Abstract

A ring laser with a beam splitter, an external reverse wave suppressor mirror, and an internal two-way coupling mirror exhibits both traveling wave and standing wave characteristics. A simple 1-D model of this hybrid laser is used to find the frequencies and losses of the steady state longitudinal modes, which depend on the mirror reflectivities and separations in an intricate manner. Adjustment of the suppressor position on a fine scale yields markedly different results for various coarse positions if the internal mirror provides adequate coupling into the reverse wave.

© 1990 Optical Society of America

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References

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  1. F. Aronowitz, R. J. Collins, “Mode Coupling Due to Backscatter in a He–Ne Traveling-Wave Ring Laser,” Appl. Phys. Lett. 9, 55–58 (1966).
    [CrossRef]
  2. W. R. Christian, L. Mandel, “Frequency Dependence of a Ring Laser with Backscattering,” Phys. Rev. A 34, 3932–3939 (1986).
    [CrossRef] [PubMed]
  3. G. V. Perevedentseva, P. A. Khandokhin, Y. A. Khanin, “Theory of a Single-Frequency Solid-State Laser,” Sov. J. Quantum Electron. 10, 71–74 (1980).
    [CrossRef]
  4. F. R. Faxvog, “Modes of Unidirectional Ring Laser,” Opt. Lett. 5, 285–287 (1980).
    [CrossRef] [PubMed]
  5. F. Aronowitz, R. J. Collins, “Lock-In and Intensity-Phase Interaction in the Ring Laser,” J. Appl. Phys. 41, 130–141 (1970).
    [CrossRef]
  6. M. M. Tehrani, L. Mandel, “Coherence Theory of the Ring Laser,” Phys. Rev. A. 17, 677–693 (1978).
    [CrossRef]
  7. R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “Unidirectional Unstable Ring Lasers,” Appl. Opt. 12, 1140–1144 (1973).
    [CrossRef] [PubMed]
  8. R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “Asymmetric Unstable Traveling-Wave Resonators,” IEEE J. Quantum Electron. QE 10, 279–289 (1974).
    [CrossRef]
  9. S. M. Jarrett, J. F. Young, “High-Efficiency Single-Frequency CW Ring Dye Laser,” Opt. Lett. 4, 176–178 (1979).
    [CrossRef] [PubMed]
  10. J. M. Bernard, R. A. Chodzko, H. Mirels, “Reverse-Wave Suppressor Mirror Effects on CW HF Unstable Ring Laser Performance,” Appl. Opt. 25, 666–671 (1986).
    [CrossRef] [PubMed]
  11. H. Mirels, R. A. Chodzko, J. M. Bernard, R. R. Geidt, J. G. Coffer, “Reverse Wave Suppression in Unstable Ring Resonator,” Appl. Opt. 23, 4509–4517 (1984).
    [CrossRef] [PubMed]
  12. W. P. Latham, “Analysis of Reverse Wave Suppression in Multiline Chemical Lasers,” Proc. Soc. Photo-Opt. Instrum. Eng. 642, 28–35 (1986).
  13. A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986), Chap. 11.
  14. W. W. Rigrod, “Homogeneously Broadened CW Lasers with Uniform Distributed Loss,” IEEE J Quantum Electron. QE 14, 377–381 (1978).
    [CrossRef]
  15. S. M. Rinaldi, “Semiclassical Theory of Injected Lasers with Arbitrary Stable and Unstable Resonators,” PhD Dissertation, AFIT/DS/PH/87-3, U.S. Air Force Institute of Technology, Wright-Patterson AFB, OH 45433.
  16. J. A. Benda, W. J. Fader, G. E. Palma, “The Influence of Coupled Resonator Configurations on Supermode Discrimination,” Proc. Soc. Photo-Opt. Instrum. Eng. 642, 42–50 (1986).

1986

W. R. Christian, L. Mandel, “Frequency Dependence of a Ring Laser with Backscattering,” Phys. Rev. A 34, 3932–3939 (1986).
[CrossRef] [PubMed]

W. P. Latham, “Analysis of Reverse Wave Suppression in Multiline Chemical Lasers,” Proc. Soc. Photo-Opt. Instrum. Eng. 642, 28–35 (1986).

J. A. Benda, W. J. Fader, G. E. Palma, “The Influence of Coupled Resonator Configurations on Supermode Discrimination,” Proc. Soc. Photo-Opt. Instrum. Eng. 642, 42–50 (1986).

J. M. Bernard, R. A. Chodzko, H. Mirels, “Reverse-Wave Suppressor Mirror Effects on CW HF Unstable Ring Laser Performance,” Appl. Opt. 25, 666–671 (1986).
[CrossRef] [PubMed]

1984

1980

F. R. Faxvog, “Modes of Unidirectional Ring Laser,” Opt. Lett. 5, 285–287 (1980).
[CrossRef] [PubMed]

G. V. Perevedentseva, P. A. Khandokhin, Y. A. Khanin, “Theory of a Single-Frequency Solid-State Laser,” Sov. J. Quantum Electron. 10, 71–74 (1980).
[CrossRef]

1979

1978

W. W. Rigrod, “Homogeneously Broadened CW Lasers with Uniform Distributed Loss,” IEEE J Quantum Electron. QE 14, 377–381 (1978).
[CrossRef]

M. M. Tehrani, L. Mandel, “Coherence Theory of the Ring Laser,” Phys. Rev. A. 17, 677–693 (1978).
[CrossRef]

1974

R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “Asymmetric Unstable Traveling-Wave Resonators,” IEEE J. Quantum Electron. QE 10, 279–289 (1974).
[CrossRef]

1973

1970

F. Aronowitz, R. J. Collins, “Lock-In and Intensity-Phase Interaction in the Ring Laser,” J. Appl. Phys. 41, 130–141 (1970).
[CrossRef]

1966

F. Aronowitz, R. J. Collins, “Mode Coupling Due to Backscatter in a He–Ne Traveling-Wave Ring Laser,” Appl. Phys. Lett. 9, 55–58 (1966).
[CrossRef]

Aronowitz, F.

F. Aronowitz, R. J. Collins, “Lock-In and Intensity-Phase Interaction in the Ring Laser,” J. Appl. Phys. 41, 130–141 (1970).
[CrossRef]

F. Aronowitz, R. J. Collins, “Mode Coupling Due to Backscatter in a He–Ne Traveling-Wave Ring Laser,” Appl. Phys. Lett. 9, 55–58 (1966).
[CrossRef]

Benda, J. A.

J. A. Benda, W. J. Fader, G. E. Palma, “The Influence of Coupled Resonator Configurations on Supermode Discrimination,” Proc. Soc. Photo-Opt. Instrum. Eng. 642, 42–50 (1986).

Bernard, J. M.

Buczek, C. J.

R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “Asymmetric Unstable Traveling-Wave Resonators,” IEEE J. Quantum Electron. QE 10, 279–289 (1974).
[CrossRef]

R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “Unidirectional Unstable Ring Lasers,” Appl. Opt. 12, 1140–1144 (1973).
[CrossRef] [PubMed]

Chenausky, P. P.

R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “Asymmetric Unstable Traveling-Wave Resonators,” IEEE J. Quantum Electron. QE 10, 279–289 (1974).
[CrossRef]

R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “Unidirectional Unstable Ring Lasers,” Appl. Opt. 12, 1140–1144 (1973).
[CrossRef] [PubMed]

Chodzko, R. A.

Christian, W. R.

W. R. Christian, L. Mandel, “Frequency Dependence of a Ring Laser with Backscattering,” Phys. Rev. A 34, 3932–3939 (1986).
[CrossRef] [PubMed]

Coffer, J. G.

Collins, R. J.

F. Aronowitz, R. J. Collins, “Lock-In and Intensity-Phase Interaction in the Ring Laser,” J. Appl. Phys. 41, 130–141 (1970).
[CrossRef]

F. Aronowitz, R. J. Collins, “Mode Coupling Due to Backscatter in a He–Ne Traveling-Wave Ring Laser,” Appl. Phys. Lett. 9, 55–58 (1966).
[CrossRef]

Fader, W. J.

J. A. Benda, W. J. Fader, G. E. Palma, “The Influence of Coupled Resonator Configurations on Supermode Discrimination,” Proc. Soc. Photo-Opt. Instrum. Eng. 642, 42–50 (1986).

Faxvog, F. R.

Freiberg, R. J.

R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “Asymmetric Unstable Traveling-Wave Resonators,” IEEE J. Quantum Electron. QE 10, 279–289 (1974).
[CrossRef]

R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “Unidirectional Unstable Ring Lasers,” Appl. Opt. 12, 1140–1144 (1973).
[CrossRef] [PubMed]

Geidt, R. R.

Jarrett, S. M.

Khandokhin, P. A.

G. V. Perevedentseva, P. A. Khandokhin, Y. A. Khanin, “Theory of a Single-Frequency Solid-State Laser,” Sov. J. Quantum Electron. 10, 71–74 (1980).
[CrossRef]

Khanin, Y. A.

G. V. Perevedentseva, P. A. Khandokhin, Y. A. Khanin, “Theory of a Single-Frequency Solid-State Laser,” Sov. J. Quantum Electron. 10, 71–74 (1980).
[CrossRef]

Latham, W. P.

W. P. Latham, “Analysis of Reverse Wave Suppression in Multiline Chemical Lasers,” Proc. Soc. Photo-Opt. Instrum. Eng. 642, 28–35 (1986).

Mandel, L.

W. R. Christian, L. Mandel, “Frequency Dependence of a Ring Laser with Backscattering,” Phys. Rev. A 34, 3932–3939 (1986).
[CrossRef] [PubMed]

M. M. Tehrani, L. Mandel, “Coherence Theory of the Ring Laser,” Phys. Rev. A. 17, 677–693 (1978).
[CrossRef]

Mirels, H.

Palma, G. E.

J. A. Benda, W. J. Fader, G. E. Palma, “The Influence of Coupled Resonator Configurations on Supermode Discrimination,” Proc. Soc. Photo-Opt. Instrum. Eng. 642, 42–50 (1986).

Perevedentseva, G. V.

G. V. Perevedentseva, P. A. Khandokhin, Y. A. Khanin, “Theory of a Single-Frequency Solid-State Laser,” Sov. J. Quantum Electron. 10, 71–74 (1980).
[CrossRef]

Rigrod, W. W.

W. W. Rigrod, “Homogeneously Broadened CW Lasers with Uniform Distributed Loss,” IEEE J Quantum Electron. QE 14, 377–381 (1978).
[CrossRef]

Rinaldi, S. M.

S. M. Rinaldi, “Semiclassical Theory of Injected Lasers with Arbitrary Stable and Unstable Resonators,” PhD Dissertation, AFIT/DS/PH/87-3, U.S. Air Force Institute of Technology, Wright-Patterson AFB, OH 45433.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986), Chap. 11.

Tehrani, M. M.

M. M. Tehrani, L. Mandel, “Coherence Theory of the Ring Laser,” Phys. Rev. A. 17, 677–693 (1978).
[CrossRef]

Young, J. F.

Appl. Opt.

Appl. Phys. Lett.

F. Aronowitz, R. J. Collins, “Mode Coupling Due to Backscatter in a He–Ne Traveling-Wave Ring Laser,” Appl. Phys. Lett. 9, 55–58 (1966).
[CrossRef]

IEEE J Quantum Electron.

W. W. Rigrod, “Homogeneously Broadened CW Lasers with Uniform Distributed Loss,” IEEE J Quantum Electron. QE 14, 377–381 (1978).
[CrossRef]

IEEE J. Quantum Electron.

R. J. Freiberg, P. P. Chenausky, C. J. Buczek, “Asymmetric Unstable Traveling-Wave Resonators,” IEEE J. Quantum Electron. QE 10, 279–289 (1974).
[CrossRef]

J. Appl. Phys.

F. Aronowitz, R. J. Collins, “Lock-In and Intensity-Phase Interaction in the Ring Laser,” J. Appl. Phys. 41, 130–141 (1970).
[CrossRef]

Opt. Lett.

Phys. Rev. A

W. R. Christian, L. Mandel, “Frequency Dependence of a Ring Laser with Backscattering,” Phys. Rev. A 34, 3932–3939 (1986).
[CrossRef] [PubMed]

Phys. Rev. A.

M. M. Tehrani, L. Mandel, “Coherence Theory of the Ring Laser,” Phys. Rev. A. 17, 677–693 (1978).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

W. P. Latham, “Analysis of Reverse Wave Suppression in Multiline Chemical Lasers,” Proc. Soc. Photo-Opt. Instrum. Eng. 642, 28–35 (1986).

Proc. Soc. Photo-Opt. Instrum. Eng.

J. A. Benda, W. J. Fader, G. E. Palma, “The Influence of Coupled Resonator Configurations on Supermode Discrimination,” Proc. Soc. Photo-Opt. Instrum. Eng. 642, 42–50 (1986).

Sov. J. Quantum Electron.

G. V. Perevedentseva, P. A. Khandokhin, Y. A. Khanin, “Theory of a Single-Frequency Solid-State Laser,” Sov. J. Quantum Electron. 10, 71–74 (1980).
[CrossRef]

Other

A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986), Chap. 11.

S. M. Rinaldi, “Semiclassical Theory of Injected Lasers with Arbitrary Stable and Unstable Resonators,” PhD Dissertation, AFIT/DS/PH/87-3, U.S. Air Force Institute of Technology, Wright-Patterson AFB, OH 45433.

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Figures (17)

Fig. 1
Fig. 1

Hybrid ring laser with standing wave and traveling wave feedback.

Fig. 2
Fig. 2

Phasor diagram of Eq. (24) for interpreting solution periodicity during suppressor position fine-tuning.

Fig. 3
Fig. 3

Map of regions of dominant feedback type (shaded areas). The dashed line separates regions of different solution periodicity. The ordinate is r 1 2 and the abscissa is R = r b 2 r 1 r 2 / t b 2 .

Fig. 4
Fig. 4

Net gains G1 and G2 of the two solutions of Example 1 as functions of the position fine-tuning parameter φ, for several values of the reverse-wave suppressor reflectivity r2. (r1 = 0.2, rb = 0.5, Δ = 0).

Fig. 5
Fig. 5

As in Fig. 4, showing change in periodicity in φ for larger r2.

Fig. 6
Fig. 6

Values of θ for the two solutions corresponding to Fig. 4.

Fig. 7
Fig. 7

Values of θ for the two solutions corresponding to Fig. 5.

Fig. 8
Fig. 8

Ratio ρ of field amplitudes for first example.

Fig. 9
Fig. 9

Net gains G1 and G2 for Example 2. (r1 = 0.5, rb = 0.9759, Δ = 0)

Fig. 10
Fig. 10

Values of θ corresponding to Fig. 9.

Fig. 11
Fig. 11

Relationship of periodic dependence of frequency on φ to the neighboring longitudinal modes, Example 2 with r2 = 0.25.

Fig. 12
Fig. 12

Mode gains for Example 3 vs r2, showing a bifurcation and a singularity in G1.

Fig. 13
Fig. 13

Mode frequencies for Example 3, showing a reverse bifurcation and a frequency discontinuity at the gain singularity.

Fig. 14
Fig. 14

Effect of the coarse position parameter Δ on the fine-tuning curves. The θ1 curve for Δ = 0 and r2 = 0.2 from Fig. 6 is reproduced at the top.

Fig. 15
Fig. 15

Effect of Δ on mode gains corresponding to Fig. 14. Note the loss of mode separation for Δ = 3.

Fig. 16
Fig. 16

Effect of Δ on the fine-tuning curves. The θ1 curve for Δ = 0 and r2 = 0.05 from Fig. 10 is reproduced as a solid line.

Fig. 17
Fig. 17

Effect of Δ on mode gains corresponding to Fig. 16. Note multiple low-loss modes for modest values of Δ.

Equations (41)

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L = L 1 + 1 + L 2 + .
[ E 1 E 2 ] = S [ E 3 E 4 ] ,
S 11 = ( i t 1 ) ( i t b ) exp [ i k ( L 1 + 1 + L 2 ) ] ,
S 12 = ( i t b ) 2 r 1 exp [ i 2 k ( L 2 + 1 ) ] + r b 2 r 2 exp [ i 2 k ( L 2 + 2 ) ] ,
S 21 = r 1 exp [ i 2 k L 1 ] ,
S 22 = ( i t b ) ( i t 1 ) exp [ i k ( L 2 + 1 + L 1 ) ] ,
E 3 = E 1 G f exp [ i k ] ,
E 4 = E 2 G r exp [ i k ] ,
E 1 = - t 1 t b G f E 1 exp [ i k L ] + r eff G r E 2 exp [ i k ( 2 L 2 + 2 1 + ) ] ,
E 2 = - t 1 t b G r E 2 exp [ i k L ] + r 1 G f E 1 exp [ i k ( 2 L 1 + ) ] ,
r eff = r b 2 r 2 exp [ i 2 k ( 2 - 1 ) ] - t b 2 r 1
[ 1 + t 1 t b G f exp ( i k L ) ] [ 1 + t 1 t b G r exp ( i k L ) ] - r 1 r eff G f G r exp ( i 2 k L ) = 0.
1 I f d I f d z = - 1 I r d I r d z .
I 1 I 4 = I 2 I 3 ,
G f 2 = I 3 / I 1 = I 4 / I 2 = G r 2 = G 2 .
k L = ( 2 n + 1 ) π + θ ,
ω = k c = [ ( 2 n + 1 ) π + θ ] c / L .
f = 2 k ( 2 - 1 ) = 2 k L Δ
Δ = ( m + φ / 2 π ) / ( 2 n + 1 ) ,
f = m 2 π + φ + 2 θ Δ .
R = r b 2 r 1 r 2 / t b 2
r eff = ( t b 2 / r 1 ) [ R exp ( i f ) - r 1 2 ] ,
[ exp ( - i θ ) - t 1 t b G ] 2 - ( G t b ) 2 [ R exp ( i f ) - r 1 2 ] = 0 ,
R exp ( i f ) - r 1 2 = B exp ( i β )
B cos ( β ) = R cos ( f ) - r 1 2 ,
B sin ( β ) = R sin ( f ) ,
B = [ R 2 + r 1 4 - 2 R r 1 2 cos ( f ) ] 1 / 2 0 ,
G t b [ t 1 B 1 / 2 exp ( i β / 2 ) ] = exp ( - i θ ) .
t 1 B 1 / 2 exp ( i β / 2 ) = D exp ( i δ ) ,
D cos ( δ ) = t 1 B 1 / 2 cos ( β / 2 ) ,
D sin ( δ ) = B 1 / 2 sin ( β / 2 ) ,
D = [ t 1 2 + B 2 t 1 B 1 / 2 cos ( β / 2 ) ] 1 / 2 0 ,
G t b D exp ( i δ ) = exp ( - i θ ) ,
G = 1 / ( t b D ) ,
θ = - δ .
Γ = TWF / SWF = t 1 / B 1 / 2 ,
F = ( R 2 + 2 r 1 2 - 1 ) / 2 R r 1 2
ρ = E 1 / E 2 = t b B 1 / 2 / r 1 = t 1 t b / r 1 Γ .
ρ 14 = E 1 / E 4 = ρ / G
ρ 23 = E 2 / E 3 = ρ G
ρ 0 = t 1 exp ( - i θ ) - t b G [ 1 + r 1 r 2 exp ( i f ) ] / ( G r 1 t 2 ) .

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