Abstract

A mathematical model of a four-sided folded planar ring resonator is established. The model can be modified into a triangular ring resonator, a square ring resonator, and a four-sided folded ring resonator, all of which are widely used for ring laser gyroscopes by changing certain design parameters such as incident angle Ai and side ratio H. By use of the extended matrix formulation, the optical axis perturbation, including optical axis decentration and optical axis tilt, in those planar ring resonators is analyzed in detail resulting in some novel findings. It has been determined that the longer the mirror radius, the larger the mode volume, the higher the sensitivity of optical axis decentration and the lower the sensitivity of optical axis tilt. The same mirror misalignment value, mostly the misalignment induced by optical axis decentration in the x and y components, has the conventional ratio of 1:[cos(Ai)]2 for the symmetrical points of the resonator. Details of the effect of Ai and H on the optical axis tilt have also been determined. The difference in optical axis tilt between different kinds of ring resonator is disclosed. The sensitivity of optical axis tilt was found to undergo singular rapid change along with the right edge of the second stable area. This singular behavior is useful for those resonators that have a small incident angle, such as Ai=15°, because those resonators have a second stable region. These interesting findings are important for cavity design, cavity improvement, and alignment of planar ring resonators.

© 2007 Optical Society of America

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  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, 61-86 (1985).
    [CrossRef]
  2. K. Andringa, "Laser gyroscope," U. S. patent 3,741,657 (26 June 1973).
  3. G. E. Stedman, "Ring-laser tests of fundamental physics and geophysics," Rep. Prog. Phys. 60, 615-688 (1997).
    [CrossRef]
  4. A. E. Siegman, "Laser beams and resonators: the 1960s," IEEE J. Special Top. Quantum Electron. 20, 100-110 (1999).
  5. J. A. Arnaud, "Degenerate optical cavities. II. Effect of misalignments," Appl. Opt. 8, 1909-1917 (1969).
  6. G. B. Altshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, "Analysis of misalignment sensitivity of ring laser resonators," Sov. J. Quantum Electron. 7, 857-859 (1977).
    [CrossRef]
  7. I. W. Smith, "Optical resonator axis stability and instability from first principles," in Fiber Optic and Laser Sensors I, E. L. Moore and O. G. Ramer, eds., Proc. SPIE 412, 203-206 (1983).
  8. A. L. Levit and V. M. Ovchinnikov, "Stability of a ring resonator with a nonplane axial contour," J. Appl. Spectrosc. (USSR) 40, 657-660 (1984).
    [CrossRef]
  9. S. C. Sheng, "Optical-axis perturbation singularity in an out-of-plane ring resonator," Opt. Lett. 19, 683-685 (1994).
  10. A. H. Paxton and W. H. Latham, Jr., "Ray matrix method for the analysis of optical resonators with image rotation," in 1985 International Lens Design Conference, D. T. Moore and W. H. Taylor, eds., Proc. SPIE 554, 159-163 (1985).
  11. A. H. Paxton and W. P. Latham, Jr., "Unstable resonators with 90° beam rotation," Appl. Opt. 25, 2939-2946 (1986).
  12. R. Rodloff, "A laser gyro with optimized resonator geometry," IEEE J. Quantum Electron. QE-23, 438-445 (1987).
    [CrossRef]
  13. H. R. Bilger and G. E. Stedman, "Stability of planar ring lasers with mirror misalignment," Appl. Opt. 26, 3710-3716 (1987).
  14. M. L. Stitch and M. Bass, eds., Laser Handbook (North-Holland, 1985), Vol. 4, Chap. 3, pp. 229-332.
  15. A. E. Siegman, Lasers (University Science, 1986), Chap. 15.
  16. A. Gerrard and J. M. Burch, Introduction of Matrix Methods in Optics (Wiley, 1975).
  17. O. Svelto, Principles of Lasers, 4th ed. (Springer, 1998), translated by D. C. Hanna.
  18. J. Yuan, X. W. Long, L. M. Liang, B. Zhang, F. Wang, and H. C. Zhao, "Nonplanar ring resonator modes: generalized Gaussian beams," Appl. Opt. 46, 2980-2989 (2007).
    [CrossRef]

2007 (1)

1999 (1)

A. E. Siegman, "Laser beams and resonators: the 1960s," IEEE J. Special Top. Quantum Electron. 20, 100-110 (1999).

1998 (1)

O. Svelto, Principles of Lasers, 4th ed. (Springer, 1998), translated by D. C. Hanna.

1997 (1)

G. E. Stedman, "Ring-laser tests of fundamental physics and geophysics," Rep. Prog. Phys. 60, 615-688 (1997).
[CrossRef]

1994 (1)

1987 (2)

H. R. Bilger and G. E. Stedman, "Stability of planar ring lasers with mirror misalignment," Appl. Opt. 26, 3710-3716 (1987).

R. Rodloff, "A laser gyro with optimized resonator geometry," IEEE J. Quantum Electron. QE-23, 438-445 (1987).
[CrossRef]

1986 (2)

1985 (3)

M. L. Stitch and M. Bass, eds., Laser Handbook (North-Holland, 1985), Vol. 4, Chap. 3, pp. 229-332.

A. H. Paxton and W. H. Latham, Jr., "Ray matrix method for the analysis of optical resonators with image rotation," in 1985 International Lens Design Conference, D. T. Moore and W. H. Taylor, eds., Proc. SPIE 554, 159-163 (1985).

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, 61-86 (1985).
[CrossRef]

1984 (1)

A. L. Levit and V. M. Ovchinnikov, "Stability of a ring resonator with a nonplane axial contour," J. Appl. Spectrosc. (USSR) 40, 657-660 (1984).
[CrossRef]

1983 (1)

I. W. Smith, "Optical resonator axis stability and instability from first principles," in Fiber Optic and Laser Sensors I, E. L. Moore and O. G. Ramer, eds., Proc. SPIE 412, 203-206 (1983).

1977 (1)

G. B. Altshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, "Analysis of misalignment sensitivity of ring laser resonators," Sov. J. Quantum Electron. 7, 857-859 (1977).
[CrossRef]

1975 (1)

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

1973 (1)

K. Andringa, "Laser gyroscope," U. S. patent 3,741,657 (26 June 1973).

1969 (1)

Altshuler, G. B.

G. B. Altshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, "Analysis of misalignment sensitivity of ring laser resonators," Sov. J. Quantum Electron. 7, 857-859 (1977).
[CrossRef]

Andringa, K.

K. Andringa, "Laser gyroscope," U. S. patent 3,741,657 (26 June 1973).

Arnaud, J. A.

Bass, M.

M. L. Stitch and M. Bass, eds., Laser Handbook (North-Holland, 1985), Vol. 4, Chap. 3, pp. 229-332.

Bilger, H. R.

Burch, J. M.

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

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, 61-86 (1985).
[CrossRef]

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, 61-86 (1985).
[CrossRef]

Gerrard, A.

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

Isyanova, E. D.

G. B. Altshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, "Analysis of misalignment sensitivity of ring laser resonators," Sov. J. Quantum Electron. 7, 857-859 (1977).
[CrossRef]

Karasev, V. B.

G. B. Altshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, "Analysis of misalignment sensitivity of ring laser resonators," Sov. J. Quantum Electron. 7, 857-859 (1977).
[CrossRef]

Latham, W. H.

A. H. Paxton and W. H. Latham, Jr., "Ray matrix method for the analysis of optical resonators with image rotation," in 1985 International Lens Design Conference, D. T. Moore and W. H. Taylor, eds., Proc. SPIE 554, 159-163 (1985).

Latham, W. P.

Levit, A. L.

A. L. Levit and V. M. Ovchinnikov, "Stability of a ring resonator with a nonplane axial contour," J. Appl. Spectrosc. (USSR) 40, 657-660 (1984).
[CrossRef]

G. B. Altshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, "Analysis of misalignment sensitivity of ring laser resonators," Sov. J. Quantum Electron. 7, 857-859 (1977).
[CrossRef]

Liang, L. M.

Long, X. W.

Ovchinnikov, V. M.

A. L. Levit and V. M. Ovchinnikov, "Stability of a ring resonator with a nonplane axial contour," J. Appl. Spectrosc. (USSR) 40, 657-660 (1984).
[CrossRef]

G. B. Altshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, "Analysis of misalignment sensitivity of ring laser resonators," Sov. J. Quantum Electron. 7, 857-859 (1977).
[CrossRef]

Paxton, A. H.

A. H. Paxton and W. P. Latham, Jr., "Unstable resonators with 90° beam rotation," Appl. Opt. 25, 2939-2946 (1986).

A. H. Paxton and W. H. Latham, Jr., "Ray matrix method for the analysis of optical resonators with image rotation," in 1985 International Lens Design Conference, D. T. Moore and W. H. Taylor, eds., Proc. SPIE 554, 159-163 (1985).

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, 61-86 (1985).
[CrossRef]

Rodloff, R.

R. Rodloff, "A laser gyro with optimized resonator geometry," IEEE J. Quantum Electron. QE-23, 438-445 (1987).
[CrossRef]

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, 61-86 (1985).
[CrossRef]

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, 61-86 (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, 61-86 (1985).
[CrossRef]

Sharlai, S. F.

G. B. Altshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, "Analysis of misalignment sensitivity of ring laser resonators," Sov. J. Quantum Electron. 7, 857-859 (1977).
[CrossRef]

Sheng, S. C.

Siegman, A. E.

A. E. Siegman, "Laser beams and resonators: the 1960s," IEEE J. Special Top. Quantum Electron. 20, 100-110 (1999).

A. E. Siegman, Lasers (University Science, 1986), Chap. 15.

Smith, I. W.

I. W. Smith, "Optical resonator axis stability and instability from first principles," in Fiber Optic and Laser Sensors I, E. L. Moore and O. G. Ramer, eds., Proc. SPIE 412, 203-206 (1983).

Stedman, G. E.

G. E. Stedman, "Ring-laser tests of fundamental physics and geophysics," Rep. Prog. Phys. 60, 615-688 (1997).
[CrossRef]

H. R. Bilger and G. E. Stedman, "Stability of planar ring lasers with mirror misalignment," Appl. Opt. 26, 3710-3716 (1987).

Stitch, M. L.

M. L. Stitch and M. Bass, eds., Laser Handbook (North-Holland, 1985), Vol. 4, Chap. 3, pp. 229-332.

Svelto, O.

O. Svelto, Principles of Lasers, 4th ed. (Springer, 1998), translated by D. C. Hanna.

Wang, F.

Yuan, J.

Zhang, B.

Zhao, H. C.

Appl. Opt. (4)

IEEE J. Quantum Electron. (1)

R. Rodloff, "A laser gyro with optimized resonator geometry," IEEE J. Quantum Electron. QE-23, 438-445 (1987).
[CrossRef]

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

A. E. Siegman, "Laser beams and resonators: the 1960s," IEEE J. Special Top. Quantum Electron. 20, 100-110 (1999).

J. Appl. Spectrosc. (1)

A. L. Levit and V. M. Ovchinnikov, "Stability of a ring resonator with a nonplane axial contour," J. Appl. Spectrosc. (USSR) 40, 657-660 (1984).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (2)

A. H. Paxton and W. H. Latham, Jr., "Ray matrix method for the analysis of optical resonators with image rotation," in 1985 International Lens Design Conference, D. T. Moore and W. H. Taylor, eds., Proc. SPIE 554, 159-163 (1985).

I. W. Smith, "Optical resonator axis stability and instability from first principles," in Fiber Optic and Laser Sensors I, E. L. Moore and O. G. Ramer, eds., Proc. SPIE 412, 203-206 (1983).

Rep. Prog. Phys. (1)

G. E. Stedman, "Ring-laser tests of fundamental physics and geophysics," Rep. Prog. Phys. 60, 615-688 (1997).
[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, 61-86 (1985).
[CrossRef]

Sov. J. Quantum Electron. (1)

G. B. Altshuler, E. D. Isyanova, V. B. Karasev, A. L. Levit, V. M. Ovchinnikov, and S. F. Sharlai, "Analysis of misalignment sensitivity of ring laser resonators," Sov. J. Quantum Electron. 7, 857-859 (1977).
[CrossRef]

Other (5)

K. Andringa, "Laser gyroscope," U. S. patent 3,741,657 (26 June 1973).

M. L. Stitch and M. Bass, eds., Laser Handbook (North-Holland, 1985), Vol. 4, Chap. 3, pp. 229-332.

A. E. Siegman, Lasers (University Science, 1986), Chap. 15.

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

O. Svelto, Principles of Lasers, 4th ed. (Springer, 1998), translated by D. C. Hanna.

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Figures (13)

Fig. 1
Fig. 1

Four-sided folded planar ring cavity used for the analysis with (a) H = 1 ; (b) H < 1 ; (c) H > 1 ; 1, 2, curved mirror; 3, 4, plane mirror; Ai, incident angle.

Fig. 2
Fig. 2

Several special cases of a four-sided folded planar ring cavity: (a) regular triangular ring cavity with H = 0 , A i = 30 ° , and L 1 = L 2 = L 3 ; (b) regular triangular ring cavity with H = , A i = 30 ° , and L 2 = L 3 = L 4 ; (c) square ring cavity with A i = 45 ° and L 1 = L 2 = L 3 = L 4 .

Fig. 3
Fig. 3

Schematic diagram of an optical-axis as it passes through the diaphragm under different kinds of optical axis perturbation such as (a) decentration, (b) tilt, (c) decentration and tilt. The center line represents the longitudinal axis of the diaphragm; the solid and broken lines represent the optical axis before and after perturbation.

Fig. 4
Fig. 4

Sensitivity of optical axis decentration versus L / R and Ai: (a) SD x 5 x 1 and (b) SD y 5 y 1 .

Fig. 5
Fig. 5

Sensitivity of optical axis decentration versus LR with different incident angles: (a) Ai = 15° and (b) Ai = 45°. Solid curve, the value of the y component including SD y 5 y 1 and SD y 6 y 1 ; dashed curve, the value of the x component including SD x 5 x 1 and SD x 6 x 1 .

Fig. 6
Fig. 6

Sensitivity of optical axis tilt ST x 5 x 1 with incident angles of 15, 22.5, 30, and 45 deg versus LR and with different values for H: (a) H = 0 , (b) H = 1 , (c) H = 10 .

Fig. 7
Fig. 7

Sensitivity of optical axis tilt ST x 6 x 1 with incident angles of 15, 22.5, 30, and 45 deg versus LR and with different values for H: (a) H = 0, (b) H = 1, (c) H = 10.

Fig. 8
Fig. 8

Sensitivity of optical axis tilt ST x 5 x 3 with incident angles of 15, 22.5, 30, and 45 deg versus LR and with different values for H: (a) H = 0, (b) H = 1, (c) H = 10.

Fig. 9
Fig. 9

Sensitivity of optical axis tilt ST x 6 x 3 with incident angles of 15, 22.5, 30, and 45 deg versus LR and with different values for H: (a) H = 0 , (b) H = 1 , (c) H = 10 .

Fig. 10
Fig. 10

Sensitivity of optical axis tilt with H = 0 , 1, and 10 and Ai = 30° versus L / R : (a) ST x 5 x 1 and (b) ST x 6 x 1 .

Fig. 11
Fig. 11

Sensitivity of optical axis tilt with H = 0 , 1, and 10 and A i = 30 ° versus L / R : (a) ST x 5 x 3 and (b) ST x 6 x 3 .

Fig. 12
Fig. 12

(a) Sensitivity of optical axis decentration including SD x 5 x 1 (broken curve) and SD y 5 y 1 (solid curve) versus L / R . (b) Sensitivity of optical axis tilt including ST x 5 x 1 (broken curve) and ST y 5 y 1 (solid curve) versus L / R . (c) The stability function ( A + D / 2 ) in both the x and the y directions of the ring resonator versus LR. H = 1, Ai = 30°, and LR ranges from 0 to 12. The dash–dot lines in (c) represent the critical values of −1 and 1.

Fig. 13
Fig. 13

(a) Sensitivity of optical axis decentration including SD x 5 x 1 (broken curve) and SD y 5 y 1 (solid curve) versus LR. (b) Sensitivity of optical axis tilt including ST x 5 x 1 (broken curve) and ST y 5 y 1 (solid curve) versus LR. (c) The stability function ( A + D / 2 ) in both the x and the y directions of the ring resonator versus L / R . H = 1, A i = 15 ° , LR ranges from 0 to 8. The dash–dot lines in (c) represent the critical values of −1 and 1.

Equations (18)

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L 1 = L ( 1 + H ) [ 1 + 1 / cos ( 2 A i ) ] ,
L 4 = L 1 H = L H ( 1 + H ) [ 1 + 1 / cos ( 2 A i ) ] ,
L 2 = L 3 = ( L 1 + L 4 ) 2 cos ( 2 A i ) = 1 2 L cos ( 2 A i ) + 1 .
( r o x r o x r o y r o y 1 ) = ( A x B x 0 0 0 C x D x 0 0 ξ x 0 0 A y B y 0 0 0 C y D y ξ y 0 0 0 0 1 ) ( r i x r i x r i y r i y 1 ) ,
M ( P i ) = [ 1 0 0 0 0 2 R i × cos A i 1 0 0 2 θ x i 0 0 1 0 0 0 0 2 × cos A i R i 1 2 θ y i 0 0 0 0 1 ] .
M ( L ) = 0 [ 1 L 0 0 0 0 1 0 0 0 0 0 1 L 0 0 0 0 1 0 0 0 0 0 1 ] .
M = M i .
( r x r x r y r y 1 ) = M ( r x r x r y r y 1 ) .
[ M 11 1 M 12 M 13 M 14 M 21 M 22 1 M 23 M 24 M 31 M 32 M 33 1 M 34 M 41 M 42 M 43 M 44 1 ] × [ r x r x r y r y ] = [ M 15 M 25 M 35 M 45 ] .
S D ( x y ) i ( x y ) j = 1 L [ r ( x y ) i θ ( x y ) j ] ,
S T ( x y ) i ( x y ) j = 1 L [ r ( x y ) i θ ( x y ) j ] ,
r ( x y ) i , r ( x y ) i
1 < ( A + D / 2 ) < 1 ,
SD x 5 x 1 = SD x 6 x 1 = SD x 5 x 3 = 1 2 cos ( A i ) L / R ,
SD y 5 y 1 = SD y 6 y 1 = SD y 5 y 3 = 1 2 1 ( L / R ) cos ( A i ) ,
SD x 6 x 3 = 1 2 [ L / R cos ( A i ) cos ( A i ) cos ( 2 A i ) ] ( L / R ) [ cos ( 2 A i ) + 1 ] ,
SD y 6 y 3 = 1 2 [ ( L / R ) cos ( A i ) 1 cos ( 2 A i ) ] ( L / R ) ( cos ( 2 A i ) + 1 ) cos ( A i ) .
SD y 5 y 1 SD x 5 x 1 = SD y 6 y 1 SD x 6 x 1 = SD y 5 y 3 SD x 5 x 3 = 1 [ cos ( A i ) ] 2 .

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