Abstract

We present a variant of the method of Fox and Li [Bell Syst. Tech. J. 40, 453 (1961); Proc. IEEE 51, 80 (1963)] dedicated to intracavity laser beam shaping for resonators containing an arbitrary number of amplitude and phase diffractive optics. Contrary to Fox and Li, the starting point is the desired field. The latter is injected into the usual sequence of lenses representing just a single round trip, and the optimization process iterates until the input and the output fields match as much as possible. We illustrate this technique by deriving a simple model for generating single cylindrical TEMp0 modes, thanks to a π-phase plate placed inside a plano-concave cavity. The experimental validation attests an excellent agreement with numerical predictions.

© 2011 Optical Society of America

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    [CrossRef]

2011 (1)

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331-1334 (2011).
[CrossRef]

2010 (1)

2009 (2)

I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Opt. Express 17, 15891-15903 (2009).
[CrossRef] [PubMed]

T. Godin, M. Fromager, B. Päivänranta, N. Passilly, G. Boudebs, E. Cagniot, and K. Ait-Ameur, “Considerations about z-scan sensitivity improvement: theory versus experiments,” Appl. Phys. B 95, 579-587 (2009).
[CrossRef]

2008 (2)

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516 (2008).
[CrossRef]

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

2007 (1)

J. Chao, B. Li, Y. Cheng, and Y. Wang, “Simulation of optical field in laser resonators cavity by eigenvector method,” Opt. Laser Technol. 39, 490-499 (2007).
[CrossRef]

2006 (1)

2005 (1)

2002 (2)

K. Ait-Ameur, “Effects of a phase aperture on the fundamental mode of a hard-apertured cavity,” J. Mod. Opt. 49, 1157-1168(2002).
[CrossRef]

G. Martel, C. Labbé, F. Sanchez, M. Fromager, and K. Ait-Ameur, “Non-Gaussian fundamental laser mode oscillation in end-pumped Nd:YVO4 microchip laser,” Opt. Commun. 201, 117-127(2002).
[CrossRef]

2001 (1)

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325-386 (2001).
[CrossRef]

2000 (1)

1999 (1)

1997 (1)

J. Seurin and S. L. Chuang, “Discrete Bessel transform and beam propagation method for modeling of vertical-cavity surface-emitting lasers,” J. Appl. Phys. 82, 2007-2016(1997).
[CrossRef]

1996 (1)

L. Ingber, “Adaptive simulated annealing (ASA): lessons learned,” Contr. Cybernet. 25, 33-54 (1996).

1994 (1)

1991 (1)

1988 (1)

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

1983 (1)

G. Stephan and M. Trumper, “Inhomogeneity effects in a gas laser,” Phys. Rev. A 28, 2344-2362 (1983).
[CrossRef]

1970 (1)

1968 (1)

A. G. Fox and T. Li, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron. 4, 460-465 (1968).
[CrossRef]

1963 (1)

A. G. Fox and T. Li, “Modes in a maser interferometer with curved mirrors,” Proc. IEEE 51, 80-89 (1963).
[CrossRef]

1961 (1)

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

Aagedal, H.

Ait-Ameur, K.

E. Cagniot, M. Fromager, T. Godin, M. Traïche, N. Passilly, B. Päivänranta, and K. Ait-Ameur, “Cascades of π-phase plates: a transparent diffractive focusing system,” J. Opt. Soc. Am. A 27, 1647-1654 (2010).
[CrossRef]

T. Godin, M. Fromager, B. Päivänranta, N. Passilly, G. Boudebs, E. Cagniot, and K. Ait-Ameur, “Considerations about z-scan sensitivity improvement: theory versus experiments,” Appl. Phys. B 95, 579-587 (2009).
[CrossRef]

G. Martel, C. Labbé, F. Sanchez, M. Fromager, and K. Ait-Ameur, “Non-Gaussian fundamental laser mode oscillation in end-pumped Nd:YVO4 microchip laser,” Opt. Commun. 201, 117-127(2002).
[CrossRef]

K. Ait-Ameur, “Effects of a phase aperture on the fundamental mode of a hard-apertured cavity,” J. Mod. Opt. 49, 1157-1168(2002).
[CrossRef]

Aït-Ameur, K.

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331-1334 (2011).
[CrossRef]

Bélanger, P. A.

Bencheikh, A.

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331-1334 (2011).
[CrossRef]

Boudebs, G.

T. Godin, M. Fromager, B. Päivänranta, N. Passilly, G. Boudebs, E. Cagniot, and K. Ait-Ameur, “Considerations about z-scan sensitivity improvement: theory versus experiments,” Appl. Phys. B 95, 579-587 (2009).
[CrossRef]

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

Cagniot, E.

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331-1334 (2011).
[CrossRef]

E. Cagniot, M. Fromager, T. Godin, M. Traïche, N. Passilly, B. Päivänranta, and K. Ait-Ameur, “Cascades of π-phase plates: a transparent diffractive focusing system,” J. Opt. Soc. Am. A 27, 1647-1654 (2010).
[CrossRef]

T. Godin, M. Fromager, B. Päivänranta, N. Passilly, G. Boudebs, E. Cagniot, and K. Ait-Ameur, “Considerations about z-scan sensitivity improvement: theory versus experiments,” Appl. Phys. B 95, 579-587 (2009).
[CrossRef]

Chao, J.

J. Chao, B. Li, Y. Cheng, and Y. Wang, “Simulation of optical field in laser resonators cavity by eigenvector method,” Opt. Laser Technol. 39, 490-499 (2007).
[CrossRef]

Chen, D.

Cheng, Y.

J. Chao, B. Li, Y. Cheng, and Y. Wang, “Simulation of optical field in laser resonators cavity by eigenvector method,” Opt. Laser Technol. 39, 490-499 (2007).
[CrossRef]

Chuang, S. L.

J. Seurin and S. L. Chuang, “Discrete Bessel transform and beam propagation method for modeling of vertical-cavity surface-emitting lasers,” J. Appl. Phys. 82, 2007-2016(1997).
[CrossRef]

Collins, S. A.

Davidson, N.

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

A. A. Ishaaya, N. Davidson, and A. A. Friesem, “Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser,” Opt. Express 13, 4952-4962(2005).
[CrossRef] [PubMed]

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325-386 (2001).
[CrossRef]

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Continuous-phase elements can improve laser beam quality,” Opt. Lett. 25, 939-941 (2000).
[CrossRef]

Elkin, N.

N. Elkin, A. Napartovich, D. Vysotsky, and V. Troshchieva, “Round-trip operator technique applied for optical resonators with dispersion elements,” in Numerical Methods and Applications, Vol. 4310 of Lecture Notes in Computer Science (Springer Berlin, 2007), pp. 542-549.
[CrossRef]

Forbes, A.

Fox, A. G.

A. G. Fox and T. Li, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron. 4, 460-465 (1968).
[CrossRef]

A. G. Fox and T. Li, “Modes in a maser interferometer with curved mirrors,” Proc. IEEE 51, 80-89 (1963).
[CrossRef]

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

Friesem, A. A.

Fromager, M.

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331-1334 (2011).
[CrossRef]

E. Cagniot, M. Fromager, T. Godin, M. Traïche, N. Passilly, B. Päivänranta, and K. Ait-Ameur, “Cascades of π-phase plates: a transparent diffractive focusing system,” J. Opt. Soc. Am. A 27, 1647-1654 (2010).
[CrossRef]

T. Godin, M. Fromager, B. Päivänranta, N. Passilly, G. Boudebs, E. Cagniot, and K. Ait-Ameur, “Considerations about z-scan sensitivity improvement: theory versus experiments,” Appl. Phys. B 95, 579-587 (2009).
[CrossRef]

G. Martel, C. Labbé, F. Sanchez, M. Fromager, and K. Ait-Ameur, “Non-Gaussian fundamental laser mode oscillation in end-pumped Nd:YVO4 microchip laser,” Opt. Commun. 201, 117-127(2002).
[CrossRef]

Godin, T.

E. Cagniot, M. Fromager, T. Godin, M. Traïche, N. Passilly, B. Päivänranta, and K. Ait-Ameur, “Cascades of π-phase plates: a transparent diffractive focusing system,” J. Opt. Soc. Am. A 27, 1647-1654 (2010).
[CrossRef]

T. Godin, M. Fromager, B. Päivänranta, N. Passilly, G. Boudebs, E. Cagniot, and K. Ait-Ameur, “Considerations about z-scan sensitivity improvement: theory versus experiments,” Appl. Phys. B 95, 579-587 (2009).
[CrossRef]

Hasman, E.

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325-386 (2001).
[CrossRef]

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Continuous-phase elements can improve laser beam quality,” Opt. Lett. 25, 939-941 (2000).
[CrossRef]

Hasnaoui, A.

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331-1334 (2011).
[CrossRef]

Ingber, L.

L. Ingber, “Adaptive simulated annealing (ASA): lessons learned,” Contr. Cybernet. 25, 33-54 (1996).

Ishaaya, A. A.

Jackel, S.

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

Ji, P.

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516 (2008).
[CrossRef]

Labbé, C.

G. Martel, C. Labbé, F. Sanchez, M. Fromager, and K. Ait-Ameur, “Non-Gaussian fundamental laser mode oscillation in end-pumped Nd:YVO4 microchip laser,” Opt. Commun. 201, 117-127(2002).
[CrossRef]

Leger, J. R.

Li, B.

J. Chao, B. Li, Y. Cheng, and Y. Wang, “Simulation of optical field in laser resonators cavity by eigenvector method,” Opt. Laser Technol. 39, 490-499 (2007).
[CrossRef]

Li, T.

A. G. Fox and T. Li, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron. 4, 460-465 (1968).
[CrossRef]

A. G. Fox and T. Li, “Modes in a maser interferometer with curved mirrors,” Proc. IEEE 51, 80-89 (1963).
[CrossRef]

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

Litvin, I. A.

Liu, W.

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516 (2008).
[CrossRef]

Lumer, Y.

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

Machavariani, G.

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

Martel, G.

G. Martel, C. Labbé, F. Sanchez, M. Fromager, and K. Ait-Ameur, “Non-Gaussian fundamental laser mode oscillation in end-pumped Nd:YVO4 microchip laser,” Opt. Commun. 201, 117-127(2002).
[CrossRef]

Meir, A.

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

Moshe, I.

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

Napartovich, A.

N. Elkin, A. Napartovich, D. Vysotsky, and V. Troshchieva, “Round-trip operator technique applied for optical resonators with dispersion elements,” in Numerical Methods and Applications, Vol. 4310 of Lecture Notes in Computer Science (Springer Berlin, 2007), pp. 542-549.
[CrossRef]

Oron, R.

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325-386 (2001).
[CrossRef]

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Continuous-phase elements can improve laser beam quality,” Opt. Lett. 25, 939-941 (2000).
[CrossRef]

Päivänranta, B.

E. Cagniot, M. Fromager, T. Godin, M. Traïche, N. Passilly, B. Päivänranta, and K. Ait-Ameur, “Cascades of π-phase plates: a transparent diffractive focusing system,” J. Opt. Soc. Am. A 27, 1647-1654 (2010).
[CrossRef]

T. Godin, M. Fromager, B. Päivänranta, N. Passilly, G. Boudebs, E. Cagniot, and K. Ait-Ameur, “Considerations about z-scan sensitivity improvement: theory versus experiments,” Appl. Phys. B 95, 579-587 (2009).
[CrossRef]

Paré, C.

Passilly, N.

E. Cagniot, M. Fromager, T. Godin, M. Traïche, N. Passilly, B. Päivänranta, and K. Ait-Ameur, “Cascades of π-phase plates: a transparent diffractive focusing system,” J. Opt. Soc. Am. A 27, 1647-1654 (2010).
[CrossRef]

T. Godin, M. Fromager, B. Päivänranta, N. Passilly, G. Boudebs, E. Cagniot, and K. Ait-Ameur, “Considerations about z-scan sensitivity improvement: theory versus experiments,” Appl. Phys. B 95, 579-587 (2009).
[CrossRef]

Sanchez, F.

G. Martel, C. Labbé, F. Sanchez, M. Fromager, and K. Ait-Ameur, “Non-Gaussian fundamental laser mode oscillation in end-pumped Nd:YVO4 microchip laser,” Opt. Commun. 201, 117-127(2002).
[CrossRef]

Seurin, J.

J. Seurin and S. L. Chuang, “Discrete Bessel transform and beam propagation method for modeling of vertical-cavity surface-emitting lasers,” J. Appl. Phys. 82, 2007-2016(1997).
[CrossRef]

Stephan, G.

G. Stephan and M. Trumper, “Inhomogeneity effects in a gas laser,” Phys. Rev. A 28, 2344-2362 (1983).
[CrossRef]

Tervo, J.

Traïche, M.

Troshchieva, V.

N. Elkin, A. Napartovich, D. Vysotsky, and V. Troshchieva, “Round-trip operator technique applied for optical resonators with dispersion elements,” in Numerical Methods and Applications, Vol. 4310 of Lecture Notes in Computer Science (Springer Berlin, 2007), pp. 542-549.
[CrossRef]

Trumper, M.

G. Stephan and M. Trumper, “Inhomogeneity effects in a gas laser,” Phys. Rev. A 28, 2344-2362 (1983).
[CrossRef]

Turunen, J.

Vahimaa, P.

Vallius, T.

Vysotsky, D.

N. Elkin, A. Napartovich, D. Vysotsky, and V. Troshchieva, “Round-trip operator technique applied for optical resonators with dispersion elements,” in Numerical Methods and Applications, Vol. 4310 of Lecture Notes in Computer Science (Springer Berlin, 2007), pp. 542-549.
[CrossRef]

Wang, Y.

J. Chao, B. Li, Y. Cheng, and Y. Wang, “Simulation of optical field in laser resonators cavity by eigenvector method,” Opt. Laser Technol. 39, 490-499 (2007).
[CrossRef]

Wang, Z.

Wen, J. J.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

Wyrowski, F.

Yang, J.

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516 (2008).
[CrossRef]

Zeitner, U. D.

Appl. Opt. (1)

Appl. Phys. B (1)

T. Godin, M. Fromager, B. Päivänranta, N. Passilly, G. Boudebs, E. Cagniot, and K. Ait-Ameur, “Considerations about z-scan sensitivity improvement: theory versus experiments,” Appl. Phys. B 95, 579-587 (2009).
[CrossRef]

Bell Syst. Tech. J. (1)

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

Contr. Cybernet. (1)

L. Ingber, “Adaptive simulated annealing (ASA): lessons learned,” Contr. Cybernet. 25, 33-54 (1996).

IEEE J. Quantum Electron. (1)

A. G. Fox and T. Li, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron. 4, 460-465 (1968).
[CrossRef]

J. Acoust. Soc. Am. (2)

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516 (2008).
[CrossRef]

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

J. Appl. Phys. (1)

J. Seurin and S. L. Chuang, “Discrete Bessel transform and beam propagation method for modeling of vertical-cavity surface-emitting lasers,” J. Appl. Phys. 82, 2007-2016(1997).
[CrossRef]

J. Mod. Opt. (1)

K. Ait-Ameur, “Effects of a phase aperture on the fundamental mode of a hard-apertured cavity,” J. Mod. Opt. 49, 1157-1168(2002).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331-1334 (2011).
[CrossRef]

G. Martel, C. Labbé, F. Sanchez, M. Fromager, and K. Ait-Ameur, “Non-Gaussian fundamental laser mode oscillation in end-pumped Nd:YVO4 microchip laser,” Opt. Commun. 201, 117-127(2002).
[CrossRef]

Opt. Express (2)

Opt. Laser Technol. (1)

J. Chao, B. Li, Y. Cheng, and Y. Wang, “Simulation of optical field in laser resonators cavity by eigenvector method,” Opt. Laser Technol. 39, 490-499 (2007).
[CrossRef]

Opt. Lett. (3)

Opt. Mater. (1)

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

Phys. Rev. A (1)

G. Stephan and M. Trumper, “Inhomogeneity effects in a gas laser,” Phys. Rev. A 28, 2344-2362 (1983).
[CrossRef]

Proc. IEEE (1)

A. G. Fox and T. Li, “Modes in a maser interferometer with curved mirrors,” Proc. IEEE 51, 80-89 (1963).
[CrossRef]

Prog. Opt. (1)

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325-386 (2001).
[CrossRef]

Other (1)

N. Elkin, A. Napartovich, D. Vysotsky, and V. Troshchieva, “Round-trip operator technique applied for optical resonators with dispersion elements,” in Numerical Methods and Applications, Vol. 4310 of Lecture Notes in Computer Science (Springer Berlin, 2007), pp. 542-549.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Equivalent sequence for a single round trip.

Fig. 2
Fig. 2

Modeling of the equivalent sequence using ABCD matrices.

Fig. 3
Fig. 3

Single TEM 10 mode ( d = 99 % , first zero).

Fig. 4
Fig. 4

Single TEM 20 mode ( d = 99 % , first zero).

Fig. 5
Fig. 5

Single TEM 30 mode ( d = 99 % , first zero).

Fig. 6
Fig. 6

Single TEM 30 mode ( d = 99 % , second zero).

Fig. 7
Fig. 7

Single TEM 30 mode ( d = 95 % , first zero).

Fig. 8
Fig. 8

M 2 measurement for a single TEM 10 mode ( M 2 = 3.04 ).

Fig. 9
Fig. 9

M 2 measurement for a single TEM 20 mode ( M 2 = 5.04 ).

Fig. 10
Fig. 10

M 2 measurement for a single TEM 30 mode ( M 2 = 7.38 ).

Tables (1)

Tables Icon

Table 1 Results of the Optimization Process

Equations (24)

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g = 1 L R c ,
τ ( ρ a ) = { 1 , 0 ρ a , + 1 , ρ > a ,
( A 0 B 0 C 0 D 0 ) = ( 1 d 0 1 ) ,
( A 1 B 1 C 1 D 1 ) = ( 1 L d 0 1 ) × ( 1 0 1 f 1 ) × ( 1 L d 0 1 ) .
Ψ p ( r , 0 ) = C exp ( r 2 w 0 2 ) L p ( 2 r 2 w 0 2 ) ,
C = 2 π 1 w 0 ,
w 0 = Γ × w ¯ 0 ,
w ¯ 0 = λ L π ( g 1 g ) 1 2 .
0 + exp ( μ t 2 ) L n ( ν t 2 ) J 0 ( t x ) t d t = ( μ ν ) n 2 μ n + 1 exp ( x 2 4 μ ) L n ( ν x 2 4 μ ( ν μ ) ) ,
E 0 ( r , z ) = C j k B 0 exp ( j k z ) exp ( j k D 0 r 2 2 B 0 ) ( Q α ) p 2 Q p + 1 × exp ( k 2 r 2 4 Q B 0 2 ) L p ( α k 2 r 2 4 Q ( α Q ) B 0 2 ) ,
α = 2 w 0 2 ,
Q = 1 w 0 2 + j k A 0 2 B 0 .
circ ( ρ a ) n = 0 N 1 F n exp ( G n a 2 ρ 2 ) ,
{     n 1 < N , ( F n 1 ( 1 ) , G n 1 ( 1 ) ) = ( F n 1 , G n 1 ) , ( F N ( 1 ) , G N ( 1 ) ) = ( 1 2 , 0 ) ,
E 1 ( r , z ) = C ( Q α ) p 2 Q p + 1 exp ( j k z ) j k B 0 j k B 1 exp ( j k D 1 r 2 2 B 1 ) × ( 2 ) n 1 = 0 N F n 1 ( 1 ) ( Q n 1 ( 1 ) α ( 1 ) ) p 2 ( Q n 1 ( 1 ) ) p + 1 exp ( k 2 r 2 4 Q n 1 ( 1 ) B 1 2 ) × L p ( α ( 1 ) k 2 r 2 4 Q n 1 ( 1 ) ( α ( 1 ) Q n 1 ( 1 ) ) B 1 2 ) ,
α ( 1 ) = α k 2 4 Q ( α Q ) B 0 2 ,
Q ( 1 ) = k 2 4 Q B 0 2 + j k 2 ( D 0 B 0 + A 1 B 1 ) ,
    n 1 N , Q n 1 ( 1 ) = Q ( 1 ) + G n 1 ( 1 ) a 2 .
E 2 ( r , z ) = C ( Q α ) p 2 Q p + 1 exp ( j k z ) j k B 0 j k B 1 j k B 0 exp ( j k D 0 r 2 2 B 0 ) × ( 2 ) 2 n 1 = 0 N F n 1 ( 1 ) ( Q n 1 ( 1 ) α ( 1 ) ) p 2 ( Q n 1 ( 1 ) ) p + 1 n 2 = 0 N F n 2 ( 2 ) ( Q n 2 ( 2 ) α ( 2 ) ) p 2 ( Q n 2 ( 2 ) ) p + 1 × exp ( k 2 r 2 4 Q n 2 ( 2 ) B 0 2 ) L p ( α ( 2 ) k 2 r 2 4 Q n 2 ( 2 ) ( α ( 2 ) Q n 2 ( 2 ) ) B 0 2 ) ,
α ( 2 ) = α ( 1 ) k 2 4 Q n 1 ( 1 ) ( α ( 1 ) Q n 1 ( 1 ) ) B 1 2 ,
Q ( 2 ) = k 2 4 Q n 1 ( 1 ) B 1 2 + j k 2 ( D 1 B 1 + A 0 B 0 ) ,
    n 2 N , Q n 2 ( 2 ) = Q ( 2 ) + G n 2 ( 2 ) a 2 .
ε p = i I 1 ( i ) I 2 ( i ) i I 1 ( i ) × 100 .
w ( z d ) = w 0 [ 1 + ( z d z R ) 2 ] 1 2 ,

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