Abstract

To apply annular output beams emitted from an unstable resonator to a multiple-pass cell (MPC) for Raman conversion, we studied the mode-matching condition of non-Gaussian beams to a MPC using beam propagation analysis based on Laguerre–Gaussian functions. During transits of the MPC, the radial profile of an annular beam changes between annular and Airy patterns. Although such behavior indicates that it is impossible to achieve complete mode matching of an annular beam, we found a quasi-mode-matching condition under which the variation of beam size was minimized. The above theoretical analysis was verified experimentally using a CO2 laser beam prepared for a para-hydrogen Raman laser.

© 1997 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. P. Rabinowitz, A. Stein, R. Brickman, A. Kaldor, “Stimulated rotational Raman scattering from para-H2 pumped by a CO2 TEA laser,” Opt. Lett. 3, 147–148 (1978).
    [CrossRef] [PubMed]
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    [CrossRef]
  6. A. Owyoung, C. W. Patterson, R. S. Mcdowell, “Cw stimulated Raman gain spectroscopy of the ν1 fundamental of methane,” Chem. Phys. Lett. 59, 156–162 (1978).
    [CrossRef]
  7. D. R. Herriott, H. J. Schulte, “Folded optical delay lines,” Appl. Opt. 4, 883–889 (1965).
    [CrossRef]
  8. R. G. Wenzel, “Oscillation of Gaussian beam parameters in periodic lens waveguides,” Opt. Commun. 43, 89–92 (1982).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  12. A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Appl. Opt. 20, 1933–1935 (1981).
  13. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.
  14. H. Kogelnik, T. Li, “Laser beams and resonators,” Bell Syst. Tech. J. 40, 453–488 (1961).
  15. B. Perry, R. O. Brickman, A. Stein, E. B. Treacy, P. Rabinowitz, “Controllable pulse compression in a multiple-pass-cell Raman laser,” Opt. Lett. 5, 288–290 (1980).
    [CrossRef] [PubMed]
  16. J. T. Luxon, D. E. Parker, “Higher-order CO2 laser beam spot size and depth of focus determined,” Appl. Opt. 20, 1933–1935 (1981).
    [CrossRef] [PubMed]
  17. R. L. Phillips, L. C. Andrews, “Spot size and divergence for Laguerre Gaussian beams of any order,” Appl. Opt. 22, 643–644 (1983).
    [CrossRef] [PubMed]
  18. F. Gori, M. Santarsiero, A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
    [CrossRef]
  19. Y. Kokubun, S. Tamura, “Precise recursive formula for calculating spot size in optical waveguides and accurate evaluation of splice loss,” Appl. Opt. 34, 6862–6873 (1995).
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  20. K. Midorikawa, H. Tashiro, Y. Aoki, K. Ohashi, K. Nagasaka, K. Toyoda, S. Namba, “Output performance of a liquid-N2-cooled para-H2 Raman laser,” J. Appl. Phys. 57, 1504–1508 (1985).
    [CrossRef]

1995 (1)

1991 (1)

F. Gori, M. Santarsiero, A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

1989 (1)

1985 (2)

K. Midorikawa, H. Tashiro, Y. Aoki, K. Nagasaka, K. Toyoda, S. Namba, “Room-temperature operation of a para-H2 rotational Raman laser,” Appl. Phys. Lett. 47, 1033–1035 (1985).
[CrossRef]

K. Midorikawa, H. Tashiro, Y. Aoki, K. Ohashi, K. Nagasaka, K. Toyoda, S. Namba, “Output performance of a liquid-N2-cooled para-H2 Raman laser,” J. Appl. Phys. 57, 1504–1508 (1985).
[CrossRef]

1983 (1)

1982 (1)

R. G. Wenzel, “Oscillation of Gaussian beam parameters in periodic lens waveguides,” Opt. Commun. 43, 89–92 (1982).
[CrossRef]

1981 (2)

1980 (2)

1978 (3)

1965 (2)

D. R. Herriott, H. J. Schulte, “Folded optical delay lines,” Appl. Opt. 4, 883–889 (1965).
[CrossRef]

A. E. Siegman, “Unstable optical resonator for laser applications,” Proc. IEEE 53, 277–287 (1965); IEEE. J. Quantum Electron. QE-20, 1933–1935 (1981).
[CrossRef]

1964 (1)

1961 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Bell Syst. Tech. J. 40, 453–488 (1961).

Andrews, L. C.

Aoki, Y.

K. Midorikawa, H. Tashiro, Y. Aoki, K. Ohashi, K. Nagasaka, K. Toyoda, S. Namba, “Output performance of a liquid-N2-cooled para-H2 Raman laser,” J. Appl. Phys. 57, 1504–1508 (1985).
[CrossRef]

K. Midorikawa, H. Tashiro, Y. Aoki, K. Nagasaka, K. Toyoda, S. Namba, “Room-temperature operation of a para-H2 rotational Raman laser,” Appl. Phys. Lett. 47, 1033–1035 (1985).
[CrossRef]

Brickman, R.

Brickman, R. O.

Byer, R. L.

Duncan, A.

Fox, A. G.

Gori, F.

F. Gori, M. Santarsiero, A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

Hall, D. R.

Herriott, D. R.

Kaldor, A.

Kogelnik, H.

D. R. Herriott, H. Kogelnik, R. Kompfner, “Off-axis paths in spherical mirror interferometers,” Appl. Opt. 3, 523–526 (1964).
[CrossRef]

H. Kogelnik, T. Li, “Laser beams and resonators,” Bell Syst. Tech. J. 40, 453–488 (1961).

Kokubun, Y.

Kompfner, R.

Li, T.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Appl. Opt. 20, 1933–1935 (1981).

H. Kogelnik, T. Li, “Laser beams and resonators,” Bell Syst. Tech. J. 40, 453–488 (1961).

Luxon, J. T.

Mcdowell, R. S.

A. Owyoung, C. W. Patterson, R. S. Mcdowell, “Cw stimulated Raman gain spectroscopy of the ν1 fundamental of methane,” Chem. Phys. Lett. 59, 156–162 (1978).
[CrossRef]

Midorikawa, K.

K. Midorikawa, H. Tashiro, Y. Aoki, K. Nagasaka, K. Toyoda, S. Namba, “Room-temperature operation of a para-H2 rotational Raman laser,” Appl. Phys. Lett. 47, 1033–1035 (1985).
[CrossRef]

K. Midorikawa, H. Tashiro, Y. Aoki, K. Ohashi, K. Nagasaka, K. Toyoda, S. Namba, “Output performance of a liquid-N2-cooled para-H2 Raman laser,” J. Appl. Phys. 57, 1504–1508 (1985).
[CrossRef]

Nagasaka, K.

K. Midorikawa, H. Tashiro, Y. Aoki, K. Ohashi, K. Nagasaka, K. Toyoda, S. Namba, “Output performance of a liquid-N2-cooled para-H2 Raman laser,” J. Appl. Phys. 57, 1504–1508 (1985).
[CrossRef]

K. Midorikawa, H. Tashiro, Y. Aoki, K. Nagasaka, K. Toyoda, S. Namba, “Room-temperature operation of a para-H2 rotational Raman laser,” Appl. Phys. Lett. 47, 1033–1035 (1985).
[CrossRef]

Namba, S.

K. Midorikawa, H. Tashiro, Y. Aoki, K. Nagasaka, K. Toyoda, S. Namba, “Room-temperature operation of a para-H2 rotational Raman laser,” Appl. Phys. Lett. 47, 1033–1035 (1985).
[CrossRef]

K. Midorikawa, H. Tashiro, Y. Aoki, K. Ohashi, K. Nagasaka, K. Toyoda, S. Namba, “Output performance of a liquid-N2-cooled para-H2 Raman laser,” J. Appl. Phys. 57, 1504–1508 (1985).
[CrossRef]

Ohashi, K.

K. Midorikawa, H. Tashiro, Y. Aoki, K. Ohashi, K. Nagasaka, K. Toyoda, S. Namba, “Output performance of a liquid-N2-cooled para-H2 Raman laser,” J. Appl. Phys. 57, 1504–1508 (1985).
[CrossRef]

Owyoung, A.

A. Owyoung, C. W. Patterson, R. S. Mcdowell, “Cw stimulated Raman gain spectroscopy of the ν1 fundamental of methane,” Chem. Phys. Lett. 59, 156–162 (1978).
[CrossRef]

Parker, D. E.

Patterson, C. W.

A. Owyoung, C. W. Patterson, R. S. Mcdowell, “Cw stimulated Raman gain spectroscopy of the ν1 fundamental of methane,” Chem. Phys. Lett. 59, 156–162 (1978).
[CrossRef]

Perry, B.

Phillips, R. L.

Rabinowitz, P.

Santarsiero, M.

F. Gori, M. Santarsiero, A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

Schulte, H. J.

Siegman, A. E.

A. E. Siegman, “Unstable optical resonator for laser applications,” Proc. IEEE 53, 277–287 (1965); IEEE. J. Quantum Electron. QE-20, 1933–1935 (1981).
[CrossRef]

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.

Sona, A.

F. Gori, M. Santarsiero, A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

Stein, A.

Tamura, S.

Tashiro, H.

K. Midorikawa, H. Tashiro, Y. Aoki, K. Ohashi, K. Nagasaka, K. Toyoda, S. Namba, “Output performance of a liquid-N2-cooled para-H2 Raman laser,” J. Appl. Phys. 57, 1504–1508 (1985).
[CrossRef]

K. Midorikawa, H. Tashiro, Y. Aoki, K. Nagasaka, K. Toyoda, S. Namba, “Room-temperature operation of a para-H2 rotational Raman laser,” Appl. Phys. Lett. 47, 1033–1035 (1985).
[CrossRef]

Toyoda, K.

K. Midorikawa, H. Tashiro, Y. Aoki, K. Nagasaka, K. Toyoda, S. Namba, “Room-temperature operation of a para-H2 rotational Raman laser,” Appl. Phys. Lett. 47, 1033–1035 (1985).
[CrossRef]

K. Midorikawa, H. Tashiro, Y. Aoki, K. Ohashi, K. Nagasaka, K. Toyoda, S. Namba, “Output performance of a liquid-N2-cooled para-H2 Raman laser,” J. Appl. Phys. 57, 1504–1508 (1985).
[CrossRef]

Treacy, E. B.

Trutna, W. R.

Wenzel, R. G.

R. G. Wenzel, “Oscillation of Gaussian beam parameters in periodic lens waveguides,” Opt. Commun. 43, 89–92 (1982).
[CrossRef]

Witteman, W. J.

W. J. Witteman, The CO2 laser (Springer-Verlag, Berlin, 1987), Chap. 6.

Xin, J. G.

Appl. Opt. (8)

Appl. Phys. Lett. (1)

K. Midorikawa, H. Tashiro, Y. Aoki, K. Nagasaka, K. Toyoda, S. Namba, “Room-temperature operation of a para-H2 rotational Raman laser,” Appl. Phys. Lett. 47, 1033–1035 (1985).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Bell Syst. Tech. J. 40, 453–488 (1961).

Chem. Phys. Lett. (1)

A. Owyoung, C. W. Patterson, R. S. Mcdowell, “Cw stimulated Raman gain spectroscopy of the ν1 fundamental of methane,” Chem. Phys. Lett. 59, 156–162 (1978).
[CrossRef]

J. Appl. Phys. (1)

K. Midorikawa, H. Tashiro, Y. Aoki, K. Ohashi, K. Nagasaka, K. Toyoda, S. Namba, “Output performance of a liquid-N2-cooled para-H2 Raman laser,” J. Appl. Phys. 57, 1504–1508 (1985).
[CrossRef]

Opt. Commun. (2)

R. G. Wenzel, “Oscillation of Gaussian beam parameters in periodic lens waveguides,” Opt. Commun. 43, 89–92 (1982).
[CrossRef]

F. Gori, M. Santarsiero, A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

Opt. Lett. (3)

Proc. IEEE (1)

A. E. Siegman, “Unstable optical resonator for laser applications,” Proc. IEEE 53, 277–287 (1965); IEEE. J. Quantum Electron. QE-20, 1933–1935 (1981).
[CrossRef]

Other (2)

W. J. Witteman, The CO2 laser (Springer-Verlag, Berlin, 1987), Chap. 6.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.

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Figures (10)

Fig. 1
Fig. 1

(a) Schematic view of the MPC consisting of mirrors with a curvature Rm separated by a distance L. The reflecting spots are uniformly distributed on mirrors around the rims. The numbers from 0 to 25 indicate spot numbers. A laser beam enters the MPC through the hole in Mirror 1, is multiply reflected and focused between Mirrors 1 and 2, and exits from the MPC through the hole in Mirror 2. (b) The coordinate in the MPC. ω m0 is the beam waist size of the fundamental mode of the MPC and R m is the curvature of the mirrors.

Fig. 2
Fig. 2

Variation of spot size during MPC propagation for a top-hat beam. The spot size is normalized to that of the fundamental mode of the MPC. The calculated spot sizes for ω m0/a are (a) 1.20, (b) 0.85, (c) 0.60. Even and odd spot numbers correspond to z = L/2 (on Mirror 2) and - L/2 (on Mirror 1), respectively.

Fig. 3
Fig. 3

Standard deviation of normalized spot sizes on mirrors and at the transit center together with their maximum and minimum limits for a top-hat beam. The spot size is normalized to that of the fundamental mode of the MPC.

Fig. 4
Fig. 4

Standard deviation of a normalized spot size on mirrors and at the transit center together with their maximum and minimum limits for an annular beam with M = 2.0. The spot size is normalized to that of the fundamental mode of the MPC.

Fig. 5
Fig. 5

Optimum aperture, with which the standard deviation of the normalized spot size for an annular beam is minimized, as a function of magnification factor.

Fig. 6
Fig. 6

Optical arrangement for use of a non-Gaussian beam in a MPC under a quasi-mode-matched condition.

Fig. 7
Fig. 7

Schematic presentation of relative radial intensity profile variations outside and inside the MPC. The beam, which is initially an annular beam with M = 2 and φ(0) = 0 at the output coupler of an unstable resonator, changes but reproduces its profiles because of repeated focusing. The beam propagation is presented in terms of exp(-iΦ), which reappears for each π increase.

Fig. 8
Fig. 8

Experimental setup for measurements of intensity profiles in the MPC.

Fig. 9
Fig. 9

Comparison between measured and calculated radial intensity profiles just behind the output coupler and at a distance of 10 m from the output coupler.

Fig. 10
Fig. 10

Radial intensity profiles measured on the MPC mirrors and the corresponding calculated profile.

Equations (28)

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ejω0=0Er, 0Lj2r2ω02exp-r2ω024rω02dr,
Er, z=1ωz2π exp-r2ωz2jejω0Lj2r2ωz2×exp-i2j+1φzexp-ikz+kr22qz,
ωz=ω01+λzπω0221/2,
qz=z1+πω02λz2,
φz=tan-1z/zR,
zR=πω02λ.
ql=-1ql+2R-1,
zb=l-ql1+λqlπω2l2.
ω0=ωl1+πω2lλql21/2=ω01+λlπω0221/21+πω02λ2-1ql+2R21+λlπω0221/2,
φl=φl.
ωz=ω01+λz-zbπω0221/2,
qz=z-zb1+πω02λz-zb2.
φz=φl+tan-1zbzR-tan-1z-zbzR,
zR=πω02λ.
Er, z=1ωz2π×exp-r2ωz2jejω0Lj2r2ωz2×exp-i2j+1φz×exp-ik2l-z+kr22qz.
ωm0=λL2π2RmL-11/21/2.
ωmz=ωm01+λzπωm0221/2,
qmz=z1+πωm02λz2,
zm=πωm02λ.
Inr, z=Enr, z2=2πωmz2 exp-2r2ωmz2jejω0×Lj2r2ωmz2exp-i2j+1φnz.
θ=2πK/N,
cos θ=1-L/Rm.
tanτ2=L2zm.
τ=θ.
φnz=φ10+n-1θ+tan-1z/zRn=oddφ10+n-1θ-tan-1z/zRn=even.
1-1e2=0ωe2πrIr, zdr02πrIr, zdr.
d1=f±ω0ωm0f2-f02,
d2=f±ωm0ω0f2-f02,

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