## Abstract

The formation of the given low loss and large beamwidth doughnut-like fundamental mode of stable resonator by an intracavity flexible mirror is discussed. The mirror is a bimorph one with one round and two ring controlling electrodes. An inverse propagation method is used to determine the appropriate shape of the controlled mirror. The mirror reproduces the shape with minimal RMS error by combining weights of experimentally measured response functions of the mirror sample. The voltages applied to each mirror electrode are calculated. The calculations are carried out for industrial CW CO_{2} laser.

© Optical Society of America

## 1. Introduction

The idea of exploiting an aspherical resonator to improve the laser beam parameters is fairy old [1–3]. For various reasons, in particular technical ones, this approach has never been very popular. Advances in technology now permit to solve some of these difficulties. For example, to make graded-phase mirrors (GPM) [4–8] diamond turning machining [9], a process that permits submicron precision, is now used.

The experimental results with GPM used in pulsed TEA and CW CO_{2} lasers have shown an increase of 50% of monomode energy output as compared to the output of a conventional semiconfocal resonator [6–8].

But the GPM can only serve for the specific application they were design for: every change of laser parameters needs its own unique mirror. On the other hand with the help of just one flexible controlled mirror one can form a number of the given laser outputs. It is also possible to compensate different phase distortions coursed for example by thermal deformations of laser mirrors or by aberrations of active medium. Such phase distortions would be able to destroy the given laser output intensity distribution and to predict such distortions completely is not possible because some of them depends for example on the pumping power, the inhomogeneity of active medium and so on. That is why for solving different tasks it is more universal and perspective to use flexible controlled mirrors.

In this work we propose to use phase-conjugate resonator with intracavity flexible bimorph mirror for the given intensity distribution formation.

## 2. Adaptive mirror design

We used a flexible mirror composed of a semipassive bimorph element [10–12], made in Scietific Research Center for Technological Lasers of Russian Academy of Sciences. The construction of adaptive mirror is shown in Fig.1. It consists of copper plate (Fig.1) firmly glued to a plane actuator disc. The last one is made of two piezoceramic discs, soldered together and polarized normally to their surfaces. The thickness of each piezoceramic disc is 0.35 mm while the thickness of the copper plate is 2.5 mm. The interface between the two piezoceramic discs contains a continuous conducting “ground’ electrode. Another continuous conducting electrode between the piezodisc and the copper plate “e1” is used to control the curvature of the whole mirror. Two controlling electrodes “e2”, “e3” having a form of concentric rings were attached to the outer surface of the piezodisc.

The 3-D surface profiles of deformation of the corrector with applied voltage 40 V and 20 V to the electrodes (so called “response functions”) are given in Fig.2 and were measured using a modified Fizeau interferometer. In order to check the behavior of mirror electrodes such interferograms were analyzed and expansion coefficients of the response functions of all electrodes in terms of the Zernike polynomials were found.

## 3. Results and discussions

The geometry of studied stable adaptive mirror resonator is shown in Fig.3. The cavity consists of plane output coupler and active mirror separated from the coupler at the distance L=2m. Such geometry corresponds to the industrial continuos discharge CO_{2} laser ILGN-704 produced by “Istok”, Fryazino, Russia.

Azimutal symmetry can be assumed which allows us to use the one dimentional Huygens-Fresnel integral equations [13–14] and to calculate the back-propagation of the desired laser beam Ψ_{1}(*r*
_{1}) from output coupler through all resonator’s elements to adaptive mirror:

where γ_{i} is the eigenvalue and Ψ_{i}(*r _{i}*) is the eigenmode of the resonator, r

_{i}, are radial coordinates, i=1 is related to plane output mirror of the diameter 2b, i=2 - to the active one of the diameter 2a,

Here J_{0} is the Bessel function of zero order, A, B, D are the constants determined by the ABCD ray matrix of the laser resonator. We consider empty resonator, so A=1, B=L, C=0, D=1.

In the plane of adaptive mirror the wavefront of the desired laser beam was extracted and served to determine the appropriate shape of the bimorph flexible mirror. Such surface profile was reconstructed with minimal RMS error by combining with appropriate weights the experimentally measured response functions of the mirror (see Fig.2). The weights correspond to the voltages applied to each mirror electrode. Convergence to the given initial field distribution was calculated by the Fox and Li iterative method of successive approximations [13–14].

#### 3.2. Formation of a doughnut-like beam.

It was shown that a doughnut-like laser beam has less nonlinear distortions while propagating in nonlinear and turbulent medium than other forms of laser beams [15–16]. That is why for more effective laser beam energy transportation it is preferable to form a doughnut-like form of a beam intensity [15–16]. Traditional way to get it is to cut the central part of a gaussian fundamental mode. One may see that in this case we have high enough additional power losses. In this chapter it is shown that using intracavity active mirror it is possible to form doughnut-like beam without any additional power losses and even with less diffraction losses than traditional pure gaussian fundamental mode has.

The initial field distribution of the doughnut-like beam on the plane coupler is chosen as Ψ(r)= (r+0.1)^{2}Exp(-((r+0.1)/3.1)^{4}). (It was found out empirically that the small decentration value, for example 0.1, had to be added for the diffraction didn’t change drastically the desired doughnut-like intensity shape of fundamental mode during Fox-Li numerical calculations.) Main parameters of the laser resonator (Fig.3) are: Fresnel numbers N_{1}= b^{2}/(Bλ)=1, N_{2}= a^{2}/(Bλ)=4.7 and geometrical factor is G =(1-L/R_{2})=0.5 where λ=10.6 μm is wavelength, R_{2} = 4m is the radius of curvature of active mirror and L=2m is the length of the resonator cavity.

Fig.4 (a) represents the phase distribution of the doughnut-like beam (lilac curve) propagated back at the distance L=2m from the output plane coupler (Fig.3). Blue curve in Fig.4(a) illustrates the phase profile of the active mirror reproducing the phase shape of laser beam with RMS error 0.7%. Fig.4(c) shows intensity distributions on the plane coupler. Lilac curve corresponds to the given initial relative intensity profile, blue - profile produced with flexible mirror. For a comparison we plotted the doughnut-like fundamental mode formed by GPM, which reproduces the shape of the given laser beam phase ideally.

Applying the active mirror (blue curve in Fig.4(c)) the diffraction losses per transit decrease by a factor of 1.4 in a comparison with pure gaussian beam of the resonator with the same parameters but with pure spherical back mirror. In this case the diffraction losses of first order mode TEM_{01} were more than in two times higher.

The far field pattern of such intensity distribution contains about 96% of total energy in the main kern (blue curve in Fig.5,6) that makes this intensity profile very attractive for industrial applications. For a comparison we plotted far field pattern for a pure gaussian fundamental mode of the same resonator but with spherical back mirror (lilac curve). More detailed ends of the intensities in far field zone are shown in Fig.6.

The voltages applied to each electrode to form doughnut-like fundamental mode are given in Table1.

#### 4.2. Formation of a doughnut-like beam Ψ(r)=(r+0.01)^{1/2} Exp(-((r+0.01)/3.1)^{4})

Fig.7(a) represents the exact phase distribution to be reconstructed (lilac curve) and the phase profile of active mirror- blue curve. RMS error of the approximation the is 0.4%. Fig.7(b) shows how the intracavity flexible mirror can form intensity distributions on the plane coupler. One may mention that due to diffraction intensity offset |ψ(0,φ)|^{2} appears. The far-field patterns are very close to the previous case of formation shown in Fig.5,6.

The mode volume of the output intensity distribution increases by a factor of 1.2 in comparison with pure gaussian fundamental mode TEM_{00} while diffraction losses per transit decrease in 1.8 times. The voltages applied to each electrode are given in Table1.

## 4. Conclusion

The diffraction analysis presented in this paper has confirmed the possibility to form interesting for laser energy transportation systems special form of intensity distribution of fundamental mode in near-field zone by flexible mirror. Such diffraction analysis contains:

- Inverse-propagation calculation,
- Approximation of laser beam phase shape with experimentally measured response functions of the sample of the semipassive bimorph flexible mirror having two ring and one round controlling electrodes,
- Minimization of the RMS error of such approximation,
- Numerical Fox and Li simulations.

It has been shown that it is possible to form doughnut-like beams without any additional power losses and even with less diffraction losses and increased mode volume than pure gaussian fundamental mode of the resonator with the same dimensions but with spherical mirror.

## Acknowledgments

The authors wish to thank Prof. P.-A. Bélanger and Dr. C.Paré, both from Laval University, Quebec, Canada for their fruitful collaboration in the beginning of this work. We are also grateful to prof. V.I Shmal’gauzen from Moscow State University for the comments of the work and R. Van Neste from Fiso Technologies, Inc., Quebec, Canada for valuable discussions and for his thesis kindly presented to the authors.

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