## Abstract

A theoretical framework is presented for calculating three-dimensional resonator modes of both stable and unstable laser resonators. The resonant modes of an optical resonator are computed using a kernel formulation of the resonator round-trip Huygens–Fresnel diffraction integral. To substantiate the validity of this method, both stable and unstable resonator mode results are presented. The predicted lowest loss and higher order modes of a semi-confocal stable resonator are in agreement with the analytic formulation. Higher order modes are determined for an asymmetrically aberrated confocal unstable resonator, whose lowest loss unaberrated mode is consistent with published results. The three-dimensional kernel method provides a means to evaluate multi-mode configurations with two-dimensional aberrations that cannot be decomposed into one-dimensional representations.

© 2011 OSA

## 1. Introduction

Unstable optical resonators [1,2] have found wide application for high power lasers. Unstable resonators possess major advantages with regard to mode volume (for relatively short resonators), transverse mode control, single-mode operation (i.e., near diffraction limited output), energy extraction (i.e., uniformity of illumination seen by the gain), and far-field brightness. The success of unstable resonators is realized within the regime where the laser configuration exhibits moderate gain per pass and a Fresnel number larger than a few times unity [3]. Diffraction effects in unstable resonators create a population of transverse modes, which at a particular equivalent Fresnel number, may possess mode loss degeneracy. It cannot be over emphasized that the numerical determination of the properties of the transverse mode populations has evolved into sophisticated calculation techniques with mathematical and computational complexities [4–7].

One would like to be able to calculate the three-dimensional mode properties in a straightforward fashion. The typical method for determining the three-dimensional mode structure of a rectangular aperture is to take the product of the strip resonator modes describing each axes (as mentioned [3,4,8]), with care being taken to mix all possible permutations. Certain higher order aberrations common in laser resonators (e.g., astigmatism and coma) cannot be decomposed into one-dimensional aperture representations. The method presented in this paper provides a direct means for determining both unity and non-unity aspect ratio modes with either symmetric or asymmetric aberrations. Furthermore, any arbitrary aperture shape or aberration can be investigated. An additional strength of this method is that it not only predicts the three-dimensional aberrated lowest loss mode properties but also determines the aberrated higher order modes in a single calculation.

High power laser systems having volume constrains and a large gain aperture necessitate operation at large equivalent Fresnel numbers, which enhances the possibility of multi-mode operation. The determination of single-mode operation is further complicated by aberrations common to less than perfect cooling schemes in solid-state lasers. The question of interest in these applications is whether a specific aberrated resonator design will operate with single transverse mode discrimination. The importance arises in the belief that good beam quality requires single-mode operation. For cases where multi-mode operation occurs, the methods presented here will allow for the determination of the higher order field distributions and their impact on beam quality.

## 2. Three-dimensional Huygens-Fresnel kernel formulation

We present here a numerical procedure for determining the three-dimensional transverse mode properties of an optical resonator using the Huygens-Fresnel integral equation [8]. The Huygens-Fresnel kernel is generated using Gaussian quadrature integration [9] to convert the round-trip resonator diffraction integral into an equivalent matrix eigenproblem. The round-trip formulation allows for an exact eigenvalue solution. The strength of this procedure lies in its straightforward simplicity of implementation and generality with regard to the type of resonant optical cavity. This method has the advantage (over traditional iterative calculational procedures that predict only the lowest loss transverse mode [8]) in that it determines the lowest loss and higher order resonator modes of the eigenproblem. To solve the eigenproblem, the authors use standard linear algebra routines via LAPACK.

The resonator diffraction integral requires that the optical amplitude distribution reproduces itself to within a constant after making a round-trip in the resonator. The constant is the eigenvalue associated with the amplitude distribution, or eigenmode. The eigenvalue equation for an optical resonator round-trip has the general form:

*γ*is the eigenvalue associated with the complex amplitude (eigenvector)

*U*,

*λ*is the wavelength of the circulating light,$\rho \left({x}_{1},{y}_{1}\right)$ is the output aperture function cross sectional area, [${A}_{x},{B}_{x},{C}_{x}$;${A}_{y},{B}_{y},{C}_{y}$] are elements of the round-trip resonator ABCD characteristic matrix, and [${x}_{1},{y}_{1}$;${x}_{2},{y}_{2}$] are the three dimensional spatial variables at the source (subscript 1) and observation planes (subscript 2). The round-trip ABCD matrix for a laser resonator:

*L*is the length of the resonator, ${g}_{n}$are the generalized resonator parameters defined as ${g}_{1,2}=1-L/{R}_{1,2}$, and ${R}_{n}$ are the radii of curvature of the resonator mirrors. Equation (1) can be simplified by normalization of the spatial variables by the output aperture radii

*a*and

*b*of the

*x*and

*y*dimension as ${\xi}_{1}={x}_{1}/a$, ${\xi}_{2}={x}_{2}/a$, ${\eta}_{1}={y}_{1}/b$, and ${\eta}_{2}={y}_{2}/b$. The normalized form of the diffraction integral:

*K*is the resonator kernel broken into

*x*and

*y*components, with eigenmodes${\Phi}_{u,v}=U({x}_{2},{y}_{2})$, and${\Phi}_{i,j}=U({x}_{1},{y}_{1})$. A further simplification is made by letting $\alpha =N(i-1)+j$ and $\beta =N(u-1)+v$ for $i=1,j=1\dots M$;$i=2,j=1\dots M$; $\dots i=N,j=1\dots M$ and $u=1,v=1\dots M$; $u=2,v=1\dots M$; $\dots u=N,v=1\dots M$. Application of these substitutions to Eq. (4) provides the eigenvalue equation

*Z*is the phase aberration, and $W={w}_{i}\left({\xi}_{i}\right)\cdot {w}_{j}\left({\eta}_{j}\right)$.

*W*provides a means to study variable reflectivity mirrors [10]; otherwise, the values of

*w*are unity over the mirror diameter representing a hard edge.

## 3. Aberrated optical resonator studies

To demonstrate the generality of the kernel formulation, we calculated the first nine three-dimensional mode intensity distributions of a semi-confocal stable resonator. The semi-confocal stable resonator configuration selected is described by the following parameters:${g}_{1}$ = 1,${g}_{2}$ = 0.5,*L* = 10 m, and*λ* = 1 μm. Results of the three-dimensional kernel calculation are shown in Fig. 1
. The three-dimensional mode order is from left to right, top to bottom, or inferred from the number of nulls along each axis. The beam waist of the fundamental mode (${w}_{0}$) is 1.03 times the analytic value. The mode beam waist ratios along each axis ($w/{w}_{0}$) are 1.03, 1.50, 1.79, which are in reasonable agreement with the analytic formulation [11] considering the mesh size used. The axial intensity plots are presented adjacent to the modes they represent. The amplitude of the local maxima is sensitive to the aperture diameter used to calculate the Fresnel number. The calculation used a diameter of 4${w}_{0}$. Note that the lowest loss and higher order stable resonator modes are determined for a particular working two-dimensional aperture. The calculation does not require the implementation of an artificial aperture, obscuration, or segmented mirror (i.e., added artificial loss) to individually determine activate higher order modes as would be required in a Fox–Li type calculation. This numerical method more accurately describes the physics of large aperture stable resonators.

The case of most widespread practical importance in high power laser oscillators is the positive branch confocal unstable resonator [12]. To further the validity of this method, we will now present graphically the results of the kernel calculation of a confocal unstable resonator. When$L>>\lambda $, the resonator can be characterized by the round-trip geometrical magnification *m* and the equivalent Fresnel number${N}_{eq}$. The equivalent Fresnel number [2] is defined by ${N}_{eq}=[(m-1)/(2{m}^{2})]\cdot [(ab)/(\lambda L)]$and the fractional power loss per round-trip for a particular eigenmode is given by${P}_{LOSS}=1-{\left|\gamma \right|}^{2}$.

For the unstable resonator calculation we chose a two-dimensional aperture based on the strip resonators reviewed in Rensch and Chester [13] having resonator parameters:${g}_{1}$ = 0.852,${g}_{2}$ = 1.21,*m* = 1.42, and${N}_{eq}$ = 0.52. We present the transverse mode resulting from an aberrated cavity. Unstable resonator mode intensities are normalized to unity and length scales are normalized consistent with Ref [13]. The four optical distortions studied include: tilt (*x*), focus ($1-2{x}^{2}$), astigmatism ($2xy$), and coma ($-2x+3x{y}^{2}+3{x}^{3}$), shown in Fig. 2
.

Figure 3
shows the lowest loss mode calculated using the Fox–Li iterative method via a fast Fourier transform for comparison with the Huygens–Fresnel kernel results detailed in Fig. 4
. The two methods show good agreement. The higher order modes are presented in Fig. 4(ii)-(v) for the unaberrated and aberrated cases. For the kernel calculations, the field beyond the output coupler mirror was determined by a round-trip Huygens–Fresnel integral propagation of the resultant eigenmode. The magnitudes of the aberrations were kept small in this study to ensure the validity of a thin sheet approximation. The starting and ending reference plane (and phase sheet) can be located anywhere within the cavity via a modification of Eq. (2). Comparison of the lowest loss kernel and Fox–Li ${P}_{LOSS}$ are in agreement to within 2x10^{−2}.

The authors chose a low equivalent Fresnel number square resonator with identical resonator parameters along each axis since it greatly simplifies the interpretation of the higher order mode progression. For example, in the unaberrated case (Fig. 4(a)) the next lowest lost mode (Fig. 4(a)(ii)) is a mixture of the fundamental mode along the vertical and the next lowest loss mode along the horizontal (evident by the central node, which is consistent with orthogonality). Evaluation of the first five unaberrated modes (Fig. 4(a)) showed a deviated from orthogonality [4] by less than 10^{−9}.

To further demonstrate the generality of the kernel method, a high Fresnel number rectangular aperture confocal unstable resonator is evaluated. The aperture is composed of two strip resonators cases reviewed in Rensch and Chester [13] having resonator parameters:${g}_{1}$ = 0.852,${g}_{2}$ = 1.21,*m* = 1.42, with a vertical${N}_{eq}$ = 6.25 and a horizontal${N}_{eq}$ = 3.12. Along with the higher order mode progress, the lowest loss mode is compared with the Fox-Li method in Fig. 5
.

## 4. Conclusion

The Huygens–Fresnel kernel formulation presented provides a framework for directly calculating the three-dimensional laser resonator mode properties for both stable and unstable resonators. The numerical procedure has the capability to provide not only the lowest loss mode but also the higher order modes simultaneously. The novelty of the explicit three-dimensional formulation is that it provides a means to evaluate multi-mode configurations with two-dimensional aberrations that cannot be decomposed into one-dimensional representations. The straightforward numerical procedure directly provides the three-dimensional transverse mode properties for both unity and non-unity aspect ratio mode profiles at both small and large Fresnel number unstable resonators.

The kernel formulation presented allows for asymmetrical phase distortions, commonly observed in high power laser medium, and provides a means to determine the effects of multi-mode operation on beam quality. This work can be used directly as an aid in designing laser resonators.

## References and links

**1. **A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE **53**(3), 277–287 (1965). [CrossRef]

**2. **A. E. Siegman and R. W. Arrathoon, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quantum Electron. **3**(4), 156–163 (1967). [CrossRef]

**3. **A. E. Siegman, “Unstable optical resonators,” Appl. Opt. **13**(2), 353–367 (1974). [CrossRef] [PubMed]

**4. **A. E. Siegman and H. Y. Miller, “Unstable optical resonator loss calculations using the prony method,” Appl. Opt. **9**(12), 2729–2736 (1970). [CrossRef] [PubMed]

**5. **R. L. Sanderson and W. Streifer, “Unstable laser resonator modes,” Appl. Opt. **8**(10), 2129–2136 (1969). [CrossRef] [PubMed]

**6. **P. Horwitz, “Asymptotic theory of unstable resonator modes,” J. Opt. Soc. Am. **63**(12), 1528–1543 (1973). [CrossRef]

**7. **L. W. Chen and L. B. Felsen, “Coupled-mode theory of unstable resonators,” IEEE J. Quantum Electron. **9**(11), 1102–1113 (1973). [CrossRef]

**8. **A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. **40**, 453–488 (1961).

**9. **P. J. Davis and I. Polonsky, “Numerical Interpolation, Differentiation, and Integration,” in *Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables*, M. Abramowitz and I. A. Stegun, eds. (Dover, 1972) pp. 875–924.

**10. **V. Magni, G. Valentini, and S. De Silvestri, “Recent developments in laser resonator design,” Opt. Quantum Electron. **23**(9), 1105–1134 (1991). [CrossRef]

**11. **H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. **5**(10), 1550–1567 (1966). [CrossRef] [PubMed]

**12. **W. F. Krupke and W. R. Sooy, “Properties of an unstable confocal resonator CO2 laser system,” IEEE J. Quantum Electron. **5**(12), 575–586 (1969). [CrossRef]

**13. **D. B. Rensch and A. N. Chester, “Iterative diffraction calculations of transverse mode distributions in confocal unstable laser resonators,” Appl. Opt. **12**(5), 997–1010 (1973). [CrossRef] [PubMed]