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.

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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).<br>
    [CrossRef] [PubMed]
  2. J. Zehetner, "Highly efficient diode-pumped elliptical mode Nd:YLF laser," Opt. Commun. 117, 273-267 (1995).<br>
    [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).<br>
    [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.<br>
  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.<br>
  6. A.E. Siegman, Lasers, (University Science Books, Mill Valley, California, 1986) 15.<br>
  7. A.E. Siegman, Lasers, (University Science Books, Mill Valley, California, 1986) 19.1<br>
  8. J.A. Arnaud and H. Kogelnik, "Gaussian Light beams with General Astigmatism," Appl. Opt. 8, 1687 (1969).<br>
    [CrossRef] [PubMed]
  9. A.W. Greynolds, Propagation of generally astigmatic Gaussian beams along skew ray paths, Proc. SPIE 679, 129 (1986)<br>
    [CrossRef]
  10. A.W. Greynolds, Vector formulation of the ray-equivalent method for general Gaussian beam propagation, Proc. SPIE 560, 33 (1985)<br>
  11. J.A. Arnaud, "Nonorthogonal Optical Waveguides and Resonators," Bell Syst. Tech. J. 49, 2311 (1970).<br>
  12. J.A. Arnaud, "Hamiltonian theory of beam mode propagation," Prog. Opt. XI, 249 (1973).<br>
  13. J. Serna and G. Nemes, "Decoupling of coherent Gaussian beams with general astigmatism," Opt. Lett. 18, 1774 (1993).<br>
    [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).<br>
    [CrossRef]
  15. B. L, S. Xu, G. Feng and B. Cai, "Astigmatic resonator analysis using an eigenray vector concept," Optik 99, 158 (1992).<br>
  16. B. L, S. Xu, Y. Hu and B. Cai, "Matrix representation of three-dimensional astigmatic resonators," Opt. Quantum. Electron. 24, 619 (1992).<br>
    [CrossRef]
  17. K. Kubodera and K. Otsuka, "Single-transverse-mode LiNdP 4 O 12 slab waveguide laser," J. Appl. Phys. 50, 653 (1979).
    [CrossRef]

Other (17)

F. Krausz, J. Zehetner, T. Brabec and E. Winter, "Elliptic-mode cavity for diode-pumped lasers," Opt. Lett. 16, 1496-1498 (1991).<br>
[CrossRef] [PubMed]

J. Zehetner, "Highly efficient diode-pumped elliptical mode Nd:YLF laser," Opt. Commun. 117, 273-267 (1995).<br>
[CrossRef]

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).<br>
[CrossRef] [PubMed]

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.<br>

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.<br>

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

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

J.A. Arnaud and H. Kogelnik, "Gaussian Light beams with General Astigmatism," Appl. Opt. 8, 1687 (1969).<br>
[CrossRef] [PubMed]

A.W. Greynolds, Propagation of generally astigmatic Gaussian beams along skew ray paths, Proc. SPIE 679, 129 (1986)<br>
[CrossRef]

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

J.A. Arnaud, "Nonorthogonal Optical Waveguides and Resonators," Bell Syst. Tech. J. 49, 2311 (1970).<br>

J.A. Arnaud, "Hamiltonian theory of beam mode propagation," Prog. Opt. XI, 249 (1973).<br>

J. Serna and G. Nemes, "Decoupling of coherent Gaussian beams with general astigmatism," Opt. Lett. 18, 1774 (1993).<br>
[CrossRef] [PubMed]

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).<br>
[CrossRef]

B. L, S. Xu, G. Feng and B. Cai, "Astigmatic resonator analysis using an eigenray vector concept," Optik 99, 158 (1992).<br>

B. L, S. Xu, Y. Hu and B. Cai, "Matrix representation of three-dimensional astigmatic resonators," Opt. Quantum. Electron. 24, 619 (1992).<br>
[CrossRef]

K. Kubodera and K. Otsuka, "Single-transverse-mode LiNdP 4 O 12 slab waveguide laser," J. Appl. Phys. 50, 653 (1979).
[CrossRef]

Supplementary Material (2)

» Media 1: MOV (4237 KB)     
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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]

Figure 2.
Figure 2.

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

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]

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.

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.

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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