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1 2 Laser Beam Shaping by Means 3 4 6 5 of Flexible Mirrors 6 7 8 T. Yu. Cherezova, and A. V. Kudryashov 9 10 CONTENTS 11 12 13 I. Introduction ...... 167 14 II. Different Types of Bimorph Mirrors ...... 168 15 III. Formation of a Desired Laser Beam at the Laser Resonator 16 Output by Means of an Intracavity Flexible Corrector...... 175 17 IV. Main Results ...... 176 ð ¼ Þ 18 A. Formation of a super-Gaussian Beam of the Fourth Order n 4 ...... 176 ðn ¼ Þ 19 B. Formation of a super-Gaussian Beam of the Sixth Order 6 ...... 178 ðn ¼ Þ 20 C. Formation of a super-Gaussian Beam of the Eighth Order 8 ...... 179 21 D. Optimization of Laser Parameters...... 181 22 V. Formation of an Annular Beam...... 181 23 VI. Experimental Formation of a super-Gaussian Beam by Means of Bimorph Flexible Mirror...... 183 24 3þ 25 VII. YAG:Nd Laser. Formation of a super-Gaussian Output 26 Beam — Numerical Results ...... 188 27 VIII. Conclusion...... 189 28 Acknowledgements...... 189 29 References...... 189 30 31 32 33 I. INTRODUCTION 34 35 The widespread use of lasers in technological processes highlights the problem of controlling 36 the laser beam parameters. For example, metalcutting processes typically demand the tightest 1 37 beam focus possible. On the other hand, metalhardening processes require a laser beam with the 2 38 most uniform transverse irradiance distribution possible. For several laser applications in material 39 processing and manufacturing, nonlinear conversion of a laser beam to a shape with uniform 1,2 40 rectangular cross-section is often desirable. Examples of such uniform beams include highly 3 4,5 6,7 41 multimode laser beams, flattened Gaussian beams, and super-Gaussian beams. 42 In general the specified laser irradiance distribution can be formed in different ways — both 8 9 43 extracavity and intracavity. The main advantages of intracavity shaping include not only the 44 ability to form the desired irradiance structure, but also the ability to increase the laser output 7 45 power. One of the most well-known intracavity approaches is to apply graded reflectivity mirrors. 46 However, such mirrors introduce large intrinsic power losses and thus are suitable only in lasers 47 with high-gain active media and generally with unstable resonators. Another intracavity approach is 10,11 48 to use graded-phase mirrors. However, such mirrors can only serve in the specific applications 49 for which they were designed; every change of laser parameters requires its own unique mirror. A 50 single flexible controlled mirror, on the other hand, can form a number of desired laser outputs. 51 Flexible controlled mirrors can also compensate for various phase distortions caused, for example, ARTICLE IN PRESS

168 Laser Beam Shaping Applications

52 by thermal deformations of resonator mirrors or by aberrations of active media. Uncontrolled phase 53 distortions can destroy the desired laser output distribution. Accurate prediction of phase distortions 54 is not possible because they can depend, for example, on the laser pumping power, the inhomo- 55 geneity of the active medium, and so on. For general intracavity beamshaping tasks it is thus easier 56 and more universal to use flexible controlled mirrors. 57 As the key element of any adaptive optical system, the flexible corrector and its properties 58 determine the performance of the whole system. Demands on the wavefront corrector element 59 include the following: 60 61 † A wide range of surface deformation 62 † A small number of control actuators 63 † Efficiency of reproducing wavefront aberrations 64 † Temperature stability of the surface figure 65 † The ability, if necessary, to conjugate with a wavefront sensor (essential for closed-loop 66 applications) 67 † Simplicity of fabrication and application 68 † Low cost 69 70 Bimorph mirrors are generally the most suitable correctors to satisfy these demands. It has 71 been shown that semipassive bimorph mirrors with 13 actuators effectively reproduce loworder 72 wavefront aberrations with large amplitudes.12 For example, bimorph mirrors can theoretically 73 reproduce aberrations with root-mean-square (RMS) amplitudes of the following: defocus: 0.3%, 74 Q1 astigmatism: 0.7%, coma: 5%, and spherical aberration: 6%. 75 On the other hand, deformable bimorph mirrors are not standard optical elements. They 76 are relatively thin and consist of several different material layers with different properties. As a 77 result, there are no standard optical technologies to produce bimorph mirrors. Special methods 78 of piezoceramic treatment, surface polishing, reflecting coating deposition, and so on, have to be 79 developed to produce highquality bimorph correctors. Some applications of bimorph correctors 80 in lasers and imaging optical systems require an especially wide range of deformation and high 81 stability of the mirror surface. Some of these problems are considered in the next section. 82 83 84 II. DIFFERENT TYPES OF BIMORPH MIRRORS 85 86 Pure bimorph mirrors consist of two comparatively thin piezoceramic plates polarized in opposite 87 directions. Manufacturing issues with these pure bimorph flexible mirrors have led to the 12,13 88 development of so-called semipassive bimorph correctors. A traditional semipassive bimorph 89 mirror consists of two joined plates: a comparatively thick passive glass or metal substrate and a 90 thin active piezoceramic plate (see Figure 6.1). The operational concept for the semipassive 91 bimorph corrector is similar to that for the pure bimorph case, but its sensitivity is lower. 92 Application of an electrical signal to the electrodes of the piezoceramic plate causes tension of the 93 piezodisc due to the inverse piezoelectric effect. The piezodisc thus expands in the radial direction. 94 The bonded substrate prevents free expansion, and this results in bending of the reflective surface. 95 To reproduce different types of aberrations with the help of such a corrector, the outer electrode 96 is divided among several controlling electrodes. The size as well as the number of controlling 97 electrodes depends upon the type of the aberrations to be corrected. Sometimes it is useful to 98 introduce into the mirror design an additional piezodisc with one round electrode for reproducing 99 defocus (Figure 6.1). 100 Table 6.1 shows the main characteristics for piezoceramic materials that are currently 101 commercially available. Piezoceramic material PKR-7M has the largest value of the piezomodule 102 d31: This material is thus the most sensitive one for the development of bimorph mirrors. ARTICLE IN PRESS

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103 Substrate 104 105 106 Common 107 electrode 108 Electrode for 109 defocus control Piezo-discs 110 Electrodes for low order 111 aberration reproduction 112 113 8 1 114 115 9 116 7 2 117 12 10 13 118 6 3 119 11 120 121 5 4 122 123 FIGURE 6.1 Design of a semipassive bimorph mirror. 124 125 126 The drawbacks of PKR-7M include a low Curie temperature and the fact that it is a soft 127 segnetomaterial. Since the Curie temperature defines the stability of material properties at high 128 temperatures, a low Curie temperature demands low-temperature technologies for ceramic 129 Q2 treatment, bonding of the mirror components, and coating deposition. 130 Figure 6.2 represents a comparison of the sensitivity of three experimental bimorph mirrors 131 with different types of piezoceramic material. All mirrors include a glass substrate of 2.5 mm 132 thickness and 40 mm aperture and PKR6, PIC151, or PKR-7M ceramic discs of 0.3 mm thickness 133 and 40 mm aperture. The same þ100 V potential was applied to the electrodes of the mirrors. 134 Surface deformation dynamics were recorded by a Shack-Hartmann sensor every 50 msec. 135 Sensitivity for the PKR-7M ceramic mirror was 1.4 and 2 times higher than for mirrors with PIC151 136 and PKR6 ceramics, respectively. For PKR-7M, initial measurements upon application of the 137 control voltage showed 3.2 mm of deformation. This increased to 3.33 mm in 2 sec, then stayed 138 constant for the remainder of the measurement. This indicates that the creep effect for PKP-7M 139 does not exceed 4%. 140 141 TABLE 6.1 142 The Main Characteristics of Piezoceramic Materials 143 12 26 21 144 Material d31 3 10 (C/N) Tc (8C) Hardness a (3 10 K ) Firm Manufacturer 145 , 146 ZTS-19 170 290 hard 3 ION, Russia PKR-6 195 300 middle 1–3 Ultrasound Ltd, Russia 147 PKR-7M 350 175 soft 1–3 Ultrasound Ltd, Russia 148 PZT-5H 275 193 soft 1.5 Morgan Matroc, UK 149 PZT-5A 171 365 middle 1.5–2 Morgan Matroc, UK 150 PIC-151 210 250 middle — Physik Instrumente, Germany 151 P1 94 305 185 soft — Quartz & Silica, France 152 PCM-33A 262 — soft — Matsushita Electric, Japan 153 ARTICLE IN PRESS

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154 4 155 PKR-7M 156 157 3 158 PIC151 159 m 2 160 m PKR6 161 162 1 163 164 165 0 0 2000 4000 6000 8000 10000 12000 166 t, ms 167 168 FIGURE 6.2 Deformation of bimorph mirrors made from different piezoceramic materials vs. time. 169 170 171 Typical ceramic hysteresis curves exhibit a “butterfly” pattern (Figure 6.3). Depolarization 172 or even reverse polarization phenomena occur at potentials opposite to the polarization direction 173 voltage. We measured the threshold negative voltage at which depolarization begins for PKR-7M, 174 which equals 2500 V per mm of ceramic. To avoid unpredictable behavior of the mirror defor- 175 mation, the range of negative control voltage should be restricted. For example, the range of control 176 voltage for a 0.4 mm thickness ceramic could be varied from 2200 to þ300 V. The response of 177 the bimorph actuator remains linear within this range (Figure 6.4). 178 One of the important characteristics of correctors is the temperature stability of the mirror 179 reflecting surface. Surface instability is basically caused by the difference between the thermal 180 expansion coefficients for substrate material a1 and piezoceramic material a2: In the simplest cases, 181 the instability behavior manifests itself in additional defocus deformation of the mirror surface due 182 to changes in ambient temperature. 183 Application of an additional voltage to the electrodes of the piezodisc VDT ¼ðt=d31Þ 184 ða1 2 a2ÞDT can compensate for thermal deformation. This limits the dynamic range of the control 185 26 21 voltage. For example, for a copper substrate (a1 ¼ 15.9 £ 10 K ), almost 20% of the maximal 186 187 188 ∆ 189 L 190 191 virgin curve 192 193 194 195 196 197 198 0 Voltage 199 Waffle type 200 cooling 201 structure 202 203 204 FIGURE 6.3 Response of a PZT actuator to a bipolar drive voltage. ARTICLE IN PRESS

Laser Beam Shaping by Means of Flexible Mirrors 171

205 3 7.5 206 Z3 P-V Focus P_V 207 208 2 5.0 209 210 211 1 2.5 212 213 214 0 0.0 215 216 − − 217 1 2.5 218 Voltage 219 −2 −5.0 220 −250 −200 −150 −100 −50 0 50 100 150 200 250 221 222 FIGURE 6.4 Hysteresis curve for a 40 mm bimorph mirror. The y-axis represents the value of mirror 223 deformation P–V (peak-to-valley) measured in mm; the x-axis represents the voltage given in volts. 224 225 226 range of the control voltage would be used to compensate for thermal deformation caused by a 5 8C 227 change in temperature. 228 The main criterion for the choice of a substrate material is the thermal expansion value a1, 26 21 229 which should be close to a2 for the chosen piezoceramic material (,3 £ 10 K ). Table 6.2 230 shows the parameters of optical materials, which are used in the manufacturing of experimental 231 samples of bimorph mirrors. Piezoceramic lead–zinconate–titanate (PZT) is the most suitable 232 material for a mirror substrate. In this case, we can ensure equality of thermal expansion 233 coefficients, leading to an ideal mirror for thermal stability. However, there can also be problems 234 with polishing ceramic materials that have a tendency to crumble. Ceramic material PKR-7M, 235 which is produced under special heat baking technology, allows polishing of the surface to 236 optical quality. 237 Another method for thermally stabilizing the mirror shape is water-cooling. Water-cooling 238 is very important in high power laser applications, where the mirror is heated by the laser beam. 239 In this case, channels for cooling water circulation have to be formed inside the thin substrate 240 241 242 TABLE 6.2 243 Parameters of Materials for Mirror Substrates 244 Parameters 245 Index of a 246 thermal The module Thermal 247 Density, g Index of expansion, of the Young, conductivity, Specific thermal 3 26 21 29 248 Material (kg/m ) refraction a (3 10 K ) E(Pa3 10 ) b (W/(m K)) capacity (J/(kg K)) 249 Quartz glasses 2.2 1.46 0.55 98 1.36 733 250 KU1, KU2, KU 251 Optical glass LK-5 2.27 1.4846 3.5 69 1.2 — 252 Piezoceramics PZT 7.4–7.7 — 1–3 46–90 1.4 400 253 Monocristalline Si 2.33 3.3 2.54 126–131 150 700 254 Copper Cu 8.96 — 15.9 129.5 401 385 255 ARTICLE IN PRESS

172 Laser Beam Shaping Applications

256 of the mirror (see Figure 6.5). We are producing such cooled mirrors from copper, because copper 257 is a good material for machining, diffusion welding, and optical polishing. Such cooled mirrors 258 are used for formation and correction of continuous wave (CW) CO2 laser beams with powers 13 259 up to 5 kW. 260 Figure 6.6 shows experimental samples of bimorph mirrors that we have developed for different 261 applications. Mirror apertures varies from 30 to 100 mm. The number of control electrodes on 262 the mirrors varies from 17 to 33. The mirror surfaces were deposited with multilayer dielectric 263 and protected silver coatings. These coated mirrors were subjected to beam irradiances up to 12 2 264 ,10 W/cm without observable damage. 265 266 267 268 269 270 271 272 channels 273 for water delivery 274 275 piezodisks 276 Waffle type cooling structure 277 278 279 280 281 282 283 284 285 286 287 288 289 (a) 290 291 292 293 294 “e2” 295 “e3” 296 297 “e1” 298 299 300 301 302 303 (b) 304 305 FIGURE 6.5 Sample of cooling structure for a water-cooled bimorph mirror; (b) layout of mirror electrodes 306 (e1 is underneath e2 and e3). ARTICLE IN PRESS

Laser Beam Shaping by Means of Flexible Mirrors 173

307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 FIGURE 6.6 Photograph of experimental bimorph mirrors. 324 325 The 30 mm aperture bimorph mirror shown in Figure 6.6 is used for the formation of the near- 326 field laser beam distribution for the last amplifier of the ATLAS Ti:Sa laser at MPQ, Garching, 327 Germany.14 The homogenizing of the irradiance distribution produced by this flexible mirror 328 allowed pulse energy to increase from 0.5 to 1.5 J. The next mirror, shown in Figure 6.6, having an 329 80 mm aperture and 33 control actuators, is used in the same laser for far-field wavefront correction. 330 Closed-loop control of the mirror produced an almost diffraction-limited spot at the focal plane of 331 the nonaxial parabolic mirror. 332 In addition to their utility for extracavity beam correction and formation, bimorph mirrors are 333 very useful in intracavity applications. The following sections detail an investigation of bimorph 334 correctors for intracavity laser beam formation. For intracavity applications, we suggest the use 335 of the well-developed water-cooled flexible mirror designs (Figure 6.5) to avoid any undesirable 336 surface thermal deformations. As Figure 6.5 illustrates, such mirror designs contain two piezodiscs, 337 where the interface between the piezoceramic discs contains a continuous conducting ‘ground’ 338 electrode. Another continuous conducting electrode between the piezodisc and the copper plate e1is 339 used to control the overall curvature of the entire mirror. Two controlling electrodes, e2 and e3, 340 having the form of concentric rings, were attached to the outer surface of the piezodisc. The 341 response function for each mirror electrode (the deformation of the mirror surface while applying 342 voltage to each mirror electrode) is shown in Figure 6.7 and was measured using a modified Fizeau 343 interferometer.15 Three-dimensional profiles of the response functions are shown in Figure 6.8. 344 These response functions will be used in the intracavity procedures described in this chapter. 345 346 347 348 349 350 351 352 353 Q4 354 355 FIGURE 6.7 Level map of response functions of three electrodes: (a) common focus–defocus electrode e1 356 ðP 2 V ¼ 0.81 m for applied voltage 20 V); (b) second ring electrode e3 ðP 2 V ¼ 0.35 m for applied voltage 357 40 V); (c) first ring electrode e2 ðP 2 V ¼ 0.79 m for applied voltage 40 V). ARTICLE IN PRESS

174 Laser Beam Shaping Applications

358 359 360 361 362 363 364 −0.50 365 1.00 1.00 366 0.80 0.80 367 0.60 0.60 0.40 0.40 368 0.20 0.20 0.00 0.00 369 −0.20 −0.20 − −0.40 370 0.40 −0.60 −0.60 371 −0.80 −0.80 (a) −1.00 −1.00 372 373 374 375 376 377 378 −0.00 379 380 −0.10 381 −0.20 382 1.00 1.00 383 0.80 0.80 0.60 0.60 384 0.40 0.40 385 0.20 0.20 0.00 0.00 − 386 −0.20 0.20 − −0.40 0.40 − 387 −0.60 0.60 − −0.80 0.80 388 − (b) −1.00 1.00 389 390 391 392 393 394 395 −0.00 396

397 −0.50 398 399 1.00 1.00 0.80 0.80 400 0.60 0.60 0.40 0.40 401 0.20 0.20 0.00 0.00 402 −0.20 −0.20 403 −0.40 −0.40 −0.60 −0.60 404 −0.80 −0.80 (c) − −1.00 405 1.00 406 FIGURE 6.8 Surface profiles of an adaptive mirror: (a) for common focus–defocus electrode e1 — applied 407 voltage is 20 V, (b) for second ring e3 — applied voltage is 40 V, (c) for first ring of electrodes e2 — applied 408 voltage is 40 V. ARTICLE IN PRESS

Laser Beam Shaping by Means of Flexible Mirrors 175

409 III. FORMATION OF A DESIRED LASER BEAM AT THE LASER RESONATOR 410 OUTPUT BY MEANS OF AN INTRACAVITY FLEXIBLE CORRECTOR 411 412 For many industrial applications it is desirable to have the laser generating a single, mostly 413 fundamental mode and to have good power extraction at the same time. Gaussian beams 414 are relatively narrow and therefore result in poor energy extraction. We would thus expect that 415 the formation of a wider super-Gaussian irradiance distribution inside the laser cavity would lead to 416 an increase in the active medium gain extraction. 417 The algorithm for the desired irradiance formation as well as the numerical results will be 418 discussed using a stable laser cavity (Figure 6.9). Such a cavity corresponds to the industrial 419 continuous discharge CO2 laser ILGN-704 produced by Istok (Fryazino, Russia). The geometry of 420 the resonator consists of a plane output coupler and an active mirror separated from the coupler 421 by the distance L ¼ 2m. 422 Azimutal symmetry can be assumed, which allows us to use the one-dimensional Huygens– 16,17 423 Fresnel integral equations to calculate the amplitude of a mode in the empty laser resonator;

424 ðb 425 g2C2ðr2Þ¼ K1ðr1; r2ÞC1ðr1Þr1dr1 ð6:1Þ 426 0 427 ða 428 g1C1ðr1Þ¼ K2ðr2; r1ÞC2ðr2Þr2dr2 ð6:2Þ 0 429

430 where gi is the eigenvalue and CiðriÞ is the eigenmode of the resonator, ri is a radial co-ordinate, 431 i ¼ 1 indicates a plane output mirror of diameter 2b; and i ¼ 2 indicates an active mirror of 432 diameter 2a: These lead to: 433  j r r jk 434 1 2 2 2 K1ðr1; r2Þ¼ Jl k exp 2 Ar1 þ Dr2 ð6:3Þ 435 B B 2B  436 j r r jk K ðr ; r Þ¼ J k 1 2 exp 2 Ar2 þ Dr2 exp ð jkw ðr ÞÞ ð6:4Þ 437 2 2 1 B l B 2B 1 2 mirror 2 438 439 Here, Jl is the Bessel function of order l (we take into account only the lowest transverse mode with 440 l ¼ 0), and A, B, C, and D are the constants determined by the ABCD ray matrix of the laser 441 resonator. We consider an empty resonator, so that A ¼ 1; B ¼ L; C ¼ 0; and D ¼ 1: 442 The algorithm to form the desired irradiance distribution is the so-called “inverse propagation 11,18 –21 443 method” described in several references. The desired output field distribution C1ðr; wÞ is 444 specified on the output mirror. The back-propagation of the laser beam through all the resonator’s 445 446 bimorph mirror 447 coupler 448 449 450 451 Ψ 2 2a 2b Ψ 452 1 453 454 455 456 L 457 458

459 FIGURE 6.9 Schematic setup of the CW CO2 laser with adaptive mirror. ARTICLE IN PRESS

176 Laser Beam Shaping Applications

460 elements to the active corrector is calculated using the Huygens-Fresnel integral equations 461 (Equation 6.1 and Equation 6.2). In the plane of the active corrector, the wavefront is calculated to 462 determine the appropriate mirror phase profile wmirrorðrÞ: 463 464 wmirrorðrÞ¼2wbeamðrÞð6:5Þ 465 466 An ideal corrector (graded-phase mirror) could completely reconstruct such a phase 11,18 –21 467 profile. In our case, bimorph mirrors can approximate the necessary phase profile with 468 some small degree of error. This RMS error can be calculated using the experimentally measured 469 response functions of the mirror. In other words: 470 ! Xi¼3 2 471 ZðrÞ 2 UiFiðrÞ ÿ! min ð6:6Þ 472 i¼1 473

474 where ZðrÞ is the profile to be reconstructed, FiðrÞ are response functions of the flexible mirror 475 electrodes given in Figures 6.7 and 6.8, and Ui are weight coefficients corresponding to the 476 voltages applied to each electrode. 477 The left side of Equation 6.6 has a minimum when its first derivative equals zero: 478 ! 2 479 › Xi¼3 480 ZðrÞ 2 UiFiðrÞ ¼ 0 ð6:7Þ ›Ki ¼ 481 i 1 482 Equation 6.7 then determines the applied voltages to approximate the necessary shape of the 483 laser beam. 484 The procedure described above gives us the reconstructed mirror surface wmirrorðrÞ which is 485 substituted into Equation 6.4. To solve the integral equations (Equation 6.1 and Equation 6.2) we 486 used the Fox and Li iterative method of successive approximations16,17 to take into account edge 487 diffraction as well as nonideal reproduction of the necessary phase profile by the bimorph flexible 488 mirror. 489 490 491 IV. MAIN RESULTS 492 The main parameters of the laser resonator (Figure 6.9) are the Fresnel numbers N ¼ b2=ðBlÞ and 493 1 N ¼ a2=ðBlÞ; and the geometrical factor G ¼ð1 2 L=R Þ; where l ¼ 10.6 mm is wavelength, 494 2 2 R ¼ 4 m is the radius of curvature of the active mirror, and L ¼ 2 m is the length of the resonator 495 2 cavity. The initial field distribution on the planar output coupler is chosen as CðrÞ¼expð2ðr=WÞnÞ; 496 where n determines the order of the super-Gaussian function and W is the beam waist. The 497 particular beam waist is chosen according to the methods of moments for laser beams21: 498  499 lL 1 þ G 1=2 500 w2 ¼ M2 ð6:8Þ 2p 1 2 G 501 502 where M2 is the beam quality factor: 503 504 ðwuÞ M2 ¼ mode ð6:9Þ 505 ð Þ wu TEM00 506 507 A. FORMATION OF A SUPER-GAUSSIAN BEAM OF THE FOURTH ORDER (n 5 4) 508 509 From Equation 6.8 and Equation 6.9, the super-Gaussian beam waist is calculated: W ¼ 3.1 mm. 510 Figure 6.10(a) represents the phase distribution of a super-Gaussian beam (curve 1) ARTICLE IN PRESS

Laser Beam Shaping by Means of Flexible Mirrors 177

511 2 512 1 1.5 513 2 m 514 m 1 515 0.5

516 Phase, 517 0 518 0.0 0.2 0.4 0.6 0.8 1.0 −0.5 1.0 0.8 0.6 0.4 0.2 519 − − − − − 1 520 (a) r/a 521 522 1.0 523 0.8 524 525 0.6 526 527 0.4 528 0.2

529 Normalized intensity 530 0.0 531 0.0 0.2 0.4 0.6 0.8 1.0 1.0 0.8 0.6 0.4 0.2

532 − − − − − 533 (b) r/b 534 535 FIGURE 6.10 Formation of a super-Gaussian beam of the fourth order, N1 ¼ 1; N2 ¼ 4:7; G ¼ 0:5 : (a) 1 536 (solid curve) — the phase profile of the laser beam to be reconstructed and 2 (dashed) — the phase profile of the 537 flexible mirror; (b) normalized irradiance distributions on plane coupler: 1 — Gaussian irradiance distribution, 538 corresponding to the same geometry of resonator but with a spherical mirror, 2 — the desired super-Gaussian 539 irradiance profile, 3 — irradiance formed by graded-phase mirror, 4 (dashed) — by bimorph flexible one. 540 541 back-propagated through the resonator starting at the output coupler and going a distance of 542 L ¼ 2 m to the adaptive mirror. Curve 2 (Figure 6.10(a)) illustrates the phase profile of the active 543 mirror reproducing the phase shape of laser beam with an RMS error of 0.3%. Figure 6.10(b) 544 shows the beam irradiance distribution at the output coupler for various resonator conditions. 545 Curve 1 corresponds to the fundamental Gaussian mode of the same resonator, but with a pure 546 spherical mirror. Curve 2 (solid) shows the desired super-Gaussian relative irradiance profile, 547 while curve 3 shows the profile produced with an ideal corrector (graded-phase mirror with no 548 deviation from the necessary phase profile). Finally, curve 4 corresponds to the irradiance 549 distribution in the resonator with an adaptive mirror. 550 One may see from Figure 6.10(b) that applying the active corrector (curve 4) increases the 551 output mode volume by 1.5 times in comparison with a pure Gaussian beam (curve 1). At the same 552 time, diffraction losses per transit decrease by 1.7 times. The voltages applied to each electrode 553 were calculated from Equation 6.7 and are given in Table 6.3. 554 Users of lasers often dislike super-Gaussian irradiance distributions for their side lobes 555 in far-field patterns. However, the shaped super-Gaussian distribution is not exactly the super- 556 Gaussian function: its form has been changed by edge diffraction and by the nonideal behavior 557 of the active mirror in forming the necessary phase profile. That is why the side lobes of the beam 558 formed by an active mirror contain only 2% of the total energy (dashed curve 3 in Figure 6.11). This 559 irradiance profile is thus very attractive for industrial applications. For comparison, curve 2 in 560 Figure 6.11(a) and (b) represents the far-field pattern for a super-Gaussian beam formed by an ideal 561 (graded-phase) corrector. ARTICLE IN PRESS

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562 TABLE 6.3 563 Voltages (V) Applied to the Electrodes of a Bimorph 564 Flexible Mirror 565 566 n, order of the given super- 567 Gaussian beam formation e1 e2 e3 568 4 264 2116 71 569 6 2107 2300 2240 570 8 243 2214 2171 571 572 573 574 B. FORMATION OF A SUPER-GAUSSIAN BEAM OF THE SIXTH ORDER (n 5 6) 575 In this case, the beam waist of the desired super-Gaussian distribution is chosen as 3.3 mm. 576 Figure 6.12(a) represents the exact phase distribution of the desired super-Gaussian beam and 577 the phase profile of a flexible mirror (RMS error of the approximation is about 0.1%). Figure 6.12(b) 578 shows irradiance distributions formed by the flexible mirror at the output coupler. The far-field 579 results are very close to those for the fourth-order super-Gaussian beam represented in Figure 580 6.11(a) and (b). Mirror electrode voltages calculated from Equation 6.7 are presented in Table 6.2. 581 For this case, the output mode volume increases by a factor of 1.3 in comparison with a pure 582 Gaussian beam while diffraction losses per transit decrease by 1.5 times. 583 584 585 586 1.0 587 0.8 588 589 0.6 590 1-3 0.4 591 592 0.2

593 normalized intensity 594 0.0 595 0.0 0.3 0.6 0.9 (a) 596 normalized coordinate 597 0.016 598 599 0.012 600 1 601 0.008 602 603 0.004 2 604 605 normalized intensity 3 0 606 0.6 0.9 607 (b) normalized coordinate 608 609 FIGURE 6.11 Formation of a super-Gaussian beam profile of the fourth order. (a) Normalized irradiance 610 distributions in the far-field zone: 1 — Gaussian beam, 2 — beam formed by ideal corrector (graded-phase 611 mirror), 3 (dashed curve) — beam formed by active mirror; (b) fragment near the edge of the same irradiance 612 distributions. ARTICLE IN PRESS

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613 4.0 614 0.0 615 0.0 0.2 0.4 0.6 0.8 1.0 − 1.0 0.8 0.6 0.4 0.2 616 4.0 − − − − − 617 −8.0 618 Phase, m km 619 −12.0 620 −16.0 621 (a) r/a 622 1.2 623 624 1.0 2 625 0.8 626 0.6 1 627 0.4 628 0.2

629 Normalized intensity 0.0 630 0.0 0.2 0.4 0.6 0.8 1.0 631 1.0 0.8 0.6 0.4 0.2 − − − − − 632 (b) r/b 633 634 FIGURE 6.12 Formation of a super-Gaussian beam profile of the sixth order, N1 ¼ 1; N2 ¼ 14:1; G ¼ 0:5: (a) 635 (solid) — phase profile of laser beam to be reconstructed and (dashed) — phase profile of bimorph mirror; (b) 636 normalized irradiance distributions: 1 — Gaussian TEM00 mode for the same resonator but with a pure 637 spherical bimorph mirror, 2 — sixth order super-Gaussian beam formed by the bimorph flexible mirror. 638 639 C. FORMATION OF A SUPER-GAUSSIAN BEAM OF THE EIGHTH ORDER (n 5 8) 640 For this case, the beam waist of the desired eighth-order super-Gaussian fundamental mode is equal 641 to 3.5 mm. The main results are given in Figure 6.13(a) and (b). The output mode volume increased 642 1.6 times in comparison with a pure Gaussian beam, while diffraction losses per transit decrease by 643 a factor of 1.7. Although the flexible mirror error in producing the desired phase shape is relatively 644 low (RMS error , 0.1%), it could not exactly reproduce the two main humps (Figure 6.13(a)). 645 646 However, if the controlling electrode’s position could be matched to the co-ordinates of the two 647 local maxima (Figure 6.13(a), solid curve), such an active corrector would be able to reproduce the 648 shape more accurately. Applied mirror electrode voltages are again given in Table 6.2. 649 It should be mentioned that active mirrors sometimes produce a smoother phase profile than 650 is desirable for higher-order modes. This situation can lead to higher diffraction losses for the 651 TEM00 mode than for the TEM01 mode, which causes the laser to resonate in the TEM01 or some 652 combination of higher-order modes. In contrast, an ideal corrector, such as a graded-phase mirror, 653 tends to disturb the higher-order modes to a greater extent, which causes higher diffraction losses 18 654 and thus suppresses their amplitude in the cavity. This is confirmed by Figure 6.14, which 655 shows the irradiance distribution of a TEM01 mode on the output coupler where an eighth-order 656 super-Gaussian beam is desired. The TEM01 mode, shaped by an ideal corrector (curve 1 in 657 Figure 6.14) and by an active corrector (curve 2), is shown. One may see the irradiance of 658 the TEM01 mode in the first case is more distorted; hence, its diffraction losses are higher. 659 As an example, for the previously given resonator parameters and a bimorph active corrector, 660 diffraction losses per transit of the super-Gaussian mode formed by a bimorph mirror are l l 25 l l 25 661 ð1 2 g1g2 Þ¼4:4 £ 10 %; while for the TEM01 the losses are ð1 2 g1g2 Þ¼2 £ 10 %: 662 To increase the discrimination between transverse modes one needs to perform an optimization 663 procedure. ARTICLE IN PRESS

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664 2.5 665 666 2 667 1.5 668 669 1

670 m km Phase, 671 0.5 672 0 673 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0

674 − − − − − 675 (a) r/a 676 1 677 1.0 2 678 0.8 3 679 680 0.6 681 682 0.4 683 0.2 684 Normalized intensity 685 0.0

686 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 − − − − − 687 (b) r/b 688 689 FIGURE 6.13 Formation of a super-Gaussian beam profile of order eight, N ¼ 1; N ¼ 4:7; G ¼ 0:5: 690 1 2 (a) (solid) — the phase profile of the given laser beam at the position of bimorph mirror and (dashed) — phase 691 profile of the mirror; (b) normalized irradiance distributions: 1 (solid) — the desired super-Gaussian beam, 692 2 (dotted) — irradiance produced with a graded-phase mirror, 3 (dashed) — irradiance produced with bimorph 693 flexible mirror. 694 695 696 697 698 699 700 1.20 701 1.00 702 703 0.80 1 704 0.60 2 705 0.40 706 0.20 707 normalized intensity 708 0.00 709 0.00 0.29 0.57 0.86 710 r/b 711

712 FIGURE 6.14 Normalized irradiance distribution of TEM01 mode while forming an eighth-order super- 713 Gaussian fundamental mode: 1 — ideal corrector (such as a graded-phase mirror), 2 — bimorph flexible 714 mirror. ARTICLE IN PRESS

Laser Beam Shaping by Means of Flexible Mirrors 181

715 1.00 716 0.90 717 0.80 718 0.70 Series 1 Series 2 719 0.60 Series 3 720 0.50 0.40 Series 4 721 0.30 Series 5 722

normalized intensity 0.20 723 0.10 724 0.00 725 0.00 0.29 0.57 0.86 726 r/b 727 728 FIGURE 6.15 Normalized irradiance profiles on the plane output coupler for the sixth-order super-Gaussian 729 fundamental mode. Series 1: the desired beam; Series 2: N2 ¼ 4:7; Series 3: N2 ¼ 6:8; Series 4: N2 ¼ 10 (very 730 close to the desired beam except at the center); Series 5: N2 ¼ 14:1; N1 ¼ 1; G ¼ 0:5: 731 732 D. OPTIMIZATION OF LASER PARAMETERS 733 734 The optimization of the laser resonator parameters N2 (Fresnel number) and G (geometric 735 factor) was carried out in order to produce the closest match to the desired irradiance 736 distribution. Figure 6.15 shows the normalized output irradiance profiles produced by varying 737 N2 from 4.7 to 14.1 ðN1 ¼ 1; G ¼ 0:5Þ: Note that for the graded-phase mirror, the best result is 738 reached when the Fresnel number N2 is a maximum. This is understandable: increasing N2 739 corresponds to enlarging the size of an aperture in front of the graded-phase mirror (see 740 Figure 6.9), which causes a more complete approximation of the incident resonator mode shape 741 by this mirror. As a result, the output profile is closer to what is desired. In contrast, for an 742 active corrector with given electrode axial positions, an increase in N2 does not necessarily lead 743 to better performance. For example, one may see from Figure 6.15 that for N2 ¼ 10 (series 4) 744 the obtained irradiance distribution is very close to the desired one. Also, a rather good approxi- 745 mation to the desired super-Gaussian curve was obtained for N2 ¼ 4:7 (series 2, Figure 6.15). 746 However, for the highest Fresnel number N2 ¼ 14:1 (series 5, Figure 6.15) the shape of the 747 obtained output profile is not as good as for lower Fresnel numbers. This is because the fixed 748 electrodes may have difficulty replicating a larger incident mode profile, and in fact may have 749 to make compromises in the center of the mirror where the majority of the incident mode 750 amplitude is concentrated. 751 Another parameter that must be taken into consideration is diffraction losses per transit. 752 We calculated such losses for the eighth-order super-Gaussian fundamental mode and the results 753 are presented in Figure 6.16. The curve has a minimum for N2 ¼ 6:8: In that case, the bimorph 754 flexible mirror does the best job of reproducing the resonator mode shape. At this minimum point, 755 the diffraction losses of TEM01 mode are approximately ten times higher than the TEM00 mode. 756 The obtained irradiance curves do not depend upon the variation of G; because the active mirror 757 can reproduce focus (or defocus) very well (mainly by applying a voltage to the common focus– 758 defocus electrode; Figures 6.7 and 6.8). 759 760 V. FORMATION OF AN ANNULAR BEAM 761 762 To solve various tasks in modern laser physics and nonlinear optics, it is sometimes necessary to 763 have beam profiles that differ from a uniform profile. For example, to deliver high-power laser 764 radiation through the atmosphere, it is desirable to form an annular laser beam with a planar phase 765 distribution because it has less nonlinear distortion in nonlinear and turbulent media than other ARTICLE IN PRESS

182 Laser Beam Shaping Applications

766 6.00E-05 767 5.00E-05 768 4.00E-05 769 770 3.00E-05 771 2.00E-05 772 1.00E-05 losses per transit, % losses per transit, 773 0.00E+00 774 2.3 3.2 4.7 6.8 10.6 14.4

775 N2 776 777 FIGURE 6.16 Diffraction losses per transit for an eighth-order super-Gaussian fundamental mode in a 778 resonator with an active corrector. N1 ¼ 1; G ¼ 0:5: 779 780 forms of laser beams.22,23 The traditional way to form such a beam is to obscure the central part of a 781 Gaussian fundamental mode. However, in this case we have additional power losses. As is shown 782 below, an intracavity active mirror allows us to form an annular beam without any additional power 783 losses and with less diffraction losses than for a traditional Gaussian fundamental mode. 784 The desired initial field distribution for the annular beam at the plane output coupler of a 785 4 laser resonator (Figure 6.9) is chosen as C1ðr1Þ¼ {ðr þ 0:1Þ=3:1}2 expð2ððr þ 0:1Þ=3:1Þ Þ: A small 786 decentration value, in this case 0.1 mm, had to be added so that diffraction did not drastically change 787 the desired annular irradiance shape of the fundamental mode during Fox–Li numerical 788 16,17 2 calculations. The main parameters of the laser resonator were Fresnel numbers N1 ¼ b =ðBlÞ¼ 789 2 1; N2 ¼ a =ðBlÞ¼4:7; and geometrical factor G ¼ð1 2 L=R2Þ¼0:5; where R2 ¼ 4 m is the 790 radius of curvature of the flexible mirror and L ¼ 2 m is the length of the resonator cavity. 791 Figure 6.17(a) represents the phase distribution of the annular beam propagated back to the 792 flexible mirror at the distance L ¼ 2 m from the output plane mirror (gray curve). The smoother, 793 794 2.0 795 1.0 796 0.0 m 797 m − 0.0 0.2 0.4 0.6 0.8 1.0 1.0 1.0 0.8 0.6 0.4 0.2

798 − − − − − −2.0 799 Phase, −3.0 800 −4.0 801 −5.0 802 r/a 803 804 1.2 805 1.0 0.8 806 0.6 807 0.4 808 0.2 809 Normalized intensity 0.0 810 −0.9 −0.7−0.5−0.3−0.2 0.0 0.2 0.3 0.5 0.7 0.8 811 r/b 812

813 FIGURE 6.17 Formation of a doughnut-like beam: N1 ¼ 1; N2 ¼ 4:7; G ¼ 0:5 : (a) (gray curve) the phase 814 profile of the laser mode to be reconstructed and (black) the phase profile of the bimorph flexible mirror; (b) 815 normalized irradiance distributions on the plane output coupler: (black) the desired irradiance profile and 816 (gray) irradiance profile formed by the flexible mirror. ARTICLE IN PRESS

Laser Beam Shaping by Means of Flexible Mirrors 183

817 0.7 818 0.6 819 820 0.5 821 0.4 822 0.3

823 Intensity 0.2 824 825 0.1 826 0.0 827 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 828 Normalized coordinate 829 830 FIGURE 6.18 Irradiance distributions in the far-field: (black) far-field pattern of a Gaussian fundamental 831 mode, (gray) far-field pattern of the doughnut-like beam. 832 833 black curve of Figure 6.17(a) illustrates the phase profile of the bimorph flexible mirror 834 reproducing the phase shape of the laser beam with RMS error 0.7%. Figure 6.17(b) shows 835 irradiance distributions on the plane output coupler. The black curve corresponds to the desired 836 irradiance profile and the gray curve to the profile formed by the intracavity flexible mirror. The 837 l l diffraction losses of the annular beam were estimated as d ¼ 1 2 g1g2 ; where g1; g2 are the 838 eigenvalues defined from Equation 6.1 and Equation 6.2. Such losses decreased by a factor of 839 1.4 in comparison with a Gaussian TEM00 mode. 840 The far-field pattern of this annular beam contains about 96% of its total energy in the main 841 lobe (as shown by the gray curve in Figure 6.18). With a beam quality factor M2 ¼ 1:2; this 842 irradiance profile is very attractive for industrial applications. For comparison we plotted the far- 843 field pattern for a Gaussian fundamental mode from the same resonator. The edges of the far-field 844 intensities are shown in Figure 6.19 in expanded scale, showing a small amount of energy in 845 the second diffraction ring for the annular beam. The voltages applied to each electrode to form 846 the annular fundamental mode are given in Table 6.4. 847 848 849 VI. EXPERIMENTAL FORMATION OF A SUPER-GAUSSIAN BEAM 850 BY MEANS OF BIMORPH FLEXIBLE MIRROR 851 The experimental formation of a specified beam, namely a super-Gaussian of fourth and sixth 852 orders, was performed using an industrial fast axial flow continuous-discharge CO2 laser with 853 854 855 0.03 856 857 0.02 858 859 860

Intensity 0.01 861 862 863 0.00 864 0.5 0.6 0.7 0.8 0.9 1.0 865 Normalized coordinate 866 867 FIGURE 6.19 The fragment near the beam edge for the irradiance distributions shown in Figure 6.18. ARTICLE IN PRESS

184 Laser Beam Shaping Applications

868 869 TABLE 6.4 870 Voltages (V) Applied to the Electrodes of a Flexible Mirror 871 to Form an Annular Beam 872 e1 e2 e3 e4 873 874 247.5 2254 0 0.7 875 876 877 a stable resonator, produced by IPLIT, Russian Academy of Sciences, model TLA-600.24 The laser 878 resonator (Figure 6.20) consists of a plane output coupler (7), a CO2 gain tube (6), convex (5) and 879 concave (4) mirrors, and the bimorph flexible mirror (2). 880 First of all, in order to model the behavior of the cavity with our flexible mirror and to determine 881 the mirror electrode voltages to form super-Gaussian beams of fourth and sixth orders, we perform 882 all of the numerical calculations described in Equation 6.1 to Equation 6.7. However, in the case 883 of a real laser having an active medium (not just an empty resonator as it was before), we must 884 take into account active medium saturation caused by the intense beam. That is why the kernels of 885 integral Equation 6.1 and Equation 6.2, given by Equation 6.3 and Equation 6.4, have the additional 886 multiplier HðIÞ taking into account this effect: 887  888 j r r jk K ðr ; r Þ¼2p J k 1 2 exp 2 ðAr2 þ Dr2Þ HðIÞð6:10Þ 889 1 1 2 lB 0 B 2B 1 2 890 891 and  892 j r r jk K ðr ; r Þ¼2p J k 1 2 exp 2 ðAr2 þ Dr2Þ exp ðjkw ðr ÞÞHðIÞð6:11Þ 893 2 2 1 lB 0 B 2B 1 2 mirror 2 894 895 where 0 1 896 B g L C 897 HðIÞ¼B1 þ 0 am C ð6:12Þ @ IðrÞ A 898 21þ 899 Is 900 901 902 12 11 8 7 903 6 5 904 905 906 4 907 9 908 3 909 2 1 910 10 911 912 913 914 FIGURE 6.20 Schematic of the experimental setup to form a super-Gaussian TEM00 mode: 1 — variable- 915 voltage power supply for controlling mirror electrodes, 2 — semipassive bimorph mirror, 3 — diaphragm, 4 — 916 concave mirror R ¼ 2200 mm, 5 — convex mirror R ¼ 2800 mm, 6 — active CO2 gain medium, 7 — ZnSe 917 output mirror with coefficient of reflectivity 69%, 8 — LBA-2A (laser beam analyzer), 9 — oscilloscope, 10 — 918 computer, 11 — lens f ¼ 275 mm, 12 — MAC-2 (mode analyze computer). ARTICLE IN PRESS

Laser Beam Shaping by Means of Flexible Mirrors 185

21 24 919 Here g0 is the small-signal gain coefficient; g0 ¼ 110 cm for this particular type of laser. Lam is 920 the length of the active medium ðLam ¼ 80 cm), IðrÞ is the transverse irradiance distribution of 2 921 the laser beam, and Is is the saturation irradiance (Is ¼ 110 W/mm ). If the irradiance of the laser 922 beam IðrÞ becomes comparable with Is (saturation irradiance of the active medium), then the overall 923 irradiance of the laser beam becomes lower. 924 The main resonator parameters for the CO2 laser shown in Figure 6.20 are Fresnel numbers 2 2 925 N1 ¼ b =lB ¼ 0:66 and N2 ¼ a =lB ¼ 6:47; and stability factor G ¼ 0:51 (here b ¼ 8 mm, the 926 radius of the plane output mirror; a ¼ 25 mm, the radius of bimorph mirror; l ¼ 10.6 mm, the 927 wavelength; and B ¼ 9104 mm, the effective length of the resonator). Parameters of the super- 928 Gaussian function CðrÞ¼expð2ðr=WÞnÞ are chosen as W ¼ 4.8 mm and W ¼ 5.1 mm for n ¼ 4 929 and n ¼ 6; respectively. The particular beam waists were chosen according to the theory of 21 930 moments and are well described in Be´langer and Pare´. For this laser, we can assume 931 HðIÞ¼constant, since the irradiance of the beam is much smaller than the saturation irradiance Is of 932 the active medium. The voltages at the flexible mirror electrodes needed to form the super-Gaussian 933 beams were calculated according to the algorithm described earlier in Chapter 3. 934 Figure 6.21 shows the evolution of the fourth-order super-Gaussian beam at the surface 935 of output mirror as a function of the round trip number, based on numerical calculations. Curve 936 number 1 represents the original beam, while the other curves show beam profiles after 937 a corresponding number of round trip passes calculated from Equation 6.2 to Equation 6.5. Given 938 the resonator parameters mentioned previously in this section, we need to make about 130 iterations 27 939 to reach a convergence point where successive calculations differ by less than 10 . 940 Figure 6.22(a) and (b) show the main results of the calculations: the Gaussian (curve 1) and 941 super-Gaussian of the fourth- and sixth-order fundamental mode irradiance distributions (curve 2) 942 on the plane output mirror. The far-field pattern for the calculated fourth-order super-Gaussian 943 beam is given in Figure 6.23(a). As one may see, the fourth-order super-Gaussian increases the peak 944 value of far-field irradiance by 1.6 times compared to the Gaussian TEM00 mode of the resonator. 945 At the same time, however, the super-Gaussian causes side lobes in the far-field irradiance 946 distribution. However, they are not very significant, so that the experimental M2 factor for the 947 fourth-order super-Gaussian mode is M2 ¼ 1:36: We would like to mention that for an “ideal” 948 fourth-order super-Gaussian beam having the same near-field beam waist v ¼ 4.8 mm, the M2- 949 factor is generally higher ðM2 ¼ 1:46Þ; as illustrated by curve 2 in Figure 6.11. In this case, it is seen 950 951 952 1.0 953 3 0.9 2 954 0.8 955 4-11 0.7 956 957 0.6 958 0.5 1 959 0.4 960 0.3

961 Normalized intensity 0.2 962 0.1 963 0.0 964 −1.0 −0.5 0.0 0.5 1.0 965 r/b 966 967 FIGURE 6.21 Evolution of the fourth-order super-Gaussian beam at the surface of the output mirror: 968 1 — initial beam, 2 — the beam after one round trip, 3 — after two round trips, 4 — after three round trips, etc., 969 11 — after ten round trips. ARTICLE IN PRESS

186 Laser Beam Shaping Applications

970 1.0 971 0.8 972 2 973 0.6 974 0.4 975 3 976 0.2

977 Normalized intensity 1 0.0 978 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 979 (a) r/b 980 981 1.0 982 983 0.8 2 984 0.6 985 986 0.4 987 31 988 Normalized intensity 0.2 989 990 0.0 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 991 (b) r/b 992 993 994 FIGURE 6.22 Formation of the super-Gaussian fundamental mode: (a) super-Gaussian fourth-order mode. 1 — Gaussian mode, 2 — theoretically obtained super-Gaussian fourth-order mode, 3 — experimentally 995 obtained super-Gaussian fourth-order mode; (b) super-Gaussian sixth-order mode. 1 — Gaussian mode, 996 2 — theoretically obtained super-Gaussian sixth-order mode, 3 — experimentally obtained super-Gaussian 997 sixth-order mode. 998 999 1000 from Figure 6.11 that the width of the far-field beam increases because of the side lobes. For sixth- 1001 order super-Gaussian beams the difference is higher: the beam formed by the flexible mirror has 2 2 1002 M ¼ 1:38; while the ideal beam has M ¼ 1:8: So, the fact that the resonator with a corrector does 1003 not ideally reproduce the super-Gaussian function in the near-field (Figure 6.22) results in a positive 2 2 1004 effect in the far-field: the side lobes are reduced which improves the M -factor. We define the M - 25 2 1005 factor according to the international standard ISO 11146 as M ¼ pd0u=ð4lÞ; where l is the 1006 wavelength, d0 is the near-field waist diameter calculated as the second moment of the irradiance 1007 distribution at the waist location (in our case at the plane of the output resonator mirror) 1008 ÐÐ pffiffi 2 1009 ÐÐr Iðr; zÞr dr dw d0 ¼ 2 2 ; 1010 Iðr; zÞr dr dw 1011 1012 and u is the divergence angle defined as u ¼ df =f ; where f is the focal length of lens and df is the 1013 beam width defined as the second moment of the focal plane irradiance distribution 1014 ÐÐ pffiffi 2 1015 ÐÐr If ðr; zÞr dr dw If ðr; zÞ : df ¼ 2 2 : 1016 If ðr; zÞr dr dw 1017 1018 Now we can come back to the experiment itself. To apply voltages to the electrodes of the 1019 flexible mirror we used a variable-voltage power supply (Figure 6.20; 1). The near-field irradiance 1020 distribution was observed with the help of a laser beam analyzer (LBA-2A; Figure 6.20; 8), and ARTICLE IN PRESS

Laser Beam Shaping by Means of Flexible Mirrors 187

1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 FIGURE 6.23 Far-field pattern of laser beam: (a) theoretically calculated Gaussian and fourth-order super- 1058 Gaussian modes; (b) experimentally obtained Gaussian TEM00 mode; (c) experimentally obtained super- 1059 Gaussian fourth-order TEM00 mode. 1060 1061 1062 the far-field pattern (in the focal plane of lens; Figure 6.20; 11; f ¼ 275 mm) was analyzed by a 1063 mode analyze computer (MAC-2; Figure 6.20; 12). The result of experimentally forming a fourth- 1064 order super-Gaussian beam is presented in Figure 6.22(a), curve 3. The total power of the formed 1065 super-Gaussian beam was about 10% higher than the power contained in the Gaussian beam, and ^ 1066 the waist was widened by 1.26 0.05 times (the calculations showed that it should have 1067 increased by 1.29 times). In the focal plane of lens (Figure 6.20; 11; far-field) the experimental 1068 irradiance profile for the Gaussian mode is given by Figure 6.23(b). The peak value of the far- 1069 field irradiance distribution for the fourth-order super-Gaussian beam (Figure 6.23(c)) is 1.6 times 1070 higher than that for the Gaussian beam. This fact is in good agreement with the theory as shown in 1071 Figure 6.23(a). The shape of the far-field pattern becomes narrower, but the side lobes that should ARTICLE IN PRESS

188 Laser Beam Shaping Applications

1072 1 3 5 1073 2 1074 4 1075 1076 1077 1078 FIGURE 6.24 Schematic of a telescopic-type stable resonator for a YAG:Nd3þ laser with a wide-aperture 1079 mirror: 1 — output coupler, 2 — active medium, 3 — thermal lens, 4 — meniscus, 5 — bimorph flexible mirror. 1080 1081 1082 exist are not distinguishable from the noise level. The near-field irradiance distribution for the 1083 experimentally formed sixth-order super-Gaussian beam is shown in Figure 6.22(b), curve 3. In 1084 this case we observed a 12% total power increase in comparison with the Gaussian fundamental 1085 mode. The observed far-field pattern for the formed sixth-order super-Gaussian mode is very 1086 similar to the one for the fourth order mode. 1087 1088 1089 VII. YAG:Nd31 LASER. FORMATION OF A SUPER-GAUSSIAN 1090 OUTPUT BEAM — NUMERICAL RESULTS 1091 1092 This section presents calculations aimed at investigating the formation of specified irradiance 3þ 1093 profiles for a YAG:Nd laser. Active mirrors tend to have rather large-apertures — their diameter 1094 is 20 mm or larger. It is difficult to use such deformable mirrors in the cavities of industrial CW 1095 solid-state lasers because of the relatively small beam apertures in stable resonators. To solve this 1096 problem, we suggest expanding the beam inside the laser cavity up to the diameter of the adaptive 26 1097 mirror by using a meniscus on the one end of the gain element (Figure 6.24). At the same time, 1098 the bimorph flexible mirror has a concave spherical profile. Such a laser resonator permits the use 1099 of wideaperture mirrors without any supplementary optical elements and therefore without 1100 undesirable loss. We considered the case of forming a super-Gaussian output with a resonator 2 2 1101 having the main parameters of Fresnel numbers N1 ¼ b \Bl ¼ 0:3 and N2 ¼ a \Bl ¼ 12:3; and 1102 geometric factor G ¼ 0:58: Here, 2a ¼ 20 mm is the diameter of the bimorph deformable mirror, 1103 2b ¼ 6 mm is the diameter of the plane output mirror, l ¼ 1.06 mm is the wavelength, and 1104 B ¼ 6200 mm is the effective length of the telescopic type of resonator (Figure 6.24). A, B, and D 1105 are the elements of the ray ABCD matrix for the laser resonator. 1106 1107 1108 1.2 1109 1110 1.0 2 1111 0.8 1112 0.6 1113 1 1114 0.4 1115 1116 intensity Nomalized 0.2 1117 0.0 1118 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 1119 r/b 1120 1121 FIGURE 6.25 Formation of super-Gaussian fundamental modes at the output of a stable resonator in a 1122 YAG:Nd3þ laser: 1 — Gaussian mode, 2 — fourth-order super-Gaussian modes. ARTICLE IN PRESS

Laser Beam Shaping by Means of Flexible Mirrors 189

1123 Figure 6.25 shows the results of the diffraction calculations. Curve 1 represents the Gaussian 1124 fundamental TEM00 mode at the resonator output while curve 2 shows the fourth-order super- 1125 Gaussian beam profile. Calculations show that the mode volume for the super-Gaussian beam 1126 increases by a factor of 2.1 to 2.2, diffraction losses decrease by 1.1 to 1.2 times, and the far- 1127 field peak irradiance increases by a factor of two in comparison with the Gaussian TEM00 mode. 1128 The mode volume was estimated as the average of transverse irradiance distributions calculated 1129 at five points inside the cavity and at the mirrors as well. Power losses d were calculated as l l 1130 d ¼ 1 2 g1g2 : 1131 1132 1133 VIII. CONCLUSION 1134 3þ This chapter has shown the ability to form a specified irradiance output in both YAG:Nd and CO2 1135 laser resonators by means of an intracavity flexible mirror. The experiment with a CW CO2 laser 1136 has shown that while remaining in the TEM00 regime, we were able to increase the total output 1137 power by 10 to 12% and to increase the peak value of far-field irradiance by 1.6 times in comparison 1138 with a Gaussian fundamental mode. This work opens the possibility of “intelligent” flexible lasers 1139 that generate irradiance distributions specified by the user. 1140 1141 1142 ACKNOWLEDGEMENTS 1143 The authors would like to thank personnel from the Group of Adaptive Optics for Industry and Medicine of 1144 the Institute of Laser and Information Technologies, Russian Academy of Sciences, for their help in fabricating 1145 and testing the mirrors and in carrying out experiments. Also, we are very grateful to Prof. L. N. Kaptsov and 1146 Prof. S. S. Chesnokov from Moscow State University for their helpful discussions. Finally, we wish to thank Scott 1147 C. Holswade for his assistance in editing the manuscript. 1148 1149 REFERENCES 1150 1151 1. Abil’sitov, G., ed., Technological Lasers, Moscow, 1991 (in Russian). 1152 2. Koebner, H., ed., Industrial Applications of Lasers, Wiley-Interscience, New York, 1988. 1153 3. Borghi, R., and Santarsiero, M., Modal structure analysis for a class of axially symmetric flat-topped laser beams, IEEE J. Quantum Electron., 35, 745–750, 1999. 1154 4. Santarsiero, M., and Borghi, R., Correspondence between super-Gaussian and flattened Gaussian 1155 beams, J. Opt. Soc. Am A, 16, 188–190, 1999. 1156 5. Gory, F., Flattened Gaussian beams, Opt. Comm., 107, 335–341, 1994. 1157 6. Bollanti, S., Lazzaro, P. Di., Murra, D., and Torre, A., Analytical propagation of super-gaussian-like 1158 beams in the far-field, Opt. Comm., 138, 35–39, 1997. 1159 7. De Silvestri, S., Magni, V., Svelto, O., and Valentini, G., Lasers with super-Gaussian mirrors, IEEE 1160 J. Quantum Electron., 26, 1500–1509, 1990. 1161 8. Dainty, J. C., Koryabin, A. V., and Kudryashov, A. V., Low-order adaptive deformable mirror, Appl. 1162 Opt, 37(21), 4663–4668, 1998. 1163 9. Anan’ev, Yu. A., In Laser Resonators and the Beam Divergence Problem, Higler, A., ed., 1992, 1164 Bristol. 1165 10. McClure, E. R., Manufactures turn precision optics with diamond, Laser Focus World, 27, 95–105, 1991. 1166 11. van Neste, R., Pare´, C., Lachance, R. L., and Be´langer, P. A., Graded-phase mirror resonator with a 1167 super-Gaussian output in a CW-CO2 laser, IEEE J. Quantum Electron., 30, 2663–2669, 1994. 1168 12. Kudryashov, A., and Shmalhausen, V., Semipassive bimorph flexible mirrors for atmospheric adaptive 1169 optics application, Opt. Engng., 35, 3064–3073, 1996. 1170 13. Kudryashov, A. V., and Samarkin, V. V., Control of high power CO2 laser beam by adaptive optical 1171 elements, Opt. Comm., 118, 317–322, 1995. 1172 14. Baumhacker, H., Witte, K.-J., Stehbeck, H., Kudryashov, A., and Samarkin, V., In Use of deformable 1173 mirrors in the 8-TW TiS-laser ATLAS, Love, Gordon, ed., World Scientific, pp. 28–31, 2000. ARTICLE IN PRESS

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1174 15. Kudryashov, A. V., and Seliverstov, A. V., Adaptive stabilized interferometer with laser diode, Opt. 1175 Comm., 120, 239–244, 1995. 1176 16. Fox, A. G., and Li, T., Resonant modes in a maser interferometer, The Bell Syst. Tech. J., 40, 453–488, 1177 1961. 17. Li, T., Diffraction loss and selection of modes in maser resonators with circular mirrors, The Bell Syst. 1178 Tech. J., May, 917–932, 1965. 1179 18. Pare´, C., and Be´langer, P. A., Custom laser resonators using graded-phase mirrors: circular geometry, 1180 IEEE J. Quantum Electron., 30, 1141–1148, 1994. 1181 19. van Neste, R., Pare´, C., Lachance, R. L., and Be´langer, P. A., Graded-phase mirror resonator with a 1182 super-Gaussian output in a CW-CO2 laser, IEEE J. Quantum Electron., 30, 2663–2669, 1994. 1183 20. Be´langer, P. A., Pare´, C., Lachance, R. L., and van Neste, R., U.S. patent 5,255,283, 1993. 1184 21. Be´langer, P. A., and Pare´, C., Optical resonators using graded phase mirrors, Opt. Lett., 16, 1185 1057–1059, 1991. 1186 22. Zakharova, G., Karamzin, Yu. N., and Trofimov, V. A., Some problems of optical radiation nonlinear 1187 distortions compensation. Blooming and random phase distortions of profiled beams, Atmospheric 1188 Optics and Climate, 8, 706–712, 1995. 23. Ahmanov, S. A., Vorontsov, M. A., Kandidov, V. P., Syhorykov, A. P., and Chesnokov, S. S., Thermal 1189 self-action of light beams and methods of its compensation, Izv. Visshih Uchebnih Vavedenii, XXIII, 1190 1–37, 1980 (in Russian). 1191 Q3 24. Galushkin, M. G., Golubev, V. S., Zavalov, Yu. N., Zavalova, V. Ye., and Panchenko, V. Ya., 1192 Enhancement of small-scale optical nonuniformities in active medium of high-power CW FAF CO2 1193 laser, In Optical Resonators—Science and Engineering, Kossowsky, R. et al., eds., Kluwer Academic 1194 Publishers, New York, pp. 289–300, 1998. 1195 25. International Standard ISO 11146, Optics and optical instruments, Lasers and laser related equipment, 1196 Test methods for laser beam parameters: Beam widths, divergence angle and beam propagation factor. 1197 Document ISO/TC 172/SC 9/WC, 1995. 1198 26. Cherezova, T. Yu., Kaptsov, L. N., and Kudryashov, A. V., CW industrial rod YAG:Nd3þ laser with 1199 an intracavity active bimorph mirror, Appl. Opt., 35, 2554–2561, 1996. 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 ARTICLE IN PRESS

1225 Author Queries 1226 1227 JOB NUMBER: 9325 1228 Title: Laser Beam Shaping by Means of Flexible Mirrors 1229 1230 Q1 Please check the definitions for the acronyms have been inserted correctly. 1231 1232 Q2 Please confirm ‘Segnetomaterial’ is correct. 1233 Q3 Please provide complete list of author names for Ref. 24 1234 Q4 Please check whether Figure 6.7 is ok. 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275