Intracavity Laser Beam Control and Formation


2. Control of the high power CO2 laser beam

2.1. Receiving the Q-switch regime of CO2 laser

Traditionally, laser resonators includes optical elements with rigidly fixed surface profiles intended for obtaining specific beam output quality. But when you need to change some output parameters of laser beam (for example get different output mode configuration or even change the regime of laser generation) you need to reconstruct all laser cavity or alter the block of power supply. Such operations are rather expensive and sometimes takes a long period of time. That is why the use of mirrors with controllable flexible surface in laser is of large interest. This gives the opportunity for creation of flexibly tunable resonator of a new type.

As it was mentioned before theoretical estimations3, 4 showed the possibility to change the beam structure by minor surface deformation (less than wavelength) of flexible resonator mirror. And the possibility of intracavity compensation of the simplest aberrations of the wavefront to increase beam quality have been demonstrated in experiments5, 6, 7.

Industrial CW CO2 laser TL-5 developed and manufactured in the NICTL RAN 22 is characterized by high beam quality (divergence of less then 1 mrad at 55 mm output beam diameter) and output power 5kW. Fig. 8 shows the scheme of the laser resonator.

Scheme of CO2 - laser resonator
Fig.8. Scheme of CO2 - laser resonator

The unstable confocal resonator with magnification M = 2 consists of two spherical mirrors with radii of curvature R1 = 26 m and R2 = -13 m and 4 flat folding mirrors between them. Resonator length was L = 6.5 m. One of the flat mirrors was replaced by the cooled flexible mirror based on bimorph piezoelement. The output beam of this laser had the shape of a ring with outside aperture of 55 mm and inside aperture - 25 mm. Well known parameters of this laser cavity g1 and g2, taking into a consideration deformable mirror could be written in the following way:

formula,

where R is the radius of curvature of flexible mirror under control voltage, L=6.5 m - the length of laser cavity. For specified resonator parameter (with 1/R=0) we have g1 =1.5 and g2 =0.75.

When a static voltage +300 V was applied to all electrodes of the flexible mirror radius of curvature of its surface R changed from plus to minus 200 m. The value g2 was changing correspondingly in the range of 0.69-0.82. In the dynamic regime of mirror action the cosine voltage with frequency in the range of 1 - 10 kHz was applied to mirror electrodes. At the mirror resonance frequency - 3.8 kHz the deformation of mirror increased up to R= +50 m and g2 value was changed from 0.49 to 1.0. The magnification M of unstable non-confocal resonator could be given as :

formula.

Fig.9 shows the dependence of resonator transmittance T = 1-1/M2 upon g2 for g1 = 1.5. One can see that when g2 changes from g2 < 0.67 (the case of stable resonator) to 1.0 the transmittance T varies from 0 and 0.93.

formula
Fig.9. Transmittance T of laser cavity vs g2 in case g1=1.5

So, changing the curvature of the flexible mirror surface by applying cosine voltage to the control electrodes one could change the transmittance of the telescopic unstable resonator with the same frequency and modulation of the laser output beam power could be obtained. The cosine voltage of frequency up to 10 kHz and amplitude up to 300V was applied for this purpose. Under control voltage of more than 1 kHz the period of the mirror surface oscillation could be compared with the life-time of the CO2 laser active medium. Thus the Q-switch regime of laser generation could be obtained.

In experiments the dependence between output beam power and control voltage amplitude and frequency was studied. The periodical laser generation regime with 100% power modulation depth was obtained at near-resonance frequencies - 3.8 and 7.6 kHz. The pulse peak power exceeded the average power threefold. The average power drop was insignificant (<10%). Fig.10a,10b demonstrate output beam pulses and control voltage pulses applied to deformable mirror at resonance frequencies. Here one can see that even 120 V at 3.8 kHz and 80 V at 7.6 kHz mirror control voltage is enough to get 100% power modulation.

Oscillograms of output beam power Pout and control signal Ucont at frequency 3.8 kHz Oscillograms of output beam power Pout and control signal Ucont at frequency 7.6 kHz
Fig.9. Oscillograms of output beam power Pout and control signal Ucont at frequency 3.8 (a) and 7.6 (b) kHz

Together with the power modulation the aperture of the output laser beam also changed from 35 to 65 mm at the plane where the convex resonator mirror was placed. At the same time changing the curvature of active mirror we managed to vary the beam convergence angle of industrial laser TL-5 in the range of 2 mrad. Such change of beam convergence by flexible mirror in resonator can be used in material processing for dynamic focusing of laser radiation. In different technological processes such as deep penetration welding or thick material processing the displacement of the laser focal spot along the beam axis is needed to concentrate energy in the developing area. This focal spot displacement usually was realized by moving the focusing lens along the laser beam axis or using extracavity active mirror24. The technique of change the beam convergence angle using flexible mirror within the resonator is simple and provides the focal spot displacement in a wide range with low deformation of the mirror surface.

Thus the possibility to obtain the pulse-periodical generation and Q-switch regime of continuous pumped industrial laser was demonstrated. The use of the intracavity active mirrors do not require any significant modification of the whole laser. At the same time it broaden the possible spheres of technological lasers application and can improve different technological processes.

2.2. Super-Gaussian laser intensity output formation by means of adaptive optics

It is usually desirable for laser technology to have a laser device operating in a single (mostly fundamental) mode for a good quality output beam and providing an efficient power extraction at the same time. A gaussian beam is usually relatively narrow and results in a poor energy extraction from the gain medium. Exponential field distribution profile with the order more than 2 is expected to provide better energy extraction (in comparison with the higher standard gaussian profile) while being less affected by aperturing since it has a large beam size and a fast decreasing of intensity near the edges.

One of such special intensity distributions (super-gaussian) can be formed by intracavity graded reflectivity13, 25, and graded-phase mirrors12, 26, 27. The experimental results have shown a significant increase of the monomode laser energy output compared with the output of a conventional semiconfocal resonator12, 27, 28. But the correctors have a serious drawback: since the mirror surface profile is rigid, every change of laser parameters needs its own special mirror.

In this work we suggest to overcome this problem by the use of an intracavity flexible bimorph mirror inside laser cavity for the given intensity distribution formation.

2.2.1. Basic principles

The geometry of adaptive mirror resonator is shown in Fig.11.

Schematic setup
Fig.11. Schematic setup of the CW CO2 laser with adaptive mirror

It consists of plane output coupler and active mirror separated corresponds to the industrial continuos discharge CO2 laser from the coupler at the distance L = 2m. Such geometry ILGN-704 produced by "Istok", Fryazino, Russia. We considered, that the active mirror would be a bimorph water cooled one with 3 rings of electrodes, as shown on fig. 3.2 b - all 8 electrodes of the middle and the outer rings were connected and thus formed the rings of electrodes.

Azimutal symmetry can be assumed which allows us to use the one-dimentional Huygens-Fresnel integral equations29, 30 to calculate the amplitude of a mode in the empty laser resonator, namely

formula, (3.1)

formula, (3.2)

where γi is the eigenvalue and Ψi(ri) is the eigenmode of the resonator, ri 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,

formula. (3.3)

formula

Here Jl is the Bessel function of order l (we take into account only the lowest transverse mode with l=0), 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.

The algorithm of the calculation of formation of the given fundamental mode intensity distribution is so called "inverse propagation method" described in 12, 26. The initial field distribution Ψ1(r,φ) is specified on the output mirror. The back-propagation of the laser beam through all resonator's elements to active corrector is calculated using Huygens-Fresnel integral Eqs. (3.1, 3.2). In the plane of the active corrector the wavefront is calculated and served to determine the appropriate mirror phase profile φmirror:

φmirror=-φbeam.

Such phase profile of laser beam could be completely reconstructed by ideal corrector (graded-phase mirror)12, 26. In our case bimorph mirror can approximate the necessary phase profile with some error. Such minimal RMS error was calculated using the experimentally measured response functions of the mirror. In other words:

formula, (3.4)

where Z(r) is the profile to be reconstructed, Fi(r) are response functions of adaptive mirror electrodes, Ui are weight coefficients corresponding to the voltages applied to each electrode.

Left side of the Eq.(3.4) has a minimum when it's first derivative equals to 0:

formula. (3.5)

From the Eq.(3.5) the applied voltages to approximate the necessary shape of laser beam were determined.

Described above procedure gives us reconstructed mirror surface φmirror(r) which was substituted into Eq.(3.3). To solve integral equations (3.1), (3.2) we used the Fox and Li iterative method of successive approximations29, 30, to take into account edge diffraction and non-ideal reproduction of the necessary phase profile of the laser beam by adaptive mirror.

2.2.2. Main resultss

Main parameters of the laser resonator (Fig.11) are: Fresnel numbers N1 = b2/(Bλ), N2= a2/(Bλ) and geometrical factor is G = (1-L/R2) where λ = 10.6 μm is wavelength, R2 = 4m is the radius of curvature of active mirror and L = 2m is the length of the resonator cavity. The initial field distribution on the plane coupler is chosen as Ψ(r)=Exp(-(r/w))², where n determines the order of super-gaussian function and w is the beam waist. The particular beam waist was calculated according to the methods of moments of laser beam31:

formula. (3.6)

is the beam quality factor formula, defined for a super-gaussian beam of the order of n in 31 as:

formula, (3.7)

Γ(x) is the Gamma function.


2.2.2.1. Formation of a super-gaussian beam of the 4-th order (n=4)

The super-gaussian beam waist was calculated from Eqs.(3.6, 3.7): w=3.1 mm. Fig.12(a) represents the phase distribution of super-gaussian beam (curve 1 (thin)) propagated through resonator at the distance L=2m from the output plane coupler to adaptive mirror. Curve 2 (thick-dotted) (Fig.12(a)) illustrates the phase profile of the active mirror reproducing with RMS error = 0.3% the phase shape of laser beam. Fig.12(b) shows intensity distributions of laser beam on the plane coupler.

Formation of a super-gaussian beam of the 4-th order
Fig.12. Formation of a super-gaussian beam of the 4-th order, N1=1, N2=4.7, G=0.5. a) the phase profile on the active mirror; (b) normalized intensity distributions on plane coupler

Curve 1 corresponds to the fundamental gaussian mode of the same resonator but with pure spherical mirror. Curve 2 (thin-solid) shows the given super-gaussian relative intensity profile, curve 3 (thick-dotted) - profile produced with ideal corrector (graded-phase mirror), approximating completely the necessary phase profile of the laser beam, and the curve 4 (thick) corresponds to the intensity distribution in the resonator with adaptive mirror.

One may see from Fig.12(b) that applying the active corrector (curve 4) the mode volume of the output intensity distribution increases in 1.5 times in comparison with the pure gaussian beam (curve 1) while diffraction losses per transit decrease in 1.7 times.

Users of lasers often dislike top-hat intensity distributions for their side lobes in far field patterns. Such side lobes contain about 14% of the total energy26. However, a super-gaussian function is an apodized top-hat distribution; the distinction is important since the smoother edge implies a reduction of higher spatial frequencies. Hence the far field pattern with reasonable n (n < 10) should show an important reduction of the side lobes. Moreover, the formed super-gaussian distribution is not exactly the super-gaussian function: it's form has been changed by edge diffraction and by limited possibility of active mirror to form the necessary phase profile. That is why the side lobes of the beam formed by active mirror contain only 2% of the total energy (dashed curve 3 on Fig.13) that makes this intensity profile very attractive for industrial applications. For a comparison curve 2 in Fig.13(a),(b) represents the far field pattern for the super-gaussian beam formed by an ideal (graded-phase) corrector.

Formation of a super-gaussian beam profile of the 4-th order
Fig.13. Formation of a super-gaussian beam profile of the 4-th order. (a) Intensity distributions in far-field zone: 1 - gaussian beam, 2 - beam formed by ideal corrector (graded-phase mirror), 3(thin curve) - beam formed by active mirror; (b) fragment of the same intensity distributions

2.2.2.2. Formation of a super-gaussian beam of the 6-th order (n=6)

The beam waist of the given initial distribution is 3.3 mm. Fig.14(a) represents the exact phase distribution of the given super-gaussian beam and the phase profile of active mirror (RMS error of the approximation is about 0.1%). Fig.14(b) shows how the active mirror can form intensity distributions on the plane coupler. The far-field results are very close to the case of formation of the 4-th order of the super-gaussian beam represented in Fig.13(a)(b).

Formation of a super-gaussian beam profile of the 6-th order
Fig.14. Formation of a super-gaussian beam profile of the 6-th order. N1=1, N2=14.1, G=0.5. (a) solid - the phase profile of laser beam to be reconstructed and dotted - phase profile of active mirror; (b) normalized intensity distributions: 1 (thick) - the given initial super-gaussian beam, 2 (thin-dotted) - intensity formed by active corrector

The mode volume of output intensity distribution increases for this case by a factor of 1.3 in comparison with pure gaussian beam while diffraction losses per transit decrease in 1.5 times.

The diffraction analysis presented in this paper has confirmed the possibility to form a wide class of interesting for technology intensity distributions of the fundamental mode in the near-field zone by flexible mirror. Such diffraction analysis contains an inverse-propagation calculation, the approximation of laser beam phase with experimentally measured response functions of the sample of the semipassive bimorph flexible mirror with three ring controlling electrodes and numerical Fox and Li simulations. It has been shown that in all cases mode volume of formed intensity distributions increased in 1.3-1.6 times while diffraction losses per transit decreased in 1.5-1.7 times in comparison with pure gaussian mode of the resonator with the same parameters but with spherical mirror. Sure, in case of formation of the super-gaussian modes we loose in beam quality in comparison with the gaussian beam profile: for example, in our case for the 4-th order of super-gaussian beam M2=1.1. But the output radiance defined as

formula

will also increase in 1.07 - 1.25 times (depending on the type of the order of the super-gaussian laser beam).

These results may be improved by making some optimisation procedure - varying the parameters of a resonator or the radii of the electrodes r1 and r2 of the bimorph corrector. The obtained far field patterns of formed modes are also very suited for technological applications.

he experiments carried out in NICTL proved the correctness of these calculations and were in good agreement with the theoretical results. As it was shown in the experiments we were able to increase the output power of TEM00 mode about 1.1 times, and increase the peak intensity in far field (in the focus of lens) in 1.7 times32.



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