Abstract

The transverse-mode profiles for a plano–concave Nd:YVO4 microchip laser near threshold are examined, both experimentally and theoretically, and in both the near and the far field. We study in particular the transition from dominant quadratic index guiding to dominant gain guiding. The modal profiles change dramatically in this transition, and the agreement between experiment and the theoretical model is excellent.

© 2001 Optical Society of America

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References

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  1. J. J. Zayhowski and A. Mooradian, “Single-frequency microchip Nd lasers,” Opt. Lett. 14, 24–26 (1990).
    [CrossRef]
  2. J. J. Zayhowski, “Q-switched microchip lasers find real-world application,” Laser Focus World 35, August 1999, pp. 129–136).
  3. C. Becher and K.-J. Boller, “Low-intensity-noise operation of Nd:YVO4 microchip lasers by pump-noise suppression,” J. Opt. Soc. Am. B 16, 286–295 (1999).
    [CrossRef]
  4. N. J. van Druten, Y. Lien, C. Serrat, S. S. R. Oemrawsingh, M. P. van Exter, and J. P. Woerdman, “Laser with thresholdless intensity fluctuations,” Phys. Rev. A 62, 053808, 1–9 (2000).
    [CrossRef]
  5. J. J. Zayhowski, “Thermal guiding in microchip lasers,” in Advanced Solid-State Lasers, H. P. Jenssen and G. Dubé, eds., OSA Proceedings Series 6 (Optical Society of America, Washington, D.C., 1991), pp. 9–13.
  6. G. K. Harkness and W. J. Firth, “Transverse modes of microchip solid state lasers,” J. Mod. Opt. 39, 2023–2037 (1992).
    [CrossRef]
  7. T. Y. Fan, “Aperture guiding in quasi-three-level lasers,” Opt. Lett. 19, 554–556 (1994).
    [CrossRef] [PubMed]
  8. S. Longhi, “Theory of transverse modes in end-pumped microchip lasers,” J. Opt. Soc. Am. B 11, 1098–1107 (1994).
    [CrossRef]
  9. S. Longhi, G. Cerullo, S. Taccheo, V. Magni, and P. Laporta, “Experimental observation of transverse effects in microchip solid-state lasers,” Appl. Phys. Lett. 65, 3042–3044 (1994).
    [CrossRef]
  10. S. Longhi and P. Laporta, “Longitudinal-transverse mode interplay and conical emission in microchip lasers,” J. Opt. Soc. Am. B 12, 1511–1515 (1995).
    [CrossRef]
  11. F. Sanchez and A. Chardon, “Pump size optimization in microchip lasers,” Opt. Commun. 136, 405–409 (1997).
    [CrossRef]
  12. A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
    [CrossRef]
  13. C. Serrat, M. P. van Exter, N. J. van Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
    [CrossRef]
  14. Y.-J. Cheng, C. G. Fanning, and A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
    [CrossRef] [PubMed]
  15. A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
    [CrossRef] [PubMed]
  16. A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
    [CrossRef] [PubMed]
  17. M. P. van Exter, N. J. van Druten, A. M. van der Lee, S. M. Dutra, G. Nienhuis, and J. P. Woerdman, “Semi-classical dynamics of excess quantum noise,” Phys. Rev. A 63, 043801, 1–13 (2001).
    [CrossRef]
  18. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
  19. H. G. Danielmeyer, “Effects of drift and diffusion of excited states on spatial hole burning and laser oscillation,” J. Appl. Phys. 42, 3125–3132 (1971).
    [CrossRef]
  20. J. J. Zayhowski, “The effects of spatial hole burning and energy diffusion on the single-mode operation of standing-wave lasers,” IEEE J. Quantum Electron. 26, 2052–2057 (1990).
    [CrossRef]

2001 (1)

M. P. van Exter, N. J. van Druten, A. M. van der Lee, S. M. Dutra, G. Nienhuis, and J. P. Woerdman, “Semi-classical dynamics of excess quantum noise,” Phys. Rev. A 63, 043801, 1–13 (2001).
[CrossRef]

2000 (1)

N. J. van Druten, Y. Lien, C. Serrat, S. S. R. Oemrawsingh, M. P. van Exter, and J. P. Woerdman, “Laser with thresholdless intensity fluctuations,” Phys. Rev. A 62, 053808, 1–9 (2000).
[CrossRef]

1999 (4)

J. J. Zayhowski, “Q-switched microchip lasers find real-world application,” Laser Focus World 35, August 1999, pp. 129–136).

C. Becher and K.-J. Boller, “Low-intensity-noise operation of Nd:YVO4 microchip lasers by pump-noise suppression,” J. Opt. Soc. Am. B 16, 286–295 (1999).
[CrossRef]

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
[CrossRef]

C. Serrat, M. P. van Exter, N. J. van Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
[CrossRef]

1997 (1)

F. Sanchez and A. Chardon, “Pump size optimization in microchip lasers,” Opt. Commun. 136, 405–409 (1997).
[CrossRef]

1996 (1)

Y.-J. Cheng, C. G. Fanning, and A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[CrossRef] [PubMed]

1995 (1)

1994 (3)

T. Y. Fan, “Aperture guiding in quasi-three-level lasers,” Opt. Lett. 19, 554–556 (1994).
[CrossRef] [PubMed]

S. Longhi, “Theory of transverse modes in end-pumped microchip lasers,” J. Opt. Soc. Am. B 11, 1098–1107 (1994).
[CrossRef]

S. Longhi, G. Cerullo, S. Taccheo, V. Magni, and P. Laporta, “Experimental observation of transverse effects in microchip solid-state lasers,” Appl. Phys. Lett. 65, 3042–3044 (1994).
[CrossRef]

1992 (1)

G. K. Harkness and W. J. Firth, “Transverse modes of microchip solid state lasers,” J. Mod. Opt. 39, 2023–2037 (1992).
[CrossRef]

1990 (2)

J. J. Zayhowski and A. Mooradian, “Single-frequency microchip Nd lasers,” Opt. Lett. 14, 24–26 (1990).
[CrossRef]

J. J. Zayhowski, “The effects of spatial hole burning and energy diffusion on the single-mode operation of standing-wave lasers,” IEEE J. Quantum Electron. 26, 2052–2057 (1990).
[CrossRef]

1989 (2)

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
[CrossRef] [PubMed]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[CrossRef] [PubMed]

1971 (1)

H. G. Danielmeyer, “Effects of drift and diffusion of excited states on spatial hole burning and laser oscillation,” J. Appl. Phys. 42, 3125–3132 (1971).
[CrossRef]

Becher, C.

Boller, K.-J.

Cerullo, G.

S. Longhi, G. Cerullo, S. Taccheo, V. Magni, and P. Laporta, “Experimental observation of transverse effects in microchip solid-state lasers,” Appl. Phys. Lett. 65, 3042–3044 (1994).
[CrossRef]

Chardon, A.

F. Sanchez and A. Chardon, “Pump size optimization in microchip lasers,” Opt. Commun. 136, 405–409 (1997).
[CrossRef]

Cheng, Y.-J.

Y.-J. Cheng, C. G. Fanning, and A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[CrossRef] [PubMed]

Conroy, R. S.

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
[CrossRef]

Danielmeyer, H. G.

H. G. Danielmeyer, “Effects of drift and diffusion of excited states on spatial hole burning and laser oscillation,” J. Appl. Phys. 42, 3125–3132 (1971).
[CrossRef]

Dutra, S. M.

M. P. van Exter, N. J. van Druten, A. M. van der Lee, S. M. Dutra, G. Nienhuis, and J. P. Woerdman, “Semi-classical dynamics of excess quantum noise,” Phys. Rev. A 63, 043801, 1–13 (2001).
[CrossRef]

Fan, T. Y.

Fanning, C. G.

Y.-J. Cheng, C. G. Fanning, and A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[CrossRef] [PubMed]

Firth, W. J.

G. K. Harkness and W. J. Firth, “Transverse modes of microchip solid state lasers,” J. Mod. Opt. 39, 2023–2037 (1992).
[CrossRef]

Friel, G. J.

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
[CrossRef]

Harkness, G. K.

G. K. Harkness and W. J. Firth, “Transverse modes of microchip solid state lasers,” J. Mod. Opt. 39, 2023–2037 (1992).
[CrossRef]

Kemp, A. J.

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
[CrossRef]

Laporta, P.

S. Longhi and P. Laporta, “Longitudinal-transverse mode interplay and conical emission in microchip lasers,” J. Opt. Soc. Am. B 12, 1511–1515 (1995).
[CrossRef]

S. Longhi, G. Cerullo, S. Taccheo, V. Magni, and P. Laporta, “Experimental observation of transverse effects in microchip solid-state lasers,” Appl. Phys. Lett. 65, 3042–3044 (1994).
[CrossRef]

Lien, Y.

N. J. van Druten, Y. Lien, C. Serrat, S. S. R. Oemrawsingh, M. P. van Exter, and J. P. Woerdman, “Laser with thresholdless intensity fluctuations,” Phys. Rev. A 62, 053808, 1–9 (2000).
[CrossRef]

Longhi, S.

Magni, V.

S. Longhi, G. Cerullo, S. Taccheo, V. Magni, and P. Laporta, “Experimental observation of transverse effects in microchip solid-state lasers,” Appl. Phys. Lett. 65, 3042–3044 (1994).
[CrossRef]

Mooradian, A.

Nienhuis, G.

M. P. van Exter, N. J. van Druten, A. M. van der Lee, S. M. Dutra, G. Nienhuis, and J. P. Woerdman, “Semi-classical dynamics of excess quantum noise,” Phys. Rev. A 63, 043801, 1–13 (2001).
[CrossRef]

Oemrawsingh, S. S. R.

N. J. van Druten, Y. Lien, C. Serrat, S. S. R. Oemrawsingh, M. P. van Exter, and J. P. Woerdman, “Laser with thresholdless intensity fluctuations,” Phys. Rev. A 62, 053808, 1–9 (2000).
[CrossRef]

Sanchez, F.

F. Sanchez and A. Chardon, “Pump size optimization in microchip lasers,” Opt. Commun. 136, 405–409 (1997).
[CrossRef]

Serrat, C.

N. J. van Druten, Y. Lien, C. Serrat, S. S. R. Oemrawsingh, M. P. van Exter, and J. P. Woerdman, “Laser with thresholdless intensity fluctuations,” Phys. Rev. A 62, 053808, 1–9 (2000).
[CrossRef]

C. Serrat, M. P. van Exter, N. J. van Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
[CrossRef]

Siegman, A. E.

Y.-J. Cheng, C. G. Fanning, and A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[CrossRef] [PubMed]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
[CrossRef] [PubMed]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[CrossRef] [PubMed]

Sinclair, B. D.

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
[CrossRef]

Taccheo, S.

S. Longhi, G. Cerullo, S. Taccheo, V. Magni, and P. Laporta, “Experimental observation of transverse effects in microchip solid-state lasers,” Appl. Phys. Lett. 65, 3042–3044 (1994).
[CrossRef]

van der Lee, A. M.

M. P. van Exter, N. J. van Druten, A. M. van der Lee, S. M. Dutra, G. Nienhuis, and J. P. Woerdman, “Semi-classical dynamics of excess quantum noise,” Phys. Rev. A 63, 043801, 1–13 (2001).
[CrossRef]

van Druten, N. J.

M. P. van Exter, N. J. van Druten, A. M. van der Lee, S. M. Dutra, G. Nienhuis, and J. P. Woerdman, “Semi-classical dynamics of excess quantum noise,” Phys. Rev. A 63, 043801, 1–13 (2001).
[CrossRef]

N. J. van Druten, Y. Lien, C. Serrat, S. S. R. Oemrawsingh, M. P. van Exter, and J. P. Woerdman, “Laser with thresholdless intensity fluctuations,” Phys. Rev. A 62, 053808, 1–9 (2000).
[CrossRef]

C. Serrat, M. P. van Exter, N. J. van Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
[CrossRef]

van Exter, M. P.

M. P. van Exter, N. J. van Druten, A. M. van der Lee, S. M. Dutra, G. Nienhuis, and J. P. Woerdman, “Semi-classical dynamics of excess quantum noise,” Phys. Rev. A 63, 043801, 1–13 (2001).
[CrossRef]

N. J. van Druten, Y. Lien, C. Serrat, S. S. R. Oemrawsingh, M. P. van Exter, and J. P. Woerdman, “Laser with thresholdless intensity fluctuations,” Phys. Rev. A 62, 053808, 1–9 (2000).
[CrossRef]

C. Serrat, M. P. van Exter, N. J. van Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
[CrossRef]

Woerdman, J. P.

M. P. van Exter, N. J. van Druten, A. M. van der Lee, S. M. Dutra, G. Nienhuis, and J. P. Woerdman, “Semi-classical dynamics of excess quantum noise,” Phys. Rev. A 63, 043801, 1–13 (2001).
[CrossRef]

N. J. van Druten, Y. Lien, C. Serrat, S. S. R. Oemrawsingh, M. P. van Exter, and J. P. Woerdman, “Laser with thresholdless intensity fluctuations,” Phys. Rev. A 62, 053808, 1–9 (2000).
[CrossRef]

C. Serrat, M. P. van Exter, N. J. van Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
[CrossRef]

Zayhowski, J. J.

J. J. Zayhowski, “Q-switched microchip lasers find real-world application,” Laser Focus World 35, August 1999, pp. 129–136).

J. J. Zayhowski and A. Mooradian, “Single-frequency microchip Nd lasers,” Opt. Lett. 14, 24–26 (1990).
[CrossRef]

J. J. Zayhowski, “The effects of spatial hole burning and energy diffusion on the single-mode operation of standing-wave lasers,” IEEE J. Quantum Electron. 26, 2052–2057 (1990).
[CrossRef]

Appl. Phys. Lett. (1)

S. Longhi, G. Cerullo, S. Taccheo, V. Magni, and P. Laporta, “Experimental observation of transverse effects in microchip solid-state lasers,” Appl. Phys. Lett. 65, 3042–3044 (1994).
[CrossRef]

IEEE J. Quantum Electron. (3)

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
[CrossRef]

C. Serrat, M. P. van Exter, N. J. van Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999).
[CrossRef]

J. J. Zayhowski, “The effects of spatial hole burning and energy diffusion on the single-mode operation of standing-wave lasers,” IEEE J. Quantum Electron. 26, 2052–2057 (1990).
[CrossRef]

J. Appl. Phys. (1)

H. G. Danielmeyer, “Effects of drift and diffusion of excited states on spatial hole burning and laser oscillation,” J. Appl. Phys. 42, 3125–3132 (1971).
[CrossRef]

J. Mod. Opt. (1)

G. K. Harkness and W. J. Firth, “Transverse modes of microchip solid state lasers,” J. Mod. Opt. 39, 2023–2037 (1992).
[CrossRef]

J. Opt. Soc. Am. B (3)

Laser Focus World (1)

J. J. Zayhowski, “Q-switched microchip lasers find real-world application,” Laser Focus World 35, August 1999, pp. 129–136).

Opt. Commun. (1)

F. Sanchez and A. Chardon, “Pump size optimization in microchip lasers,” Opt. Commun. 136, 405–409 (1997).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (4)

N. J. van Druten, Y. Lien, C. Serrat, S. S. R. Oemrawsingh, M. P. van Exter, and J. P. Woerdman, “Laser with thresholdless intensity fluctuations,” Phys. Rev. A 62, 053808, 1–9 (2000).
[CrossRef]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
[CrossRef] [PubMed]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[CrossRef] [PubMed]

M. P. van Exter, N. J. van Druten, A. M. van der Lee, S. M. Dutra, G. Nienhuis, and J. P. Woerdman, “Semi-classical dynamics of excess quantum noise,” Phys. Rev. A 63, 043801, 1–13 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

Y.-J. Cheng, C. G. Fanning, and A. E. Siegman, “Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes,” Phys. Rev. Lett. 77, 627–630 (1996).
[CrossRef] [PubMed]

Other (2)

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

J. J. Zayhowski, “Thermal guiding in microchip lasers,” in Advanced Solid-State Lasers, H. P. Jenssen and G. Dubé, eds., OSA Proceedings Series 6 (Optical Society of America, Washington, D.C., 1991), pp. 9–13.

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Figures (9)

Fig. 1
Fig. 1

(a) Cavity configuration considered in the theoretical model: a longitudinally homogeneous system of length L, with a transverse quadratic index guide ΔnR and a Gaussian gain guide Δng. The arrows in this figure indicate the output of the microchip laser through the end mirror with reflectivity Rm. Arrangements (b) and (c) are equivalent configurations13 as long as the longitudinal variation of the mode profile may be neglected. (b) Monolithic cavity with a gain guide Δng and a curved surface. (c) Cavity configuration used in the experiments: a crystal gain medium with a transverse Gaussian gain profile and a separate concave mirror, with radius of curvature Rc and reflectivity Rm.

Fig. 2
Fig. 2

Transverse profile of the real (dashed curve) and imaginary (dotted curve) part of the guiding potential VΔn, for wg/w0=0.3, and for the detuning Δω, where the modal gain needed to reach threshold is lowest. The transverse mode profile of the lowest-loss eigenmode (solid curve, with the vertical scale adjusted in each plot), is shown for comparison. Left, Im μ=0.36 (corresponding to Rm=98% in the experiment); the parabolic part (the mirror curvature) of the real index guide dominates. Right, Im μ=6.3 (Rm=70%); the Gaussian guide dominates, as it confines the mode to a region where the parabolic index guide is relatively small. Note the difference in vertical scale (for V) between the two plots.

Fig. 3
Fig. 3

Experimental setup of the microchip laser. The plano–concave configuration is longitudinally pumped by a titanium–sapphire laser at 808 nm. The crystal is ∼200 µm thick with a refractive index of 2.2, and the air gap has a length of 0.1 mm. The radius of curvature of the output coupler is 200 mm. We can vary the reflectivity by replacing the output coupler with another one with the same radius of curvature.

Fig. 4
Fig. 4

Left, theoretical calculation of the threshold of the modes against the cavity detuning for various reflectivities of the outcoupling mirror. Right, experimental results that show the laser intensity output against the cavity detuning. The cavity detuning is measured in terms of the voltage ΔV on the piezo-electric transducer, normalized to the voltage required for scanning a full free spectral range ΔVFSR. When in the calculation the laser jumps from one mode to the next, the experiment should show a small discontinuity as well.

Fig. 5
Fig. 5

Left, near-field and, right, far-field mode profile at reflectivity Rm=98%, for the cavity detuning corresponding to the lowest threshold. The solid curves are the experimental data, the dashed curves the result of the theoretical model. The index guide dominates (cf. Fig. 2), and the profile deviates very little from the profile of the lowest-order Laguerre–Gaussian mode LG0 of the purely parabolic index guide (both in the near field and in the far field). The LG0 mode is shown as the dotted curve in the near field; in the far field the LG0 profile is not shown, as it would overlap the curve of the theoretical model.

Fig. 6
Fig. 6

Left, near-field and, right, far-field mode profiles at reflectivity Rm=94%. Three modes are found in the experiment (solid curves). These profiles match the results from the theoretical model (dashed curves) quite well. In contrast, both deviate from the three lowest-order Laguerre–Gaussian TEM modes of the parabolic index guide (LGN, dotted curves); the near-field rings are weaker, and the far-field rings are stronger than those of the LGN modes, and the intensity between rings does not go to zero.

Fig. 7
Fig. 7

Left, near-field and, right, far-field mode profiles at reflectivity Rm=90%. Four modes are found in the experiment (solid curves), and the profiles agree well with the results from the theoretical model (dashed curves), both in the near field and in the far field. For the higher-order modes the deviations from Laguerre–Gaussian modes are large: the near-field rings are strongly suppressed while the far-field rings become quite strong compared with the central peak. A direct comparison with the LGN modes (dotted curves) is shown for the first three modes.

Fig. 8
Fig. 8

Near-field and far-field profiles for Rm=80% mirror reflectivity. The experimental data (solid curves) are again reproduced well by the theoretical model (dashed curves). The near-field profile changes only in width, and we show only one profile. The far-field profile changes much more drastically; a number of representative examples are shown. The value of the cavity detuning 2ΔL/λat (cf. Fig. 4) of the theoretical curves is indicated in the graphs of the far-field profiles. The lowest-order Laguerre–Gaussian mode LG0 of the purely parabolic index guide is also shown for comparison (dotted curves).

Fig. 9
Fig. 9

Near-field and far-field profiles for mirror reflectivity Rm=70%. The gain guide dominates (cf. Fig. 2) and is stronger than for Rm=80%; the behavior shown in Fig. 8 becomes even more pronounced. The near-field profile changes only in width as the cavity detuning is changed, and only one example is shown. Representative examples of the far-field modal profiles illustrate how the maximum intensity moves off axis as the cavity detuning is changed. The value of the cavity detuning (2ΔL/λat, cf. Fig. 4) is indicated for each far-field profile.

Tables (1)

Tables Icon

Table 1 Overview of the Experimental Parameters and the Corresponding Values of the Derived Theoretical Parameters

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

r2+2kz2Δωωat+Δnn0u=-2ikz uz,
r2=2r2+1r r+1r2 2ϕ2
E(r)=u(r)exp-i(ωt-kzz).
ΔnR(r)=-2n0r2kz2w04,
Δng(r, Δω)=-cg(r, Δω)2ωat,
g(r, Δω)=g0 exp(-2r2/wg2) i-Δω/γat1+(Δω/γat)2.
-12ρ2+2ρ2+z0gu=2iz0uz-in0Δωcu,
ρ2=2ρ2+1ρ ρ+1ρ2 2ϕ2,
uμ(ρ, z)=uμ(ρ, 0)exp[i(n0Δω/c-μ/2z0)z].
RmE0(z+2L)=E0(z).
μ=2z0kz+n0Δωc-πmL-i4L ln Rm,
2ΔLλat=-ΔωωFSR+m˜+L0 Re μ2πz0,
I(θ)0u(ρ)J0(kzw0ρθ)dρ2,
n0=n1n2n1L1+n2L2n2L1+n1L21/2,
L=L1n1+L2n2(n1L1+n2L2)1/2,
w02=2kz LRcn01/2.

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