Abstract

A technique for the modal decomposition of laser beams allows an analysis of laser oscillators perturbed by rod spherical aberration. Beam quality can be degraded or improved, depending on whether the beam scale increases or decreases with lens power. Efficient power extraction is problematic, however.

© 2002 Optical Society of America

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References

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  1. A. Stein, “Thermooptically perturbed resonators,” IEEE J. Quantum Electron. QE-10, 427–434 (1974).
    [CrossRef]
  2. N. Hodgson, H. Weber, “Influence of spherical aberration of the active medium on the performance of Nd:YAG lasers,” IEEE J. Quantum Electron. 29, 2497–2507 (1993).
    [CrossRef]
  3. D. C. Brown, “Nonlinear thermal distortion in YAG rod amplifiers,” IEEE J. Quantum Electron. 34, 2383–2392 (1998).
    [CrossRef]
  4. R. Martinez-Herrero, P. M. Mejias, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally-induced spherical aberration in laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
    [CrossRef]
  5. D. C. Brown, “Heat, fluorescence, and stimulated-emission power densities and fractions in Nd:YAG,” IEEE J. Quantum Electron. 34, 560–571 (1998).
    [CrossRef]
  6. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Amer. A 3, 1227–1246 (1986).
    [CrossRef]
  7. C. J. Kennedy, “A model for variation of laser power with M2,” Appl. Opt. 41, 4341–4346 (2002).
    [CrossRef] [PubMed]
  8. U. Ganiel, A. Hardy, “Eigenmodes of optical resonators with mirrors having Gaussian reflectivity profiles,” Appl. Opt. 15, 2145–2149 (1976).
    [CrossRef] [PubMed]
  9. S. Jackel, I. Moshe, A. Kaufman, R. Lavi, R. Lallouz, “High-energy Nd:Cr:GSGG lasers based on phase and polarization conjugated multiple-pass amplifiers,” Opt. Eng. 36, 2031–2036 (1997).
    [CrossRef]

2002 (1)

1998 (2)

D. C. Brown, “Nonlinear thermal distortion in YAG rod amplifiers,” IEEE J. Quantum Electron. 34, 2383–2392 (1998).
[CrossRef]

D. C. Brown, “Heat, fluorescence, and stimulated-emission power densities and fractions in Nd:YAG,” IEEE J. Quantum Electron. 34, 560–571 (1998).
[CrossRef]

1997 (1)

S. Jackel, I. Moshe, A. Kaufman, R. Lavi, R. Lallouz, “High-energy Nd:Cr:GSGG lasers based on phase and polarization conjugated multiple-pass amplifiers,” Opt. Eng. 36, 2031–2036 (1997).
[CrossRef]

1995 (1)

R. Martinez-Herrero, P. M. Mejias, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally-induced spherical aberration in laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
[CrossRef]

1993 (1)

N. Hodgson, H. Weber, “Influence of spherical aberration of the active medium on the performance of Nd:YAG lasers,” IEEE J. Quantum Electron. 29, 2497–2507 (1993).
[CrossRef]

1986 (1)

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Amer. A 3, 1227–1246 (1986).
[CrossRef]

1976 (1)

1974 (1)

A. Stein, “Thermooptically perturbed resonators,” IEEE J. Quantum Electron. QE-10, 427–434 (1974).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Amer. A 3, 1227–1246 (1986).
[CrossRef]

Brown, D. C.

D. C. Brown, “Nonlinear thermal distortion in YAG rod amplifiers,” IEEE J. Quantum Electron. 34, 2383–2392 (1998).
[CrossRef]

D. C. Brown, “Heat, fluorescence, and stimulated-emission power densities and fractions in Nd:YAG,” IEEE J. Quantum Electron. 34, 560–571 (1998).
[CrossRef]

Ganiel, U.

Hardy, A.

Hodgson, N.

R. Martinez-Herrero, P. M. Mejias, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally-induced spherical aberration in laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
[CrossRef]

N. Hodgson, H. Weber, “Influence of spherical aberration of the active medium on the performance of Nd:YAG lasers,” IEEE J. Quantum Electron. 29, 2497–2507 (1993).
[CrossRef]

Jackel, S.

S. Jackel, I. Moshe, A. Kaufman, R. Lavi, R. Lallouz, “High-energy Nd:Cr:GSGG lasers based on phase and polarization conjugated multiple-pass amplifiers,” Opt. Eng. 36, 2031–2036 (1997).
[CrossRef]

Kaufman, A.

S. Jackel, I. Moshe, A. Kaufman, R. Lavi, R. Lallouz, “High-energy Nd:Cr:GSGG lasers based on phase and polarization conjugated multiple-pass amplifiers,” Opt. Eng. 36, 2031–2036 (1997).
[CrossRef]

Kennedy, C. J.

Lallouz, R.

S. Jackel, I. Moshe, A. Kaufman, R. Lavi, R. Lallouz, “High-energy Nd:Cr:GSGG lasers based on phase and polarization conjugated multiple-pass amplifiers,” Opt. Eng. 36, 2031–2036 (1997).
[CrossRef]

Lavi, R.

S. Jackel, I. Moshe, A. Kaufman, R. Lavi, R. Lallouz, “High-energy Nd:Cr:GSGG lasers based on phase and polarization conjugated multiple-pass amplifiers,” Opt. Eng. 36, 2031–2036 (1997).
[CrossRef]

Martinez-Herrero, R.

R. Martinez-Herrero, P. M. Mejias, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally-induced spherical aberration in laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
[CrossRef]

Mejias, P. M.

R. Martinez-Herrero, P. M. Mejias, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally-induced spherical aberration in laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
[CrossRef]

Moshe, I.

S. Jackel, I. Moshe, A. Kaufman, R. Lavi, R. Lallouz, “High-energy Nd:Cr:GSGG lasers based on phase and polarization conjugated multiple-pass amplifiers,” Opt. Eng. 36, 2031–2036 (1997).
[CrossRef]

Stein, A.

A. Stein, “Thermooptically perturbed resonators,” IEEE J. Quantum Electron. QE-10, 427–434 (1974).
[CrossRef]

Weber, H.

R. Martinez-Herrero, P. M. Mejias, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally-induced spherical aberration in laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
[CrossRef]

N. Hodgson, H. Weber, “Influence of spherical aberration of the active medium on the performance of Nd:YAG lasers,” IEEE J. Quantum Electron. 29, 2497–2507 (1993).
[CrossRef]

Appl. Opt. (2)

IEEE J. Quantum Electron. (5)

A. Stein, “Thermooptically perturbed resonators,” IEEE J. Quantum Electron. QE-10, 427–434 (1974).
[CrossRef]

N. Hodgson, H. Weber, “Influence of spherical aberration of the active medium on the performance of Nd:YAG lasers,” IEEE J. Quantum Electron. 29, 2497–2507 (1993).
[CrossRef]

D. C. Brown, “Nonlinear thermal distortion in YAG rod amplifiers,” IEEE J. Quantum Electron. 34, 2383–2392 (1998).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally-induced spherical aberration in laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
[CrossRef]

D. C. Brown, “Heat, fluorescence, and stimulated-emission power densities and fractions in Nd:YAG,” IEEE J. Quantum Electron. 34, 560–571 (1998).
[CrossRef]

J. Opt. Soc. Amer. A (1)

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Amer. A 3, 1227–1246 (1986).
[CrossRef]

Opt. Eng. (1)

S. Jackel, I. Moshe, A. Kaufman, R. Lavi, R. Lallouz, “High-energy Nd:Cr:GSGG lasers based on phase and polarization conjugated multiple-pass amplifiers,” Opt. Eng. 36, 2031–2036 (1997).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Input m = 5 helicoid after five round trips of resonator with a quartic phase screen. Curve fit to m = 5 helicoid shown above shows mode is stable to within 1 part in 1000. Curve fit function, y = 1.000669 [2(x/0.8821024)2]5 exp[-2(x/0.8821024)2]; R = 0.9999959. Input scale size was 0.8844 mm.

Fig. 2
Fig. 2

Input m = 10 helicoid after five round trips of resonator with a quartic phase screen. Curve fit to m = 10 helicoid shown above shows mode is stable to within 1 part in 1000. Curve fit function, y = 0.9903306[2(x/0.8836908)2]5 exp[-2(x/0.8836908)2]; R = 0.9995867. Input scale size was 0.8844 mm.

Fig. 3
Fig. 3

Measured wave front distortion in a 450-W pump chamber with a 5-mm diameter rod and the spherical component removed. Vertical scale is in microns. Zernike radius = 2.0 mm. Zernike coefficients in microns: Z 20 = -0.08, Z 21= -9.433 (not plotted), Z 22 = 0.419, Z 30 = 0.088, Z 31 = -0.187, Z 32 = -0.06, Z 33 = 0.012, Z 40 = 0.054, Z 41 = -0.006, Z 42 = 0.449 (γ = 0.195), Z 43 = 0.116, and Z 44 = 0.067.

Fig. 4
Fig. 4

Optical resonator model used in simulations. The analytic plane, unless otherwise specified, lies immediately to the right of the laser rod. Mode scale sizes and wave front curvatures are determined at this plane.

Fig. 5
Fig. 5

Solutions to mode scale size satisfying both Eq. (6) and Eq. (7). U-shaped curve is a graph of Eq. (6) having a width of the inverse of the cavity length. It is displaced to the right by the value of the end mirror curvature C. Equation (7), represented by a family of curves with helicoid mode index as the parameter, radiating from (Δ, 0), modes of index 10, 30, 80, and 160, are illustrated. The dashed rectangle of width Δγ shows the range of possible values of D that fall within the radius of the rod with aberration coefficient γ. Where the radiating curves cross the U-shaped curve are solutions for effective mode lens power and scale size. From zero to two solutions are possible numerically, but these are often eliminated by falling outside the range Δγ. Clearly, a critical value of m exists, above which there is no solution. Dual solutions are possible with markedly differing scale size, but the same index.

Fig. 6
Fig. 6

Numerical simulation mode profile results of cases of operation where U-graph slope is positive. Conditions are as in Table 1. Back mirror curvature, C = 0. Axial lens power Δ is varied between profiles: a, Δ = 0.039; b, Δ = 0.03999; c, Δ = 0.042; d, Δ = 0.044; e, Δ = 0.046.

Fig. 7
Fig. 7

Experimental demonstration of the virtual obscuration effect in an aberrated oscillator. Resonator as in Fig. 5. L = 25 cm, C = 0. Image of rod is projected onto CCD camera through a flat back mirror. Drive current to 80 bar pump chamber is (a) 21.5, (b) 22, and (c) 22.5 amps. Output power is (a) 277, (b) 187, and (c) 101 W.

Fig. 8
Fig. 8

Numerical simulation results showing mode profiles of resonators operated with negative U-graph slopes. Conditions are as in Table 1. Axial lens power, Δ = 0.04. Back mirror curvature is varied between profiles: a, C = 0.03; b, C = 0.035; c, C = 0.036; d, C = 0.037; e, C = 0.038; f, C = 0.039; and g, C = 0.0399.

Fig. 9
Fig. 9

Overall graph of all simulations showing power versus mode quality.

Fig. 10
Fig. 10

Comparison of results of mode quality versus aberration coefficient for two resonator designs with intensity moment theory and helicoid mode theory. Parameters chosen for the illustrated two cases are as in Table 1, with back mirror curvature chosen to make dω/dD > 0 (C = 0) and dω/dD < 0 (C = 0.037) when the axial rod focal power Δ was = 0.04.

Tables (1)

Tables Icon

Table 1 Simulation Parameters

Equations (17)

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Imr=2πω2m!2 r2ω2m exp-2 r2ω2.
Dmr=Δ1-γrmb2,
γ=11+Z213Z42brz2,
a=Δγb2.
12L2C-Dm1-4LDm+4LC.
ωm=λLπ11-Dm-CLDm-CL1/21/2
ωm=Δ-Dm2b2mγΔ1/2,
Dm=182Δ+3L+6C-4Δ-C2+9L-4Δ-C1L1/2.
ωm=λLπ14-Dm-CLDm-CL1/21/2,
Cm=C-Dm,
Ir=PmImr.
gnPn=02π 2π 0bInr×g0 rdr1+1Ismn PmImr+PnInr=αn,
ϕx=exp-ikax4,
a=Δγb2
M2=2πλ x2-4K2a2x22-2Kax21L-2Δ+ΔL-Δ21/2,
ϕx=exp-ikax4+y4.
ϕx=exp-ikax4+2x2y2+y4.

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