Abstract

On the basis of the finite-difference time domain method, we have calculated the temporal evolution of the electromagnetic field inside a waveguide system that is bounded by perfectly conducting walls. Part of the waveguide is filled with an active medium whose amplifying properties resulting from stimulated emission processes are modeled by introducing a frequency-dependent conductivity to Maxwell’s equations. An electric noise current that is randomly distributed throughout the gain region is used to model spontaneous emission processes. The feedback provided by the reflections from the walls of the waveguide and the interfaces between the active part and the adjacent vacuum leads to lasing for a sufficiently strong gain medium. Besides this conventional feedback, we also use rough walls for the part of the waveguide that is filled with the gain medium and investigate the influence of this roughness on the output intensity, the frequency spectrum of the emitted light, and the lasing threshold.

© 2004 Optical Society of America

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References

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  1. M. P. van Albada, M. B. van der Mark, and A. Lagendijk, “Experiments on weak localization of light and their interpretation” in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1990), pp. 97–136.
  2. H. Cao, “Lasing in random media,” Waves Random Media 13, R1–R39 (2003).
    [CrossRef]
  3. V. S. Letokhov, “Light generation by a scattering medium with a negative resonant absorption,” Sov. Phys. JETP 26, 835–840 (1968).
  4. H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Wu, and R. P. H. Chang, “Ultraviolet lasing in resonators formed by scattering in semiconductor polycrystalline films,” Appl. Phys. Lett. 73, 3656–3658 (1998).
    [CrossRef]
  5. H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
    [CrossRef]
  6. D. Wiersma, “The smallest random laser,” Nature 406, 132–133 (2000).
    [CrossRef] [PubMed]
  7. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  8. A. A. Maradudin and T. Michel, “The transverse correlation length for randomly rough surfaces,” J. Stat. Phys. 58, 485–501 (1990).
    [CrossRef]
  9. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  10. A. Taflove, Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 1998).
  11. S. C. Hagness, R. M. Joseph, and A. Taflove, “Subpicosecond electrodynamics of distributed Bragg reflector microlasers: Results from finite-difference time domain simulations,” Radio Sci. 31, 931–941 (1996).
    [CrossRef]
  12. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
    [CrossRef]
  13. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 1995).

2003 (1)

H. Cao, “Lasing in random media,” Waves Random Media 13, R1–R39 (2003).
[CrossRef]

2000 (1)

D. Wiersma, “The smallest random laser,” Nature 406, 132–133 (2000).
[CrossRef] [PubMed]

1999 (1)

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

1998 (1)

H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Wu, and R. P. H. Chang, “Ultraviolet lasing in resonators formed by scattering in semiconductor polycrystalline films,” Appl. Phys. Lett. 73, 3656–3658 (1998).
[CrossRef]

1996 (1)

S. C. Hagness, R. M. Joseph, and A. Taflove, “Subpicosecond electrodynamics of distributed Bragg reflector microlasers: Results from finite-difference time domain simulations,” Radio Sci. 31, 931–941 (1996).
[CrossRef]

1994 (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

1990 (1)

A. A. Maradudin and T. Michel, “The transverse correlation length for randomly rough surfaces,” J. Stat. Phys. 58, 485–501 (1990).
[CrossRef]

1968 (1)

V. S. Letokhov, “Light generation by a scattering medium with a negative resonant absorption,” Sov. Phys. JETP 26, 835–840 (1968).

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

Berenger, J.-P.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Cao, H.

H. Cao, “Lasing in random media,” Waves Random Media 13, R1–R39 (2003).
[CrossRef]

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Wu, and R. P. H. Chang, “Ultraviolet lasing in resonators formed by scattering in semiconductor polycrystalline films,” Appl. Phys. Lett. 73, 3656–3658 (1998).
[CrossRef]

Chang, R. P. H.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Wu, and R. P. H. Chang, “Ultraviolet lasing in resonators formed by scattering in semiconductor polycrystalline films,” Appl. Phys. Lett. 73, 3656–3658 (1998).
[CrossRef]

Hagness, S. C.

S. C. Hagness, R. M. Joseph, and A. Taflove, “Subpicosecond electrodynamics of distributed Bragg reflector microlasers: Results from finite-difference time domain simulations,” Radio Sci. 31, 931–941 (1996).
[CrossRef]

Ho, S. T.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Wu, and R. P. H. Chang, “Ultraviolet lasing in resonators formed by scattering in semiconductor polycrystalline films,” Appl. Phys. Lett. 73, 3656–3658 (1998).
[CrossRef]

Joseph, R. M.

S. C. Hagness, R. M. Joseph, and A. Taflove, “Subpicosecond electrodynamics of distributed Bragg reflector microlasers: Results from finite-difference time domain simulations,” Radio Sci. 31, 931–941 (1996).
[CrossRef]

Letokhov, V. S.

V. S. Letokhov, “Light generation by a scattering medium with a negative resonant absorption,” Sov. Phys. JETP 26, 835–840 (1968).

Maradudin, A. A.

A. A. Maradudin and T. Michel, “The transverse correlation length for randomly rough surfaces,” J. Stat. Phys. 58, 485–501 (1990).
[CrossRef]

Michel, T.

A. A. Maradudin and T. Michel, “The transverse correlation length for randomly rough surfaces,” J. Stat. Phys. 58, 485–501 (1990).
[CrossRef]

Ong, H. C.

H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Wu, and R. P. H. Chang, “Ultraviolet lasing in resonators formed by scattering in semiconductor polycrystalline films,” Appl. Phys. Lett. 73, 3656–3658 (1998).
[CrossRef]

Seelig, E. W.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

Taflove, A.

S. C. Hagness, R. M. Joseph, and A. Taflove, “Subpicosecond electrodynamics of distributed Bragg reflector microlasers: Results from finite-difference time domain simulations,” Radio Sci. 31, 931–941 (1996).
[CrossRef]

Wang, Q. H.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

Wiersma, D.

D. Wiersma, “The smallest random laser,” Nature 406, 132–133 (2000).
[CrossRef] [PubMed]

Wu, J. Y.

H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Wu, and R. P. H. Chang, “Ultraviolet lasing in resonators formed by scattering in semiconductor polycrystalline films,” Appl. Phys. Lett. 73, 3656–3658 (1998).
[CrossRef]

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

Zhao, Y. G.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Wu, and R. P. H. Chang, “Ultraviolet lasing in resonators formed by scattering in semiconductor polycrystalline films,” Appl. Phys. Lett. 73, 3656–3658 (1998).
[CrossRef]

Appl. Phys. Lett. (1)

H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Wu, and R. P. H. Chang, “Ultraviolet lasing in resonators formed by scattering in semiconductor polycrystalline films,” Appl. Phys. Lett. 73, 3656–3658 (1998).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

J. Comput. Phys. (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. Stat. Phys. (1)

A. A. Maradudin and T. Michel, “The transverse correlation length for randomly rough surfaces,” J. Stat. Phys. 58, 485–501 (1990).
[CrossRef]

Nature (1)

D. Wiersma, “The smallest random laser,” Nature 406, 132–133 (2000).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

Radio Sci. (1)

S. C. Hagness, R. M. Joseph, and A. Taflove, “Subpicosecond electrodynamics of distributed Bragg reflector microlasers: Results from finite-difference time domain simulations,” Radio Sci. 31, 931–941 (1996).
[CrossRef]

Sov. Phys. JETP (1)

V. S. Letokhov, “Light generation by a scattering medium with a negative resonant absorption,” Sov. Phys. JETP 26, 835–840 (1968).

Waves Random Media (1)

H. Cao, “Lasing in random media,” Waves Random Media 13, R1–R39 (2003).
[CrossRef]

Other (4)

M. P. van Albada, M. B. van der Mark, and A. Lagendijk, “Experiments on weak localization of light and their interpretation” in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1990), pp. 97–136.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

A. Taflove, Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 1998).

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 1995).

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Figures (9)

Fig. 1
Fig. 1

Layout of the two-dimensional waveguide system with width d. The origin of the coordinate system is at the center of the bottom surface. The electromagnetic field vectors of an s-polarized field are indicated at the bottom left. The active region of length L features spontaneous emission processes with random frequencies ω as well as stimulated emission processes with central frequency ω0 (illustrated in the insets), and is bounded on its left and right by two vacuum regions of length L. The waveguide system is terminated on its left and right ends by a PML that absorbs the electromagnetic field without reflection. The entire system is bounded by a perfect conductor. The bottom surface of the active region is defined by a profile function ζ1(x); while the top surface is defined by a profile function ζ2(x).

Fig. 2
Fig. 2

Flux Fx(x0, t) in the x direction across x=x0 of a waveguide system of width d.

Fig. 3
Fig. 3

Electric and magnetic field components of the Yee grid in the staircase approximation inside a region that is bounded by a perfect electric conductor.

Fig. 4
Fig. 4

Output intensities IxV1 and IxV2 of the laser into the left and right vacuum regions for three different choices of the wall roughness as functions of the pumping intensity σ0. Solid lines interpolating circles, flat walls; dashed lines interpolating squares, asymmetric sawtooth profile with a=696 nm, H=30 nm, s=0.2; dotted lines interpolating diamonds, random rough profile with a=100 nm, δ=20 nm.

Fig. 5
Fig. 5

Spectrum of the flux into the right vacuum region V2 for the laser (σ0=6.5×1013 s-1) with flat walls (solid curve), with an asymmetric sawtooth profile with a=696 nm, H=30 nm, s=0.2 (dashed curve), and with randomly rough walls with a=100 nm, δ=20 nm (dashed–dotted curves). The frequency dependence of the amplification described by the real part of the conductivity [Eq. (10)] is indicated by the dotted curve.

Fig. 6
Fig. 6

Contour plots of the electric field Ez inside the waveguide system with d=0.5 µm after t=6 ps. The pumping strength is σ0=1.5×1013 s-1. Length scale in micrometers. Active region from -3.13 µm to +3.13 µm, vacuum regions from ±3.13 µm to ±9.36 µm, PML from ±9.36 µm to ±10.89 µm. Panel (a), asymmetric sawtooth profile, a=696 nm, H=30 nm, s=0.2: black part, Ez=-7 statvolt/cm; white part, Ez=7 statvolt/cm. Panel (b), flat walls: black part, Ez=-10-5 statvolt/cm; white part, Ez=10-5 statvolt/cm; gray part, Ez0.

Fig. 7
Fig. 7

Spatially averaged fluxes FxV1 and FxV2 in 1011 ergs/cm2 s into the left (gray area or curve) and right (black area or curve) vacuum regions as functions of time. Panel (a) displays the whole simulation time of 4 ps. Panel (b) displays a short section of 0.02 ps of the steady output.

Fig. 8
Fig. 8

Output intensities IxV1 and IxV2 of the laser without index steps into the left and right vacuum regions for three different choices of the wall roughness as functions of the pumping intensity σ0. Solid lines interpolating circles, flat walls; dashed lines interpolating squares, asymmetric sawtooth profile with a=696 nm, H=30 nm, s=0.2; dotted lines interpolating diamonds, random rough profile with a=100 nm, δ=20 nm.

Fig. 9
Fig. 9

Spectrum of the flux into the right vacuum region for the laser without index steps with planar walls (solid curve), walls of asymmetric sawtooth profile with a=696 nm, H=30 nm, s=0.2 (dashed curve), and randomly rough walls with a=100 nm, δ=20 nm (dashed–dotted lines). The amplification described by the real part of the conductivity [Eq. (10)] is indicated by the dotted curve.

Equations (38)

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ζ(x)ζ(x)=δ2 exp[-(x-x)2/a2],
ζ(x)=H1+2xsa-sax0H1-2x(1-s)a0x(1-s)a
δ=1a -sa(1-s)adxζ2(x)1/2=H1/3.
Γ(x)=1+cosh (KL/2)cosh(Kx)+cosh (KL/2),
Ez(x, t)y=-1c Hx(x, t)t,
-Ez(x, t)x=-1c tHy(x, t),
Hy(x, t)x-Hx(x, t)y=c Ez(x, t)t+4πcJz(x, t).
Jz(x, t)=Jzst(x, t)+Jzsp(x, t).
Jzst(x, t)=-+dtσˆ(x, t-t)Ez(x, t).
σ(x, ω)=-i σ0S[Ez0(x)]2T2×1ω-ω0+iT2 +1ω+ω0+iT2 .
S[Ez0(x)]=1+cn|Ez0(x)|28πIs-1.
(1+ω02T22)Jzst(x, t)+2T2 tJzst(x, t)+T22 2t2Jzst(x, t)
=S[Ez0(x)]σ0Ez(x, t)+S[Ez0(x)]σ0T2 tEz(x, t).
Gz(x, t)=tJzst(x, t).
(1+ω02T22)Jzst(x, t)+2T2Gz(x, t)+T22 tGz(x, t)
=S[Ez0(x)]σ0Ez(x, t)+S[Ez0(x)]σ0T2 tEz(x, t).
S(x, t)=c4π[E(x, t)×H(x, t)]
I(x)=c8πn|Ez0(x)|2.
Fx(x0, t)=c4πd 0ddy Re[-Ez(x0, y, t)Hy(x0, y, t)].
Ix=1/2 Fx0.
Ez(x0, ω)=1d -+dt0ddyEz(x0, y, t)exp(iωt).
EV1(x, y)=l=1(exp(iklx)+Rl exp(-iklx))sinlπdy,
EG(x, y)=l=1(T1l exp(iqlx)+T2l exp(-iqlx))sinlπdy,
EV2(x, y)=l=1Cl exp(iklx)sinlπdy,
T1m=2km(km+qm)exp(-iqmL)(km+qm)2-exp(iqmL)(km-qm)2.
exp(-iqL)(k+q)2-exp(iqL)(k-q)2=0.
Hx|i,jn+1/2=Hx|i,jn-1/2-cΔtΔy{Ez|i,j+1/2n-Ez|i,j-1/2n},
Hy|i,jn+1/2=Hy|i,jn-1/2+cΔtΔx{Ez|i+1/2,jn-Ez|i-1/2,jn},
Ez|i,jn+1=Ez|i,jn+cΔt Hy|i+1/2,jn+1/2-Hy|i-1/2,jn+1/2Δx-Hx|i,j+1/2n+1/2-Hx|i,j-1/2n+1/2Δy-2πΔt{Jzst|i,jn+1+Jzst|i,jn+Jzsp}.
Jzst|i,jn+1=Jzst|i,jn+Δt2{Gz|i,jn+1+Gz|i,jn},
Gz|i,jn+1
=c1Hy|i+1/2,jn+1/2-Hy|i-1/2,jn+1/2Δx-Hx|i,j+1/2n+1/2-Hx|i,j-1/2n+1/2Δy+c2Ez|i,jn+c3Gz|i,jn+c4Jzst|i,jn.
c1=4cΔtS|i,jσ0(2T2+Δt)β,
c2=8σ0ΔtS|i,jβ,
c3=-{8T2(Δt-T2)+(Δt)2[2(1+ω02T22)+4πσ0S|i,j(Δt+2T2)]}/β,
c4=-{4Δt[2(1+ω02T22)+4πσ0S|i,j(2T2+Δt)]}/β,
β=8T2(Δt+T2)+(Δt)2[2(1+ω02T22)+4πσ0S|i,j(2T2+Δt)],
S|i,j=1+cn|Ez0|i,j|28πIs-1

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