Abstract

A new kind of laser beam, called a decentered elliptical Gaussian beam (DEGB), is defined by a tensor method. The propagation formula for a DEGB passing through an axially nonsymmetrical paraxial optical system is derived through vector integration. The derived formula can be reduced to the formula for a fundamental elliptical Gaussian beam and a decentered Gaussian beam under certain conditions. As an example application of the derived formula, the propagation characteristics of a DEGB in free space are calculated and discussed. As another example we study the properties of a generalized laser beam array constructed by use of a DEGB as the fundamental mode.

© 2002 Optical Society of America

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References

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  1. L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12, 2423–2441 (1973).
    [CrossRef]
  2. A. R. Al-Rashed, B. E. A. Saleh, “Decentered Gaussian beams,” Appl. Opt. 34, 6819–6825 (1995).
    [CrossRef] [PubMed]
  3. C. Palma, “Decentered Gaussian beams, ray bundles, and Bessel-Gaussian beams,” Appl. Opt. 36, 1116–1120 (1997).
    [CrossRef] [PubMed]
  4. B. Lü, H. Ma, “Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185–194 (1999).
    [CrossRef]
  5. B. Lü, H. Ma, “Coherent and incoherent off-axis Hermite-Gaussian beam combinations,” Appl. Opt. 39, 1279–1289 (2000).
    [CrossRef]
  6. B. Lü, H. Ma, “The beam quality in coherent and incoherent combinations of one-dimensional off-axis Hermite-Gaussian beams,” Optik (Stuttgart) 111, 269–272 (2000).
  7. 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]
  8. 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).
  9. 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]
  10. 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]
  11. 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]
  12. 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]
  13. J. D. Strohschein, H. J. J. Seguin, C. E. Capjack, “Beam propagation constant for a radial laser array,” Appl. Opt. 37, 1045–1048 (1998).
    [CrossRef]
  14. B. Lü, H. Ma, “Beam propagation properties of radial lasers arrays, ” J. Opt. Soc. Am. A 17, 2005–2009 (2000).
    [CrossRef]

2000 (4)

1999 (1)

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

1998 (1)

1997 (1)

1996 (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]

1995 (1)

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).

1973 (1)

L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12, 2423–2441 (1973).
[CrossRef]

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.

Baker, H. 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]

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]

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]

Capjack, C. E.

Casperson, L. W.

L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12, 2423–2441 (1973).
[CrossRef]

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.

Hall, D. R.

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]

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]

Lin, Q.

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, “The beam quality in coherent and incoherent combinations of one-dimensional off-axis Hermite-Gaussian beams,” Optik (Stuttgart) 111, 269–272 (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 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]

Ma, H.

B. Lü, H. Ma, “The beam quality in coherent and incoherent combinations of one-dimensional off-axis Hermite-Gaussian beams,” Optik (Stuttgart) 111, 269–272 (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 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]

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]

Palma, C.

Saleh, B. E. A.

Seguin, H. J. J.

J. D. Strohschein, H. J. J. Seguin, C. E. Capjack, “Beam propagation constant 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]

Strohschein, J. D.

J. D. Strohschein, H. J. J. Seguin, C. E. Capjack, “Beam propagation constant 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]

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.

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).

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]

Appl. Opt. (5)

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. Opt. Soc. Am. A (2)

Opt. Commun. (2)

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

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]

Optik (Stuttgart) (2)

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).

B. Lü, H. Ma, “The beam quality in coherent and incoherent combinations of one-dimensional off-axis Hermite-Gaussian beams,” Optik (Stuttgart) 111, 269–272 (2000).

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 (3)

Fig. 1
Fig. 1

Three-dimensional relative intensity distributions of DEGBs on planes of various propagation distances: (a) z = 0, (b) z = 1.6z x , (c) z = 4z x , (d) z = 8z x .

Fig. 2
Fig. 2

Three-dimensional relative intensity distributions and corresponding contour graphs of the phase-locked beam array on planes of several propagation distances: (a) z = 0, (b) z = 2z x , (c) z = 10z x , (d) z = 20z x .

Fig. 3
Fig. 3

Three-dimensional relative intensity distributions and corresponding contour graphs of the non-phase-locked beam array on planes of several propagation distances: (a) z = 0, (b) z = 2z x , (c) z = 10z x , (d) z = 20z x .

Equations (24)

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Er1=exp-ik2r1TQ1-1r1,
Q1- 1=q1xx- 1q1xy- 1q1xy- 1q1yy- 1.
Er1=exp-ik2r1- r0TQ1-1r1- r0,
Q1- 1=q1- 1000,
Ex1, 0= exp-ik2q1x1-xd- ix02.
E2r2=-in1λdetB1/2exp-ikl0× E1r1exp-ikl1dr1,
l1=12r1r2Tn1B-1A-n1B-1n2C-DB- 1An2DB- 1 r1r2,
r2r2=ABCD r1r1.
Er2=detA+BQ1-11/2 exp-ikl0×exp-ik2r2TQ2- 1r2exp-ik2r0TQ1+A- 1B- 1r0expikr0TAQ1+ B-1r2,
Q2- 1=C+ DQ1-1A+BQ1-1- 1.
B- 1AT=B- 1A, -B-1T=C-DB- 1A, DB- 1T=DB- 1.
E2r2=detA+BQ1-11/2× exp-ikl0exp-ik2r2TQ2- 1r2.
Ex2= a+b/q1-1/ 2 exp-ik2q2x2- ax02-ikcx0x2+ ik2 acx02exp-ikl0,
q2=aq1+bcq1+d;
x2x2=abcd x1x1.
A=1001, B=z00z, C=0000, D=1001.
Q1- 1=-0.201i-0.0503i-0.0503i-0.0895im-1.
Er1, 0=n=0N- 1 Enr1n, 0,
Enr1n, 0=exp-ik2r1n- r0TQ1-1r1n- r0,
r1n=cos αsin α-sin αcos αr1=x1 cos α+y1 sin αy1 cos α-x1 sin α,
Enr2n=detA+BQ1-1-1/ 2×exp-ikl0exp-ik2r2nTQ2-1r2n× exp-ik2r0TQ1+A- 1B- 1r0×expikr0TAQ1+B- 1r2n,
r2n=x2 cos α+y2 sin αy2 cos α-x2 sin α.
I=E*r2Er2.
I=n= 0N-1 In=n=0N-1 En*r2nEnr2n.

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