Abstract

A new approach to modeling the spatial intensity profile from Porro prism resonators is proposed based on rotating loss screens to mimic the apex losses of the prisms. A numerical model based on this approach is presented which correctly predicts the output transverse field distribution found experimentally from such resonators.

© 2007 Optical Society of America

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References

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  1. G. Gould, S. Jacobs, P. Rabinowitz and T. Shultz, "Crossed Roof Prism Interferometer," Appl. Opt. 1, 533-534 (1962).
    [CrossRef]
  2. I. Kuo and T. Ko, "Laser resonators of a mirror and corner cube reflector: analysis by the imaging method," Appl. Opt. 23, 53-56 (1984).
    [CrossRef] [PubMed]
  3. G. Zhou and L.W. Casperson, "Modes of a laser resonator with a retroreflecting roof mirror," Appl. Opt. 20, 3542-3546 (1981).
    [CrossRef] [PubMed]
  4. J. Lee and C. Leung, "Beam pointing direction changes in a misaligned Porro prism resonator," Appl. Opt. 27, 2701-2707 (1988).
    [CrossRef] [PubMed]
  5. Y. A. Anan’ev, V. I. Kuprenyuk, V. V. Sergeev and V. E. Sherstobitov, "Investigation of the properties of an unstable resonator using a dihedral corner reflector in a continuous-flow cw CO2 laser," Sov. J. Quantum Electron. 7, 822-824 (1977).
    [CrossRef]
  6. I. Singh, A. Kumar and O. P. Nijhawan, "Design of a high-power Nd:YAG Q-switched laser cavity," Appl. Opt. 34, 3349-3351 (1995).
    [CrossRef] [PubMed]
  7. N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005), Chap. 17.
  8. Y. Z. Virnik, V. B. Gerasimov, A. L. Sivakov and Y. M. Treivish, "Formation of fields in resonators with a composite mirror consisting of inverting elements," Sov. J. Quantum Electron 17, 1040-1043 (1987).
    [CrossRef]
  9. T. A. Anan’ev, "Unstable prism resonators," Sov. J. Quantum Electron 3, 58-59 (1973).
  10. A. E. Siegman, H. Y. Miller, "Unstable optical resonator loss calculations using Prony Method," Appl. Opt. 9, 2729-2736 (1970).
    [CrossRef] [PubMed]

1995 (1)

1988 (1)

1987 (1)

Y. Z. Virnik, V. B. Gerasimov, A. L. Sivakov and Y. M. Treivish, "Formation of fields in resonators with a composite mirror consisting of inverting elements," Sov. J. Quantum Electron 17, 1040-1043 (1987).
[CrossRef]

1984 (1)

1981 (1)

1977 (1)

Y. A. Anan’ev, V. I. Kuprenyuk, V. V. Sergeev and V. E. Sherstobitov, "Investigation of the properties of an unstable resonator using a dihedral corner reflector in a continuous-flow cw CO2 laser," Sov. J. Quantum Electron. 7, 822-824 (1977).
[CrossRef]

1973 (1)

T. A. Anan’ev, "Unstable prism resonators," Sov. J. Quantum Electron 3, 58-59 (1973).

1970 (1)

1962 (1)

Anan’ev, T. A.

T. A. Anan’ev, "Unstable prism resonators," Sov. J. Quantum Electron 3, 58-59 (1973).

Anan’ev, Y. A.

Y. A. Anan’ev, V. I. Kuprenyuk, V. V. Sergeev and V. E. Sherstobitov, "Investigation of the properties of an unstable resonator using a dihedral corner reflector in a continuous-flow cw CO2 laser," Sov. J. Quantum Electron. 7, 822-824 (1977).
[CrossRef]

Casperson, L.W.

Gerasimov, V. B.

Y. Z. Virnik, V. B. Gerasimov, A. L. Sivakov and Y. M. Treivish, "Formation of fields in resonators with a composite mirror consisting of inverting elements," Sov. J. Quantum Electron 17, 1040-1043 (1987).
[CrossRef]

Gould, G.

Jacobs, S.

Ko, T.

Kumar, A.

Kuo, I.

Kuprenyuk, V. I.

Y. A. Anan’ev, V. I. Kuprenyuk, V. V. Sergeev and V. E. Sherstobitov, "Investigation of the properties of an unstable resonator using a dihedral corner reflector in a continuous-flow cw CO2 laser," Sov. J. Quantum Electron. 7, 822-824 (1977).
[CrossRef]

Lee, J.

Leung, C.

Miller, H. Y.

Nijhawan, O. P.

Rabinowitz, P.

Sergeev, V. V.

Y. A. Anan’ev, V. I. Kuprenyuk, V. V. Sergeev and V. E. Sherstobitov, "Investigation of the properties of an unstable resonator using a dihedral corner reflector in a continuous-flow cw CO2 laser," Sov. J. Quantum Electron. 7, 822-824 (1977).
[CrossRef]

Sherstobitov, V. E.

Y. A. Anan’ev, V. I. Kuprenyuk, V. V. Sergeev and V. E. Sherstobitov, "Investigation of the properties of an unstable resonator using a dihedral corner reflector in a continuous-flow cw CO2 laser," Sov. J. Quantum Electron. 7, 822-824 (1977).
[CrossRef]

Shultz, T.

Siegman, A. E.

Singh, I.

Sivakov, A. L.

Y. Z. Virnik, V. B. Gerasimov, A. L. Sivakov and Y. M. Treivish, "Formation of fields in resonators with a composite mirror consisting of inverting elements," Sov. J. Quantum Electron 17, 1040-1043 (1987).
[CrossRef]

Treivish, Y. M.

Y. Z. Virnik, V. B. Gerasimov, A. L. Sivakov and Y. M. Treivish, "Formation of fields in resonators with a composite mirror consisting of inverting elements," Sov. J. Quantum Electron 17, 1040-1043 (1987).
[CrossRef]

Virnik, Y. Z.

Y. Z. Virnik, V. B. Gerasimov, A. L. Sivakov and Y. M. Treivish, "Formation of fields in resonators with a composite mirror consisting of inverting elements," Sov. J. Quantum Electron 17, 1040-1043 (1987).
[CrossRef]

Zhou, G.

Appl. Opt. (6)

Sov. J. Quantum Electron (2)

Y. Z. Virnik, V. B. Gerasimov, A. L. Sivakov and Y. M. Treivish, "Formation of fields in resonators with a composite mirror consisting of inverting elements," Sov. J. Quantum Electron 17, 1040-1043 (1987).
[CrossRef]

T. A. Anan’ev, "Unstable prism resonators," Sov. J. Quantum Electron 3, 58-59 (1973).

Sov. J. Quantum Electron. (1)

Y. A. Anan’ev, V. I. Kuprenyuk, V. V. Sergeev and V. E. Sherstobitov, "Investigation of the properties of an unstable resonator using a dihedral corner reflector in a continuous-flow cw CO2 laser," Sov. J. Quantum Electron. 7, 822-824 (1977).
[CrossRef]

Other (1)

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005), Chap. 17.

Supplementary Material (3)

» Media 1: AVI (3288 KB)     
» Media 2: AVI (3489 KB)     
» Media 3: AVI (3415 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

A typical Porro prism based Nd:YAG laser with passive Q-switch, showing the following optical elements: Porro prisms (elements a and h); intra-cavity lenses (elements b and g); a beamsplitter cube (element c); a quarter wave plate (element d), and a passive Q- switch (element e).(interactive pdf 117 KB)

Fig. 2.
Fig. 2.

Illustration of the effect of phase and intensity screens on an incident field.

Fig. 3.
Fig. 3.

(a) - (e): Evolution of a ray as it is reflected back and forth in the resonator, for starting Porro angle α = 60°. After 3 round trips the pattern is complete (e) and starts to repeat. (f) - (j): Equivalent case but with α = 30°, and now taking 6 round trips for completion.

Fig. 4.
Fig. 4.

The apexes of two Porro prisms at angles ϕ1 and ϕ2. Initially the apex of PP 1 is in the horizontal plane (a), but after successive reflections about the inverting edges of the two prisms the apex will appear to be rotating about the circle: (b) 1 pass, (c) 2 passes and (d) 3 passes.

Fig. 5.
Fig. 5.

Plot of the discrete set of angles a that give rise to a petal pattern, with the corresponding number of petals to be observed. Data calculated for m ∈ [1,100] and i ∈ [1,50].

Fig. 6.
Fig. 6.

Photograph of assembled laser. The beamsplitter cube and one of the Porro prisms can be made out on the left of the assembly.

Fig. 7.
Fig. 7.

The analytical model depiction of finitely sub–divided fields in (a) and (b), and an infinitely sub–divided field in (c). Numerically this results in a pattern with (d) 10 petals (movie 3.3 MB), (e) 14 petals (movie 3.5 MB) and (f) no petals (movie 3.4 MB). The corresponding experimentally observed output is shown in (g) - (i).

Fig. 8.
Fig. 8.

Analytically calculated sub–division of the field using Eqs. (4) and (7) (top row), with corresponding petal patterns calculated numerically using this model (bottom row).

Fig. 9.
Fig. 9.

Plot of the round-trip loss as a function of the number of petals as predicted by the numerical model.

Fig. 10.
Fig. 10.

The transverse field distribution, with (a) two and (b) one pulse. The angle between the Porro prisms is 13° (giving 14 spots).

Tables (1)

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Table 1. Petal pattern observations: theory and experiment

Equations (15)

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U out ( x , y ) = U in x y t x y = U in x y A x y exp ( i φ x y ) ,
θ 1 ( n ) = ( 1 ) n + 1 2 ,
ϕ 1 ( n ) = i = 0 n θ 1 ( i ) = α 2 [ 1 ( 1 ) n ( 1 + 2 n ) ] .
ν 1 ( n ) = cos ϕ 1 ( n ) sin ϕ 1 ( n ) sin ϕ 1 ( n ) cos ϕ 1 ( n ) 1 0 .
θ 2 ( n ) = ( 1 ) n 2 ;
ϕ 2 ( n ) = α α 2 [ 1 ( 1 ) n ( 1 + 2 n ) ] ;
ν 2 ( n ) = ( cos ϕ 2 ( n ) sin ϕ 2 ( n ) sin ϕ 2 ( n ) cos ϕ 2 ( n ) )    ( 1 0 ) .
ν 1 ( 0 ) = 1 0 ,
ν 2 ( 0 ) = cos α sin α ,
cos ϕ 1 ( n ) sin ϕ 1 ( n ) sin ϕ 1 ( n ) cos ϕ 1 ( n ) = 1 0 0 1 .
ϕ 1 ( n ) = α 2 [ 1 ( 1 ) n ( 1 + 2 n ) ] = i 2 π ,
α = i π m ,
α j = 2 π N .
N = j 2 π α .
ψ = 2 π N = α j .

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