Abstract

We discuss the formation of a specified super-Gaussian intensity distribution of a fundamental mode by means of an intracavity controlled mirror, which is a water-cooled bimorph flexible mirror equipped with four controlling electrodes. Analysis has confirmed the possibility to form fourth-, sixth-, and eighth-order super-Gaussian intensity distributions at the output of the stable resonators of industrial cw CO2 and YAG:Nd3+ lasers. We present the results of the experimental formation of fourth-order and sixth-order super-Gaussian fundamental modes at the output of a cw CO2 laser by means of an intracavity flexible mirror. We observed an increase in power up to 12% and an enlargement of the peak value of the far-field intensity by as much as 1.6 times that with a Gaussian TEM00 mode of the cw CO2 laser.

© 2001 Optical Society of America

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References

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  1. G. Abil’sitov, Technological Lasers (Mashinostroinie, Moscow, 1991), in Russian.
  2. H. Koechner, Industrial Applications of Lasers (Wiley-Interscience, New York, 1988).
  3. R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
    [CrossRef]
  4. M. Santarsiero, R. Borghi, “Correspondence between super-Gaussian and flattened Gaussian beams,” J. Opt. Soc. Am. A 16, 188–190 (1999).
    [CrossRef]
  5. F. Gory, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  6. S. Bollanti, P. Di Lazzaro, D. Murra, A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun. 138, 35–39 (1997).
    [CrossRef]
  7. S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
    [CrossRef]
  8. A. V. Goncharsky, V. V. Popov, V. V. Stepanov, Introduction to Computer Optics (Moscow State University, Moscow, 1991), in Russian.
  9. Yu. A. Anan’ev, Laser Resonators and the Beam Divergence Problem (Institute of Physics, London, 1992).
  10. E. R. McClure, “Manufacturers turn precision optics with diamond,” Laser Focus World 27, 95–105 (1991).
  11. R. van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, “Graded-phase mirror resonator with a super-Gaussian output in a CW-CO2 laser,” IEEE J. Quantum Electron. 30, 2663–2669 (1994).
    [CrossRef]
  12. A. V. Kudryashov, V. V. Samarkin, “Control of high power CO2 laser beam by adaptive optical elements,” Opt. Commun. 118, 317–322 (1995).
    [CrossRef]
  13. A. V. Kudryashov, A. V. Seliverstov, “Adaptive stabilized interferometer with laser diode,” Opt. Commun. 120, 239–244 (1995).
    [CrossRef]
  14. M. G. Galushkin, V. S. Golubev, Yu. N. Zavalov, V. Ye. Zavalova, V. Ya. Panchenko, “Enhancement of small-scale optical nonuniformities in active medium of high-power CW FAF CO2 laser,” in Optical Resonators: Science and Engineering, R. Kossowsky, M. Jelinek, J. Novak, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1998), pp. 289–300.
  15. A. Yariv, Quantum Electronics (Wiley, New York, 1989).
  16. T. Li, “Diffraction loss and selection of modes in maser resonators with circular mirrors,” Bell Syst. Tech. J. 44, 917–932 (1965).
    [CrossRef]
  17. A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
    [CrossRef]
  18. A. G. Fox, T. Li, “Effect of gain saturation on the oscillating modes of optical masers,” IEEE J. Quantum Electron. QE-2, 774–783 (1966).
    [CrossRef]
  19. P. A. Bélanger, C. Paré, “Optical resonators using graded phase mirrors,” Opt. Lett. 16, 1057–1059 (1991).
    [CrossRef]
  20. International Standard ISO 11146, “Optics and optical instruments, lasers and laser related equipment, Test methods for laser beam parameters: Beam widths, divergence angle, and beam propagation factor,” (Swiss Association for Standardization, Geneva, Switzerland, 1995).
  21. T. Yu. Cherezova, L. N. Kaptsov, A. V. Kudryashov, “Cw industrial rod YAG:Nd3+ laser with an intracavity active bimorph mirror,” Appl. Opt. 35, 2554–2561 (1996).
    [CrossRef] [PubMed]
  22. B. A. Boley, J. H. Weiner, Theory of the Thermal Stresses (Wiley, New York, 1963).
  23. I. A. Borodina, M. A. Vorontsov, “The effect of the mirror thermal deformations on the spatial structure of laser radiation,” J. Atmos. Oceanic Optics 1, 79–85 (1988), in Russian.

1999

R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
[CrossRef]

M. Santarsiero, R. Borghi, “Correspondence between super-Gaussian and flattened Gaussian beams,” J. Opt. Soc. Am. A 16, 188–190 (1999).
[CrossRef]

1997

S. Bollanti, P. Di Lazzaro, D. Murra, A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun. 138, 35–39 (1997).
[CrossRef]

1996

1995

A. V. Kudryashov, V. V. Samarkin, “Control of high power CO2 laser beam by adaptive optical elements,” Opt. Commun. 118, 317–322 (1995).
[CrossRef]

A. V. Kudryashov, A. V. Seliverstov, “Adaptive stabilized interferometer with laser diode,” Opt. Commun. 120, 239–244 (1995).
[CrossRef]

1994

F. Gory, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

R. van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, “Graded-phase mirror resonator with a super-Gaussian output in a CW-CO2 laser,” IEEE J. Quantum Electron. 30, 2663–2669 (1994).
[CrossRef]

1991

E. R. McClure, “Manufacturers turn precision optics with diamond,” Laser Focus World 27, 95–105 (1991).

P. A. Bélanger, C. Paré, “Optical resonators using graded phase mirrors,” Opt. Lett. 16, 1057–1059 (1991).
[CrossRef]

1990

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

1988

I. A. Borodina, M. A. Vorontsov, “The effect of the mirror thermal deformations on the spatial structure of laser radiation,” J. Atmos. Oceanic Optics 1, 79–85 (1988), in Russian.

1966

A. G. Fox, T. Li, “Effect of gain saturation on the oscillating modes of optical masers,” IEEE J. Quantum Electron. QE-2, 774–783 (1966).
[CrossRef]

1965

T. Li, “Diffraction loss and selection of modes in maser resonators with circular mirrors,” Bell Syst. Tech. J. 44, 917–932 (1965).
[CrossRef]

1961

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Abil’sitov, G.

G. Abil’sitov, Technological Lasers (Mashinostroinie, Moscow, 1991), in Russian.

Anan’ev, Yu. A.

Yu. A. Anan’ev, Laser Resonators and the Beam Divergence Problem (Institute of Physics, London, 1992).

Bélanger, P. A.

R. van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, “Graded-phase mirror resonator with a super-Gaussian output in a CW-CO2 laser,” IEEE J. Quantum Electron. 30, 2663–2669 (1994).
[CrossRef]

P. A. Bélanger, C. Paré, “Optical resonators using graded phase mirrors,” Opt. Lett. 16, 1057–1059 (1991).
[CrossRef]

Boley, B. A.

B. A. Boley, J. H. Weiner, Theory of the Thermal Stresses (Wiley, New York, 1963).

Bollanti, S.

S. Bollanti, P. Di Lazzaro, D. Murra, A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun. 138, 35–39 (1997).
[CrossRef]

Borghi, R.

R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
[CrossRef]

M. Santarsiero, R. Borghi, “Correspondence between super-Gaussian and flattened Gaussian beams,” J. Opt. Soc. Am. A 16, 188–190 (1999).
[CrossRef]

Borodina, I. A.

I. A. Borodina, M. A. Vorontsov, “The effect of the mirror thermal deformations on the spatial structure of laser radiation,” J. Atmos. Oceanic Optics 1, 79–85 (1988), in Russian.

Cherezova, T. Yu.

De Silvestri, S.

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

Di Lazzaro, P.

S. Bollanti, P. Di Lazzaro, D. Murra, A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun. 138, 35–39 (1997).
[CrossRef]

Fox, A. G.

A. G. Fox, T. Li, “Effect of gain saturation on the oscillating modes of optical masers,” IEEE J. Quantum Electron. QE-2, 774–783 (1966).
[CrossRef]

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Galushkin, M. G.

M. G. Galushkin, V. S. Golubev, Yu. N. Zavalov, V. Ye. Zavalova, V. Ya. Panchenko, “Enhancement of small-scale optical nonuniformities in active medium of high-power CW FAF CO2 laser,” in Optical Resonators: Science and Engineering, R. Kossowsky, M. Jelinek, J. Novak, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1998), pp. 289–300.

Golubev, V. S.

M. G. Galushkin, V. S. Golubev, Yu. N. Zavalov, V. Ye. Zavalova, V. Ya. Panchenko, “Enhancement of small-scale optical nonuniformities in active medium of high-power CW FAF CO2 laser,” in Optical Resonators: Science and Engineering, R. Kossowsky, M. Jelinek, J. Novak, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1998), pp. 289–300.

Goncharsky, A. V.

A. V. Goncharsky, V. V. Popov, V. V. Stepanov, Introduction to Computer Optics (Moscow State University, Moscow, 1991), in Russian.

Gory, F.

F. Gory, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

Kaptsov, L. N.

Koechner, H.

H. Koechner, Industrial Applications of Lasers (Wiley-Interscience, New York, 1988).

Kudryashov, A. V.

T. Yu. Cherezova, L. N. Kaptsov, A. V. Kudryashov, “Cw industrial rod YAG:Nd3+ laser with an intracavity active bimorph mirror,” Appl. Opt. 35, 2554–2561 (1996).
[CrossRef] [PubMed]

A. V. Kudryashov, V. V. Samarkin, “Control of high power CO2 laser beam by adaptive optical elements,” Opt. Commun. 118, 317–322 (1995).
[CrossRef]

A. V. Kudryashov, A. V. Seliverstov, “Adaptive stabilized interferometer with laser diode,” Opt. Commun. 120, 239–244 (1995).
[CrossRef]

Lachance, R. L.

R. van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, “Graded-phase mirror resonator with a super-Gaussian output in a CW-CO2 laser,” IEEE J. Quantum Electron. 30, 2663–2669 (1994).
[CrossRef]

Li, T.

A. G. Fox, T. Li, “Effect of gain saturation on the oscillating modes of optical masers,” IEEE J. Quantum Electron. QE-2, 774–783 (1966).
[CrossRef]

T. Li, “Diffraction loss and selection of modes in maser resonators with circular mirrors,” Bell Syst. Tech. J. 44, 917–932 (1965).
[CrossRef]

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Magni, V.

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

McClure, E. R.

E. R. McClure, “Manufacturers turn precision optics with diamond,” Laser Focus World 27, 95–105 (1991).

Murra, D.

S. Bollanti, P. Di Lazzaro, D. Murra, A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun. 138, 35–39 (1997).
[CrossRef]

Panchenko, V. Ya.

M. G. Galushkin, V. S. Golubev, Yu. N. Zavalov, V. Ye. Zavalova, V. Ya. Panchenko, “Enhancement of small-scale optical nonuniformities in active medium of high-power CW FAF CO2 laser,” in Optical Resonators: Science and Engineering, R. Kossowsky, M. Jelinek, J. Novak, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1998), pp. 289–300.

Paré, C.

R. van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, “Graded-phase mirror resonator with a super-Gaussian output in a CW-CO2 laser,” IEEE J. Quantum Electron. 30, 2663–2669 (1994).
[CrossRef]

P. A. Bélanger, C. Paré, “Optical resonators using graded phase mirrors,” Opt. Lett. 16, 1057–1059 (1991).
[CrossRef]

Popov, V. V.

A. V. Goncharsky, V. V. Popov, V. V. Stepanov, Introduction to Computer Optics (Moscow State University, Moscow, 1991), in Russian.

Samarkin, V. V.

A. V. Kudryashov, V. V. Samarkin, “Control of high power CO2 laser beam by adaptive optical elements,” Opt. Commun. 118, 317–322 (1995).
[CrossRef]

Santarsiero, M.

R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
[CrossRef]

M. Santarsiero, R. Borghi, “Correspondence between super-Gaussian and flattened Gaussian beams,” J. Opt. Soc. Am. A 16, 188–190 (1999).
[CrossRef]

Seliverstov, A. V.

A. V. Kudryashov, A. V. Seliverstov, “Adaptive stabilized interferometer with laser diode,” Opt. Commun. 120, 239–244 (1995).
[CrossRef]

Stepanov, V. V.

A. V. Goncharsky, V. V. Popov, V. V. Stepanov, Introduction to Computer Optics (Moscow State University, Moscow, 1991), in Russian.

Svelto, O.

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

Torre, A.

S. Bollanti, P. Di Lazzaro, D. Murra, A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun. 138, 35–39 (1997).
[CrossRef]

Valentini, G.

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

van Neste, R.

R. van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, “Graded-phase mirror resonator with a super-Gaussian output in a CW-CO2 laser,” IEEE J. Quantum Electron. 30, 2663–2669 (1994).
[CrossRef]

Vorontsov, M. A.

I. A. Borodina, M. A. Vorontsov, “The effect of the mirror thermal deformations on the spatial structure of laser radiation,” J. Atmos. Oceanic Optics 1, 79–85 (1988), in Russian.

Weiner, J. H.

B. A. Boley, J. H. Weiner, Theory of the Thermal Stresses (Wiley, New York, 1963).

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1989).

Zavalov, Yu. N.

M. G. Galushkin, V. S. Golubev, Yu. N. Zavalov, V. Ye. Zavalova, V. Ya. Panchenko, “Enhancement of small-scale optical nonuniformities in active medium of high-power CW FAF CO2 laser,” in Optical Resonators: Science and Engineering, R. Kossowsky, M. Jelinek, J. Novak, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1998), pp. 289–300.

Zavalova, V. Ye.

M. G. Galushkin, V. S. Golubev, Yu. N. Zavalov, V. Ye. Zavalova, V. Ya. Panchenko, “Enhancement of small-scale optical nonuniformities in active medium of high-power CW FAF CO2 laser,” in Optical Resonators: Science and Engineering, R. Kossowsky, M. Jelinek, J. Novak, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1998), pp. 289–300.

Appl. Opt.

Bell Syst. Tech. J.

T. Li, “Diffraction loss and selection of modes in maser resonators with circular mirrors,” Bell Syst. Tech. J. 44, 917–932 (1965).
[CrossRef]

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

IEEE J. Quantum Electron.

A. G. Fox, T. Li, “Effect of gain saturation on the oscillating modes of optical masers,” IEEE J. Quantum Electron. QE-2, 774–783 (1966).
[CrossRef]

R. van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, “Graded-phase mirror resonator with a super-Gaussian output in a CW-CO2 laser,” IEEE J. Quantum Electron. 30, 2663–2669 (1994).
[CrossRef]

R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
[CrossRef]

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

J. Atmos. Oceanic Optics

I. A. Borodina, M. A. Vorontsov, “The effect of the mirror thermal deformations on the spatial structure of laser radiation,” J. Atmos. Oceanic Optics 1, 79–85 (1988), in Russian.

J. Opt. Soc. Am. A

Laser Focus World

E. R. McClure, “Manufacturers turn precision optics with diamond,” Laser Focus World 27, 95–105 (1991).

Opt. Commun.

F. Gory, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

S. Bollanti, P. Di Lazzaro, D. Murra, A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun. 138, 35–39 (1997).
[CrossRef]

A. V. Kudryashov, V. V. Samarkin, “Control of high power CO2 laser beam by adaptive optical elements,” Opt. Commun. 118, 317–322 (1995).
[CrossRef]

A. V. Kudryashov, A. V. Seliverstov, “Adaptive stabilized interferometer with laser diode,” Opt. Commun. 120, 239–244 (1995).
[CrossRef]

Opt. Lett.

Other

International Standard ISO 11146, “Optics and optical instruments, lasers and laser related equipment, Test methods for laser beam parameters: Beam widths, divergence angle, and beam propagation factor,” (Swiss Association for Standardization, Geneva, Switzerland, 1995).

M. G. Galushkin, V. S. Golubev, Yu. N. Zavalov, V. Ye. Zavalova, V. Ya. Panchenko, “Enhancement of small-scale optical nonuniformities in active medium of high-power CW FAF CO2 laser,” in Optical Resonators: Science and Engineering, R. Kossowsky, M. Jelinek, J. Novak, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1998), pp. 289–300.

A. Yariv, Quantum Electronics (Wiley, New York, 1989).

A. V. Goncharsky, V. V. Popov, V. V. Stepanov, Introduction to Computer Optics (Moscow State University, Moscow, 1991), in Russian.

Yu. A. Anan’ev, Laser Resonators and the Beam Divergence Problem (Institute of Physics, London, 1992).

G. Abil’sitov, Technological Lasers (Mashinostroinie, Moscow, 1991), in Russian.

H. Koechner, Industrial Applications of Lasers (Wiley-Interscience, New York, 1988).

B. A. Boley, J. H. Weiner, Theory of the Thermal Stresses (Wiley, New York, 1963).

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Figures (10)

Fig. 1
Fig. 1

Photograph of the mirror sample. Each of eight electrode segments were connected to reproduce a ringlike electrode.

Fig. 2
Fig. 2

Construction of the bimorph deformable mirror and a schematic of its electrodes.

Fig. 3
Fig. 3

Experimental setup of the formation of the super-Gaussian TEM00 mode: 1, block of mirror electrode control; 2, semipassive bimorph mirror; 3, diaphragm; 4, concave mirror R = 2200 mm; 5, convex mirror R = -800 mm; 6, active medium of CO2; 7, ZnSe output mirror with a 69% coefficient of reflectivity; 8, LBA-2A (laser beam analyzer); 9, oscilloscope; 10, computer; 11, lens f = 275 mm; 12, MAC-2 (mode analyzer computer).

Fig. 4
Fig. 4

Evolution of the fourth-order super-Gaussian beam at the surface of an output mirror: 1, initial beam; 2, beam after one round trip; 3, beam after two round trips; 4, beam after three round trips, etc.; 11, beam after ten round trips.

Fig. 5
Fig. 5

(a) Formation of the fourth-order super-Gaussian fundamental mode: 1, Gaussian; 2, theoretically obtained; 3, experimentally obtained. (b) Formation of the sixth-order super-Gaussian fundamental mode: 1, Gaussian; 2, theoretically obtained; 3, experimentally obtained.

Fig. 6
Fig. 6

Far-field patterns of a laser beam: (a) theory of the fourth-order super-Gaussian modes, (b) experimental results of the Gaussian TEM00 mode, (c) experimental results of the super-Gaussian fourth-order TEM00 mode.

Fig. 7
Fig. 7

Normalized far-field intensity distributions: (a) 1, fourth-order super-Gaussian formed by a flexible mirror (with near-field waist ω = 4.8 mm); 2, ideal fourth-order super-Gaussian mode (ω = 4.8 mm); 3, Gaussian beam (ω = 4.8 mm). (b) Fragment of the intensity distributions in (a).

Fig. 8
Fig. 8

Layout of a telescope-type stable resonator of a YAG:Nd3+ laser with a wide-aperture mirror: 1, output coupler; 2, active medium; 3, thermal lens; 4, meniscus; 5, bimorph flexible mirror.

Fig. 9
Fig. 9

Formation of the super-Gaussian fundamental modes at the output of a YAG:Nd3+ laser stable resonator: 1, Gaussian mode; 2, fourth-order super-Gaussian mode.

Fig. 10
Fig. 10

Super-Gaussian TEM00 mode at the plane output coupler of a telescope-type YAG:Nd3+ laser stable resonator: 1, without thermal deformations of the output coupler; 2, with distortions of the intensity distribution caused by thermal deformation of the output mirror; 3, with phase aberration corrections caused by thermal deformations with a bimorph flexible mirror.

Tables (2)

Tables Icon

Table 1 Voltages Applied to Electrodes of Flexible Mirrors to Form Super-Gaussian Beams

Tables Icon

Table 2 Voltages Applied to Electrodes of Flexible Mirrors to Form Super-Gaussian TEM00 Modes at the Output of a YAG:Nd3+ Laser

Equations (16)

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

Ψ1r1=E0 exp-r1ως,
γ2Ψ2r2=0b K1r1, r2Ψ1r1r1dr1,
K1r1, r2=2π jλB J0k r1r2Bexp-jk2BAr12+Dr22HI;
HI=1+g0Lam21+IrIs,
γ1Ψ1r1=0a K2r2, r1Ψ2r2r2dr2,
K2r2, r1=2π jλB J0k r1r2B×exp-jk2BAr12+Dr22×expjkφmirrorr2HI.
d0=22 r2Ir, zrdrdφ Ir, zrdrdφ;
Ifr, z, df=22 r2Ifr, zrdrdφ Ifr, zrdrdφ.
ΔT=0,
χTzz=z0/2=k1Ir,
Tr0, z=0,
Tr0, -z0/2=0.
Tr, z=k=1 J0μk0rAkshγkz+Bkchγkz,
ΔU-Ur2+11-2νEr-21+ν1-ν α Tr=0, ΔW+11-νEz-21+ν1-2ν α Tz=0,
E=Ur+Ur+Wz.
Wr, z=αz01+νk=1 J0μk0r×Bkshγkz+Akchγkz/γk+6αr01-ν1-r2k=1 J1μk0×Akγkchγk/2-2shγk/2/γk3.

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