We have found and corrected a number of errors in the analysis of anisotropy of acousto-optic figure of merit (AOFM) performed in our recent work [1] for the case of anisotropic acousto-optic (AO) diffraction in ${\mathrm{LiNbO}}_3$ crystals. The errors have slipped in because we have not taken correctly into account the changes in the polarization of diffracted optical wave occurring due to changing orientation of the interaction plane under its rotation around the principal $X$, $Y$ and $Z$ axes, as well as due to changing incidence angle of the incoming optical wave and changing diffraction angle.

In Ref. [1], we have incorrectly calculated the electric field of the diffracted optical wave for the cases of diffractions occurring in the interaction plane $XZ$ rotated around $X$ (or $Z$) axis by the angle ${\varphi }_X$ (or ${\varphi }_Z$). The same is true of the case of interaction plane $YZ$ rotated around $Y$ axis by the angle ${\varphi }_Y$. As a consequence, Eqs. (19), (21), and (23) of Ref. [1] include some errors. Their correct versions are as follows:

Here $E_1$ and $E_2$ are the electric field components of the diffracted wave, $\theta$ and $\gamma$ the angles that describe propagation directions of the incident and diffracted waves with respect to the $X$, $X^{\prime }$ or $Y$ axes. Then the relation $E=\sqrt{E_1^2+E_2^2}$ and the appropriate relations for the effective elasto-optic coefficients (EEC), analogues of Eqs. (25)–(33) in Ref. [1], can be written as

The errors present in the relations (19) and (21)–(33) of Ref. [1] have affected the main results and their interpretation. In particular, the mended versions of Figs. 2–4 of Ref. [1] should be as follows:

 

Fig. 2. Dependences of AOFM (a), (c), (e), (g) and EEC (b), (d), (f), (h) on $\theta +\gamma$ angle for the type VII of AO interactions occurring at ${\varphi }_Y=40\mathrm{deg}$ (a)–(d) and ${\varphi }_Y=140\mathrm{deg}$ (e)–(h).

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Fig. 3. Dependences of AOFM (a) and EEC (b) on $\theta +\gamma$ angle for the type VIII of AO interactions occurring at ${\varphi }_X=40\mathrm{deg}$.

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Fig. 4. Dependences of AOFM (a), (c) and EEC (b), (d) on $\theta +\gamma$ angle for the type IX of AO interactions occurring at ${\varphi }_Y=10\mathrm{deg}$.

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The corrected data of our numerical analyses are gathered in the improved version of Table 1.

Table 1. Geometries of AO Interaction Providing Maximal AOFMs at Different Types of Anisotropic Interactions in ${\mathsf{LiNbO}}_{\mathsf{3}}$ Crystals

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Finally, the sections “3. Results and discussion” and “4. Conclusions” of Ref. [1] are to be revised as follows.

3. RESULTS AND DISCUSSION

Figures 2–4 present the dependences of the AOFM and the effective EEC on the angle $\theta +\gamma$ for the types VII, VII, and IX of AO interactions. Here we restrict our consideration to those interaction planes for which the maximum AOFM values can be reached. The maximal AOFM for the type VII of AO interactions with the quasi-longitudinal AW ($2.6\times {10}^{-15}{\mathrm{s}}^3/\mathrm{kg}$) is peculiar for the diffraction in the interaction plane $Y{Z}^{\prime }$ rotated around the $Y$ axis by 40 and 140 deg (see Fig. 2 and Table 1). This kind of diffraction can be observed at two different orientations of the interaction plane, instead of a single orientation as suggested in Ref. [1]:

For the type VIII of AO interactions with the AW QT1 (i.e., the AW polarized in the plane of interaction), the maximum AOFM value ($12.9\times {10}^{-15}{\mathrm{s}}^3/\mathrm{kg}$) is achieved in the $X{Z}^{\prime }$ interaction plane rotated around the $X$ axis by 40 or 220 deg (see Fig. 3 and Table 1). Then the AW propagates under the angle 182.6 deg in respect to $X$ axis, while the incident optical and diffracted optical waves, propagate at 280.0 deg and 260 deg, respectively, to the $X$ axis. These diffractions can be implemented at AW frequencies ($f_a\approx 4.29\mathrm{GHz}$). Comparing Fig. 3(a) with Fig. 3(b) one concludes that the AOFM anisotropy is mainly caused by the anisotropy of the effective EEC.

In case of the type IX of AO interactions with the AW QT2 polarized perpendicular to the interaction plane, the maximum AOFM is equal to $12.6\times {10}^{-15}{\mathrm{s}}^3/\mathrm{kg}$ (see Fig. 4 and Table 1). This value of AOFM is achieved in $Y{Z}^{\prime }$ plane rotated around $Y$ axis on the angle 10 deg. The incident optical wave propagates at 20.0 or 200 deg with respect to the $Y$ axis, while the diffracted one at 29.0 or 209.0 deg with respect to the same axis. Then the AW propagates at the angle 100.8 deg with respect to the $Y$ axis. This kind of AO diffraction can be implemented at the AW frequency $f_a=2.03\mathrm{GHz}$. Similarly to the cases of types VIII and IX of AO interactions, here the AOFM anisotropy is determined by the anisotropy of the EEC (see Figs. 3 and 4).

Let us compare our results with those known from the earlier literature on the subject. As already mentioned above, the maximal AOFM value obtained in the work [2] is equal to $20\times {10}^{-15}{\mathrm{s}}^3/\mathrm{kg}$ for the diffraction implemented in the $YZ$ plane under propagation of the fast quasi-transverse AW at the angle 60 deg with respect to the $Y$ axis. In our notation, this kind of AO diffraction corresponds to the type VIII, which is realized in the $YZ$ plane. Notice that the VIII type of interaction cannot be realized in $YZ$ plane, since the EEC in this plane is equal to zero [see Eq. (9)].

As already mentioned, the AOFM value equal to $22\times {10}^{-15}{\mathrm{s}}^3/\mathrm{kg}$ has been reported in the work [3] for the interaction in the $YZ$ plane with the slow AW that propagates at the angle 120 deg with respect to the $Y$ axis and is polarized parallel to the $X$ axis. This kind of AO diffraction corresponds to our interaction type IX. According to our data, the AOFM in this interaction plane acquires its maximum equal to $12.4\times {10}^{-15}{\mathrm{s}}^3/\mathrm{kg}$ when we deal with interaction with the AW propagating at the angle 100.8 deg with respect to the $Y$ axis. However, the AOFM value obtained by us is almost two times smaller than that reported in Ref. [3]. Of course, this difference can be caused, at least, by different constitutive coefficients (the elastooptic coefficients and the refractive indices) used in our calculations and in the study [3]. Nonetheless, it is essentially large and, most probably, should involve some other factors.

Since, according to our analysis, the maximal AOFM value in $YZ$ interaction plane is $12.4\times {10}^{-15}{\mathrm{s}}^3/\mathrm{kg}$, the AOFM value obtained in Ref. [4] ($15.9\times {10}^{-15}{\mathrm{s}}^3/\mathrm{kg}$) is the closest to our results. Notice that the both AOFMs correspond to the type IX of AO interactions in the $YZ$ plane. On the other hand, according to our results, the slow AW then propagates at the angle $-78\mathrm{deg}$ with respect to the $Y$ axis, instead of $-46.5\mathrm{deg}$ as declared in Ref. [4]. The difference between our calculation results and those presented in Ref. [4] can be caused by different elastooptic coefficients used. For example, we have the values $p_{33}=0.118$ [4] and $p_{33}=0.141$ in our recent work [5], $p_{41}=-0.109$ in Ref. [4] and $p_{41}=-0.051$ in Ref. [5], while for the coefficient the values $p_{14}=-0.052$ and 0.057 have been reported in Refs. [4] and [5], respectively. Nonetheless, no relations for the EEC have been presented in Ref. [4], which makes it impossible to clarify the reasons for the difference of the experimental geometries. Besides, we note that the AOFM value obtained using our method with a set of elastooptic and the other constitutive coefficients used in Ref. [4] is equal to $11.2\times {10}^{-15}{\mathrm{s}}^3/\mathrm{kg}$ but not $15.9\times {10}^{-15}{\mathrm{s}}^3/\mathrm{kg}$ for the interaction geometry mentioned in Ref. [4]. This difference is caused by the inaccuarate value of AW velocity taken in Ref. [4], i.e., 3536 m/s instead 3967 m/s.

Now let us compare the results of our analysis with the experimental data available in the literature. Unfortunately, the experimental data on the relative AOFM values under the conditions of anisotropic diffraction in ${\mathrm{LiNbO}}_3$ is poor. As far as we know, the only results are available for the case of collinear diffraction when all the three AWs propagate along the directions close to the optic axis [6]. Then the quasi-transverse AW $v_{31}$ perturbs the refractive indices and induces the elastooptic coefficient $p_{14}$, while the incident and diffracted optical waves have the polarizations parallel to the $X$ and $Y$ axes, respectively. In our notation, this diffraction belongs to the type VIII (${\varphi }_Z=0\mathrm{deg}$ and $\theta =\theta +\gamma \approx 90\mathrm{deg}$). The AOFM obtained in Ref. [6] is equal to $2.92\times {10}^{-15}{\mathrm{s}}^3/\mathrm{kg}$. Notice that, exactly under the condition $\theta =\theta +\gamma =90\mathrm{deg}$, the anisotropic diffraction is impossible in principle, since we deal with the direction of the optic axis. Following from Eq. (7), the elastooptic coefficient under this interaction geometry is equal to the EOC $p_{14}$, as it has been noticed in Ref. [6]. According to our data [5], the coefficient $p_{14}$ is equal to $0.057\pm 0.004$, whereas Ref. [6] reports the EEC equal to 0.070, basing on the Dixon–Cohen method. Hence, accounting for the experimental errors $(\pm 10\%)$ testifies a good agreement between our analytical results and the experimental data [6].

4. CONCLUSIONS

In the present work we have developed a method for the analysis of AOFM anisotropy, which is valid for the case of anisotropic diffractions in the crystals belonging to the point symmetry groups 3 m, 32 and $\overline{3}\mathrm{m}$. We have performed our analysis on the example of ${\mathrm{LiNbO}}_3$ crystals. The relations for the EEC and the AOFM have been obtained for the three types of anisotropic AO interactions. We have shown that the maximal AOFM proper for the type VII of AO interactions with the quasi-longitudinal AW is equal to $2.6\times {10}^{-15}{\mathrm{s}}^3/\mathrm{kg}$. This type of interaction is realized when in the $Y{Z}^{\prime }$ plane inclined at the angles ${\varphi }_Y=40$ and 140 deg in respect to $YZ$ plane. At the type VIII of AO interactions with the AW QT1, which is polarized in the plane of interaction, the maximum AOFM ($12.9\times {10}^{-15}{\mathrm{s}}^3/\mathrm{kg}$) is achieved in the interaction plane $X{Z}^{\prime }$ rotated around the $X$ axis by 40 or 220 deg. Then the AW propagates at the angle 182.6 deg with respect to the $X$ axis, while the incident optical and diffracted optical waves, propagate at 280.0 deg and 260 deg, respectively, to the $X$ axis. For the type IX of AO interactions, with the AW termed as QT2 and polarized perpendicular to the interaction plane $Y{Z}^{\prime }$ rotated on the angle 10 deg around $Y$ axis the maximum AOFM is equal to $12.6\times {10}^{-15}{\mathrm{s}}^3/\mathrm{kg}$. It should be noted that principal maximum AOFM obtained for VIII and IX types of interaction as well as maximum value of AOFM within $YZ$ interaction plane for IX type of interaction are the same with accounting of the total errors of measurements and the mean value of error of calculation which is about 25%.