## Abstract

The aberration sensitivity of unstable-cavity geometries is studied by incorporating the change of the ray path by the intra-cavity aberrations in geometric-optic approximation. The first order non-linear correction in a positive branch, confocal unstable cavity is obtained analytically. In particular, as the optic field passes through an aberration plane in which the higher order aberrations are presented, more higher order aberrations are induced on the wavefront of optic field, and the original transverse modes are mixed. This mode mixing is studied by introducing non-constant propagation matrices. Similar to the linear results, we find that the non-linear corrections to aberration sensitivity decreases when geometric magnification or aberration orders increase.

©2008 Optical Society of America

## 1. Introduction

In heat capacity solid-state lasers (HCSSL) [1], the beam suffers a time-dependent wavefront distortion because the pump beam is no longer to be homogenized to approach an ideal one. This wavefront distortion is cumulated linearly in time and becomes unexpectedly worse quickly. This kind of laser systems favors an intra-cavity adaptive optic (ICAO) system to correct the wavefront distortions [2]. However, in unstable-cavity geometries, the aberration of the output beam does not coincide with the proper aberration inside the resonator. The relationship between the proper aberration and the output beam aberration, so-called *aberration sensitivity* [3], is a fundamental parameter to depict the resonator. In order to compensate the proper aberration via an ICAO, the proper aberration has to be solved from the measured wavefront of laser field by means of the aberration sensitivity. The usual way to do it is to use the linear geometric-optic approximation [4, 5], which assumes that the origin ray path is not changed by the aberration plane. This assumption, however, is not enough as there are larger intra-cavity aberrations. In this paper, therefore, we study the aberration sensitivity of unstable-cavity geometries by taking into consideration of the change of the ray path by the aberration plane. It actually requires us to do the non-linear corrections. The scalar-wave diffraction integral formulation has been proposed to perform a rigorous non-perturbation investigation in Refs. [6, 7], in which the pure phase-tilt and the pure phase-curvature aberrations have been studied. The limitation of this method is that the calculation is rather complicated, and usually the analytical results can be obtained only for few intra-cavity aberration planes and for lower order aberrations. To overcome that limitation, in this paper we develop an alternative method to catch the non-linear corrections, where the pure geometric-optic approximation is still used.

Assume that there are *m* intra-cavity aberration planes perpendicular to the optic axis with the axis position {*z _{i}*|

*i*=1,2, …,

*m*} respectively. To simplify, we only consider the phase inhomogeneity in a given meridional line (with transverse coordinate

*x*and origin at the optic axis), which can be expanded in powers of

*x*as

The same form is presented in phase structure of the output beam at the outcoupling aperture plane,

The basic assumption of the linear approximation indicates that the intra-cavity aberrations do not induce the mode mixing. In other words, the *j*-th order aberration at the outcoupler plane is entirely determined by the same order aberrations in various intra-cavity planes,

where the coefficient *ν*̄* _{j}* describe the

*j*-th order aberration sensitivity. In general, a wavefront mode mixing should be induced after the optic field pass through the aberration plane. Then Eq. (3) should be generalized to a complicated nonlinear relation,

where the sensitivity coefficient *ν _{jk}* can be expanded in powers of

*α*

^{in},

The main purpose of this paper is to calculate the first order non-linear coefficient *c*
_{1jk}.

The difference between Eq. (3) and (4) can also be understood from viewpoint of geometric optics: in the linear approximation, since the ray paths coincide with those followed in the ideal unperturbed cavity, the optical lengths are changed only by additional phase shift introduced by aberration (1). Whereas in the non-linear treatment, the optical lengths receive additional contributions from the change of the ideal ray paths. These contributions, of course, are order of *α*
^{in}. The essential complication of the non-linear treatment is that the additional phase shift introduced by given aberration plane depends on the history of ray paths, or explicit positions on various aberration planes where the ray has passed through. Therefore, it is impossible to obtain a complete non-perturbation result by a study using the geometric optic method, and an analytical result can not be obtained even though the physical optic method is used. However, since *α*
^{in}s are smaller quantities essentially, we can pay attention to the lower order contributions only. Then the geometric optic method can still be used to find the *α*
^{in} corrections order by order. In this paper the first order non-linear correction is found by using the propagation matrix method, and the mode mixing is studied via introducing a set of non-constant propagation matrices.

## 2. Non-linear corrections to aberration sensitivity

Let us consider a ray parallel to the optic axis and is incident on outcoupler at transverse coordinate *x*′. After *N* round-trip iterations, it will be outputted at *x*=*M ^{N}^{x}*′ and parallel to the optic axis yet if there are no any aberrations inside the resonator, where

*M*is the magnification of resonator and

*N*satisfies

*M*′>

^{N}x*r*

_{0}>

*M*

^{N-1}

*x*′ with

*r*

_{0}the radius of outcoupler. However, as certain intra-cavity aberrations are incorporated, the ray path must be changed by the aberrations (Fig.1). Let

*β*be the first diagonal element of propagation matrix describing the ray pass through the aberration plane from left to right after

^{L}_{pk}*p*round-trip iterations, and

*β*be one from right to left. Then a ray travelling around

^{R}_{pk}*N*iterations in resonator will get the following additional phase shift:

where *α _{jk}*=

*α*(

_{j}*z*). Therefore, we can obtain the total phase shift of a ray running out the cavity at

_{k}*x*via substituting

*x*′ in Eq. (6) by

Use *R*
_{1} and *R*
_{2} as matrices to describe the reflection of ray by concave mirror and outcouple mirror respectively, and *P _{i}*(

*i*=1,2, …,

*m*+1) as matrices to describe the propagation of ray between two aberration planes (to specify the cavity mirrors as aberration planes too). Here we need further introduce a set of matrix

*K*(

_{i}*i*=1,2, …,

*m*) to describe the refraction of ray by the aberration planes:

where *δ _{i}* is proportional to

*α*

^{in}

*and we assume it is constant in this section. A matrix representing path of a ray, which is reflected by the outcoupler first, then pass through the*

_{i}*k*-th aberration plane from left to right, is as follows,

Our basic observation is that *δ _{i}*s are small quantities. Up to

*O*(

*δ*), the elements of

*T*can solved as

_{k}^{L}$${B}_{k}\simeq {z}_{k}+\sum _{i=1}^{k}{z}_{i}\left({z}_{k}-{z}_{i}\right){\delta}_{i},$$

$${C}_{k}\simeq \sum _{i=1}^{k}{\delta}_{i}+\frac{M-1}{L}{D}_{k},\phantom{\rule{.5em}{0ex}}{D}_{k}\approx 1+\sum _{i=1}^{k}{z}_{i}{\delta}_{i},$$

where *z*
_{0}=0 and *z*
_{m+1}=*L* with *L* the length of resonator.

After *p* round-trip iterations, the ray depicted by Fig. 1 passes through the *k*-th aberration plane from left side and right side respectively at the transverse positions given by

where *Q* is the round-trip matrix,

$$=\left(\begin{array}{cc}M+\Delta +\Upsilon +\left(2M-1\right)\Omega +\frac{\Delta -2\Upsilon}{M}& L\left(1+\frac{1}{M}\right)+\frac{2\Upsilon L}{M}+2\Omega L\\ -\frac{\Upsilon}{\mathrm{ML}}\left({M}^{2}-2M+2\right)+\frac{{M}^{2}+1}{\mathrm{ML}}\Delta +\frac{M-1}{L}\Omega & \frac{1+\Delta}{M}+\Delta +\Omega -\Upsilon \end{array}\right),$$

with

The expression of *Q ^{p}* can be calculated by diagonalization of

*Q*. Then up to

*O*(

*δ*) we have

$${\beta}_{\mathrm{pk}}^{L}={\lambda}_{1}^{p}{A}_{k}-s\frac{{\lambda}_{1}^{p}-{\lambda}_{2}^{p}}{M-1}+O\left({\delta}^{2}\right),$$

where *λ*
_{1} and *λ*
_{2} are eigenvalues of *Q*:

and

Actually we apply a ring-like beam with the circular aperture radius *r*
_{0}, so that the incident coordinate *x*′ has to satisfy

${r}_{0}<\underset{N\to \infty}{{\displaystyle \mathrm{lim}}}{\beta}_{\mathrm{pk}}^{L,R}x\text{'}<M{r}_{0}.$.

Furthermore, in order to keep the resonator unstable, we require *β ^{R}_{N}*=

*λ*

^{N}_{1}>1. Finally, the total

*O*(

*δ*) phase shift at wavefront of the output beam is

$$-{\left(1+\frac{\left(M-1\right){z}_{k}}{L}\right)}^{j-1}\frac{\mathrm{js}}{M-1}-\mathrm{Mjs}\frac{{z}_{k}}{L}\}+O\left({\delta}^{2}\right).$$

The sensitivity coefficients *ν _{jk}* can be easily found from Eq. (17). In particular, the linear result [5] can be obtained by taking

*δ*=0 in the above Eq.:

_{i}An important conclusion from Eq. (17) is that, to take proper values of *α*
^{in} in any given aberration plane can cancel the accumulative aberrations from other aberration planes. This is theoretical foundation to perform the intra-cavity adaptive compensation. However, we shall see in section 4 that this conclusion is not exact for higher order aberrations as the mode mixing is induced.

## 3. Examples for pure phase-curvature aberration

In the first example we consider that there are four spherical lenses inside the resonator, with the focal length *f*. The distances between them and outcoupler are *L*/5, 2*L*/5, 3*L*/5 and 4*L*/5 (*L*≪*f*) respectively. Then we have *δ _{i}*=-1/

*f*and up to

*O*(

*L*/

*f*) the total aberration coefficient on wavefront of output beam is

$$a=\frac{13{M}^{2}+4M+3}{5\left({M}^{2}-1\right)},$$

$$b=\frac{46{M}^{4}+3066{M}^{3}+3767{M}^{2}+666M+546}{125\left(M+1\right){\left(M-1\right)}^{3}}.$$

Here the coefficient *b* denotes the first order non-linear correction. It should be further pointed out that, in this case the condition that the resonator keeps as unstable one is

This condition can not be produced by the linear approximation.

The variations of coefficients *a* and *b* with the magnification *M* are given in Fig. 2. Since in the solid-state lasers the thermal focal length can easily achieve several hundred times of cavity length, the non-linear correction is important for smaller magnification (e.g., *M*<2.0). Moreover, both of *a* and *b* decrease with the magnification, and the non-linear correction to aberration sensitivity decreases rapidly.

In the second example we put two spherical lenses with focal length *f* and *f*′ in resonator at axis position *L*/5 and 4*L*/5 respectively. Up to the first order correction, the total aberration coefficient is

If we require that the induced aberration from two lenses cancels each other, we have

In an ICAO system, this Eq. denotes the relationship between focal length of the deformable mirror (DM) and one of proper intra-cavity aberration as they do not locate at the same position. But we are interested in the relationship betweenDM focal length and the measurable curvature radius of wavefront of output beam, *R _{c}*=-1/

*α*

^{out}:

Here we use the fact that DM is flat before the value of *R _{c}* is detected. In Fig. 3 we show the magnification dependence of the linear and quadratic coefficients in Eq. (23). Again, this result show that the non-linear correction can not be ignored for smaller magnification. For example, as

*M*=1.4,

*f*=200

*L*, the non-linear term in Eq. (23) yields about 17% contribution. It should be pointed out that, since the non-linear coefficient in Eq. (23) depends on

*a*and

*c*that are functions of position(s) of proper aberration plane(s), these position(s) must be known in advance in order to perform non-linear compensation more precisely. Furthermore, if there are

*m*>1 proper effective lenses (planes with pure phase-curvature aberration), the focal length of

*m*-1 lenses has to be known besides of their positions and measured curvature radius of wavefront of output beam. This is impossible in experiment. However, for general designs it is possible to identify all thermal lenses that possess the same focal length approximately and to compensate the non-linear correction.

## 4. Higher order aberration and mode mixing

If there is an phase aberration *ϕ*=*σx ^{k}* with order

*k*>2, the parameter in its refracted matrix (8) is coordinate dependent, i.e.,

*δ*≃2

*σx*

^{k-2}. Then the coordinate dependent propagation matrices have to be introduced and geometrical optic approximation would be hard to use in general. However, as

*σ*is small enough, the problem can be solved in powers of

*σ*order by order. The method can mimic in part physical optics and hence, give an approximation on mode mixing phenomena.

Let *λ _{p}*>1 be an eigenvalue of matrix

*Q*, which describes the (

_{p}*p*+1)-th round-trip of a ray incident on outcoupler and depends on transverse incident coordinate

*x*

*. Then up to the first order of*

_{p}*σ*,

*x*is magnified to

_{p}*x*

_{p+1}≃

*λ*

*. For sake of convenience, here we consider the case that all meridional planes possess the same order aberration of*

_{p}x_{p}*j*+2. Then up to order of

*κ*(

*κ*=

*σx*′

^{j}with

*x*′→0 the initial incident coordinate), the eigenvalues of

*Q*are

_{p}$${\lambda}_{2p}=M\left[1-2{M}^{j\left(p-1\right)}{x\text{'}}^{j}\tilde{\omega}+O\left({\kappa}^{2}\right)\right],$$

where

$${\Delta}_{0}=2L\sum _{i=1}^{m}{\sigma}_{i},\phantom{\rule{.5em}{0ex}}{\Omega}_{0}=2\sum _{i=1}^{m}{z}_{i}{\sigma}_{i},\phantom{\rule{.5em}{0ex}}{\Upsilon}_{0}=2\sum _{i=1}^{m}{z}_{i}\left(1-\frac{{z}_{i}}{L}\right){\sigma}_{i}.$$

Moreover, the matrix defined by Eq. (9) and Eq. (10) depend on the initial coordinate, and hence the number of round-trip too:

where

Then we have

where *s*
_{0} is given via replacing Δ, Ω and γ in *s* (Eq. (16)) by Δ_{0}, Ω_{0} and γ_{0} given by Eq. (25), respectively. Inserting Eq. (28) into Eq. (6) and replacing *x*κ by

we obtain the total wavefront distortion of output beam as

where *a _{k}* can be obtained from Eq. (18), and

$$-\frac{{s}_{0}{t}_{k}\left(M+1\right)}{\left({M}^{2\left(j+1\right)}-1\right)\left({M}^{j+2}-1\right)}\left({t}_{k}^{j}+{M}^{2\left(j+1\right)}\right).$$

Equation (30) tells us that, as a beam passes through an aberration plane contains the aberration of order of *k*>2, the aberration of order of 2*k*-2 can be induced on beam wavefront. This induced aberration causes the spatial oscillation of field with a higher frequency, and leads the perturbation modes mix each other. Furthermore, as an aberration of order of *k* is introduced on DM to compensate the same order proper aberration, the induced higher order aberration can not be compensated automatically, because DM itself induces higher order aberration too. It indicates that the compensation for the aberration of order 2*k*-2 actually depends on how the aberration of order *k* is compensated. Therefore, the compensation of higher order aberration has to be performed order by order in aberration order, by taking into account the induced aberration by DM.

Let us consider a simple example, in which an aberration plane is located at the center of resonator:

Then the total wavefront distortion of output beam is given by

where

$$+\frac{k}{{M}^{2\left(k-1\right)}-1}\left\{{M}^{2\left(k-1\right)}-\frac{{\left(M+1\right)}^{k-1}}{{2}^{k}}\frac{5{M}^{2}+M+2}{{M}^{k}-1}\left[\frac{1}{M+1}+\frac{1}{2}{M}^{2\left(k-1\right)}\right]\right\}.$$

In Fig. 4 and Fig. 5 we show the values of *a _{k}* and

*b*as the functions of magnification for several lower order aberrations. It can be concluded that, if

_{k}*Lσ*is not small enough, the induced aberration can not be ignored for smaller magnification. Furthermore, if an aberration at a given order

*k*is order to the wavelength of lasers

*λ*, the total wavefront distortion of output beam can be rewritten approximately as

where *N _{f}*=

*r*

^{2}

_{0}/

*Lλ*is the Fresnel number with

*r*

_{0}the radius of outcoupler. This Eq. indicates the induced aberrations can be suppressed by larger Fresnel number.

## 5. Conclusions

The geometric optic method is proposed to extract the first order non-linear correction in aberration sensitivity of a positive branch, confocal unstable cavity. In this study, the aberration receives the contributions from not only the additional phase shift as the rays pass through the aberration planes, but also the path refraction of rays by the aberration planes. Usually, if solid-state lasers use the unstable resonators, the magnification is not large enough. Then serious thermal lens can induce a significant non-linear corrections. In particular, although in HCSSL the intra-cavity adaptive optic system can greatly reduce the aberration of output beam, the proper thermal lenses in gain mediums can not be reduced at all. Then the non-linear corrections become important in a long-time running HCSSL. On the other hand, higher order (*k*>2) proper aberrations will induce more higher order aberrations on the wavefront of optic field. It leads to mixing of the perturbation modes. We study this mode mixing phenomena by introducing the non-constant propagation matrices, and show that the induced aberration is order of 2*k*-2 at the first order non-linear correction. Therefore, a rather complicated process is needed in order to perform exact intra-cavity compensation to higher order aberrations.

In this paper, we investigate only the case of one-dimension aberrations. This study can be directly generalized to two dimension case with axial symmetry. For more general case, the azimuthal rotation is generated besides of radial refraction after the rays pass through the aberration planes. This will require the 4×4 propagation matrices to be used and lead to much more complicate calculations.

## Acknowledgment

This work was partly supported by the China Academy of Engineering Physics, under Grant No. 2007B09006.

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