Abstract

Gain-saturation gratings, induced by the interaction between the active medium and standing waves in laser cavities, influence the performance of lasers. The modulation depth of the grating is reduced by diffusion of the excitation energy in most host materials, but gain saturation has been modeled only in the limit of strong or weak diffusion. We present an analytical solution of the diffusion equation for two-level systems at an arbitrary diffusion rate. It is shown that for many rare-earth dopants the migration process is not well described by either the weak- or the strong-diffusion approximation.

© 1998 Optical Society of America

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  1. H. G. Danielmeyer, “Effects of drift and diffusion of excited states on spatial hole burning and laser oscillation,” J. Appl. Phys. 42, 3125–3132 (1971).
    [CrossRef]
  2. M. Sargent III, “Laser saturation grating phenomena,” Appl. Phys. 9, 127–141 (1976).
    [CrossRef]
  3. S. J. Frisken, “Transient Bragg reflection gratings in erbium-doped fiber amplifiers,” Opt. Lett. 17, 1776–1778 (1992).
    [CrossRef] [PubMed]
  4. X. Pan, H. Olesen, B. Tromborg, and H. E. Lassen, “Analytic description of the standing wave effects in DFB lasers,” Proc. Inst. Electr. Eng. Part J 139, 189–193 (1992).
  5. A. Mecozzi, “Cavity standing-wave and gain compression coefficient in semiconductor lasers,” Opt. Lett. 19, 640–642 (1994).
    [CrossRef] [PubMed]
  6. B. Jaskorzynska, E. V. Vanin, S. Helmfrid, and A. Asseh, “Gain saturation and pump depletion in high-efficiency distributed-feedback rare-earth-doped lasers,” Opt. Lett. 21, 1366–1368 (1996).
    [CrossRef] [PubMed]
  7. G. P. Agrawal and M. Lax, “Effects of interference on gain saturation in laser resonators,” J. Opt. Soc. Am. B 69, 1717–1719 (1979).
    [CrossRef]
  8. G. P. Agrawal and M. Lax, “Analytic evaluation of interference effects on laser output in a Fabry–Perot resonator,” J. Opt. Soc. Am. B 71, 515–519 (1981).
    [CrossRef]
  9. J. J. Zayhowski, “The effects of spatial hole burning and energy diffusion on the single-mode operation of standing-wave lasers,” IEEE J. Quantum Electron. 26, 2052–2057 (1990).
    [CrossRef]
  10. A. Mossakowska, P. Szczepanski, and W. Wolinski, “Modulation bandwidth of waveguide distributed feedback lasers,” IEEE J. Quantum Electron. 30, 230–234 (1994).
    [CrossRef]
  11. M. Horowitz, R. Daisy, B. Fischer, and J. Zyskind, “Narrow-linewidth, singlemode erbium-doped fibre laser with intracavity wave mixing in saturable absorber,” Electron. Lett. 30, 648–649 (1994).
    [CrossRef]
  12. M. Horowitz, R. Daisy, B. Fischer, and J. L. Zyskind, “Linewidth-narrowing mechanisms in lasers by nonlinear wave mixing,” Opt. Lett. 19, 1406–1408 (1994).
    [CrossRef] [PubMed]
  13. Y. Cheng, J. T. Kringlebotn, W. H. Loh, R. I. Laming, and D. N. Payne, “Stable single-frequency traveling-wave fiber loop laser with integral saturable-absorber-based tracking narrow-band filter,” Opt. Lett. 20, 875–877 (1995).
    [CrossRef] [PubMed]
  14. S. Helmfrid and K. Tatsuno, “Stable single-mode operation of intracavity-doubled diode-pumped Nd:YVO4 lasers: theoretical study,” J. Opt. Soc. Am. B 11, 436–445 (1994).
    [CrossRef]
  15. K. O. Hill and A. Watanabe, “Envelope gain saturation in distributed feedback lasers,” Appl. Opt. 14, 950–961 (1975).
    [CrossRef] [PubMed]
  16. V. A. French and R. C. Powell, “Laser-induced grating measurements of energy migration in Tm:Ho:YAG,” Opt. Lett. 16, 666–668 (1991).
    [CrossRef] [PubMed]
  17. M. A. Noginov, H. J. Caulfield, P. Venkateswarlu, and M. Mahdi, “Study of migration in Cr:Er:YSGG using an upconversion light induced grating technique,” Opt. Mater. 5, 97–103 (1996).
    [CrossRef]
  18. H. E. Lassen, H. Olesen, and B. Tromborg, “Gain compression and asymmetric gain due to the Bragg grating induced by the standing waves in Fabry–Perot lasers,” IEEE Photon. Technol. Lett. 1, 261–263 (1989).
    [CrossRef]

1996

M. A. Noginov, H. J. Caulfield, P. Venkateswarlu, and M. Mahdi, “Study of migration in Cr:Er:YSGG using an upconversion light induced grating technique,” Opt. Mater. 5, 97–103 (1996).
[CrossRef]

B. Jaskorzynska, E. V. Vanin, S. Helmfrid, and A. Asseh, “Gain saturation and pump depletion in high-efficiency distributed-feedback rare-earth-doped lasers,” Opt. Lett. 21, 1366–1368 (1996).
[CrossRef] [PubMed]

1995

1994

A. Mecozzi, “Cavity standing-wave and gain compression coefficient in semiconductor lasers,” Opt. Lett. 19, 640–642 (1994).
[CrossRef] [PubMed]

M. Horowitz, R. Daisy, B. Fischer, and J. L. Zyskind, “Linewidth-narrowing mechanisms in lasers by nonlinear wave mixing,” Opt. Lett. 19, 1406–1408 (1994).
[CrossRef] [PubMed]

S. Helmfrid and K. Tatsuno, “Stable single-mode operation of intracavity-doubled diode-pumped Nd:YVO4 lasers: theoretical study,” J. Opt. Soc. Am. B 11, 436–445 (1994).
[CrossRef]

A. Mossakowska, P. Szczepanski, and W. Wolinski, “Modulation bandwidth of waveguide distributed feedback lasers,” IEEE J. Quantum Electron. 30, 230–234 (1994).
[CrossRef]

M. Horowitz, R. Daisy, B. Fischer, and J. Zyskind, “Narrow-linewidth, singlemode erbium-doped fibre laser with intracavity wave mixing in saturable absorber,” Electron. Lett. 30, 648–649 (1994).
[CrossRef]

1992

X. Pan, H. Olesen, B. Tromborg, and H. E. Lassen, “Analytic description of the standing wave effects in DFB lasers,” Proc. Inst. Electr. Eng. Part J 139, 189–193 (1992).

S. J. Frisken, “Transient Bragg reflection gratings in erbium-doped fiber amplifiers,” Opt. Lett. 17, 1776–1778 (1992).
[CrossRef] [PubMed]

1991

1990

J. J. Zayhowski, “The effects of spatial hole burning and energy diffusion on the single-mode operation of standing-wave lasers,” IEEE J. Quantum Electron. 26, 2052–2057 (1990).
[CrossRef]

1989

H. E. Lassen, H. Olesen, and B. Tromborg, “Gain compression and asymmetric gain due to the Bragg grating induced by the standing waves in Fabry–Perot lasers,” IEEE Photon. Technol. Lett. 1, 261–263 (1989).
[CrossRef]

1981

G. P. Agrawal and M. Lax, “Analytic evaluation of interference effects on laser output in a Fabry–Perot resonator,” J. Opt. Soc. Am. B 71, 515–519 (1981).
[CrossRef]

1979

G. P. Agrawal and M. Lax, “Effects of interference on gain saturation in laser resonators,” J. Opt. Soc. Am. B 69, 1717–1719 (1979).
[CrossRef]

1976

M. Sargent III, “Laser saturation grating phenomena,” Appl. Phys. 9, 127–141 (1976).
[CrossRef]

1975

1971

H. G. Danielmeyer, “Effects of drift and diffusion of excited states on spatial hole burning and laser oscillation,” J. Appl. Phys. 42, 3125–3132 (1971).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal and M. Lax, “Analytic evaluation of interference effects on laser output in a Fabry–Perot resonator,” J. Opt. Soc. Am. B 71, 515–519 (1981).
[CrossRef]

G. P. Agrawal and M. Lax, “Effects of interference on gain saturation in laser resonators,” J. Opt. Soc. Am. B 69, 1717–1719 (1979).
[CrossRef]

Asseh, A.

Caulfield, H. J.

M. A. Noginov, H. J. Caulfield, P. Venkateswarlu, and M. Mahdi, “Study of migration in Cr:Er:YSGG using an upconversion light induced grating technique,” Opt. Mater. 5, 97–103 (1996).
[CrossRef]

Cheng, Y.

Daisy, R.

M. Horowitz, R. Daisy, B. Fischer, and J. Zyskind, “Narrow-linewidth, singlemode erbium-doped fibre laser with intracavity wave mixing in saturable absorber,” Electron. Lett. 30, 648–649 (1994).
[CrossRef]

M. Horowitz, R. Daisy, B. Fischer, and J. L. Zyskind, “Linewidth-narrowing mechanisms in lasers by nonlinear wave mixing,” Opt. Lett. 19, 1406–1408 (1994).
[CrossRef] [PubMed]

Danielmeyer, H. G.

H. G. Danielmeyer, “Effects of drift and diffusion of excited states on spatial hole burning and laser oscillation,” J. Appl. Phys. 42, 3125–3132 (1971).
[CrossRef]

Fischer, B.

M. Horowitz, R. Daisy, B. Fischer, and J. L. Zyskind, “Linewidth-narrowing mechanisms in lasers by nonlinear wave mixing,” Opt. Lett. 19, 1406–1408 (1994).
[CrossRef] [PubMed]

M. Horowitz, R. Daisy, B. Fischer, and J. Zyskind, “Narrow-linewidth, singlemode erbium-doped fibre laser with intracavity wave mixing in saturable absorber,” Electron. Lett. 30, 648–649 (1994).
[CrossRef]

French, V. A.

Frisken, S. J.

Helmfrid, S.

Hill, K. O.

Horowitz, M.

M. Horowitz, R. Daisy, B. Fischer, and J. Zyskind, “Narrow-linewidth, singlemode erbium-doped fibre laser with intracavity wave mixing in saturable absorber,” Electron. Lett. 30, 648–649 (1994).
[CrossRef]

M. Horowitz, R. Daisy, B. Fischer, and J. L. Zyskind, “Linewidth-narrowing mechanisms in lasers by nonlinear wave mixing,” Opt. Lett. 19, 1406–1408 (1994).
[CrossRef] [PubMed]

Jaskorzynska, B.

Kringlebotn, J. T.

Laming, R. I.

Lassen, H. E.

X. Pan, H. Olesen, B. Tromborg, and H. E. Lassen, “Analytic description of the standing wave effects in DFB lasers,” Proc. Inst. Electr. Eng. Part J 139, 189–193 (1992).

H. E. Lassen, H. Olesen, and B. Tromborg, “Gain compression and asymmetric gain due to the Bragg grating induced by the standing waves in Fabry–Perot lasers,” IEEE Photon. Technol. Lett. 1, 261–263 (1989).
[CrossRef]

Lax, M.

G. P. Agrawal and M. Lax, “Analytic evaluation of interference effects on laser output in a Fabry–Perot resonator,” J. Opt. Soc. Am. B 71, 515–519 (1981).
[CrossRef]

G. P. Agrawal and M. Lax, “Effects of interference on gain saturation in laser resonators,” J. Opt. Soc. Am. B 69, 1717–1719 (1979).
[CrossRef]

Loh, W. H.

Mahdi, M.

M. A. Noginov, H. J. Caulfield, P. Venkateswarlu, and M. Mahdi, “Study of migration in Cr:Er:YSGG using an upconversion light induced grating technique,” Opt. Mater. 5, 97–103 (1996).
[CrossRef]

Mecozzi, A.

Mossakowska, A.

A. Mossakowska, P. Szczepanski, and W. Wolinski, “Modulation bandwidth of waveguide distributed feedback lasers,” IEEE J. Quantum Electron. 30, 230–234 (1994).
[CrossRef]

Noginov, M. A.

M. A. Noginov, H. J. Caulfield, P. Venkateswarlu, and M. Mahdi, “Study of migration in Cr:Er:YSGG using an upconversion light induced grating technique,” Opt. Mater. 5, 97–103 (1996).
[CrossRef]

Olesen, H.

X. Pan, H. Olesen, B. Tromborg, and H. E. Lassen, “Analytic description of the standing wave effects in DFB lasers,” Proc. Inst. Electr. Eng. Part J 139, 189–193 (1992).

H. E. Lassen, H. Olesen, and B. Tromborg, “Gain compression and asymmetric gain due to the Bragg grating induced by the standing waves in Fabry–Perot lasers,” IEEE Photon. Technol. Lett. 1, 261–263 (1989).
[CrossRef]

Pan, X.

X. Pan, H. Olesen, B. Tromborg, and H. E. Lassen, “Analytic description of the standing wave effects in DFB lasers,” Proc. Inst. Electr. Eng. Part J 139, 189–193 (1992).

Payne, D. N.

Powell, R. C.

Sargent III, M.

M. Sargent III, “Laser saturation grating phenomena,” Appl. Phys. 9, 127–141 (1976).
[CrossRef]

Szczepanski, P.

A. Mossakowska, P. Szczepanski, and W. Wolinski, “Modulation bandwidth of waveguide distributed feedback lasers,” IEEE J. Quantum Electron. 30, 230–234 (1994).
[CrossRef]

Tatsuno, K.

Tromborg, B.

X. Pan, H. Olesen, B. Tromborg, and H. E. Lassen, “Analytic description of the standing wave effects in DFB lasers,” Proc. Inst. Electr. Eng. Part J 139, 189–193 (1992).

H. E. Lassen, H. Olesen, and B. Tromborg, “Gain compression and asymmetric gain due to the Bragg grating induced by the standing waves in Fabry–Perot lasers,” IEEE Photon. Technol. Lett. 1, 261–263 (1989).
[CrossRef]

Vanin, E. V.

Venkateswarlu, P.

M. A. Noginov, H. J. Caulfield, P. Venkateswarlu, and M. Mahdi, “Study of migration in Cr:Er:YSGG using an upconversion light induced grating technique,” Opt. Mater. 5, 97–103 (1996).
[CrossRef]

Watanabe, A.

Wolinski, W.

A. Mossakowska, P. Szczepanski, and W. Wolinski, “Modulation bandwidth of waveguide distributed feedback lasers,” IEEE J. Quantum Electron. 30, 230–234 (1994).
[CrossRef]

Zayhowski, J. J.

J. J. Zayhowski, “The effects of spatial hole burning and energy diffusion on the single-mode operation of standing-wave lasers,” IEEE J. Quantum Electron. 26, 2052–2057 (1990).
[CrossRef]

Zyskind, J.

M. Horowitz, R. Daisy, B. Fischer, and J. Zyskind, “Narrow-linewidth, singlemode erbium-doped fibre laser with intracavity wave mixing in saturable absorber,” Electron. Lett. 30, 648–649 (1994).
[CrossRef]

Zyskind, J. L.

Appl. Opt.

Appl. Phys.

M. Sargent III, “Laser saturation grating phenomena,” Appl. Phys. 9, 127–141 (1976).
[CrossRef]

Electron. Lett.

M. Horowitz, R. Daisy, B. Fischer, and J. Zyskind, “Narrow-linewidth, singlemode erbium-doped fibre laser with intracavity wave mixing in saturable absorber,” Electron. Lett. 30, 648–649 (1994).
[CrossRef]

IEEE J. Quantum Electron.

J. J. Zayhowski, “The effects of spatial hole burning and energy diffusion on the single-mode operation of standing-wave lasers,” IEEE J. Quantum Electron. 26, 2052–2057 (1990).
[CrossRef]

A. Mossakowska, P. Szczepanski, and W. Wolinski, “Modulation bandwidth of waveguide distributed feedback lasers,” IEEE J. Quantum Electron. 30, 230–234 (1994).
[CrossRef]

IEEE Photon. Technol. Lett.

H. E. Lassen, H. Olesen, and B. Tromborg, “Gain compression and asymmetric gain due to the Bragg grating induced by the standing waves in Fabry–Perot lasers,” IEEE Photon. Technol. Lett. 1, 261–263 (1989).
[CrossRef]

J. Appl. Phys.

H. G. Danielmeyer, “Effects of drift and diffusion of excited states on spatial hole burning and laser oscillation,” J. Appl. Phys. 42, 3125–3132 (1971).
[CrossRef]

J. Opt. Soc. Am. B

S. Helmfrid and K. Tatsuno, “Stable single-mode operation of intracavity-doubled diode-pumped Nd:YVO4 lasers: theoretical study,” J. Opt. Soc. Am. B 11, 436–445 (1994).
[CrossRef]

G. P. Agrawal and M. Lax, “Effects of interference on gain saturation in laser resonators,” J. Opt. Soc. Am. B 69, 1717–1719 (1979).
[CrossRef]

G. P. Agrawal and M. Lax, “Analytic evaluation of interference effects on laser output in a Fabry–Perot resonator,” J. Opt. Soc. Am. B 71, 515–519 (1981).
[CrossRef]

Opt. Lett.

Opt. Mater.

M. A. Noginov, H. J. Caulfield, P. Venkateswarlu, and M. Mahdi, “Study of migration in Cr:Er:YSGG using an upconversion light induced grating technique,” Opt. Mater. 5, 97–103 (1996).
[CrossRef]

Proc. Inst. Electr. Eng. Part J

X. Pan, H. Olesen, B. Tromborg, and H. E. Lassen, “Analytic description of the standing wave effects in DFB lasers,” Proc. Inst. Electr. Eng. Part J 139, 189–193 (1992).

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Figures (4)

Fig. 1
Fig. 1

Magnitude of the gain envelope, in units of the corresponding value at infinite diffusion, as function of normalized intensity (T0/Usat)|E±|2 and with normalized diffusion constant 4k2T0D as a parameter. We assume that the amplitudes of the forward- and the backward-going waves are equally large, |E+|=|E-|.

Fig. 2
Fig. 2

Modulation depth of the first-order gain grating as a function of normalized intensity and with normalized diffusion as a parameter. The amplitudes of the forward- and backward-going waves are assumed to be equally large.

Fig. 3
Fig. 3

Modulation depth of higher-order gain gratings as a function of grating order m for different values of normalized intensity (T0/Usat)|E±|2. The normalized diffusion constant 4k2T0D is (a) 0 and (b) 2.

Fig. 4
Fig. 4

Relative increase of the gain-saturation coefficient of the forward-going wave owing to scattering on the induced gain grating, (g˜+-g0)/(g˜+D=-g0)-1, as function of normalized intensity and with normalized diffusion constant as a parameter. The amplitudes of the forward- and the backward-going waves are assumed to be equally large.

Tables (1)

Tables Icon

Table 1 Examples of Normalized Diffusion Constants for Some Rare Earths a

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

Is=|E+ exp(iωst-ikz)+E- exp(iωst+ikz)|2=|E+|2+|E-|2+[E+E-* exp(-2ikz)+c.c.],
D 2gz2-g-g0T0
-g |E+|2+|E-|2+[E+E-* exp(-2ikz)+c.c.]Usat=0,
g(z)=g˜0+m=11/2[g˜2m exp(-2ikmz)+c.c.]=g˜0+m=1y˜m[Bm exp(-2ikmz)+c.c.],
C0g˜0+|B|2y˜1=g0,
g˜0+2C1y˜1+|B|2y˜2=0,
y˜m-1+2Cmy˜m+|B|2y˜m+1=0,(m=2, 3, ),
vm=detg0|B|20002C1|B|200012C2|B|200012Cm-1|B|20012Cm,
wm=detC0|B|20012C1|B|200012C2|B|200012Cm-1|B|20012Cm.
ξm-2Cmξm-1+|B|2ξm-2=0,
ξm=αj=0mFj+βj=0mLj.
vm=g0(C1+C12-|B|2)2(C0+C02-|B|2)C12-|B|2×j=0m(Cj+Cj2-|B|2)-g0(C1-C12-|B|2)2(C0-C02-|B|2)C12-|B|2×j=0m(Cj-Cj2-|B|2),
wm=C0(C1+C12-|B|2)-|B|22(C0+C02-|B|2)C12-|B|2×j=0m(Cj+Cj2-|B|2)-C0(C1-C12-|B|2)-|B|22(C0-C02-|B|2)C12-|B|2×j=0m(Cj-Cj2-|B|2),
g˜0=g0(C1+C12-|B|2)C0(C1+C12-|B|2)-|B|2.
limm j=0m(Cj-Cj2-|B|2)j=0m(Cj+Cj2-|B|2)=0.
g˜0D==g0/C0=g01+T0Usat (|E+|2+|E-|2)-1.
g˜2=2y˜1B=-2g0BC0(C1+C12-|B|2)-|B|2.
g˜2m=-2g0BmC0(C1+C12-|B|2)-|B|2×j=2m -1Cj2-|B|2+Cj(m=2, 3, ).
dE+dzgain=1/2(g˜0E++1/2g˜2E-)=1/2g˜+E+,
dE-dzgain=-1/2(g˜0E-+1/2g˜2*E+)=-1/2g˜-E-,
dIp(z)dz=-αL+(g0max-g˜0) σa(ωp)+σe(ωp)σa(ωs)+σe(ωs)Ip(z),
D 2N(z, t)z2-{R[N(z, t)]-R[N0(z, t)]}
-1Γ G[N(z, t)]S(z, t)=0,
N2t=-N1t=D×2N2-N2T1+σa(ωs)ωs Is+σa(ωp)ωp IpN1-σe(ωs)ωs Is+σe(ωp)ωp IpN2.
g(r, t)=σe(ωs)N2(r, t)-σa(ωs)N1(r, t).
gt=D×2g-g-g0T0-g IsUsat,

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