Abstract

The diffractive effects of a single laser rod in an unstable super-Gaussian coupled cavity are modeled for a range of cavity configurations, with an intracavity, zero-thickness aperture. After fundamental mode propagation through a maximally flat output coupler, beam quality (M2) and far-field power loss values are related. Beam quality is most sensitive to cavity magnification and aperture Fresnel number, both correlated to the aperture-equivalent Fresnel number. In contrast, variation of M2 with aperture position is sufficiently conservative to predict the intensity profile of a solid-state laser with a typical gain length, in good agreement with experimental data.

© 2005 Optical Society of America

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References

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  1. W. Koechner, “Optical resonators: unstable resonators,” in Solid State Laser Engineering, 4th ed., A. L. Schawlow, A. E. Siegman, T. Tamir, H. K. V. Lotsch, eds. (Springer-Verlag, New York, 1996), pp. 262–272.
  2. A. E. Siegman, “An introduction to lasers, wave optics and Gaussian beams, complex paraxial wave optics, and unstable optical resonators,” in Lasers, A. Kelly, ed. (University Science Books, Sausalito, Calif., 1986), pp. 44, 626–635, 777–785, 858–890.
  3. M. S. Bowers, S. E. Moody, “Numerical solutions of the exact cavity equations of motion for an unstable optical resonator,” Appl. Opt. 29, 3905–3915 (1990).
    [CrossRef] [PubMed]
  4. S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13, 201–203 (March1988).
    [CrossRef] [PubMed]
  5. A. E. Siegman, Handbook of Laser Beam Propagation and Beam Quality Formulas Using the Spatial-Frequency and Intensity-Moments Analyses, draft version, 2 July 1991 (Edward L. Ginzton Laboratory, Stanford University, Palo Alto, Calif., personal communication).
  6. J. W. Goodman, “Fresnel and Fraunhofer Diffraction,” in Introduction to Fourier Optics, 2nd ed., S. W. Director, ed. (McGraw-Hill, New York, 1996), pp. 63–95.
  7. M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1997).
    [CrossRef]
  8. C. F. Maes, E. M. Wright, “Mode properties of an external cavity laser with Gaussian gain,” Opt. Lett. 29, 229–231 (2004).
    [CrossRef] [PubMed]
  9. A. E. Siegman, “Laser without photons—or should it be lasers with too many photons?” Appl. Phys. B 60, 247–257 (1995).
    [CrossRef]

2004 (1)

1997 (1)

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1997).
[CrossRef]

1995 (1)

A. E. Siegman, “Laser without photons—or should it be lasers with too many photons?” Appl. Phys. B 60, 247–257 (1995).
[CrossRef]

1990 (1)

1988 (1)

Bowers, M. S.

De Silvestri, S.

Goodman, J. W.

J. W. Goodman, “Fresnel and Fraunhofer Diffraction,” in Introduction to Fourier Optics, 2nd ed., S. W. Director, ed. (McGraw-Hill, New York, 1996), pp. 63–95.

Koechner, W.

W. Koechner, “Optical resonators: unstable resonators,” in Solid State Laser Engineering, 4th ed., A. L. Schawlow, A. E. Siegman, T. Tamir, H. K. V. Lotsch, eds. (Springer-Verlag, New York, 1996), pp. 262–272.

Laporta, P.

Lindberg, A. M.

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1997).
[CrossRef]

Maes, C. F.

Magni, V.

Moody, S. E.

Siegman, A. E.

A. E. Siegman, “Laser without photons—or should it be lasers with too many photons?” Appl. Phys. B 60, 247–257 (1995).
[CrossRef]

A. E. Siegman, Handbook of Laser Beam Propagation and Beam Quality Formulas Using the Spatial-Frequency and Intensity-Moments Analyses, draft version, 2 July 1991 (Edward L. Ginzton Laboratory, Stanford University, Palo Alto, Calif., personal communication).

A. E. Siegman, “An introduction to lasers, wave optics and Gaussian beams, complex paraxial wave optics, and unstable optical resonators,” in Lasers, A. Kelly, ed. (University Science Books, Sausalito, Calif., 1986), pp. 44, 626–635, 777–785, 858–890.

Svelto, O.

Thijssen, M. S.

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1997).
[CrossRef]

van Eijkelenborg, M. A.

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1997).
[CrossRef]

Woerdman, J. P.

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1997).
[CrossRef]

Wright, E. M.

Appl. Opt. (1)

Appl. Phys. B (1)

A. E. Siegman, “Laser without photons—or should it be lasers with too many photons?” Appl. Phys. B 60, 247–257 (1995).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, J. P. Woerdman, “Resonance of quantum noise in an unstable cavity laser,” Phys. Rev. Lett. 77, 4314–4317 (1997).
[CrossRef]

Other (4)

W. Koechner, “Optical resonators: unstable resonators,” in Solid State Laser Engineering, 4th ed., A. L. Schawlow, A. E. Siegman, T. Tamir, H. K. V. Lotsch, eds. (Springer-Verlag, New York, 1996), pp. 262–272.

A. E. Siegman, “An introduction to lasers, wave optics and Gaussian beams, complex paraxial wave optics, and unstable optical resonators,” in Lasers, A. Kelly, ed. (University Science Books, Sausalito, Calif., 1986), pp. 44, 626–635, 777–785, 858–890.

A. E. Siegman, Handbook of Laser Beam Propagation and Beam Quality Formulas Using the Spatial-Frequency and Intensity-Moments Analyses, draft version, 2 July 1991 (Edward L. Ginzton Laboratory, Stanford University, Palo Alto, Calif., personal communication).

J. W. Goodman, “Fresnel and Fraunhofer Diffraction,” in Introduction to Fourier Optics, 2nd ed., S. W. Director, ed. (McGraw-Hill, New York, 1996), pp. 63–95.

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Figures (5)

Fig. 1
Fig. 1

Equivalent-lens waveguide view of an SGC unstable cavity with inserted aperture to model solid-state gain-medium diffraction. The wave is propagated from the OC eigenplane (simulated by grid GOC) to the aperture plane (GAp) and the HR plane (GHR) and back, through segments S3 and S4.

Fig. 2
Fig. 2

Beam quality versus cavity magnification for n = 2, NGa = 8.5 and d/L = 0.6, using two methods: (a) M2 (measured with a 13.7-μm sampling interval) and (b) DFE analysis. (c) Neq integer positions in the same NAp, M range.

Fig. 3
Fig. 3

(a) M2 versus M for n = 6, NGa = 8.5, d/L = 0.6; measured with a 13.7-μm sampling interval. Normalized output intensity profiles from configurations in (b) the OC-dominated eigenmode region at point b and (c) the aperture-dominated eigenmode region at point c.

Fig. 4
Fig. 4

M2 versus d/L for two (M, n, NGa, NAp) sets, measured with a 15-μm sampling interval. Insets showing normalized intensity profiles, exiting configurations K and L, are similar owing to the preservation of the Neq value. The M2 value at d/L = 0.6 is 20% less than the corresponding point in Fig. 2(a), owing to a 10% sampling interval difference.

Fig. 5
Fig. 5

Simulated output with rod HR face diffraction (solid curve) compared with Nd:YAG near-field profile (circle size indicates data transcription uncertainty) for n = 2.8; wm = 2.2 mm; HR radius of curvature = −3 m; Dm = 6.4 mm; rod length = 76.2 mm, located near HR; and L = 375 mm5.

Tables (1)

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Table 1 Normalized Input Parameters for a Positive-Branch, Confocal Unstable Resonator2 (SGC)

Equations (4)

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U p + 1 ( w m ξ , w m η ) = S p U p ( w m ξ , w m η ) ,
S p = d ξ d η ρ p ( ξ ,     η ) K p ( ξ ,     η ,     ξ ,     η ) .
b min / 2 a ( 2 a / λ ) 1 / 3 .
d / L = G OC / ( G OC + G HR ) .

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