Abstract

We present a theoretical and experimental investigation of a new family of variable reflectivity mirrors, based on frustrated total internal reflection and interference effects. These mirrors are sensitive to frequency in the sense that their transverse reflectivity distribution changes significantly as a function of the frequency of light. The mirror reported here shows total power reflectivity changes of 50% within 8.0 GHz. The mirror was tested as the output coupler of an unstable resonator in a Nd:YAG laser working in the free-running regime. This configuration was compared with the standard stable multimode resonator configuration. The beam divergence was reduced by more than 1 order of magnitude and the output power was reduced by only 10%. The laser resonator mode competition that is due to the frequency-dependent mirror reflectivity distribution is discussed.

© 1999 Optical Society of America

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References

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    [CrossRef]
  2. C. Giuri, M. R. Perrone, D. Flori, A. Piegari, S. Scaglione, “Phase shift of stepwise reflectivity profile mirrors,” Appl. Opt. 36, 2495–2498 (1997).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  5. Y. Takenaka, M. Kusumoto, K. Yasui, “A 5 kW CO2 laser using a novel negative-branch unstable resonator with a phase-unifying output coupler,” IEEE J. Quantum Electron. 28, 1855–1858 (1992).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  11. M. Keselbrener, S. Ruschin, B. Lissak, A. Gover, “Numerical studies of resonators with on-axis holes for FEL applications,” Nucl. Instrum. Methods A 304, 782–786 (1991).
    [CrossRef]
  12. S. Ruschin, T. Hurvits, M. Keselbrener, “Properties of the transverse eigenmode set in optical resonators with apertures: comment,” J. Opt. Soc. Am. A 13, 1287–1290 (1996).
    [CrossRef]
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    [CrossRef]
  14. M. Keselbrener, S. Ruschin, “Characterization of solid state laser with time dependent pump,” presented at the Tenth Meeting on Optical Engineering, 2–6 March 1997, Jerusalem. Program and abstracts available from Electro-Optics Industries, Kyriat Weizmann P.O. Box 1165, Rehovot 76111, Israel.
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]

1997 (1)

1996 (1)

1993 (1)

1992 (1)

Y. Takenaka, M. Kusumoto, K. Yasui, “A 5 kW CO2 laser using a novel negative-branch unstable resonator with a phase-unifying output coupler,” IEEE J. Quantum Electron. 28, 1855–1858 (1992).
[CrossRef]

1991 (1)

M. Keselbrener, S. Ruschin, B. Lissak, A. Gover, “Numerical studies of resonators with on-axis holes for FEL applications,” Nucl. Instrum. Methods A 304, 782–786 (1991).
[CrossRef]

1990 (2)

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Laser with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

X. Yasui, S. Yagi, M. Tanaka, “Negative-branch unstable resonator with a phase unifying output coupler for high Nd:YAG lasers,” Appl. Opt. 29, 1277–1280 (1990).
[CrossRef] [PubMed]

1987 (1)

1985 (1)

1981 (2)

1980 (1)

1977 (1)

1975 (1)

U. Ganiel, A. Hardy, Y. Silberberg, “Stability of optical laser resonator with mirrors of Gaussian reflectivity profiles, which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
[CrossRef]

1966 (1)

A. G. Fox, T. Li, “Effects of gain saturation on the oscillating modes of optical masers,” IEEE J. Quantum Electron. QE-2, 774–783 (1966).
[CrossRef]

1961 (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Agrawal, G. P.

Armandillo, E.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Byer, R. L.

De Silvestri, S.

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Laser with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Radially variable reflectivity output coupler of novel design for unstable resonators,” Opt. Lett. 12, 84–86 (1987).
[CrossRef] [PubMed]

Eggleston, J. M.

Emiliani, G.

Ermakov, A. A.

S. G. Lukishova, S. A. Kovtonuk, A. A. Ermakov, P. P. Pashinin, E. E. Plavtov, A. S. Svakhin, A. A. Golubsky, “Dielectric films deposition with cross-section variable thickness for amplitude filters on the basis of frustrated total internal reflection,” in Optical Thin Films and Applications, R. Herrmann, ed. Proc. SPIE1270, 260–271 (1990).
[CrossRef]

Flori, D.

Fox, A. G.

A. G. Fox, T. Li, “Effects of gain saturation on the oscillating modes of optical masers,” IEEE J. Quantum Electron. QE-2, 774–783 (1966).
[CrossRef]

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Ganiel, U.

U. Ganiel, A. Hardy, Y. Silberberg, “Stability of optical laser resonator with mirrors of Gaussian reflectivity profiles, which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
[CrossRef]

Giuliani, G.

Giuri, C.

Golubsky, A. A.

S. G. Lukishova, S. A. Kovtonuk, A. A. Ermakov, P. P. Pashinin, E. E. Plavtov, A. S. Svakhin, A. A. Golubsky, “Dielectric films deposition with cross-section variable thickness for amplitude filters on the basis of frustrated total internal reflection,” in Optical Thin Films and Applications, R. Herrmann, ed. Proc. SPIE1270, 260–271 (1990).
[CrossRef]

Gover, A.

M. Keselbrener, S. Ruschin, B. Lissak, A. Gover, “Numerical studies of resonators with on-axis holes for FEL applications,” Nucl. Instrum. Methods A 304, 782–786 (1991).
[CrossRef]

Hardy, A.

A. Hardy, “Gaussian modes of resonators containing saturable gain medium,” Appl. Opt. 19, 3830–3836 (1980).
[CrossRef] [PubMed]

U. Ganiel, A. Hardy, Y. Silberberg, “Stability of optical laser resonator with mirrors of Gaussian reflectivity profiles, which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
[CrossRef]

Hodgson, N.

N. Hodgson, W. Weber, Optical Resonator: Fundamentals, Advanced Concepts and Applications (Springer-Verlag, New York, 1997).
[CrossRef]

Hurvits, T.

Keselbrener, M.

S. Ruschin, T. Hurvits, M. Keselbrener, “Properties of the transverse eigenmode set in optical resonators with apertures: comment,” J. Opt. Soc. Am. A 13, 1287–1290 (1996).
[CrossRef]

M. Keselbrener, S. Ruschin, B. Lissak, A. Gover, “Numerical studies of resonators with on-axis holes for FEL applications,” Nucl. Instrum. Methods A 304, 782–786 (1991).
[CrossRef]

M. Keselbrener, S. Ruschin, “Characterization of solid state laser with time dependent pump,” presented at the Tenth Meeting on Optical Engineering, 2–6 March 1997, Jerusalem. Program and abstracts available from Electro-Optics Industries, Kyriat Weizmann P.O. Box 1165, Rehovot 76111, Israel.

Kovtonuk, S. A.

S. G. Lukishova, S. A. Kovtonuk, A. A. Ermakov, P. P. Pashinin, E. E. Plavtov, A. S. Svakhin, A. A. Golubsky, “Dielectric films deposition with cross-section variable thickness for amplitude filters on the basis of frustrated total internal reflection,” in Optical Thin Films and Applications, R. Herrmann, ed. Proc. SPIE1270, 260–271 (1990).
[CrossRef]

Kusumoto, M.

Y. Takenaka, M. Kusumoto, K. Yasui, “A 5 kW CO2 laser using a novel negative-branch unstable resonator with a phase-unifying output coupler,” IEEE J. Quantum Electron. 28, 1855–1858 (1992).
[CrossRef]

Laporta, P.

Lax, M.

Li, T.

A. G. Fox, T. Li, “Effects of gain saturation on the oscillating modes of optical masers,” IEEE J. Quantum Electron. QE-2, 774–783 (1966).
[CrossRef]

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Lissak, B.

M. Keselbrener, S. Ruschin, B. Lissak, A. Gover, “Numerical studies of resonators with on-axis holes for FEL applications,” Nucl. Instrum. Methods A 304, 782–786 (1991).
[CrossRef]

Lukishova, S. G.

S. G. Lukishova, S. A. Kovtonuk, A. A. Ermakov, P. P. Pashinin, E. E. Plavtov, A. S. Svakhin, A. A. Golubsky, “Dielectric films deposition with cross-section variable thickness for amplitude filters on the basis of frustrated total internal reflection,” in Optical Thin Films and Applications, R. Herrmann, ed. Proc. SPIE1270, 260–271 (1990).
[CrossRef]

Magni, V.

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Laser with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Radially variable reflectivity output coupler of novel design for unstable resonators,” Opt. Lett. 12, 84–86 (1987).
[CrossRef] [PubMed]

Pashinin, P. P.

S. G. Lukishova, S. A. Kovtonuk, A. A. Ermakov, P. P. Pashinin, E. E. Plavtov, A. S. Svakhin, A. A. Golubsky, “Dielectric films deposition with cross-section variable thickness for amplitude filters on the basis of frustrated total internal reflection,” in Optical Thin Films and Applications, R. Herrmann, ed. Proc. SPIE1270, 260–271 (1990).
[CrossRef]

Perrone, M. R.

Piegari, A.

Plavtov, E. E.

S. G. Lukishova, S. A. Kovtonuk, A. A. Ermakov, P. P. Pashinin, E. E. Plavtov, A. S. Svakhin, A. A. Golubsky, “Dielectric films deposition with cross-section variable thickness for amplitude filters on the basis of frustrated total internal reflection,” in Optical Thin Films and Applications, R. Herrmann, ed. Proc. SPIE1270, 260–271 (1990).
[CrossRef]

Ruschin, S.

S. Ruschin, T. Hurvits, M. Keselbrener, “Properties of the transverse eigenmode set in optical resonators with apertures: comment,” J. Opt. Soc. Am. A 13, 1287–1290 (1996).
[CrossRef]

M. Keselbrener, S. Ruschin, B. Lissak, A. Gover, “Numerical studies of resonators with on-axis holes for FEL applications,” Nucl. Instrum. Methods A 304, 782–786 (1991).
[CrossRef]

M. Keselbrener, S. Ruschin, “Characterization of solid state laser with time dependent pump,” presented at the Tenth Meeting on Optical Engineering, 2–6 March 1997, Jerusalem. Program and abstracts available from Electro-Optics Industries, Kyriat Weizmann P.O. Box 1165, Rehovot 76111, Israel.

Scaglione, S.

Siegman, A. E.

Silberberg, Y.

U. Ganiel, A. Hardy, Y. Silberberg, “Stability of optical laser resonator with mirrors of Gaussian reflectivity profiles, which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
[CrossRef]

Svakhin, A. S.

S. G. Lukishova, S. A. Kovtonuk, A. A. Ermakov, P. P. Pashinin, E. E. Plavtov, A. S. Svakhin, A. A. Golubsky, “Dielectric films deposition with cross-section variable thickness for amplitude filters on the basis of frustrated total internal reflection,” in Optical Thin Films and Applications, R. Herrmann, ed. Proc. SPIE1270, 260–271 (1990).
[CrossRef]

Svelto, O.

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Laser with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Radially variable reflectivity output coupler of novel design for unstable resonators,” Opt. Lett. 12, 84–86 (1987).
[CrossRef] [PubMed]

Takenaka, Y.

Y. Takenaka, M. Kusumoto, K. Yasui, “A 5 kW CO2 laser using a novel negative-branch unstable resonator with a phase-unifying output coupler,” IEEE J. Quantum Electron. 28, 1855–1858 (1992).
[CrossRef]

Tanaka, M.

Valentini, G.

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Laser with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

Weber, W.

N. Hodgson, W. Weber, Optical Resonator: Fundamentals, Advanced Concepts and Applications (Springer-Verlag, New York, 1997).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Yagi, S.

Yasui, K.

Y. Takenaka, M. Kusumoto, K. Yasui, “A 5 kW CO2 laser using a novel negative-branch unstable resonator with a phase-unifying output coupler,” IEEE J. Quantum Electron. 28, 1855–1858 (1992).
[CrossRef]

Yasui, X.

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

IEEE J. Quantum Electron. (3)

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Laser with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

Y. Takenaka, M. Kusumoto, K. Yasui, “A 5 kW CO2 laser using a novel negative-branch unstable resonator with a phase-unifying output coupler,” IEEE J. Quantum Electron. 28, 1855–1858 (1992).
[CrossRef]

A. G. Fox, T. Li, “Effects of gain saturation on the oscillating modes of optical masers,” IEEE J. Quantum Electron. QE-2, 774–783 (1966).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Nucl. Instrum. Methods A (1)

M. Keselbrener, S. Ruschin, B. Lissak, A. Gover, “Numerical studies of resonators with on-axis holes for FEL applications,” Nucl. Instrum. Methods A 304, 782–786 (1991).
[CrossRef]

Opt. Commun. (1)

U. Ganiel, A. Hardy, Y. Silberberg, “Stability of optical laser resonator with mirrors of Gaussian reflectivity profiles, which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
[CrossRef]

Opt. Lett. (4)

Other (4)

N. Hodgson, W. Weber, Optical Resonator: Fundamentals, Advanced Concepts and Applications (Springer-Verlag, New York, 1997).
[CrossRef]

M. Keselbrener, S. Ruschin, “Characterization of solid state laser with time dependent pump,” presented at the Tenth Meeting on Optical Engineering, 2–6 March 1997, Jerusalem. Program and abstracts available from Electro-Optics Industries, Kyriat Weizmann P.O. Box 1165, Rehovot 76111, Israel.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

S. G. Lukishova, S. A. Kovtonuk, A. A. Ermakov, P. P. Pashinin, E. E. Plavtov, A. S. Svakhin, A. A. Golubsky, “Dielectric films deposition with cross-section variable thickness for amplitude filters on the basis of frustrated total internal reflection,” in Optical Thin Films and Applications, R. Herrmann, ed. Proc. SPIE1270, 260–271 (1990).
[CrossRef]

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Figures (17)

Fig. 1
Fig. 1

(a) Fox–Smith interferometer based on frustrated total internal reflection. (b) VRM based on this principle.

Fig. 2
Fig. 2

Power reflectivity as a function of wavelength shift (in angstroms) for different widths h 0. The zero corresponds to λ = 1.06 µm. The prism side dimension is d = 12.7 mm, giving a free spectral range of approximately 8.0 GHz.

Fig. 3
Fig. 3

(a) Power reflectivity (R) and transmissivity (T) of the parallel air gap interface (TE polarization). (b) Corresponding phase shift.

Fig. 4
Fig. 4

One-dimensional mirror reflectivities at different wavelengths. Here the prism interface is spherical, with a curvature of 10.0 m, and is at a distance of h 0 = 0.1 µm from the second prism. The wavelength shift between two plots approximately is 600 MHz. The full horizontal scale axis is 12.7 mm.

Fig. 5
Fig. 5

Experimental setup for characterization of the graded interferometric wave mirror.

Fig. 6
Fig. 6

Theoretical (upper) and experimental (lower) two-dimensional distributions of the power transmissivity as a function of light frequency. The frequency step between each plot is approximately 600 MHz.

Fig. 7
Fig. 7

Absolute power reflectivity as a function of frequency: (a) experimental and (b) theoretical results.

Fig. 8
Fig. 8

(a) Experimental laser resonator configuration and (b) equivalent cavity for the Fox–Li simulation.

Fig. 9
Fig. 9

Gain aperture with a super-Gaussian distribution of order n = 4 and width ω0 = 0.00250 m.

Fig. 10
Fig. 10

Lowest mode losses as a function of frequency for three different radii of curvature of the prism interface.

Fig. 11
Fig. 11

(a) Empty resonator loss and (b) a super-Gaussian distribution of the medium aperture.

Fig. 12
Fig. 12

Calculated reflectivity (continuous curve) and transmissivity (dashed curve) of the frequency-dependent GRM: (a) squared amplitudes and (b) the phase. Note that in (b) the two curves are identical.

Fig. 13
Fig. 13

Solution of the Fox–Li calculation of lowest loss mode in an empty cavity: (a) power and (b) phase distribution of the near field at the output coupler plane inside (continuous curve) and outside (dashed curve) the resonator.

Fig. 14
Fig. 14

Near-field distributions: (a) stable multimode (measured), (b) unstable resonator including the frequency-dependent GRM (measured), (c) unstable resonator with the frequency-dependent GRM (calculated). The near-field diameter distribution is approximately 5.0 mm in the three cases. In the calculated profile we introduced astigmatism by different scaling of the x and y axes (1:2).

Fig. 15
Fig. 15

Intracavity power distribution (a) without and (b) with gain.

Fig. 16
Fig. 16

Measured far-field distributions: (left) stable multimode and (right) unstable resonator with a VRM. The far-field view is approximately 10 mrad in the two cases. The full scale in both pictures corresponds to an angular spread of 10 mrad.

Fig. 17
Fig. 17

Transverse intensity distribution calculated with near-field propagation. The calculation was performed for six propagation distances. The plot (3) also includes the experimentally measured intensity distribution.

Equations (10)

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

rbk=tt expiδ1-r2 expiδ,
δ=-2πλ02dn3.
r=r12+r23 exp2iβ1+r12r23 exp2iβ,
t=t12t23 expiβ1+r12r23 exp2iβ,
β=2πλ0 n2h cos θ2,
n12=n2/n1,
cos θ2=isin2 θ1n122-11/2
sin θ2=sin θ1/n12,
tr=exp jkr2/R2,
tr=expgl1=expg0l11+I1r+I2rIs,

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