Abstract

Several findings on the modes of a nonplanar ring resonator have been carried out by utilizing the matrix method of generalized Gaussian beams. The widely used nonplanar ring resonator—a four-equal-sided nonplanar ring cavity with two curvature mirrors and two planar mirrors—has been chosen as an example. The overall stability maps of this typical nonplanar ring cavity with L∕R ranging from 0 to ∞ are calculated and described, while the total cavity length is L and the radius of the curvature mirrors is R. The stability map has also been found to have asymmetry with total image rotation ρ ranging from 0° to 360° and 0<L/R<2. It has also been found that there still exist several stable regions in the extended region with L/R>2. The Gaussian modes of the nonplanar ring cavity have been found to have different characteristics with different design parameters such as ρ and L/R. The azimuth angles of the major and minor axes of the spot size have been found to be variable with different design parameters such as ρ and L/R and variable under different location of the nonplanar cavity. These interesting findings are totally different from the behavior of conventional planar stable resonators and are important to the cavity designs of nonplanar ring resonators.

© 2007 Optical Society of America

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References

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  1. M. V. Berry, "The geometric phase," Sci. Am. 259, 26-32 (1988).
    [CrossRef]
  2. A. H. Paxton and W. P. Latham, Jr., "Unstable resonators with 90 degree beam rotation," Appl. Opt. 25, 2939-2945 (1986).
    [CrossRef] [PubMed]
  3. A. H. Paxton, "Unstable ring resonator with an intracavity prism beam expander," IEEE J. Quantum Electron. QE-23, 241-244 (1987).
    [CrossRef]
  4. S. Holswade, R. Riviere, C. A. Huguley, C. M. Clayton, and G. C. Dente, "Experimental evaluation of a unstable ring resonator with 90 degrees beam rotation: HiQ experimental results," Appl. Opt. 27, 4396-4406 (1988).
    [CrossRef] [PubMed]
  5. W. P. Latham, Jr., A. H. Paxton, and G. C. Dente, "Laser with 90-degree beam rotation," in Optical Resonators, Proc. SPIE 1224, 265-282 (1990).
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    [CrossRef] [PubMed]
  7. I. W. Smith and T. Dorschner, "Electromagnetic wave ring resonator," U.S. patent 4,110,045 (29 August 1978).
  8. V. E. Sanders and D. Z. Anderson, "Isotropic nonplanar ring resonator," U.S. patent 4,247,832 (27 January 1981).
  9. T. A. Dorschner, "Nonplanar rings for laser gyroscopes," in Fiber Optic and Laser Sensors, Proc. SPIE 412, 192-202 (1983).
  10. H. Statz, T. A. Dorschner, M. Holtz, and I. W. Smith, "The multioscillator ring laser gyroscope," in Laser Handbook, M. L. Stitch and M. Bass, eds. (North-Holland, 1985), Vol. 4, Chap. 3.
  11. G. J. Martin, S. C. Gillespie, and C. H. Volk, "Small ZLGreg Triax technology," in Proceedings of the AIAA Guidance Navigation and Control Conference (AIAA, 1996), pp. 1-8.
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    [CrossRef]
  13. J. A. Arnaud, "Nonorthogonal optical waveguides and resonators," Bell Syst. Tech. J. 49, 2311-2348 (1970).
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    [CrossRef]
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  16. G. Nemes and J. Serna, "The ten physical parameters associated with a full general astigmatic beam: a Gauss-Schell model," in Laser Beam and Optics Characterization (LBOC4), A. Giesen and M. Morin, eds. (VDI Technologiezentrum, 1997), pp. 92-105.
  17. G. Nemes and J. Serna, "Laser beam characterization with use of second order moments: an overview," in DPSS Lasers: Applications and Issues (OSA TOPS Volume 17), M. W. Dowley, ed. (Optical Society of America, 1998), pp. 200-207.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  21. A. E. Siegman, Lasers (University Science, 1986), Chap. 16.
  22. O. Svelto, Principles of Lasers, 4th ed. (Kluwer Academic, 1998).
  23. A. Gerrard and J. M. Burch, Introduction of Matrix Methods in Optics (Wiley, 1975), pp. 97-178.

2000 (1)

A. E. Siegman, "Laser beams and resonators: beyond the 1960s," IEEE J. Sel. Top. Quantum Electron. 6, 1389-1399 (2000).
[CrossRef]

1988 (2)

1987 (1)

A. H. Paxton, "Unstable ring resonator with an intracavity prism beam expander," IEEE J. Quantum Electron. QE-23, 241-244 (1987).
[CrossRef]

1986 (1)

1985 (1)

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, "The ring laser gyro," Rev. Mod. Phys. 57, 81-86 (1985).
[CrossRef]

1974 (1)

J. A. Arnaud, "Optical resonators in the approximation of Gauss," Proc. IEEE 62, 1561-1570 (1974).
[CrossRef]

1972 (1)

1971 (1)

1970 (1)

J. A. Arnaud, "Nonorthogonal optical waveguides and resonators," Bell Syst. Tech. J. 49, 2311-2348 (1970).

Anderson, D. Z.

V. E. Sanders and D. Z. Anderson, "Isotropic nonplanar ring resonator," U.S. patent 4,247,832 (27 January 1981).

Arnaud, J. A.

J. A. Arnaud, "Optical resonators in the approximation of Gauss," Proc. IEEE 62, 1561-1570 (1974).
[CrossRef]

J. A. Arnaud, "Nonorthogonal optical waveguides and resonators," Bell Syst. Tech. J. 49, 2311-2348 (1970).

Bergmann, E. E.

Berry, M. V.

M. V. Berry, "The geometric phase," Sci. Am. 259, 26-32 (1988).
[CrossRef]

Burch, J. M.

A. Gerrard and J. M. Burch, Introduction of Matrix Methods in Optics (Wiley, 1975), pp. 97-178.

Chow, W. W.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, "The ring laser gyro," Rev. Mod. Phys. 57, 81-86 (1985).
[CrossRef]

Clayton, C. M.

Dente, G. C.

Dorschner, T.

I. W. Smith and T. Dorschner, "Electromagnetic wave ring resonator," U.S. patent 4,110,045 (29 August 1978).

Dorschner, T. A.

H. Statz, T. A. Dorschner, M. Holtz, and I. W. Smith, "The multioscillator ring laser gyroscope," in Laser Handbook, M. L. Stitch and M. Bass, eds. (North-Holland, 1985), Vol. 4, Chap. 3.

T. A. Dorschner, "Nonplanar rings for laser gyroscopes," in Fiber Optic and Laser Sensors, Proc. SPIE 412, 192-202 (1983).

Gea-Banacloche, J.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, "The ring laser gyro," Rev. Mod. Phys. 57, 81-86 (1985).
[CrossRef]

Gerrard, A.

A. Gerrard and J. M. Burch, Introduction of Matrix Methods in Optics (Wiley, 1975), pp. 97-178.

Gillespie, S. C.

G. J. Martin, S. C. Gillespie, and C. H. Volk, "The Litton 11 cm Triaxial Zero-Lock Gyro," in Proceedings of PLANS '96 Symposium (1996), pp. 49-55.
[CrossRef]

G. J. Martin, S. C. Gillespie, and C. H. Volk, "Small ZLGreg Triax technology," in Proceedings of the AIAA Guidance Navigation and Control Conference (AIAA, 1996), pp. 1-8.

Holswade, S.

Holtz, M.

H. Statz, T. A. Dorschner, M. Holtz, and I. W. Smith, "The multioscillator ring laser gyroscope," in Laser Handbook, M. L. Stitch and M. Bass, eds. (North-Holland, 1985), Vol. 4, Chap. 3.

Huguley, C. A.

Jacobs, G. B.

Latham, W. P.

A. H. Paxton and W. P. Latham, Jr., "Unstable resonators with 90 degree beam rotation," Appl. Opt. 25, 2939-2945 (1986).
[CrossRef] [PubMed]

W. P. Latham, Jr., A. H. Paxton, and G. C. Dente, "Laser with 90-degree beam rotation," in Optical Resonators, Proc. SPIE 1224, 265-282 (1990).

Martin, G. J.

G. J. Martin, S. C. Gillespie, and C. H. Volk, "The Litton 11 cm Triaxial Zero-Lock Gyro," in Proceedings of PLANS '96 Symposium (1996), pp. 49-55.
[CrossRef]

G. J. Martin, S. C. Gillespie, and C. H. Volk, "Small ZLGreg Triax technology," in Proceedings of the AIAA Guidance Navigation and Control Conference (AIAA, 1996), pp. 1-8.

Nemes, G.

G. Nemes and J. Serna, "Laser beam characterization with use of second order moments: an overview," in DPSS Lasers: Applications and Issues (OSA TOPS Volume 17), M. W. Dowley, ed. (Optical Society of America, 1998), pp. 200-207.

G. Nemes, "Synthesis of general astigmatic optical systems, the detwisting procedure and the beam quality factors of general astigmatic beams," in Laser Beam Characterization, H. Weber, N. Reng, L. Ludtke, and P. M. Mejias, eds. (Festkorper-Laser-Institute Berlin GmbH, 1994), pp. 93-104.

G. Nemes and J. Serna, "The ten physical parameters associated with a full general astigmatic beam: a Gauss-Schell model," in Laser Beam and Optics Characterization (LBOC4), A. Giesen and M. Morin, eds. (VDI Technologiezentrum, 1997), pp. 92-105.

Paxton, A. H.

A. H. Paxton, "Unstable ring resonator with an intracavity prism beam expander," IEEE J. Quantum Electron. QE-23, 241-244 (1987).
[CrossRef]

A. H. Paxton and W. P. Latham, Jr., "Unstable resonators with 90 degree beam rotation," Appl. Opt. 25, 2939-2945 (1986).
[CrossRef] [PubMed]

W. P. Latham, Jr., A. H. Paxton, and G. C. Dente, "Laser with 90-degree beam rotation," in Optical Resonators, Proc. SPIE 1224, 265-282 (1990).

Pedrotti, L. M.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, "The ring laser gyro," Rev. Mod. Phys. 57, 81-86 (1985).
[CrossRef]

Riviere, R.

Sanders, V. E.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, "The ring laser gyro," Rev. Mod. Phys. 57, 81-86 (1985).
[CrossRef]

V. E. Sanders and D. Z. Anderson, "Isotropic nonplanar ring resonator," U.S. patent 4,247,832 (27 January 1981).

Schleich, W.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, "The ring laser gyro," Rev. Mod. Phys. 57, 81-86 (1985).
[CrossRef]

Scully, M. O.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, "The ring laser gyro," Rev. Mod. Phys. 57, 81-86 (1985).
[CrossRef]

Serna, J.

G. Nemes and J. Serna, "Laser beam characterization with use of second order moments: an overview," in DPSS Lasers: Applications and Issues (OSA TOPS Volume 17), M. W. Dowley, ed. (Optical Society of America, 1998), pp. 200-207.

G. Nemes and J. Serna, "The ten physical parameters associated with a full general astigmatic beam: a Gauss-Schell model," in Laser Beam and Optics Characterization (LBOC4), A. Giesen and M. Morin, eds. (VDI Technologiezentrum, 1997), pp. 92-105.

Siegman, A. E.

A. E. Siegman, "Laser beams and resonators: beyond the 1960s," IEEE J. Sel. Top. Quantum Electron. 6, 1389-1399 (2000).
[CrossRef]

A. E. Siegman, Lasers (University Science, 1986), Chap. 16.

Smith, I. W.

H. Statz, T. A. Dorschner, M. Holtz, and I. W. Smith, "The multioscillator ring laser gyroscope," in Laser Handbook, M. L. Stitch and M. Bass, eds. (North-Holland, 1985), Vol. 4, Chap. 3.

I. W. Smith and T. Dorschner, "Electromagnetic wave ring resonator," U.S. patent 4,110,045 (29 August 1978).

Statz, H.

H. Statz, T. A. Dorschner, M. Holtz, and I. W. Smith, "The multioscillator ring laser gyroscope," in Laser Handbook, M. L. Stitch and M. Bass, eds. (North-Holland, 1985), Vol. 4, Chap. 3.

Svelto, O.

O. Svelto, Principles of Lasers, 4th ed. (Kluwer Academic, 1998).

Volk, C. H.

G. J. Martin, S. C. Gillespie, and C. H. Volk, "Small ZLGreg Triax technology," in Proceedings of the AIAA Guidance Navigation and Control Conference (AIAA, 1996), pp. 1-8.

G. J. Martin, S. C. Gillespie, and C. H. Volk, "The Litton 11 cm Triaxial Zero-Lock Gyro," in Proceedings of PLANS '96 Symposium (1996), pp. 49-55.
[CrossRef]

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

J. A. Arnaud, "Nonorthogonal optical waveguides and resonators," Bell Syst. Tech. J. 49, 2311-2348 (1970).

IEEE J. Quantum Electron. (1)

A. H. Paxton, "Unstable ring resonator with an intracavity prism beam expander," IEEE J. Quantum Electron. QE-23, 241-244 (1987).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

A. E. Siegman, "Laser beams and resonators: beyond the 1960s," IEEE J. Sel. Top. Quantum Electron. 6, 1389-1399 (2000).
[CrossRef]

Proc. IEEE (1)

J. A. Arnaud, "Optical resonators in the approximation of Gauss," Proc. IEEE 62, 1561-1570 (1974).
[CrossRef]

Rev. Mod. Phys. (1)

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, "The ring laser gyro," Rev. Mod. Phys. 57, 81-86 (1985).
[CrossRef]

Sci. Am. (1)

M. V. Berry, "The geometric phase," Sci. Am. 259, 26-32 (1988).
[CrossRef]

Other (13)

G. Nemes, "Synthesis of general astigmatic optical systems, the detwisting procedure and the beam quality factors of general astigmatic beams," in Laser Beam Characterization, H. Weber, N. Reng, L. Ludtke, and P. M. Mejias, eds. (Festkorper-Laser-Institute Berlin GmbH, 1994), pp. 93-104.

G. Nemes and J. Serna, "The ten physical parameters associated with a full general astigmatic beam: a Gauss-Schell model," in Laser Beam and Optics Characterization (LBOC4), A. Giesen and M. Morin, eds. (VDI Technologiezentrum, 1997), pp. 92-105.

G. Nemes and J. Serna, "Laser beam characterization with use of second order moments: an overview," in DPSS Lasers: Applications and Issues (OSA TOPS Volume 17), M. W. Dowley, ed. (Optical Society of America, 1998), pp. 200-207.

I. W. Smith and T. Dorschner, "Electromagnetic wave ring resonator," U.S. patent 4,110,045 (29 August 1978).

V. E. Sanders and D. Z. Anderson, "Isotropic nonplanar ring resonator," U.S. patent 4,247,832 (27 January 1981).

T. A. Dorschner, "Nonplanar rings for laser gyroscopes," in Fiber Optic and Laser Sensors, Proc. SPIE 412, 192-202 (1983).

H. Statz, T. A. Dorschner, M. Holtz, and I. W. Smith, "The multioscillator ring laser gyroscope," in Laser Handbook, M. L. Stitch and M. Bass, eds. (North-Holland, 1985), Vol. 4, Chap. 3.

G. J. Martin, S. C. Gillespie, and C. H. Volk, "Small ZLGreg Triax technology," in Proceedings of the AIAA Guidance Navigation and Control Conference (AIAA, 1996), pp. 1-8.

G. J. Martin, S. C. Gillespie, and C. H. Volk, "The Litton 11 cm Triaxial Zero-Lock Gyro," in Proceedings of PLANS '96 Symposium (1996), pp. 49-55.
[CrossRef]

W. P. Latham, Jr., A. H. Paxton, and G. C. Dente, "Laser with 90-degree beam rotation," in Optical Resonators, Proc. SPIE 1224, 265-282 (1990).

A. E. Siegman, Lasers (University Science, 1986), Chap. 16.

O. Svelto, Principles of Lasers, 4th ed. (Kluwer Academic, 1998).

A. Gerrard and J. M. Burch, Introduction of Matrix Methods in Optics (Wiley, 1975), pp. 97-178.

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Figures (11)

Fig. 1
Fig. 1

Four-equal-sided nonplanar ring cavity used in the numerical analysis.

Fig. 2
Fig. 2

(Color online) Folder angle β and image rotation ρ versus incidence angle θ.

Fig. 3
Fig. 3

(Color online) Traditional stability map for four-equal-sided nonplanar ring resonator. The number of unstable transverse directions is given for L / R ranging from 0 to 2 and image rotation produced by the nonplanar arrangement of the mirrors. In region 0, the resonator is stable, with (in general) elliptical isophotes. In region 1, the major axis of the isophotes ellipse is infinite, and in region 2 both axes are infinite. The shaded regions are stable.

Fig. 4
Fig. 4

(Color online) Extended stability map for four-equal-sided nonplanar ring resonator. The number of unstable transverse directions is given for L / R ranging from 2 to 8 and image rotation produced by the nonplanar arrangement of the mirrors. In region 0, the resonator is stable, with (in general) elliptical isophotes. In region 1, the major axis of the isophotes ellipse is infinite, and in region 2 both axes are infinite. The shaded regions are stable.

Fig. 5
Fig. 5

(Color online) Normalized Gaussian beam size parameter 4 L / k w 2 versus L / R for the total image rotation of (a) ρ = 0 ° , (b) ρ = 180 ° .

Fig. 6
Fig. 6

(Color online) Normalized Gaussian beam size parameter 4 L / k w 2 versus L / R and L / R is ranging (a) from 0 to 2 and ρ = 90 ° , (b) from 3 to 4 and ρ = 90 ° , (c) from 0 to 2 and ρ = 270 ° , and (d) from 3.5 to 6 and ρ = 270 ° .

Fig. 7
Fig. 7

(Color online) Azimuth angle of the Gaussian beam radius in major and minor axes versus L / R , with the image rotation of (a) ρ = 0 ° , (b) ρ = 180 ° , (c) ρ = 90 ° , (d) ρ = 270 ° .

Fig. 8
Fig. 8

(Color online) Gaussian beam radius versus azimuth angle with the image rotation of (a) ρ = 0 ° , (b) ρ = 180 ° , (c) ρ = 90 ° , (d) ρ = 270 ° (The unit of angles is degree, and the unit of w is meters).

Fig. 9
Fig. 9

(Color online) Azimuth angle of the major axis at point X 1 versus ratio1 with different L / R and different total image rotation: (a) ρ = 0 ° , (b) ρ = 90 ° , (c) ρ = 180 ° , (d) ρ = 270 ° .

Fig. 10
Fig. 10

(Color online) Azimuth angle of the major axis at point X 2 versus ratio2 with different L∕R and different total image rotation: (a) ρ = 0 ° , (b) ρ = 90 ° , (c) ρ = 180 ° , (d) ρ = 270 ° .

Fig. 11
Fig. 11

(Color online) Azimuth angle of the major axis at point X 3 versus ratio3 with different L / R and different total image rotation: (a) ρ = 0 ° , (b) ρ = 90 ° , (c) ρ = 180 ° , (d) ρ = 270 ° .

Equations (24)

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

E ( r , ϕ , z ) = E 0 1 q exp [ i ( k z w t ) ] exp ( i k r 2 2 q ) ,
E = E 0 ( q x q y ) 1 / 2 exp [ i ( k z w t ) ] × exp ( i k x 2 2 q x ) exp ( i k y 2 2 q y ) .
E = E 0 exp [ i ( k z w t ) ] [ ( z z 1 ) ( z z 2 ) ] 1 / 2 × exp [ i k r 2 ( a + b cos ( 2 ϕ ) + c sin ( 2 ϕ ) ) / 4 ] ,
a = ( z z 1 ) 1 + ( z z 2 ) 1 ,
b = [ ( z z 1 ) 1 ( z z 2 ) 1 ] cos 2 θ ,
c = [ ( z z 1 ) 1 ( z z 2 ) 1 ] sin 2 θ ,
R ( ϕ ) = 2 Re ( a + b cos 2 ϕ + c sin 2 ϕ ) ,
1 w 2 = 1 4 k Im ( a + b cos 2 ϕ + c sin 2 ϕ ) ,
4 k w 2 = Im a ± ( Im b ) 2 + ( Im c ) 2 ,
ϕ w = π 4 ± π 4 + 1 2 arctan Im c Im b .
d a 2 b 2 c 2 ,
α a / d ,
β b / d ,
γ c / d ,
δ 1 / d .
a = α d ,
b = β d ,
c = γ d ,
d = 1 / δ .
[ α β γ δ 1 ] = M i [ α β γ δ 1 ] ,
M ( l ) [ 1 0 0 0 1 2 l 0 1 0 0 0 0 0 1 0 0 l 0 0 1 1 4 l 2 0 0 0 0 1 ] .
M ( θ ) [ 1 0 0 0 0 0 cos 2 θ sin 2 θ 0 0 0 sin 2 θ cos 2 θ 0 0 0 0 0 1 0 0 0 0 0 1 ] .
M = Π M i .
M [ α β γ δ 1 ] = λ [ α β γ δ 1 ] .

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