Abstract

A new kind of laser beam called the decentered elliptical Hermite–Gaussian beam (DEHGB) is defined by use of a tensor method. The propagation formula of the DEHGB passing through a nonsymmetrical paraxial optical system is derived through vector integration. The derived formula can be easily reduced to the propagation formula of an aligned elliptical Hermite–Gaussian beam and that of a decentered elliptical Gaussian beam under certain conditions. By use of this formula, the propagation characteristics of the DEHGB through free space are presented graphically. As application examples, we construct a generalized laser array using the DEHGB as the fundamental mode. We also obtain the decentered elliptical flattened Gaussian beam by expressing it as superposition of a series of DEHGBs by using polynomial expansion.

© 2003 Optical Society of America

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References

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  1. A. E. Siegman, “Hermite–gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
    [CrossRef]
  2. W. H. Carter, “Spot size and divergence for Hermite Gaussian beams of any order,” Appl. Opt. 19, 1027–1029 (1980).
    [CrossRef] [PubMed]
  3. P. A. Belanger, “Packetlike solutions of the homogeneous-wave equation,” J. Opt. Soc. Am. A 1, 723–724 (1984).
    [CrossRef]
  4. V. N. Smirnov, G. A. Strokovskii, “On diffraction of optical Hermite–Gaussian beams from a diaphragm,” Opt. Spectrosc. 76, 912–919 (1994).
  5. E. Zauderer, “Complex argument Hermite–Gaussian and Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 3, 465–469 (1986).
    [CrossRef]
  6. S. Saghafi, C. J. R. Sheppard, “Near field and far field of elegant Hermite–Gaussian and Laguerre–Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
    [CrossRef]
  7. S. Saghafi, C. J. R. Sheppard, J. A. Piper, “Characteris-ing elegant and standard Hermite–Gaussian beam modes,” Opt. Commun. 191, 173–179 (2001).
    [CrossRef]
  8. A. A. Tovar, L. W. Casperson, “Production and propagation of Hermite–sinusoidal-Gaussian laser beams,” J. Opt. Soc. Am. A 15, 2425–2432 (1998).
    [CrossRef]
  9. H. Laabs, C. Gao, H. Weber, “Twisting to three-dimensional Hermite–Gaussian beams,” J. Mod. Opt. 46, 709–719 (1999).
  10. Y. Cai, Q. Lin, “The elliptical Hermite–Gaussian beam and its propagation through paraxial systems,” Opt. Commun. 207, 139–147 (2002).
    [CrossRef]
  11. L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12, 2423–2441 (1973).
    [CrossRef]
  12. A. R. Al-Rashed, B. E. A. Saleh, “Decentered Gaussian beams,” Appl. Opt. 34, 6819–6825 (1995).
    [CrossRef] [PubMed]
  13. C. Palma, “Decentered Gaussian beams, ray bundles, and Bessel Gauss beams,” Appl. Opt. 36, 1116–1120 (1997).
    [CrossRef] [PubMed]
  14. B. Lü, H. Ma, “Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185–194 (1999).
    [CrossRef]
  15. B. Lü, H. Ma, “Coherent and incoherent off-axis Hermite–Gaussian beam combinations,” Appl. Opt. 39, 1279–1289 (2000).
    [CrossRef]
  16. M. L. Luis, M. Y. Omel, J. J. D. Joris, “Incoherent su-perposition of off-axis polychromatic Hermite–Gaussian modes,” J. Opt. Soc. Am. A 19, 1572–1582 (2002).
    [CrossRef]
  17. P. J. Cronin, P. Török, P. Varga, C. Cogswell, “High-aperture diffraction of a scalar, off-axis Gaussian beam,” J. Opt. Soc. Am. A 17, 1556–1564 (2000).
    [CrossRef]
  18. J. A. Arnaud, H. Kogelnik, “Gaussian light beams with general astigmatism,” Appl. Opt. 8, 1687–1693 (1969).
    [CrossRef] [PubMed]
  19. W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. 62, 1195–1201 (1972).
    [CrossRef]
  20. Q. Lin, Y. Cai, “Decentered elliptical Gaussian beam,” Appl. Opt. 41, 4336–4340 (2002).
    [CrossRef] [PubMed]
  21. Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67–72 (1990).
  22. J. Alda, S. Wang, E. Bernabeu, “Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams,” Opt. Commun. 80, 350–352 (1991).
    [CrossRef]
  23. P. Baues, “Huygens’ principle in homogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-electronics 1, 37–44 (1969).
    [CrossRef]
  24. K. M. Abramski, A. D. Colley, H. J. Baker, D. R. Hall, “High-power two-dimensional waveguide CO2 laser arrays,” IEEE J. Quantum Electron. 32, 340–349 (1996).
    [CrossRef]
  25. H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, E. F. Yelden, “Propagation characteristics of coherent array beam from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
    [CrossRef]
  26. W. D. Bilida, J. D. Strohschein, H. J. J. Seguin, “High-power 24-channel radial array slab rf-excited carbon dioxide laser,” in Gas and Chemical Lasers and Applications II, R. C. Sze, E. A. Dorko, eds., Proc. SPIE2987, 13–21 (1997).
    [CrossRef]
  27. J. D. Strohschein, H. J. J. Seguin, C. E. Capjack, “Beam propagation constants for a radial laser array,” Appl. Opt. 37, 1045–1048 (1998).
    [CrossRef]
  28. B. Lü, H. Ma, “Beam propagation properties of radial lasers arrays,” J. Opt. Soc. Am. A 17, 2005–2009 (2000).
    [CrossRef]
  29. B. Lü, H. Ma, “Beam combination of a radial laser array: Hermite–Gaussian model,” Opt. Commun. 15, 395–403 (2000).
  30. F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “Intensity-based model analysis of partially coherent beams with Hermite–Gaussian modes,” Opt. Lett. 23, 989–991 (1998).
    [CrossRef]
  31. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  32. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
    [CrossRef]

2002 (3)

2001 (1)

S. Saghafi, C. J. R. Sheppard, J. A. Piper, “Characteris-ing elegant and standard Hermite–Gaussian beam modes,” Opt. Commun. 191, 173–179 (2001).
[CrossRef]

2000 (4)

1999 (2)

B. Lü, H. Ma, “Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185–194 (1999).
[CrossRef]

H. Laabs, C. Gao, H. Weber, “Twisting to three-dimensional Hermite–Gaussian beams,” J. Mod. Opt. 46, 709–719 (1999).

1998 (4)

1997 (1)

1996 (3)

K. M. Abramski, A. D. Colley, H. J. Baker, D. R. Hall, “High-power two-dimensional waveguide CO2 laser arrays,” IEEE J. Quantum Electron. 32, 340–349 (1996).
[CrossRef]

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, E. F. Yelden, “Propagation characteristics of coherent array beam from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[CrossRef]

1995 (1)

1994 (2)

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

V. N. Smirnov, G. A. Strokovskii, “On diffraction of optical Hermite–Gaussian beams from a diaphragm,” Opt. Spectrosc. 76, 912–919 (1994).

1991 (1)

J. Alda, S. Wang, E. Bernabeu, “Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams,” Opt. Commun. 80, 350–352 (1991).
[CrossRef]

1990 (1)

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67–72 (1990).

1986 (1)

1984 (1)

1980 (1)

1973 (2)

1972 (1)

1969 (2)

P. Baues, “Huygens’ principle in homogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-electronics 1, 37–44 (1969).
[CrossRef]

J. A. Arnaud, H. Kogelnik, “Gaussian light beams with general astigmatism,” Appl. Opt. 8, 1687–1693 (1969).
[CrossRef] [PubMed]

Abramski, K. M.

K. M. Abramski, A. D. Colley, H. J. Baker, D. R. Hall, “High-power two-dimensional waveguide CO2 laser arrays,” IEEE J. Quantum Electron. 32, 340–349 (1996).
[CrossRef]

Alda, J.

J. Alda, S. Wang, E. Bernabeu, “Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams,” Opt. Commun. 80, 350–352 (1991).
[CrossRef]

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67–72 (1990).

Al-Rashed, A. R.

Arnaud, J. A.

Bagini, V.

Baker, H. J.

K. M. Abramski, A. D. Colley, H. J. Baker, D. R. Hall, “High-power two-dimensional waveguide CO2 laser arrays,” IEEE J. Quantum Electron. 32, 340–349 (1996).
[CrossRef]

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, E. F. Yelden, “Propagation characteristics of coherent array beam from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

Baues, P.

P. Baues, “Huygens’ principle in homogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-electronics 1, 37–44 (1969).
[CrossRef]

Belanger, P. A.

Bernabeu, E.

J. Alda, S. Wang, E. Bernabeu, “Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams,” Opt. Commun. 80, 350–352 (1991).
[CrossRef]

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67–72 (1990).

Bilida, W. D.

W. D. Bilida, J. D. Strohschein, H. J. J. Seguin, “High-power 24-channel radial array slab rf-excited carbon dioxide laser,” in Gas and Chemical Lasers and Applications II, R. C. Sze, E. A. Dorko, eds., Proc. SPIE2987, 13–21 (1997).
[CrossRef]

Borghi, R.

Cai, Y.

Q. Lin, Y. Cai, “Decentered elliptical Gaussian beam,” Appl. Opt. 41, 4336–4340 (2002).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, “The elliptical Hermite–Gaussian beam and its propagation through paraxial systems,” Opt. Commun. 207, 139–147 (2002).
[CrossRef]

Capjack, C. E.

Carter, W. H.

Casperson, L. W.

Cogswell, C.

Colley, A. D.

K. M. Abramski, A. D. Colley, H. J. Baker, D. R. Hall, “High-power two-dimensional waveguide CO2 laser arrays,” IEEE J. Quantum Electron. 32, 340–349 (1996).
[CrossRef]

Cronin, P. J.

Gao, C.

H. Laabs, C. Gao, H. Weber, “Twisting to three-dimensional Hermite–Gaussian beams,” J. Mod. Opt. 46, 709–719 (1999).

Gori, F.

Guattari, G.

Hall, D. R.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, E. F. Yelden, “Propagation characteristics of coherent array beam from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

K. M. Abramski, A. D. Colley, H. J. Baker, D. R. Hall, “High-power two-dimensional waveguide CO2 laser arrays,” IEEE J. Quantum Electron. 32, 340–349 (1996).
[CrossRef]

Hornby, A. M.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, E. F. Yelden, “Propagation characteristics of coherent array beam from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

Joris, J. J. D.

Kogelnik, H.

Laabs, H.

H. Laabs, C. Gao, H. Weber, “Twisting to three-dimensional Hermite–Gaussian beams,” J. Mod. Opt. 46, 709–719 (1999).

Lin, Q.

Y. Cai, Q. Lin, “The elliptical Hermite–Gaussian beam and its propagation through paraxial systems,” Opt. Commun. 207, 139–147 (2002).
[CrossRef]

Q. Lin, Y. Cai, “Decentered elliptical Gaussian beam,” Appl. Opt. 41, 4336–4340 (2002).
[CrossRef] [PubMed]

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67–72 (1990).

Lü, B.

B. Lü, H. Ma, “Beam propagation properties of radial lasers arrays,” J. Opt. Soc. Am. A 17, 2005–2009 (2000).
[CrossRef]

B. Lü, H. Ma, “Beam combination of a radial laser array: Hermite–Gaussian model,” Opt. Commun. 15, 395–403 (2000).

B. Lü, H. Ma, “Coherent and incoherent off-axis Hermite–Gaussian beam combinations,” Appl. Opt. 39, 1279–1289 (2000).
[CrossRef]

B. Lü, H. Ma, “Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185–194 (1999).
[CrossRef]

Luis, M. L.

Ma, H.

B. Lü, H. Ma, “Coherent and incoherent off-axis Hermite–Gaussian beam combinations,” Appl. Opt. 39, 1279–1289 (2000).
[CrossRef]

B. Lü, H. Ma, “Beam combination of a radial laser array: Hermite–Gaussian model,” Opt. Commun. 15, 395–403 (2000).

B. Lü, H. Ma, “Beam propagation properties of radial lasers arrays,” J. Opt. Soc. Am. A 17, 2005–2009 (2000).
[CrossRef]

B. Lü, H. Ma, “Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185–194 (1999).
[CrossRef]

Morley, R. J.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, E. F. Yelden, “Propagation characteristics of coherent array beam from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

Omel, M. Y.

Pacileo, A. M.

Palma, C.

Piper, J. A.

S. Saghafi, C. J. R. Sheppard, J. A. Piper, “Characteris-ing elegant and standard Hermite–Gaussian beam modes,” Opt. Commun. 191, 173–179 (2001).
[CrossRef]

Saghafi, S.

S. Saghafi, C. J. R. Sheppard, J. A. Piper, “Characteris-ing elegant and standard Hermite–Gaussian beam modes,” Opt. Commun. 191, 173–179 (2001).
[CrossRef]

S. Saghafi, C. J. R. Sheppard, “Near field and far field of elegant Hermite–Gaussian and Laguerre–Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

Saleh, B. E. A.

Santarsiero, M.

Seguin, H. J. J.

J. D. Strohschein, H. J. J. Seguin, C. E. Capjack, “Beam propagation constants for a radial laser array,” Appl. Opt. 37, 1045–1048 (1998).
[CrossRef]

W. D. Bilida, J. D. Strohschein, H. J. J. Seguin, “High-power 24-channel radial array slab rf-excited carbon dioxide laser,” in Gas and Chemical Lasers and Applications II, R. C. Sze, E. A. Dorko, eds., Proc. SPIE2987, 13–21 (1997).
[CrossRef]

Sheppard, C. J. R.

S. Saghafi, C. J. R. Sheppard, J. A. Piper, “Characteris-ing elegant and standard Hermite–Gaussian beam modes,” Opt. Commun. 191, 173–179 (2001).
[CrossRef]

S. Saghafi, C. J. R. Sheppard, “Near field and far field of elegant Hermite–Gaussian and Laguerre–Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

Siegman, A. E.

Smirnov, V. N.

V. N. Smirnov, G. A. Strokovskii, “On diffraction of optical Hermite–Gaussian beams from a diaphragm,” Opt. Spectrosc. 76, 912–919 (1994).

Strohschein, J. D.

J. D. Strohschein, H. J. J. Seguin, C. E. Capjack, “Beam propagation constants for a radial laser array,” Appl. Opt. 37, 1045–1048 (1998).
[CrossRef]

W. D. Bilida, J. D. Strohschein, H. J. J. Seguin, “High-power 24-channel radial array slab rf-excited carbon dioxide laser,” in Gas and Chemical Lasers and Applications II, R. C. Sze, E. A. Dorko, eds., Proc. SPIE2987, 13–21 (1997).
[CrossRef]

Strokovskii, G. A.

V. N. Smirnov, G. A. Strokovskii, “On diffraction of optical Hermite–Gaussian beams from a diaphragm,” Opt. Spectrosc. 76, 912–919 (1994).

Taghizadeh, M. R.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, E. F. Yelden, “Propagation characteristics of coherent array beam from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

Török, P.

Tovar, A. A.

Varga, P.

Wang, S.

J. Alda, S. Wang, E. Bernabeu, “Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams,” Opt. Commun. 80, 350–352 (1991).
[CrossRef]

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67–72 (1990).

Weber, H.

H. Laabs, C. Gao, H. Weber, “Twisting to three-dimensional Hermite–Gaussian beams,” J. Mod. Opt. 46, 709–719 (1999).

Yelden, E. F.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, E. F. Yelden, “Propagation characteristics of coherent array beam from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

Zauderer, E.

Appl. Opt. (8)

IEEE J. Quantum Electron. (2)

K. M. Abramski, A. D. Colley, H. J. Baker, D. R. Hall, “High-power two-dimensional waveguide CO2 laser arrays,” IEEE J. Quantum Electron. 32, 340–349 (1996).
[CrossRef]

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, E. F. Yelden, “Propagation characteristics of coherent array beam from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

J. Mod. Opt. (2)

S. Saghafi, C. J. R. Sheppard, “Near field and far field of elegant Hermite–Gaussian and Laguerre–Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

H. Laabs, C. Gao, H. Weber, “Twisting to three-dimensional Hermite–Gaussian beams,” J. Mod. Opt. 46, 709–719 (1999).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (7)

Opt. Commun. (6)

B. Lü, H. Ma, “Beam combination of a radial laser array: Hermite–Gaussian model,” Opt. Commun. 15, 395–403 (2000).

J. Alda, S. Wang, E. Bernabeu, “Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams,” Opt. Commun. 80, 350–352 (1991).
[CrossRef]

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

Y. Cai, Q. Lin, “The elliptical Hermite–Gaussian beam and its propagation through paraxial systems,” Opt. Commun. 207, 139–147 (2002).
[CrossRef]

S. Saghafi, C. J. R. Sheppard, J. A. Piper, “Characteris-ing elegant and standard Hermite–Gaussian beam modes,” Opt. Commun. 191, 173–179 (2001).
[CrossRef]

B. Lü, H. Ma, “Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185–194 (1999).
[CrossRef]

Opt. Lett. (1)

Opt. Spectrosc. (1)

V. N. Smirnov, G. A. Strokovskii, “On diffraction of optical Hermite–Gaussian beams from a diaphragm,” Opt. Spectrosc. 76, 912–919 (1994).

Optik (Stuttgart) (1)

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Stuttgart) 85, 67–72 (1990).

Opto-electronics (1)

P. Baues, “Huygens’ principle in homogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-electronics 1, 37–44 (1969).
[CrossRef]

Other (1)

W. D. Bilida, J. D. Strohschein, H. J. J. Seguin, “High-power 24-channel radial array slab rf-excited carbon dioxide laser,” in Gas and Chemical Lasers and Applications II, R. C. Sze, E. A. Dorko, eds., Proc. SPIE2987, 13–21 (1997).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Three-dimensional relative intensity distribution of the DEHGB with different decentered parameter r0 on the plane of z=0. (a) r0T=(0 0), (b) r0T=(2+0.1i 2+0.1i), (c) r0T=(2+0.5i 2+0.5i), (d) r0T=(2+2i 2+2i).

Fig. 2
Fig. 2

Three-dimensional relative intensity distribution of the DEHGB with m=4 on planes of different propagation distances. (a) z=0, (b) z=1.6zx, (c) z=3zx, (d) z=6zx.

Fig. 3
Fig. 3

Evolution of the ratio of the long axis and short axis of the beam spot in free-space propagation.

Fig. 4
Fig. 4

Three-dimensional relative intensity distribution of the beam array for the phase-locked case on planes of different propagation distances. (a) z=0, (b) z=zx, (c) z=2zx, (d) z=20zx.

Fig. 5
Fig. 5

Three-dimensional relative intensity distribution of the beam array for the non-phase-locked case on planes of different propagation distances. (a) z=0, (b) z=zx, (c) z=2zx, (d) z=20zx.

Fig. 6
Fig. 6

Three-dimensional intensity distribution of the decentered elliptical flattened Gaussian beam with N=5 on the plane of several propagation distances. (a) z=0, (b) z=10 zx.

Equations (45)

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

E(r1)=E0exp-ik2r1TQ-1r1,
Q-1=qxx-1qxy-1qxy-1qyy-1.
Ep(r1)=exp-ik2r1TQe-1r1Hp[(ikr1TQh-1r1)1/2],p=0, 1, 2, 3,
Ep(r1)=exp-ik2 (r1-r0)TQe-1(r1-r0)×Hp{[ik(r1-r0)TQh-1(r1-r0)]1/2},
p=0, 1, 2, 3,,
Qe-1=Qh-1=-iλ/πw0x2-iλ/πw0xy2-iλ/πw0xy2-iλ/πw0y2=-0.2011i-0.0503i-0.0503i-0.0895i(m)-1.
E2(r2)=-in1λ[det(B)]1/2exp(-ikl0)×E1(r1)exp(-ikl1)dr1,
l1=12r1r2Tn1B-1A-n1B-1n2(C-DB-1A)n2DB-1r1r2,
r2r2=ABCDr1r1.
E2p(r2)=[det(A+BQe1-1)]-1/2[1-2/det(B-1AQh1+Qe1-1Qh1)]p/2exp(-ikl0)×exp-ik2r2TQe2-1r2×exp-ik2r0T(Qe1+A-1B)-1r0×exp[ikr0T(AQe1+B)-1r2]×Hp{[1-2/det(B-1AQh1+Qe1-1Qh1)]-1/2×[ik(r2-r0)TQh2-1(r2-r0)]1/2},
Qe2-1=(C+DQe1-1)(A+BQe1-1)-1,
Qh2-1=(A+BQe1-1)-1T(AQh1+BQe1-1Qh1)-1.
-exp-(x-b)22aHp(x)dx
=2πa(1-2a)p/2Hpb1-2a.
(B-1A)T=B-1A,(-B-1)T=(C-DB-1A),
(DB-1)T=DB-1.
E2p(r2)=[det(A+BQe1-1)]-1/2[1-2/det(B-1AQh1+Qe1-1Qh1)]p/2exp(-ikl0)×exp-ik2r2TQe2-1r2×Hp{[1-2/det(B-1AQh1+Qe1-1Qh1)]-1/2×(ikr2TQh2-1r2)1/2}.
E2p(r2)=[det(A+BQe1-1)]-1/2×exp(-ikl0)exp-ik2r2TQe2-1r2×exp-ik2r0T(Qe1+A-1B)-1r0×exp[ikr0T(AQe1+B)-1r2].
Q1e-1=Q1h-1=q1-1000,
Ep(x2)=(a+b/q1)-1/2aq1-baq1+bp/2exp(-ikl0)×exp-ik2q2 (x2-axr)2-ikcxrx2+ik2 acxr2Hp2(x2-axr)w0(a2-b2/q12)1/2,
x2x2=abcdx1x1.
A=1001,B=z00z,
C=0000,D=1001.
rxy=Im[Qexx-1]Im[Qeyy-1].
E(r1, 0)=n=0N-1Enp(r1n, 0),
Enp(r1n, 0)=exp-ik2 (r1n-r0)TQe-1(r1n-r0)×Hp{[ik(r1n-r0)TQh-1(r1n-r0)]1/2},
p=0,1,2,3,,
r1n=x1cos α+y1sin αy1cos α-x1sin α,
α=nα0(n=0, 1, 2,, N-1),
α0=2π/N.
Enp(r2n)=[det(A+BQe1-1)]-1/2[1-2/det(B-1AQh1+Qe1-1Qh1)]p/2exp(-ikl0)×exp-ik2r2nTQe2-1r2n×exp-ik2r0T(Qe1+A-1B)-1r0×exp[ikr0T(AQe1+B)-1r2n]×Hp{[1-2/det(B-1AQh1+Qe1-1Qh1)]-1/2×[ik(r2n-r0)TQh2-1(r2n-r0)]1/2},
r2n=x2cos α+y2sin αy2cos α-x2sin α,
α=nα0(n=0, 1, 2,, N-1),
α0=2π/N.
I=E*(r2)E(r2).
I=n=0N-1In=n=0N-1Enp*(r2n)Enp(r2n).
E(r1)=exp-i k2r1TQ1,N-1r1n=0N1n!i k2r1TQ1,N-1r1n,
E(r1)=exp-i k2 (r1-r0)TQ1,N-1(r1-r0)×n=0N1n!i k2 (r1-r0)TQ1,N-1(r1-r0)n.
[(r1-r0)TQ1,N-1(r-r0)]n
=(2n)!23nm=0n1(n-m)!(2m)!×H2m[2|Q1,N-1/2(r1-r0)|],
E(r1, 0)=exp-i k2 (r1-r0)TQ1,N-1(r1-r0)×n=0N1(2n)!m=nN123m(2m)!(m-n)!m!×H2m[|(ikQ1,N-1)1/2(r1-r0)|].
E(r2)=A0[det(A+BQ1,N-1)]-1/2exp-ik2r2TQ2,N-1r2×exp(-ikl0)exp-ik2r0T(Q1,N+A-1B)-1r0×exp[ikr0T(AQ1,N+B)-1r2]n=0N1(2n)!×m=nN123m(2m)!(m-n)!m! [1-2/det(B-1AQ1,N+I)]m×H2m{[1-2/det(B-1AQ1,N+I)]-1/2×[ik(r2-r0)T(A+BQ1,N-1)-1T(AQ1,N+B)-1(r2-r0)]1/2},
Q2,N-1=(C+DQ1,N-1)(A+BQ1,N-1)-1.
Q1,N-1=-1.2085i-0.1342i-0.1342i-0.0895i(m)-1,
λ=632.8 nm,N=5,andr0T=(2 2).

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