We examine the optical resonator composed of two astigmatic elements, in a twisted configuration. These cavities have mode cross-sections with principal axes that rotate on propagation. Explicit cavity mode equations are derived for the case of identical mirrors. Such a resonator is appropriate for a solid-state laser that is end-pumped with the output of a laser-diode array brought to a line focus. We present a simple analysis of the significance of rotational misalignment, which effects the pump-to-mode power coupling, beam quality, and cavity stability.
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Astigmatic resonators have been used in solid-state lasers that are end-pumped with a beam brought to a line-focus [1–5]. This strategy optimises the overlap between an elliptical cavity mode and the highly elongated pumped region within the laser crystal and therefore delivers excellent pump-to-cavity mode power coupling and beam quality. The cylindrical mirrors or lenses in these resonators are generally configured so that their principal axes are aligned. As Gaussian beams are separable, it is then possible to find Gaussian beam modes for the two orthogonal directions independently . The first frame of the animation in figure 1 shows a resonator composed of two identical cylindrical end mirrors, orientated at 90 degrees to each other. Since the principal axes are parallel, it is straightforward to find the resonator mode.
When one of the cylindrical mirrors is rotated, as in subsequent frames of the animation in Figure 1, the principal axes of the beam profile rotate with propagation. For a beam to be a mode of such a resonator, the surfaces of constant phase of the beam must match the end mirrors of the resonator . As a result, the principal axes of the phase of the resonator mode at the ends of the resonator must match those of the cylindrical end mirrors. In practice, optics are never perfectly aligned, so it is important to consider a twisted resonator.
The resonator with orthogonal cylindrical mirrors has a mode with simple astigmatism (SA). The beam waists in the x and y directions are separated and located at the end mirrors. When the mirrors are rotated obliquely, the resonator mode has a more general form of astigmatism. This type of beam has been considered by Arnaud and Kogelnik  in the context of the passage of Gaussian beams through a series of cylindrical lenses that are not aligned orthogonally. The introduction of a rotating coordinate systems for the phase and irradiance distribution enabled Arnaud and Kogelnik to obtain expressions for Gaussian beams with general astigmatism (GA) analogous to those for beams with SA. Properties peculiar to Gaussian beams with GA were also discussed in reference .
Greynolds [9,10] considered the propagation of Gaussian beams with GA through an asymmetric optical system, based on ray tracing. Arnaud [11,12] presented a theoretical analysis of nonorthogonal waveguides and resonators. Serna and Nemes  derived expressions for the position, orientation and strength of a cylindrical lens needed to transform a Gaussian beam with GA to one with SA. Golovin et al.  used an eigenvalue analysis of the resonator’s round-trip transfer matrix to find the transverse modes for a nonplanar ring resonator. Lu et al. [15,16] also discuss various nonorthogonal cavities.
While there evidently exists an extensive body of theoretical work on astigmatic resonators, the recent emergence of such resonators in real world laser-diode end-pumped devices requires that the theory be extended to cover experimentally important cases. These include the conditions for cavity stability and the spatial behaviour of the cavity mode on rotation of an end cylindrical mirror (rotational misalignment). These influence both the pump-to-mode power coupling and the beam quality. The methods used here can be adapted to similar resonator designs, as required.
Section 2 introduces a convenient set of generalised equations for Gaussian beams with GA. The resonator mode equations are presented in Section 3. Section 4 explores the spatial properties of such modes.
2. A generalised Gaussian beam
As discussed in the introduction, nonorthogonal cavities have modes whose principal axes rotate on propagation. To realise this, the scalar field distribution in the transverse plane located at z=0 must include cross terms in the transverse variables:
where , j = √-1, , and Φ and M are 2 × 2 real, symmetric matrices that contain transverse phase and amplitude information respectively. Cross terms in x and y enable us to have a beam with elliptical or hyperbolic contours of equal phase and amplitude. For physical beams, however, the amplitude contours must be elliptical.
The field distribution after propogation through a distance z can be obtained by applying the Fresnel transform, ie
where and I is the 2 × 2 identity matrix. By using
3. Mode for resonator bounded by two identical cylindrical mirrors
As previously discussed, the resonator mode must have surfaces of constant phase that match the end cylindrical mirrors of the resonator. Thus we know exactly what the phase terms in equations (1) and (4) must be for the field to be the corresponding mode. The problem is now reduced to finding the intensity distribution at the first mirror that will give the phase distribution demanded by the second mirror. By equating the inverse of the phase distribution at the second mirror and the inverse of the phase terms in Eq. (4), three coupled second-order equations in three unknowns are obtained. (Working with the inverses greatly simplifies the algebra.) We have considered the special case of cylindrical end mirrors of equal radius of curvature R separated by a distance L. Assuming identical end mirrors simplifies the algebra while still containing the essential features. The first mirror is fixed, being active in the x direction. The second mirror is rotated some angle t around the optical axis, being active in the y direction when t = 0. On solving the associated equations, we obtain the solution for the intensity distribution at the first mirror:
Equation (4) can now be used to find the transverse profile of the mode in any plane.
Since ζ must be real, t and α = L/R must satisfy
4. Resonator mode properties
Of particular importance, is the behaviour of the cavity mode when an end mirror is rotated around the optical axis. Spatial changes of the cavity mode effect both the pump-to-mode power coupling and the quality of the laser output , especially in cases where an elliptical cavity mode is used to optimize the spatial overlap with a highly elongated pump volume [1–5].
The animation in Figure 3 shows irradiance and phase contour plots along the beam within a resonator composed of two cylindrical mirrors oriented at π/4 radians to each other. Note that the principal axes of the amplitude rotate by more than π/4 radians during propagation, and that the principal axes of the phase and amplitude distributions are not aligned. The intensity distribution is always elliptical for the lowest order mode, but the principal axes of the intensity at each mirror do not coincide with those of the mirrors when the system is rotationally misaligned. We find that the rotation angle of the elliptical intensity foot print is a monotonically increasing function of the mirror rotation angle. The rotational angle of the intensity distribution is plotted against mirror rotation in Figure 4, for various values of α=L/R. When α falls in the range [1.0,2.0], the curves are plotted only for values of t in which the resonator is stable. The non-linearity in these plots is reflected in the meandering of the principal axes of the beam footprint at the top mirror in the animation in Figure 1. The fact that s reaches π/2 when t does is consistent with the fact that the average velocity of rotation of the beam footprint at the bottom mirror matches that of the top mirror.
The aspect ratio of the intensity distribution at the mirrors is plotted against mirror rotation in Figure 5, for various values of α=L/R. The curves for α ∈ (0.0,1.0) are reasonably well behaved, having a local maximum at t=0, and a singularity at t=π/2. The curves for α ∈ [1.0,2.0] however, have singularities at some t>0 and t=π/2. It can be seen from Figure 5, that an orthogonal resonator (t = 0) with an aspect ratio of 10, can be created by choosing α=0.99. This resonator would not require critical rotational alignment of its elements, because the aspect ratio is insensitive to small changes in t (due to the local maximum), as is the rotation of the intensity distribution (s~0.54t for small values of t, when α~1.)
The lowest order optical mode for the resonator composed of two misaligned identical cylindrical mirrors is characterised by Eqs. (4) and (5). The simplicity of the unique solution is surprising given the complexity of the intermediate results in solving the associated coupled non-linear equations. The method presented here can be readily extended to more general nonorthogonal resonators, although it may then be necessary to rely more heavily on numerical solutions. (The next step of non-identical cylindrical mirrors involves only minor complications.)
The developed resonator mode equations were used to investigate spatial effects of rotational misalignment, which affects cavity stability, the pump-to-mode power coupling and beam quality in end-pumped solid-state lasers. The sensitivity of the cavity mode to mirror rotational misalignment was found to be manageable for α=L/R slightly less than 1.0. These resonators were found to be geometrically stable with respect to rotational misalignment for all angles whenever α is less than 1.0, and for a non-zero interval of mirror rotation angles for a between 1.0 and 2.0. Due to the highly elliptical cavity mode, the resonator composed of two cylindrical end-mirrors is appropriate for a solid-state laser that is end-pumped with a beam brought to a line focus.
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