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Intracavity adaptive correction of an unstable standing-wave resonator based on reconstruction matrix optimization

Chao Yang, Shiqing Ma, Lingxi Kong, Mengmeng Zhao, Chunxuan Su, Ruiyan Jin, Ping Yang, Shuai Wang, Boheng Lai, and Bing Xu

Open Access Open Access

Abstract

Due to the existence of the expanding beam portion in the positive branch confocal unstable resonator, the laser passes through the intracavity deformable mirror (DM) twice with different apertures, which makes it complicated to calculate the required compensation surface of the DM. In this paper, an adaptive compensation method for intracavity aberrations based on reconstruction matrix optimization is proposed to solve this problem. A collimated probe laser of 976 nm and a Shack-Hartmann wavefront sensor (SHWFS) are introduced from the outside of the resonator to detect intracavity aberrations. The feasibility and effectiveness of this method are verified by numerical simulations and the passive resonator testbed system. By adopting the optimized reconstruction matrix, the control voltages of the intracavity DM can be directly calculated from the SHWFS slopes. After compensation by the intracavity DM, the beam quality β of the annular beam coupled out from the scraper is improved from 6.2 times diffraction limit to 1.6 times diffraction limit.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The positive branch confocal unstable resonator (PBCUR) is widely used in the field of high-energy lasers due to its large mode volume and good transverse mode discrimination ability, which is easy to achieve high-power fundamental mode oscillation [1]. However, affected by many factors such as quantum defects of the activated gain medium, high-energy resonators inevitably generate dynamic thermal-induced wavefront distortion during lasing operation, resulting in degradation of output beam quality [2]. Adaptive optics (AO) technology can detect and correct the distorted wavefront in real-time, which is an effective method to improve the beam quality [3,4]. Compared with the extracavity AO system [5], the intracavity AO system has several advantages, such as a more compact structure and the ability to directly control the resonator’s mode. Intracavity adaptive optics in unstable resonators has attracted a lot of attention over the past few decades [6–8]. Nevertheless, there is no one-to-one correspondence between the intracavity phase that can be detected and the phase that needs to be applied as compensation. The laser goes through the intracavity deformable mirror (DM) twice with different apertures, which results in a “non-conjugate” correction problem, i.e., the desired DM surface is not conjugate to the intracavity aberration. Thus, it is difficult to directly calculate the required compensation surface of the intracavity DM.

In general, there are two typical ways to solve this problem. The first method is the search-based optimization algorithm. Related research progresses have been widely reported in the past decades [9,10]. This and similar approaches regard the resonator as a “black box”, and then iteratively calculate the control signal of the intracavity DM by continuously optimizing the far-field beam quality or output power. Accordingly, the underlying mechanism of compensation for intracavity aberrations is ignored. These methods can obtain good compensation results with a simple and compact structure, especially when the intracavity aberrations change slowly. However, the optimization algorithm usually requires multiple iterations, which results in slow convergence. What’s more, the optimization algorithm tends to converge to a local extremum. The final convergence results also strongly depend on the selection of algorithm parameters, initial voltage, and optimization criteria. Consequently, it is not always possible to obtain the desired correction results when there are rapidly changing aberrations inside the resonator. The second method is based on geometrical-optics approximation [11–13]. Because this method is derived under the condition of small aberration approximation, it is only suitable for the case of small intracavity aberrations. Compared with the search-based optimization method, the geometrical model does not require multiple iterations. Although it is faster, it is difficult to obtain a good correction result in the case of large intracavity aberrations. Furthermore, this method needs to determine the coordinate of the intracavity aberrations along the optical axis of the resonator. In the high-power unstable resonator, there is often more than one plane that can introduce intracavity aberrations. In this case, implementing intracavity aberrations correction becomes extremely complicated.

To solve the problem of “non-conjugate” correction of intracavity aberrations inside PBCUR, an adaptive compensation method based on reconstruction matrix optimization is proposed in this paper, combined with the round-trip detection of the probe beam. More specifically, a collimated probe beam of 976 nm is injected from the outside of the resonator. After a round-trip propagation, the probe beam enters the Shack-Hartmann wavefront sensor (SHWFS) positioned outside the resonator. The probe beam passes through the gain medium and the DM twice during one round-trip propagation. The oscillating laser of 1064 nm and the probe beam propagates in the common ray path along the optical axis of the resonator. If the wavefront of the output probe beam can still recover the plane, the accumulated aberration of both the probe beam and the oscillating laser will be zero during a round-trip transmission, which indicates that the intracavity aberrations have been effectively compensated. Assuming the existence of expanding beam portion in the resonator and the requirement of the conjugate distance of SHWFS are ignored simultaneously, then the reconstruction matrix between the SHWFS slope of the output probe beam and the voltage of the DM could be established. In this paper, the reconstruction matrix can be updated by a gradient descent algorithm [14], which takes the root mean square (RMS) value of the output probe beam wavefront as a reference. At last, the control signal of the intracavity DM can be easily obtained through the optimized reconstruction matrix. It is worth noting that the proposed correction strategy is well compatible with the existing AO systems.

The rest of the paper is arranged as follows. In section 2, we first briefly describe the overall system layout and the principle of the adaptive correction method for intracavity aberrations based on reconstruction matrix optimization. Next, a numerical simulation platform for adaptive correction of intracavity aberrations is established, and numerical analysis results are presented to illustrate the feasibility and effectiveness of the proposed method in section 3. Afterwards, the initial experimental results of the passive resonator testbed system are shown in section 4. At last, some discussions and conclusions are given in section 5.

2. Method

2.1 System layout

A PBCUR with an intracavity AO system based on the round-trip detection of probe beam is depicted in Fig. 1(a). It is mainly composed of the following three components: the positive branch confocal unstable resonator, the output power and beam quality detection system, and the intracavity AO system. As shown in Fig. 1(b), the concave mirror (M1) and the convex mirror (M2) are placed confocally, so the plane wave and the spherical wave with a specific radius of curvature can oscillate in the resonator. The radii of curvature of the M1 and M2 are 6 m and -4 m, respectively, the cavity length is 1 m, and the geometric magnification is 1.5. The gain medium used in the simulation is an Nd: YAG thin disk, with the thickness of 2 mm, the diameter of 30 mm, and the doping area aperture of 24 mm. As shown in Fig. 1(a), the annular output beam of the resonator is divided into two parts by a beam splitter (BS3) with a transmission of 1%@1064 nm. The reflected part of the beam enters the power meter, and the weakly transmitted part of the beam enters the beam quality detection system composed of a convex lens with a focal length of 900 mm and a CCD camera.

 figure: Fig. 1.

Fig. 1. (a) The schematic diagram of adaptive aberration compensation inside a positive branch confocal unstable resonator based on the round-trip detection of the probe beam. (b) The schematic diagram of the positive branch confocal unstable resonator.

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The intracavity AO system mainly consists of a deformable mirror (DM) located in the resonator, a wavefront controller located outside the resonator, a Shack-Hartmann wavefront sensor (SHWFS), and a collimated probe laser of 976 nm. To detect intracavity aberrations, a pair of beam splitters BS1 and BS2 are introduced. The BS1 and BS2 are highly reflective at 1064 nm under 45°. The ratio of transmission and reflectivity of BS1 and BS2 at 976 nm with an incident angle of 45° is about 5:5. The collimated probe beam of 976 nm passes through the beam splitter BS1, and then the transmitted portion is incident on the convex mirror M2. After being reflected by the convex mirror M2, it becomes a divergent spherical wave, sequentially passes the scraper, BS1, BS2, gain medium and DM, and finally reaches the concave mirror M1. Next, after being reflected by the concave mirror M1, it becomes a parallel beam, goes through the DM and gain medium in turn, leaves the resonator via BS2, and eventually enters SHWFS placed outside the resonator. The ray path of the probe beam shown in Fig. 1(a) can be unfolded according to the lens sequence, as shown in Fig. 2. The DM is equivalent to a phase screen located at the intracavity aberration compensation plane. The aberration introduced by the gain medium is also equivalent to a phase screen located at the intracavity aberration introduction plane.

 figure: Fig. 2.

Fig. 2. The unfolded lens equivalent of the probe beam (BS1 and BS2 are not shown for simplicity).

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Using SHWFS to detect the wavefront of the output probe beam can not only provide feedback for the intracavity DM but also measure intracavity aberrations in real-time to evaluate the performance of compensation for intracavity aberrations. The relative position between the sub-apertures of the SHWFS and the deformable mirror actuators is demonstrated in Fig. 3. The actuators of the piezoelectric (PZT) DM are arranged in a square array. The stroke and pitch of the actuator are ±2.5 µm and 3 mm, respectively. The number of valid actuators is 52. The sub-aperture size (object surface) of SHWFS is 1.5 mm × 1.5 mm. The sub-aperture is arranged in a square array. The number of pixels contained in each sub-aperture is 32 × 32. The number of active sub-apertures is 156, and the bit depth of the Hartmann camera is 10 bits.

 figure: Fig. 3.

Fig. 3. The configuration between the sub-apertures and the actuators.

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2.2 Principle of adaptive compensation for intracavity aberrations

The principle of adaptive compensation for intracavity aberrations is based on reconstruction matrix optimization, combined with the round-trip detection of the probe beam. The probe beam is used to detect intracavity aberrations and provides feedback for the intracavity DM. As shown in Fig. 2, the collimated probe beam goes through the intracavity aberration introduction plane and the intracavity aberration compensation plane (DM) during a round-trip propagation. If the wavefront of the output probe beam can still maintain the plane, it means the accumulated aberration of the probe beam is zero after one round-trip propagation. The oscillating laser and the probe beam travel back and forth in the common ray path along the optical axis of the resonator. Therefore, the accumulated aberration of the oscillating laser during each round-trip oscillation is also zero, indicating that the intracavity aberrations have been effectively compensated.

Firstly, the existence of a non-parallel ray path in the resonator and the requirement of the conjugate distance of SHWFS are overlooked simultaneously. The interaction matrix between SHWFS slope and DM voltage can be measured according to the “push-pull” method [15]. More specifically, by pushing and pulling each actuator at a time, the SHWFS response is recorded after the probe beam one round-trip through the resonator. For an intracavitary AO system with n sub-apertures and m actuators (setting n = 156 and m = 52), the interaction matrix M can be expressed as:

$${{\mathbf G}_{2n \times 1}} = {{\mathbf M}_{2n \times \textrm{m}}}{{\mathbf U}_{m \times 1}},$$
where G2n × 1 and Um × 1 are the SHWFS slopes and DM voltages, respectively. Singular value decomposition (SVD) is employed to calculate the pseudo-inverse matrix of M, which is the so-called reconstruction matrix R. Therefore, the matrix R satisfies the following expression:
$${{\mathbf U}_{m \times 1}} = {{\mathbf R}_{m \times 2n}}{{\mathbf G}_{2n \times 1}}.$$

For the convenience of the following description, the matrix R obtained by the SVD algorithm is denoted as Rpush-pull. The reconstruction matrix R obtained based on the above approximation is difficult to accurately describe the relationship between the DM voltage and the SHWFS slope. The matrix R can be updated by employing a gradient descent algorithm. From the images acquired by SHWFS, the wavefront can be reconstructed using the model wavefront reconstruction algorithm [16,17]. The RMS value of the corrected residual wavefront of the probe beam is used as the performance metric J of the gradient descent algorithm. The chain rule is used to calculate the partial derivative of the performance metric J to the matrix R, namely,

$$\frac{{\partial J}}{{\partial {\mathbf R}}} = \frac{{\partial J}}{{\partial {\mathbf U}}} \times \frac{{\partial {\mathbf U}}}{{\partial {\mathbf R}}} = \frac{{\partial J}}{{\partial {\mathbf U}}} \times {{\mathbf G}^\textrm{T}}.$$

In Eq. (3), the partial derivative of the performance metric J to the control voltage U is the item to be determined. After the small random disturbance voltage ΔU={δu1, δu2,…, δu52} that satisfies the Bernoulli distribution is applied to the DM, the variation of the performance metric J can be calculated. The gradient of the performance metric J to the control voltage U can be approximately estimated by Eqs. (4) and (5) [18].

$$\begin{array}{l} {J_ + } = J[{\mathbf U} + \Delta {\mathbf U}]\\ {J_ - } = J[{\mathbf U} - \Delta {\mathbf U}], \end{array}$$
$${\left( {\frac{{\partial J}}{{\partial {\mathbf U}}}} \right)_i} \approx ({{J_ + } - {J_ - }} )\delta {u_i}/{({\delta {u_i}} )^2},i = 1,2,\ldots ,52.$$

Therefore, based on the principle of gradient descent, the update expression of the matrix R is shown as follows:

$${\mathbf R} = {\mathbf R} - \alpha \frac{{\partial J}}{{\partial {\mathbf U}}}{{\mathbf G}^\textrm{T}},$$
where α is called the learning rate. To speed up the convergence, the matrix Rpush-pull is utilized as the initial value of the gradient descent algorithm.

For bidirectional disturbances, the following relationship can be easily obtained:

$$\begin{array}{l} {\mathbf U} + {\Delta \mathbf U} = {\mathbf R}{{\mathbf G}_{\textrm{pos}}}\\ {\mathbf U} - {\Delta \mathbf U} = {\mathbf R}{{\mathbf G}_{\textrm{neg}}}, \end{array}$$
where Gpos and Gneg are the corresponding SHWFS slopes after positive and negative disturbance voltages are applied on the actuators, respectively. From Eq. (7), the following relation can be derived:
$${\Delta \mathbf G} = ({{{\mathbf G}_{\textrm{pos}}} - {{\mathbf G}_{\textrm{neg}}}} )/2.$$

The disturbance voltage ΔU and the corresponding disturbance slope ΔG approximately satisfy ΔU = R×ΔG, which is a constraint condition for the matrix R. During the gradient estimation, the matrix R can be identified from multiple samples (ΔG, ΔU) using the recursive least squares (RLS) algorithm [19]. It is easy to find that Eq. (2) is a linear model. To prevent this model from drifting over time and affecting the generalization relationship between the driving voltage and the SHWFS slope, the reconstruction matrix R can be constrained according to the identification results of the RLS algorithm.

The process of the RLS algorithm is briefly described as follows. The recursive relationship of R(t) is given by:

$${\mathbf R}(t) = {\mathbf R}(t - 1) + {\mathbf e}(t){{\mathbf G}^\textrm{T}}(t){\mathbf P}(t),$$
where the notation “(t)” in Eq. (9) denotes the discrete-time instant t. The error vector e(t) is given by:
$${\mathbf e}(t) = \Delta {\mathbf U}(t) - {\mathbf R}(t - 1)\Delta {\mathbf G}(t).$$

The gain vector K(t) can be expressed as:

$${\mathbf K}(t) = \frac{{{\mathbf P}(t - 1)\Delta {\mathbf G}(t)}}{{\Delta {{\mathbf G}^\textrm{T}}(t){\mathbf P}(t - 1)\Delta {\mathbf G}(t) + \mu }},$$
where µ is the forgetting factor of the RLS algorithm. The RLS algorithm with a forgetting factor µ can gradually weaken the influence of the old samples and can pay more attention to the new samples. The forgetting factor µ belongs to the interval (0, 1). The parameter µ is of great importance to identify the reconstruction matrix. In practice, the parameter µ can be adjusted to get the desired correction result. The recursive expression of P(t) with K(t) can be written as:
$${\mathbf P}(t) = {{[{\mathbf P}(t - 1) - {\mathbf K}(t)\Delta {{\mathbf G}^\textrm{T}}(t){\mathbf P}(t - 1)]} / \mu }.$$

The initial values of R(t) and P(t) are

$$\begin{array}{l} {\mathbf R}(0) = {{\mathbf R}_{\textrm{push - pull}}},\\ {\mathbf P}(0) = {\delta ^{ - 1}}{\mathbf I}(0 < \delta \le 1), \end{array}$$
where I is an identity matrix of order 2n. The matrix Rpush-pull is utilized as the initial value of the RLS algorithm.

Both the identification matrix denoted R1 and the optimization matrix denoted R2 are combined to generate the final matrix R, shown as follows:

$${\mathbf R} = ({1 - w} )\times {{\mathbf R}_\textrm{1}} + w \times {{\mathbf R}_2},$$
where w is the weight factor. The implementation of this proposed adaptive compensation method for intracavity aberrations is briefly described in Algorithm 1.

oe-31-5-7825-i001

3. Simulation results

To verify the effectiveness of the proposed method, numerical simulations are carried out. The self-reproduction mode of resonators is analyzed using an iterative algorithm based on fast Fourier transform and scalar diffraction theory [20–22]. In the simulation, the maximum number of iterations T is set to 1000, the learning rate α is set to 0.01, the proportional parameter (pia) and integral parameter (pib) of the PI controller are set to 0.999 and -0.2, respectively, the weight factor w is set 0.9, and the forgetting factor µ is set to 0.96. The initial light field is a plane wave with uniform amplitude over a circular aperture (setting the diameter of 14 mm). The iterative calculation continues until a stable-state light field distribution is obtained. Table 1 gives the distances between these optical elements within the PBCUR in Fig. 2.

Tables Icon

Table 1. The distances between these optical elements in Fig. 2

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The compensation surface generated by a 52-element continuous surface and discrete actuators DM is given by:

$${\phi _{\textrm{DM}}}(x,y) = \sum\limits_{i = 1}^{52} {{u_i}{f_i}(x,y)} ,$$
where ui is the voltage of the i-th actuator, and fi(x, y) is the influence function of the i-th actuator. The influence function fi(x, y) is approximated by a Gaussian model [23] shown as follows:
$${f_i}(x,y) = \exp [\ln (\omega ) \times {(\sqrt {{{(x - {x_i})}^2} + {{(y - {y_i})}^2}} /d)^a}],$$
where a is the super-Gaussian index, ω is the coupling value of actuators, d is the normalized distance between adjacent actuators, and (xi, yi) are the position coordinates of the i-th actuator. The parameters a and ω were determined using a multi-physics model built in ANSYS. Eventually, the parameters a and ω are set to 2.2 and 0.2, respectively. Therefore, the transmittance function of DM can be written as
$${T_{\textrm{DM}}}(x,y) = \exp [ik{\phi _{\textrm{DM}}}(x,y)],$$
where k = 2π/λ is the wavenumber.

The intracavitary aberrations can be roughly divided into two types. The first type is the static aberration caused by alignment and manufacturing errors of all optical components. The second type is the thermally induced dynamic aberrations during the pumping process. These static aberrations can be corrected easily, so only dynamic aberrations generated during laser operation are considered in this paper. The temporal and spatial characteristics of intracavity aberrations are closely related to the parameters of the resonator, the state of the gain medium module, and the level of the pump power. To reduce the complexity of the model while approaching the real physical situation, the aberration caused by the gain medium is modeled as a phase screen in the numerical simulation. The phase screen is characterized by the linear combination of the first 36-term Zernike polynomials [24]. It should be pointed out that the piston term in the Zernike mode can be compensated by adjusting the cavity length. Furthermore, the intracavity tip-tilt aberrations can be individually corrected by the tip-tilt mirror (TTM). So the tip-tilt aberration will not be considered in this paper. Therefore, the Zernike mode coefficients of Z1, Z2, and Z3 are set to zero. Finally, the transmission function of the phase screen (excluding piston, tip/tilt items) is depicted as:

$$\begin{array}{l} {\phi _{\textrm{Gain}}}(x,y) = \sum\limits_{i = \textrm{4}}^{36} {{a_i}{Z_i}(x,y)} ,\\ {T_{\textrm{Gain}}}(x,y) = \exp [i{\phi _{\textrm{Gain}}}(x,y)], \end{array}$$
where Zi(x, y) and ai are the i-th order Zernike polynomial and its corresponding coefficient, respectively.

The gain medium aberration is generated according to the mean and variance of the Zernike coefficients, which are derived from the experimental data of a thin disk laser in our previous measurement. The corresponding coefficients’ amplitude ranges are given in Table 2. We use these Zernike coefficients to generate a phase screen for convenience, as it does not affect the effectiveness of the proposed method.

Tables Icon

Table 2. Amplitude ranges of 4th-36th Zernike modes coefficient

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A randomly selected sample of simulation correction results of combined aberrations is presented in Fig. (4). The wavefront aberration of the intracavity gain medium is shown in Fig. 4(a), and the corresponding RMS value and the peak-to-valley (PV) value are 0.38 µm and 2.37 µm, respectively. The wavefront phases of the probe beam after a round-trip transmission along the optical axis of the resonator before and after the adaptive correction, are shown in Figs. 4(b) and 4(d), respectively. After intracavity aberrations correction, the RMS value of the residual wavefront error is 0.02 µm, which is rather close to the plane phase. The far-field intensity distributions corresponding to Figs. 4(b) and 4(d) are shown in Figs. 4(c) and 4(e), respectively. After intracavity aberrations compensation, the concentration of far-field intensity distribution is considerably improved. The β factor of the probe beam is defined as the square root ratio of the far-field area of the real beam to that of the ideal beam [25]. Here, the area contains 84% of total power in the far-field. The beam quality factor β is reduced from 8.2 to 1.2. It should be noted that the accumulated aberration of the probe beam after one round-trip propagation along the optical axis of the resonator is almost zero, indicating that the intracavity aberrations have been well compensated.

 figure: Fig. 4.

Fig. 4. The correction results of the probe beam of 976 nm. (a) The aberration distribution of intracavity gain medium. (b) and (c) are the near-field wavefront and far-field intensity before correction, respectively. (d) and (e) are the near-field wavefront and far-field intensity after correction, respectively.

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Next, the correction results of the oscillating laser are also presented. Figures 5(a) and 5(c) are the normalized near-field intensity distributions of the annular beam coupled out from the scraper before and after correction, respectively. The annular beam coupled out from the resonator has an inside aperture of 14 mm and an outside aperture of 21 mm. The uniformity of the near-field pattern can be evaluated by the F factor, which is defined as F = Imean/Imax. The larger the F factor is, the better the uniformity of the near-field pattern is. After the control loop of intracavity AO system is closed, the F factor is improved from 0.18 to 0.47, and the uniformity of the near-field pattern is obviously improved. This can be attributed to the fact that the DM can effectively correct intracavity aberrations before they diffract into output intensity fluctuations. The β factor of the oscillating laser is defined as the square root ratio of the far-field area of the real beam to that of the ideal beam. In this paper, the area contains 65% of total energy in the far-field. As shown in Figs. 5(b) and 5(d), the peak of the far-field pattern of the annular beam increased from 406 analog digital units (ADU) to 3910 ADU. Meanwhile, the beam quality factor β is reduced from 6.6 to 1.3. Therefore, the DM can effectively restore the near-diffraction-limited near-field intensity profile, which contributes to a better far-field correction.

 figure: Fig. 5.

Fig. 5. The correction results of the oscillating laser of 1064 nm. (a) and (b) are the normalized near-field intensity profile and the far-field pattern before correction, respectively. (c) and (d) are the normalized near-field intensity profile and the far-field pattern after correction, respectively.

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4. Experimental results of the passive resonator testbed system

To further demonstrate the effectiveness of the proposed correction strategy, an experimental setup of passive resonator testbed system is built as the first step toward a high-average power resonator, which is shown in Fig. 6(a). As a proof-of-concept experiment, the gain medium is replaced by a static aberration plate in our passive resonator testbed system. The aberration profile of the phase plate is presented in Fig. 6(b), and its RMS value and PV value are 0.33 µm and 1.79 µm, respectively. The amplitude of the static phase plate is within the correction range of the intracavity DM. The experimental setup consists of a probe laser (976 nm), two beam splitters (BS1 and BS2), a deformable mirror (DM), a static phase plate, two cavity mirrors (M1 and M2), a multi-function sensor (including a high-speed Shack-Hartmann wavefront sensor and a far-field sensor), and a low-cost desktop-based wavefront controller. The schematic diagram of the multi-function sensor is displayed in Fig. 6(c). The parameters of the experimental system are in accordance with the parameters in section 3. The closed-loop control system is developed based on Qt and C++ language and runs on the Win10 operating system. A traditional digital proportional-integral (PI) controller is used during the experiments [26]. The SHWFS is calibrated by directly detecting the probe beam with a plane phase, which is used as the closed-loop control system reference.

 figure: Fig. 6.

Fig. 6. The experimental setup. (a) The experimental setup of passive resonator testbed system. (b) Aberration distribution of the static phase plate. (c) The schematic diagram of the multi-function sensor.

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The experimental results of the closed-loop correction of an aberrated passive resonator (without gain medium) are demonstrated in Fig. 7. The uncorrected far-field pattern in Fig. 7(a) has a residual wavefront error RMS of 0.67 µm, and a calculated beam quality factor β of 7.3. With this as the starting point for the controller, the output in Fig. 7(b) has a residual wavefront error RMS of 0.03 µm and a beam quality factor β of 1.6 after 19 control loop iterations. After closed-loop compensation, the beam quality factor β is greatly reduced. The evolution curve of the RMS wavefront error is displayed in Fig. 7(c). The insets are examples of the wavefront distributions of the output probe beam before and after one typical correction. Figure 7(c) implies that the algorithm converges after about 19 iterations. To evaluate the performance of this correction strategy, the wavefront of the output probe beam is decomposed on Zernike polynomials basis, as shown in Fig. 7(d). After correction, the magnitude of the Zernike polynomial coefficients is significantly reduced.

 figure: Fig. 7.

Fig. 7. The experimental results of passive resonator testbed system. (a) The uncorrected far-field pattern of probe beam after introducing a phase plate. (b) The corrected far-field pattern of probe beam after 19 controller iterations. (c) The RMS value of wavefront error as a function of the controller iteration number (The insets are the wavefronts of the output probe beam before and after correction). (d) Zernike mode decomposition of the wavefront error.

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The surface shape of the intracavity DM can be calculated from the combination of converged actuator voltages and influence function. In our passive resonator testbed system, the resonator’s mode can be analyzed through the numerical model established in section 3. Figure 8 is the simulation results of the oscillating laser. Figures 8(a) and 8(b) are the normalized near-field intensity profile and the far-field pattern before correction, respectively. Figures 8(c) and 8(d) are the normalized near-field intensity profile and the far-field pattern after correction, respectively. With the assistance of the intracavity AO system, the F factor was improved from 0.23 to 0.44, and the uniformity of the near-field pattern is greatly improved. Meanwhile, the peak of the far-field pattern of the annular beam increased from 524 ADU to 3376 ADU. The beam quality factor β is reduced from 6.2 to 1.6. The above experimental results demonstrate the effectiveness of the adaptive correction of intracavity aberrations. Compared with the search-based optimization method, the mechanism of intracavity aberrations compensation is taken into account in our method. Hence the convergence is faster. The traditional geometrical model is only suitable for the case of small intracavity aberrations and needs to determine the position of intracavity aberrations. The method proposed in this paper does not have similar limitations.

 figure: Fig. 8.

Fig. 8. The simulation correction results of the oscillating laser of 1064 nm. (a) and (b) are the normalized near-field intensity profile and the far-field pattern before correction, respectively. (c) and (d) are the normalized near-field intensity profile and the far-field pattern after correction, respectively.

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5. Conclusion and discussion

In conclusion, an adaptive correction method for intracavity aberrations based on reconstruction matrix optimization is proposed to solve the problem that it is difficult to directly calculate the control signal of the intracavity DM due to the existence of expanding ray path in the PBCUR. What’s more, during the reconstruction matrix optimization process, the RMS value of the wavefront of the probe beam after one round trip along the optical axis of the resonator is used as a reference. A numerical simulation platform for adaptive correction of intracavity aberrations is established. The control voltages of the intracavity DM can be directly calculated from the SHWFS slopes by using the optimized reconstruction matrix. After compensation, the beam quality β is improved from 6.2 times diffraction limit to 1.6 times diffraction limit. The effectiveness and feasibility of the proposed method are verified by numerical simulations and passive resonator testbed experiments. We believe this method is of significance and can be applied in the areas of high-power unstable resonators with an intracavity AO system.

Funding

National Natural Science Foundation of China (11704382, 61805251, 61875203, 62105336); Youth Innovation Promotion Association of the Chinese Academy of Sciences (Y2021103).

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.

References

1. K. Yasui and J.-i. Nishimae, “Beam-mode calculations of a strongly pumped solid-state rod laser with an unstable resonator,” Opt. Lett. 19(8), 560–562 (1994). [CrossRef]  

2. B. Tang, T. Zhou, D. Wang, and M. Li, “Optical distortions in end-pumped zigzag slab lasers,” Appl. Opt. 54(10), 2693–2702 (2015). [CrossRef]  

3. Y. Guo, Q. Peng, Y. Bo, Z. Chen, Y. Li, L. Zhang, C. Shao, L. Yuan, B. Wang, J. Xu, J. Xu, H. Gao, Y. Xu, B. Lai, C. Su, S. Ma, and T. Cheng, “24.6 kW near diffraction limit quasi-continuous-wave Nd:YAG slab laser based on a stable–unstable hybrid cavity,” Opt. Lett. 45(5), 1136–1139 (2020). [CrossRef]  

4. Y. Guo, L. Zhong, L. Min, J. Wang, Y. Wu, K. Chen, K. Wei, and C. Rao, “Adaptive optics based on machine learning: a review,” Opto-Electron Adv. 5(7), 200082 (2022). [CrossRef]  

5. L. Sun, Y. Guo, C. Shao, Y. Li, Y. Zheng, C. Sun, X. Wang, and L. Huang, “10.8 kW, 2.6 times diffraction limited laser based on a continuous wave Nd:YAG oscillator and an extra-cavity adaptive optics system,” Opt. Lett. 43(17), 4160–4163 (2018). [CrossRef]  

6. R. H. Freeman, R. J. Freiberg, and H. R. Garcia, “Adaptive laser resonator,” Opt. Lett. 2(3), 61–63 (1978). [CrossRef]  

7. K. E. Oughstun, J. M. Spinhirne, and D. Anafi, “Intracavity adaptive optics. 4: Comparison of theory and experiment,” Appl. Opt. 23(10), 1529–1541 (1984). [CrossRef]  

8. N. L. Kai, L. H. Randall, M. J. Erik, D. Clifford Brent, N. F. Scott, and M. B. James, “Intracavity adaptive correction of a 10-kW solid state heat-capacity laser,” Proc.SPIE 5333, 53–61 (2004). [CrossRef]  

9. J. M. Spinhirne, D. Anafi, R. H. Freeman, and H. R. Garcia, “Intracavity adaptive optics. 1: Astigmatism correction performance,” Appl. Opt. 20(6), 976–984 (1981). [CrossRef]  

10. P. Yang, M. Ao, Y. Liu, B. Xu, and W. Jiang, “Intracavity transverse modes controlled by a genetic algorithm based on Zernike mode coefficients,” Opt. Express 15(25), 17051–17062 (2007). [CrossRef]  

11. K. E. Oughstun, “Aberration sensitivity of unstable-cavity geometries,” J. Opt. Soc. Am. A 3(8), 1113–1141 (1986). [CrossRef]  

12. K. E. Oughstun, “Intracavity adaptive optic compensation of phase aberrations. I: Analysis,” J. Opt. Soc. Am. 71(7), 862–872 (1981). [CrossRef]  

13. X.-J. Wang, “Non-linear corrections to aberration sensitivity of unstable-cavity geometries,” Opt. Express 16(26), 21223–21232 (2008). [CrossRef]  

14. M. A. Vorontsov, G. W. Carhart, and J. C. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. 22(12), 907–909 (1997). [CrossRef]  

15. J. Wenhan and L. Huagui, “Hartmann-Shack wavefront sensing and wavefront control algorithm,” Proc.SPIE 1271, 82–93 (1990). [CrossRef]  

16. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69(7), 972–977 (1979). [CrossRef]  

17. G. Dai, “Modal wave-front reconstruction with Zernike polynomials and Karhunen–Loève functions,” J. Opt. Soc. Am. A 13(6), 1218–1225 (1996). [CrossRef]  

18. S. Ma, P. Yang, B. Lai, C. Su, W. Zhao, K. Yang, R. Jin, T. Cheng, and B. Xu, “Adaptive Gradient Estimation Stochastic Parallel Gradient Descent Algorithm for Laser Beam Cleanup,” Photonics 8(5), 165 (2021). [CrossRef]  

19. S. A. U. Islam and D. S. Bernstein, “Recursive Least Squares for Real-Time Implementation [Lecture Notes],” IEEE Control Syst. 39(3), 82–85 (2019). [CrossRef]  

20. A. Fox and L. Tingye, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron. 4(7), 460–465 (1968). [CrossRef]  

21. 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]  

22. E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14(8), 1874–1889 (1975). [CrossRef]  

23. P. Yang, Y. Ning, X. Lei, B. Xu, X. Li, L. Dong, H. Yan, W. Liu, W. Jiang, L. Liu, C. Wang, X. Liang, and X. Tang, “Enhancement of the beam quality of non-uniform output slab laser amplifier with a 39-actuator rectangular piezoelectric deformable mirror,” Opt. Express 18(7), 7121–7130 (2010). [CrossRef]  

24. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976). [CrossRef]  

25. B. Lai, L. Dong, S. Chen, G. Tang, W. Liu, S. Wang, X. He, K. Yang, P. Yang, B. Xu, C. Wang, X. Liu, Q. Pang, and Y. Liu, “Hybrid adaptive optics system for a solid-state zigzag masteroscillator power amplifier laser system,” IEEE Photon. Technol. Lett. 14(9), 091402 (2016). [CrossRef]  

26. C. Yang, B. Xu, B. Lai, C. Su, S. Ma, R. Jin, S. Wang, and P. Yang, “Intra-Cavity Adaptive Correction of a Unidirectional Nd:YAG Slab Unstable Travelling-Wave Resonator,” IEEE Photonics Technol. Lett. 33(23), 1333–1336 (2021). [CrossRef]  

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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Figures (8)
Fig. 1.
Fig. 1. (a) The schematic diagram of adaptive aberration compensation inside a positive branch confocal unstable resonator based on the round-trip detection of the probe beam. (b) The schematic diagram of the positive branch confocal unstable resonator.

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Fig. 2.
Fig. 2. The unfolded lens equivalent of the probe beam (BS1 and BS2 are not shown for simplicity).

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Fig. 3.
Fig. 3. The configuration between the sub-apertures and the actuators.

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Fig. 4.
Fig. 4. The correction results of the probe beam of 976 nm. (a) The aberration distribution of intracavity gain medium. (b) and (c) are the near-field wavefront and far-field intensity before correction, respectively. (d) and (e) are the near-field wavefront and far-field intensity after correction, respectively.

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Fig. 5.
Fig. 5. The correction results of the oscillating laser of 1064 nm. (a) and (b) are the normalized near-field intensity profile and the far-field pattern before correction, respectively. (c) and (d) are the normalized near-field intensity profile and the far-field pattern after correction, respectively.

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Fig. 6.
Fig. 6. The experimental setup. (a) The experimental setup of passive resonator testbed system. (b) Aberration distribution of the static phase plate. (c) The schematic diagram of the multi-function sensor.

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Fig. 7.
Fig. 7. The experimental results of passive resonator testbed system. (a) The uncorrected far-field pattern of probe beam after introducing a phase plate. (b) The corrected far-field pattern of probe beam after 19 controller iterations. (c) The RMS value of wavefront error as a function of the controller iteration number (The insets are the wavefronts of the output probe beam before and after correction). (d) Zernike mode decomposition of the wavefront error.

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Fig. 8.
Fig. 8. The simulation correction results of the oscillating laser of 1064 nm. (a) and (b) are the normalized near-field intensity profile and the far-field pattern before correction, respectively. (c) and (d) are the normalized near-field intensity profile and the far-field pattern after correction, respectively.

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Tables (2)

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Table 1. The distances between these optical elements in Fig. 2

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Table 2. Amplitude ranges of 4th-36th Zernike modes coefficient

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Equations (18)

Equations on this page are rendered with MathJax. Learn more.

G 2 n × 1 = M 2 n × m U m × 1 ,
U m × 1 = R m × 2 n G 2 n × 1 .
∂ J ∂ R = ∂ J ∂ U × ∂ U ∂ R = ∂ J ∂ U × G T .
J + = J [ U + Δ U ] J − = J [ U − Δ U ] ,
( ∂ J ∂ U ) i ≈ ( J + − J − ) δ u i / ( δ u i ) 2 , i = 1 , 2 , … , 52.
R = R − α ∂ J ∂ U G T ,
U + Δ U = R G pos U − Δ U = R G neg ,
Δ G = ( G pos − G neg ) / 2.
R ( t ) = R ( t − 1 ) + e ( t ) G T ( t ) P ( t ) ,
e ( t ) = Δ U ( t ) − R ( t − 1 ) Δ G ( t ) .
K ( t ) = P ( t − 1 ) Δ G ( t ) Δ G T ( t ) P ( t − 1 ) Δ G ( t ) + μ ,
P ( t ) = [ P ( t − 1 ) − K ( t ) Δ G T ( t ) P ( t − 1 ) ] / μ .
R ( 0 ) = R push - pull , P ( 0 ) = δ − 1 I ( 0 < δ ≤ 1 ) ,
R = ( 1 − w ) × R 1 + w × R 2 ,
ϕ DM ( x , y ) = ∑ i = 1 52 u i f i ( x , y ) ,
f i ( x , y ) = exp ⁡ [ ln ⁡ ( ω ) × ( ( x − x i ) 2 + ( y − y i ) 2 / d ) a ] ,
T DM ( x , y ) = exp ⁡ [ i k ϕ DM ( x , y ) ] ,
ϕ Gain ( x , y ) = ∑ i = 4 36 a i Z i ( x , y ) , T Gain ( x , y ) = exp ⁡ [ i ϕ Gain ( x , y ) ] ,
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