Abstract

Optical fibers in which gain-guiding effects are significant or even dominant compared with conventional index guiding may become of practical interest for future high-power single-mode fiber lasers. I derive the propagation characteristics of symmetrical slab waveguides and cylindrical optical fibers having arbitrary amounts of mixed gain and index guiding, assuming a single uniform transverse profile for both the gain and the refractive-index steps. Optical fibers of this type are best characterized by using a complex-valued v˜-squared parameter in place of the real-valued v parameter commonly used to describe conventional index-guided optical fibers.

© 2003 Optical Society of America

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References

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  1. S. E. Miller, “Light propagation in generalized lens-like media,” Bell Syst. Tech. J. 44, 2017–2064 (1964).
    [CrossRef]
  2. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).
  3. H.-G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, UK, 1977).
  4. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  5. B. E. A. Saleh, M. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
  6. K. Okamoto, Fundamentals of Optical Waveguides (Academic, New York, 2000).
  7. T. L. Paoli, “Waveguiding in a stripe-geometry junction laser,” IEEE J. Quantum Electron. QE-13, 662–668 (1977).
    [CrossRef]
  8. W. Streifer, R. D. Burnham, D. R. Scifres, “An analytic study of (GaAl)As gain guided lasers at threshold,” IEEE J. Quantum Electron. QE-18, 856–864 (1982).
    [CrossRef]
  9. G. P. Agrawal, Semiconductor Lasers: Past, Present, and Future (AIP Press, Woodbury, N.Y., 1995)
  10. L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
    [CrossRef]
  11. B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
    [CrossRef]
  12. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif.1986).
  13. A.-A. R. Al-Rashed, B. E. A. Saleh, “Decentered Gaussian beams,” Appl. Opt. 34, 6819–6825 (1995).
    [CrossRef] [PubMed]
  14. A. K. Chan, C. P. Lai, H. F. Taylor, “Antiguiding index profiles in broad stripe semiconductor lasers for high-power,” IEEE J. Quantum Electron. 24, 489–495 (1988).
    [CrossRef]
  15. F. Salin, J. Squier, “Gain guiding in solid-state lasers,” Opt. Lett. 17, 1352–1354 (1992).
    [CrossRef] [PubMed]
  16. A. E. Siegman, “High power purely gain guided fiber la-sers,” presented at the Annual Meeting of the Optical Society of America, Long Beach, California, October 16, 2001.
  17. L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
    [CrossRef]
  18. L. W. Casperson, S. D. Lunnam, “Gaussian modes in high loss laser resonators,” Appl. Opt. 14, 1193–1199 (1975).
    [CrossRef] [PubMed]
  19. U. Ganiel, A. Hardy, Y. Silberberg, “Stability of optical laser resonators with mirrors of gaussian reflectivity profile, which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
    [CrossRef]
  20. A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
    [CrossRef]
  21. M. Nazarathy, A. Hardy, J. Shamir, “Generalized mode theory of conventional and phase-conjugate resonators,” J. Opt. Soc. Am. 73, 576–586 (1983).
    [CrossRef]
  22. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).
  23. A. E. Siegman, “Does evanescent gain exist?” manuscript in preparation, available from the author, Stanford University, Stanford, California (siegman@stanford.edu).
  24. N. S. Kapany, Fiber Optics (Academic, New York, 1967).
  25. J. Midwinter, Optical Fibers for Transmission (Wiley, New York, 1979).

1995 (1)

1992 (1)

1988 (1)

A. K. Chan, C. P. Lai, H. F. Taylor, “Antiguiding index profiles in broad stripe semiconductor lasers for high-power,” IEEE J. Quantum Electron. 24, 489–495 (1988).
[CrossRef]

1983 (2)

B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

M. Nazarathy, A. Hardy, J. Shamir, “Generalized mode theory of conventional and phase-conjugate resonators,” J. Opt. Soc. Am. 73, 576–586 (1983).
[CrossRef]

1982 (1)

W. Streifer, R. D. Burnham, D. R. Scifres, “An analytic study of (GaAl)As gain guided lasers at threshold,” IEEE J. Quantum Electron. QE-18, 856–864 (1982).
[CrossRef]

1977 (1)

T. L. Paoli, “Waveguiding in a stripe-geometry junction laser,” IEEE J. Quantum Electron. QE-13, 662–668 (1977).
[CrossRef]

1975 (3)

U. Ganiel, A. Hardy, Y. Silberberg, “Stability of optical laser resonators with mirrors of gaussian reflectivity profile, which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
[CrossRef]

A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

L. W. Casperson, S. D. Lunnam, “Gaussian modes in high loss laser resonators,” Appl. Opt. 14, 1193–1199 (1975).
[CrossRef] [PubMed]

1974 (1)

L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
[CrossRef]

1968 (1)

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

1964 (1)

S. E. Miller, “Light propagation in generalized lens-like media,” Bell Syst. Tech. J. 44, 2017–2064 (1964).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Semiconductor Lasers: Past, Present, and Future (AIP Press, Woodbury, N.Y., 1995)

Al-Rashed, A.-A. R.

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Burnham, R. D.

W. Streifer, R. D. Burnham, D. R. Scifres, “An analytic study of (GaAl)As gain guided lasers at threshold,” IEEE J. Quantum Electron. QE-18, 856–864 (1982).
[CrossRef]

Casperson, L. W.

L. W. Casperson, S. D. Lunnam, “Gaussian modes in high loss laser resonators,” Appl. Opt. 14, 1193–1199 (1975).
[CrossRef] [PubMed]

L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
[CrossRef]

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

Chan, A. K.

A. K. Chan, C. P. Lai, H. F. Taylor, “Antiguiding index profiles in broad stripe semiconductor lasers for high-power,” IEEE J. Quantum Electron. 24, 489–495 (1988).
[CrossRef]

Ganiel, U.

U. Ganiel, A. Hardy, Y. Silberberg, “Stability of optical laser resonators with mirrors of gaussian reflectivity profile, which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
[CrossRef]

Hardy, A.

M. Nazarathy, A. Hardy, J. Shamir, “Generalized mode theory of conventional and phase-conjugate resonators,” J. Opt. Soc. Am. 73, 576–586 (1983).
[CrossRef]

U. Ganiel, A. Hardy, Y. Silberberg, “Stability of optical laser resonators with mirrors of gaussian reflectivity profile, which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
[CrossRef]

Kapany, N. S.

N. S. Kapany, Fiber Optics (Academic, New York, 1967).

Lai, C. P.

A. K. Chan, C. P. Lai, H. F. Taylor, “Antiguiding index profiles in broad stripe semiconductor lasers for high-power,” IEEE J. Quantum Electron. 24, 489–495 (1988).
[CrossRef]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Lunnam, S. D.

Midwinter, J.

J. Midwinter, Optical Fibers for Transmission (Wiley, New York, 1979).

Miller, S. E.

S. E. Miller, “Light propagation in generalized lens-like media,” Bell Syst. Tech. J. 44, 2017–2064 (1964).
[CrossRef]

Nazarathy, M.

Newstein, M.

B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides (Academic, New York, 2000).

Paoli, T. L.

T. L. Paoli, “Waveguiding in a stripe-geometry junction laser,” IEEE J. Quantum Electron. QE-13, 662–668 (1977).
[CrossRef]

Perry, B. N.

B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

Rabinowitz, P.

B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

Saleh, B. E. A.

Salin, F.

Scifres, D. R.

W. Streifer, R. D. Burnham, D. R. Scifres, “An analytic study of (GaAl)As gain guided lasers at threshold,” IEEE J. Quantum Electron. QE-18, 856–864 (1982).
[CrossRef]

Shamir, J.

Siegman, A. E.

A. E. Siegman, “High power purely gain guided fiber la-sers,” presented at the Annual Meeting of the Optical Society of America, Long Beach, California, October 16, 2001.

A. E. Siegman, “Does evanescent gain exist?” manuscript in preparation, available from the author, Stanford University, Stanford, California (siegman@stanford.edu).

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif.1986).

Silberberg, Y.

U. Ganiel, A. Hardy, Y. Silberberg, “Stability of optical laser resonators with mirrors of gaussian reflectivity profile, which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
[CrossRef]

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Squier, J.

Streifer, W.

W. Streifer, R. D. Burnham, D. R. Scifres, “An analytic study of (GaAl)As gain guided lasers at threshold,” IEEE J. Quantum Electron. QE-18, 856–864 (1982).
[CrossRef]

Taylor, H. F.

A. K. Chan, C. P. Lai, H. F. Taylor, “Antiguiding index profiles in broad stripe semiconductor lasers for high-power,” IEEE J. Quantum Electron. 24, 489–495 (1988).
[CrossRef]

Teich, M.

B. E. A. Saleh, M. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Unger, H.-G.

H.-G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, UK, 1977).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Yariv, A.

A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

Yeh, P.

A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

Bell Syst. Tech. J. (1)

S. E. Miller, “Light propagation in generalized lens-like media,” Bell Syst. Tech. J. 44, 2017–2064 (1964).
[CrossRef]

IEEE J. Quantum Electron. (4)

T. L. Paoli, “Waveguiding in a stripe-geometry junction laser,” IEEE J. Quantum Electron. QE-13, 662–668 (1977).
[CrossRef]

W. Streifer, R. D. Burnham, D. R. Scifres, “An analytic study of (GaAl)As gain guided lasers at threshold,” IEEE J. Quantum Electron. QE-18, 856–864 (1982).
[CrossRef]

L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
[CrossRef]

A. K. Chan, C. P. Lai, H. F. Taylor, “Antiguiding index profiles in broad stripe semiconductor lasers for high-power,” IEEE J. Quantum Electron. 24, 489–495 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

U. Ganiel, A. Hardy, Y. Silberberg, “Stability of optical laser resonators with mirrors of gaussian reflectivity profile, which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
[CrossRef]

A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

Other (12)

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif.1986).

G. P. Agrawal, Semiconductor Lasers: Past, Present, and Future (AIP Press, Woodbury, N.Y., 1995)

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

H.-G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, UK, 1977).

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

B. E. A. Saleh, M. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

K. Okamoto, Fundamentals of Optical Waveguides (Academic, New York, 2000).

A. E. Siegman, “High power purely gain guided fiber la-sers,” presented at the Annual Meeting of the Optical Society of America, Long Beach, California, October 16, 2001.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

A. E. Siegman, “Does evanescent gain exist?” manuscript in preparation, available from the author, Stanford University, Stanford, California (siegman@stanford.edu).

N. S. Kapany, Fiber Optics (Academic, New York, 1967).

J. Midwinter, Optical Fibers for Transmission (Wiley, New York, 1979).

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Figures (11)

Fig. 1
Fig. 1

Schematic representation of a step-profile optical fiber or a one-dimensional slab waveguide in which the core or central slab has both a real refractive-index step Δn and a gain or loss coefficient step Δα with respect to the lossless refractive-index n0 in the cladding.

Fig. 2
Fig. 2

Alternative forms of the gain and loss coefficient profiles for gain-guided waveguides or fibers (top row) and for loss-guided waveguides and fibers (bottom row).

Fig. 3
Fig. 3

Values of the w˜ parameter for propagating modes in symmetric slab waveguides or gain-guided fibers will be limited to the right half of the complex w˜ plane as shown in this figure, with purely index-guided modes located along the positive real axis. The quadrants above or below this axis correspond to modes having a mixture of index guiding or antiguiding plus gain or loss guiding.

Fig. 4
Fig. 4

Allowed regions in the u˜ plane for the modes of a symmetric slab waveguide. Each of the ovate regions in the upper plot corresponds to the full w˜ plane in Fig. 3, and the thinner dashed contours in the lower plot correspond directly to the circular arcs in Fig. 3. The heavier dashed lines indicated the pure gain-guided or loss-guided cases.

Fig. 5
Fig. 5

Propagation regions for the lowest- and higher-order modes of a slab waveguide transformed into the complex v˜2 plane. The x and y axes are labeled Δn and Δα, but the numerical scales correspond to the associated values of v˜2.

Fig. 6
Fig. 6

Allowed regions in the complex u˜ plane for the LP01 and LP11 modes of gain- or loss-guided step-profile fibers.

Fig. 7
Fig. 7

Mode boundaries and mode propagation regions for the LP01 and LP11 modes of a cylindrical gain-guided step-profile in the complex (v˜2 or Δn, Δα) plane. The roughly circular dashed arcs correspond to semicircles in the complex w˜ plane as shown in Fig. 3.

Fig. 8
Fig. 8

The purely real values of wr=Re[w˜] plotted versus the purely real value of ur/umax along the positive real axis in Figs. 3 and 5. The values of ur are normalized to umax=π/2 for the m=1 slab waveguide case and to umax=j01 for the LP01 fiber mode case.

Fig. 9
Fig. 9

Upper plot: Contours of constant confinement factor Γ in the complex v˜2 plane for the lowest-order (m=1) mode of a slab waveguide. Lower plot: Confinement factor Γ versus magnitude of the v˜ parameter for slab waveguides with conventional index guiding and with pure gain guiding.

Fig. 10
Fig. 10

Upper plot: Contours of constant confinement factor Γ in the complex v˜2 plane for the LP01 mode of a cylindrical fiber. Lower plot: Confinement factor Γ versus magnitude of the v˜ parameter for conventional index-guided LP01 mode, pure gain-guided LP01 mode, and pure gain-guided LP11 mode.

Fig. 11
Fig. 11

Relative magnitudes and directions of the k and g vectors for the inhomogeneous plane-wave components inside and outside a gain-guided and index-antiguided slab waveguide operating in the lowest-order or m=1 mode.

Equations (23)

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v2πaλ[(n0+Δn)2-n02]1/22πaλ(2n0Δn)1/2,
2π(n0+Δn)λ+jΔα=2π(n0+Δn+j(λ/2π)Δα)λ=2π(n0+Δn˜)λ,
v˜2=2πaλ2[(n0+Δn˜)2-n02]2πaλ22n0Δn+j λ2π Δα.
E˜(x)=cos(u˜x/a)|x| acos(u˜)×exp[-w˜(|x|-a)/a]|x| a,
(β˜a)2=(ka)2(n0+Δn˜)2-u˜2=(ka)2n02+w˜2
β˜akn0a+w˜22kn0a=kn0a+12kn0a(wr2-wi2+2jwrwi),
w˜2+u˜2=v˜2,
w˜=u˜ tan u˜forsymmetricorodd-mmodes
w˜=-u˜tan u˜ forantisymmetricoreven-mmodes.
E˜01(r)=J0(u˜r/a)ra[J0(u˜)/K0(w˜)]×K0(w˜r/a)ra,
E˜11(r)=J1(u˜r/a)ra[J1(u˜)/K1(w˜)]×K1(w˜r/a)ra,
u˜J1(u˜)J0(u˜)=w˜K1(w˜)K0(w˜)(LP01modes),
u˜J1(u˜)J1(u˜)=-w˜K0(w˜)K1(w˜)(LP11modes).
limr Km(w˜r/a)πa2w˜exp-w˜ra,
E˜(r)=exp(-jkr+gr).
Ecore(x)exp[-j(βz+urx)+(gz+uix)]
Eclad(x)exp[-j(βz+wix)+(gz-wrx)]
wr=ursin 2ur-uisinh 2uicos 2ur-cosh 2ui,
wi=uisin 2ur+ursinh 2uicos 2ur-cosh 2ui,
(β˜a)2=(ka)2n02+w˜2(modalpropagationconstant),
(β˜pwa)2=(ka)2n02+v˜2(plane-wavepropagationconstant).
ΓIm β˜Im β˜pwwrwivrvi,
Γm=1=(ursin 2ur-uisinh 2ui)(uisin 2ur+ursinh 2ui)2urui+2uruicos 2urcosh 2ui+(ur2-ui2)sin 2ursinh 2ui,

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