Abstract

The forward and backward waves in a standing-wave laser resonator are coupled by the nonlinear gain. Proper inclusion of interference effects leads to significant changes in intensity relative to the usual approximation.

© 1979 Optical Society of America

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References

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  1. A. E. Siegman and E. A. Sziklas, “Mode calculations in unstable resonators with flowing saturable gain: 1. Hermite Gaussian expansions,” Appl. Opt. 13, 2775–2792 (1974); E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain: 2. Fast Fourier Transform method,” Appl. Opt. 14, 1874–1889 (1974).
    [CrossRef] [PubMed]
  2. G. T. Moore and R. J. McCarthy, “Lasers with unstable resonators in the geometrical optics limit,” J. Opt. Soc. Am. 67, 221–227 (1977); G. T. Moore and R. J. McCarthy, “Theory of Modes in a loaded strip confocal resonator,” J. Opt. Soc. Am. 67, 228–241 (1977).
    [CrossRef]
  3. Yu. N. Karamzin and Yu. B. Konev, “Numerical investigation of the operation of unstable telescopic resonators allowing for diffraction and saturation in the active medium,” Sov. J. Quantum Electron. 5, 144–148 (1975).
    [CrossRef]
  4. V. S. Rogov and M. M. Rikenglaz, “Numerical investigation of the influence of optical inhomogeneities of the active medium on the operation of an unstable telescopic resonator,” Sov. J. Quantum Electron. 1, 18–21 (1977).
    [CrossRef]
  5. W. W. Rigrod, “Saturation effects in high gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
    [CrossRef]
  6. M. Lax. Brandeis University Summer Institute in Theoretical Physics Lectures (Gordon and Breach, New York, 1966); H. Haken, Handbuch der Physik XXV/2C (Springer-Verlag, Berlin, 1969); M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Massachusetts, 1974).
  7. H. Kogelnik and C. V. Shank, “Coupled wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972). See also H. C. Casey and M. B. Panish, Heterostructure Lasers, Part A (Academic, New York, 1978), Sec. 2.10 and references cited therein.
    [CrossRef]
  8. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  9. A. E. Siegman, “Unstable optical resonators,” Appl. Opt. 13, 353–367 (1974).
    [CrossRef] [PubMed]

1977 (2)

G. T. Moore and R. J. McCarthy, “Lasers with unstable resonators in the geometrical optics limit,” J. Opt. Soc. Am. 67, 221–227 (1977); G. T. Moore and R. J. McCarthy, “Theory of Modes in a loaded strip confocal resonator,” J. Opt. Soc. Am. 67, 228–241 (1977).
[CrossRef]

V. S. Rogov and M. M. Rikenglaz, “Numerical investigation of the influence of optical inhomogeneities of the active medium on the operation of an unstable telescopic resonator,” Sov. J. Quantum Electron. 1, 18–21 (1977).
[CrossRef]

1975 (2)

Yu. N. Karamzin and Yu. B. Konev, “Numerical investigation of the operation of unstable telescopic resonators allowing for diffraction and saturation in the active medium,” Sov. J. Quantum Electron. 5, 144–148 (1975).
[CrossRef]

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1974 (2)

1972 (1)

H. Kogelnik and C. V. Shank, “Coupled wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972). See also H. C. Casey and M. B. Panish, Heterostructure Lasers, Part A (Academic, New York, 1978), Sec. 2.10 and references cited therein.
[CrossRef]

1965 (1)

W. W. Rigrod, “Saturation effects in high gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[CrossRef]

Karamzin, Yu. N.

Yu. N. Karamzin and Yu. B. Konev, “Numerical investigation of the operation of unstable telescopic resonators allowing for diffraction and saturation in the active medium,” Sov. J. Quantum Electron. 5, 144–148 (1975).
[CrossRef]

Kogelnik, H.

H. Kogelnik and C. V. Shank, “Coupled wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972). See also H. C. Casey and M. B. Panish, Heterostructure Lasers, Part A (Academic, New York, 1978), Sec. 2.10 and references cited therein.
[CrossRef]

Konev, Yu. B.

Yu. N. Karamzin and Yu. B. Konev, “Numerical investigation of the operation of unstable telescopic resonators allowing for diffraction and saturation in the active medium,” Sov. J. Quantum Electron. 5, 144–148 (1975).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

M. Lax. Brandeis University Summer Institute in Theoretical Physics Lectures (Gordon and Breach, New York, 1966); H. Haken, Handbuch der Physik XXV/2C (Springer-Verlag, Berlin, 1969); M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Massachusetts, 1974).

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

McCarthy, R. J.

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Moore, G. T.

Rigrod, W. W.

W. W. Rigrod, “Saturation effects in high gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[CrossRef]

Rikenglaz, M. M.

V. S. Rogov and M. M. Rikenglaz, “Numerical investigation of the influence of optical inhomogeneities of the active medium on the operation of an unstable telescopic resonator,” Sov. J. Quantum Electron. 1, 18–21 (1977).
[CrossRef]

Rogov, V. S.

V. S. Rogov and M. M. Rikenglaz, “Numerical investigation of the influence of optical inhomogeneities of the active medium on the operation of an unstable telescopic resonator,” Sov. J. Quantum Electron. 1, 18–21 (1977).
[CrossRef]

Shank, C. V.

H. Kogelnik and C. V. Shank, “Coupled wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972). See also H. C. Casey and M. B. Panish, Heterostructure Lasers, Part A (Academic, New York, 1978), Sec. 2.10 and references cited therein.
[CrossRef]

Siegman, A. E.

Sziklas, E. A.

Appl. Opt. (2)

J. Appl. Phys. (2)

H. Kogelnik and C. V. Shank, “Coupled wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972). See also H. C. Casey and M. B. Panish, Heterostructure Lasers, Part A (Academic, New York, 1978), Sec. 2.10 and references cited therein.
[CrossRef]

W. W. Rigrod, “Saturation effects in high gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[CrossRef]

J. Opt. Soc. Am. (1)

Phys. Rev. A (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Sov. J. Quantum Electron. (2)

Yu. N. Karamzin and Yu. B. Konev, “Numerical investigation of the operation of unstable telescopic resonators allowing for diffraction and saturation in the active medium,” Sov. J. Quantum Electron. 5, 144–148 (1975).
[CrossRef]

V. S. Rogov and M. M. Rikenglaz, “Numerical investigation of the influence of optical inhomogeneities of the active medium on the operation of an unstable telescopic resonator,” Sov. J. Quantum Electron. 1, 18–21 (1977).
[CrossRef]

Other (1)

M. Lax. Brandeis University Summer Institute in Theoretical Physics Lectures (Gordon and Breach, New York, 1966); H. Haken, Handbuch der Physik XXV/2C (Springer-Verlag, Berlin, 1969); M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Massachusetts, 1974).

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Figures (2)

FIG. 1
FIG. 1

Intensity variation along the resonator axis in the geometrical-optics approximation. The cavity is 7.3 m long and the region between 3.3 m and 5.3 m consists of the gain medium. In this region the right- and left-going waves increase in intensity. In the free space outside the gain medium, the right-going wave decreases in intensity while the left-going wave does not because it travels parallel to the resonator axis in the confocal geometry adopted here. The full curve is obtained using the modified gain formula, Eq. (8), where the interference effects have been incorporated. For comparison the dashed curve is obtained by using the conventional gain formula, Eq. (1). The calculation is done for the on-resonance case Ω = 0.

FIG. 2
FIG. 2

See Fig. 1 caption; off-resonance case Ω = 1.

Equations (12)

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g ( I ) = g 0 ( 1 + i Ω ) 1 + Ω 2 + I ,
g 0 = n p 2 D w ω k L ћ γ a b 0 c ; D w = λ a γ a λ b γ b ,
( 2 + k 2 ) ψ = i k g 0 ( 1 + i Ω ) 1 + Ω 2 + | ψ | 2 ψ ,
ψ = n odd ψ n e i n k z ,
n e i n k z ( k 2 ( 1 n 2 ) ψ n + 2 i n k ψ n z 2 + 2 ψ n z 2 + T 2 ψ n ) = i k g 0 ( 1 + i Ω ) m ψ m e i m k z 1 + Ω 2 + | p ψ p e i p k z | 2 ,
| ψ n / ψ | ( g 0 / k ) , | n | 1 .
( 2 i k z + T 2 ) ψ R = i k g R ψ R ,
( 2 i k z + T 2 ) ψ L = i k g L ψ L ,
g μ = g 0 ( 1 + i Ω ) ( a 2 b 2 ) 1 / 2 [ 1 a ( a 2 b 2 ) 1 / 2 2 | ψ μ | 2 ] ; ( μ = R , L ) ,
a = 1 + Ω 2 + | ψ R | 2 + | ψ L | 2 , b = 2 | ψ R | | ψ L | .
d I R d z + 2 ( M 1 ) d + ( M 1 ) z I R = Re ( g R ) I R ,
d I L d z + 2 ( M 1 ) d + ( M 1 ) ( d z ) I L = Re ( g L ) I L ,