Abstract

Unstable resonators with 90° beam rotation are discussed. Geometric properties of the forward and reverse modes are derived, and diffractive-optical properties are described. An equation for the equivalent Frensel number is obtained. Plots of eigenvectors of the resonator integral equation are included, and the equivalence of these resonators to a type of strip resonator is established.

© 1986 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric Study of Apertured Focused Gaussian Beams,” Appl. Opt. 11, 565 (1972).
    [CrossRef] [PubMed]
  2. T. R. Ferguson, M. E. Smithers, “Optical Resonators with Nonuniform Magnification,” J. Opt. Soc. Am. A1, 653 (1984).
    [CrossRef]
  3. M. E. Smithers, T. R. Ferguson, “Unstable Optical Resonators with Linear Magnification,” Appl. Opt. 23, 3718 (1984).
    [CrossRef] [PubMed]
  4. Yu. A. Ananev, Optical Resonators and the Problem of Divergence of Laser Emissions (Izdatelstvo, Moscow, 1979).
  5. Yu. A. Ananev, “Unstable Laser Resonator for Low-Gain Media,” Sov. Tech. Phys. Lett. 4, 150 (1978).
  6. V. N. Kuprenyuk, V. E. Semenov, L. D. Smirnova, V. E. Sherstobitov, “Wave-Approximation Calculation of an Unstable Resonator with Field Rotation,” Sov. J. Quantum Electron. 13, 1613 (1983).
    [CrossRef]
  7. Yu. K. Danileiko, V. A. Lobachev, “New Rotating-Field Resonator for Lasers,” Sov. J. Quantum Electron. 4, 389 (1974).
    [CrossRef]
  8. D. L. Bullock, D. N. Mansell, S. G. Forbes, “Azimuthal Mode Control for Lasers,” U.S. Patent4,011,523 (8, Mar.1977).
  9. A. H. Paxton, W. P. Latham, “Ray Matrix Method for the Analysis of Optical Resonators with Image Rotation,” Proc. Soc. Photo-Opt. Instrum. Eng. 554, 159 (1985).
  10. J. A. Arnaud, “Degenerate Optical Cavities,” Appl. Opt. 8, 189 (1969).
    [CrossRef] [PubMed]
  11. P. Hoffmann, “Confocal Unstable Optical Resonator with Asymmetrical Magnification,” Opt. Lett. 6, 598 (1981).
    [CrossRef] [PubMed]
  12. C. Cason, R. W. Jones, J. F. Perkins, “Unstable Optical Resonators with Tilted Spherical Mirrors,” Opt. Lett. 2, 145 (1978).
    [CrossRef] [PubMed]
  13. A. E. Siegman, “A Canonical Formulation for Analyzing Multielement Unstable Resonators,” IEEE J. Quantum Electron. QE-12, 35 (1976).
    [CrossRef]
  14. A. Gerrard, J. M. Burch, Introduction of Matrix Methods in Optics (Gersham Press, Old Woking, Surrey, England, 1975).
  15. A. E. Siegman, Lasers (University Science Books, Mill Valley, California, 1986).
  16. E. A. Sziklas, A. E. Siegman, “Mode Calculation in Unstable Resonators with Flowing Saturable Gain. 2: Fast Fourier Transform Method,” Appl. Opt. 14, 1874 (1975).
    [CrossRef] [PubMed]
  17. R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “Asymmetric Unstable Traveling-Wave Resonators,” IEEE J. Quantum Electron. QE-10, 279 (1974).
    [CrossRef]
  18. G. C. Dente, C. L. Clayton {U.S. Air Force Weapons Laboratory; private communications} regarding the injection of a non-resonant beam in the forward direction if the usual combination of scraper and suppressor mirrors is used.
  19. A. E. Siegman, R. Arrathoon, “Modes in Unstable Optical Resonators and Lens Waveguides,” IEEE J. Quantum Electron. QE-3, 156 (1967).
    [CrossRef]
  20. A. E. Siegman, H. Y. Miller, “Unstable Optical Resonator Loss Calculations using the Prony Method,” Appl. Opt. 9, 2729 (1970).
    [CrossRef] [PubMed]
  21. Yu. A. Ananev, “Unstable Resonators and Their Applications,” Sov. J. Quantum Electron. 1, 565 (1972).
    [CrossRef]
  22. A. H. Paxton, T. C. Salvi, “Unstable Optical Resonator with Self Imaging Aperture,” Opt. Commun. 26, 305 (1978).
    [CrossRef]

1985 (1)

A. H. Paxton, W. P. Latham, “Ray Matrix Method for the Analysis of Optical Resonators with Image Rotation,” Proc. Soc. Photo-Opt. Instrum. Eng. 554, 159 (1985).

1984 (2)

T. R. Ferguson, M. E. Smithers, “Optical Resonators with Nonuniform Magnification,” J. Opt. Soc. Am. A1, 653 (1984).
[CrossRef]

M. E. Smithers, T. R. Ferguson, “Unstable Optical Resonators with Linear Magnification,” Appl. Opt. 23, 3718 (1984).
[CrossRef] [PubMed]

1983 (1)

V. N. Kuprenyuk, V. E. Semenov, L. D. Smirnova, V. E. Sherstobitov, “Wave-Approximation Calculation of an Unstable Resonator with Field Rotation,” Sov. J. Quantum Electron. 13, 1613 (1983).
[CrossRef]

1981 (1)

1978 (3)

A. H. Paxton, T. C. Salvi, “Unstable Optical Resonator with Self Imaging Aperture,” Opt. Commun. 26, 305 (1978).
[CrossRef]

C. Cason, R. W. Jones, J. F. Perkins, “Unstable Optical Resonators with Tilted Spherical Mirrors,” Opt. Lett. 2, 145 (1978).
[CrossRef] [PubMed]

Yu. A. Ananev, “Unstable Laser Resonator for Low-Gain Media,” Sov. Tech. Phys. Lett. 4, 150 (1978).

1976 (1)

A. E. Siegman, “A Canonical Formulation for Analyzing Multielement Unstable Resonators,” IEEE J. Quantum Electron. QE-12, 35 (1976).
[CrossRef]

1975 (1)

1974 (2)

R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “Asymmetric Unstable Traveling-Wave Resonators,” IEEE J. Quantum Electron. QE-10, 279 (1974).
[CrossRef]

Yu. K. Danileiko, V. A. Lobachev, “New Rotating-Field Resonator for Lasers,” Sov. J. Quantum Electron. 4, 389 (1974).
[CrossRef]

1972 (2)

Yu. A. Ananev, “Unstable Resonators and Their Applications,” Sov. J. Quantum Electron. 1, 565 (1972).
[CrossRef]

D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric Study of Apertured Focused Gaussian Beams,” Appl. Opt. 11, 565 (1972).
[CrossRef] [PubMed]

1970 (1)

1969 (1)

1967 (1)

A. E. Siegman, R. Arrathoon, “Modes in Unstable Optical Resonators and Lens Waveguides,” IEEE J. Quantum Electron. QE-3, 156 (1967).
[CrossRef]

Ananev, Yu. A.

Yu. A. Ananev, “Unstable Laser Resonator for Low-Gain Media,” Sov. Tech. Phys. Lett. 4, 150 (1978).

Yu. A. Ananev, “Unstable Resonators and Their Applications,” Sov. J. Quantum Electron. 1, 565 (1972).
[CrossRef]

Yu. A. Ananev, Optical Resonators and the Problem of Divergence of Laser Emissions (Izdatelstvo, Moscow, 1979).

Arnaud, J. A.

Arrathoon, R.

A. E. Siegman, R. Arrathoon, “Modes in Unstable Optical Resonators and Lens Waveguides,” IEEE J. Quantum Electron. QE-3, 156 (1967).
[CrossRef]

Avizonis, P. V.

Buczek, C. J.

R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “Asymmetric Unstable Traveling-Wave Resonators,” IEEE J. Quantum Electron. QE-10, 279 (1974).
[CrossRef]

Bullock, D. L.

D. L. Bullock, D. N. Mansell, S. G. Forbes, “Azimuthal Mode Control for Lasers,” U.S. Patent4,011,523 (8, Mar.1977).

Burch, J. M.

A. Gerrard, J. M. Burch, Introduction of Matrix Methods in Optics (Gersham Press, Old Woking, Surrey, England, 1975).

Cason, C.

Chenausky, P. P.

R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “Asymmetric Unstable Traveling-Wave Resonators,” IEEE J. Quantum Electron. QE-10, 279 (1974).
[CrossRef]

Clayton, C. L.

G. C. Dente, C. L. Clayton {U.S. Air Force Weapons Laboratory; private communications} regarding the injection of a non-resonant beam in the forward direction if the usual combination of scraper and suppressor mirrors is used.

Danileiko, Yu. K.

Yu. K. Danileiko, V. A. Lobachev, “New Rotating-Field Resonator for Lasers,” Sov. J. Quantum Electron. 4, 389 (1974).
[CrossRef]

Dente, G. C.

G. C. Dente, C. L. Clayton {U.S. Air Force Weapons Laboratory; private communications} regarding the injection of a non-resonant beam in the forward direction if the usual combination of scraper and suppressor mirrors is used.

Ferguson, T. R.

T. R. Ferguson, M. E. Smithers, “Optical Resonators with Nonuniform Magnification,” J. Opt. Soc. Am. A1, 653 (1984).
[CrossRef]

M. E. Smithers, T. R. Ferguson, “Unstable Optical Resonators with Linear Magnification,” Appl. Opt. 23, 3718 (1984).
[CrossRef] [PubMed]

Forbes, S. G.

D. L. Bullock, D. N. Mansell, S. G. Forbes, “Azimuthal Mode Control for Lasers,” U.S. Patent4,011,523 (8, Mar.1977).

Freiberg, R. J.

R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “Asymmetric Unstable Traveling-Wave Resonators,” IEEE J. Quantum Electron. QE-10, 279 (1974).
[CrossRef]

Gerrard, A.

A. Gerrard, J. M. Burch, Introduction of Matrix Methods in Optics (Gersham Press, Old Woking, Surrey, England, 1975).

Hoffmann, P.

Holmes, D. A.

Jones, R. W.

Korka, J. E.

Kuprenyuk, V. N.

V. N. Kuprenyuk, V. E. Semenov, L. D. Smirnova, V. E. Sherstobitov, “Wave-Approximation Calculation of an Unstable Resonator with Field Rotation,” Sov. J. Quantum Electron. 13, 1613 (1983).
[CrossRef]

Latham, W. P.

A. H. Paxton, W. P. Latham, “Ray Matrix Method for the Analysis of Optical Resonators with Image Rotation,” Proc. Soc. Photo-Opt. Instrum. Eng. 554, 159 (1985).

Lobachev, V. A.

Yu. K. Danileiko, V. A. Lobachev, “New Rotating-Field Resonator for Lasers,” Sov. J. Quantum Electron. 4, 389 (1974).
[CrossRef]

Mansell, D. N.

D. L. Bullock, D. N. Mansell, S. G. Forbes, “Azimuthal Mode Control for Lasers,” U.S. Patent4,011,523 (8, Mar.1977).

Miller, H. Y.

Paxton, A. H.

A. H. Paxton, W. P. Latham, “Ray Matrix Method for the Analysis of Optical Resonators with Image Rotation,” Proc. Soc. Photo-Opt. Instrum. Eng. 554, 159 (1985).

A. H. Paxton, T. C. Salvi, “Unstable Optical Resonator with Self Imaging Aperture,” Opt. Commun. 26, 305 (1978).
[CrossRef]

Perkins, J. F.

Salvi, T. C.

A. H. Paxton, T. C. Salvi, “Unstable Optical Resonator with Self Imaging Aperture,” Opt. Commun. 26, 305 (1978).
[CrossRef]

Semenov, V. E.

V. N. Kuprenyuk, V. E. Semenov, L. D. Smirnova, V. E. Sherstobitov, “Wave-Approximation Calculation of an Unstable Resonator with Field Rotation,” Sov. J. Quantum Electron. 13, 1613 (1983).
[CrossRef]

Sherstobitov, V. E.

V. N. Kuprenyuk, V. E. Semenov, L. D. Smirnova, V. E. Sherstobitov, “Wave-Approximation Calculation of an Unstable Resonator with Field Rotation,” Sov. J. Quantum Electron. 13, 1613 (1983).
[CrossRef]

Siegman, A. E.

A. E. Siegman, “A Canonical Formulation for Analyzing Multielement Unstable Resonators,” IEEE J. Quantum Electron. QE-12, 35 (1976).
[CrossRef]

E. A. Sziklas, A. E. Siegman, “Mode Calculation in Unstable Resonators with Flowing Saturable Gain. 2: Fast Fourier Transform Method,” Appl. Opt. 14, 1874 (1975).
[CrossRef] [PubMed]

A. E. Siegman, H. Y. Miller, “Unstable Optical Resonator Loss Calculations using the Prony Method,” Appl. Opt. 9, 2729 (1970).
[CrossRef] [PubMed]

A. E. Siegman, R. Arrathoon, “Modes in Unstable Optical Resonators and Lens Waveguides,” IEEE J. Quantum Electron. QE-3, 156 (1967).
[CrossRef]

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

Smirnova, L. D.

V. N. Kuprenyuk, V. E. Semenov, L. D. Smirnova, V. E. Sherstobitov, “Wave-Approximation Calculation of an Unstable Resonator with Field Rotation,” Sov. J. Quantum Electron. 13, 1613 (1983).
[CrossRef]

Smithers, M. E.

T. R. Ferguson, M. E. Smithers, “Optical Resonators with Nonuniform Magnification,” J. Opt. Soc. Am. A1, 653 (1984).
[CrossRef]

M. E. Smithers, T. R. Ferguson, “Unstable Optical Resonators with Linear Magnification,” Appl. Opt. 23, 3718 (1984).
[CrossRef] [PubMed]

Sziklas, E. A.

Appl. Opt. (5)

IEEE J. Quantum Electron. (3)

R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “Asymmetric Unstable Traveling-Wave Resonators,” IEEE J. Quantum Electron. QE-10, 279 (1974).
[CrossRef]

A. E. Siegman, R. Arrathoon, “Modes in Unstable Optical Resonators and Lens Waveguides,” IEEE J. Quantum Electron. QE-3, 156 (1967).
[CrossRef]

A. E. Siegman, “A Canonical Formulation for Analyzing Multielement Unstable Resonators,” IEEE J. Quantum Electron. QE-12, 35 (1976).
[CrossRef]

J. Opt. Soc. Am. (1)

T. R. Ferguson, M. E. Smithers, “Optical Resonators with Nonuniform Magnification,” J. Opt. Soc. Am. A1, 653 (1984).
[CrossRef]

Opt. Commun. (1)

A. H. Paxton, T. C. Salvi, “Unstable Optical Resonator with Self Imaging Aperture,” Opt. Commun. 26, 305 (1978).
[CrossRef]

Opt. Lett. (2)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

A. H. Paxton, W. P. Latham, “Ray Matrix Method for the Analysis of Optical Resonators with Image Rotation,” Proc. Soc. Photo-Opt. Instrum. Eng. 554, 159 (1985).

Sov. J. Quantum Electron. (3)

V. N. Kuprenyuk, V. E. Semenov, L. D. Smirnova, V. E. Sherstobitov, “Wave-Approximation Calculation of an Unstable Resonator with Field Rotation,” Sov. J. Quantum Electron. 13, 1613 (1983).
[CrossRef]

Yu. K. Danileiko, V. A. Lobachev, “New Rotating-Field Resonator for Lasers,” Sov. J. Quantum Electron. 4, 389 (1974).
[CrossRef]

Yu. A. Ananev, “Unstable Resonators and Their Applications,” Sov. J. Quantum Electron. 1, 565 (1972).
[CrossRef]

Sov. Tech. Phys. Lett. (1)

Yu. A. Ananev, “Unstable Laser Resonator for Low-Gain Media,” Sov. Tech. Phys. Lett. 4, 150 (1978).

Other (5)

Yu. A. Ananev, Optical Resonators and the Problem of Divergence of Laser Emissions (Izdatelstvo, Moscow, 1979).

D. L. Bullock, D. N. Mansell, S. G. Forbes, “Azimuthal Mode Control for Lasers,” U.S. Patent4,011,523 (8, Mar.1977).

A. Gerrard, J. M. Burch, Introduction of Matrix Methods in Optics (Gersham Press, Old Woking, Surrey, England, 1975).

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

G. C. Dente, C. L. Clayton {U.S. Air Force Weapons Laboratory; private communications} regarding the injection of a non-resonant beam in the forward direction if the usual combination of scraper and suppressor mirrors is used.

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 (17)

Fig. 1
Fig. 1

Locations X that a ray of the mode of an unstable resonator with beam rotation pierces a transverse plane during a number of passes. (The beam rotation angle is ϕb, and the resonator magnification is M).

Fig. 2
Fig. 2

Ring resonator with 90° beam rotation. A rectangular output beam is scraped off one side of the resonator beam.

Fig. 3
Fig. 3

Cross section of the beam at the scraper plane.

Fig. 4
Fig. 4

Cross section of the beam of a resonator with different magnification in the two transverse directions.

Fig. 5
Fig. 5

Twisted loop with a square cross section has a topology similar to the ring resonator with 90° beam rotation.

Fig. 6
Fig. 6

Unfolded resonator.

Fig. 7
Fig. 7

Unfolded and untwisted resonator—two passes.

Fig. 8
Fig. 8

Intensity of bare-cavity lowest loss mode at the scraper–mirror plane. (The optical axis is at X = 0.0. The dashed line just to the left of the optical axis is the scraper edge location. The other two dashed lines are at the geometric edges of the beam.)

Fig. 9
Fig. 9

Phase of lowest loss bare-cavity mode.

Fig. 10
Fig. 10

Mode intensity at output plane, bare cavity.

Fig. 11
Fig. 11

Mode phase at output plane, bare cavity.

Fig. 12
Fig. 12

Intensity of second lowest loss mode.

Fig. 13
Fig. 13

Phase of second lowest loss mode.

Fig. 14
Fig. 14

Intensity of second lowest loss mode.

Fig. 15
Fig. 15

Phase of second lowest loss mode.

Fig. 16
Fig. 16

Untwisted and unfolded resonator. (Dashed line is geometric edge of reverse mode.)

Fig. 17
Fig. 17

Equivalent Fresnel number is determined from the distance between the edge of the scraped forward wave front and the wave front of the resonator converging wave.

Equations (40)

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

R ( A x B x P Q d x C x D x R S θ x T U A y B y d y V W C y D y θ y 0 0 0 0 1 ) = ( Θ x p d x θ x q Θ y d y θ y 0     0 0     0 1 ) ,
( cos ϕ 0 - sin ϕ 0 0 0 cos ϕ 0 - sin ϕ 0 sin ϕ 0 cos ϕ 0 0 0 sin ϕ 0 cos ϕ 0 0 0 0 0 1 ) .
V = ( r x r x r y r y 1 ) ,
R V λ = λ V λ ,
λ = M x , 1 M x ,             M y ,             1 M y ,             1.
λ = ± i M x M y ,             ± i / M x M y ,             1.
R V A = V A .
( Θ x 0 0 Θ y ) ( 0 - 1 1 0 ) ( V x V y ) = λ ( V x V y ) ,
V x = ( r x r x ) , V y = ( r y r y ) .
- Θ x V y = λ V x ;
Θ y V x = λ V y .
- Θ y Θ x V y = λ 2 V y .
R = ( C M x C L x - S M x - S L x α x D x 0 C / M x 0 - S / M x α x / M x S M y S L y C M y C L y α y D y 0 S / M y 0 C / M y α y / M y 0 0 0 0 1 ) ,
L 1 + L 2 = L A S = sin ϕ r , C = cos ϕ r , L x = M x ( L 1 + L 2 ) + L B + L C / M x , L y = M y ( L 1 + L 2 ) + L B + L C / M y , D x = M x L 2 + L B + L C / M x , D y = M y L 2 + L B + L C / M y .
V 1 = r x = { a x [ D x ( 1 + M x M y ) - L x - L y M x M y ] + α y M x [ L y - L x - D y ( 1 + M x M y ) ] } / ( 1 + M x M y ) 2 ,
V x = r y = { α x M y [ L y - L x + D x ( 1 + M x M y ) ] + α y [ D y ( 1 + M x M y ) - L y - L x M x M y ) ] } / ( 1 + M x M y ) 2 ;
V 2 = r x = ( α x M y - α y ) / ( 1 + M x M y ) ,
V 4 = r y = ( α x - α y M x ) / ( 1 + M x M y ) .
w ( x , y ) = K ( x , y , x , y ) w ( x , y ) d x d y ,
K ( x , y , x , y ) = K 1 ( x , x ) K 2 ( y , y ) .
w ( x , y ) = u ( x ) v ( y ) .
v ( y ) = - a K 1 ( y , x ) u ( x ) d x ,
u ( x ) = - K 2 ( - x , y ) v ( y ) d y .
u ( x ) = - a C ( x , x ) u ( x ) d x ,
C ( x , x ) = - K 2 ( - x , y ) K 1 ( y , x ) d y ,
λ i u i ( x ) = - a C ( x , x ) u i ( x ) d x ,
v i ( y ) = - a K 1 ( y , x ) u i ( x ) d x .
L eq = L 1 + L 2 ( 1 / M x + 1 / M x 2 M y ) + L 3 / M x 2 + L 4 / M x 2 M y 2 .
L eq = L 1 + L 2 / M x + L 4 / M x 2 .
L eq = L A ( 1 + 1 / M x 2 ) + L B ( 1 / M x + 1 / M x 2 M y ) + L c ( 1 / M x 2 + 1 / M x 2 M y 2 ) ,
L eq = L A + L B / M x + L c / M x 2 .
C ( x , x ) = ( 1 / i M λ L eq ) exp [ i π λ L eq ( x + x / M ) 2 ] ,
K 1 ( y , x ) = ( 1 / i M x λ L eq ) exp [ i π λ L eq ( x + y / M x ) 2 ] .
N = a 2 / λ L eq .
λ i v i = P v i ,
P j k = ( f / i M ) 1 / 2 exp [ i π f [ ( j - n ) / M + k - n ] 2 ] , f = N / n 2 .
N = a 2 / λ L eq
R r = M x 2 M y 2 L eq / ( M x 2 M y 2 - 1 ) .
N eq = d / λ .
N eq = a 2 ( M x 2 M y 2 - 1 ) 2 λ M x 2 M y 2 L eq .

Metrics