Abstract

A birefringent crystal with a surface curvature ρ provides a radial variation of polarization retardation. One or more such radial birefringent elements in combination with a polarizer form a radial intensity filter (RIF). RIF’s permit the generation of a variety of radial transmittance and reflectance profiles that are useful in resonator and spatial filter applications.

© 1981 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2343–2376 (1970).
    [Crossref]
  2. L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
    [Crossref]
  3. U. Ganiel and Y. Silberberg, “Stability of optical resonators with an active medium,” Appl. Opt. 14, 306–309 (1975).
    [Crossref] [PubMed]
  4. A. Yariv and P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
    [Crossref]
  5. G. Giuliani, Y. K. Park, and R. L. Byer, “The radial birefringent element and its application to laser resonator design,” presented at the Eleventh International Quantum Electronics Conference, Boston, Mass., June 1980.
  6. G. Giuliani, Y. K. Park, and R. L. Byer, “The radial birefringent elements and its application to a Nd:YAG resonator,” Opt. Lett. 5, 491–493 (1980).
    [Crossref] [PubMed]
  7. N. G. Vorkhimov, “Selection of axial modes in open resonators,” Radio Eng. Electron. Phys. 10, 469 (1975).
  8. N. Kumagei, H. Mori, and T. Shiozawa, “Resonant modes in a Fabry–Perot resonator consisting of non-uniform reflectors,” Electron. Commun. Jpn. 49, 1 (1966).
  9. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).
  10. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975).
  11. A. E. Siegman, “A canonical formulation for analyzing multielement unstable resonators,” IEEE J. Quantum Electron. QE-1235 (1976).
    [Crossref]
  12. A. J. Campillo, J. E. Pearson, S. L. Sharpiro, and N. J. Terrell, “Fresnel diffraction effects in the design of high power laser systems,” Appl. Phys. Lett. 23, 85 (1973).
    [Crossref]
  13. B. J. Feldman and S. J. Gitomer, “Annular lens soft aperture for high power laser systems,” Appl. Opt. 15, 1379–1380 (1976).
    [Crossref] [PubMed]
  14. R. C. Jones, “A new calculus for treatment of optical systems,” J. Opt. Soc. Am. 32, 486–493 (1942).
    [Crossref]
  15. J. M. Beckers, “Achromatic linear retarders,” Appl. Opt. 10, 973–975 (1971).
    [Crossref] [PubMed]

1980 (1)

1976 (2)

A. E. Siegman, “A canonical formulation for analyzing multielement unstable resonators,” IEEE J. Quantum Electron. QE-1235 (1976).
[Crossref]

B. J. Feldman and S. J. Gitomer, “Annular lens soft aperture for high power laser systems,” Appl. Opt. 15, 1379–1380 (1976).
[Crossref] [PubMed]

1975 (3)

U. Ganiel and Y. Silberberg, “Stability of optical resonators with an active medium,” Appl. Opt. 14, 306–309 (1975).
[Crossref] [PubMed]

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

N. G. Vorkhimov, “Selection of axial modes in open resonators,” Radio Eng. Electron. Phys. 10, 469 (1975).

1974 (1)

L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
[Crossref]

1973 (1)

A. J. Campillo, J. E. Pearson, S. L. Sharpiro, and N. J. Terrell, “Fresnel diffraction effects in the design of high power laser systems,” Appl. Phys. Lett. 23, 85 (1973).
[Crossref]

1971 (1)

1970 (1)

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2343–2376 (1970).
[Crossref]

1966 (1)

N. Kumagei, H. Mori, and T. Shiozawa, “Resonant modes in a Fabry–Perot resonator consisting of non-uniform reflectors,” Electron. Commun. Jpn. 49, 1 (1966).

1942 (1)

Beckers, J. M.

Byer, R. L.

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

G. Giuliani, Y. K. Park, and R. L. Byer, “The radial birefringent element and its application to laser resonator design,” presented at the Eleventh International Quantum Electronics Conference, Boston, Mass., June 1980.

Campillo, A. J.

A. J. Campillo, J. E. Pearson, S. L. Sharpiro, and N. J. Terrell, “Fresnel diffraction effects in the design of high power laser systems,” Appl. Phys. Lett. 23, 85 (1973).
[Crossref]

Casperson, L. W.

L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
[Crossref]

Feldman, B. J.

Ganiel, U.

Gitomer, S. J.

Giuliani, G.

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

G. Giuliani, Y. K. Park, and R. L. Byer, “The radial birefringent element and its application to laser resonator design,” presented at the Eleventh International Quantum Electronics Conference, Boston, Mass., June 1980.

Jones, R. C.

Kumagei, N.

N. Kumagei, H. Mori, and T. Shiozawa, “Resonant modes in a Fabry–Perot resonator consisting of non-uniform reflectors,” Electron. Commun. Jpn. 49, 1 (1966).

Mori, H.

N. Kumagei, H. Mori, and T. Shiozawa, “Resonant modes in a Fabry–Perot resonator consisting of non-uniform reflectors,” Electron. Commun. Jpn. 49, 1 (1966).

Park, Y. K.

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

G. Giuliani, Y. K. Park, and R. L. Byer, “The radial birefringent element and its application to laser resonator design,” presented at the Eleventh International Quantum Electronics Conference, Boston, Mass., June 1980.

Pearson, J. E.

A. J. Campillo, J. E. Pearson, S. L. Sharpiro, and N. J. Terrell, “Fresnel diffraction effects in the design of high power laser systems,” Appl. Phys. Lett. 23, 85 (1973).
[Crossref]

Sharpiro, S. L.

A. J. Campillo, J. E. Pearson, S. L. Sharpiro, and N. J. Terrell, “Fresnel diffraction effects in the design of high power laser systems,” Appl. Phys. Lett. 23, 85 (1973).
[Crossref]

Shiozawa, T.

N. Kumagei, H. Mori, and T. Shiozawa, “Resonant modes in a Fabry–Perot resonator consisting of non-uniform reflectors,” Electron. Commun. Jpn. 49, 1 (1966).

Siegman, A. E.

A. E. Siegman, “A canonical formulation for analyzing multielement unstable resonators,” IEEE J. Quantum Electron. QE-1235 (1976).
[Crossref]

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).

Silberberg, Y.

Terrell, N. J.

A. J. Campillo, J. E. Pearson, S. L. Sharpiro, and N. J. Terrell, “Fresnel diffraction effects in the design of high power laser systems,” Appl. Phys. Lett. 23, 85 (1973).
[Crossref]

Vorkhimov, N. G.

N. G. Vorkhimov, “Selection of axial modes in open resonators,” Radio Eng. Electron. Phys. 10, 469 (1975).

Yariv, A.

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

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975).

Yeh, P.

A. Yariv and 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, 2343–2376 (1970).
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

A. J. Campillo, J. E. Pearson, S. L. Sharpiro, and N. J. Terrell, “Fresnel diffraction effects in the design of high power laser systems,” Appl. Phys. Lett. 23, 85 (1973).
[Crossref]

Bell Syst. Tech. J. (1)

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2343–2376 (1970).
[Crossref]

Electron. Commun. Jpn. (1)

N. Kumagei, H. Mori, and T. Shiozawa, “Resonant modes in a Fabry–Perot resonator consisting of non-uniform reflectors,” Electron. Commun. Jpn. 49, 1 (1966).

IEEE J. Quantum Electron. (2)

L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
[Crossref]

A. E. Siegman, “A canonical formulation for analyzing multielement unstable resonators,” IEEE J. Quantum Electron. QE-1235 (1976).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

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

Opt. Lett. (1)

Radio Eng. Electron. Phys. (1)

N. G. Vorkhimov, “Selection of axial modes in open resonators,” Radio Eng. Electron. Phys. 10, 469 (1975).

Other (3)

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975).

G. Giuliani, Y. K. Park, and R. L. Byer, “The radial birefringent element and its application to laser resonator design,” presented at the Eleventh International Quantum Electronics Conference, Boston, Mass., June 1980.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

(a) Schematic of a resonator described by the ABCD matrix with reference plane and Gaussian reflector indicated. (b) Schematic of a simple resonator that includes a Gaussian reflector. The mirror has radius of curvature R, and the cavity length is fixed at L = 1 m. The cavity g parameter is given by g = (1 + L/R), and B is given by (g + 1)L. At 1 μm, λB is approximately 1 mm.

Fig. 2
Fig. 2

Normalized spot size, wm/(λB)1/2, as a function of g for various values of Nm = wm2/(λB), where wm is the spot size of the Gaussian reflector.

Fig. 3
Fig. 3

The perturbation stability factor F as a function of the cavity g parameter for various values of Nm.

Fig. 4
Fig. 4

Loss factor L, which is due to the presence of a Gaussian filter, with a generalized Fresnel number Nm, as a function of g.

Fig. 5
Fig. 5

Schematic of a radial intensity filter as an end mirror of a laser resonator. P, RBE, and M are the polarizer, the radial birefringent element with thickness d0 and curvature ρ, and the mirror, respectively.

Fig. 6
Fig. 6

Examples of the reflectance profiles at 1 μm generated with the single-element RIF’s as shown schematically in Fig. 5, with center thickness d0 and radius of curvature ρ. For all profiles ra is 3 mm.

Fig. 7
Fig. 7

(a) Schematic of a two-element RIF as an element in a laser resonator. P and M are the polarizer and the mirror, respectively. The two RBE’s in the RIF have thicknesses d1 and d2 and curvatures ρ1 and ρ2. (b) The angles θ1 and θ2 are measured from the principal axes of the RBE’s to the principal axis of the polarizer.

Fig. 8
Fig. 8

Examples of reflectance profiles that may be generated by a two-element RIF. One element is a half-wave plate at r = 0 and a quarter-wave plate at r = ra. The other is a quarter-wave plate at r = 0 and a half-wave plate at r = ra. The angles θ1 and θ2 are given in parentheses.

Fig. 9
Fig. 9

Wavelength dependence of uncompensated and compensated single-element RIF’s near 1 μm. The RBE has a center thickness of 0.344 mm and a curvature of −25 cm; the compensating plate is flat and has a thickness of 0.344 mm. Both elements are made from crystal quartz with Δn = 0.008725 at 1 μm. (a) The center reflectance of the uncompensated RIF as a function of wavelength about 1 μm. (b) The aperture distance ra as a function of wavelength for the uncompensated RIF about 1 μm. (c) The center reflectance of the compensated RIF as a function of wavelength about 1 μm. Note the wavelength scale change from the uncompensated RIF. (d) The aperture distance ra as a function of wavelength around 1 μm for the compensated RIF.

Fig. 10
Fig. 10

The wavelength dependence of a two-element broadband RIF (see Table 1), without the compensating plates at θ1 = 45° and θ2 = −15°. (a) Reflectance profile at the 1-μm design wavelength. (b) Reflectance at the center of the RIF, r = 0, as a function of wavelength around 1 μm. (c) The aperture distance as a function of wavelength around 1 μm. (d) The reflectance at the aperture distance as a function of wavelength around 1 μm. Note the scale used for reflectance.

Fig. 11
Fig. 11

When a Gaussian variable reflector is included in a laser resonator, both confined (stable) and unconfined (unstable) resonaors produce TEM00 mode outputs. If a knife edge cuts one side of each intracavity beam, then the outputs from the two resonators are manifestly different. (a) The confined resonator has a symmetrically clipped output. (b) The unconfined resonator has an output that is clipped on one side only.

Tables (1)

Tables Icon

Table 1 Two-Element Radial Intensity Filter Designs

Equations (40)

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

1 q = 1 R - i λ π w 2 ,
1 q o = C + D / q i A + B / q i .
R = R o exp ( - 2 r 2 w m 2 ) .
| 1 0 - i λ π w m 2 1 | .
1 w o 2 = 1 w m 2 + 1 w i 2 .
| A - i λ B π w m 2 B C - i λ D π w m 2 D | = | A B C D | ,
1 q = C + D / q A + B / q .
1 q = D - A 2 B + i λ 2 π w m 2 ± 1 B [ ( D + A 2 - i λ B 2 π w m 2 ) 2 - 1 ] 1 / 2 .
F = A + B / q .
F ± = | g - i 2 π N m ± [ ( g - i 2 π N m ) 2 - 1 ] 1 / 2 | ,
= 1 - 1 F 2 .
| cos [ α ( r ) ] + i sin [ α ( r ) ] cos ( 2 θ ) i sin [ α ( r ) ] sin ( 2 θ ) i sin [ α ( r ) ] sin ( 2 θ ) cos [ α ( r ) ] - i sin [ α ( r ) ] cos ( 2 θ ) | ,
R = cos 2 [ 2 α ( r ) ] + sin 2 [ 2 α ( r ) ] cos 2 ( 2 θ ) .
2 α ( r ) = 2 π Δ n λ d ( r ) ,
d ( r ) = d 0 - ρ [ 1 - ( 1 - r 2 ρ 2 ) 1 / 2 ] ,
d ( r ) = d 0 - r 2 2 ρ ,
R = cos 2 ( 2 θ ) + sin 2 ( 2 θ ) cos 2 [ 2 π Δ n λ ( d 0 - r 2 2 ρ ) ] .
R θ = 45° = cos 2 [ 2 π Δ n λ ( d 0 - r 2 2 ρ ) ] .
d 0 = λ 2 π Δ n ( arc cos { ± [ R ( 0 ) ] 1 / 2 } + n π ) ,
ρ = π Δ n r a 2 λ 1 arc cos { [ R ( 0 ) ] 1 / 2 } ± π / 2 .
w m 2 = r a 2 arc cos [ R ( 0 ) 1 / 2 e ] - arc cos [ R ( 0 ) 1 / 2 ] π 2 - arc cos [ R ( 0 ) 1 / 2 ] .
α i ( r ) = π Δ n i λ ( d i - r 2 2 ρ i ) ,             i = 1 , 2.
R ( r ) = P 2 + Q 2 ,
P = cos ( 2 α 1 ) cos ( 2 α 2 ) - sin ( 2 α 1 ) sin ( 2 α 2 ) cos ( 2 θ 1 - 2 θ 2 ) , Q = [ cos 2 ( α 1 ) cos ( 2 θ 2 ) - sin 2 ( α 1 ) cos ( 4 θ 1 - 2 θ 2 ) ] sin ( 2 α 2 ) + sin ( 2 α 1 ) cos ( 2 α 2 ) cos ( 2 θ 1 ) .
α 1 ( r ) = π 4 ( 2 - r 2 r a 2 ) , α 2 ( r ) = π 4 ( 1 + r 2 r a 2 ) .
R ( r = r a ) = cos 2 ( 2 θ 1 ) ,
R ( r = 0 ) = cos 2 [ 2 ( 2 θ 1 - θ 2 ) ] .
2 α ( r ) = 2 π Δ n λ 0 ( d 0 - r 2 2 ρ ) ( 1 - Δ λ λ 0 ) ,
2 α ( r ) = 2 π Δ n λ 0 ( d 0 - d c - r 2 2 ρ ) ( 1 - Δ λ λ 0 ) .
M = ( 1 + w 2 w m 2 ) 1 / 2 ,
B q = [ C B + D ( B / q ) ] ( A + B / q ) .
B q = D - ( A + B / q ) * F 2 ,
F = ( 1 + w 2 w m 2 ) 1 / 2 .
D i = | D i x D i y | ,             D r = | D r x D r y | ,             D t = | D t x D t y | .
P ¯ = | 1 0 0 0 | ,             Ī = | 1 0 0 1 | .
D r = PM ¯ ( r ) IM ¯ ( r ) P ¯ D i ,
R = D r x * D r x + D r y * D r y D i x * D i x + D i y * D i y .
R = cos 2 [ 2 α ( r ) ] + sin 2 [ 2 α ( r ) ] cos 2 ( 2 θ ) .
T = 1 - R
T = 1 - sin 2 [ 2 α ( r ) ] sin 2 ( 2 θ ) ,