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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]
Fig. 1. Schematic configuration of beam spot radii and wavefront radii of curvature versus the optical axis in a laser resonator. To display the realization of astigmatism compensation for Gaussian beam intuitively, the curves in the xz (sagittal) and yz (tangential) planes are sketched together in the yz plane, and the optical axis z is straightened.
We assume that there is a Gaussian beam waist in the Brewster cell, with a beam waist radius ω0 and a distance Lej 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]
where 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 Lj away from the boundary in free space, as shown in Fig. 1, the propagation matrices can be described asAfter propagating through an optical system described by its ray matrix Tjx, the parameter qj of a Gaussian beam a distance Lj 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 ω0 and a distance Lej from the boundary, can be equivalent to two separate ones propagating in free space in the orthogonal planes, shown as
where Ljxis the distance the equivalent waist is from the curved mirror Fj. 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 Lj 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 Fj is placed at a distance Lj 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 Fj.
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 Fj can be obtained:
where ${z_{0t}} = {z_0}/{n^3}$, ${z_{0s}} = {z_0}/n$. The beam spot sizes are not altered when Gaussian beams pass through a thin lens. Therefore, to eliminate astigmatism after Fj, 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 $R_{jt}^{\prime} = R_{js}^{\prime}$ yieldsThere the two distinct roots for Eq. (14) are
Inserting Eqs. (7), 8, and 11 into Eq. (16), one observes that rj>0 because n>0. In the following, we verify that θj 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 rj>0, $\sqrt {{{({f_j}{r_j})}^2} + 4} > 2$ and $\sqrt {{{({f_j}{r_j})}^2} + 4} > {f_j}{r_j}$, if fj>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 fj<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, θj has only one physical solution in the case that $0 < - {f_j}{r_j} + \sqrt {{{({f_j}{r_j})}^2} + 4} < 2$ when fj>0, because θj should satisfy $\cos {\theta _j} \in (0,1)$.Then, the unique solution for Eq. (15) can be used to obtain
with the prerequisite that fj>0. Obviously, the value of θj 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 Fj; in other words, the astigmatisms are eliminated completely after Fj.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 obtainThe 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 Lj between the curved mirror Fj 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 Fj. 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 ω0 at the center of the crystal is chosen as 200 µm. The curved mirrors M1 and M2 with the focal lengths f1=f2=50 mm are used. According to the first astigmatism compensation equation Eq. (10), M1 and M2 are located at the distances L1=L2=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 θ1=θ2=37.19° of M1 and M2 based on the second astigmatism compensation Eq. (17). According to Eqs. (21) and 22, we obtain that a flat mirror M3 is at distance L10=63.3 mm from M1, and a curved mirror M4 with focal length f4=150 mm is a distance L2+=L20+L20+=360.2 mm from M2. 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.
Fig. 3. Laser beam spot radii versus optical axis z (zero point from M3, following M1, M2 and M4) intracavity. Astigmatisms of the Gaussian beam at the arms between M3 and M1, M2 and M4, and inside the Brewster crystal are completely compensated.
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.
Fig. 4. Plots of the Gaussian beam intensity distribution. Beam spot positions are: (a), (e) at center of Brewster crystal; (b), (f) at M2; (c), (g) at the position of Gaussian beam waist in the arm between M2 and M4 in the tangential plane; (d), (h) at M4. The parameters of the upper four plots are given previously and meet our expressions (10) and (17), while the lower four with slight changes to θ1=θ2=42.19° and L1=L2=71.6mmdo not meet our expressions.
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 Fj, 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.
Fig. 5. Phase (red) and the reciprocal of wavefront radii of curvature (blue) differences of Gaussian beams in the tangential and sagittal planes versus the optical axis z.
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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