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Abstract
Based on the matrices of elastic and elasto-optic coefficients, the extreme surfaces of the spatial distribution of the acousto-optic (AO) figure-of-merit М2 for langasite and calcium-tantalum gallosilicate crystals are constructed and analyzed. The determined maximal achievable values of М2 for these crystals are commensurate and twice the maximal value of М2 for quartz. The most effective AO interaction geometries and geometry of cuts for optimized AO cells are given. The investigated crystals, due to the temperature stability, excellent optical quality, resistance to environmental aggression and high optical damage threshold can be used for AO modulators of the ultraviolet spectral range.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Acousto-optic (AO) modulators based on fused or crystalline quartz are used to control optical radiation in the ultraviolet spectral region (see, e.g. [1]). This material is characterized by a set of characteristics required for acousto-optical devices: wide spectral transmission range including ultraviolet (UV) region, resistance to strong electromagnetic radiation, high thermal stability of physical characteristics, high mechanical strength and resistance to environmental aggression. In addition, crystalline quartz is a strong piezoelectric, which allows to create acoustic waves on its own piezoelectric effect, that significantly simplifies the design of the AO cell, reduces its mass and dimensions. The main disadvantages of quartz are its low acousto-optical efficiency (AO figure-of-merit М2 ∼ 2·10−15 s3/kg [2]) and relatively high attenuation of longitudinal acoustic waves [3]. Therefore, the task of finding effective AO materials for the control of optical radiation in the UV region of the spectrum is relevant.
Crystals of the langasite group [4–6] have a unique combination of physical properties important for the application of these crystals. Particularly, they are characterized by the absence of pyroelectric effect and phase transitions up to melting temperatures (1300…1500 °C [7]), temperature stability of physical characteristics, including elastic coefficients [8,9], as well as large piezoelectric coefficients and coefficients of electromechanical coupling [7]. These characteristics have made crystals of the langasite group popular for the development of acoustoelectronic devices [10–13]. Since these crystals reveal a high threshold of optical damage ∼ 1 Gw/cm2 [13] (an order of magnitude higher than the one for model electro- and acousto-optic material LiNbO3), record low attenuation of acoustic waves (several times lower than in quartz [3]), transparency in the deep UV region [7], it is advisable to study the possibility of using these crystals in acousto-optic devices.
Langasite La3Ga5SiO14 (LGS) is the most well-known gallosilicate which, however, is not without its shortcomings in terms of optical quality (langasite reveals strips of growth) and the high cost of the crystal [14]. Therefore, the efforts of researchers were aimed at obtaining of other compounds that belong to the group of langasite (there are about a hundred crystals in this group). A positive example of the synthesis was the catangasite Ca3TaGa3Si2O14 (CTGS), which predominate over the langasite in some important characteristics important for acousto-optics [15–17]. Namely, CTGS crystals reveal higher thermal stability of dielectric and piezoelectric constants, they are colorless, characterized by high optical quality and short-wavelength limit of transparency in the deep UV region (λ = 260 nm; this is ∼ 100 nm lower than for LGS) [7,13,18,19]. In [17] it was shown that CTGS outperforms LGS crystals also in terms of photoelastic characteristics, on the basis of which the acousto-optic efficiency of the crystal can be found. According to preliminary estimates, the AO figure-of-merit M2 of LGS and CTGS crystals, calculated on the basis of the largest values of the elastic-optic coefficients, is equal to 0.89 and 1.79·10−15 s3/kg, respectively [17,20]. However, this is not enough to fully characterize the AO properties of the crystal: it is necessary to investigate the spatial anisotropy of elastic-optical and acousto-optic effects and find the geometries with the maximum values of these effects. There are various ways to represent the spatial anisotropy of physical effects induced by external fields [21–25]. However, the most informative is the method of analyzing of spatial anisotropy of the AO figure-of-merit M2 using extreme surfaces [26–28]. Therefore, it is advisable to carry out a full analysis of M2 anisotropy in LGS and CTGS crystals for all possible variants of light wave interaction with acoustic wave in order to determine the maximum achievable value of the AO figure-of-merit M2 in these crystals and obtain optimal sample geometry (AO cells) for use in acousto-optic devices.
2. Method of calculation
As is known, the efficiency of Bragg diffraction of light on an acoustic wave is described by the acousto-optic figure-of-merit M2 determined by the general expression [26,29]:
For each direction of the acoustic wave normal $\vec{a}$ the unit vector of its polarization ${\vec{f}_q}$ is determined from Christoffel equation [31]:
The value of the AO figure-of-merit M2, as it is seen from formulas (1) – (3), depends on the directions of polarization of incident and diffracted light waves, which depend, in turn, on the direction of their propagation and directions of propagation and polarization of the acoustic wave.
The objective of our analysis is to determine such directions of propagation of light and acoustic waves (geometry of AO interaction), which correspond to the highest value of M2 for a given crystalline material. The basis of this analysis, as in [26,27], is the construction and study of the properties of extreme surfaces that visualize the anisotropy of the AO effect in crystals. Such surfaces reflect all possible maxima of M2 for the wave vector of an incident light wave passing all possible directions within a full spherical angle.
In general, the extreme surface is constructed as follows. Firstly we specify the type of incident and diffracted light waves (ordinary or extraordinary) as well as the type of acoustic wave (quasi-longitudinal, quasi-transversal fast or quasi-transversal slow). Let the possible directions of the incident light wave vector in the crystal can change in the full spherical angle, i.e. correspond to the polar and azimuthal angles θ ∈ [0; π], φ ∈ [0; 2π]. For surface construction, these angles are changed within the mentioned ranges with the small enough step (1° in our calculations). For each direction of the wave vector, i.e. for each pair of angles (θ, φ), we define the optimal direction of the acoustic wave, i.e. the direction that corresponds to the maximum of M2. At that it is taken into account that the most effective diffraction, under other constant conditions, is realized if the following relation (momentum conservation law) is fulfilled [30]:
where ${\vec{k}_\nu }$, ${\vec{k}_\mu }$ and $\vec{K}$ are the wave vectors of electromagnetic diffracted (ν), electromagnetic incident (µ) and acoustic waves, respectively. The condition (4) allows to significantly reduce the searching region of possible directions of $\vec{K}$. Indeed, the vector ${\vec{k}_\mu }$ (θ,φ) ends in some point A of the incident wave vector surface. Let’s construct the acoustic wave vector surface with the center in the point A. In accordance with the method of construction, each point of this surface corresponds to the sum ${\vec{k}_\mu } + \vec{K}$ (because of the symmetry of the wave vector surface, the case ${\vec{k}_\mu }$–$\vec{K}$ is automatically taken into account). The constructed acoustic wave vector surface crosses the diffracted wave vector surface along some line (note that the incident and diffracted wave vector surfaces coincide in the case of isotropic diffraction and differ for anisotropic one). The points of this line determine all possible directions of $\vec{K}$ satisfied the condition (4). Note that for the case of anisotropic diffraction, when the one of the wave vector surfaces corresponds to ordinary, and the other – to extraordinary wave, the condition (4) may not be fulfilled for some directions at low enough acoustic frequencies and, correspondingly, low lengths of $\vec{K}$. In such cases the distances between the incident and diffracted wave vector surfaces are higher than |$\vec{K}$| and the mentioned intersection is absent. In other cases we calculate the values of M2 along the line of intersection and determine the highest of them for the given θ and φ, M2 max(θ,φ). Repeating this calculation for all θ and φ, we obtain the dependence M2 max(θ,φ) represented by extreme surface: the direction of each radius-vector of this surface is determined by θ,φ and the length of this radius-vector is equal to M2 max. Obviously, the values of M2 max are, in general, different for different θ and φ; the highest of them, $M_{2\max }^{extr}$ is a global maximum, determination of which is the aim of our analysis. At that, because of symmetry of interaction and crystal, few equivalent global maxima are observed, determined by different angles θmax, φmax. It should also be noted, that although the determination of $M_{2\max }^{extr}$ does not require the mandatory construction of extreme surface ($M_{2\max }^{extr}$ is determined during its calculation), such a construction gives additional useful information. Particularly, the extreme surfaces allow to reveal the alternative maxima that can be more convenient from the design or technological point of view. The analysis of the extreme surfaces also allows to estimate the relative significance of the obtained maxima and the changes of its value caused by deviation of the light wave directions from the given one.3. Results and discussion
To construct the extreme surfaces of the AO figure-of-merit M2, the parameters of LGS and CTGS crystals are taken from [5,16,17,32–36] (see Table 1). All calculations were carried out for the light wavelength λ of 632.8 nm and the acoustic wave frequency f of 100 MHz.
Table 1. Parameters of LGS and CTGS crystals required for М2 calculation
Extreme surfaces are calculated and constructed for the cases of isotropic (types o → o and e → e) and anisotropic (types o → e and e → o) diffraction on acoustic waves of different polarization: quasi-longitudinal (q = 3), quasi-transverse fast (q = 2) and quasi-transverse slow (q = 1); symbols o and e denote ordinary and extraordinary light waves. These surfaces are shown in Figs. 1–3, and the results of optimization (the highest values of М2 for different types of AO interaction) are presented in Tables 2, 3. The tables show the positions of only one of the maxima М2, and the positions of the other maxima can be obtained using the symmetry operations of the point group $\bar{3}m$ (the symmetry of the LGS and CTGS crystals (32) supplemented by the center of inversion in accordance with the symmetry of interaction). Because the extreme surfaces for anisotropic diffractions of type o → e or e → o for acoustic frequencies of about 1 GHz are visually similar and, in addition, the maximal achievable values of М2 for these cases are the same (the proof of this general feature was given [26]), the results only for o → e diffraction are shown in Figs. 1–3 and Tables 2, 3.
Fig. 1. The extreme surfaces of the AO figure-of-merit М2 for LGS and CTGS crystals for the case of isotropic o → o diffraction (all in units 10–15 s3/kg); q = 3, q = 2, q = 1 denote the quasi-longitudinal and two quasi-transversal acoustic waves.
Fig. 2. The extreme surfaces of the AO figure-of-merit М2 for LGS and CTGS crystals for the case of isotropic e → e diffraction (all in units 10–15 s3/kg).
Fig. 3. The extreme surfaces of the AO figure-of-merit М2 for LGS and CTGS crystals for the case of anisotropic o → e diffraction (all in units 10–15 s3/kg).
Table 2. The results of optimization for LGS crystal (λ = 633 nm, f = 100 MHz)a
Table 3. The results of optimization for CTGS crystal (λ = 633 nm, f = 100 MHz)
As it is seen from Figs. 1 and 2, visually the extreme surfaces of М2 for LGS and CTGS crystals are not as similar as, for example, the surfaces of piezoelectric and elastic-optic effects [20]. They differ especially for the case of anisotropic diffraction (Fig. 3). This is due to the fact that these crystals differ significantly in the magnitude of the elastic constants (see Table 1). It is the elastic modulus Сmn that determine the magnitude and spatial anisotropy of the velocities of acoustic waves V in the crystal, and, consequently, make a significant contribution to the anisotropy of М2 (see formula (1)).
In general, the AO figure-of-merit М2 of catangasite crystals is higher (see Tables 2, 3) and only for the case of isotropic е → е diffraction on the transversal acoustic wave the values of М2 for LGS are higher than the ones for CTGS. The value of M2 for CTGS crystals for diffraction on a quasi-longitudinal acoustic wave is twice or an order of magnitude higher than the value of М2 for LGS crystals. However, the highest achievable value of М2 for CTGS – 4.1·10−15 s3/kg (е → е diffraction on the fast quasi-transversal acoustic wave) is 13% lower than the one found for LGS – 4.7·10−15 s3/kg (diffraction of the same type) (the corresponding rows are marked by bold in Tables 2, 3). At the same time, the obtained value for CTGS is twice as high as M2 for quartz. Despite the maximal achievable value of M2 for CTGS is slightly lower than the one for LGS, catangasite crystals are superior to langasite in optical quality and lower short-wavelength limit of the spectral transmission region. Therefore, they can undoubtedly be used as sensitive elements for AO light modulators in the UV spectral region (the advantages of CTGS crystals over quartz used for AO light modulation in the UV region are detailed in the Introduction).
In Fig. 4 it is shown the variants of optimized AO cells from LGS and CTGS crystals and AO interaction geometry, i.e. directions of propagation and polarization of acoustic and optical waves. For LGS the figure is drawn on the basis of data of Table 2 (highlighted), and for CTGS based on optimized values from Table 3 (highlighted) using the symmetry operation of the $\bar{3}m$ point group (rotation around the X3 axis by 120°); the direction of light propagation in the AO cell is tied to one of the main crystal physics axis X2.
Fig. 4. The geometry of the crystal sample and the orientation of the vectors of acoustic and electromagnetic waves that ensure the highest value of the AO figure-of-merit M2: a) for the crystal of langasite, b) for the crystal of catangasite. Here a and f are the directions of propagation and polarization of the acoustic wave, k is the direction of propagation of the incident light wave and i – its polarization.
4. Conclusions
Extreme surfaces of the spatial distribution of the AO figure-of-merit M2 are constructed on the basis of matrices of elastic-optic coefficients of langasite (LGS) and catangasite (CTGS) crystals. The analysis of the anisotropy of these surfaces is carried out and the values of the maxima of these surfaces and the corresponding geometries of the AO interaction (directions of propagation and polarization of light and acoustic waves) are determined.
The investigated LGS and CTGS crystals have rather high maximum values of the AO figure-of-merit M2 = 4.7·10−15 s3/kg та M2 = 4.1·10−15 s3/kg, respectively, and can be used as sensitive elements in AO modulators for UV spectral region, because these values of the M2 are twice the value of M2 for quartz crystals used in acousto-optic devices.
The previously determined [17,20] maximum values of M2 for these crystals on the basis of the highest elastic-optic coefficients or maximum values of the elastic-optic effect are significantly lower. In particular, in [20] it was found that the maxima of M2 are 0.89 and 1.79·10−15 s3/kg for LGS and CTGS, respectively. Thus, based on the analysis of the spatial anisotropy of the extreme surfaces of the AO figure-of-merit M2, the geometries of the AO interaction with significantly higher M2 values were found: ∼ 5 times for LGS and 2.3 times for CTGS. This confirms the effectiveness of the method of analysis of the anisotropy of the AO effect on the basis of extreme surfaces of M2, which take into account the anisotropy of all necessary parameters described by tensors of higher ranks, see formulas (2) and (3).
Based on the geometries of the AO interaction, which correspond to the above mentioned maximum values of М2, the geometries of the samples are proposed, which will have high AO efficiency.
Funding
H2020 Marie Skłodowska-Curie Actions (grant agreement No 778156); National Research Foundation of Ukraine (Project 2020.02/0211).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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