Abstract

A detailed study of phase-locked and non-phase-locked radial laser arrays is presented. The closed-form propagation expressions for the beamlets and the resulting beam are given, which enable us to study beam propagation properties of radial laser arrays for both phase-locked and non-phase-locked cases. Numerical calculation examples are given to illustrate the application of our analytical results and the differences between phase-locked and non-phase-locked radial arrays.

© 2000 Optical Society of America

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References

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  1. H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, E. F. Yelden, “Propagation characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
    [CrossRef]
  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]
  3. E. F. Yelden, H. J. J. Seguin, C. E. Capjack, S. K. Nikumb, “Multichannel slab discharge for CO2 laser excitation,” Appl. Phys. Lett. 58, 693–695 (1991).
    [CrossRef]
  4. 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, H. R. C. Sze, E. A. Dorko, eds., Proc. SPIE2987, 13–21 (1997).
    [CrossRef]
  5. A. E. Siegman, “How to (maybe) measure the laser beam quality,” in Diode Pumped Solid State Lasers: Applications and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 184–199.
  6. 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]
  7. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [CrossRef]
  8. C. Palma, “Decentered Gaussian beams, ray bundles, and Bessel–Gaussian beams,” Appl. Opt. 36, 1116–1120 (1997).
    [CrossRef] [PubMed]
  9. B. Lü, H. Ma, “Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185–194 (1999).
    [CrossRef]

1999

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

1998

1997

1996

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, E. F. Yelden, “Propagation characteristics of coherent array beams 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]

1991

E. F. Yelden, H. J. J. Seguin, C. E. Capjack, S. K. Nikumb, “Multichannel slab discharge for CO2 laser excitation,” Appl. Phys. Lett. 58, 693–695 (1991).
[CrossRef]

1970

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]

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 beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

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, H. R. C. Sze, E. A. Dorko, eds., Proc. SPIE2987, 13–21 (1997).
[CrossRef]

Capjack, C. E.

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]

E. F. Yelden, H. J. J. Seguin, C. E. Capjack, S. K. Nikumb, “Multichannel slab discharge for CO2 laser excitation,” Appl. Phys. Lett. 58, 693–695 (1991).
[CrossRef]

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]

Collins, S. A.

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 beams 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 beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

Lü, B.

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, “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 beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

Nikumb, S. K.

E. F. Yelden, H. J. J. Seguin, C. E. Capjack, S. K. Nikumb, “Multichannel slab discharge for CO2 laser excitation,” Appl. Phys. Lett. 58, 693–695 (1991).
[CrossRef]

Palma, C.

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]

E. F. Yelden, H. J. J. Seguin, C. E. Capjack, S. K. Nikumb, “Multichannel slab discharge for CO2 laser excitation,” Appl. Phys. Lett. 58, 693–695 (1991).
[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, H. R. C. Sze, E. A. Dorko, eds., Proc. SPIE2987, 13–21 (1997).
[CrossRef]

Siegman, A. E.

A. E. Siegman, “How to (maybe) measure the laser beam quality,” in Diode Pumped Solid State Lasers: Applications and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 184–199.

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, H. 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 beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

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 beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

E. F. Yelden, H. J. J. Seguin, C. E. Capjack, S. K. Nikumb, “Multichannel slab discharge for CO2 laser excitation,” Appl. Phys. Lett. 58, 693–695 (1991).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

E. F. Yelden, H. J. J. Seguin, C. E. Capjack, S. K. Nikumb, “Multichannel slab discharge for CO2 laser excitation,” Appl. Phys. Lett. 58, 693–695 (1991).
[CrossRef]

IEEE J. Quantum Electron.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, E. F. Yelden, “Propagation characteristics of coherent array beams 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]

J. Opt. Soc. Am.

Opt. Commun.

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

Other

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, H. R. C. Sze, E. A. Dorko, eds., Proc. SPIE2987, 13–21 (1997).
[CrossRef]

A. E. Siegman, “How to (maybe) measure the laser beam quality,” in Diode Pumped Solid State Lasers: Applications and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 184–199.

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Figures (6)

Fig. 1
Fig. 1

Schematic illustration of the radial laser array.

Fig. 2
Fig. 2

Irradiance distributions (arbitrary units) (upper figure) and isophote diagrams (lower figure) of the resulting beam focused by a lens for the non-phase-locked case. The calculation parameters are N=12, w0x/w0y=2, r/w0x=1.0, Nf=5. (a) Δz=-1, (b) Δz=-0.1, (c) Δz=0.

Fig. 3
Fig. 3

Irradiance distributions (arbitrary units) (upper figure) and isophote diagrams (lower figure) of the resulting beam focused by a lens for the phase-locked case. The calculation parameters are the same as those in Fig. 2. (a) Δz=-1, (b) Δz=-0.1, (c) Δz=0.

Fig. 4
Fig. 4

Irradiance distributions (arbitrary units) of the resulting beam focused by a lens for the non-phase-locked case. The calculation parameters are N=12, w0x/w0y=2, r/w0x=4.0, Nf=5. (a) Δz=-1, (b) Δz=-0.1, (c) Δz=0.

Fig. 5
Fig. 5

Irradiance distributions (arbitrary units) of the resulting beam focused by a lens for the phase-locked case. The calculation parameters are the same as those in Fig. 4. (a) Δz=-1, (b) Δz=-0.1, (c) Δz=0.

Fig. 6
Fig. 6

M2 factor of the resulting beam as a function of inverse radial fill factor r/w0x. Solid curve, phase locked case; dashed curve, non-phase-locked case. The calculation parameters are N=12, w0x/w0y=2.

Equations (32)

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E0(x, y, 0)=exp-(x-r)2w0x2-y2w0y2.
E0(x, y, z)=iλBE0(x0, y0, 0)×exp-ik2B [A(x02+y02)-2(x0x+y0y)+D(x2+y2)]dx0dy0,
k=2π/λ.
E0(x, y, z)=1[(A+B/q0x)(A+B/q0y)]1/2×exp-ik2qx (x-x1d)2-ik1xx+iφx×exp-ik2qy y2,
qj=Aq0j+BCq0j+D,
q0j=iπw0j2λ,
x1d=Ar,
1x=Cr,
φx=kAC2 r2.
xx cos α+y sin α,
y-x sin α+y cos α,
En(x, y, 0)=exp-(x cos α+y sin α-r)2w0x2-(y cos α-x sin α)2w0y2,
α=nα0,(n=0, 1, 2, N-1),α0=2π/N.
En(x, y, z)=1[(A+B/q0x)(A+B/q0y)]1/2×exp-ik2qx (x cos α+y sin α-x1d)2-ik1x(x cos α+y sin α)+iφx×exp-ik2qy (y cos α-x sin α)2.
E(x, y, z)=n=0N-1En(x, y, z),
I(x, y, z)=E*(x, y, z)E(x, y, z),
I(x, y, z)=n=0N-1In(x, y, z),
In(x, y, z)=En*(x, y, z)En(x, y, z).
σr2=14 (w0x2+w0y2+4r2)
σp2=116π21w0x2+1w0y2;
Mr2=2πσr2σp2=w0x2w0y1+w0yw0x2×1+w0yw0x2+4rw0x21/2.
ABCD=-Δzf(1+Δz)-1f1,
Δz=z-ff.
E0(x, y, Δz)
=(w0y/w0x)μδχexp-μβδ (x+rΔz)2+2μxr+μr2Δz-μγχ y2,
x=xw0x,y=yw0x,r=rw0x.
Nf=w0x2/λf,
μ=iπNf,
β=1-iπNf,
γ=1-iπNf (w0y/w0x)2,
δ=1+βΔz,
χ=1+γΔz.

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