIR method to take into account multiple reflection effects and discuss relative errors that can be caused by using the simple MZIR analysis. We find that the results using the simple MZIR analysis are identical to the simple Teng-Man method provided the ratio of $r_{13}$ to $r_{33}$ obtained from the simple MZIR analysis is also used in the simple Teng-Man analysis, which requires an assumed value for this ratio (often 1/3). For the simple MZIR method, we also provide corrections to mathematical formulas for phase, effective EO coefficients, and the ratio of $r_{13}$ to $r_{33}$ given previously in the literature.

2. Theory

2.1 Rigorous Expressions for the rigorous MZIR model

The experimental setup for the MZIR method is shown in Fig. 1 [14, 18, 19]. A thin glass plate in the reference arm is rotated to produce a phase shift relative to the laser light reflected from the multilayered sample (glass/TCO/NLO/gold). Interference intensity data without applying voltage to the sample can be collected from a photo-detector as a function of phase angle, Ω_m, created by the rotating glass plate. The dc intensity is expressed as

 

Fig. 1 Schematic of a MZIR experimental setup. PD is a photo-detector. Laser is continuous-wave operating at a communication wavelength of 1.3 or 1.55 μm.

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Application of a voltage to the sample changes the refractive indices and thickness of the sample, thus inducing a change in both the magnitude and the phase of the reflection coefficient r_m. Similar to the dc intensity curve, the modulated intensity is collected from the lock-in amplifier under the application of AC voltage to the sample as a function of the phase Ω_m created by the glass plate. To first order in V, the modulated intensity $I_M^{(m)}$is obtained by differentiating $I_{dc}^{(m)}$ to get

From simple curve fitting of $I_{dc}^{(m)}$ and $I_M^{(m)}$, we can obtain A, B, δA, δB, and δΨ for s- and p- polarized light. We note that Eqs. (2) and (4) are quite similar to those in the rigorous Teng-Man method [7]. We define ${\tilde{B}}_m=I_0{r}_m{t}_{gm}^{*}=B_m\exp \left[i\left({\Psi }_m-{\Omega }_m\right)\right]$ from which calculation of $\delta {\tilde{B}}_m/{\tilde{B}}_m$gives

 

Fig. 2 Multilayered structures in (a) the rigorous model and (b) the simple model.

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Performing a Taylor expansion to first order, the right hand side of Eq. (5) can be written in the form

Equation (10) can thus be written as a linear function of the real and imaginary parts of the EO coefficients in the form

As described in Ref. 19, the converse piezoelectric coefficient d₃₃ is determined by a separate measurement with the sample flipped and the back electrode acting as a mirror while the modulating voltage is applied. In this case, there is only modulation of the phase of the reflected beam given by

Equation (5) together with Eq. (13) and d₃₃ from Eq. (16) can be written in the matrix forms

A single angle of incidence measurement at 45° is usually performed in MZ interferometry, but it is possible, though cumbersome, to take data at multiple angles as demonstrated in Ref. 18. This analysis can also be applied to the MZIT method that uses a poled organic thin film as a rotating/fixed plate in a transmission scheme, in which case a multi-angle data set can be obtained by rotating the sample [17]. Multiple angle data sets can be analyzed by implementing least square fitting as described in Ref. 7. For the rigorous analysis, it is necessary to know detailed sample information such as complex refractive indices and thicknesses for each layer, as illustrated in Fig. 2(a).

2.2 Simple MZIR Model

Typically, in the simple model, the properties of the TCO layer and multilayered structure of the sample are ignored as shown in Fig. 2(b). The reflectance at the air/film interface is ignored and the metal electrode is considered as a perfect electric conductor (PEC). Assuming the anisotropic refractive indices are complex numbers and ignoring reflection at the air/NLO film interface, the reflection coefficients from the sample (air/NLO/PEC) are given by [20]

The modulated phase and reflectivity in the simple MZIR method can be derived based on Eqs. (17) and (18). We use a lowercase h, $\psi$, and b for the simple model instead of H, $\Psi$, and $B$, respectively, which include the effect of all layers. From Eq. (14), we obtain

Outside the absorption band where ${\kappa }_{o,e}\ll 1$, we can approximate Eqs. (20) and (21) to obtain expressions for $\delta {\psi }_m$ and $\delta b/b$ in the form

Subtracting $\delta {\psi }_p$ from $\delta {\psi }_s$ to compare with $\delta {\psi }_{sp}$ of the simple Teng-Man method, we get

2.3 Comparison to previous MZIR simple model equations

For the case that s_j3 = 0, we derive the expressions in Eqs. (22) and (23) for the voltage induced phase change by differentiation of ${\beta }_m=k_0{n}_m\cos ({\alpha }_m)$ to get

The first term in Eq. (26) agrees with that given in Norwood, et al. [14], but the corresponding term in Ref. 19, which was drawn from Ref. 18, is not correct. The origin of the error in Ref. 18 is speculative, but might be traced to Eq. (10) in Han and Wu [22]. The path length in Eq. (10) of Han and Wu corresponding to s in Ref. 18 and s_EO in Ref. 19 is not useful for phase determination because the separate parts of the expression correspond to different refractive indices; therefore the full expression cannot be multiplied by a single refractive index to provide a valid optical path length ${\displaystyle \int nds}$. However, the final expression [Eq. (20) in Ref. 22] for the modulated s-p phase difference is correct. The details of Eq. (26) from the point of view of diagrammatic phase delays in the thin film are given in Appendix A.

The second term in Eq. (26) was given incorrectly as $2k_0{n}_m\left(\delta d/\cos {\alpha }_m\right)$in Ref. 19. We also discuss the correct form using diagrammatic phase delays in Appendix A. We note that when the internal angle is 45°, the converse piezoelectric contribution to the phase change in Eq. (26) can differ by about a factor of four compared to the previous result in Ref. 19.

Following Ref. 14, further evaluation of the first term in Eq. (26) requires determination of effective EO coefficients$r_{eff}^{(m)}$ satisfying$\delta n_m=-\left(1/2\right)n_m^3{r}_{eff}^{(m)}E$. The expression for $r_{eff}^{(s)}$is straightforward with$r_{eff}^{(s)}=r_{13}$. For the p-polarization case, however, expressions for $r_{eff}^{(m)}$ reported previously in the literature [8, 14, 17] vary. The expressions for the effective EO coefficients in Refs. 8, 14, and 17 are for the case of a fixed propagation direction and the expression obtained in Ref. 14 also ignores the walk-off angle in an anisotropic medium. These issues are discussed in Appendix B. By simply differentiating Eq. (9) for the case that a field is applied and both the refractive index $n_p$ and the internal angle ${\alpha }_p$ change in response to changes in the anisotropic indices n_o and n_e, we obtain an expression for the effective EO coefficient in the form

3. Results: numerical estimation of errors from use of the simple analysis

In the previous section, we have discussed simple and rigorous analyses of the MZIR method. Also, we have shown that the modulated phase retardation $\delta {\psi }_{sp}$ from difference between $\delta {\psi }_s$ and $\delta {\psi }_p$ in the simple MZIR method is identical to that in the simple Teng-Man method when the same ratio of r₁₃ to r₃₃ is used. This implies that results from the simple MZIR method might be erroneous when multiple reflection effects are ignored. In this section, we simulate the relative error of r₁₃ and r₃₃ resulting from use of the simple analysis of the MZIR method. As in Ref. 7, the relative error is defined as $\left(r_{j3}^{SM}-r_{j3}\right)/r_{j3}$, where $r_{j3}^{SM}$ is the EO coefficient estimated from the simple model analysis.

We used anisotropic refractive indices (n_o, n_e) = (1.73, 1.79) at the wavelength of 1320 nm and a 150 nm thick Abrisa^® ITO/glass substrate (the ITO refractive index is about 1 + 0.2i at 1320 nm). The EO coefficients are assumed to be 100 pm/V and 300 pm/V for r₁₃ and r₃₃, respectively. First, modulated phases of s- and p- waves, $\delta {\psi }_s$ and $\delta {\psi }_p$, are calculated from the modulated reflection coefficients using Eqs. (5) and (10). From the modulated phases, the EO coefficients by the simple model are obtained from Eqs. (22) and (23) ignoring the converse piezoelectric effects. Figure 3(a) shows errors in the EO coefficients calculated by the simple model analysis based on Eqs. (22) and (23). It should be emphasized that the r₃₃ by the simple model analysis of MZIR will be exactly same as that by the simple Teng-Man method when the ratio of r₁₃ to r₃₃ obtained from the MZIR method is used in the simple Teng-Man analysis. Figure 3(b) shows errors when a 45 nm thick ITO (Thin Film Devices, Inc., with refractive index about 1.2 + 0.15i at 1320 nm) was used to compare with a thicker Abrisa ITO. It shows much smaller error in r₃₃ compared with the Abrisa ITO because of much lower reflectivity from the ITO layer that reduces overall multiple reflection effects.

 

Fig. 3 Plot of the error in r₁₃ (black) and r₃₃ (red) by the simple MZIR method for the case of thick ITO. Error in r₃₃ (blue dot) by the simple Teng-Man method is plotted for comparison. Anisotropic index of refraction (1.73, 1.79) was used for the NLO thin film. For a TCO layer, (a) a 150 nm thick Abrisa ITO substrate and (b) a 45 nm thick TFD ITO substrate were used for this simulation.

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Figure 4 shows the simulated ratio of r₃₃ to r₁₃ as a function of thickness of the nonlinear polymer for the case of thick ITO (Abrisa) and thin ITO (TFD). When the thickness is less than ~0.2 μm, the ratios for both cases are larger than 10, which is much higher than the correct value of 3 that was assumed in the simulation. For the thickness larger than ~0.2 μm, the simulated ratio for the case of Abrisa ITO shows a large cyclic variation from approximately 0 to 8, as expected in the r₃₃ and r₁₃ from Fig. 3(a), whereas the ratio for the TFD ITO is much smaller than that for the Abrisa ITO because of lower absorption of the TFD ITO at the operating wavelength of 1320 nm. Nevertheless, even with the less reflective TFD ITO, values of the ratio $r_{33}/r_{13}$extracted from experimental results using the simple models can vary ± 33% from the correct value. These considerations point out the difficulty in obtaining reliable r-coefficient ratios if the multilayer nature of the sample is neglected.

 

Fig. 4 Ratio of r₃₃ to r₁₃ for the case of 150 nm thick Abrisa ITO and 45 nm thick TFD ITO at the wavelength of 1320 nm resulting from simple analysis of MZIR when the correct value is 3.

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4. Conclusions

In this paper, we have analyzed the Mach-Zehnder interferometer method that specifically uses a poled polymer thin film as one of the reflective mirrors for determination of EO coefficients of NLO polymer thin films. Here we have referred to this as the MZIR method. We discussed in detail corrections to previous expressions in the literature for the simple analysis of MZIR. We found that the simple MZIR model yields identical results to the simple Teng-Man method when the ratio of r₁₃ to r₃₃ obtained from the simple MZIR analysis is used in the simple Teng-Man measurement, indicating that the simple MZIR method can be erroneous because of the multiple reflection effects. This means that, although the MZIR method has the advantage of separate determination of r₁₃ and r₃₃, the simple MZIR method cannot be viewed as an independent measurement to check simple Teng-Man measurements.

We have also presented mathematical expressions for phase modulation including converse piezoelectric effects as well as for the effective EO coefficient $r_{eff}^{(p)}$ in the simple model. These provide corrections to earlier expressions in the literature. If the light reflected once through the sample is detected and other reflections are blocked, thick films/crystals such as LiNbO₃ can be analyzed using the simple MZIR method because the thickness is large enough to avoid multiple reflections. However, it is not generally possible to avoid multiple reflection effects in a thin NLO film, because reflections from interfaces in multilayer structures overlap. We have also discussed the relative error caused by ignoring multiple reflection effects. Depending on the optical properties of NLO and TCO such as refractive index and thickness, the error in using the simple model can be large (>100%) in both r₁₃ and r₃₃ and oscillates from positive and negative, as can be seen in the simple Teng-Man method as discussed in Ref. 7.

Appendix A: calculation of the modulated phase term for reflection from a thin slab

To further justify the simple model equations discussed above compared to Refs. 18 and 19, here we follow the diagrammatic approach in Ref. 18 and derive the modulated phase term for reflection in the three different cases illustrated in Fig. 5 : (a) back-side reflection from gold for the converse piezoelectric effect, (b) front-side reflection through the NLO film with only an EO effect, and (c) front-side reflection through the NLO film with only a converse piezoelectric effect.

 

Fig. 5 Reflection with a voltage bias across the film. (a) Back-side reflection from metal (gold) surface. Horizontal dashed line represents a voltage-induced thickness change. (b) Front-side reflection from a glass-film-gold with only an EO effect. (c) Front-side reflection with only a converse piezoelectric effect.

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First, the phase change for reflection from the back side of the EO sample with a voltage modulation is considered. In the ray traces shown in Fig. 5(a), beams b₁ and b₂ are the reflections at the air-gold interfaces with and without a voltage-induced thickness change, respectively. Then, since the optical paths are always in air, the path length difference is simply $2\overline{AB}-\overline{AD}$ and the phase change is given by

(29)$$\delta \psi =k_0{n}_{air}\left(2\overline{AB}-\overline{AD}\right)=2k_0\cos \theta \delta d,$$

where $\overline{AB}=\delta d/\cos \theta$and $\overline{AD}=2\overline{AB}{\sin }^2\theta$. This agrees with our Eq. (16) above.

Second, consider the phase change from reflection in a glass-film-gold structure with only an EO effect. In Fig. 5(b), beam b₁ is the reflection at the glass-film interface, beam b₂ is the single pass trace through the film reflected off the gold, and beam ${b^{\prime }}_2$ is the single pass trace after application of a voltage. The phase difference between beam b₂ and the voltage-deviated beam ${b^{\prime }}_2$ is given by

(30)$$\begin{array}{c}\delta {\psi }_m=k_0\left[{n^{\prime }}_m\left(\overline{A{B}^{\prime }}+\overline{B^{\prime }{C}^{\prime }}\right)-n_m\left(\overline{AB}+\overline{BC}\right)-n_g\overline{D{D}^{\prime }}\right]\\ =k_0\left\{\left[{n^{\prime }}_m\left(\overline{A{B}^{\prime }}+\overline{B^{\prime }{C}^{\prime }}\right)-n_g\overline{A{D}^{\prime }}\right]-\left[n_m\left(\overline{AB}+\overline{BC}\right)-n_g\overline{AD}\right]\right\}\\ =2k_{0}d\left({n^{\prime }}_m\cos {{\alpha }^{\prime }}_m-n_m\cos {\alpha }_m\right)=2k_{0}d\delta \left(n_m\cos {\alpha }_m\right),\end{array}$$

where $n_g\sin {\theta }_1=n_m\sin {\alpha }_m={n^{\prime }}_m\sin {{\alpha }^{\prime }}_m$ (Snell’s law) has been used, along with the relations $\overline{AB}=\overline{BC}=d/\cos {\alpha }_m$and $\overline{AD}=2d\tan {\alpha }_m\sin {\theta }_1$. Here ${n^{\prime }}_m$ and ${{\alpha }^{\prime }}_m$ are the refractive index and the internal angle of incidence with a voltage bias. This then leads directly to the first term on the right hand side in Eq. (26) upon use of$\delta {\alpha }_m=-\tan {\alpha }_m\left(\delta n_m/n_m\right)$.

Referring to Fig. 5(c), the induced phase change when the applied voltage only induces a change in thickness with no EO effect is similarly calculated to be

(31)$$\begin{array}{c}\delta {\psi }_m={\psi }_m({b^{\prime }}_2)-{\psi }_m(b_2)\\ =k_0\left[n_m\left(\overline{A{B}^{\prime }}+\overline{B^{\prime }{C}^{\prime }}\right)-n_m\left(\overline{AB}+\overline{BC}\right)-n_g\overline{D{D}^{\prime }}\right]\\ =k_0\left[n_m\left(\overline{A{B}^{\prime }}+\overline{B^{\prime }{C}^{\prime }}\right)-n_m\left(\overline{AB}+\overline{BC}\right)-n_g\left(\overline{A{D}^{\prime }}-\overline{AD}\right)\right]\\ =k_0\left[n_m\left(\frac{2d^{\prime }}{\cos {\alpha }_m}\right)-n_m\left(\frac{2d}{\cos {\alpha }_m}\right)-n_g\left(2d^{\prime }\tan {\alpha }_m\sin {\theta }_1-2d\tan {\alpha }_m\sin {\theta }_1\right)\right]\\ =2k_0{n}_m\cos {\alpha }_m\delta d,\end{array}$$

which agrees with the second term on the right in Eq. (26). One can derive the same result assuming the thickness change $\delta d$is all at the top of the film rather than at the bottom.

Appendix B: Effective EO coefficient for p-polarization

Here we derive the effective electro-optic coefficient appropriate for p-polarized light propagating at non-normal incidence through a poled polymer film. Electro-optic coefficients are typically defined in connection with the electric impermeability tensor$\overleftrightarrow{B}={\overleftrightarrow{\epsilon }}_0{}^{-1}$ [23]. For the case of a poled polymer, $\overleftrightarrow{\epsilon }={\epsilon }_0\text{diag}\left(\begin{array}{ccc}{n}_o^2& n_o^2& n_e^2\end{array}\right)$, so

(32)$$\overleftrightarrow{B}=\text{diag}\left[\begin{array}{ccc}\frac{1}{n_o^2}& \frac{1}{n_o^2}& \frac{1}{n_e^2}\end{array}\right].$$

To first order in the applied field, E_k, electro-optically induced changes in the refractive index are represented by $\delta B_{pq}=r_{pq,k}{E}_k$, where (k, p, q = 1, 2, 3) [23]. The linear electro-optic coefficients $r_{pq,k}$ are symmetric in the first two indices and usually written using reduced 6 × 3 tensor notation $r_{sk}$, where $pq\to s$according to (11→1, 22→2, 33→3, 23→4, 31→5, 12→6). Thus, corresponding to Eq. (32), we have the usual $\delta \left(1/n_o^2\right)=\delta B_{11}=r_{13}{E}_3$ and $\delta \left(1/n_e^2\right)=\delta B_{33}=r_{33}{E}_3$ when the applied field is in the z-direction only.

Expressions for effective electro-optic coefficients for s-and p- polarizations are desired in the form

(33)$$\delta \left(\frac{1}{n_m^2}\right)=r_{eff}^{(m)}{E}_3,$$

recalling that m = s or p. Because $n_s=n_o$, the case of s-polarization is straightforward and $r_{eff}^{(s)}=r_{13}$. For the case of p-polarization, we recall the well-known expression [24] resulting from Maxwell’s equations for a plane electric wave $E_{\omega }{e}^{i\left(k\cdot r-\omega t\right)}$propagating in the direction $\widehat{k}$with constant $E_{\omega }$ through an anisotropic material,

(34)$$n^2\left[E_{\omega }-\widehat{k}\left(E_{\omega }\cdot \widehat{k}\right)\right]-\frac{1}{{\epsilon }_0}{D}_{\omega }=0,$$

where D_ω is the electric displacement vector and n is the refractive index appropriate to the propagation direction and polarization. In the absence of nonlinear effects, $D_{\omega }=\overleftrightarrow{\epsilon }\cdot E_{\omega }$. As is well-known, $D_{\omega }\cdot \widehat{k}=0$, but in general $E_{\omega }\cdot \widehat{k}\ne 0$. If we insert$E_{\omega }={\overleftrightarrow{\epsilon }}^{-1}\cdot D_{\omega }=\left(1/{\epsilon }_0\right)\overleftrightarrow{B}\cdot D_{\omega }$, define $\widehat{u}$ to be a unit vector in the direction of D_ω, and then take the dot product of Eq. (34) with $\widehat{u}$, we get

(35)$$\widehat{u}\cdot \overleftrightarrow{B}\cdot \widehat{u}=\frac{1}{n^2},$$

having used $\widehat{u}\cdot \widehat{k}=0$. Note that the substitution $\widehat{u}=\left(1/n\right)\left(\begin{array}{ccc}x& y& z\end{array}\right)$gives the well-known equation for the index ellipsoid. For p-polarization, we have ${\widehat{k}}_p=\left(\begin{array}{ccc}\sin {\alpha }_p& 0& \cos {\alpha }_p\end{array}\right)$ and ${\widehat{u}}_p=\left(\begin{array}{ccc}-\cos {\alpha }_p& 0& \sin {\alpha }_p\end{array}\right)$, which when inserted in Eq. (35) together with Eq. (32) gives Eq. (9).

For the case of a bulk material when the direction of propagation is fixed during the application of an electric field so that α_p is constant, we have

(36)$$\delta \left(\frac{1}{n_p^2}\right)={\widehat{u}}_p\cdot \delta \overleftrightarrow{B}\cdot {\widehat{u}}_p={\cos }^2{\alpha }_p\delta B_{11}+{\sin }^2{\alpha }_p\delta B_{33}\text{(}{\alpha }_p\text{fixed)}\text{.}$$

Thus,

(37)$${\left[r_{eff}^{(p)}\right]}_{{\alpha }_p=\text{constant}}=r_{13}{\cos }^2{\alpha }_p+r_{33}{\sin }^2{\alpha }_p.$$

This is the expression used in Refs. 8 and 17, but is not appropriate for a finite thickness film when the external angle of incidence is fixed under the application of an applied field because a change in the refractive index, by Snell’s law, changes the internal angle α_p and thus ${\widehat{u}}_p$. Consequently, for both Teng-Man and MZIR we have

(38)$$\begin{array}{c}\delta \left(\frac{1}{n_p^2}\right)=\delta {\widehat{u}}_p\cdot \overleftrightarrow{B}\cdot {\widehat{u}}_p+{\widehat{u}}_p\cdot \delta \overleftrightarrow{B}\cdot {\widehat{u}}_p+{\widehat{u}}_p\cdot \overleftrightarrow{B}\cdot \delta {\widehat{u}}_p\\ =\frac{n_o^2}{n_p^2}\left[r_{13}{\cos }^2{\alpha }_p+r_{33}{\sin }^2{\alpha }_p\right]E_3,\end{array}$$

having used $\delta {\alpha }_p=\left(1/2\right)n_p^2\tan {\alpha }_p\delta \left(1/n_p^2\right)$, which follows from Snell’s law, together with $\left(1/n_e^2-1/n_o^2\right)n_p^2{\sin }^2{\alpha }_p=1-n_p^2/n_o^2$, which follows from Eq. (9). From this result, we get the appropriate expression for $r_{eff}^{(p)}$given in Eq. (27).

The MZIR analysis used in Ref. 19 follows that of Ref. 14 and relies on yet a different expression for $r_{eff}^{(p)}$that was derived from the nonlinear polarization, $P_{\omega }^{NL}$, where $P_{\omega j}^{NL}=-{\epsilon }_0{n}_j^4{r}_{j3}{E}_{\omega j}{E}_3^{}$, (j=1,3) for p-polarization [25]. To examine this approach, we substitute $D_{\omega }=\overleftrightarrow{\epsilon }\cdot E_{\omega }+P_{\omega }{}^{NL}$ in Eq. (34) and take the dot product of the resulting equation with ${\widehat{e}}_p$, where $E_{\omega }=E_{\omega }{\widehat{e}}_p$to get

(39)$$n_p^2\left[{\widehat{e}}_p\cdot E_{\omega }-{\widehat{e}}_p\cdot \widehat{k}\left(E_{\omega }\cdot \widehat{k}\right)\right]-\frac{1}{{\epsilon }_0}{\widehat{e}}_p\cdot \overleftrightarrow{\epsilon }\cdot E_{\omega }=\frac{1}{{\epsilon }_0}{\widehat{e}}_p\cdot P_{\omega }^{NL},$$

where ${\widehat{e}}_p=\left[\begin{array}{ccc}-\cos \left({\alpha }_p-\gamma \right)& 0& \sin \left({\alpha }_p-\gamma \right)\end{array}\right]$ and γ is the walk-off angle [26] between D_ω and E_ω, i.e., $D_{\omega }\cdot E_{\omega }=D_{\omega }{E}_{\omega }\cos \gamma$. Separating out the zero-applied field contribution in Eq. (39), we find

(40)$$\delta \left(n_p^2\right)=-\frac{1}{{\epsilon }_0}\frac{{\widehat{e}}_p\cdot P_{\omega }^{NL}}{E_{\omega }{\cos }^2\gamma }=-\left[n_o^4{r}_{13}\frac{{\cos }^2\left({\alpha }_p-\gamma \right)}{{\cos }^2\gamma }+n_e^4{r}_{33}\frac{{\sin }^2\left({\alpha }_p-\gamma \right)}{{\cos }^2\gamma }\right]E_3,$$

which, after comparison with $\delta n_p=-\frac{1}{2}{n}_p^3{r}_{eff}^{(p)}{E}_3$, implies

(41)$${\left[r_{eff}^{(p)}\right]}_{{\alpha }_p=\text{constant}}=\frac{1}{n_p^4}\left[n_o^4{r}_{13}\frac{{\cos }^2\left({\alpha }_p-\gamma \right)}{{\cos }^2\gamma }+n_e^4{r}_{33}\frac{{\sin }^2\left({\alpha }_p-\gamma \right)}{{\cos }^2\gamma }\right],$$

noting that α_p is constant in this derivation. If one ignores the walk-off angle, as done in Ref. 14 and puts $\gamma =0$in this expression, the result agrees with that given in Ref. 14. However, ignoring the walk-off angle implies $n_o\approx n_e\approx n_p$. Finally we note that in Appendix A of Ref. 26 it was discussed that the relations

(42)$$\cos \left({\alpha }_p-\gamma \right)={\left[\frac{n_p}{n_o}\right]}^2\cos \gamma \cos {\alpha }_p\text{and}\sin \left({\alpha }_p-\gamma \right)={\left[\frac{n_p}{n_e}\right]}^2\cos \gamma \sin {\alpha }_p,$$

follow directly from $n_p^2\left[{\widehat{e}}_p-\widehat{k}\left({\widehat{e}}_p\cdot \widehat{k}\right)\right]-\left(1/{\epsilon }_0\right)\overleftrightarrow{\epsilon }\cdot {\widehat{e}}_p=0$. These relations turn Eq. (40) into Eq. (37), so the approach using the nonlinear polarization is consistent with using the electric impermeability tensor. But, again, this is for the case that the applied field does not have the effect of also changing the angle α_p. The appropriate $r_{eff}^{(p)}$ for both the MZIR and Teng-Man sample structures is given by Eq. (27), which can also be derived starting from Eq. (38) with considerably more complication in algebra.

Acknowledgments

The authors gratefully acknowledge discussions with Professor Robert A. Norwood at the Optical Sciences Center of the University of Arizona.

References and links

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