Abstract

We consider an active cavity defined by a geometrically symmetric ring of N mirrors, each of which may have an arbitrary radius. The beam steering imposed by all possible mirror misalignment parameters is shown to diverge if the radii of curvature Ri of the mirrors satisfy two constraints of the form |κN({Ri})| = 2. By comparison, the standard conditions for the stability of rays departing from the design path in a perfectly aligned ring have the form |κN({Ri})| ≤ 2, so that only approximately half of the critical radii for the latter stability conditions is also critical radii for the former. The detailed solutions identify the most sensitive misalignment parameters regarding beam steering. Certain beam steering effects vanish for resonators with κN = −2; this can be of practical importance, as, for example, the N = 2 confocal resonator.

© 1987 Optical Society of America

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References

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  1. W. F. Krupke, W. R. Sooy, “Properties of an Unstable Confocal Resonator CO2 Laser System,” IEEE J. Quantum Electron. QE-5575 (1969).
    [CrossRef]
  2. R. J. Freiberg, A. S. Halsted, “Properties of Low Order Transverse Modes in Ar+ Lasers,” Appl. Opt. 8, 355 (1969).
    [CrossRef] [PubMed]
  3. N. Yamauchi, “Resonant Modes in a Ring Laser Resonator and their Deformations by Mirror Rearrangements,” Electron. Commun. Jpn. 57-C, 92 (1974).
  4. R. Hauck, H. P. Kortz, H. Weber, “Misalignment Sensitivity of Optical Resonators,” Appl. Opt. 19, 598 (1980).
    [CrossRef] [PubMed]
  5. A. D. Schnurr, “Unstable Resonator Misalignment in Ring and Linear Toroidal Resonators,” Appl. Opt. 22, 298 (1983).
    [CrossRef] [PubMed]
  6. V. Magni, “Resonators for Solid State Lasers with Large-Volume Fundamental Mode and High Alignment Stability,” Appl. Opt. 25, 107 (1986).
    [CrossRef] [PubMed]
  7. E. F. Ishchenko, E. F. Reshetin, “Sensitivity to Misalignment of an Optical Ring Resonator,” Opt. Spektrosk. 46, 202 (1979)[Opt. Spectrosc. USSR 46, 366 (1979)].
  8. I. W. Smith, “Optical Resonator Axis Stability and Instability from First Principles,” Proc. Soc. Photo-Opt. Instrum. Eng. 412, 203 (1983).
  9. T. Habib, M.Sc. Thesis, Oklahoma State U. (1986).
  10. A. E. Siegman, Lasers (University Science Books, Mill Hill Valley, CA, 1986), p. 607.
  11. J. A. Arnaud, “Degenerate Optical Cavities. 2: Effect of Misalignments,” Appl. Opt. 8, 1909 (1969).
    [CrossRef] [PubMed]
  12. J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1981).
  13. S. A. Collins, “Analysis of Optical Resonators Involving Focusing Elements,” Appl. Opt. 3, 1263 (1964).
    [CrossRef]
  14. H. R. Bilger, “Modelling a Large Ring Resonator Gyroscope,” Report: AFOSR grant 84-0058 (1985).
  15. T. F. Ewanizky, “Ray Transfer Matrix Approach to Unstable Resonator Analysis,” Appl. Opt. 18, 724 (1979).
    [CrossRef] [PubMed]

1986 (1)

1983 (2)

A. D. Schnurr, “Unstable Resonator Misalignment in Ring and Linear Toroidal Resonators,” Appl. Opt. 22, 298 (1983).
[CrossRef] [PubMed]

I. W. Smith, “Optical Resonator Axis Stability and Instability from First Principles,” Proc. Soc. Photo-Opt. Instrum. Eng. 412, 203 (1983).

1980 (1)

1979 (2)

E. F. Ishchenko, E. F. Reshetin, “Sensitivity to Misalignment of an Optical Ring Resonator,” Opt. Spektrosk. 46, 202 (1979)[Opt. Spectrosc. USSR 46, 366 (1979)].

T. F. Ewanizky, “Ray Transfer Matrix Approach to Unstable Resonator Analysis,” Appl. Opt. 18, 724 (1979).
[CrossRef] [PubMed]

1974 (1)

N. Yamauchi, “Resonant Modes in a Ring Laser Resonator and their Deformations by Mirror Rearrangements,” Electron. Commun. Jpn. 57-C, 92 (1974).

1969 (3)

1964 (1)

Arnaud, J. A.

Bilger, H. R.

H. R. Bilger, “Modelling a Large Ring Resonator Gyroscope,” Report: AFOSR grant 84-0058 (1985).

Collins, S. A.

Ewanizky, T. F.

Freiberg, R. J.

Habib, T.

T. Habib, M.Sc. Thesis, Oklahoma State U. (1986).

Halsted, A. S.

Hauck, R.

Ishchenko, E. F.

E. F. Ishchenko, E. F. Reshetin, “Sensitivity to Misalignment of an Optical Ring Resonator,” Opt. Spektrosk. 46, 202 (1979)[Opt. Spectrosc. USSR 46, 366 (1979)].

Kortz, H. P.

Krupke, W. F.

W. F. Krupke, W. R. Sooy, “Properties of an Unstable Confocal Resonator CO2 Laser System,” IEEE J. Quantum Electron. QE-5575 (1969).
[CrossRef]

Magni, V.

Reshetin, E. F.

E. F. Ishchenko, E. F. Reshetin, “Sensitivity to Misalignment of an Optical Ring Resonator,” Opt. Spektrosk. 46, 202 (1979)[Opt. Spectrosc. USSR 46, 366 (1979)].

Schnurr, A. D.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Hill Valley, CA, 1986), p. 607.

Smith, I. W.

I. W. Smith, “Optical Resonator Axis Stability and Instability from First Principles,” Proc. Soc. Photo-Opt. Instrum. Eng. 412, 203 (1983).

Sooy, W. R.

W. F. Krupke, W. R. Sooy, “Properties of an Unstable Confocal Resonator CO2 Laser System,” IEEE J. Quantum Electron. QE-5575 (1969).
[CrossRef]

Verdeyen, J. T.

J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1981).

Weber, H.

Yamauchi, N.

N. Yamauchi, “Resonant Modes in a Ring Laser Resonator and their Deformations by Mirror Rearrangements,” Electron. Commun. Jpn. 57-C, 92 (1974).

Appl. Opt. (7)

Electron. Commun. Jpn. (1)

N. Yamauchi, “Resonant Modes in a Ring Laser Resonator and their Deformations by Mirror Rearrangements,” Electron. Commun. Jpn. 57-C, 92 (1974).

IEEE J. Quantum Electron. (1)

W. F. Krupke, W. R. Sooy, “Properties of an Unstable Confocal Resonator CO2 Laser System,” IEEE J. Quantum Electron. QE-5575 (1969).
[CrossRef]

Opt. Spektrosk. (1)

E. F. Ishchenko, E. F. Reshetin, “Sensitivity to Misalignment of an Optical Ring Resonator,” Opt. Spektrosk. 46, 202 (1979)[Opt. Spectrosc. USSR 46, 366 (1979)].

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

I. W. Smith, “Optical Resonator Axis Stability and Instability from First Principles,” Proc. Soc. Photo-Opt. Instrum. Eng. 412, 203 (1983).

Other (4)

T. Habib, M.Sc. Thesis, Oklahoma State U. (1986).

A. E. Siegman, Lasers (University Science Books, Mill Hill Valley, CA, 1986), p. 607.

J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1981).

H. R. Bilger, “Modelling a Large Ring Resonator Gyroscope,” Report: AFOSR grant 84-0058 (1985).

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Figures (3)

Fig. 1
Fig. 1

Deviations κN(β), κN(γ) and stabilities Sn(β) [= detDN(β)], SN(γ) in the in-plane (p) and sagittal (s) directions, for a resonator with N = 3 equivalent mirrors of radius R at the corners of an equilateral triangle of side l. Where the deviation is +2, −2, we have positive and negative criticality respectively. Both curves should be within these limits for optical path stability for a circulating beam deviating from the ideal ray in a perfectly aligned ring, while beam steering effects due to mirror misalignment are likely to be minimized if stability is maximized, and certainly are infinite for zero stability.

Fig. 2
Fig. 2

Deviations and stabilities as for Fig. 1, except that one mirror is an optical flat (R3 = ∞). Note the (hatched) pass bands for stable laser action in the parameter l/R.

Fig. 3
Fig. 3

Deviations and stabilities as for Fig. 1, now for a square resonator of side l with four equivalent mirrors.

Tables (1)

Tables Icon

Table I Coefficients in the Solutions of Difference Equations for Beam Stability in a Perfectly Aligned Ring. For Γ read B in the Case of in-plane Difference Equations

Equations (26)

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( s ̂ I + 1 , I + s ̂ I 1 , I ) × n I = 0 .
γ I + 1 + γ I 1 = Γ I γ I , β I + 1 + β I 1 = B I β I ,
Γ i 2 ( 1 l s / R i ) , B i 2 [ l / ( s R i ) 1 ] .
γ i 1 + γ i + 1 = Γ i γ i , i = 2 , 3 , , N 1 , γ N 1 + γ 1 = Γ N γ N ,
γ N + γ 2 = Γ 1 γ 1 ,
γ i 1 + γ i + 1 = Γ i γ i , i = 2 , 3 , , N 1 , γ N 1 + γ 1 = Γ N γ N ,
γ 1 + γ 1 = κ N ( γ ) γ 1 ,
κ N ( γ ) c N + 2 d N
M N 1 ( γ ) [ Γ 2 1 1 Γ 3 1 1 Γ 4 . . . . 1 Γ N 1 1 1 Γ N ] .
κ N ( γ ) = ( 1 ) N + 1 { Γ 1 det [ M N 1 ( γ ) ] + 11 + N 1 , N 1 } ,
κ N ( γ ) 2 = ( 1 ) N det D N ( γ ) ,
D N ( γ ) [ Γ 1 1 1 Γ 2 1 1 Γ 3 1 1 Γ 4 1 . . . . 1 1 Γ N 1 1 Γ N ] ,
det M N 1 ( γ ) = Γ 2 11 + 12 = Γ 2 det M N 2 ( γ ) det M N 3 ( γ ) ,
det D N + 1 ( γ ) + det D N 1 ( γ ) + Γ det D N ( γ ) = ( 1 ) N ( 4 2 Γ ) ,
κ N + 1 ( γ ) + κ N 1 ( γ ) = Γ κ N ( γ )
| κ N ( γ ) | 2 | κ N ( β ) | .
r i = ( ξ i , β i + η i , γ i + ζ i ) , n i = ( 1 , β i / R i + φ i , γ i / R i θ i ) ,
D N ( γ ) γ = G ,
κ N ( γ ) = 2 and / or κ N ( β ) = 2 .
0 + ( 1 l / R 1 ) ( 1 l / R 2 ) 1 .
G i = 2 s l θ i ( ζ i + 1 + ζ i 1 2 ζ i ) , i = 2 l φ i / s + ( c / s ) ( ξ i + 1 ξ i 1 ) ( η i + 1 + η i 1 + 2 η i ) .
γ = [ Γ 2 2 2 Γ 1 ] G / ( 4 Γ 1 Γ 2 ) .
γ 1 γ 2 = [ 1 g 2 g 1 g 2 1 ] G 1 [ 1 g 1 g 1 g 2 1 ] G 2 .
γ = [ Γ 2 1 Γ + 1 Γ + 1 Γ + 1 Γ 2 1 Γ + 1 Γ + 1 Γ + 1 Γ 2 1 ] G / ( 3 Γ + 2 Γ 3 ) .
γ = [ 2 Γ 2 Γ 2 Γ Γ 2 Γ 2 Γ 2 2 Γ 2 Γ 2 Γ Γ 2 Γ 2 Γ 2 ] G / [ Γ ( Γ 2 4 ) ] .
γ = [ 4 ( 1 Γ ) 2 Γ 4 2 Γ 2 Γ 2 Γ ( 1 Γ ) 2 Γ 2 Γ 4 2 Γ 4 ( 1 Γ ) 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ ( 1 Γ ) ] G / [ 4 Γ 2 8 Γ ] ,

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