Abstract

A mirror with a Gaussian reflectivity profile and a complementary transmission has been designed. This nonabsorptive smooth mirror is based on radially varying the thickness of a high-refractive-index dielectric deposited on a transparent substrate. Prototypes have been fabricated for use around 10 μm. The reflection and transmission profiles of one of them have been measured together with the modifications of the reflected and transmitted wave fronts.

© 1985 Optical Society of America

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Errata

Pierre Lavigne, Nathalie McCarthy, and Jean-Guy Demers, "Design and characterization of complementary Gaussian reflectivity mirrors: erratum," Appl. Opt. 24, 4278-4278 (1985)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-24-24-4278

References

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  1. S. N. Vlasov, V. I. Talanov, “Selection of Axial Modes in Open Resonators,” Radio Eng. Electron. Phys. 10, 469 (1965).
  2. N. G. Vakhimov, “Open Resonators with Mirrors Having Variable Reflection Coefficients,” Radio Eng. Electron. Phys. 10, 1439 (1965).
  3. N. Kumagai, H. Mori, T. Shiozawa, “Resonant Modes in a Fabry-Perot Resonator Consisting of Nonuniform Reflectors,” Electron. Commun. Jpn. 49, 1 (1966).
  4. Y. Suematsu, K. Iga, H. Nagashima, T. Kitano, “A Light Beam Waveguide Using Gaussian-Mode Filters Together with Lenses,” Electron. Commun. Jpn. 51-B, 67 (1968).
  5. H. Zucker, “Optical Resonators with Variable Reflectivity Mirrors,” Bell Syst. Tech. J. 49, 2349 (1970).
  6. A. E. Siegman, “Unstable Optical Resonator for Laser Applications,” Proc. IEEE 53, 277 (1965).
    [CrossRef]
  7. N. McCarthy, P. Lavigne, “Optical Resonators with Gaussian Reflectivity Mirrors: Output Beam Characteristics,” Appl. Opt. 23, 3845 (1984).
    [CrossRef] [PubMed]
  8. G. Giuliani, Y. K. Park, R. L. Byer, “Radial Birefringent Element and its Application to Laser Resonator Design,” Opt. Lett. 5, 491 (1980).
    [CrossRef] [PubMed]
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), 808 pp.
  10. C. P. Wang, “Performance of a Laser Beam Wavefront Sensor,” Rev. Sci. Instrum. 54, 469 (1983).
    [CrossRef]

1984

1983

C. P. Wang, “Performance of a Laser Beam Wavefront Sensor,” Rev. Sci. Instrum. 54, 469 (1983).
[CrossRef]

1980

1970

H. Zucker, “Optical Resonators with Variable Reflectivity Mirrors,” Bell Syst. Tech. J. 49, 2349 (1970).

1968

Y. Suematsu, K. Iga, H. Nagashima, T. Kitano, “A Light Beam Waveguide Using Gaussian-Mode Filters Together with Lenses,” Electron. Commun. Jpn. 51-B, 67 (1968).

1966

N. Kumagai, H. Mori, T. Shiozawa, “Resonant Modes in a Fabry-Perot Resonator Consisting of Nonuniform Reflectors,” Electron. Commun. Jpn. 49, 1 (1966).

1965

S. N. Vlasov, V. I. Talanov, “Selection of Axial Modes in Open Resonators,” Radio Eng. Electron. Phys. 10, 469 (1965).

N. G. Vakhimov, “Open Resonators with Mirrors Having Variable Reflection Coefficients,” Radio Eng. Electron. Phys. 10, 1439 (1965).

A. E. Siegman, “Unstable Optical Resonator for Laser Applications,” Proc. IEEE 53, 277 (1965).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), 808 pp.

Byer, R. L.

Giuliani, G.

Iga, K.

Y. Suematsu, K. Iga, H. Nagashima, T. Kitano, “A Light Beam Waveguide Using Gaussian-Mode Filters Together with Lenses,” Electron. Commun. Jpn. 51-B, 67 (1968).

Kitano, T.

Y. Suematsu, K. Iga, H. Nagashima, T. Kitano, “A Light Beam Waveguide Using Gaussian-Mode Filters Together with Lenses,” Electron. Commun. Jpn. 51-B, 67 (1968).

Kumagai, N.

N. Kumagai, H. Mori, T. Shiozawa, “Resonant Modes in a Fabry-Perot Resonator Consisting of Nonuniform Reflectors,” Electron. Commun. Jpn. 49, 1 (1966).

Lavigne, P.

McCarthy, N.

Mori, H.

N. Kumagai, H. Mori, T. Shiozawa, “Resonant Modes in a Fabry-Perot Resonator Consisting of Nonuniform Reflectors,” Electron. Commun. Jpn. 49, 1 (1966).

Nagashima, H.

Y. Suematsu, K. Iga, H. Nagashima, T. Kitano, “A Light Beam Waveguide Using Gaussian-Mode Filters Together with Lenses,” Electron. Commun. Jpn. 51-B, 67 (1968).

Park, Y. K.

Shiozawa, T.

N. Kumagai, H. Mori, T. Shiozawa, “Resonant Modes in a Fabry-Perot Resonator Consisting of Nonuniform Reflectors,” Electron. Commun. Jpn. 49, 1 (1966).

Siegman, A. E.

A. E. Siegman, “Unstable Optical Resonator for Laser Applications,” Proc. IEEE 53, 277 (1965).
[CrossRef]

Suematsu, Y.

Y. Suematsu, K. Iga, H. Nagashima, T. Kitano, “A Light Beam Waveguide Using Gaussian-Mode Filters Together with Lenses,” Electron. Commun. Jpn. 51-B, 67 (1968).

Talanov, V. I.

S. N. Vlasov, V. I. Talanov, “Selection of Axial Modes in Open Resonators,” Radio Eng. Electron. Phys. 10, 469 (1965).

Vakhimov, N. G.

N. G. Vakhimov, “Open Resonators with Mirrors Having Variable Reflection Coefficients,” Radio Eng. Electron. Phys. 10, 1439 (1965).

Vlasov, S. N.

S. N. Vlasov, V. I. Talanov, “Selection of Axial Modes in Open Resonators,” Radio Eng. Electron. Phys. 10, 469 (1965).

Wang, C. P.

C. P. Wang, “Performance of a Laser Beam Wavefront Sensor,” Rev. Sci. Instrum. 54, 469 (1983).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), 808 pp.

Zucker, H.

H. Zucker, “Optical Resonators with Variable Reflectivity Mirrors,” Bell Syst. Tech. J. 49, 2349 (1970).

Appl. Opt.

Bell Syst. Tech. J.

H. Zucker, “Optical Resonators with Variable Reflectivity Mirrors,” Bell Syst. Tech. J. 49, 2349 (1970).

Electron. Commun. Jpn.

N. Kumagai, H. Mori, T. Shiozawa, “Resonant Modes in a Fabry-Perot Resonator Consisting of Nonuniform Reflectors,” Electron. Commun. Jpn. 49, 1 (1966).

Y. Suematsu, K. Iga, H. Nagashima, T. Kitano, “A Light Beam Waveguide Using Gaussian-Mode Filters Together with Lenses,” Electron. Commun. Jpn. 51-B, 67 (1968).

Opt. Lett.

Proc. IEEE

A. E. Siegman, “Unstable Optical Resonator for Laser Applications,” Proc. IEEE 53, 277 (1965).
[CrossRef]

Radio Eng. Electron. Phys.

S. N. Vlasov, V. I. Talanov, “Selection of Axial Modes in Open Resonators,” Radio Eng. Electron. Phys. 10, 469 (1965).

N. G. Vakhimov, “Open Resonators with Mirrors Having Variable Reflection Coefficients,” Radio Eng. Electron. Phys. 10, 1439 (1965).

Rev. Sci. Instrum.

C. P. Wang, “Performance of a Laser Beam Wavefront Sensor,” Rev. Sci. Instrum. 54, 469 (1983).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), 808 pp.

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Figures (9)

Fig. 1
Fig. 1

Configuration of the proposed complementary Gaussian reflectivity mirror.

Fig. 2
Fig. 2

Dielectric thickness profiles and corresponding intensity reflectivity curves for (A) germanium (n = 4.0) on a perfectly matched AR-coating–substrate pair (air–Ge–air); (B) germanium on a NaCl (n = 1.49) substrate; (C) germanium on a germanium substrate AR coated with ZnS (n = 2.20).

Fig. 3
Fig. 3

Relative radial phase shift of (A) the transmitted wave front and (B) the reflected wave front. These curves were calculated for the ideal air–Ge–air case (…), the air–Ge–ZnS–Ge situation (—) with neq > n1, and the air–Ge–NaF–NaCl situation (---) with neq < n1. The values of the refractive indices used in those calculations are n(Ge) = 4.00, n(NaF) = 1.20, n(ZnS) = 2.2, and n(NaCl) = 1.49. It is assumed that the second surface of the substrate does not contribute to the reflected light.

Fig. 4
Fig. 4

Variation of the radial reflectivity profile at wavelengths λ different from the design value λ0 for the air–Ge–air configuration: (A) λ < λ0; (B) λ > λ0.

Fig. 5
Fig. 5

Variation of the HWHM value ρ1/2 normalized to the design HWHM value ρ 1 / 2 0 as a function of Δλ/λ0 for the air–Ge–air configuration.

Fig. 6
Fig. 6

Gaussian reflectivity mirror prototype. The substrate is a 12.5-cm diam Ge meniscus. The graded coating is on the convex side. The design radius wint is 1.2 cm.

Fig. 7
Fig. 7

Layout of the 10-μm interferometer which was used to characterize one of the Gaussian reflectivity mirror prototypes. The solid and dashed lines represent the configuration adopted to characterize the reflected and transmitted beams, respectively.

Fig. 8
Fig. 8

Radial reflectivity and transmittivity curves of a 5.0-cm diam complementary Gaussian mirror (Ge–ZnS–Ge–ZnS). The dashed curve was calculated assuming a Gaussian reflectivity profile.

Fig. 9
Fig. 9

Radial phase variations introduced by the shaped Ge layer of a 5.0-cm diam complementary plane Gaussian mirror on (A) the transmitted and (B) the reflected light. Mirror configuration: Ge–ZnS–Ge–ZnS. The reference curves represent the phase variations obtained with (A) a plane-parallel NaCl window and (B) a plane totally reflecting mirror.

Equations (12)

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ε 2 ( ρ ) = λ 0 4 π n 2 cos 1 { 1 2 r 12 r eq [ ( 1 r 12 2 ) ( 1 r eq 2 ) 1 R ( ρ ) r 12 2 r eq 2 1 ] } ,
r eq = r 23 + r 34 exp ( i 4 π n 3 ε 3 / λ 0 ) 1 + r 23 r 34 exp ( i 4 π n 3 ε 3 / λ 0 ) ,
r i j = n i n j n i + n j .
R ( ρ ) = R 0 exp ( ρ 2 / w int 2 ) ,
R max 0 = ( 2 r 12 1 + r 12 2 ) 2 .
Φ ( ρ ) = ϕ 2 ( ρ ) ϕ 2 ( 0 ) + ϕ air ( ρ ) ,
ϕ air ( ρ ) = 2 k [ ε ( ρ ) ε ( 0 ) ] ,
u r u i = r 12 + r eq exp [ i 4 π n 2 ε 2 ( ρ ) / λ 0 ] 1 + r 12 r eq exp [ i 4 π n 2 ε 2 ( ρ ) / λ 0 ] ,
ϕ 2 ( ρ ) = tan 1 [ imaginary ( u r / u i ) real ( u r / u i ) ] .
t ( ρ ) = [ 1 R 0 exp ( ρ 2 w int 2 ) ] 1 / 2 exp [ i Ψ ( ρ ) ] ,
Ψ ( ρ ) = ψ 2 ( 0 ) ψ 2 ( ρ ) ψ air ( ρ ) ,
u t u i = 1 + u r u i .

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