Abstract

We extend the cavity equations of motion for lasers with arbitrary output coupling [ Appl. Phys. Lett. 36, 412 ( 1980)] to include an externally injected signal. The external signal is initially introduced into the cavity through the output coupler and enters the formulation through the boundary condition on the cavity electric field. A transformation is performed so that the injected signal is eliminated from the boundary condition and replaced by a driving-polarization term in the equations of motion for the cavity field that develops as a result of laser gain. We illustrate the use of this new formulation by applying it to an injection-seeded, gain-switched, standing-wave pulsed oscillator. The intracavity intensities as functions of axial position and time are compared for single-frequency and multiple-frequency operation.

© 1994 Optical Society of America

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References

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  1. A. E. Siegman, Lasers (University Science, Mill Valley, Calf., 1986), Chap. 29.
  2. Y. K. Park, G. Guiliani, and R. L. Byer, “Single axial mode operation of a Q-switched Nd:YAG oscillator by injection seeding,” IEEE J. Quantum Electron. QE-20, 117–125 (1984).
    [CrossRef]
  3. J.-L. Lachambre, P. Lavigne, G. Otis, and M. Noël, “Injection locking and mode selection in TEA-CO2laser oscillators,” IEEE J. Quantum Electron. QE-12, 756–764 (1976).
    [CrossRef]
  4. T. D. Raymond and A. V. Smith, “Injection-seeded titanium-doped-sapphire laser,” Opt. Lett. 16, 33–35 (1991).
    [CrossRef] [PubMed]
  5. I. J. Bigio and M. Slatkine, “Injection-locking unstable resonator excimer lasers,” IEEE J. Quantum Electron. QE-19, 1426–1436 (1983).
    [CrossRef]
  6. L. E. Erickson and A. Szabo, “Spectral narrowing of dye laser output by injection of monochromatic radiation into the laser cavity,” Appl. Phys. Lett. 18, 433–435 (1971).
    [CrossRef]
  7. J. E. Bjorkholm and H. G. Danielmeyer, “Frequency control of a pulsed optical parametric oscillator by radiation injection,” Appl. Phys. Lett. 15, 171–173 (1969).
    [CrossRef]
  8. A. E. Siegman, “Exact cavity equations for lasers with large output coupling,” Appl. Phys. Lett. 36, 412–414 (1980).
    [CrossRef]
  9. M. S. Bowers and S. E. Moody, “Numerical solution of the exact cavity equations of motion for an unstable optical resonator,” Appl. Opt. 29, 3905–3915 (1990).
    [CrossRef] [PubMed]
  10. Ref. 1, Chap. 24 .
  11. Ref. 1, p. 951 .
  12. A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical system. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
    [CrossRef] [PubMed]
  13. T. D. Raymond, A. V. Smith, and P. Esherick, “Dual longitudinal mode Nd:YAG laser,” in Advanced Solid-State Lasers, A. A. Pinto and T. Y. Fan, eds., Vol. 15 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1993), pp. 169–171.

1991 (1)

1990 (1)

1989 (1)

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical system. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[CrossRef] [PubMed]

1984 (1)

Y. K. Park, G. Guiliani, and R. L. Byer, “Single axial mode operation of a Q-switched Nd:YAG oscillator by injection seeding,” IEEE J. Quantum Electron. QE-20, 117–125 (1984).
[CrossRef]

1983 (1)

I. J. Bigio and M. Slatkine, “Injection-locking unstable resonator excimer lasers,” IEEE J. Quantum Electron. QE-19, 1426–1436 (1983).
[CrossRef]

1980 (1)

A. E. Siegman, “Exact cavity equations for lasers with large output coupling,” Appl. Phys. Lett. 36, 412–414 (1980).
[CrossRef]

1976 (1)

J.-L. Lachambre, P. Lavigne, G. Otis, and M. Noël, “Injection locking and mode selection in TEA-CO2laser oscillators,” IEEE J. Quantum Electron. QE-12, 756–764 (1976).
[CrossRef]

1971 (1)

L. E. Erickson and A. Szabo, “Spectral narrowing of dye laser output by injection of monochromatic radiation into the laser cavity,” Appl. Phys. Lett. 18, 433–435 (1971).
[CrossRef]

1969 (1)

J. E. Bjorkholm and H. G. Danielmeyer, “Frequency control of a pulsed optical parametric oscillator by radiation injection,” Appl. Phys. Lett. 15, 171–173 (1969).
[CrossRef]

Bigio, I. J.

I. J. Bigio and M. Slatkine, “Injection-locking unstable resonator excimer lasers,” IEEE J. Quantum Electron. QE-19, 1426–1436 (1983).
[CrossRef]

Bjorkholm, J. E.

J. E. Bjorkholm and H. G. Danielmeyer, “Frequency control of a pulsed optical parametric oscillator by radiation injection,” Appl. Phys. Lett. 15, 171–173 (1969).
[CrossRef]

Bowers, M. S.

Byer, R. L.

Y. K. Park, G. Guiliani, and R. L. Byer, “Single axial mode operation of a Q-switched Nd:YAG oscillator by injection seeding,” IEEE J. Quantum Electron. QE-20, 117–125 (1984).
[CrossRef]

Danielmeyer, H. G.

J. E. Bjorkholm and H. G. Danielmeyer, “Frequency control of a pulsed optical parametric oscillator by radiation injection,” Appl. Phys. Lett. 15, 171–173 (1969).
[CrossRef]

Erickson, L. E.

L. E. Erickson and A. Szabo, “Spectral narrowing of dye laser output by injection of monochromatic radiation into the laser cavity,” Appl. Phys. Lett. 18, 433–435 (1971).
[CrossRef]

Esherick, P.

T. D. Raymond, A. V. Smith, and P. Esherick, “Dual longitudinal mode Nd:YAG laser,” in Advanced Solid-State Lasers, A. A. Pinto and T. Y. Fan, eds., Vol. 15 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1993), pp. 169–171.

Guiliani, G.

Y. K. Park, G. Guiliani, and R. L. Byer, “Single axial mode operation of a Q-switched Nd:YAG oscillator by injection seeding,” IEEE J. Quantum Electron. QE-20, 117–125 (1984).
[CrossRef]

Lachambre, J.-L.

J.-L. Lachambre, P. Lavigne, G. Otis, and M. Noël, “Injection locking and mode selection in TEA-CO2laser oscillators,” IEEE J. Quantum Electron. QE-12, 756–764 (1976).
[CrossRef]

Lavigne, P.

J.-L. Lachambre, P. Lavigne, G. Otis, and M. Noël, “Injection locking and mode selection in TEA-CO2laser oscillators,” IEEE J. Quantum Electron. QE-12, 756–764 (1976).
[CrossRef]

Moody, S. E.

Noël, M.

J.-L. Lachambre, P. Lavigne, G. Otis, and M. Noël, “Injection locking and mode selection in TEA-CO2laser oscillators,” IEEE J. Quantum Electron. QE-12, 756–764 (1976).
[CrossRef]

Otis, G.

J.-L. Lachambre, P. Lavigne, G. Otis, and M. Noël, “Injection locking and mode selection in TEA-CO2laser oscillators,” IEEE J. Quantum Electron. QE-12, 756–764 (1976).
[CrossRef]

Park, Y. K.

Y. K. Park, G. Guiliani, and R. L. Byer, “Single axial mode operation of a Q-switched Nd:YAG oscillator by injection seeding,” IEEE J. Quantum Electron. QE-20, 117–125 (1984).
[CrossRef]

Raymond, T. D.

T. D. Raymond and A. V. Smith, “Injection-seeded titanium-doped-sapphire laser,” Opt. Lett. 16, 33–35 (1991).
[CrossRef] [PubMed]

T. D. Raymond, A. V. Smith, and P. Esherick, “Dual longitudinal mode Nd:YAG laser,” in Advanced Solid-State Lasers, A. A. Pinto and T. Y. Fan, eds., Vol. 15 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1993), pp. 169–171.

Siegman, A. E.

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical system. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[CrossRef] [PubMed]

A. E. Siegman, “Exact cavity equations for lasers with large output coupling,” Appl. Phys. Lett. 36, 412–414 (1980).
[CrossRef]

A. E. Siegman, Lasers (University Science, Mill Valley, Calf., 1986), Chap. 29.

Slatkine, M.

I. J. Bigio and M. Slatkine, “Injection-locking unstable resonator excimer lasers,” IEEE J. Quantum Electron. QE-19, 1426–1436 (1983).
[CrossRef]

Smith, A. V.

T. D. Raymond and A. V. Smith, “Injection-seeded titanium-doped-sapphire laser,” Opt. Lett. 16, 33–35 (1991).
[CrossRef] [PubMed]

T. D. Raymond, A. V. Smith, and P. Esherick, “Dual longitudinal mode Nd:YAG laser,” in Advanced Solid-State Lasers, A. A. Pinto and T. Y. Fan, eds., Vol. 15 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1993), pp. 169–171.

Szabo, A.

L. E. Erickson and A. Szabo, “Spectral narrowing of dye laser output by injection of monochromatic radiation into the laser cavity,” Appl. Phys. Lett. 18, 433–435 (1971).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

L. E. Erickson and A. Szabo, “Spectral narrowing of dye laser output by injection of monochromatic radiation into the laser cavity,” Appl. Phys. Lett. 18, 433–435 (1971).
[CrossRef]

J. E. Bjorkholm and H. G. Danielmeyer, “Frequency control of a pulsed optical parametric oscillator by radiation injection,” Appl. Phys. Lett. 15, 171–173 (1969).
[CrossRef]

A. E. Siegman, “Exact cavity equations for lasers with large output coupling,” Appl. Phys. Lett. 36, 412–414 (1980).
[CrossRef]

IEEE J. Quantum Electron. (3)

Y. K. Park, G. Guiliani, and R. L. Byer, “Single axial mode operation of a Q-switched Nd:YAG oscillator by injection seeding,” IEEE J. Quantum Electron. QE-20, 117–125 (1984).
[CrossRef]

J.-L. Lachambre, P. Lavigne, G. Otis, and M. Noël, “Injection locking and mode selection in TEA-CO2laser oscillators,” IEEE J. Quantum Electron. QE-12, 756–764 (1976).
[CrossRef]

I. J. Bigio and M. Slatkine, “Injection-locking unstable resonator excimer lasers,” IEEE J. Quantum Electron. QE-19, 1426–1436 (1983).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical system. II. Laser oscillators,” Phys. Rev. A 39, 1264–1268 (1989).
[CrossRef] [PubMed]

Other (4)

T. D. Raymond, A. V. Smith, and P. Esherick, “Dual longitudinal mode Nd:YAG laser,” in Advanced Solid-State Lasers, A. A. Pinto and T. Y. Fan, eds., Vol. 15 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1993), pp. 169–171.

A. E. Siegman, Lasers (University Science, Mill Valley, Calf., 1986), Chap. 29.

Ref. 1, Chap. 24 .

Ref. 1, p. 951 .

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Figures (11)

Fig. 1
Fig. 1

Model for a standing-wave cavity with an externally injected signal. The gain medium is traversed twice during a cavity round trip.

Fig. 2
Fig. 2

Schematic of a ring-laser cavity with an externally injected signal. The gain medium is traversed once during a single pass through the cavity.

Fig. 3
Fig. 3

Standing-wave oscillator, unfolded to show a complete round trip. The gain medium is traversed twice during a round trip. The rightmost gain medium is the mirror image of the leftmost gain medium.

Fig. 4
Fig. 4

Predicted output pulse shape and frequency shift for an injection-seeded, gain-switched oscillator. The gain medium is located at the center of the cavity, gol = 2.7, and the pump FWHM is 15 round-trip times. The injected signal is detuned 0.5 rad/T from the power oscillator resonance frequency.

Fig. 5
Fig. 5

Predicted output power spectrum for single-frequency operation. The parameters used are the same as in Fig. 4 and Table 1.

Fig. 6
Fig. 6

Intracavity intensity as a function of the unfolded cavity axial distance. Different curves are for different basis set sizes used in the Fourier expansion. The intensity distribution is a snapshot taken at the peak of the output pulse. Parameters are the same as in Fig. 4 and Table 1.

Fig. 7
Fig. 7

Intracavity intensity at different times during the laser pulse shown in Fig. 4. The time between the different snap-shots of intensity is half of a round-trip time.

Fig. 8
Fig. 8

Predicted (a) output pulse shape and (b) output power spectrum for the oscillator, with parameters the same as in Fig. 4 but with the injected signal detuned to midway between two cavity resonances of the oscillator.

Fig. 9
Fig. 9

Snapshots of the intracavity intensity as a function of unfolded axial distance for the multimode operation. The time interval between snapshots is 0.2 cavity round-trip time. The arrows indicate the direction of propagation.

Fig. 10
Fig. 10

Predicted (a) output pulse shapes and (b) power spectra with the gain medium centered and at the output mirror. The injected signal is tuned to a cavity resonance. Other parameters are given in Table 1 with gol = 6, and the pump FWHM is one cavity round-trip time.

Fig. 11
Fig. 11

Snapshots of the intracavity intensity at time intervals of 0.2 cavity round-trip time for the results shown in Fig. 10 with the gain medium at the output mirror. The arrow indicates the direction of propagation.

Tables (1)

Tables Icon

Table 1 Parameters for the Numerical Solutions

Equations (29)

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( 2 z 2 - μ 0 σ t - μ 0 2 t 2 ) E ( z , t ) = μ 0 2 p ( z , t ) t 2 ,
E ( z , t ) = [ 2 ( μ 0 ) 1 / 2 ] 1 / 2 Re { E ˜ ( z , t ) exp [ i ( ω 0 t - k 0 z ) ] } , p ( z , t ) = [ 2 ( μ 0 ) 1 / 2 ] 1 / 2 Re { p ˜ ( z , t ) exp [ i ( ω 0 t - k 0 z ) ] } ,
E i ( t ) = [ 2 ( μ 0 ) 1 / 2 ] 1 / 2 Re { E ˜ i exp ( i ω i t ) } = [ 2 ( μ 0 ) 1 / 2 ] 1 / 2 Re { E ˜ i exp [ i ( ω i - ω 0 ) t ] exp ( i ω 0 t ) } .
2 ω 0 L / c = 2 π q 0 ,
( z + α 0 c + 1 c t ) E ˜ ( z , t ) = - i k 0 2 p ˜ ( z , t ) ,
E ˜ ( 0 , t ) = r ^ E ˜ ( 2 L , t ) + i t ^ E ˜ i exp [ i ( ω i - ω 0 ) t ] .
E ˜ ( z , t ) = E ˜ L ( z , t ) + E ˜ inj ( z , t ) ,
( z + α 0 c + 1 c t ) E ˜ inj ( z , t ) = 0 ,
E ˜ inj ( 0 , t ) = r ^ E ˜ inj ( 2 L , t ) + i t ^ E ˜ i exp [ i ( ω i - ω 0 ) t ] .
E ˜ inj ( z , t ) = i t ^ E ˜ i exp ( - α 0 z / c ) exp [ i Δ ω ( t - z / c ) ] 1 - r ^ exp ( - α 0 2 L / c ) exp ( - i Δ ω 2 L / c ) ,
( z + α 0 c + 1 c t ) E ˜ L ( z , t ) = - i k 0 2 p ˜ ( z , t ) .
E ˜ L ( 0 , t ) = r ^ E ˜ L ( 2 L , t ) ,
E ˜ L ( z , t ) = r ^ exp ( - z 2 L ln r ^ ) A ˜ ( z , t ) ,
( z + α 0 c - 1 2 L ln r ^ + 1 c t ) A ˜ ( z , t ) = - i k 0 2 r ^ exp ( z 2 L ln r ^ ) p ˜ ( z , t ) .
A ˜ ( 0 , t ) = A ˜ ( 2 L , t ) .
A ˜ ( z , t ) = q A ˜ q ( t ) exp ( - i π q z / L ) .
( d d t - c 2 L ln r ^ + α 0 - i π q c L ) A ˜ q ( t ) = - i k 0 c 2 P ˜ q ( t ) ,
P ˜ q ( t ) = 1 2 L r ^ 0 2 L d z exp ( z 2 L ln r ^ ) p ˜ ( z , t ) exp ( i π q z L ) .
p ˜ ( z , t ) = i k 0 g ( z , t ) E ˜ ( z , t )
[ z + α 0 c - 1 2 g ( z , t ) + 1 c t ] E ˜ L ( z , t ) = 1 2 g ( z , t ) E ˜ inj ( z , t ) .
( d d t - c 2 L ln r ^ + α 0 - i π q c L ) A ˜ q ( t ) = c 2 q G ˜ q , q ( t ) A ˜ q ( t ) + c 2 P ˜ q i ( t ) ,
G ˜ q , q ( t ) = 1 2 L 0 2 L d z g ( z , t ) exp [ i π z ( q - q ) L ]
P ˜ q i ( t ) = 1 2 L r ^ 0 2 L d z exp ( z 2 L ln r ^ ) g ( z , t ) E ˜ inj ( z , t ) × exp ( i π q z L ) .
g ( z , t ) t = - g ( z , t ) [ I ( z , t ) + I ( 2 L - z , t ) ] E sat + R p ( t ) ,
R p ( t ) = 1 τ p ( 4 ln 2 π ) 1 / 2 g 0 exp [ - 4 ( ln 2 ) ( t - t p ) 2 / τ p 2 ] .
g ( z , t ) = g ( 2 L - z , t )
G q , q ( t ) = 1 L 0 L d z g ( z , t ) cos [ π z ( q - q ) L ] .
E ˜ out ( t ) = r ^ E ˜ i exp [ i ( ω i - ω o ) t ] + i t ^ [ E ˜ L ( 2 L , t ) + E ˜ inj ( 2 L , t ) ] .
E out ( t ) i t ^ E ˜ ( 2 L , t ) = i t ^ q A ˜ q ( t ) ,

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