Abstract

Self-induced transparency (SIT) pulses induce a traveling population inversion in two-level atoms. As a rule, the active medium in which the soliton travels has to be homogeneous. Here, we study the effect of a spatially disordered modulation in the refractive index profile that may lead to Anderson localizations. The interplay between the ultrashort SIT pulse, a nonlinear effect, and this kind of disorder-induced mode exhibits intriguing features. Once the SIT pulse is confined in the spatially confined regions, they act as closed cavities for the SIT population inversion. A positive optical feedback mechanism can be thus activated and, as a result, a two-level laserlike emission can be obtained.

© 2012 Optical Society of America

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    [CrossRef]

2011

2010

A. Szameit, Y. V. Kartashov, P. Zeil, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, V. A. Vysloukh, and L. Torner, “Wave localization at the boundary of disordered photonic lattices,” Opt. Lett. 35, 1172–1174 (2010).
[CrossRef]

Claudio Conti, Elena D’Asaro, Salvatore Stivala, Alessandro Busacca, and Gaetano Assanto, “Parametric self-trapping in the presence of randomized quasi phase matching,” Opt. Lett. 35, 3760–3762 (2010).
[CrossRef]

V. Folli and C. Conti, “Frustrated brownian motion of nonlocal solitary waves,” Phys. Rev. Lett. 104, 193901 (2010).
[CrossRef]

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, “Bistability of anderson localized states in nonlinear random media,” Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef]

N. N. Rosanov, V. V. Kozlov, and S. Wabnitz, “Maxwell-Drude-Bloch dissipative few-cycle optical solitons,” Phys. Rev. A 81, 043815 (2010).
[CrossRef]

M. Bamba, S. Pigeon, and C. Ciuti, “Quantum squeezing generation versus photon localization in a disordered planar microcavity,” Phys. Rev. Lett. 104, 213604 (2010).
[CrossRef]

R. G. S. El-Dardiry, A. P. Mosk, O. L. Muskens, and A. Lagendijk, “Experimental studies on the mode structure of random lasers,” Phys. Rev. A 81, 043830 (2010).
[CrossRef]

2009

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, “Phase diagram and complexity of mode-locked lasers: from order to disorder,” Phys. Rev. Lett. 102, 083901 (2009).
[CrossRef]

S. Gentilini, A. Fratalocchi, L. Angelani, G. Ruocco, and C. Conti, “Ultrashort pulse propagation and the anderson localization,” Opt. Lett. 34, 130–132 (2009).
[CrossRef]

2008

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: theory, 3d+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

D. S. Wiersma, “The physics and applications of random lasers,” Nat. Phys. 4, 359–367 (2008).
[CrossRef]

C. Conti and A. Fratalocchi, “Dynamic light diffusion, Anderson localization and lasing in disordered inverted opals: 3D ab-initio Maxwell-Bloch computation,” Nat. Phys. 4, 794–798 (2008).
[CrossRef]

C. López, “Anderson localization of light: A little disorder is just right,” Nat. Phys. 4, 755–756 (2008).
[CrossRef]

2007

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef]

2005

I. V. Mel’nikov and J. S. Aitchison, “Gap soliton memory in a resonant photonic crystal,” Appl. Phys. Lett. 87201111 (2005).
[CrossRef]

2004

B. I. Mantsyzov, I. V. Mel’nikov, and J. S. Aitchison, “Dynamic control over optical solitons in a resonant photonic crystal,” IEEE J. Sel. Top. Quantum Electron. 10, 893–899 (2004).
[CrossRef]

2003

H. Cao, X. Jiang, Y. Ling, J. X. Xu, and C. M. Soukoulis, “Mode repulsion and mode coupling in random lasers,” Phys. Rev. B 67, 161101R (2003).
[CrossRef]

2000

1995

R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. A 52, 3082–3094 (1995).
[CrossRef]

1990

A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Y. M. Sklyarov, “Present state of self-induced transparency theory, Phys. Rep. 191, 1–108 (1990).
[CrossRef]

1987

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489(1987).
[CrossRef]

1984

S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53, 2169–2172 (1984).
[CrossRef]

1971

G. L. Lamb, “Analytical descriptions of ultrashort optical pulse propagation in a resonant medium,” Rev. Mod. Phys. 43, 99–124 (1971).
[CrossRef]

1969

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

1968

V. S. Lethokov, “Generation of light by a scattering medium with negative resonance absorption,” Sov. Phys. JETP 26, 835–840 (1968).

1967

S. L. McCall and E. L. Hahn, “Self-induced transparency by pulsed coherent light,” Phys. Rev. Lett. 18, 908–911 (1967).
[CrossRef]

1966

R. V. Ambartsumyan, N. G. Basov, P. G. Kryukov, and S. Lethokov, “5A10(b)—a laser with a nonresonant feedback,” IEEE J. Quantum Electron. 2, 442–446 (1966).
[CrossRef]

1917

A. Einstein, “On the quantum theory of radiation (quantentheorie der strahlung),” Phys. Z. 18, 121–128 (1917).

Aitchison, J. S.

I. V. Mel’nikov and J. S. Aitchison, “Gap soliton memory in a resonant photonic crystal,” Appl. Phys. Lett. 87201111 (2005).
[CrossRef]

B. I. Mantsyzov, I. V. Mel’nikov, and J. S. Aitchison, “Dynamic control over optical solitons in a resonant photonic crystal,” IEEE J. Sel. Top. Quantum Electron. 10, 893–899 (2004).
[CrossRef]

Allen, L.

L. Allen and L. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, 1975).

Ambartsumyan, R. V.

R. V. Ambartsumyan, N. G. Basov, P. G. Kryukov, and S. Lethokov, “5A10(b)—a laser with a nonresonant feedback,” IEEE J. Quantum Electron. 2, 442–446 (1966).
[CrossRef]

Angelani, L.

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, “Phase diagram and complexity of mode-locked lasers: from order to disorder,” Phys. Rev. Lett. 102, 083901 (2009).
[CrossRef]

S. Gentilini, A. Fratalocchi, L. Angelani, G. Ruocco, and C. Conti, “Ultrashort pulse propagation and the anderson localization,” Opt. Lett. 34, 130–132 (2009).
[CrossRef]

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: theory, 3d+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

Arnold, J. M.

R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. A 52, 3082–3094 (1995).
[CrossRef]

Assanto, Gaetano

Bamba, M.

M. Bamba, S. Pigeon, and C. Ciuti, “Quantum squeezing generation versus photon localization in a disordered planar microcavity,” Phys. Rev. Lett. 104, 213604 (2010).
[CrossRef]

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef]

Basharov, A. M.

A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Y. M. Sklyarov, “Present state of self-induced transparency theory, Phys. Rep. 191, 1–108 (1990).
[CrossRef]

Basov, N. G.

R. V. Ambartsumyan, N. G. Basov, P. G. Kryukov, and S. Lethokov, “5A10(b)—a laser with a nonresonant feedback,” IEEE J. Quantum Electron. 2, 442–446 (1966).
[CrossRef]

Bliokh, K. Y.

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, “Bistability of anderson localized states in nonlinear random media,” Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef]

Bliokh, Y. P.

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, “Bistability of anderson localized states in nonlinear random media,” Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef]

Busacca, Alessandro

Cao, H.

H. Cao, X. Jiang, Y. Ling, J. X. Xu, and C. M. Soukoulis, “Mode repulsion and mode coupling in random lasers,” Phys. Rev. B 67, 161101R (2003).
[CrossRef]

Ciuti, C.

M. Bamba, S. Pigeon, and C. Ciuti, “Quantum squeezing generation versus photon localization in a disordered planar microcavity,” Phys. Rev. Lett. 104, 213604 (2010).
[CrossRef]

Conti, C.

V. Folli and C. Conti, “Self-induced transparency and the anderson localization of light,” Opt. Lett. 36, 2830–2832(2011).
[CrossRef]

V. Folli and C. Conti, “Frustrated brownian motion of nonlocal solitary waves,” Phys. Rev. Lett. 104, 193901 (2010).
[CrossRef]

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, “Phase diagram and complexity of mode-locked lasers: from order to disorder,” Phys. Rev. Lett. 102, 083901 (2009).
[CrossRef]

S. Gentilini, A. Fratalocchi, L. Angelani, G. Ruocco, and C. Conti, “Ultrashort pulse propagation and the anderson localization,” Opt. Lett. 34, 130–132 (2009).
[CrossRef]

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: theory, 3d+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

C. Conti and A. Fratalocchi, “Dynamic light diffusion, Anderson localization and lasing in disordered inverted opals: 3D ab-initio Maxwell-Bloch computation,” Nat. Phys. 4, 794–798 (2008).
[CrossRef]

Conti, Claudio

D’Asaro, Elena

Dodd, R. K.

R. K. Dodd, Solitons and Nonlinear Wave Equations(Academic, 1982).

Dreisow, F.

Eberly, L. H.

L. Allen and L. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, 1975).

Einstein, A.

A. Einstein, “On the quantum theory of radiation (quantentheorie der strahlung),” Phys. Z. 18, 121–128 (1917).

El-Dardiry, R. G. S.

R. G. S. El-Dardiry, A. P. Mosk, O. L. Muskens, and A. Lagendijk, “Experimental studies on the mode structure of random lasers,” Phys. Rev. A 81, 043830 (2010).
[CrossRef]

Elyutin, S. O.

A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Y. M. Sklyarov, “Present state of self-induced transparency theory, Phys. Rep. 191, 1–108 (1990).
[CrossRef]

Feynman, R.

R. Feynman, QED (Princeton University, 1988).

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef]

Folli, V.

V. Folli and C. Conti, “Self-induced transparency and the anderson localization of light,” Opt. Lett. 36, 2830–2832(2011).
[CrossRef]

V. Folli and C. Conti, “Frustrated brownian motion of nonlocal solitary waves,” Phys. Rev. Lett. 104, 193901 (2010).
[CrossRef]

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, “Phase diagram and complexity of mode-locked lasers: from order to disorder,” Phys. Rev. Lett. 102, 083901 (2009).
[CrossRef]

Fratalocchi, A.

S. Gentilini, A. Fratalocchi, L. Angelani, G. Ruocco, and C. Conti, “Ultrashort pulse propagation and the anderson localization,” Opt. Lett. 34, 130–132 (2009).
[CrossRef]

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: theory, 3d+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

C. Conti and A. Fratalocchi, “Dynamic light diffusion, Anderson localization and lasing in disordered inverted opals: 3D ab-initio Maxwell-Bloch computation,” Nat. Phys. 4, 794–798 (2008).
[CrossRef]

Freilikher, V.

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, “Bistability of anderson localized states in nonlinear random media,” Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef]

Gentilini, S.

Gogny, D. M.

R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. A 52, 3082–3094 (1995).
[CrossRef]

Hahn, E. L.

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

S. L. McCall and E. L. Hahn, “Self-induced transparency by pulsed coherent light,” Phys. Rev. Lett. 18, 908–911 (1967).
[CrossRef]

Heinrich, M.

Jiang, X.

H. Cao, X. Jiang, Y. Ling, J. X. Xu, and C. M. Soukoulis, “Mode repulsion and mode coupling in random lasers,” Phys. Rev. B 67, 161101R (2003).
[CrossRef]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489(1987).
[CrossRef]

S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53, 2169–2172 (1984).
[CrossRef]

Kartashov, Y. V.

Keil, R.

Kivshar, Y. S.

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, “Bistability of anderson localized states in nonlinear random media,” Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef]

Kozlov, V. V.

N. N. Rosanov, V. V. Kozlov, and S. Wabnitz, “Maxwell-Drude-Bloch dissipative few-cycle optical solitons,” Phys. Rev. A 81, 043815 (2010).
[CrossRef]

V. V. Kozlov and A. B. Matsko, “Stochastic theory of self-induced transparency: linearized approach,” J. Opt. Soc. Am. B 17, 1031–1038 (2000).
[CrossRef]

Kryukov, P. G.

R. V. Ambartsumyan, N. G. Basov, P. G. Kryukov, and S. Lethokov, “5A10(b)—a laser with a nonresonant feedback,” IEEE J. Quantum Electron. 2, 442–446 (1966).
[CrossRef]

Lagendijk, A.

R. G. S. El-Dardiry, A. P. Mosk, O. L. Muskens, and A. Lagendijk, “Experimental studies on the mode structure of random lasers,” Phys. Rev. A 81, 043830 (2010).
[CrossRef]

Lamb, G. L.

G. L. Lamb, “Analytical descriptions of ultrashort optical pulse propagation in a resonant medium,” Rev. Mod. Phys. 43, 99–124 (1971).
[CrossRef]

Leonetti, M.

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: theory, 3d+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

Lethokov, S.

R. V. Ambartsumyan, N. G. Basov, P. G. Kryukov, and S. Lethokov, “5A10(b)—a laser with a nonresonant feedback,” IEEE J. Quantum Electron. 2, 442–446 (1966).
[CrossRef]

Lethokov, V. S.

V. S. Lethokov, “Generation of light by a scattering medium with negative resonance absorption,” Sov. Phys. JETP 26, 835–840 (1968).

Leuzzi, L.

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, “Phase diagram and complexity of mode-locked lasers: from order to disorder,” Phys. Rev. Lett. 102, 083901 (2009).
[CrossRef]

Ling, Y.

H. Cao, X. Jiang, Y. Ling, J. X. Xu, and C. M. Soukoulis, “Mode repulsion and mode coupling in random lasers,” Phys. Rev. B 67, 161101R (2003).
[CrossRef]

López, C.

C. López, “Anderson localization of light: A little disorder is just right,” Nat. Phys. 4, 755–756 (2008).
[CrossRef]

Maimistov, A. I.

A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Y. M. Sklyarov, “Present state of self-induced transparency theory, Phys. Rep. 191, 1–108 (1990).
[CrossRef]

Mantsyzov, B. I.

B. I. Mantsyzov, I. V. Mel’nikov, and J. S. Aitchison, “Dynamic control over optical solitons in a resonant photonic crystal,” IEEE J. Sel. Top. Quantum Electron. 10, 893–899 (2004).
[CrossRef]

Matsko, A. B.

McCall, S. L.

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

S. L. McCall and E. L. Hahn, “Self-induced transparency by pulsed coherent light,” Phys. Rev. Lett. 18, 908–911 (1967).
[CrossRef]

Mel’nikov, I. V.

I. V. Mel’nikov and J. S. Aitchison, “Gap soliton memory in a resonant photonic crystal,” Appl. Phys. Lett. 87201111 (2005).
[CrossRef]

B. I. Mantsyzov, I. V. Mel’nikov, and J. S. Aitchison, “Dynamic control over optical solitons in a resonant photonic crystal,” IEEE J. Sel. Top. Quantum Electron. 10, 893–899 (2004).
[CrossRef]

Mosk, A. P.

R. G. S. El-Dardiry, A. P. Mosk, O. L. Muskens, and A. Lagendijk, “Experimental studies on the mode structure of random lasers,” Phys. Rev. A 81, 043830 (2010).
[CrossRef]

Muskens, O. L.

R. G. S. El-Dardiry, A. P. Mosk, O. L. Muskens, and A. Lagendijk, “Experimental studies on the mode structure of random lasers,” Phys. Rev. A 81, 043830 (2010).
[CrossRef]

Nolte, S.

Pigeon, S.

M. Bamba, S. Pigeon, and C. Ciuti, “Quantum squeezing generation versus photon localization in a disordered planar microcavity,” Phys. Rev. Lett. 104, 213604 (2010).
[CrossRef]

Rosanov, N. N.

N. N. Rosanov, V. V. Kozlov, and S. Wabnitz, “Maxwell-Drude-Bloch dissipative few-cycle optical solitons,” Phys. Rev. A 81, 043815 (2010).
[CrossRef]

Ruocco, G.

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, “Phase diagram and complexity of mode-locked lasers: from order to disorder,” Phys. Rev. Lett. 102, 083901 (2009).
[CrossRef]

S. Gentilini, A. Fratalocchi, L. Angelani, G. Ruocco, and C. Conti, “Ultrashort pulse propagation and the anderson localization,” Opt. Lett. 34, 130–132 (2009).
[CrossRef]

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: theory, 3d+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

Schwartz, T.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef]

Segev, M.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef]

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Figures (13)

Fig. 1.
Fig. 1.

Sketch of Δ ϵ r ( z ) as a function of the propagation distance z showing the disordered dielectric layers.

Fig. 2.
Fig. 2.

Sketch of Δ N a ( z ) as a function of the propagation distance z . Into a dielectric layer, the optically active response of the medium is absent, so the density of the polarizable atoms is N a ( z ) = 0 ; correspondingly, in the perturbed regions Δ N a ( z ) = N a .

Fig. 3.
Fig. 3.

The simulation region is 200 μm long, and the homogeneous two-level medium extends from 7.5 to 50 μm. The disordered two-level medium ( ϵ r = 1 ) extends from 50 to 150 μm; the initial population is equal to ρ 30 = 1 and randomly alternated with a dielectric nonresonant medium with a different refractive index ( ρ 30 = 0 , ϵ r = 11 ). The inset shows an enlarged region of the disordered structure.

Fig. 4.
Fig. 4.

Spectrum of the input pulse for τ = 100 fs (thin dashed curve); spectrum of a transmitted short pulse without the resonant medium showing the photonic bandgap (thick dashed curve, γ = 0 ); transmitted spectrum in the absence of a resonant medium with γ = 0.5 showing the formation of resonances in the forbidden band (continuous thin curve); a further increase of the strength of disorder γ = 1 produces a more pronounced localization of the surface Anderson states (continuous bold curve).

Fig. 5.
Fig. 5.

SIT population inversion ρ 3 in (a) the low-index ordered case ( ϵ r = 1.5 , γ = 0 ) and in (b) the high-index disordered case ( ϵ r = 11 , γ = 1 ). The modulation of the population inversion between ρ 3 = 1 and ρ 3 = 0 is related to the underlying refractive index scenario. In fact, the dielectric layers with ρ 3 = 0 are alternated with the resonant medium on spatial scales much smaller than the width of the SIT pulse. This produces the oscillations in the ρ 3 profile that jumps to unity when the resonant medium is present and the population inversion takes place, while it is zero elsewhere.

Fig. 6.
Fig. 6.

Electric field E in (a) the low-index ordered case ( ϵ r = 1.5 , γ = 0 ) and in (b) the high-index disordered case ( ϵ r = 11 , γ = 1 ).

Fig. 7.
Fig. 7.

(a) The random medium for 50 μm < z < 150 μm is created by alternating slices of the two-level sample with ϵ r = 1 with dieletric nonresonant layers with ϵ r = 11 . We show the snapshot of two launched pulses with a temporal delay of 50 fs. (b) The black dashed curve represents the spectrum of the transmitted double SIT pulses without the resonant medium with γ = 0 ; the bandgap is still observable. We also show the transmitted spectrum for one short pulse for γ = 0.5 (continuous black curve) and the transmitted spectrum for two short pulses, showing the more pronounced Anderson resonances with respect to the former case (continuous light curve).

Fig. 8.
Fig. 8.

SIT population inversion ρ 3 for two different snapshots, the case with a single pulse (thick curve) and the case with two consecutive pulses (thin curve). As can be seen, in the latter case the inversion population is more effective and mediated by Anderson states deeply located in the structure.

Fig. 9.
Fig. 9.

(a) SIT pulse trajectories calculated for γ = 0.1 , ϵ r = 6 for three different initial pulse amplitudes E 01 = 4.2186 × 10 9 V / m , E 02 = 4 E 01 , and E 03 = 8 E 01 ; (b) corresponding calculated standard deviations.

Fig. 10.
Fig. 10.

SIT trajectories for ϵ r 1 = 2.5 , ϵ r 2 = 6 , ϵ r 3 = 11 when E 0 = E 01 . Curves for ϵ r 1 , 2 are cut (they proceed as straight lines) for comparison with the ϵ r 3 case. Results are averaged over 100 disorder realizations.

Fig. 11.
Fig. 11.

(a) Temporal profile of the input SIT pulse with τ = 100 fs and E 0 = 4.2186 × 10 9 V / m , (b) transmitted pulse without disordered structure, (c) emitted signal in the presence of an highly scattering medium ( ϵ r = 11 , γ = 1 ). The horizontal scale in (b) and (c) is overlapped with that in (a).

Fig. 12.
Fig. 12.

Temporal profiles of the reflected electric field in the highly scattering case in Fig. 11(c), including the input pulse, its reflected fraction, and the subsequent two-level laser emission.

Fig. 13.
Fig. 13.

Transmitted spectrum (continuous red curve) by Anderson states compared with the input SIT pulse (dashed blue curve).

Equations (63)

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N 2 N 1 = exp [ ( E 2 E 1 ) k B T ] ,
t H y = 1 μ 0 z E x ,
t E x = 1 ϵ z H y N a p ϵ t S 1 ,
t S 1 = 1 T 2 S 1 + ω 0 S 2 ,
t S 2 = ω 0 S 1 1 T 2 S 2 + 2 p E x S 3 ,
t S 3 = 2 p E x S 2 1 T 1 ( S 3 S 30 ) ,
S 1 = ρ ^ 12 + ρ ^ 21 ,
S 2 = i ( ρ ^ 12 ρ ^ 21 ) ,
S 3 = i ( ρ ^ 11 ρ ^ 22 ) ,
ρ . 12 = ( 1 T 2 + i ω 0 ) ρ ^ 12 + i p E x ( ρ ^ 22 ρ ^ 11 ) ,
ρ ^ . 22 ρ ^ . 11 = i 2 p E x ( ρ ^ 12 ρ ^ 21 * ) ,
1 T 1 [ ρ ^ 22 ρ ^ 11 ( ρ ^ 22 0 ρ ^ 11 0 ) ] ,
P ( t ) = N a p ( ρ ^ 12 + ρ ^ 12 * ) x ^ ,
E ( r , t ) = [ A ( t , z ) e ( i ω c t k z ) + c c ] x ^ ,
P ( r , t ) = [ Λ ( t , z ) e ( i ω c t k z ) + c c ] x ^ ,
ρ ^ 12 = Λ ( t ) e i ( ω c t k z ) N a p .
ρ ^ 22 ρ ^ 11 = N ( t ) N a ,
Λ . ( t ) + [ 1 T 2 + i ( ω 0 ω c ) ] Λ ( t ) = i p 2 A ( t ) N ( t ) .
N . ( t ) + 1 T 1 ( N N 0 ) = i 2 [ ( Λ A * Λ * A ) + ( A Λ e 2 i ( ω c t k z ) A * Λ * e + 2 i ( ω c t k z ) ) ] .
z 2 E x 1 c 2 t 2 E x = 1 ϵ c 2 t 2 P .
2 i ω c c ϵ r ( A e A * e + ) + ( A e A * e + ) + i ϵ r ω c c 2 ( A . e A . * e + ) ϵ r c 2 ( A .. e + A .. * e + ) = ω c 2 ϵ 0 c 2 P ,
A ˙ ( t ) + c n z A = i ω c 2 ϵ 0 n 2 Λ .
n c t A + z A = ω c p B 0 ϵ 0 c n B ,
t B = p N a 2 B 0 A N ,
t N = B 0 p N a ( B A * + B * A ) .
[ z + ( 1 c 1 v g ) ξ ] A = κ B ,
ξ B = A N 2 ,
ξ N = ( B * A + B A * ) ,
A pulse 2 π = Ω R τ p 2 ,
A ( z , t ) = 2 β sech [ β ( ξ X ) ] exp ( i Ω ξ + i θ ) .
( 1 + Δ ϵ ϵ ) t E x = 1 ϵ z H y p ϵ [ N a + Δ N a ( z ) ] t S 1 .
t A + c n z A = i ω c 2 ϵ 0 n 2 Λ + i ω c 2 n 2 Δ ϵ r A ,
t Λ + [ 1 T 2 + i ( ω c ω 0 ) ] Λ = i p 2 A N ,
t N + 1 T 1 ( N N 0 ) = i 2 ( Λ A * Λ * A ) .
[ z + ( 1 c 1 v g ) ξ i μ ] A = κ B + S 1 ,
ξ B = 1 2 A N + S 2 ,
ξ N = ( B * A + B A * ) + S 3 ,
S 1 = Δ ϵ r ( i ω c 2 n c ) A + Δ N a k N a B , S 2 = 0 , S 3 = 0 .
A s = u E = 2 β E cosh ( w ) , B s = p E = β 2 β 2 + Ω 2 [ tanh ( w ) i Ω β ] E cosh ( w ) , N s = n = 1 + 2 β 2 β 2 + Ω 2 1 cosh 2 ( w ) ,
1 v g = 1 c + k / 2 β 2 + Ω 2 .
( z A⃗ 1 ) · I 1 = L ( A⃗ 1 ) + S⃗ p ,
I = ( 1 0 0 0 1 0 0 0 1 ) ,
L ( A⃗ ) 1 = [ i μ A 1 ( 1 c 1 v g ) ξ A 1 + k B 1 ξ B 1 + 1 2 ( A 1 N s + A s N 1 ) ξ N 1 + 2 ( B s * A 1 + B 1 * A s ) ] .
f⃗ X = X A⃗ s ,
f⃗ θ = θ A⃗ s ,
f⃗ β = β A⃗ s ,
f⃗ Ω = Ω A⃗ s ,
A⃗ 1 = f⃗ X δ X + f⃗ β δ β + f⃗ θ δ θ + f⃗ Ω δ Ω + α⃗ R ,
f ^ θ = i f⃗ β , f ^ β = i f⃗ θ , f ^ Ω = i f⃗ X , f ^ X = i f⃗ Ω X ,
( f ^ a , f⃗ b ) = N a δ a , b ,
N θ = N β = 1 3 β 2 ,
N X = N Ω = 2 π β 1 3 β .
( f ^ λ i , f⃗ λ i · I 1 ) δ λ i . = ( f ^ λ i , L 1 f⃗ λ j ) δ λ j + ( f ^ λ i , S⃗ p ) , ( f ^ μ i , f⃗ μ i · I 1 ) δ μ i . = ( f ^ μ i , L 1 f⃗ μ j ) δ μ j + ( f ^ μ i , S⃗ p ) ,
( f ^ β , L 1 f⃗ X ) = ( f ^ Ω , L 1 f⃗ θ ) = 0 , ( f ^ θ , L 1 f⃗ Ω ) = ( f ^ X , L 1 f⃗ β ) = 4 κ β 2 , ( f ^ β , f⃗ β · I 1 ) = ( f ^ θ , f⃗ θ · I 1 ) = 4 , ( f ^ X , f⃗ X · I 1 ) = ( f ^ Ω , f⃗ Ω · I 1 ) = 4 β .
V A , B ( z ) V A , B ( z ) = V A , B 2 δ ( z z ) ,
( f ^ β , S⃗ p ) = 0 ,
( f ^ Ω , S⃗ p ) = 0 ,
( f ^ X , S⃗ p ) = 2 / β V B ( z ) ,
( f ^ θ , S⃗ p ) = 4 V A ( z ) .
δ X . = k β 3 δ β + V B ( z ) 2 β 2 , δ θ . = k β 2 δ Ω + V A ( z ) ,
δ X ( z ) 2 = V B 2 z 4 β 4 .
( κ 2 N a 2 ( Δ N a ) 2 tanh 2 ( w ) + ω c 2 n 2 c 2 β 2 ( Δ ϵ r ) 2 ) 1 / 2 cosh ( w ) .
ω c n c β ( Δ ϵ r ) 2 1 / 2 1 .

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