Abstract

The analysis of the stability of the polarization modes of a quasi-isotropic laser is extended. The theory predicts the existence of catastrophes in the hysteresis loop of polarization versus frequency. Allowance is made for a weak cavity birefringence. Experiments with a He–Ne laser that operates at 3.39 μm quantitatively confirm the dependence of the hysteresis on gain, including the previously unidentified catastrophes.

© 1991 Optical Society of America

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References

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  1. For earlier (pioneering) work see the paper by D. Lenstra, “On the polarization effects in gas lasers,” Phys. Rep. 59, 299 (1980), and references therein.
    [Crossref]
  2. G. Stéphan and D. Hugon, “Light polarization of a quasi-isotropic laser with optical feedback,” Phys. Rev. Lett. 55, 703 (1985).
    [Crossref] [PubMed]
  3. A. D. May and G. Stéphan, “Stability of polarized modes in a quasi-isotropic laser,” J. Opt. Soc. Am. B 6, 2355 (1989). [There are three typographical errors in the paper. The characters fs2p9.7 should be removed from Eq. (1); the matrix V has 1 along the diagonal and −1 off the diagonal (p. 2357); and Eqs. (A6)–(A8) are each missing an overall minus sign on the right-hand side.]
    [Crossref]
  4. A. D. May and G. Stéphan, “Polarization dynamics in a quasi-isotropic laser,” J. H. Eberly, L. Mandel, and E. Wolf, eds., in Coherence and Quantum Optics VI (Plenum, New York, 1990), p. 731.
    [Crossref]
  5. W. E. Lamb, “Theory of an optical maser,” Phys. Rev. A 134, 1429 (1964).
  6. G. Stéphan, A. D. May, R. E. Mueller, and B. Aissaoui, “Competition effects in the polarization of light in a quasi-isotropic laser,” J. Opt. Soc. Am. B 4, 1276 (1987).
    [Crossref]
  7. A. preliminary report on the catastrophies has been given in P. Paddon, P. Glanznig, A. D. May, and G. Stéphan, “Comportement catastrophique du domain de bistabilité d’un laser vectoriel,” Ann. Phys. (Paris) 15, 177 (1990).
  8. See, for example, F. Aronowitz and W. L. Lim, “Positive scale factor correction in the laser gyro,” IEEE J. Quantum Electron. QE-13, 338 (1977).
    [Crossref]
  9. Liapunov exponents determine the stability of stationary solutions of nonlinear equations. The equations are linearized in departures of the variables from their stationary value. If new variables are chosen that diagonalize the linear equations, then the coefficients are the Liapunov exponents λ. Since the equations have the form d(δv)/dt= λδv, the perturbation δv will grow or decay to zero, depending on whether λ is positive or negative.
  10. The phase origin ϕf0would depend only on the frequency of the laser and the distance to the feedback element if the mirror other than the output–feedback mirror were scanned in the experiment.
  11. I. P. Konovalov, A. I. Popov, and E. D. Protsenko, “Dependence of the 5s′[1/2]10-4p′[3/2]2 Ne(3.39-μ m) line width on the composition of the mixture in a He–Ne discharge,” Opt. Spektrosk. 33, 109, 198 (1972).
  12. G. Stéphan and M. Trumper, “Macroscopic parameters and line shapes of a gas laser,” Phys. Rev. A 30, 1925 (1984).
    [Crossref]
  13. A. I. Popov and E. D. Protsenko, “Laser gain due to 5s′[1/2]10-4p′[3/2] transition in neon at λ = 3.39 μ,” Sov. J. Quantum Electron. 5, 1153 (1975).
    [Crossref]

1990 (1)

A. preliminary report on the catastrophies has been given in P. Paddon, P. Glanznig, A. D. May, and G. Stéphan, “Comportement catastrophique du domain de bistabilité d’un laser vectoriel,” Ann. Phys. (Paris) 15, 177 (1990).

1989 (1)

1987 (1)

1985 (1)

G. Stéphan and D. Hugon, “Light polarization of a quasi-isotropic laser with optical feedback,” Phys. Rev. Lett. 55, 703 (1985).
[Crossref] [PubMed]

1984 (1)

G. Stéphan and M. Trumper, “Macroscopic parameters and line shapes of a gas laser,” Phys. Rev. A 30, 1925 (1984).
[Crossref]

1980 (1)

For earlier (pioneering) work see the paper by D. Lenstra, “On the polarization effects in gas lasers,” Phys. Rep. 59, 299 (1980), and references therein.
[Crossref]

1977 (1)

See, for example, F. Aronowitz and W. L. Lim, “Positive scale factor correction in the laser gyro,” IEEE J. Quantum Electron. QE-13, 338 (1977).
[Crossref]

1975 (1)

A. I. Popov and E. D. Protsenko, “Laser gain due to 5s′[1/2]10-4p′[3/2] transition in neon at λ = 3.39 μ,” Sov. J. Quantum Electron. 5, 1153 (1975).
[Crossref]

1972 (1)

I. P. Konovalov, A. I. Popov, and E. D. Protsenko, “Dependence of the 5s′[1/2]10-4p′[3/2]2 Ne(3.39-μ m) line width on the composition of the mixture in a He–Ne discharge,” Opt. Spektrosk. 33, 109, 198 (1972).

1964 (1)

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. A 134, 1429 (1964).

Aissaoui, B.

Aronowitz, F.

See, for example, F. Aronowitz and W. L. Lim, “Positive scale factor correction in the laser gyro,” IEEE J. Quantum Electron. QE-13, 338 (1977).
[Crossref]

Glanznig, P.

A. preliminary report on the catastrophies has been given in P. Paddon, P. Glanznig, A. D. May, and G. Stéphan, “Comportement catastrophique du domain de bistabilité d’un laser vectoriel,” Ann. Phys. (Paris) 15, 177 (1990).

Hugon, D.

G. Stéphan and D. Hugon, “Light polarization of a quasi-isotropic laser with optical feedback,” Phys. Rev. Lett. 55, 703 (1985).
[Crossref] [PubMed]

Konovalov, I. P.

I. P. Konovalov, A. I. Popov, and E. D. Protsenko, “Dependence of the 5s′[1/2]10-4p′[3/2]2 Ne(3.39-μ m) line width on the composition of the mixture in a He–Ne discharge,” Opt. Spektrosk. 33, 109, 198 (1972).

Lamb, W. E.

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. A 134, 1429 (1964).

Lenstra, D.

For earlier (pioneering) work see the paper by D. Lenstra, “On the polarization effects in gas lasers,” Phys. Rep. 59, 299 (1980), and references therein.
[Crossref]

Lim, W. L.

See, for example, F. Aronowitz and W. L. Lim, “Positive scale factor correction in the laser gyro,” IEEE J. Quantum Electron. QE-13, 338 (1977).
[Crossref]

May, A. D.

A. preliminary report on the catastrophies has been given in P. Paddon, P. Glanznig, A. D. May, and G. Stéphan, “Comportement catastrophique du domain de bistabilité d’un laser vectoriel,” Ann. Phys. (Paris) 15, 177 (1990).

A. D. May and G. Stéphan, “Stability of polarized modes in a quasi-isotropic laser,” J. Opt. Soc. Am. B 6, 2355 (1989). [There are three typographical errors in the paper. The characters fs2p9.7 should be removed from Eq. (1); the matrix V has 1 along the diagonal and −1 off the diagonal (p. 2357); and Eqs. (A6)–(A8) are each missing an overall minus sign on the right-hand side.]
[Crossref]

G. Stéphan, A. D. May, R. E. Mueller, and B. Aissaoui, “Competition effects in the polarization of light in a quasi-isotropic laser,” J. Opt. Soc. Am. B 4, 1276 (1987).
[Crossref]

A. D. May and G. Stéphan, “Polarization dynamics in a quasi-isotropic laser,” J. H. Eberly, L. Mandel, and E. Wolf, eds., in Coherence and Quantum Optics VI (Plenum, New York, 1990), p. 731.
[Crossref]

Mueller, R. E.

Paddon, P.

A. preliminary report on the catastrophies has been given in P. Paddon, P. Glanznig, A. D. May, and G. Stéphan, “Comportement catastrophique du domain de bistabilité d’un laser vectoriel,” Ann. Phys. (Paris) 15, 177 (1990).

Popov, A. I.

A. I. Popov and E. D. Protsenko, “Laser gain due to 5s′[1/2]10-4p′[3/2] transition in neon at λ = 3.39 μ,” Sov. J. Quantum Electron. 5, 1153 (1975).
[Crossref]

I. P. Konovalov, A. I. Popov, and E. D. Protsenko, “Dependence of the 5s′[1/2]10-4p′[3/2]2 Ne(3.39-μ m) line width on the composition of the mixture in a He–Ne discharge,” Opt. Spektrosk. 33, 109, 198 (1972).

Protsenko, E. D.

A. I. Popov and E. D. Protsenko, “Laser gain due to 5s′[1/2]10-4p′[3/2] transition in neon at λ = 3.39 μ,” Sov. J. Quantum Electron. 5, 1153 (1975).
[Crossref]

I. P. Konovalov, A. I. Popov, and E. D. Protsenko, “Dependence of the 5s′[1/2]10-4p′[3/2]2 Ne(3.39-μ m) line width on the composition of the mixture in a He–Ne discharge,” Opt. Spektrosk. 33, 109, 198 (1972).

Stéphan, G.

A. preliminary report on the catastrophies has been given in P. Paddon, P. Glanznig, A. D. May, and G. Stéphan, “Comportement catastrophique du domain de bistabilité d’un laser vectoriel,” Ann. Phys. (Paris) 15, 177 (1990).

A. D. May and G. Stéphan, “Stability of polarized modes in a quasi-isotropic laser,” J. Opt. Soc. Am. B 6, 2355 (1989). [There are three typographical errors in the paper. The characters fs2p9.7 should be removed from Eq. (1); the matrix V has 1 along the diagonal and −1 off the diagonal (p. 2357); and Eqs. (A6)–(A8) are each missing an overall minus sign on the right-hand side.]
[Crossref]

G. Stéphan, A. D. May, R. E. Mueller, and B. Aissaoui, “Competition effects in the polarization of light in a quasi-isotropic laser,” J. Opt. Soc. Am. B 4, 1276 (1987).
[Crossref]

G. Stéphan and D. Hugon, “Light polarization of a quasi-isotropic laser with optical feedback,” Phys. Rev. Lett. 55, 703 (1985).
[Crossref] [PubMed]

G. Stéphan and M. Trumper, “Macroscopic parameters and line shapes of a gas laser,” Phys. Rev. A 30, 1925 (1984).
[Crossref]

A. D. May and G. Stéphan, “Polarization dynamics in a quasi-isotropic laser,” J. H. Eberly, L. Mandel, and E. Wolf, eds., in Coherence and Quantum Optics VI (Plenum, New York, 1990), p. 731.
[Crossref]

Trumper, M.

G. Stéphan and M. Trumper, “Macroscopic parameters and line shapes of a gas laser,” Phys. Rev. A 30, 1925 (1984).
[Crossref]

Ann. Phys. (Paris) (1)

A. preliminary report on the catastrophies has been given in P. Paddon, P. Glanznig, A. D. May, and G. Stéphan, “Comportement catastrophique du domain de bistabilité d’un laser vectoriel,” Ann. Phys. (Paris) 15, 177 (1990).

IEEE J. Quantum Electron. (1)

See, for example, F. Aronowitz and W. L. Lim, “Positive scale factor correction in the laser gyro,” IEEE J. Quantum Electron. QE-13, 338 (1977).
[Crossref]

J. Opt. Soc. Am. B (2)

Opt. Spektrosk. (1)

I. P. Konovalov, A. I. Popov, and E. D. Protsenko, “Dependence of the 5s′[1/2]10-4p′[3/2]2 Ne(3.39-μ m) line width on the composition of the mixture in a He–Ne discharge,” Opt. Spektrosk. 33, 109, 198 (1972).

Phys. Rep. (1)

For earlier (pioneering) work see the paper by D. Lenstra, “On the polarization effects in gas lasers,” Phys. Rep. 59, 299 (1980), and references therein.
[Crossref]

Phys. Rev. A (2)

G. Stéphan and M. Trumper, “Macroscopic parameters and line shapes of a gas laser,” Phys. Rev. A 30, 1925 (1984).
[Crossref]

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. A 134, 1429 (1964).

Phys. Rev. Lett. (1)

G. Stéphan and D. Hugon, “Light polarization of a quasi-isotropic laser with optical feedback,” Phys. Rev. Lett. 55, 703 (1985).
[Crossref] [PubMed]

Sov. J. Quantum Electron. (1)

A. I. Popov and E. D. Protsenko, “Laser gain due to 5s′[1/2]10-4p′[3/2] transition in neon at λ = 3.39 μ,” Sov. J. Quantum Electron. 5, 1153 (1975).
[Crossref]

Other (3)

A. D. May and G. Stéphan, “Polarization dynamics in a quasi-isotropic laser,” J. H. Eberly, L. Mandel, and E. Wolf, eds., in Coherence and Quantum Optics VI (Plenum, New York, 1990), p. 731.
[Crossref]

Liapunov exponents determine the stability of stationary solutions of nonlinear equations. The equations are linearized in departures of the variables from their stationary value. If new variables are chosen that diagonalize the linear equations, then the coefficients are the Liapunov exponents λ. Since the equations have the form d(δv)/dt= λδv, the perturbation δv will grow or decay to zero, depending on whether λ is positive or negative.

The phase origin ϕf0would depend only on the frequency of the laser and the distance to the feedback element if the mirror other than the output–feedback mirror were scanned in the experiment.

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Figures (9)

Fig. 1
Fig. 1

Schematic of experimental setup.

Fig. 2
Fig. 2

Plots of the Liapunov exponent for modes A and B as a function of the length of the laser for three values of the gain.

Fig. 3
Fig. 3

Calculated intensity profiles of mode B, corresponding to Fig. 2. The domain of bistability, H, is between the vertical parts of the intensity curves. Thus in (a) the hysteresis loop occupies almost the entire width of the gain curve.

Fig. 4
Fig. 4

Reproductions of experimental oscillograph traces, illustrating the existence of catastrophes for varying gain. Note the abrupt changes in the width of the hysteresis when the gain is slightly increased between (a) and (b) and between (b) and (c).

Fig. 5
Fig. 5

Variation of the Liapunov exponent with the length of the laser for three values of the phase origin.

Fig. 6
Fig. 6

Experimental confirmation of the existence of catastrophes for various phase origins.

Fig. 7
Fig. 7

Dependence of the width of the hysteresis loop on gain. Filled squares are experimental results. The curve was calculated by means of the theory outlined in the text.

Fig. 8
Fig. 8

Plots illustrating the sensitivity of the calculated hysteresis loop to phase origin and anisotropies. The parameters used are the same as those used for Fig. 7 except for curves ii, wherein ϕf0 = 102.6°, = 1.7 × 10−3, and i = 3.5 × 10−3 were used in (a), (b), and (c), respectively.

Fig. 9
Fig. 9

Measured round-trip gain versus tube current. Line be indicates the threshold for the bare laser, while wd indicates the threshold with the diaphragm set to limit the oscillation to the 0,0 mode.

Tables (1)

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Table 1 Input Parameters for Calculation of the Intensity and the Stability of the Modes

Equations (3)

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λ ( A , B ) = { ± 2 [ r + cos ϕ f ] } + { [ S r 2 - 4 ( i + sin ϕ f ) 2 ± 4 ( i + sin ϕ f ) S i ] 1 / 2 - S r } ,
I ( B ) = 0 2 = [ α r + 2 ( r + cos ϕ f ) ] / ( β r + θ r ) ,
α i + 2 ( i + sin ϕ f ) - ( β i + θ i ) = 0 ,

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