Abstract

Thermal blooming of a laser beam propagating in a gas-filled tube is investigated both analytically and experimentally. A self-consistent formulation taking into account heating of the gas and the resultant laser beam spreading (including diffraction) is presented. The heat equation is used to determine the temperature variation while the paraxial wave equation is solved in the eikonal approximation to determine the temporal and spatial variation of the Gaussian laser spot radius, Gouy phase (longitudinal phase delay), and wavefront curvature. The analysis is benchmarked against a thermal blooming experiment in the literature using a CO2 laser beam propagating in a tube filled with air and propane. New experimental results are presented in which a CW fiber laser (1 μm) propagates in a tube filled with nitrogen and water vapor. By matching laboratory and theoretical results, the absorption coefficient of water vapor is found to agree with calculations using MODTRAN (the MODerate-resolution atmospheric TRANsmission molecular absorption database) and HITRAN (the HIgh-resolution atmospheric TRANsmission molecular absorption database).

© 2014 Optical Society of America

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    [CrossRef]

2013

Y. Sun, F. Xi, Z. Ren, Z. Chen, Q. Peng, H. Chen, L. Guo, and Q. Lu, “Simulation for thermal blooming of the axial pipe flow,” Proc. SPIE 8904, 890411 (2013).

2012

2010

P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A 81, 013819 (2010).
[CrossRef]

2009

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102, 233902 (2009).
[CrossRef]

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J. P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J. M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J. Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaele, and J. A. Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

2005

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: a reformulated atmospheric band model with auxiliary species and practical multiple scattering options,” Proc. SPIE 5655, 88 (2005).
[CrossRef]

2003

2002

M. Kolesik, J. V. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. 89, 283902 (2002).
[CrossRef]

P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66, 046418 (2002).
[CrossRef]

1994

C. S. Liu and V. K. Tripathi, “Laser guiding in an axially nonuniform plasma channel,” Phys. Plasmas 1, 3100–3103 (1994).
[CrossRef]

1992

B. Johnson, “Thermal-blooming laboratory experiments,” Lincoln Lab. J. 5, 151 (1992).

1991

R. Holmes, R. Myers, and C. Duzy, “A linearized theory of transient laser heating in fluids,” Phys. Rev. A 44, 6862–6876 (1991).
[CrossRef]

V. V. Vorob’ev, “Thermal blooming of laser beams in the atmosphere,” Prog. Quantum Electron. 15, 1–152 (1991).
[CrossRef]

1987

A. M. Rubenchik and S. K. Turitsyn, “On self-focusing of laser beams in plasma,” Lasers Part Beams 5, 3–14 (1987).

1983

A. Gavrielides and P. Peterson, “Time-dependent thermal blooming in axial pipe flow,” Appl. Opt. 22, 3359–3365 (1983).
[CrossRef]

M. N. Kogan and A. N. Kucherov, “Investigation of thermal self-action (blooming) of intense beams in homogeneous gas flows,” Sov. Phys. J. 26, 196–201 (1983).
[CrossRef]

L. K. Forbes, “High-order series solution for unsteady thermal blooming with crosswind,” J. Appl. Math. Phys. 34, 334–346 (1983).
[CrossRef]

1982

1981

A. T. Jones and J. A. McMordie, “Thermal blooming of continuous wave laser radiation,” J. Phys. D 14, 163–172 (1981).
[CrossRef]

1979

H. Breaux, W. Evers, R. Sepucha, and C. Whitney, “Algebraic model for CW thermal-blooming effects,” Appl. Opt. 18, 2638–2644 (1979).
[CrossRef]

P. L. Eggins, J. A. McMordie, A. C. Douglas, and P. D. Roberts, “Thermal blooming of single pulse CO2 laser radiation,” J. Phys. D 12, 2063–2077 (1979).

1977

D. C. Smith, “High power laser propagation: thermal blooming,” Proc. IEEE 65, 1679–1714 (1977).
[CrossRef]

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere: II,” Appl. Phys. 14, 99–115 (1977).
[CrossRef]

1976

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

1975

1973

A. H. Aitken, J. N. Hayes, and P. B. Ulrich, “Thermal blooming of pulsed focused Gaussian laser beams,” Appl. Opt. 12, 193–197 (1973).
[CrossRef]

H. Kleiman and R. W. O’Neil, “Thermal blooming of pulsed laser radiation,” Appl. Phys. Lett. 23, 43–44 (1973).
[CrossRef]

1972

1971

F. G. Gebhardt and D. C. Smith, “Self-induced thermal distortion in the near field for a laser beam in a moving medium,” IEEE J. Quantum Electron. 7, 63–73 (1971).
[CrossRef]

1969

P. R. Longaker and M. M. Litvak, “Perturbation of the refractive index of absorbing media by a pulsed laser beam,” J. Appl. Phys. 40, 4033–4041 (1969).
[CrossRef]

A. J. Glass, “Thermal blooming in gases,” Opto-electron. 1, 174–178 (1969).
[CrossRef]

D. C. Smith, “Thermal defocusing of CO2 laser radiation in gases,” IEEE J. Quantum Electron. 5, 600–607 (1969).
[CrossRef]

1968

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10, 609–636 (1968).
[CrossRef]

1967

K. A. Breuckner and S. Jorna, “Linearized theory of laser-induced instabilities in liquids and gases,” Phys. Rev. 164, 182–193 (1967).
[CrossRef]

Acharya, P. K.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: a reformulated atmospheric band model with auxiliary species and practical multiple scattering options,” Proc. SPIE 5655, 88 (2005).
[CrossRef]

Adler-Golden, S. M.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: a reformulated atmospheric band model with auxiliary species and practical multiple scattering options,” Proc. SPIE 5655, 88 (2005).
[CrossRef]

Aitken, A. H.

Akhmanov, S. A.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10, 609–636 (1968).
[CrossRef]

Anderson, G. P.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: a reformulated atmospheric band model with auxiliary species and practical multiple scattering options,” Proc. SPIE 5655, 88 (2005).
[CrossRef]

Auwera, J. A.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J. P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J. M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J. Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaele, and J. A. Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Barbe, A.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J. P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J. M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J. Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaele, and J. A. Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Benner, D. C.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J. P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J. M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J. Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaele, and J. A. Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Berger, P. J.

Berk, A.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: a reformulated atmospheric band model with auxiliary species and practical multiple scattering options,” Proc. SPIE 5655, 88 (2005).
[CrossRef]

Bernath, P. F.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J. P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J. M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J. Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaele, and J. A. Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Bernstein, L. S.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: a reformulated atmospheric band model with auxiliary species and practical multiple scattering options,” Proc. SPIE 5655, 88 (2005).
[CrossRef]

Birk, M.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J. P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J. M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J. Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaele, and J. A. Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Boudon, V.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J. P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J. M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J. Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaele, and J. A. Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Elsevier, 2008), Section 9.6.

Breaux, H.

Breuckner, K. A.

K. A. Breuckner and S. Jorna, “Linearized theory of laser-induced instabilities in liquids and gases,” Phys. Rev. 164, 182–193 (1967).
[CrossRef]

Brown, L. R.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J. P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J. M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J. Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaele, and J. A. Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Buser, R. G.

Campargue, A.

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Tripathi, V. K.

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[CrossRef]

Turitsyn, S. K.

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102, 233902 (2009).
[CrossRef]

A. M. Rubenchik and S. K. Turitsyn, “On self-focusing of laser beams in plasma,” Lasers Part Beams 5, 3–14 (1987).

Ulrich, P. B.

Vandaele, A. C.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J. P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J. M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J. Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaele, and J. A. Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Vorob’ev, V. V.

V. V. Vorob’ev, “Thermal blooming of laser beams in the atmosphere,” Prog. Quantum Electron. 15, 1–152 (1991).
[CrossRef]

Whitney, C.

Whitney, K. G.

K. G. Whitney, G. L. Mader, and P. B. Ulrich, “SSPARAMA: a nonlinear, wave optics multipulse (and CW) steady-state propagation code with adaptive coordinates,” , (1977).

Xi, F.

Y. Sun, F. Xi, Z. Ren, Z. Chen, Q. Peng, H. Chen, L. Guo, and Q. Lu, “Simulation for thermal blooming of the axial pipe flow,” Proc. SPIE 8904, 890411 (2013).

Appl. Opt.

Appl. Phys.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere: II,” Appl. Phys. 14, 99–115 (1977).
[CrossRef]

Appl. Phys. Lett.

H. Kleiman and R. W. O’Neil, “Thermal blooming of pulsed laser radiation,” Appl. Phys. Lett. 23, 43–44 (1973).
[CrossRef]

IEEE J. Quantum Electron.

F. G. Gebhardt and D. C. Smith, “Self-induced thermal distortion in the near field for a laser beam in a moving medium,” IEEE J. Quantum Electron. 7, 63–73 (1971).
[CrossRef]

D. C. Smith, “Thermal defocusing of CO2 laser radiation in gases,” IEEE J. Quantum Electron. 5, 600–607 (1969).
[CrossRef]

J. Appl. Math. Phys.

L. K. Forbes, “High-order series solution for unsteady thermal blooming with crosswind,” J. Appl. Math. Phys. 34, 334–346 (1983).
[CrossRef]

J. Appl. Phys.

P. R. Longaker and M. M. Litvak, “Perturbation of the refractive index of absorbing media by a pulsed laser beam,” J. Appl. Phys. 40, 4033–4041 (1969).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

J. Phys. D

A. T. Jones and J. A. McMordie, “Thermal blooming of continuous wave laser radiation,” J. Phys. D 14, 163–172 (1981).
[CrossRef]

P. L. Eggins, J. A. McMordie, A. C. Douglas, and P. D. Roberts, “Thermal blooming of single pulse CO2 laser radiation,” J. Phys. D 12, 2063–2077 (1979).

J. Quant. Spectrosc. Radiat. Transfer

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J. P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J. M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J. Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaele, and J. A. Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Lasers Part Beams

A. M. Rubenchik and S. K. Turitsyn, “On self-focusing of laser beams in plasma,” Lasers Part Beams 5, 3–14 (1987).

Lincoln Lab. J.

B. Johnson, “Thermal-blooming laboratory experiments,” Lincoln Lab. J. 5, 151 (1992).

Opto-electron.

A. J. Glass, “Thermal blooming in gases,” Opto-electron. 1, 174–178 (1969).
[CrossRef]

Phys. Plasmas

C. S. Liu and V. K. Tripathi, “Laser guiding in an axially nonuniform plasma channel,” Phys. Plasmas 1, 3100–3103 (1994).
[CrossRef]

Phys. Rev.

K. A. Breuckner and S. Jorna, “Linearized theory of laser-induced instabilities in liquids and gases,” Phys. Rev. 164, 182–193 (1967).
[CrossRef]

Phys. Rev. A

R. Holmes, R. Myers, and C. Duzy, “A linearized theory of transient laser heating in fluids,” Phys. Rev. A 44, 6862–6876 (1991).
[CrossRef]

P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A 81, 013819 (2010).
[CrossRef]

Phys. Rev. E

P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66, 046418 (2002).
[CrossRef]

Phys. Rev. Lett.

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102, 233902 (2009).
[CrossRef]

M. Kolesik, J. V. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. 89, 283902 (2002).
[CrossRef]

Proc. IEEE

D. C. Smith, “High power laser propagation: thermal blooming,” Proc. IEEE 65, 1679–1714 (1977).
[CrossRef]

Proc. SPIE

Y. Sun, F. Xi, Z. Ren, Z. Chen, Q. Peng, H. Chen, L. Guo, and Q. Lu, “Simulation for thermal blooming of the axial pipe flow,” Proc. SPIE 8904, 890411 (2013).

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: a reformulated atmospheric band model with auxiliary species and practical multiple scattering options,” Proc. SPIE 5655, 88 (2005).
[CrossRef]

Prog. Quantum Electron.

V. V. Vorob’ev, “Thermal blooming of laser beams in the atmosphere,” Prog. Quantum Electron. 15, 1–152 (1991).
[CrossRef]

Sov. Phys. J.

M. N. Kogan and A. N. Kucherov, “Investigation of thermal self-action (blooming) of intense beams in homogeneous gas flows,” Sov. Phys. J. 26, 196–201 (1983).
[CrossRef]

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[CrossRef]

Other

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L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Elsevier, 2011), Section 56.

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M. Rosenbluth, “Thermal blooming in the presence of wind shear,” (1990).

W. Kaiser and M. Maier, “Stimulated Rayleigh, Brillouin and Raman spectroscopy,” in Laser Handbook, F. T. Arecchi and E. O. Schulz-DuBois, eds. (North-Holland, 1972), pp. 1077–1150.

L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Elsevier, 2011), Section 49.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Elsevier, 2008), Sections 75, 80.

R. W. Boyd, Nonlinear Optics (Elsevier, 2008), Section 9.6.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Elsevier, 2008), p. 120.

K. G. Whitney, G. L. Mader, and P. B. Ulrich, “SSPARAMA: a nonlinear, wave optics multipulse (and CW) steady-state propagation code with adaptive coordinates,” , (1977).

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Figures (5)

Fig. 1.
Fig. 1.

Spot radius of CO2 laser beam after free propagation region.

Fig. 2.
Fig. 2.

Gouy phase of CO2 laser beam after free propagation region.

Fig. 3.
Fig. 3.

Wavefront curvature of CO2 laser beam after free propagation region.

Fig. 4.
Fig. 4.

CO2 laser beam profiles with and without blooming.

Fig. 5.
Fig. 5.

Time evolution of laser beam spot radius for the thermal blooming experiment at the Naval Research Laboratory (solid circles). The curves represent the corresponding theoretical results.

Tables (1)

Tables Icon

Table 1. Parameters for Thermal Blooming Experiment with CO2 Laser Beam [25]

Equations (34)

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dρdt+ρ·v=0,
ρdvdt=p+12c(αeI)+ρg+ρν2v,
ρTdsdt=·(κT)+αI,
Ra=βg5νχκQ,
ρ¨cT22(ρ+βρT)=αe2c2I
ρTs˙=·(κT)+αI,
T˙=χ2T+(α/ρcp)I.
I(r,z,t)=I0(t)R02R2(z,t)eαze2r2/R2(z,t),
T(r,z,t)=αR024πκeαz0t+dt0I0(t0)H(tt0)R2(z,t0)dx0dy0e[(xx0)2+(yy0)2]/4χ(tt0)e2(x02+y02)/R2(z,t0)tt0,
T(r,z,t)=αR02ρcpeαz0t+dt0I0(t0)H(tt0)R2(z,t0)+8χ(tt0)e2r2R2(z,t0)+8χ(tt0).
(2+2z21c22t2)E=S,
E(r,t)=A(r,t)exp[iψ(z,t)]e^/2+c.c.,
S(r,t)=S(r,t)exp[iψ(z,t)]e^/2+c.c.,
(2k02+ω02c2+2ik0z+2iω0ct+2z21c22t2)A(r,t)=S(r,t).
(2+2ik0z)A(r,τ)=STB(r,τ)i(ω0/c)2(Imε)A(r,τ),
[k0z+(2Φ)+(Φr)r]Ao2=0,
Φz+12k0(Φr)2=2A02k0A0ω0/c2n0(no21)βT.
A0=A^R0R(z,τ)er2/R2(z,τ),
Φ=k0B(z,τ)r2/2+φ(z,τ),
2Rz2=4k02R3+2(n021)αβR02n02ρcpeαzR0τ+dτ0I0(τ0)H(ττ0)[R2(z,τ0)+8χ(ττ0)]2,
φz=2k0R2(n021)(ω0/c)αβR022n0ρcpeαz0τ+dτ0I0(τ0)H(ττ0)R2(z,τ0)+8χ(ττ0),
B=R1R/z,
R(z)=R0(1+z2/ZR2)1/2,
φ(z)=tan1(z/ZR),
B=z/(ZR2+z2),
2R(z,τ)z2=4k02R3(z,τ)+(n021)αP2πn02κTeαzR(z,τ)[1R2(z,τ)R2(z,0)+8χτ],
φ(z,τ)z=2k0R2(z,τ)+(n021)(ω0/c)αP8πn0κTeαzn[R2(z,τ)R2(z,0)+8χτ],
2R(z,τ)z24k02R3(z,τ)+4(n021)αPπn02ρcpTeαzR3(z,τ)τ.
R(z)=R0[1+(M2zZR)2]1/2,
T(r,z,t)=m=1Am(z,t)Jν(yνmr/Rtube),
Ant+χ(y0nRtube)2An=2α0RtubedrrJ0(y0nr/Rtube)I(r,t)ρcpRtube2J02(y0n).
An(z,t)=2αR02eαzρcpRtube2J02(y0n)0tdτe(y0n/Rtube)2χ(tτ)I0(τ)R2(z,τ)0RtubedrrJ0(y0nrRtube)e2r2/R2(z,τ).
2Rz2=4k02R3+(n021)αβR028n02ρcpRtube2eαzRn=1y0n2/Rtube2J02(y0n)0τdτ0I0(τ0)e(y0n2/8Rtube2)[R2(z,τ0)+8χ(ττ0)].
2Rz2=4k02R3+(n021)αP4πn02κTRtube2eαzRn=1ey0n2R2/8Rtube2J02(y0n)(1ey0n2χτ/Rtube2).

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