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Abstract
A matrix method for tensor transformations in Voigt notation known from the elasticity calculations has been applied to elasto-optical calculations. Using invariant tensor-to-matrix mapping, the second- and fourth-rank Cartesian tensor transformations and basic operations can be performed by means of matrix multiplication. This approach brings what we believe is a new method of correct tensorial operations in Voigt notation, replacing a well-known approach that allocated specific constants to some matrix elements to even up the difference between the tensor transformations and $6 \times 6$ matrix operations. This general approach also simplifies the use of elasto-optical calculations for an arbitrary crystal class in an arbitrary orientation.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
One of the serious issues that limits the efficient high-power operation of diode-pumped solid-state lasers [1,2] is the thermal effect resulting from the deposition of heat due to physical phenomena such as quantum defect, non-unity quantum efficiency, concentration quenching, and upconversion in laser amplifiers or simple absorption in other optical components. An inhomogeneous deposition of the heat or a spatially nonuniform heat removal leads to the formation of thermal gradients, which cause changes to the refractive index and also cause mechanical stresses through the thermal expansion of the material. Local mechanical stresses force the material to become birefringent via the effect, which generates a change in the polarization state of the laser beam and leads to a significant decrease in the output power and a marked change in the beam intensity profile on the polarization-sensitive optical components.
Since the elastic and elasto-optical properties of the crystals are characterized by fourth-rank tensors, the Voigt formalism mapping the tensors to matrices is usually used. The Voigt notation representing symmetrical fourth-rank tensors by $6 \times 6$ matrices and the second-rank tensors by $6 \times 1$ matrices is usually used to simplify the tensor operations. However, the mapping of the tensors to the matrices with a reduced number of elements introduces specific multiplicative constants to some matrix elements to even up the difference between the tensor and $6 \times 6$ matrix operations and transformations. These constants have been assigned according to the set of rules introduced by Nye [3], for example, for each specific tensor. The operation with these constants for more complex tensorial operations is unpractical and often leads to mistakes. To the best of our knowledge, the first “elasto-optical” paper that used the transformation of the piezo-optic matrix from crystallographic to laboratory for YAG cubic crystal transformed from $ [ {001} ] $ orientation to $ [ {111} ] $ frame was written by Lü et al. [4]. The paper, however, introduced an error in the above-mentioned multiplicative coefficients. This form of the piezo-optic matrix was then also referenced by other authors [5,6]. The correction of this mistake has been presented by Chen et al. [7], and their version of the matrix was later also used by other authors [8,9]. Unfortunately, there was another mistake in the coefficients and therefore the form of the transformed piezo-optic matrix was, again, incorrect. A summary of the numerical differences of those matrix components was provided in [10], where the correct form of the matrix was also derived. Later, the correct form of the transformed matrix also was used in [11].
Another calculation has been accomplished for cubic $ {\text{Yb:CaF}_{2}} $ crystal in [12]. The latter calculation will be further analyzed in Discussion section.
A method reducing all the addition of the multiplicative constants to the matrix multiplication operations will be introduced. The usefulness of such matrix representation of Cartesian tensor-to-matrix mapping already has been demonstrated in elasticity calculations [13] and its usage in elasto-optics will be demonstrated in this paper.
2. VOIGT NOTATION
Let us first specify the notation that allows us to distinguish unambiguously among the mathematical structures used within the article. It will be needed to distinguish between second- and fourth-rank tensors and their representation by 6-vector or $6 \times 6$ matrices. For example, let us represent the fourth-rank tensor by the overlined letter $ \overline s $ and its components as $ {s_{\textit{ijkl}}} $, where $ i,j,k,l = 1,2,3 $. In some cases, which will be described later, this tensor can be represented by a $6 \times 6$ matrix that will be denoted $ {[ s ]} $ with the components $ {[ s ]} _{\alpha \beta } $, where the Greek indices $ \alpha ,\beta = 1,2, \ldots ,6 $. The complete notation is summarized synoptically in Table 1.
Table 1. Notation Overview
Generally, the calculations with fourth-rank tensors, as well as their transformations, are rather tough and their use is impractical for numerical calculations. Fortunately, in linear elasto-optics and if body torques are absent, one deals with the fourth-rank tensors that possess both minor diagonal symmetries which, in general, are given by the relations
the minor diagonal symmetric fourth-rank tensors having only 36 independent elements out of 81. These symmetries allow us to represent them by $6 \times 6$ matrices by the mapping of the indices,It should be noted that some of the tensors (e.g., elastic stiffness $ \overline c $ and elastic compliance tensor $ \overline s $) also possess major diagonal symmetry given by
This kind of symmetry leads to the representation by a symmetrical $6 \times 6$ matrix.The symmetrical second-rank tensors are represented by ${6 \times 1}$ vectors (demonstrated on the stress tensor components $ {\sigma _{\textit{ij}}} $) as [3]
For the fourth-rank tensors, the first pair of the indices is reduced to one index, which runs from 1 to 6, according to Eq. (3). The second pair of indices is transformed in the same way.
It is important to pay attention to the fact that those matrices are not tensors and therefore their components do not transform as true tensor components. It is necessary to transform the components correctly as a component of a fourth-rank tensor. The identity symmetrical (symmetrized with respect to the minor diagonals) fourth-rank tensor $ \overline I $ is given as [14]
The basic tensor operations transferred to the matrix formalism were derived in [13] and are listed in Table 2. As can be directly seen from Table 2. all the basic tensor operations have been reduced to the matrix operations with some corrections provided by the matrices $ {[ I ]} $ and ${[I]}^{-1} $ given by Eqs. (6) and (7), respectively. These corrections represent the multiplicative factors introduced by Nye [3].
The change of the basis (coordinate system) given by the transformation of the vector components $ {a_{\textit{ij}}} $,
yields for the second-rank tensor and for the fourth-rank tensor. The transformation described by $ \hat a $ is orthogonal; i.e., its matrix obeys $ {{\hat a}^{ - 1}} = {{\hat a}^T} $. In the matrix notation, this transformation is given by and where $ {[ A ]} $ is given byGenerally, the matrix $ {[ A ]} $ is no longer orthogonal and therefore its transpose $ { {[ A ]} ^T} $ cannot be exchanged for its inverse $ { {[ A ]} ^{ - 1}} $.
Table 2. Transformation of the Basic Tensor Operations
The transformation matrix of an inverse transformation $ { {[ A ]} ^{ - 1}} $ can be obtained by taking advantage of the coordinate system transformation matrix orthogonality, so the inversion of Eq. (10) is
Applying the same procedure as in the case of the generation of the Eq. (13) transformation matrix, one getsAt this point all the necessary operations have been defined in Voigt notation and one can investigate the operations used in the elasto-optical calculations.
3. HOOKE’S LAW
According to Table 2, the generalized Hooke’s law, relating the strain $ \hat \varepsilon $ and stress $ \hat \sigma $ tensor via the elastic stiffness tensor $ \overline c $, for small deformations of the body-torques free solid can be transformed to Voigt notation as
This redefinition leads to exactly the same coefficients in the strain vector and elastic compliance matrix, which have been derived in Eq. (3). However, the fully matrix form derived here is much more suitable for numerical calculations. Another conclusion that can be drawn from the last row in Table 2 is how to evaluate the elements of the compliance tensor, since it is usually the elastic stiffness matrix which is provided for a given material in the literature. The evaluation yields
This formula can be validated by the direct inversion of elastic stiffness in the full tensor form, for example, for the m3m point group symmetry cubic crystal class. The inversion procedure for fourth-rank symmetric (possessing both a minor and major diagonal symmetry) tensor has been described in [14] using the decomposition of the fourth-rank tensor to the linear combination of three symmetric linearly independent fourth-rank tensors. It was also shown that the inverse tensor can be obtained simply by replacing each coefficient of the linear combination of these three linearly independent fourth-rank tensors by its reciprocal. This procedure provides the elements of elastic compliance tensor, which can be obtained asIf it is necessary to use the stiffness matrix $ {[ c ]} $ in some other coordinate system published in literature $ {[ {c^\prime } ]} $ the transformation needed will be, according to Eq. (12),
where $ {[ A ]} $ is the transformation matrix (13). The transformed elastic stiffness matrix then can be naturally used for the evaluation of the elastic compliance matrix in a proper coordinate system using Eq. (20).It is important to point out that the coordinate transformation in Eqs. (11) and (12) is valid only for matrices that have not been redefined and for redefined matrices that are not changed by redefinition (i.e., mechanical stress $ {\overline {\{ \sigma \}} } $, elasticity matrix $ {\overline {[ c ]} } $, elasto-optical matrix $ {\overline {[ p ]} } $, and impermeability matrix $ {\overline {\{ \eta \}} } $). If the redefinition of the matrix contains the coefficient matrix $ {[ I ]} ^{ - 1} $ then the transformation matrix $ {[ A ]} $ positioned at the same place as the coefficient matrix must be inverted and transposed (i.e., $ {[ A ]} {\{ \varepsilon \}} \to {[ A ]} ^{ - T} {\overline {\{ \varepsilon \}} } $, $ {[ A ]} {[ s ]} {[ A ]} ^T \to {[ A ]}^{ - T} {\overline {[ s ]} } {[ A ]}^{ - 1} $, and $ {[ A ]} {[ \pi ]} {[ A ]} ^T \to {[ A ]} {\overline {[ \pi ]} } {[ A ]} ^{ - 1} $, where $ {[ A ]} ^{ - T} $ denotes the transpose of $ {[ A ]} ^{ - 1} $ given by Eq. (15). (Note: The $ {\overline {[ \pi ]} } $ matrix will be defined in Section 6).
4. DIELECTRIC IMPERMEABILITY
The optical properties of the materials at optical frequencies are determined by the dielectric tensor $ { \epsilon _{\textit{ij}}} $, which reduces to the dielectric constant for an isotropic material and generally can be the function of the spatial coordinates in an inhomogeneous medium. The dielectric tensor in linear optics is defined as the transformation tensor between the electric intensity field $ {\bf E} $ and the electric displacement field $ {\bf D} $, so
It is also convenient to define the relative dielectric impermeability tensor $ {\eta _{\textit{ij}}} $ as an inverse of $ { \epsilon _{\textit{ij}}} $, so
The dielectric tensor and the dielectric impermeability tensors or optically transparent medium are due to the energy conservation, real symmetric (or Hermitian in the presence of circular birefringence) second-rank tensors. Such tensors can be represented by quadrics as
where $ i,j = 1,2,3 $. In the principal axes, those quadrics are ellipsoids. In terms of the index of refraction, the ellipsoid of wave normals or indicatrix of the refractive index is given as where the index of refraction $ n $ is defined in the principal dielectric axes system as $ {n^2} = \epsilon \unicode{x00B5}$. Taking into account that the relative magnetic permeability $\mu $ is close to unity for most of the optical materials, the index of refraction is The individual indices of refraction for a given directions are given by the lengths (or their inverted values) of the semiaxes of the above-mentioned ellipsoids. It can be shown [7] that the principal axes of the indicatrix are the same as those for the dielectric impermeability tensor. In terms of the eigenvalues $ {\Lambda _i} $ of the impermeability tensor, the indices of refraction are given as5. IMPERMEABILITY OF UNSTRESSED CRYSTAL
Henceforth, let all the quantities expressed in the laboratory frame $ ( {x,y,z} ) $ be denoted by a simple letter (e.g., $ {\{ \sigma \}} $), while the quantities considered in the crystallographic coordinate system $ ( {x^\prime,y^\prime,z^\prime} ) $ will be denoted by primed letters (e.g., $ {\{ {\sigma ^\prime} \}} $). The coordinate system transformation represented by the matrix $ {[ A ]} $ will be considered to transform the vector components from crystallographic to laboratory frame (e.g., $ {\{ \sigma \}} = {[ A ]} {\{ {\sigma ^\prime} \}} $).
Some crystals are birefringent naturally even if no mechanical stress is applied. This means that the impermeability tensor for unstressed crystal $ \eta _{\textit{ij}}^0 $ must be evaluated for various crystal classes. The simplest case is the isotropic medium or cubic crystal. In these cases the indicatrix is a sphere and therefore there is no double refraction. The impermeability tensor in this case is given by
The situation changes if one considers an optically uniaxial crystal (hexagonal, tetragonal, trigonal). In this case the indicatrix takes the form of single-axial ellipsoid. The impermeability tensor here (in principal axes, optical axis oriented in the $ z $ direction) is given as
The most complex case occurs for the biaxial crystals (orthorhombic, monoclinic, triclinic) with three different principal indices of refraction. The indicatrix in this case is represented by a biaxial ellipsoid and the impermeability tensor in the principal axes is given as
The conditions for the indicatrix principal axes for the particular crystal classes given by the crystal symmetry are summarized in Table 3 [15].
Table 3. Symmetry Constraints on the Optical Indicatrix
The impermeability tensors described above are all diagonalized in the principal axes. Hence, if the beam is not propagating along the principal axis corresponding to $ \eta _{33}^0 $ element, the tensor must be transformed to the proper coordinate system. This transformation is, naturally, not needed in the case of an isotropic or cubic medium. The transformation can be done using the matrix $ {[ A ]} $ and Eq. (11).
6. PHOTOELASTIC EFFECT
The photoelastic effect causes changes in the dielectric impermeability tensor due to the presence of the strain $ { \epsilon _{\textit{ij}}} $ (or stress $ {\sigma _{\textit{ij}}} $) in the material. It was first proposed by Friedrich Carl Alwin Pockels in 1893, whose theory is based on two basic postulates: that homogenous deformations will change the indicatrix of refractive index and that the stress within the elastic limit induces the change of the optical properties, which can be represented by the linear function of the stress tensor components [16].
The change of impermeability tensor components based on these assumptions is given as
Taking advantage of the general symmetry relations of the crystals, the Voigt notation can be used. In this case, the photoelastic and piezo-optic tensors are transformed to $ 6\times 6 $ matrices, thus reducing the number of elements generally from 81 to 36 [3]. It should be noted that in contrast to elastic tensors, elasto-optic tensors do not generally possess the major diagonal symmetry and so their representative $ 6\times 6 $ matrix does not need to be symmetrical for all crystal classes. Taking into account the transformation rules for the tensor operations from Table 2, Eq. (33) becomes
Bearing in mind that the compliance matrix is given by Eq. (20), the matrix for the piezo-optic tensor elements will be It is usually necessary to transform the piezo-optic tensor to another basis, since the crystallographic coordinate system of tabulated material properties are not always in the same coordinate system as the laboratory frame with $ z $-axis oriented along the beam propagation direction. This transformation can be done using the transformation matrix $ {[ A ]} $ defined in Eq. (13) asThe change of the dielectric impermeability $ \Delta \eta $ is given by Eq. (32). By the transformation to the matrix notation, one gets
The total impermeability value is given as the sum of unstressed crystal impermeability $ {\{ {{\eta _0}} \}} $, specified in the previous section, and the impermeability change induced by the mechanical stresses $ {\{ {\Delta \eta } \}} $ both in the laboratory coordinate system as
It should be noted here that the redefined impermeability matrix $ {\overline {\{ \eta \}} } $ is equal to $ {\{ \eta \}} $ without any multiplicative coefficients.
7. DISCUSSION
As a first check of the correctness of the result in Eq. (38), let us compare the two possible approaches. The first has been used for the derivation of the impermeability change and consists of the transformation of the piezo-optic tensor to the laboratory coordinates $ {[ \pi ]} \to {[ {\pi ^\prime} ]} $ and its double contraction with stress tensor $ {\pi _{\textit{ijkl}}}{\sigma _{kl}} $. The last step is the expression of the piezo-optic tensor as $ {p _{\textit{ijmn}}^\prime} c _{\textit{mnkl}}^{\prime { - 1}} $. On the other hand, it also can be a stress tensor, which could be transformed to a crystallographic frame $ {\{ \sigma \}} \to {\{ {\sigma ^\prime} \}} $ using $ {[ A ]}^{ - 1} {\{ \sigma \}} $, where $ {[ A ]}^{ - 1} $ is given by Eq. (13), and transformed with the piezo-optic tensor in the same frame expressed as in terms of elasto-optic and inversed elastic stiffness tensors. The whole resulting impermeability change then must be transformed back into the laboratory coordinates. Such approach was used by Koechner and Rice [17] in one of the first papers about thermally induced birefringence. Taking advantage of the operations listed in Table 2, the crystallographic coordinate impermeability change can be expressed as
Let us validate the results obtained by the matrix method in an example of the trigonal crystal symmetry, point groups 32, $ 3m $, $ \overline 3 m $ containing, for example, sapphire or BBO optical crystals. Using the method proposed by Walpole [14], one can calculate the nonzero piezo-optic tensor $ {\pi _{\textit{ijkl}}} $ components in a crystallographic frame as a function of elasticity $ {c_{\textit{ijkl}}} $ and elasto-optic $ {p_{\textit{ijkl}}} $ tensor components in the form shown in Eqs. (44):
Let us reconstruct the transformation of the elasticity and elasto-optic matrices presented in [12] for a cubic (m3m point group symmetry) $ {\text{Yb:CaF}_{2}} $ crystal using the matrix formalism. The elasticity matrix $ {[ c ]} $ and the elasto-optic matrix $ {[ p ]} $ have the same form for m3m symmetry; therefore, it is sufficient to demonstrate the calculation on just one matrix. For this purpose, let us use the elasto-optic matrix. Even if the crystallographic orientation is not uniquely specified, because just the orientation of the crystallographic c-axis into $ [ {110} ] $ and $ [ {111} ] $ is given, the orientation of at least one of the remaining axes is not provided even if the form of the transformed elasticity and elasto-optic depends on this entry. Nevertheless, the orientation of the crystallographic axes $ a, b, c $ can be deduced from the positions of zero elements in elasticity and elasto-optic matrices to be $ a \to [ {1\overline 1 0} ] $, $ b \to [ {00\overline 1 } ] $, and $ c \to [ {110} ] $ for $ [ {110} ] $ orientation, while it is $ a \to [ {1\overline 2 1} ] $, $ b \to [ {10\overline 1 } ] $, and $ c \to [ {111} ] $ for $ [ {111} ] $ orientation. These orientations correspond to transformation matrices for vector components, so
8. CONCLUSIONS
The precise operations with the second- and fourth-rank tensor quantities are the crucial issue for elasto-optical calculations. Because of the complexity and numerical severity of full-tensorial calculation, the reduced suffix representation of the symmetrical tensors is usually used. However, special numerical factors must be introduced to some components to maintain the appropriate relationships of physical quantities. Up until now, these factors were added according to the set of rules, which is relatively complicated to be used for more complex tensor operations and especially in numerical calculations. In contrast to this method, an alternative approach has been proposed in this paper. It uses a simple matrix multiplication that reduces the likelihood of mistakes and provides the possibility to derive the proper form for a dielectric impermeability tensor for any crystal class in arbitrary orientation. The correctness of the approach has been tested several times in the paper using a full tensorial approach in some special cases. A discussion of the results obtained and used in previously published literature is also provided.
We believe this method can speed up elasto-optical calculations, eliminate the common mistakes that arise due to the improper usage of the multiplicative coefficients needed in the matrix representation of the tensors, and make the calculations much easier for less symmetrical crystal classes.
Funding
Horizon 2020 Framework Programme (739573); Ministry of Education, Youth and Sports of the Czech Republic (LO1602); Large Research Infrastructure Project (LM2015086); European Regional Development Fund and the state budget of the Czech Republic (CZ.02.1.01/0.0/0.0/15_006/0000674).
Disclosures
The authors declare that there are no conflicts of interest related to this paper.
REFERENCES
1. P. Mason, M. Divoky, K. Ertel, J. Pilar, T. Butcher, M. Hanus, S. Banerjee, J. Phillips, J. Smith, M. De Vido, A. Lucianetti, C. Hernandez-Gomez, C. Edwards, T. Mocek, and J. Collier, “Kilowatt average power 100 J-level diode pumped solid state laser,” Optica 4, 438–439 (2017). [CrossRef]
2. B. LeGarrec, S. Sebban, D. Margarone, M. Precek, S. Weber, O. Klimo, G. Korn, and B. Rus, “ELI-Beamlines: Extreme light infrastructure science and technology with ultra-intense lasers,” Proc. SPIE 8962, 89620I (2014). [CrossRef]
3. J. F. Nye, Physical Properties of Crystals (Oxford University, 2011).
4. Q. Lü, U. Wittrock, and S. Dong, “Photoelastic effects in Nd:YAG rod and slab lasers,” Opt. Laser Technol. 27, 95–101 (1995). [CrossRef]
5. D. C. Brown, “Nonlinear thermal distortion in YAG rod amplifiers,” IEEE J. Quantum Electron. 34, 2383–2392 (1998). [CrossRef]
6. M. Ostermeyer, D. Mudge, P. J. Veitch, and J. Munch, “Thermally induced birefringence in Nd:YAG slab lasers,” Appl. Opt. 45, 5368–5376 (2006). [CrossRef]
7. Y. Chen, B. Chen, M. K. R. Patel, and M. Bass, “Calculation of thermal-gradient-induced stress birefringence in slab lasers–I,” IEEE J. Quantum Electron. 40, 909–916 (2004). [CrossRef]
8. H. Yang, G. Feng, and S. Zhou, “Thermal effects in high-power Nd:YAG disk-type solid state laser,” Opt. Laser Technol. 43, 1006–1015 (2011). [CrossRef]
9. O. Slezak, A. Lucianetti, M. Divoky, M. Sawicka, and T. Mocek, “Optimization of wavefront distortions and thermal-stress induced birefringence in a cryogenically-cooled multislab laser amplifier,” IEEE J. Quantum Electron. 49, 960–966 (2013). [CrossRef]
10. L. Liu, S. Guo, Q. Lu, X. Xu, J. Leng, J. Chen, and Z. Liu, “Stress-induced depolarization loss in a YAG zigzag slab,” Opt. Laser Technol. 43, 622–629 (2011). [CrossRef]
11. M. Sawicka-Chyla, M. Divoky, O. Slezak, M. De Vido, A. Lucianetti, and T. Mocek, “Numerical analysis of thermal effects in a concept of a cryogenically cooled Yb: YAG multislab 10 J/100-Hz laser amplifier,” IEEE J. Quantum Electron. 55, 1–8 (2019). [CrossRef]
12. K. Genevrier, D. N. Papadopoulos, M. Besbes, P. Camy, J.-L. Doualan, R. Moncorgé, P. Georges, and F. Druon, “Thermally-induced-anisotropy issues in oriented cubic laser crystals, the cryogenically cooled case,” Appl. Phys. B 124, 209 (2018). [CrossRef]
13. J. C. Nadeau and M. Ferrari, “Invariant tensor-to-matrix mappings for evaluation of tensorial expressions,” J. Elasticity 52, 43–61 (1998). [CrossRef]
14. L. J. Walpole, “Fourth-rank tensors of the thirty-two crystal classes: multiplication tables,” Proc. R. Soc. Lond. A 391, 149–179 (1984). [CrossRef]
15. A. Authier, ed. International Tables for Crystallography (2006), Vol. D.
16. T. S. Narasimhamurty, Photoelastic and Electrooptic Properties of Crystals (Plenum, 1982).
17. W. Koechner and D. Rice, “Effect of birefringence on the performance of linearly polarized YAG:Nd lasers,” IEEE J. Quantum Electron. 6, 557–566 (1970). [CrossRef]