1. Introduction

Axially symmetric polarized laser beams, especially with radial or azimuthal polarization, have gained considerable interest within the last years. For example, radially polarized beams can be focused to substantially higher peak intensities than beams with linear or circular polarization [1]. Laser beams with such polarization distributions may be advantageous for some laser material processing applications also. For example, radially polarized CO₂-laser beams are predicted to increase the process efficiency of sheet metal cutting compared to circular polarization by a factor of 1.5 to 2 [2]. Preliminary results obtained with a modified industrial CO₂-laser emitting up to 3 kW of radially polarized light confirm that the maximum speed for quality cuts in steel is significantly increased [3]. Several methods to generate these polarization states, either intra-cavity or externally, are described in the literature. In the presence of a strong axially symmetric thermal lensing effect, as in the case of high-power rod lasers, the resonator can be designed to discriminate between the different polarization states [4]. In the absence of strong thermal lensing, an axially symmetric polarizer can be introduced into the resonator. This component can be realized by different means, e.g. by using resonant grating waveguide mirrors (RGWM), as demonstrated for CO₂-lasers [5] and thin-disk lasers [6], or with specially designed (“GIRO”) dielectric diffraction gratings [7].

First attempts to produce a radially polarized beam inside a laser resonator with folding mirrors using a RGWM showed that the residual phase shift of the folding mirrors (only a few degrees) can degrade or even destroy the desired polarization state, depending on the actual parameters of the polarizing rear mirror, especially the reflectivity difference between radial and azimuthal polarization and the axially symmetric phase shift. To understand this, the Jones matrix formalism was extended to higher-order transverse modes [8]. In the following, we will demonstrate the application of this formalism to the calculation of the polarization states of a laser resonator. The discussion will reveal that a sufficiently large phase shift in the rear mirror is extremely beneficial for the robustness of the laser output in terms of polarization purity as well as regarding the losses inside the resonator in the presence of some phase shift in the folding mirrors.

2. Applying the extended Jones matrix formalism to resonators

The extended Jones matrix formalism for higher-order transverse modes [8] can be applied to laser resonators in the same way as the well known “standard” Jones matrix formalism by simply replacing the 2 x 2 matrices by the appropriate N x N matrices for each optical element. It is, of course, necessary to define all used matrices within the same polarization base state system. As in the “standard” Jones matrix formalism, the matrices of the elements along one round-trip, starting at the output coupler, have to be multiplied to calculate the polarization state of the output beam from the eigenvectors and the round-trip losses from the eigenvalues of the round-trip matrix. Mirrors and rotated elements have to be taken into account by using the appropriate mirror and rotation matrix, respectively. The round-trip loss factor, which is defined by the ratio of the power after one round-trip to the power at the starting point, is given by the squared absolute values ${\eta }_i={\left|\lambda {\text{}}_i\right|}^2$of the eigenvalues λ_i of the round-trip matrix. The round-trip losses ${\alpha }_i$ are obtained from the round-trip loss factor by the following simple relation: ${\alpha }_i=1-\eta {\text{}}_i$. Assuming the same amplification per round-trip for all eigenvectors, the solution belonging to the largest eigenvalue and therefore having the lowest losses will oscillate inside the resonator. If there are two or more solutions with the same round-trip losses, all of them will oscillate simultaneously. Complex or negative eigenvalues may occur; different arguments of the complex eigenvalues mean that there is a phase shift between them. Therefore, in the case of simultaneously oscillating solutions with different arguments of their eigenvalues, the polarization state is not reproduced after each round-trip.

The axially symmetric polarizer, which is introduced into the laser resonator to generate radial or azimuthal polarization, can be locally characterized by its Jones matrix:

A linear resonator consisting of this element as the rear mirror and a simple output coupler with an amplitude reflectivity r (no polarization selectivity) has the following eigenvalues:

Depending on the relative magnitude of s and t, either the radial polarization ($\left|s\right|>\left|t\right|$) or the azimuthal polarization ($\left|s\right|<\left|t\right|$) will oscillate. The two hybrid modes have the same losses, which is the average between those for radial and for azimuthal polarization. Assuming that s and t are complex quantities does not change the magnitude of the eigenvalues ${\lambda }_1$ and ${\lambda }_2$, but decreases the magnitude of the eigenvalues for the two hybrid polarizations:

This means that the discrimination of the hybrid modes increases with an increasing phase shift between s and t. The best discrimination is achieved using a phase shift of π; for values above π/2, the discrimination of the hybrid modes is better than for the other discriminated polarization (radial or azimutal).

The situation is getting more complicated when a folding mirror is added to the laser resonator as depicted in Fig. 1(a) . A conventional folding mirror, having different reflectivities for s- and p-polarization (and – in general – a phase shift between both polarizations), has the following RAH representation, which can be derived by multiplying the mirror matrix $M_{mirror,RAH}$with the matrix $M_{ABCD,RAH}$for a global Jones matrix optical element [see Eq. (10) in ref. 8] using $A=u$, $D=v$, and $B=C=0$ where u and v represent the complex amplitude reflectivities for p- and s-polarization, respectively:

 

Fig. 1 Schematic setup of the laser resonator with folding mirror and radially symmetric polarizing rear mirror (a), the reflection properties of the mirrors are described by complex amplitude reflectivities as indicated; orthogonal set of polarization states for the TEM₀₁* Hermite-Gaussian mode in RAH representation (b), the donut-shaped amplitude distribution of the electrical field is shown in gray scale, arrows indicate the local orientation of the field vector.

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The round-trip matrix for a resonator consisting of the axially symmetric polarizing rear mirror, a conventional folding mirror, and a simple output coupler (with amplitude reflectivity r), as shown in Fig. 1(a), is derived by multiplying the respective matrices for the elements:

The eigenvalues are found to be:

The eigenvectors (not normalized) have the following form:

The eigenvalues and the eigenvectors with the same index belong together, i.e. they obey the equation $M_{reso}\cdot \overrightarrow{V_i}=\lambda {\text{}}_i\cdot \overrightarrow{V_i}$. To characterize the eigenvectors by their polarization base mode composition, the following useful quantities, which correspond to the squared magnitude of the projection of the eigenvectors to the RAH base vectors, can be defined:

Given that the polarization properties of the optical components inside the resonator can be described reasonably well by the extended Jones matrix formalism presented in ref. 8 and the resonator operates in a single higher-order transverse mode with axial symmetry, the polarization mode analysis presented here can be applied to a broad range of laser resonators.

The single folding mirror discussed here can be replaced by any combination of an odd number of mirrors (folded in the same or an orthogonal plane) and several transmissive linear polarizing elements (e.g. retarders, Brewster plates etc.) that can be described by a global Jones matrix optical element [8] with $B=C=0$. If this should be insufficient to describe the actual setup (e.g. for an even number of folding mirrors or rotated folding planes etc.), the calculation described above can be adapted easily by replacing $M_{fold,RAH}$in Eq. (6) by the appropriate 4x4-matrix in RAH representation, which can either be constructed from the basic 4x4-matrices or calculated from the local 2x2-Jones matrix as described in ref. 8.

3. Example

Despite the universality of the formalism described above it may be useful to explore the influence of phase shifts in the axially symmetric polarizing rear mirror as well as in the folding mirror using a set of typical values for a specific type of laser. We decided to assume a radially polarized thin-disk laser similar to that presented in ref. 6. In this case, the laser crystal is the folding mirror. The gain (or loss) inside the active medium is not taken into account. The complex amplitude reflectivities t and v are expressed by their absolute values ($\left|t\right|$,$\left|v\right|$) and their arguments ψ and δ, respectively: $t=\left|t\right|\cdot \exp \left(i\cdot \psi \right)$, $v=\left|v\right|\cdot \exp \left(i\cdot \delta \right)$. The parameters s, $\left|t\right|$, u, and $\left|v\right|$ are kept constant; their values $s=u=\left|v\right|=\sqrt{0.999}$, $\left|t\right|=\sqrt{0.97}$, and $r=\sqrt{0.9604}$ are chosen to be typical for a thin-disk laser.

In the following, the round-trip loss factors of the four different polarization eigenmodes – and especially of the preferred, mainly radially polarized mode with the eigenvalue η₃ – as well as the degree of radial polarization (“radiality”) of the eigenvector $\overrightarrow{V_3}$ at the output coupler will be investigated in dependence of the phase shifts ψ and δ of the rear and the folding mirror, respectively. Additionally, the reflectivity difference between radial and azimuthal polarization ${\left|s\right|}^2-{\left|t\right|}^2$of the rear mirror, which is required to discriminate the unwanted polarization modes, is varied to identify its optimum value.

The highest round-trip loss factor belongs to the mode that will oscillate and therefore is essential for the efficiency of the laser. The difference between the highest and the next lower round-trip loss factor determines the mode discrimination, which should be high enough to suppress unwanted polarization modes even in the presence of small distortions, e.g. due to some depolarization inside the laser crystal. Since the phase shift of the mirrors may be subject to significant variations, e.g. due to fabrication tolerances, the round-trip loss factors as well as the purity of the polarization at the output should be robust against small changes in the phase shifts ψ and δ.

Figure 2 shows the round-trip loss factors of the polarization eigenmodes, which are the squared magnitudes of the eigenvalues ${\eta }_i={\left|\lambda {\text{}}_i\right|}^2$, in dependence of the phase shift δ in the folding mirror for three different values of the phase shift ψ = 0°, 2°, and 180° in the axially symmetric polarizing rear mirror. It can be seen that for ψ = 0° even a small phase shift δ of about 0.2° radically changes the behavior of the resonator compared to δ = 0°. At δ > 0.2°, the two solutions belonging to λ₃ and λ₄ degenerate and form a mixture of “hybrid1”-polarization and radial polarization (the degeneracy occurs in case of |u| = |v| only). This is accompanied by a significant reduction of the round-trip loss factors (~1.5%), which will lead to about 1/3 less output power. On the other hand, at ψ = 180°, even a ten times higher phase shift δ = 2° of the folding mirror does not affect the performance of the laser significantly. In this case, the eigenvalues of the predominantly hybrid solutions are close to zero – and are therefore not displayed in Fig. 2 – for all values of δ. Even a relatively small phase shift ψ of 2° relaxes the requirements for the phase shift δ of the folding mirrors significantly.

 

Fig. 2 Round-trip loss factors η_i of the thin-disk laser resonator (details in the text) in dependence of the phase shift δ in the folding mirror for three different values ψ of the phase shift in the polarizing rear mirror.

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Figure 3 shows the dependency of the squared absolute value η₃ of the eigenvalue λ₃ on the phase shift δ in the folding mirror for different phase shifts ψ of the axially symmetric polarizing rear mirror in more detail. It turns out that increasing the value of ψ continuously decreases the sensitivity to small phase shifts δ. This tendency slows down at values of ψ larger than 10°; the curves for values between 20° and 180° are almost identical. The maximum acceptable phase shift δ depends on the acceptable drop in efficiency; e.g. for a 0.5% reduction in the round-trip loss factor η₃ – corresponding to about 12.5% less output power – the allowed phase shift δ is about 4° for values of ψ between 20° and 180°.

 

Fig. 3 Round-trip loss factor η₃ of the thin-disk laser resonator (details in the text) in dependence of the phase shift δ in the folding mirror for different values ψ of the phase shift in the polarizing rear mirror.

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In Fig. 4 , the variation of the round-trip loss factor η₃, which is equal to 1 for no losses and decreases with increasing losses, is plotted in dependence of the phase shift ψ in the axially symmetrical polarizing rear mirror for different values of the phase shift δ. For δ < 0.1°, the round-trip losses η₃ are nearly independent of ψ. For larger values of δ and a phase shift ψ of less than about δ/2, the losses are substantially higher; in this range, for δ ≥ 0.5°, (denoted “region A” in the diagram) the losses are nearly independent of ψ and δ. At values of ψ above δ/2, (“region B” in the diagram) the losses start to decrease with increasing phase shift ψ. The minimum value of ψ required to obtain losses below a certain level increases with increasing values of δ. Beyond a certain value of ψ – typically in the range of 5 to 30° – (“region C” in the diagram) the losses saturate at a level that is increasing with increasing values of δ.

 

Fig. 4 Round-trip loss factor η₃ of the thin-disk laser resonator (details in the text) in dependence of the phase shift ψ in the polarizing rear mirror for different values δ of the phase shift in the folding mirror.

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Besides the losses, also the purity of the output polarization is a key factor for the laser operation. Therefore, in Fig. 5 the radiality R₃ – as defined above – for the oscillating eigenvector V₃ is plotted versus the phase shift δ in the folding mirror for different values of ψ. The diagram shows that for all values of the phase shift ψ in the axially symmetric polarizing rear mirror, the degree of radiality is monotonically dropping with increasing values of δ. The value of ψ only determines the onset of significant reductions of R₃. At small values of ψ (less than 0.5°), even a tiny phase shift δ of about 0.1° is sufficient to reduce the radiality to 0.95. At ψ = 10°, the maximum acceptable phase shift δ is about 1.2°. At ψ = 45°, this limit is increased to about 5.6°. ψ = 90° leads to a limit of δ = 12.2°, whereas ψ = 180° leads to an acceptable phase shift δ of about 22.5°. Obviously, with respect to the purity of the polarization, a phase shift ψ close to 180° is preferable whereas for the losses a value of 20° is sufficient for near optimum results. But if only a maximum value of δ = 4° is acceptable due to the losses, a value of ψ of about 33° is sufficient to keep the degree of radiality above 95%.

 

Fig. 5 Radiality R₃ of the thin-disk laser resonator (details in the text) in dependence of the phase shift δ in the folding mirror for different values of the phase shift ψ in the polarizing rear mirror.

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Complementary to Fig. 5, Fig. 6 shows the radiality R₃ in dependence of ψ for different values of δ. It shows that for all cases displayed (δ ≤ 5°), the radiality can be increased to values larger than 99% using a rear mirror with sufficiently large phase shift ψ. At δ = 5°, a phase shift ψ of at least 88° is required to achieve this high degree of purity.

 

Fig. 6 Radiality R₃ of the thin-disk laser resonator (details in the text) in dependence of the phase shift ψ in the polarizing rear mirror for different values δ of the phase shift in the folding mirror.

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In some cases it may be more convenient to plot contour lines of equal η₃ or R₃ versus ψ and δ as given in Figs. 7 and 8 . Whereas in Fig. 7 the normalized loss factor η₃(ψ, δ) / η₃(ψ = 0°, δ = 0°) is shown in the range 0° < δ < 5° and 0° < ψ < 90°, Fig. 8 shows the radiality R₃ in the same parameter range. As can be seen in Fig. 7, at ψ > 30°, the losses are virtually independent of ψ and are depending on δ roughly proportional to $1-{\delta }^2$.

 

Fig. 7 Normalized round-trip loss factor η₃ of the thin-disk laser resonator (details in the text) in dependence of the phase shifts in the polarizing rear mirror and in the folding mirror.

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Fig. 8 Radiality R₃ of the thin-disk laser resonator (details in the text) in dependence of the phase shifts in the polarizing rear mirror and in the folding mirror.

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As can be seen from Fig. 8, the radiality R₃ behaves much different: For values of δ larger than 0.5°, the phase shift ψ required in the axially symmetric polarizing rear mirror for a given radiality R₃ is nearly proportional to the phase shift δ in the folding mirror.

Figure 9 shows the loss factor versus the reflectivity difference ${\left|s\right|}^2-{\left|t\right|}^2$of the axially symmetric polarizing rear mirror for different values of the phase shift ψ in the rear mirror. The phase shift δ is kept at a fixed value of 1°. At low values of ψ (less than about 10°), the losses are strongly depending on the reflectivity difference. A sharp maximum of the losses occurs at values close to 15% reflectivity difference. At small reflectivity differences (less than 1%), the losses are relatively low. When the reflectivity difference approaches 100%, the losses are also reduced. At 100%, the losses are independent of the phase shift ψ in the rear mirror. For all smaller values, a higher phase shift ψ means lower losses. Above 90° phase shift, the losses are lowest and virtually independent of the exact value of ψ; in this range, the losses are slightly increasing with the reflectivity difference; this is significant above about 20% only. Even a phase shift of only 45° is sufficient for near optimal losses for less than about 10% reflectivity difference. To determine the performance of the laser resonator, the radiality has to be taken into account also.

 

Fig. 9 Round-trip loss factor η₃ of the thin-disk laser resonator (details in the text) in dependence of the reflectivity difference |s|² -|t|² in the polarizing rear mirror for different phase shifts ψ in the polarizing rear mirror and a fixed phase shift δ = 1° in the folding mirror.

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In Fig. 10 , the radiality is displayed in a similar manner as the loss factor in Fig. 9. As can be seen, for reflectivity differences below about 15% and a phase shift ψ of less than about 5°, the radiality is only about 90% or less, which is not desirable for most applications. For reflectivity differences above about 50%, the radiality is quite good (about 99% or higher) even for low values of ψ. At ψ = 20°, the radiality is higher than 99% independent of the reflectivity difference. For values of ψ above 45°, the radiality is approaching unity.

 

Fig. 10 Radiality R₃ of the thin-disk laser resonator (details in the text) in dependence of the reflectivity difference |s|² -|t|² in the polarizing rear mirror for different phase shifts ψ in the polarizing rear mirror and a fixed phase shift δ = 1° in the folding mirror.

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In conclusion, for low reflectivity differences, a phase shift ψ of at least about 20° is required for good results. For low values of ψ, only high reflectivity differences above 50% give satisfying results; higher values are preferable. With respect to the losses, at a sufficient phase shift ψ of at least 10°, low reflectivity differences (< 1%) are more favorable than high reflectivity differences (> 90%).

4. Summary

In summary, we have presented a general, accurate, and straightforward method to calculate the polarization eigenmodes of resonators oscillating on higher-order transverse modes including the round-trip losses of each mode. These resonators may include, for example, axially symmetrical polarizing elements as well as mirrors and rotations. The intra-cavity generation of beams with radial or azimuthal polarization is a potential application of this method. For the discussed example of a thin-disk laser resonator the variation of the phase shifts reveals some favorable parameter regimes. It turns out that sufficient axially symmetrical phase shift of the rear mirror allows to stabilize the desired pure axially symmetrical polarization state in the presence of some linear phase shift e.g. introduced by one or more folding mirrors.