Birefringence effects can have a significant influence on the polarization state as well as on the transversal mode structure of laser resonators. This work introduces a flexible, fast and fully vectorial algorithm for the analysis of resonators containing homogeneous, anisotropic optical components. It is based on a generalization of the Fox and Li algorithm by field tracing, enabling the calculation of the dominant transversal resonator eigenmode. For the simulation of light propagation through the anisotropic media, a fast Fourier Transformation (FFT) based angular spectrum of plane waves approach is introduced. Besides birefringence effects, this technique includes intra-crystal diffraction and interface refraction at crystal surfaces. The combination with numerically efficient eigenvalue solvers, namely vector extrapolation methods, ensures the fast convergence of the method. Furthermore a numerical example is presented which is in good agreement to experimental measurements.
© 2015 Optical Society of America
Anisotropic, linear and homogeneous crystals are commonly used in laser resonators. They are introduced e.g. for polarization control [1, 2], light amplification [3, 4] or transversal resonator mode shaping [5, 6] in practice. Several experimental investigations (see e.g. in [7, 8]) of real lasers showed that birefringence effects caused by anisotropic crystals inside the laser cavity might have a significant influence on the transversal mode structure and polarization state of the emitted laser light. However as discussed in Section 2 the existing state-of-the-art theoretical models for the computer-aided analysis of such birefringence effects in complex laser resonators are limited in their practical applicability. To give further insight into the theoretical understanding of laser resonator modes, we will introduce in this work a flexible and numerically efficient simulation technique, which will allow the analysis of intracavity birefringent effects on the polarization state and the transversal shape of the fundamental laser resonator mode. It should assist experimental researchers to further improve their high quality lasers.
There are mainly two effects which non-absorbing, linear, homogeneous, anisotropic crystals may produce on the resonator mode inside a laser cavity: light refraction at the crystal front and rear surfaces and light diffraction inside the crystal along the light propagation direction. The latter we will call intra-crystal diffraction in the following. We represent the dominant transversal resonator eigenmode by a fully vectorial complex amplitude V (r, ω0) = (V1, V2, …, V6)T = (EX, EY, EZ, HX, HY, HZ)T of a single monochromatic electromagnetic field, varying in space with the 3-dimensional position vector r = (ρ, Z)T = (X, Y, Z)T and in time with the angular frequency ω0. To describe the interaction of the field with the anisotropic crystal, the following dielectric tensor is used:
In Section 3, Eq. (1) is used to develop a simulation technique for the fully vectorial transversal resonator eigenmode calculation of a laser resonator including homogeneous, anisotropic media. Furthermore, major differences with respect to the state-of-the-art simulation techniques, which are reviewed in Section 2, are discussed. In Section 4 a numerical example is presented. It is in good agreement with experimental measurements from literature.
2. State-of-the-art simulation techniques for intracavity birefringence effects
In principle there are two different groups of physical models to simulate birefringence effects which are related to laser resonators: Single pass simulation approaches, which are just calculating light propagation through an anisotropic medium once, and intracavity approaches which calculate the light propagation through the anisotropic media several times, due to multiple reflections of light at the resonator mirrors.
2.1. Single pass simulation
The former one, which is mainly used in the context of laser rod analysis, is typically based on beam propagation methods (BPM) to simulate a single pass through the active laser medium. Therefore the anisotropic, coupled paraxial wave equations [9, 10]9, 11] or the finite-difference BPM [10, 12]. Here is the transversal nabla operator, k0 is the wavenumber in vacuum, n0 is the linear isotropic refractive index share of and is defined by the slowly varying envelope approximation
The advantage of these approaches is that in the dielectric tensor components εab, besides of homogeneous anisotropy, also inhomogeneous thermal- and stress- induced anisotropy, thermal lensing as well as nonlinear (isotropic and anisotropic) gain can be included. However from the birefringence analysis of a single pass, only some birefringence effects of the overall laser might be deduced. Nevertheless due to the restriction to a single pass propagation, the transversal mode structure of the resonator cannot be calculated. This calculation would require the performance of several BPM resonator round trips, which was, to our knowledge, up to now never reported for anisotropic media in literature. The reason is probably the high computational effort of the BPM techniques, caused by the limited step increment along the light propagation direction z.
2.2. Intracavity approaches
For intracavity approaches we have shown in our recent publications [13–15] that the calculation of the dominant transversal mode in a plane of interest z0 of any resonator setup can be formulated by the generalized Fox and Li algorithm in the field tracing notation
Furthermore it was discussed in detail in , that depending on the structure of the single round trip operator , Eq. (5) transforms into an eigenvalue problem with coupled polarization components so that γ1 = γ2, or degenerates into two separate eigenvalue problems, one for each polarization component. Consequently it is feasible to call γℓ eigenvalues with the corresponding eigenmodes Vℓ in the following. Also it was shown in [13, 14] that we can use the field tracing operator concept to describe a single round trip through the resonator by a sequence of optical component operators and free-space propagation operators between the optical components so that we get
Here m = 1…n represents the index of the optical component. Figure 1 illustrates the different components and propagation operators for an exemplary resonator containing several isotropic and anisotropic optical elements.
It was shown in  that, if the arbitrary plane z0 where the transversal mode should be calculated, is located in an isotropic media, the remaining field components V3, V4, V5 and V6 can be calculated on demand directly by Maxwell’s equations from the solutions V1 and V2 of Eq. (5). However this means that Eq. (5) is not restricted to resonators including isotropic media only. It is possible to use component operators in certain subdomains of the resonator which handle all three electric and even all three magnetic field components at once. However once the propagation through the anisotropic media is solved, again V1 and V2 are sufficient for a unique representation of the electromagnetic field in the isotropic media. Consequently most essential for the simulation of birefringence effects on the transversal resonator mode are the component operators describing the single pass light propagation through the anisotropic media. Obviously split-step or finite-difference BPM methods discussed above could be an option. However as already mentioned, this techniques suffer from the high numerical effort required for the sampling of the field along the propagation axis z.
In literature mainly three other techniques are used: the Jones matrix approach [17–20], mode expansion methods [21–25] or rigorous Maxwell solvers [26–28]. For the latter ones the complete round trip operator is replaced by discretization techniques of the wave or Maxwell’s equations. These techniques, like the Finite Element Method (FEM) or the finite difference time domain method (FDTD), are limited in their practical application only to micro resonator cavities, due to the high numerical effort. Methods based on the mode expansion of the electromagnetic field inside the laser cavity in combination with coupled mode theory are typically applied in literature for microscopic vertical-cavity surface-emitting lasers (VCSEL). These techniques are numerically suitable as long as the cavity geometry or intracavity components do not cause a strong coupling between the expanded modes. Dependening on the mode expansion applied, this is typically the case for radially symmetric cavities where just planar intracavity components are used (e.g. thin-film stacks) or for transversally periodic cavity structures (e.g. photonic-crystal structures). However for realistic macroscopic laser resonators with plenty of different intracavity components these mode expansion techniques suffer from a high numerical effort due to the appearance of strong mode coupling. That is probably why in the past the numerically efficient Jones matrix approach became the most popular technique for analysis of birefringence effects in laser resonators. In the field tracing notation the component operator represented by a Jones matrix can be written as:
However Jones matrices are just an idealized model of anisotropic components, assuming normal incidence plane wave illumination. Only phase shifts Φaa caused by optical path differences between the different polarization components, Fresnel transmission or reflection losses taa and simplified polarization crosstalk pab are taken into account. Intra-crystal diffraction and refraction effects at the crystal surface are neglected. Furthermore the Jones matrix approach assumes the effect of the anisotropic media on the electric field in a single infinite thin plane.
In the following chapter we will introduce a fast Fourier Transformation (FFT)-based component operator for the efficient propagation of light through homogeneous anisotropic media. In combination with Eq. (5) and Eq. (6) it represents the eigenvalue problem to be solved for the calculation of the fully vectorial transversal eigenmode V in a resonator containing one or more homogeneous anisotropic media. The algorithm for the solution of the resulting eigenvalue problem itself is based on vector extrapolation methods, which will not be discussed in detail here. It was already discussed in [15, 29, 30].
3. Component operator for the light propagation through homogeneous anisotropic media
We use an angular spectrum of plane waves approach for modeling monochromatic electromagnetic fields propagation through arbitrarily oriented anisotropic media, within the framework of the field tracing operator concept. Please note that the following component operator is a generalization of the work given by Landry and Maldonado . In  only the interaction of a single plane wave at a single plane interface between two anisotropic or one anisotropic and one isotropic media was discussed. We have extended this method to general incident beams, by combining their method with the angular spectrum of plane waves. Especially the expressions in  are formulated in the plane of incidence, where it is possible to obtain a y-invariant case. In our work, all formulations are done in a general coordinate system, which is needed to deal with general incident beams. Furthermore we show that this angular spectrum of plane waves representation of light is very suitable to rigorously propagate the light further through linear and homogeneous crystals. Figure 2 shows the task of the component operator for light propagation through a linear, homogeneous and anisotropic medium. Applied on the incident field Vi(ρ, zin) given in the plane zin in front of the front crystal surface the operator should calculate the electromagnetic field components Vt(ρ, zout) directly in the plane zout behind the rear crystal surface. Please note that inside the anisotropic medium it is more convenient to propagate not just the two electric field components EX and EY, but the three-dimensional field vector (EX, EY, EZ)T, by using the following approach. However if we assume that the light is entering the anisotropic media from an isotropic media and is leaving the crystal into an isotropic media, it is mathematically sufficient to formulate a unique input-output problem by
Consequently to solve Eq. (11) the propagated angular spectrum has to be known. Therefore in the following subsections we will discuss the light refraction and propagation of the plane waves through the homogeneous, anisotropic media to construct . In what follows the field components of each plane wave with the direction κ will be denoted as to keep their physical meaning in mind. For better readability we have skipped the variable κ in the following.
3.1. Plane wave refraction at plane interface from εi into
A plane wave component from the incident field can be extracted in Eq. (9) and expressed in its vectorial form by31], on the plane interface between isotropic and anisotropic media, an incident plane wave causes one reflected plane wave which has the form
For a plane wave with a given wavevector in an anisotropic medium, its electric field vector direction is uniquely determined. As a consequence in Eq. (16), the complex electric field vector must be expressed as , the product of magnitude αt and normalized electric field eigenvector , and the same for . In Eq. (15) and Eq. (16), the wavevectors kr, and , as well as the electric field vectors εr, and are unknown. We solve the unknown quantities in the following sequence:
- Step I – determine the unknown wavevectors kr, and by using phase matching conditions at the plane interface;
- Step II – find out the electric field eigenvectors and by solving the coupled wave equations in the form of an eigenvalue problem;
- Step III – apply boundary conditions to obtain the reflected electric field vector εr as well as the magnitudes αt and βt of the transmitted plane waves.
We present the algorithm workflow in Fig. 3, as an overview of the calculation process, while discussion in detail follows afterward.
3.1.1. Calculation of unknown wavevectors (Step I)
To calculate the unknown wavevectors, we use the phase matching condition. Consequently the transverse components of the wavevectors κ on both sides of the interface must be equal, as is illustrated in Fig. 4. Therefore there is32, 33], sometimes referred to as the Booker quartic , can be found in the form
Solving Eq. (18) in general yields four solutions with j = 1, 2, 3, 4. Consequently there are four wavevectors and four refractive indices . As Yeh argued in , only two of the four wavevectors correspond to energy transmission into +z direction. To pick them out, we need to examine the time-averaged Poynting vector associated with a wavevector . For the calculation of the associated electric field vector needs to be calculated first.
3.1.2. Determination of electric field eigenvectors (Step II)
Following the treatment in  and using the plane wave ansatz , the waveequations in a homogeneous anisotropic medium with the dielectric tensor can be formulated as an eigenvalue problem33]. According to Eq. (20), is an eigenvector of matrix corresponding to the eigenvalue . Numerically we normalize the eigenvector to as defined in Eq. (16). When calculating the eigenvalues of the 3 × 3 matrix , three values , and are obtained. One of them equals zero and is therefore non-physical, while the other two non-zero values correspond to the two refractive indices of the crystal for the direction . Since we have previously fixed the wave vector by solving Eq. (18) and determined the refractive index , we should pick up the eigenvalue to that.
Then we calculate the time-averaged Poynting vectorEq. (16) as and , and and respectively. Please note that in Eq. (16) αt and βt still are unknown, as well as εr in Eq. (15).
3.1.3. Application of boundary conditions (Step III)
In the last step the boundary conditionsEq. (23) can be written in the matrix form
Multiplying the inverse matrix on both sides of Eq. (25) leads to
Now the unknown quantities , αt and βt in Eq. (15) and Eq. (16) can be determined, since the right-hand side of Eq. (27) is known. From Eq. (26) we know that is a non-diagonal matrix, and consequently its inverse is also non-diagonal. Therefore polarization crosstalk between different electric field components may happen. As a result, the operator in Eq. (8) and the round trip operator are non-diagonal operator matrices.
3.2. Plane wave propagation inside anisotropic media
The propagation of a plane wave inside the anisotropic medium is completely governed by Eq. (16). Once the electric field vectors and are known, the plane waves can be calculated at arbitrary positions inside the anisotropic medium. For a propagation over a distance d = zout −zin, we just need to substitute z = d in Eq. (16). By applying it to all of the plane wave components, the angular spectrum of plane waves (SPW) operator in the anisotropic medium can be obtainedEq. (28), the angular spectrum for any κ consists of two parts. The propagation of these two parts follows different rules, i.e., multiplication with or . Therefore during the propagation in an anisotropic medium, the polarization state may change. Other existing propagation methods in isotropic media based on the angular spectrum of plane waves like the Fresnelintegral, can also be extended to anisotropic cases in a similar way.
3.3. Plane wave refraction at plane interface from into εt
Finally we need to deal with refraction at the rear surface to obtain the transmitted field in the isotropic medium behind. The method is completely analogous to Section 3.1. Thus we briefly give the expressions of the plane waves and the coefficients matrix.
The incident, reflected and transmitted plane waves can be expressed as
In this case, the incident and reflected plane waves are in the anisotropic medium. The two incident plane waves in Eq. (29) share the same transverse wavevector component κ, and as a consequence there is only one transmitted plane wave in the isotropic medium εt. Following the procedure in Section 3.1, we firstly determine the unknown wavevectors. The transmitted wavevector is found as with . The reflected wavevectors and need to be calculated by using the quartic in Eq. (18). Then the associated electric field eigenvectors and can be found by solving the eigenvalue-eigenvector problem in Eq. (20). Applying the boundary conditions yields four linear equations in a matrix form
Multiplying on both side of Eq. (32) results in
For the transmitted plane wave in the isotropic medium, we end up with , which could be used in subsequent field tracing procedures.
4. Numerical example: Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal
In 2006, Yonezawa et al.  introduced a laser resonator including an anisotropic c-cut Nd:YVO4 crystal, for the generation of a radially polarized transversal resonator mode. The main idea is the induction of different optical paths between an extraordinary and an ordinary beam by the birefringence of the crystal. If the distance between a spherical and a plane cavity mirror is chosen correctly, the effective optical cavity length of the ordinary beam will be in the unstable region, whereas the optical cavity length of the extraordinary beam, which is radially polarized, is in the stable region. Consequently the resonator round trip losses, which are higher for the ordinary beam than for the extraordinary beam, will damp out the ordinary one. The corresponding resonator is given in Fig. 5 and Table 1.
In their work Yonezawa et al. gave an experimental verification of the appearance of a radially polarized beam. In the following we will use the above described field tracing techniques to reproduce Yonezawa’s experimental measurement results by a computer simulation.
Therefore we have implemented the eigenvalue problem given by Eq. (5) in the optics design software VirtualLab . The round trip operator ℛ is described according to Eq. (6) by a sequence of component operators given in Table 2 and angular spectrum of plane waves (SPW) operators [16, 35] for the free space propagation between the components. For the simulation of the birefringence effects of the uniaxial Nd:YVO4 crystal the techniques introduced in Section 3 are applied. A further polarization coupling effect is achieved by the simulation of the gain including nonlinear saturation effects. The gain is approximately simulated directly after the anisotropic operator by the diagonal thin element approximation (TEA) operator [15,36,37]Eq. (36), the round trip operator ℛ is nonlinear. Furthermore the two field components in Eq. (5) are coupled by this nonlinearity.
Please note that in this numerical example an uniaxial anisotropic medium is simulated. However the above described techniques are not limited to uniaxial crystals, but can be also applied to more general anisotropic media, described by the dielectric tensor of Eq. (1). This example was just chosen due to the existence of a measured reference resonator mode in the literature and the practical relevance of Nd:YVO4 crystals for solid state lasers.
In real stable laser resonators, standing waves (axial modes) will build up in the laser cavity due to multiple interference. Consequently the absolute phase distribution of the field components of the dominant resonator mode given in an arbitrary plane of the cavity are not changing after a single resonator round trip. This means that there is a fixed phase relation between the two field components Vℓ of the resonator eigenmode. This fixed phase relation is also important for the generation of purely radially polarized light, because a change of the phase relation between the field components will change the response of the anisotropic material. Due to the sequential formulation of the round trip operator in Eq. (5) and Eq. (6) multiple interference effects between forward and backward propagating waves inside the standing-wave resonator given in Fig. 5 are not included up to now. That is why an additional operator after each round trip operation is included to ensure the fixed phase relation between the field components. The operator introduces a (lateral) constant phase shift on each field component , which is obtained after the round trip iteration j by:15].
Once the eigenvalue problem is formulated, it is solved by a vector extrapolation method, namely the minimal polynomial extrapolation (MPE) technique, which was discussed in detail in one of our previous publications . To check the convergence of this iterative eigenvalue solver, the convergence of the algorithm is monitored using the deviation term [13,15]15] using 12 round trip iterations per cycle. The corresponding convergence of the algorithm in terms of deviation is given in Fig. 6 in comparison to the power method used in . The much faster convergence of the cycled MPE algorithm in comparison to the power method in terms of required round trip iterations can be clearly seen. The calculation was performed on a personal computer with Intel Core™ i7-4700HQ (8CPUs @ 2.40GHz). The calculation time per round trip was approximately 36 seconds if 559×559 plane waves in the angular spectrum of plane waves approach given by Eq. (9) were used. Consequently the computational grid in the spatial domain consisted also of 559×559 sampling points with an equal-distant sampling pitch of 10 microns ×10 microns. The maximum RAM memory requirement during the calculation was approximately 1.3 Gigabyte. Please note that so far no effort was made by the authors to accelerate the current code implementation of the algorithms.
The intensity of the dominant transversal resonator mode in the plane of the flat outcoupling mirror is given by Figs. 7a) and 7f). Its radial polarization is illustrated in Figs. 7b) – 7e), which show the simulation results of the mode propagation through a linear polarizer with varying direction. Here a Jones matrix was used to simulate the effect of the linear polarizer. The shape and polarization properties of the dominant transversal resonator mode are in good agreement with the measured results given in Fig. 3 in .
Yonezawa et al.  also observed in their experiment that if the effective cavity length is reduced, the ordinary beam and the extraordinary beam are in the stable resonator region, ending up with an unpolarized TEM00-like fundamental mode. We can also reproduce these results in our simulation, by reducing e.g. the distance between the spherical mirror and the aperture from 180 mm to 160 mm in Fig. 5. If we apply again the cycled MPE, after 120 round trip iterations a deviation of is reached. The resulting eigenmode is given in Fig. 8 and directly shows a TEM00-like mode. However it is Ex-polarized, which seems to be different to the observation by Yoneszawa et al. . Nevertheless the simulation is indeed correct because of the fact that a monochromatic field is per definition always fully polarized according to Maxwell’s equations , and thus the eigenvalue problem given by Eq. (5) always converges to a polarized eigenmode due to the definition of the eigenvalue problem in terms of a monochromatic field. In addition please note that due to the radial symmetry of the resonator, the choice of the x–y transversal coordinate system is free. Consequently depending on the orientation of the choice of the coordinate orientation the polarization direction of the TEM00-like transversal mode also changes. It is always polarized but the polarization could be along any radial direction. Experimentally, this is also true if a high monochromaticity is obtained. But due to unavoidable fluctuations in practice, the polarization direction may change from one radial direction to the other. That leads to the result that an unpolarized field is observed in the experiment by Yonezawa et al. . Furthermore in Fig. 8b) a higher order Ey component is obtained due to the fact that in our resonator geometry the TEM00-like Ex component does not completely fill the active medium volume. Consequently in the active medium regions far away from the optical axis there is still some inversion left, which will amplify the second dominant mode shape. In the experiment this mode shape will also be unpolarized due to the above given arguments. Yonezawa et al.  do not mention the appearance of this second dominant mode shape, which might have the following reasons:
- With decreasing aperture diameter the strength of this second dominant mode decreases, because higher order mode losses are increased. The exact aperture size of the resonator used by Yonezawa et al. is not given in . So it might be that in the experiment an aperture diameter smaller than in our simulations was used, and consequently no second order mode could be observed.
- In our simulation we assumed a homogeneous pump light distribution, which is of course an approximation of the real experimental pump setup.
- In Fig. 8 we can see that the maximum intensity of the TEM00-like mode is much higher than the maximum intensity of the higher order mode. In the experiment it might be very difficult to detect the weak signal of the higher order mode.
In this work a flexible, fast and fully-vectorial approach for the transversal mode calculation of laser resonators containing linear, homogeneous and anisotropic media was introduced. It is based on a vectorial generalization of the Fox and Li algorithm by field tracing, which allows the flexible application of different simulation techniques in different subdomains of the resonator. For anisotropic subdomains a fast and rigorous simulation algorithm was presented. It is based on the FFT-based angular spectrum of plane waves approach and includes polarization crosstalk, birefringence, intra-crystal diffraction as well as refraction effects at the crystals surfaces. In combination with a vector extrapolation method the discussed techniques allow a fast and accurate investigation of birefringence effects in laser cavities. It was shown by an example that the simulation results are in good agreement with experimental results published in .
We would like to thank our colleague Ms. Olga Baladron Zorita for revising the manuscript. This work was supported by the Thuringian Ministry of Economy, Labor and Technology funded from the European Social Fund. This project has been supported by the Federal Ministry of Economic Affairs and Energy on the basis of a decision by the German Bundestag.
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