Abstract

Two laser stability criteria or lasing conditions for cavity geometry based on the standard ABCD matrix analysis and the Bilger and Stedman analysis [Appl. Opt. 26, 3710 (1987)] are reconciled. Beam steering from mirror misalignment is discussed similarly, generalizing the Bilger and Stedman analysis to nonregular polygons by extending the standard ABCD matrix analysis to 3 × 3 matrices, which facilitates the thorough design of large rectangular ring lasers and is applied to a number of existing or planned ring lasers with perimeters of 77–120 m.

© 2002 Optical Society of America

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References

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  1. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
    [CrossRef]
  2. W. W. Rigrod, “The optical ring resonator,” Bell Syst. Tech. J. 44, 907–916 (1965).
    [CrossRef]
  3. A. Siegman, Lasers (University Science, Mill Valley Calif., 1986), p. 744.
  4. H. R. Bilger, G. E. Stedman, “Stability of planar ring lasers with mirror misalignment,” Appl. Opt. 26, 3710–3716 (1987).
    [CrossRef] [PubMed]
  5. R. W. Dunn, D. E. Shabalin, R. J. Thirkettle, G. J. MacDonald, G. E. Stedman, K. U. Schreiber, “Design and initial operation of a 367-m2 rectangular ring laser,” Appl. Opt. 41, 1685–1688 (2002).
    [CrossRef] [PubMed]
  6. C. H. Rowe, K. U. Schreiber, S. J. Cooper, B. T. King, M. Poulton, G. E. Stedman, “Design and operation of a very large ring laser gyroscope,” Appl. Opt. 38, 2516–2523 (1999).
    [CrossRef]
  7. R. W. Dunn, “Design of a triangular active ring laser 13 m on a side,” Appl. Opt. 37, 6405–6409 (1998).
    [CrossRef]
  8. V. E. Zharov, S. N. Markova, M. V. Sazhin, E. D. Fedoseev, “Calculation of an optical system for a laser-controlled gyroscope,” Moscow U. Phys. Bull. 47, 86–90 (1992).
  9. A. K. Bhowmik, “Polygonal optical cavities,” Appl. Opt. 39, 3071–3075 (2000).
    [CrossRef]
  10. T. Skettrup, T. Meelby, T. K. Faerch, S. L. Frederiksen, C. Pedersen, “Triangular laser resonators with astigmatic compensation,” Appl. Opt. 39, 4306–4312 (2000).
    [CrossRef]
  11. A. Pancha, T. H. Webb, G. E. Stedman, D. P. McLeod, K. U. Schreiber, “Ring laser detection of rotations from teleseismic waves,” Geophys. Res. Lett. 27, 3553–3556 (2000).
    [CrossRef]
  12. K. U. Schreiber, G. E. Stedman, “Earth tide and tilt detection by a ring laser gyroscope,” J. Geophys. Res. Solid Earth (submitted for publication).
  13. //www.phys.canterbury.ac.nz/p̃hysges/RingStabSpot.xls .
  14. A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart & Winston, New York, 1976), pp. 72–74.
  15. G. E. Stedman, “Ring-laser tests of fundamental physics and geophysics,” Rep. Prog. Phys. 60, 1–73 (1997).
    [CrossRef]
  16. K. U. Schreiber, C. H. Rowe, D. N. Wright, S. J. Cooper, G. E. Stedman, “Precision stabilization of the optical frequency in a large ring laser gyroscope,” Appl. Opt. 37, 8371–8381 (1998).
    [CrossRef]
  17. K. U. Schreiber, M. Schneider, C. H. Rowe, G. E. Stedman, and W. Schlüter, “Stabilising the operation of a large ring laser,” in Proceedings of the Symposium on Gyro Technology 1999, H. Sorg, ed. (Universität Stuttgart, Institut A für Mechanik, Stuttgart, Germany, September1999), pp. 14.0–14.7.
  18. D. G. Glynn, Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand (personal communication, 1999).
  19. T. Muir, W. H. Metzler, “A treatise on the theory of determinants,” (New York: Dover Publications, 1960).

2002

2000

1999

1998

1997

G. E. Stedman, “Ring-laser tests of fundamental physics and geophysics,” Rep. Prog. Phys. 60, 1–73 (1997).
[CrossRef]

1992

V. E. Zharov, S. N. Markova, M. V. Sazhin, E. D. Fedoseev, “Calculation of an optical system for a laser-controlled gyroscope,” Moscow U. Phys. Bull. 47, 86–90 (1992).

1987

1965

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

W. W. Rigrod, “The optical ring resonator,” Bell Syst. Tech. J. 44, 907–916 (1965).
[CrossRef]

Bhowmik, A. K.

Bilger, H. R.

Cooper, S. J.

Dunn, R. W.

Faerch, T. K.

Fedoseev, E. D.

V. E. Zharov, S. N. Markova, M. V. Sazhin, E. D. Fedoseev, “Calculation of an optical system for a laser-controlled gyroscope,” Moscow U. Phys. Bull. 47, 86–90 (1992).

Frederiksen, S. L.

Glynn, D. G.

D. G. Glynn, Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand (personal communication, 1999).

King, B. T.

Kogelnik, H.

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

MacDonald, G. J.

Markova, S. N.

V. E. Zharov, S. N. Markova, M. V. Sazhin, E. D. Fedoseev, “Calculation of an optical system for a laser-controlled gyroscope,” Moscow U. Phys. Bull. 47, 86–90 (1992).

McLeod, D. P.

A. Pancha, T. H. Webb, G. E. Stedman, D. P. McLeod, K. U. Schreiber, “Ring laser detection of rotations from teleseismic waves,” Geophys. Res. Lett. 27, 3553–3556 (2000).
[CrossRef]

Meelby, T.

Metzler, W. H.

T. Muir, W. H. Metzler, “A treatise on the theory of determinants,” (New York: Dover Publications, 1960).

Muir, T.

T. Muir, W. H. Metzler, “A treatise on the theory of determinants,” (New York: Dover Publications, 1960).

Pancha, A.

A. Pancha, T. H. Webb, G. E. Stedman, D. P. McLeod, K. U. Schreiber, “Ring laser detection of rotations from teleseismic waves,” Geophys. Res. Lett. 27, 3553–3556 (2000).
[CrossRef]

Pedersen, C.

Poulton, M.

Rigrod, W. W.

W. W. Rigrod, “The optical ring resonator,” Bell Syst. Tech. J. 44, 907–916 (1965).
[CrossRef]

Rowe, C. H.

C. H. Rowe, K. U. Schreiber, S. J. Cooper, B. T. King, M. Poulton, G. E. Stedman, “Design and operation of a very large ring laser gyroscope,” Appl. Opt. 38, 2516–2523 (1999).
[CrossRef]

K. U. Schreiber, C. H. Rowe, D. N. Wright, S. J. Cooper, G. E. Stedman, “Precision stabilization of the optical frequency in a large ring laser gyroscope,” Appl. Opt. 37, 8371–8381 (1998).
[CrossRef]

K. U. Schreiber, M. Schneider, C. H. Rowe, G. E. Stedman, and W. Schlüter, “Stabilising the operation of a large ring laser,” in Proceedings of the Symposium on Gyro Technology 1999, H. Sorg, ed. (Universität Stuttgart, Institut A für Mechanik, Stuttgart, Germany, September1999), pp. 14.0–14.7.

Sazhin, M. V.

V. E. Zharov, S. N. Markova, M. V. Sazhin, E. D. Fedoseev, “Calculation of an optical system for a laser-controlled gyroscope,” Moscow U. Phys. Bull. 47, 86–90 (1992).

Schneider, M.

K. U. Schreiber, M. Schneider, C. H. Rowe, G. E. Stedman, and W. Schlüter, “Stabilising the operation of a large ring laser,” in Proceedings of the Symposium on Gyro Technology 1999, H. Sorg, ed. (Universität Stuttgart, Institut A für Mechanik, Stuttgart, Germany, September1999), pp. 14.0–14.7.

Schreiber, K. U.

R. W. Dunn, D. E. Shabalin, R. J. Thirkettle, G. J. MacDonald, G. E. Stedman, K. U. Schreiber, “Design and initial operation of a 367-m2 rectangular ring laser,” Appl. Opt. 41, 1685–1688 (2002).
[CrossRef] [PubMed]

A. Pancha, T. H. Webb, G. E. Stedman, D. P. McLeod, K. U. Schreiber, “Ring laser detection of rotations from teleseismic waves,” Geophys. Res. Lett. 27, 3553–3556 (2000).
[CrossRef]

C. H. Rowe, K. U. Schreiber, S. J. Cooper, B. T. King, M. Poulton, G. E. Stedman, “Design and operation of a very large ring laser gyroscope,” Appl. Opt. 38, 2516–2523 (1999).
[CrossRef]

K. U. Schreiber, C. H. Rowe, D. N. Wright, S. J. Cooper, G. E. Stedman, “Precision stabilization of the optical frequency in a large ring laser gyroscope,” Appl. Opt. 37, 8371–8381 (1998).
[CrossRef]

K. U. Schreiber, G. E. Stedman, “Earth tide and tilt detection by a ring laser gyroscope,” J. Geophys. Res. Solid Earth (submitted for publication).

K. U. Schreiber, M. Schneider, C. H. Rowe, G. E. Stedman, and W. Schlüter, “Stabilising the operation of a large ring laser,” in Proceedings of the Symposium on Gyro Technology 1999, H. Sorg, ed. (Universität Stuttgart, Institut A für Mechanik, Stuttgart, Germany, September1999), pp. 14.0–14.7.

Shabalin, D. E.

Siegman, A.

A. Siegman, Lasers (University Science, Mill Valley Calif., 1986), p. 744.

Skettrup, T.

Stedman, G. E.

R. W. Dunn, D. E. Shabalin, R. J. Thirkettle, G. J. MacDonald, G. E. Stedman, K. U. Schreiber, “Design and initial operation of a 367-m2 rectangular ring laser,” Appl. Opt. 41, 1685–1688 (2002).
[CrossRef] [PubMed]

A. Pancha, T. H. Webb, G. E. Stedman, D. P. McLeod, K. U. Schreiber, “Ring laser detection of rotations from teleseismic waves,” Geophys. Res. Lett. 27, 3553–3556 (2000).
[CrossRef]

C. H. Rowe, K. U. Schreiber, S. J. Cooper, B. T. King, M. Poulton, G. E. Stedman, “Design and operation of a very large ring laser gyroscope,” Appl. Opt. 38, 2516–2523 (1999).
[CrossRef]

K. U. Schreiber, C. H. Rowe, D. N. Wright, S. J. Cooper, G. E. Stedman, “Precision stabilization of the optical frequency in a large ring laser gyroscope,” Appl. Opt. 37, 8371–8381 (1998).
[CrossRef]

G. E. Stedman, “Ring-laser tests of fundamental physics and geophysics,” Rep. Prog. Phys. 60, 1–73 (1997).
[CrossRef]

H. R. Bilger, G. E. Stedman, “Stability of planar ring lasers with mirror misalignment,” Appl. Opt. 26, 3710–3716 (1987).
[CrossRef] [PubMed]

K. U. Schreiber, M. Schneider, C. H. Rowe, G. E. Stedman, and W. Schlüter, “Stabilising the operation of a large ring laser,” in Proceedings of the Symposium on Gyro Technology 1999, H. Sorg, ed. (Universität Stuttgart, Institut A für Mechanik, Stuttgart, Germany, September1999), pp. 14.0–14.7.

K. U. Schreiber, G. E. Stedman, “Earth tide and tilt detection by a ring laser gyroscope,” J. Geophys. Res. Solid Earth (submitted for publication).

Thirkettle, R. J.

Webb, T. H.

A. Pancha, T. H. Webb, G. E. Stedman, D. P. McLeod, K. U. Schreiber, “Ring laser detection of rotations from teleseismic waves,” Geophys. Res. Lett. 27, 3553–3556 (2000).
[CrossRef]

Wright, D. N.

Yariv, A.

A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart & Winston, New York, 1976), pp. 72–74.

Zharov, V. E.

V. E. Zharov, S. N. Markova, M. V. Sazhin, E. D. Fedoseev, “Calculation of an optical system for a laser-controlled gyroscope,” Moscow U. Phys. Bull. 47, 86–90 (1992).

Appl. Opt.

Bell Syst. Tech. J.

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

W. W. Rigrod, “The optical ring resonator,” Bell Syst. Tech. J. 44, 907–916 (1965).
[CrossRef]

Geophys. Res. Lett.

A. Pancha, T. H. Webb, G. E. Stedman, D. P. McLeod, K. U. Schreiber, “Ring laser detection of rotations from teleseismic waves,” Geophys. Res. Lett. 27, 3553–3556 (2000).
[CrossRef]

Moscow U. Phys. Bull.

V. E. Zharov, S. N. Markova, M. V. Sazhin, E. D. Fedoseev, “Calculation of an optical system for a laser-controlled gyroscope,” Moscow U. Phys. Bull. 47, 86–90 (1992).

Rep. Prog. Phys.

G. E. Stedman, “Ring-laser tests of fundamental physics and geophysics,” Rep. Prog. Phys. 60, 1–73 (1997).
[CrossRef]

Other

K. U. Schreiber, M. Schneider, C. H. Rowe, G. E. Stedman, and W. Schlüter, “Stabilising the operation of a large ring laser,” in Proceedings of the Symposium on Gyro Technology 1999, H. Sorg, ed. (Universität Stuttgart, Institut A für Mechanik, Stuttgart, Germany, September1999), pp. 14.0–14.7.

D. G. Glynn, Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand (personal communication, 1999).

T. Muir, W. H. Metzler, “A treatise on the theory of determinants,” (New York: Dover Publications, 1960).

K. U. Schreiber, G. E. Stedman, “Earth tide and tilt detection by a ring laser gyroscope,” J. Geophys. Res. Solid Earth (submitted for publication).

//www.phys.canterbury.ac.nz/p̃hysges/RingStabSpot.xls .

A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart & Winston, New York, 1976), pp. 72–74.

A. Siegman, Lasers (University Science, Mill Valley Calif., 1986), p. 744.

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Figures (2)

Fig. 1
Fig. 1

In-plane (solid curve) and out-of-plane (dashed curve) stability, dimensionless parameter S, in a UG1 geometry. The abscissa is the astigmatically corrected ROC Q (normalized by perimeter P of 77 m) common to each mirror [CCCC configuration, Eq. (21)].

Fig. 2
Fig. 2

Stability S for UG1 (upper) and UG2 (lower) in the C 1 C 2 C 2 C 1 configuration plotted as a function of δ1 and δ2. The ratios of the perimeter versus the astigmatically corrected mirror radii are shown.

Tables (2)

Tables Icon

Table 1 Relevant Parameters for the Typical Ring Laser Gyroscope Situated in the Cashmere Caverna

Tables Icon

Table 2 Design Parameters for the Ultra-G Family of Ring Lasersa

Equations (38)

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

XnABCDn; un+1 =Xnun, uxyn.
Lf=10-1/f1, MR=10-2/R1, Dd=1d01.
X-1=Y A-B-CD,
MQ= 10-2/Q1.
|A+D|=|χ ML|2.
DNγ=G,
DN= -Γ11011-Γ21001-Γ30100-ΓN.
Gi=-2l ηi/Ri+ϕi-cosπ/Nμi+1-μi-1,
Gi=2l sinπ/Nνi/Ri-θi,
G=-2lα, αif θi-νi/Ri,
κNγ -1N det DN+2; |κNγ|2.
S=A+D= -NκN.
1q= D-A2B± i2B  4- A+D2.
d= D-A2C; w0=λπ4-A+D22|C|.
κNγ2.
X¯=100EABFCD,
M¯iαi, Qi=1000102αi-2/Qi1, M¯iˆ=D¯lM¯i=1002lαi1-2l/Qil2αi-2/Qi1,
M¯iL-Iui=0.
Ai-1BiCiDi-1riri =-EiFi.
efi=Miefi-1+2αil1.
γαi=rαi.
S=2+127-2χ-χ2δ1δ2+ 143+2χ-χ2×δ12+δ22 +12χ2δ1δ2-8-δ1δ2δ1+δ2+ 116 δ12δ221-χ22.
L2L1X=1- 41-χδ1-δ28+δ1δ21-χ2-4δ1+δ2+21-χδ1-δ2, L1=P8 32-1+χ23-χδ1+δ2-δ1δ21-χ2X2δ1+δ2-1-χδ1δ2.
S=-4+12 δ1δ2δ1+δ2+ 34δ12+δ22+ 116 δ12δ22+2+ 72 δ1δ2, X=1- 4δ1-δ28+δ1δ2-6δ2-2δ1.
DN=-A1011-B1001-C0100-Z,
χML=χPAPBPCPZ, PX, l1l-X-2/l-X-1.
-1Ndet DN+2=χ 11A-2A-1 ·· 11Z-2Z-1.
s=1110, p¯I=I-110,
χP¯=χ p¯Ap¯B p¯Z=χpApZ.
C-A, -B,, -Y, -Z=det-A100001-B100001-C000000-X100001-Y100001-Z.
C-A, -B,, -Y, -Z=-ZC-A, -B,, -Y-C-A, -B,, -X.
det M+2-1N=C-A, -B,, -Y, -Z-C-B,, -Y.
P¯=-1NC-A, -B,, -Y, -Z-C-B,,-Y, -ZC-A, -B,, -Y-C -B,, -Y.
r1r1=2Mˆ2Mˆ1-1-1α2+Mˆ2α1l1=-Q1Q2-α1l/Q2+α1+α2Q1α1-α2Q21Q1+Q2-l, γ1γ2=-21-lQ122-21-lQ2-1-2lα=-Q1Q2α1+α2-α1l/Q2-Q1Q2α1+α2-α2l/Q11Q1+Q2-l.
--A22-B-1α1α2 =α1B+2α2Aα2+2α11BA-4
B-11B-21A-11A-21-1-1α2+B-11B-21α1 ×11=α1B+2α2α1B-Aα2-2α1+2α21BA-4
--A111-B111-C-1α1α2α3 =Cα1B-α1+α2+α3+Cα2+Bα3Cα1+α1-α2+CAα2+α3+α3Aα1+α2-α3+Aα2+α1B+BAα31CBA-C-A-B-2
C-11C-21B-11B-21A-11A-21 -1-1α3+C-11C-21α2+B-11B-21 α111= Cα1B-α1+α2+α3+Cα2+Bα3Cα1B-2α1+2α3-Aα2-α1B+Cα2+Bα3-BAα3CBA-C-A-B-2

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