Abstract

We examine the optical resonator composed of two astigmatic elements, in a twisted configuration. These cavities have mode cross-sections with principal axes that rotate on propagation. Explicit cavity mode equations are derived for the case of identical mirrors. Such a resonator is appropriate for a solid-state laser that is end-pumped with the output of a laser-diode array brought to a line focus. We present a simple analysis of the significance of rotational misalignment, which effects the pump-to-mode power coupling, beam quality, and cavity stability.

©1998 Optical Society of America

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References

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  1. F. Krausz, J. Zehetner, T. Brabec, and E. Winter, “Elliptic-mode cavity for diode-pumped lasers,” Opt. Lett. 16, 1496–1498 (1991).
    [Crossref] [PubMed]
  2. J. Zehetner, “Highly efficient diode-pumped elliptical mode Nd:YLF laser,” Opt. Commun. 117, 273–267 (1995).
    [Crossref]
  3. D. Kopf, U. Keller, M.A. Emanuel, R.J. Beach, and J.A. Skidmore, “1.1-W cw Cr:LiSAF laser pumped by a 1-cm diode array,” Opt. Lett. 22, 99–101 (1997).
    [Crossref] [PubMed]
  4. J.L. Blows, J.M. Dawes, and J.A. Piper, “Highly astigmatic diode pumped laser cavities for intracavity nonlinear optical conversion,” in International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), MJ3.
  5. J. Blows, J. Dawes, J. Piper, and G. Forbes, “A highly astigmatic diode end-pumped solid-state laser,” in Trends in Optics and Photonics, Vol. 10, Advanced Solid State Lasers, Clifford R. Pollock and Walter R. Bosenburg, eds. (Optical Society of America, Washington, DC, 1997), p. 376–379.
  6. A.E. Siegman, Lasers, (University Science Books, Mill Valley, California, 1986) §15.
  7. A.E. Siegman, Lasers, (University Science Books, Mill Valley, California, 1986) §19.1
  8. J.A. Arnaud and H. Kogelnik, “Gaussian Light beams with General Astigmatism,” Appl. Opt. 8, 1687 (1969).
    [Crossref] [PubMed]
  9. A.W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 679, 129 (1986)
    [Crossref]
  10. A.W. Greynolds, “Vector formulation of the ray-equivalent method for general Gaussian beam propagation,” Proc. SPIE 560, 33 (1985)
  11. J.A. Arnaud, “Nonorthogonal Optical Waveguides and Resonators,” Bell Syst. Tech. J. 49, 2311 (1970).
  12. J.A. Arnaud, “Hamiltonian theory of beam mode propagation,” Prog. Opt. XI, 249 (1973).
  13. J. Serna and G. Nemes, “Decoupling of coherent Gaussian beams with general astigmatism,” Opt. Lett. 18, 1774 (1993).
    [Crossref] [PubMed]
  14. I.V. Golovnin, A.N. Kovrigin, A.N. Konovalov, and G.D. Laptev, “Description of propagation of a Gaussian beam with general astigmatism by the ray method and applications of this method to calculation of the parameters of nonplanar ring cavities.,” Quantum Electron. 25, 436 (1995).
    [Crossref]
  15. B. Lü, S. Xu, G. Feng, and B. Cai, “Astigmatic resonator analysis using an eigenray vector concept,” Optik 99, 158 (1992).
  16. B. Lü, S. Xu, Y. Hu, and B. Cai, “Matrix representation of three-dimensional astigmatic resonators,” Opt. Quantum. Electron. 24, 619 (1992).
    [Crossref]
  17. K. Kubodera and K. Otsuka, “Single-transverse-mode LiNdP4O12 slab waveguide laser,” J. Appl. Phys. 50, 653 (1979).
    [Crossref]

1997 (1)

1995 (2)

J. Zehetner, “Highly efficient diode-pumped elliptical mode Nd:YLF laser,” Opt. Commun. 117, 273–267 (1995).
[Crossref]

I.V. Golovnin, A.N. Kovrigin, A.N. Konovalov, and G.D. Laptev, “Description of propagation of a Gaussian beam with general astigmatism by the ray method and applications of this method to calculation of the parameters of nonplanar ring cavities.,” Quantum Electron. 25, 436 (1995).
[Crossref]

1993 (1)

1992 (2)

B. Lü, S. Xu, G. Feng, and B. Cai, “Astigmatic resonator analysis using an eigenray vector concept,” Optik 99, 158 (1992).

B. Lü, S. Xu, Y. Hu, and B. Cai, “Matrix representation of three-dimensional astigmatic resonators,” Opt. Quantum. Electron. 24, 619 (1992).
[Crossref]

1991 (1)

1986 (1)

A.W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 679, 129 (1986)
[Crossref]

1985 (1)

A.W. Greynolds, “Vector formulation of the ray-equivalent method for general Gaussian beam propagation,” Proc. SPIE 560, 33 (1985)

1979 (1)

K. Kubodera and K. Otsuka, “Single-transverse-mode LiNdP4O12 slab waveguide laser,” J. Appl. Phys. 50, 653 (1979).
[Crossref]

1973 (1)

J.A. Arnaud, “Hamiltonian theory of beam mode propagation,” Prog. Opt. XI, 249 (1973).

1970 (1)

J.A. Arnaud, “Nonorthogonal Optical Waveguides and Resonators,” Bell Syst. Tech. J. 49, 2311 (1970).

1969 (1)

Arnaud, J.A.

J.A. Arnaud, “Hamiltonian theory of beam mode propagation,” Prog. Opt. XI, 249 (1973).

J.A. Arnaud, “Nonorthogonal Optical Waveguides and Resonators,” Bell Syst. Tech. J. 49, 2311 (1970).

J.A. Arnaud and H. Kogelnik, “Gaussian Light beams with General Astigmatism,” Appl. Opt. 8, 1687 (1969).
[Crossref] [PubMed]

Beach, R.J.

Blows, J.

J. Blows, J. Dawes, J. Piper, and G. Forbes, “A highly astigmatic diode end-pumped solid-state laser,” in Trends in Optics and Photonics, Vol. 10, Advanced Solid State Lasers, Clifford R. Pollock and Walter R. Bosenburg, eds. (Optical Society of America, Washington, DC, 1997), p. 376–379.

Blows, J.L.

J.L. Blows, J.M. Dawes, and J.A. Piper, “Highly astigmatic diode pumped laser cavities for intracavity nonlinear optical conversion,” in International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), MJ3.

Brabec, T.

Cai, B.

B. Lü, S. Xu, G. Feng, and B. Cai, “Astigmatic resonator analysis using an eigenray vector concept,” Optik 99, 158 (1992).

B. Lü, S. Xu, Y. Hu, and B. Cai, “Matrix representation of three-dimensional astigmatic resonators,” Opt. Quantum. Electron. 24, 619 (1992).
[Crossref]

Dawes, J.

J. Blows, J. Dawes, J. Piper, and G. Forbes, “A highly astigmatic diode end-pumped solid-state laser,” in Trends in Optics and Photonics, Vol. 10, Advanced Solid State Lasers, Clifford R. Pollock and Walter R. Bosenburg, eds. (Optical Society of America, Washington, DC, 1997), p. 376–379.

Dawes, J.M.

J.L. Blows, J.M. Dawes, and J.A. Piper, “Highly astigmatic diode pumped laser cavities for intracavity nonlinear optical conversion,” in International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), MJ3.

Emanuel, M.A.

Feng, G.

B. Lü, S. Xu, G. Feng, and B. Cai, “Astigmatic resonator analysis using an eigenray vector concept,” Optik 99, 158 (1992).

Forbes, G.

J. Blows, J. Dawes, J. Piper, and G. Forbes, “A highly astigmatic diode end-pumped solid-state laser,” in Trends in Optics and Photonics, Vol. 10, Advanced Solid State Lasers, Clifford R. Pollock and Walter R. Bosenburg, eds. (Optical Society of America, Washington, DC, 1997), p. 376–379.

Golovnin, I.V.

I.V. Golovnin, A.N. Kovrigin, A.N. Konovalov, and G.D. Laptev, “Description of propagation of a Gaussian beam with general astigmatism by the ray method and applications of this method to calculation of the parameters of nonplanar ring cavities.,” Quantum Electron. 25, 436 (1995).
[Crossref]

Greynolds, A.W.

A.W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 679, 129 (1986)
[Crossref]

A.W. Greynolds, “Vector formulation of the ray-equivalent method for general Gaussian beam propagation,” Proc. SPIE 560, 33 (1985)

Hu, Y.

B. Lü, S. Xu, Y. Hu, and B. Cai, “Matrix representation of three-dimensional astigmatic resonators,” Opt. Quantum. Electron. 24, 619 (1992).
[Crossref]

Keller, U.

Kogelnik, H.

Konovalov, A.N.

I.V. Golovnin, A.N. Kovrigin, A.N. Konovalov, and G.D. Laptev, “Description of propagation of a Gaussian beam with general astigmatism by the ray method and applications of this method to calculation of the parameters of nonplanar ring cavities.,” Quantum Electron. 25, 436 (1995).
[Crossref]

Kopf, D.

Kovrigin, A.N.

I.V. Golovnin, A.N. Kovrigin, A.N. Konovalov, and G.D. Laptev, “Description of propagation of a Gaussian beam with general astigmatism by the ray method and applications of this method to calculation of the parameters of nonplanar ring cavities.,” Quantum Electron. 25, 436 (1995).
[Crossref]

Krausz, F.

Kubodera, K.

K. Kubodera and K. Otsuka, “Single-transverse-mode LiNdP4O12 slab waveguide laser,” J. Appl. Phys. 50, 653 (1979).
[Crossref]

Laptev, G.D.

I.V. Golovnin, A.N. Kovrigin, A.N. Konovalov, and G.D. Laptev, “Description of propagation of a Gaussian beam with general astigmatism by the ray method and applications of this method to calculation of the parameters of nonplanar ring cavities.,” Quantum Electron. 25, 436 (1995).
[Crossref]

Lü, B.

B. Lü, S. Xu, G. Feng, and B. Cai, “Astigmatic resonator analysis using an eigenray vector concept,” Optik 99, 158 (1992).

B. Lü, S. Xu, Y. Hu, and B. Cai, “Matrix representation of three-dimensional astigmatic resonators,” Opt. Quantum. Electron. 24, 619 (1992).
[Crossref]

Nemes, G.

Otsuka, K.

K. Kubodera and K. Otsuka, “Single-transverse-mode LiNdP4O12 slab waveguide laser,” J. Appl. Phys. 50, 653 (1979).
[Crossref]

Piper, J.

J. Blows, J. Dawes, J. Piper, and G. Forbes, “A highly astigmatic diode end-pumped solid-state laser,” in Trends in Optics and Photonics, Vol. 10, Advanced Solid State Lasers, Clifford R. Pollock and Walter R. Bosenburg, eds. (Optical Society of America, Washington, DC, 1997), p. 376–379.

Piper, J.A.

J.L. Blows, J.M. Dawes, and J.A. Piper, “Highly astigmatic diode pumped laser cavities for intracavity nonlinear optical conversion,” in International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), MJ3.

Serna, J.

Siegman, A.E.

A.E. Siegman, Lasers, (University Science Books, Mill Valley, California, 1986) §15.

A.E. Siegman, Lasers, (University Science Books, Mill Valley, California, 1986) §19.1

Skidmore, J.A.

Winter, E.

Xu, S.

B. Lü, S. Xu, Y. Hu, and B. Cai, “Matrix representation of three-dimensional astigmatic resonators,” Opt. Quantum. Electron. 24, 619 (1992).
[Crossref]

B. Lü, S. Xu, G. Feng, and B. Cai, “Astigmatic resonator analysis using an eigenray vector concept,” Optik 99, 158 (1992).

Zehetner, J.

J. Zehetner, “Highly efficient diode-pumped elliptical mode Nd:YLF laser,” Opt. Commun. 117, 273–267 (1995).
[Crossref]

F. Krausz, J. Zehetner, T. Brabec, and E. Winter, “Elliptic-mode cavity for diode-pumped lasers,” Opt. Lett. 16, 1496–1498 (1991).
[Crossref] [PubMed]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

J.A. Arnaud, “Nonorthogonal Optical Waveguides and Resonators,” Bell Syst. Tech. J. 49, 2311 (1970).

J. Appl. Phys. (1)

K. Kubodera and K. Otsuka, “Single-transverse-mode LiNdP4O12 slab waveguide laser,” J. Appl. Phys. 50, 653 (1979).
[Crossref]

Opt. Commun. (1)

J. Zehetner, “Highly efficient diode-pumped elliptical mode Nd:YLF laser,” Opt. Commun. 117, 273–267 (1995).
[Crossref]

Opt. Lett. (3)

Opt. Quantum. Electron. (1)

B. Lü, S. Xu, Y. Hu, and B. Cai, “Matrix representation of three-dimensional astigmatic resonators,” Opt. Quantum. Electron. 24, 619 (1992).
[Crossref]

Optik (1)

B. Lü, S. Xu, G. Feng, and B. Cai, “Astigmatic resonator analysis using an eigenray vector concept,” Optik 99, 158 (1992).

Proc. SPIE (2)

A.W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 679, 129 (1986)
[Crossref]

A.W. Greynolds, “Vector formulation of the ray-equivalent method for general Gaussian beam propagation,” Proc. SPIE 560, 33 (1985)

Prog. Opt. (1)

J.A. Arnaud, “Hamiltonian theory of beam mode propagation,” Prog. Opt. XI, 249 (1973).

Quantum Electron. (1)

I.V. Golovnin, A.N. Kovrigin, A.N. Konovalov, and G.D. Laptev, “Description of propagation of a Gaussian beam with general astigmatism by the ray method and applications of this method to calculation of the parameters of nonplanar ring cavities.,” Quantum Electron. 25, 436 (1995).
[Crossref]

Other (4)

J.L. Blows, J.M. Dawes, and J.A. Piper, “Highly astigmatic diode pumped laser cavities for intracavity nonlinear optical conversion,” in International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), MJ3.

J. Blows, J. Dawes, J. Piper, and G. Forbes, “A highly astigmatic diode end-pumped solid-state laser,” in Trends in Optics and Photonics, Vol. 10, Advanced Solid State Lasers, Clifford R. Pollock and Walter R. Bosenburg, eds. (Optical Society of America, Washington, DC, 1997), p. 376–379.

A.E. Siegman, Lasers, (University Science Books, Mill Valley, California, 1986) §15.

A.E. Siegman, Lasers, (University Science Books, Mill Valley, California, 1986) §19.1

Supplementary Material (2)

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Figures (5)

Figure 1.
Figure 1. This animation shows the behaviour of a contour of equal field amplitude, for a resonator composed of two identical cylindrical mirrors, as one end mirror is rotated. The principal axes are traced in green. [Media 1]

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Figure 2.
Figure 2. The shading indicates the existence of a cavity mode. This map has a period of π rads. in t.

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Figure 3.
Figure 3. This animation depicts the transverse phase and amplitude distributions for a resonator composed of two end cylindrical mirrors orientated at π/4 rads. to each other. The slices are equally spaced in z. The resonator parameters are α = 0.97 and t = π/4. [Media 2]

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Figure 4.
Figure 4. In this series of curves, the angle of rotation s that the elliptical beam at the first mirror has rotated, is plotted as a function of the angle t of rotation of the second mirror, for different values of α=L/R.

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Figure 5.
Figure 5. In this series of curves, the aspect ratio of the elliptical beam cross-section at the mirrors, is plotted as a function of the angle t of rotation between them for different values of L/R.

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Equations (10)

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ψ ( u ¯ , 0 ) = ψ 0 exp [ − j k 2 u ¯ T · Φ · u ¯ − u ¯ T · M · u ¯ ] ,
ψ ( v ¯ , z ) = − j exp ( j k z ) z λ ∫ ∫ All space ψ ( u ¯ , 0 ) exp ( j k 2 z ( u ¯ − v ¯ ) T I ( u ¯ − v ¯ ) ) d u ¯ ,
∫ ∫ All space exp [ − u ¯ T · Q · u ¯ + b ¯ T · u ¯ ] d u ¯ = π det Q exp ( 1 4 b ¯ T · Q − 1 · b ¯ ) ,
ψ ( v ¯ , z ) = ψ 0 − j π exp [ j k ( z + 1 2 z v · v ) ] z λ ∣ M + j k 2 Φ − j k 2 z I ∣ exp [ − ( k 2 z ) 2 v ¯ T · ( M + j k 2 Φ − j k 2 z I ) − 1 · v ¯ ] .
M 11 = k α 2 L ζ [ sin 2 t + ( 1 − α ) ( 1 + ∣ cos t ∣ ) ] ,
M 22 = k α 2 L ζ ( 1 + ∣ cos t ∣ ) ∣ cos t ∣ ,
M 12 = M 21 = k α 2 L ζ sin t ∣ cos t ∣ ,
α = L R ,
ζ = sin 2 t + ( 1 − α ) ( 1 + ∣ cos t ∣ ) 2 .
∣ cos ( t ) ∣ < 2 α − 1 .

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