Abstract

A simple and versatile laser output coupler with a radially variable reflectivity profile is described. This device, based on a radially variable Fabry–Perot interferometer made of two suitable spherical mirrors, provides a variety of reflectivity profiles that are particularly useful for generating large-cross-section diffraction-limited beams with unstable resonators.

© 1987 Optical Society of America

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References

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  1. N. G. Vakhimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electr. Phys. 10, 1439–1446 (1965).
  2. H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).
  3. G. L. McAllister, W. H. Steier, W. B. Lacina, “Improved mode properties of unstable resonators with tapered reflectivity mirrors and shaped apertures,” IEEE J. Quantum Electron. QE-10, 346–355 (1974).
    [CrossRef]
  4. L. W. Casperson, S. D. Lunnam, “Gaussian modes in high loss laser resonators,” Appl. Opt. 14, 1193–1199 (1975).
    [CrossRef] [PubMed]
  5. A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
    [CrossRef]
  6. G. Giuliani, Y. K. Park, R. L. Byer, “Radial birefringent element and its application to a Nd:YAG resonator,” Opt. Lett. 5, 491–493 (1980).
    [CrossRef] [PubMed]
  7. E. Armandillo, G. Giuliani, “Achievement of large-sized TEM00mode from an excimer laser by means of a novel apoditic filter,” Opt. Lett. 10, 445–447 (1985).
    [CrossRef] [PubMed]
  8. N. McCarthy, P. Lavigne, “Large-size Gaussian mode in unstable resonators using Gaussian mirrors,” Opt. Lett. 10, 553–555 (1985).
    [CrossRef] [PubMed]
  9. W. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 325.
  10. A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE 53, 277–287 (1965).
    [CrossRef]
  11. A. E. Siegman, “Stabilizing output with unstable resonators,” Laser Focus 7(5), 42–47 (1971).
  12. N. McCarthy, P. Lavigne, “Optical resonators with Gaussian reflectivity mirrors: output beam characteristics,” Appl. Opt. 23, 3845–3850 (1984).
    [CrossRef] [PubMed]
  13. S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Radially variable reflectivity output coupler of novel design for unstable resonators,” Opt. Lett. 12, 84–87 (1987).
    [CrossRef] [PubMed]

1987 (1)

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Radially variable reflectivity output coupler of novel design for unstable resonators,” Opt. Lett. 12, 84–87 (1987).
[CrossRef] [PubMed]

1985 (2)

E. Armandillo, G. Giuliani, “Achievement of large-sized TEM00mode from an excimer laser by means of a novel apoditic filter,” Opt. Lett. 10, 445–447 (1985).
[CrossRef] [PubMed]

N. McCarthy, P. Lavigne, “Large-size Gaussian mode in unstable resonators using Gaussian mirrors,” Opt. Lett. 10, 553–555 (1985).
[CrossRef] [PubMed]

1984 (1)

N. McCarthy, P. Lavigne, “Optical resonators with Gaussian reflectivity mirrors: output beam characteristics,” Appl. Opt. 23, 3845–3850 (1984).
[CrossRef] [PubMed]

1980 (1)

G. Giuliani, Y. K. Park, R. L. Byer, “Radial birefringent element and its application to a Nd:YAG resonator,” Opt. Lett. 5, 491–493 (1980).
[CrossRef] [PubMed]

1975 (2)

L. W. Casperson, S. D. Lunnam, “Gaussian modes in high loss laser resonators,” Appl. Opt. 14, 1193–1199 (1975).
[CrossRef] [PubMed]

A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

1974 (1)

G. L. McAllister, W. H. Steier, W. B. Lacina, “Improved mode properties of unstable resonators with tapered reflectivity mirrors and shaped apertures,” IEEE J. Quantum Electron. QE-10, 346–355 (1974).
[CrossRef]

1971 (1)

A. E. Siegman, “Stabilizing output with unstable resonators,” Laser Focus 7(5), 42–47 (1971).

1970 (1)

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).

1965 (2)

N. G. Vakhimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electr. Phys. 10, 1439–1446 (1965).

A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE 53, 277–287 (1965).
[CrossRef]

Armandillo, E.

E. Armandillo, G. Giuliani, “Achievement of large-sized TEM00mode from an excimer laser by means of a novel apoditic filter,” Opt. Lett. 10, 445–447 (1985).
[CrossRef] [PubMed]

Born, W.

W. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 325.

Byer, R. L.

G. Giuliani, Y. K. Park, R. L. Byer, “Radial birefringent element and its application to a Nd:YAG resonator,” Opt. Lett. 5, 491–493 (1980).
[CrossRef] [PubMed]

Casperson, L. W.

L. W. Casperson, S. D. Lunnam, “Gaussian modes in high loss laser resonators,” Appl. Opt. 14, 1193–1199 (1975).
[CrossRef] [PubMed]

De Silvestri, S.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Radially variable reflectivity output coupler of novel design for unstable resonators,” Opt. Lett. 12, 84–87 (1987).
[CrossRef] [PubMed]

Giuliani, G.

E. Armandillo, G. Giuliani, “Achievement of large-sized TEM00mode from an excimer laser by means of a novel apoditic filter,” Opt. Lett. 10, 445–447 (1985).
[CrossRef] [PubMed]

G. Giuliani, Y. K. Park, R. L. Byer, “Radial birefringent element and its application to a Nd:YAG resonator,” Opt. Lett. 5, 491–493 (1980).
[CrossRef] [PubMed]

Lacina, W. B.

G. L. McAllister, W. H. Steier, W. B. Lacina, “Improved mode properties of unstable resonators with tapered reflectivity mirrors and shaped apertures,” IEEE J. Quantum Electron. QE-10, 346–355 (1974).
[CrossRef]

Laporta, P.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Radially variable reflectivity output coupler of novel design for unstable resonators,” Opt. Lett. 12, 84–87 (1987).
[CrossRef] [PubMed]

Lavigne, P.

N. McCarthy, P. Lavigne, “Large-size Gaussian mode in unstable resonators using Gaussian mirrors,” Opt. Lett. 10, 553–555 (1985).
[CrossRef] [PubMed]

N. McCarthy, P. Lavigne, “Optical resonators with Gaussian reflectivity mirrors: output beam characteristics,” Appl. Opt. 23, 3845–3850 (1984).
[CrossRef] [PubMed]

Lunnam, S. D.

L. W. Casperson, S. D. Lunnam, “Gaussian modes in high loss laser resonators,” Appl. Opt. 14, 1193–1199 (1975).
[CrossRef] [PubMed]

Magni, V.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Radially variable reflectivity output coupler of novel design for unstable resonators,” Opt. Lett. 12, 84–87 (1987).
[CrossRef] [PubMed]

McAllister, G. L.

G. L. McAllister, W. H. Steier, W. B. Lacina, “Improved mode properties of unstable resonators with tapered reflectivity mirrors and shaped apertures,” IEEE J. Quantum Electron. QE-10, 346–355 (1974).
[CrossRef]

McCarthy, N.

N. McCarthy, P. Lavigne, “Large-size Gaussian mode in unstable resonators using Gaussian mirrors,” Opt. Lett. 10, 553–555 (1985).
[CrossRef] [PubMed]

N. McCarthy, P. Lavigne, “Optical resonators with Gaussian reflectivity mirrors: output beam characteristics,” Appl. Opt. 23, 3845–3850 (1984).
[CrossRef] [PubMed]

Park, Y. K.

G. Giuliani, Y. K. Park, R. L. Byer, “Radial birefringent element and its application to a Nd:YAG resonator,” Opt. Lett. 5, 491–493 (1980).
[CrossRef] [PubMed]

Siegman, A. E.

A. E. Siegman, “Stabilizing output with unstable resonators,” Laser Focus 7(5), 42–47 (1971).

A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE 53, 277–287 (1965).
[CrossRef]

Steier, W. H.

G. L. McAllister, W. H. Steier, W. B. Lacina, “Improved mode properties of unstable resonators with tapered reflectivity mirrors and shaped apertures,” IEEE J. Quantum Electron. QE-10, 346–355 (1974).
[CrossRef]

Svelto, O.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Radially variable reflectivity output coupler of novel design for unstable resonators,” Opt. Lett. 12, 84–87 (1987).
[CrossRef] [PubMed]

Vakhimov, N. G.

N. G. Vakhimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electr. Phys. 10, 1439–1446 (1965).

Wolf, E.

W. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 325.

Yariv, A.

A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

Yeh, P.

A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

Zucker, H.

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).

Appl. Opt. (2)

L. W. Casperson, S. D. Lunnam, “Gaussian modes in high loss laser resonators,” Appl. Opt. 14, 1193–1199 (1975).
[CrossRef] [PubMed]

N. McCarthy, P. Lavigne, “Optical resonators with Gaussian reflectivity mirrors: output beam characteristics,” Appl. Opt. 23, 3845–3850 (1984).
[CrossRef] [PubMed]

Bell Syst. Tech. J. (1)

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).

IEEE J. Quantum Electron. (1)

G. L. McAllister, W. H. Steier, W. B. Lacina, “Improved mode properties of unstable resonators with tapered reflectivity mirrors and shaped apertures,” IEEE J. Quantum Electron. QE-10, 346–355 (1974).
[CrossRef]

Laser Focus (1)

A. E. Siegman, “Stabilizing output with unstable resonators,” Laser Focus 7(5), 42–47 (1971).

Opt. Commun. (1)

A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

Opt. Lett. (4)

G. Giuliani, Y. K. Park, R. L. Byer, “Radial birefringent element and its application to a Nd:YAG resonator,” Opt. Lett. 5, 491–493 (1980).
[CrossRef] [PubMed]

E. Armandillo, G. Giuliani, “Achievement of large-sized TEM00mode from an excimer laser by means of a novel apoditic filter,” Opt. Lett. 10, 445–447 (1985).
[CrossRef] [PubMed]

N. McCarthy, P. Lavigne, “Large-size Gaussian mode in unstable resonators using Gaussian mirrors,” Opt. Lett. 10, 553–555 (1985).
[CrossRef] [PubMed]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Radially variable reflectivity output coupler of novel design for unstable resonators,” Opt. Lett. 12, 84–87 (1987).
[CrossRef] [PubMed]

Proc. IEEE (1)

A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE 53, 277–287 (1965).
[CrossRef]

Radio Eng. Electr. Phys. (1)

N. G. Vakhimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electr. Phys. 10, 1439–1446 (1965).

Other (1)

W. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 325.

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Figures (7)

Fig. 1
Fig. 1

Schematic of the RAVI. d(0) is the central distance between the surfaces S1 and S2; ρ1 and ρ2 are the radii of curvature of S1 and S2 (negative for convex surfaces). The two radii of curvature are set such that zero reflectivity is obtained at the radial distance a.

Fig. 2
Fig. 2

Reflectivity as a function of the distance d between S1 and S2 normalized to λ/4 for different values of the reflectivity Rs of the coated surfaces. The horizontal dashed line corresponds to a reflectivity value of 0.55. For an explanation of the plus and the minus see Eq. (5) in the text.

Fig. 3
Fig. 3

Reflectivity profiles of the RAVI as a function of the radial distance r from the center normalized to the aperture distance a for different values of the reflectivity Rs of the coated surfaces. The central reflectivity R0 of the device is set equal to 0.55. Curves a and c are relative to Rs = 0.3 (i.e., ξ = 0.5), whereas curve b is relative to Rs = 0.2 (i.e., ξ = 1).

Fig. 4
Fig. 4

Reflectivity profiles of the RAVI as a function of r/ā for different values of ξ, where ā is the aperture distance corresponding to ξ = 0.5. The reflectivity Rs of the coated surfaces is equal to 0.3.

Fig. 5
Fig. 5

Plot of the ratio of the RAVI spot size wm to the aperture distance a as a function of ξ for different values of the central reflectivity R0.

Fig. 6
Fig. 6

Frequency dependence of the RAVI reflectivity profile as a function of the ratio Δν/ν0 for different values of the (integer) number n: (a) central reflectivity; (b) reflectivity at the aperture distance.

Fig. 7
Fig. 7

Frequency dependence of the RAVI aperture distance a(ν), normalized to a(ν0), as a function of the ratio Δν/ν0 for different values of n.

Equations (21)

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R = 4 R s sin 2 ( δ / 2 ) ( 1 - R s ) 2 + 4 R s sin 2 ( δ / 2 ) .
δ = 4 π λ d ( r ) ,
R M = 4 R s / ( 1 + R s ) 2 .
d ( r ) = d ( 0 ) - r 2 2 ρ ¯ ,
d ( 0 ) = ( λ / 4 ) ( 2 n ± ξ ) ,
ξ = 2 π arcsin [ R 0 ( 1 - R s ) 2 4 R s ( 1 - R 0 ) ] 1 / 2 .
Δ d = - ( λ / 4 ) ξ ,
Δ d = ( λ / 4 ) ( 2 - ξ ) .
ρ ¯ = 2 a 2 / λ ξ             ( central maximum ) ,
ρ ¯ = 2 a 2 / λ ( ξ - 2 )             ( central minimum ) .
ρ ¯ = 2 a 2 / λ .
δ = π ξ ( 1 - r 2 a 2 ) + 2 n π .
( w m a ) 2 = 1 - 2 π ξ arcsin [ ( 1 - R 0 e 2 - R 0 ) 1 / 2 sin π ξ 2 ] .
R s = 1 + ( 2 / F ) [ 1 - ( 1 + F ) 1 / 2 ] ,
F = R 0 / ( 1 - R 0 ) sin 2 ( π ξ / 2 ) .
a ( ν ) / a ( ν 0 ) = [ 1 + ( 2 n / ξ ) ( 1 - ν 0 / ν ) ] 1 / 2 .
Δ R 0 / R 0 = { π 2 ( 2 n + 1 ) 2 4 ( R 0 - 1 ) ( Δ ν / ν 0 ) 2 for ξ = 1 π ( 2 n + ξ ) t g ( π ξ / 2 ) ( 1 - R 0 ) ( Δ ν / ν 0 ) for ξ < 1 ,
Δ R a = π 2 n 2 ( R 0 1 - R 0 ) ( Δ ν / ν 0 ) 2 .
Δ a / a = ( n / ξ ) ( Δ ν / ν 0 ) .
w m = w in / ( M 2 - 1 ) 1 / 2 .
R 0 = 1 / M 2 .

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