## Abstract

We present a simple method for designing a high beam quality laser resonator, in which the astigmatism in all of the arms is facilely compensated, for the first time, to the best of our knowledge. The analytical expressions for astigmatism compensation are also derived. A folded resonator is designed using these analytical expressions, in which the astigmatic aberration of all the arms is completely eliminated. The theoretical research results show that not only the spot intensity profile’s deformation but also the phase distortion in all the arms of the laser resonator can be simultaneously and completely compensated.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

A high beam quality has become very attractive in the areas of laser material processing, printing, marking, cutting, drilling and measuring. Astigmatism is a well-known aberration, which deforms the circular transverse pattern of the output beam and limits laser performance. An astigmatism in folded and ring resonators that include Brewster-cut crystals and/or off-axis focusing elements is the most important factor contributing to the poor beam quality of lasers. How to design an astigmatism-free and high-performance resonator is a question that is attracting many researchers. Many studies focus on the elimination of astigmatisms, but they only compensated for astigmatism in one arm by using an approximate analytical solution [1,2] or the numerical method [3] in folded resonators. To compensate for astigmatism, some researchers designed a specifically shaped resonator, such as a symmetric ring resonator [4,5] or a symmetrical incident non-planar resonator [6]. In contrast with symmetric resonators or non-planar resonators, universal planar resonators are more widely used. Recently, Wen et al. reported that the two terminal arms of a universal planar folded resonator can compensate an astigmatism completely [7,8]. However, astigmatism compensation in all arms of a typical planar folded resonator has not been previously available. In terms of designing a resonator, analytical expressions are more efficient and convenient than numerical calculations.

In this letter, we present a simple method for designing a high beam quality laser resonator, and compensating for the astigmatism in all arms of a folded resonator, for the first time, to the best of our knowledge. The analytical expressions for astigmatism compensation in all the arms of a folded resonator are also derived. We use the expressions and design a folded resonator that is astigmatism-free in all the arms. The numerical calculation is also implemented to verify our expressions.

## 2. Deduction

In this section, we derive the analytical expressions for astigmatism compensation in all the arms of a folded resonator based on the theory of Gaussian beams propagation and transformation. As illustrated in Fig. 1, when a Gaussian beam is incident with Brewster’s angle upon the rhombic plates with thickness *d* and refractive index *n*, the corresponding effective geometrical path length of propagation through the Brewster cell, can be written as [1]

We assume that there is a Gaussian beam waist in the Brewster cell, with a beam waist radius *ω*_{0} and a distance *L*_{ej} separating it from the boundary. Subscripts *j*=1 and *j*=2 correspond to the left and right propagation directions, respectively (the same hereinafter). Hence the related Rayleigh length is described as ${z_0} = {{\pi \omega _0^2} / {({{\lambda / n}} )}}$, where λ is the wavelength of a Gaussian beam in vacuum. The definition of the *q* parameter is given by [9]

*R*is the wavefront radius of curvature and

*ω*is the beam spot radius. The

*q*parameter of the beam waist inside the Brewster cell can be written as where the subscript 0 indicates the beam waist (the same hereinafter). When the Gaussian beam propagates from the beam waist position inside the Brewster cell to a plane a distance

*L*

_{j}away from the boundary in free space, as shown in Fig. 1, the propagation matrices can be described as

*x = t*and

*x = s*indicate the tangential and sagittal planes (the same hereinafter), respectively;

*T*

_{Bt}and

*T*

_{Bs}denote the transformation matrices of a Gaussian beam passing through the Brewster cell boundary in the orthogonal planes, written as [10]

After propagating through an optical system described by its ray matrix *T*_{jx}, the parameter *q*_{j} of a Gaussian beam a distance *L*_{j} from the boundary is given by

Inserting Eqs. (4) and 5 into Eq. (6) yields

Based on Eq. (7), the aforementioned Gaussian beam inside the Brewster cell, with a beam waist radius *ω*_{0} and a distance *L*_{ej} from the boundary, can be equivalent to two separate ones propagating in free space in the orthogonal planes, shown as

*L*

_{jx}is the distance the equivalent waist is from the curved mirror

*F*

_{j}. The equivalent parameters are described as ${q_{0t}} = i{z_{0t}} = i{z_0}/{n^3}$ and ${L_{jt}} = {L_{ej}}/{n^3} + {L_j}$ in the tangential plane and ${q_{0s}} = i{z_{0s}} = i{z_0}/n$ and ${L_{js}} = {L_{ej}}/n + {L_j}$ in the sagittal plane.

To compensate astigmatism, the spot sizes of the equivalent Gaussian beams should become equal at a distance *L*_{j} away from the boundary. According to Eqs. (2) and 7, one obtains

Inserting Eqs. (2), 3, and 7 and solving Eq. (9) produces the solution

It is to be noted that Eq. (10) is one of the two most significant expressions in this paper and suggests that if a curved mirror *F*_{j} is placed at a distance *L*_{j} from the Brewster cell boundary, one can achieve a circular beam spot at this position. However, the wavefront radii of curvature in the orthogonal planes are not equal under this condition. Astigmatism still exists after the *F*_{j}.

In the following section, we continue deriving the second necessary and sufficient condition for compensating astigmatism. Using Eqs. (2) and 8, the wavefront radii of curvature of Gaussian beams at *F*_{j} can be obtained:

*F*

_{j}, the wavefront curvature radii of a Gaussian beam in the orthogonal planes should be identical, i.e., $R_{jt}^{\prime} = R_{js}^{\prime}$. The laws of geometrical optics can be used to relate the radii

*R*and $R^{\prime}$ as follows: where

*R*and ${R^{\prime}}$ are the radii of wavefront curvature before and after the thin lens with focal length

*f*.

It is well known that the effective focal length of an off-axis curved mirror in the tangential and sagittal planes are different and can be related to the effective focal lengths by [1]

where*θ*is the angle of incidence. Substituting Eqs. (11) and (13) into Eq. (12) and using the eqution $R_{jt}^{\prime} = R_{js}^{\prime}$ yields

There the two distinct roots for Eq. (14) are

Inserting Eqs. (7), 8, and 11 into Eq. (16), one observes that *r*_{j}>0 because *n*>0. In the following, we verify that *θ*_{j} only has a unique solution in Eq. (15). In practice, ${\theta _j} \in ( - {\pi / 2},\,\;{\pi / 2})$ restricts $\cos {\theta _j} \in (0,1)$. Due to *r*_{j}>0, $\sqrt {{{({f_j}{r_j})}^2} + 4} > 2$ and $\sqrt {{{({f_j}{r_j})}^2} + 4} > {f_j}{r_j}$, if *f*_{j}>0 or equivalently ${f_j}{r_j} > 0$.Therefore, we have $0 < - {f_j}{r_j} + \sqrt {{{({f_j}{r_j})}^2} + 4} < 2$ and $- {f_j}{r_j} - \sqrt {{{({f_j}{r_j})}^2} + 4} < - 2$; if *f*_{j}<0, or equivalently ${f_j}{r_j} < 0$, one obtains $- {f_j}{r_j} + \sqrt {{{({f_j}{r_j})}^2} + 4} > 2$ and $- 2 < - {f_j}{r_j} - \sqrt {{{({f_j}{r_j})}^2} + 4} < 0$. Hence, *θ*_{j} has only one physical solution in the case that $0 < - {f_j}{r_j} + \sqrt {{{({f_j}{r_j})}^2} + 4} < 2$ when *f*_{j}>0, because *θ*_{j} should satisfy $\cos {\theta _j} \in (0,1)$.Then, the unique solution for Eq. (15) can be used to obtain

*f*

_{j}>0. Obviously, the value of

*θ*

_{j}can be obtained simply by an inverse cosine function manipulation of Eq. (17). It is to be noted that Eq. (17) is the other significant expression in this paper, which suggests that the wavefront radii of curvature of Gaussian beams in the orthogonal planes become equal after transmitting through

*F*

_{j}; in other words, the astigmatisms are eliminated completely after

*F*

_{j}.

To date, we have obtained astigmatism compensation in the Brewster cell and the two terminal arms of the folded resonator, as shown in Fig. 1. Because there is no element in the astigmatism regions, the existence of an astigmatism outside the Brewster cell in the middle arm does not impact the property of a laser. Thus, the astigmatism of all arms in the folded cavity is compensated completely as long as the cavity parameters obey Eqs. (10) and 17.

In this section, we investigate the spot radii of beam waists and their positions in the two terminal arms. When a Gaussian beam passes through a thin lens, the relations between the beam waist parameters in the object and image spaces are given by Kogelnik [8]:

where ${l_0}$ and ${l^{\prime}_0}$ are the respective distances of the object and image beam waist distances from the lens with a focal length*f*, and their respective confocal parameters are ${b_0} = {{\pi {\omega _0}^2} / \lambda }$ and ${b^{\prime}_0} = {{\pi {{\omega ^{\prime}}_0}^2} / \lambda }$. The positions and spot radii of beam waists in the two terminal arms in only the tangential or sagittal planes are calculated owing to the completion of astigmatism compensation. We use Eqs. (8), 18, and 19 to obtain

*ω*

_{j0}and

*L*

_{j0}indicate the respective beam waist radius and the distance from the curved mirror

*F*

_{j}in the other side, the left or right arm depending on whether

*j*=1 or 2. In a linear laser cavity, the wavefront radii of curvature of Gaussian beams are individually equal to the radii of curvature of the terminal mirrors. When a flat mirror is located at the position

*L*

_{j0}, a stable folded cavity is obtained. Similarly, when a curved mirror with focal length

*f*

_{j+}is placed a distance

*L*

_{j0+}from the beam waist

*ω*

_{j0}, one also obtains a stable folded cavity.

*L*

_{j0+}and

*f*

_{j+}set a restriction as [10] where ${z_{j0}} = {{\pi \omega _{j0x}^2} / \lambda }$.

The above deduction denotes that the method for simultaneously compensating the astigmatisms of all the arms of a folded resonator is very effective and simple when it is designed by the following simple steps. First, the Gaussian beam with a determined size beam waist for a particular gain medium is given, and we can use the exact analytical Eq. (10) to directly calculate the value of the distance *L*_{j} between the curved mirror *F*_{j} and the Brewster cell boundary of the gain medium. Second, Eq. (17) can be used to make the wavefront radii of curvature of Gaussian beams in the orthogonal planes become equal after transmitting through *F*_{j}. Finally, Eq. (22) will make the Gaussian beam in the resonator satisfy the self-reproducing principle. Therefore, it has been easily fulfilled to design a folded resonator that is astigmatism-free in all the arms and high beam quality laser resonator is achieved.

## 3. Results and discussion

In this section, we use the expressions given above and design a novel folded resonator that is astigmatism-free in all the arms, as shown in Fig. 2. The thickness *d* of a Brewster cut rhomb crystal and the refractive index *n* are 3 mm and 1.986, respectively. The beam waist radius *ω*_{0} at the center of the crystal is chosen as 200 µm. The curved mirrors *M*_{1} and *M*_{2} with the focal lengths *f*_{1}=*f*_{2}=50 *mm* are used*.* According to the first astigmatism compensation equation Eq. (10), *M*_{1} and *M*_{2} are located at the distances *L*_{1}=*L*_{2}=64.6 mm from the left and right sides of the crystal, respectively. It is also straightforward to obtain the required oblique angles of incidence *θ*_{1}=*θ*_{2}=37.19° of *M*_{1} and *M*_{2} based on the second astigmatism compensation Eq. (17). According to Eqs. (21) and 22, we obtain that a flat mirror *M*_{3} is at distance *L*_{10}=63.3 *mm* from *M*_{1}, and a curved mirror *M*_{4} with focal length *f*_{4}=150 *mm* is a distance *L*_{2+}=*L*_{20}+*L*_{20+}=360.2 *mm* from *M*_{2}. To date, the design of a folded resonator that is astigmatism-free in all the arms has been finished. It is simple and pleasant to use our analytical expressions to design a folded resonator that is astigmatism-free in all the arms.

Now, we are going to implement the numerical calculation to verify that the folded resonator we have designed is astigmatism-free in all the arms. The numerical calculation based on the ABCD-law with the cavity parameters in Fig. 2 is performed, as illustrated in Fig. 3. It can be seen from Fig. 3 that the astigmatisms at the two terminal arms and inside the crystal are completely compensated. The results agree well with our theoretical predictions. Other cavity elements such as a frequency doubling crystal or saturable absorber elements can be placed in two terminal arms and there is no astigmatism in all the elements of our astigmatically compensated resonator.

Figure 4 shows the Gaussian intensity distribution contrast between the cavity parameters that do and do not satisfy the astigmatism compensation Eq. (10) and 17. It can be seen from Fig. 4 that the beam spot radii in the resonator that is astigmatism-free, as shown in Fig. 3, are all circular spots in all intracavity elements and the astigmatisms are eliminated completely. Otherwise, an astigmatism exists in the resonator, which does not satisfy our astigmatism compensation formulas.

The phase plays an important role in modern optics. In this section, we discuss the phase in the resonator that is astigmatism-free in all the arms, which we have designed. As shown in Fig. 2, the beam waist radius and position in the Brewster crystal are known. Assuming that the initial phase is equal to zero at the beam waist, we first calculate the phase differences on the optical axis of Gaussian beams between the tangential and sagittal planes along the optical axis z in the middle arm using Eqs. (8) and 23. Equation (23) is the phase shift factor formula of the fundamental mode Gaussian beam [9],

From the aforementioned discussions, if Gaussian beam waist radii and positions in the orthogonal planes become equal after passing through lens *F*_{j}, then the phase differences on the optical axis *z* can be obtained by calculating Eq. (23) involving the new beam waist parameters. Meanwhile, one also determines the differences of the reciprocal of wavefront radii of curvature ${1 / {{R_t}}} - {1 / {{R_s}}}$ versus the cavity optical axis z based on the Gaussian beam propagation property and Eq. (8).

The left and right vertical coordinates in Fig. 5 indicate the phase differences on the optical axis ${\phi _t} - {\phi _s}$ and the differences of the reciprocal of wavefront radii of curvature ${1 / {{R_t}}} - {1 / {{R_s}}}$ between the orthogonal planes, respectively. It can be seen from Fig. 5 that the phases and the radii of wavefront curvature become equal. This means that the phase distortion in the two terminal arms and the Brewster crystal of the laser cavity of Fig. 3 can be simultaneously compensated completely, and the output of a high beam quality laser beam is very easy to obtain.

## 4. Conclusion

For the first time, to the best of our knowledge, we presented a simple method to design a high beam quality laser resonator by facilely compensating the astigmatism in all the arms of a folded resonator. We also derived exact analytical expressions for astigmatic compensation in all the arms of a folded resonator. A folded resonator that is astigmatism-free in all the arms was designed using these analytical expressions. The research results show that not only the spot intensity profile’s deformation but also the phase distortion in the two terminal arms and the Brewster crystal of the laser cavity can be completely compensated.

## Funding

Guangdong Basic and Applied Basic Research Foundation (2021A1515010964); Science and Technology Innovation Commission of Shenzhen Municipality (SGDX20190919094803949, JCYJ20200109105810074, JCYJ20170412111625378).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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