Distortionless pulse-train propagation

Distortionless Pulse-Train Propagation in a Nonlinear Photonic Bandgap Structure Doped Uniformly With Inhomogeneously Broadening Two-Level Atoms

Hong-Yih Tseng and Sien Chi

Abstract—The pulse propagation in a one-dimensional non-linear photonic bandgap(PBG)structure doped uniformly with inhomogeneously broadening two-level atoms is investigated.The Maxwell–Bloch equations describing pulse propagation in such a uniformly doped PBG structure are derived first and further reduced to effective nonlinear coupled-mode equations.An exact analytic pulse-train solution to these effective coupled-mode equa-tions is obtained.Such a distortionless pulse-train solution is given by sinusoidal functions with a DC background and a modulated phase.Numerical examples of the distortionless pulse train in a silica-based PBG structure doped uniformly with Lorentzian line-shape two-level atoms are shown.

Index Terms—Maxwell–Bloch equations,photonic bandgap structure,self-induced transparency.

I.I NTRODUCTION

T HE DISTORTIONLESS propagation of light through an optical resonance medium has been widely discussed since McCall and Hahn discovered self-induced transparency (SIT)[1].The SIT is characterized by the continuous absorption and reemission of electromagnetic radiation from the resonant atoms.Thus the optical pulse propagates through the medium without loss and distortion.Because of the SIT effect,the group velocity of such a coherent pulse depends on the pulsewidth and is much less than the speed of light in the host medium. Furthermore,the SIT effect is described by the Maxwell–Bloch equations,which have distortionless pulse-train solutions given by the Jacobi elliptic functions[2]–[4].Such pulse-train propagation results from the energy of resonant atoms peri-odically oscillating between the ground state and upper state. In particular,when the Jacobi elliptic modulus is unity,the pulse-train solutions are reduced to single-pulse solutions of hyperbolic secant functions.These single pulse solutions are called SIT solitons.Both SIT solitons and periodic pulse trains have been observed in the experiments[5],[6].

More recently,a photonic bandgap(PBG)structure doped with resonant atoms has drawn considerable attention[7]–[15]. In the meantime,Ak?zbek and John have investigated the funda-mental work on SIT solitary waves in PBG materials doped uni-formly with resonant atoms[16].For example,they have found

Manuscript received January14,2002;revised March21,2002.This work was supported by the National Science Council,Taiwan,R.O.C.under Contract NSC89-2215-E-009-112.

The authors are with the Institute of Electro-Optical Engineering,National Chiao-Tung University,Hsinchu,Taiwan300,R.O.C.(e-mail:dabin.eo86g@ https://www.sodocs.net/doc/9e2922572.html,.tw).

Publisher Item Identifier S1077-260X(02)05479-5.single pulse solutions for frequency detuned far from Bragg resonance and frequency detuned near the PBG edge.How-ever,the SIT analytic solution suitably for general frequency detuning and general phase modulation in a uniformly doped PBG medium has never been found.In this paper,we study the SIT in a nonlinear PBG structure doped uniformly with inho-mogeneously broadening two-level atoms.After neglecting the high-order spatial harmonics of the material polarization,we show that the Maxwell–Bloch equations can be reduced to ef-fective nonlinear coupled-mode equations(NLCMEs).Analytic distortionless pulse-train solutions to these effective NLCMEs are obtained.It is found that even if the carrier frequency of the pulse train is inside the forbidden band,the pulse trains can propagate through the PBG structure and obey the general SIT phase modulation effect.

The paper is organized as follows:In Section II,the Maxwell–Bloch equations governing the optical pulse propagating in a uniformly doped PBG structure are derived by keeping the second derivative of electromagnetic field with respect to the propagation distance.Because this second derivative is consid-ered,our model involves the SIT-induced quadratic dispersion due to the slow-light propagation.We also take into account the material dispersion and Kerr nonlinearity of the host medium. In Section III,we solve the Bloch equations and subsequently reduce the Maxwell–Bloch equations to effective NLCMEs.The effective NLCMEs describe that pulse propagation through a uniformly doped PBG structure is equivalent to that through an effective PBG structure without dopants.In Section IV,we solve the effective NLCMEs and obtain exact pulse-train solutions given by the sinusoidal functions.It is also shown that such a pulse train obeys the general SIT phase modulation effect.In Section V,we numerically study the characteristics of the pulse trains by assuming the inhomogeneously broadening line shape of the resonant atoms is Lorentzian.In Section VI,we compare our results with the previous research and conclude this paper.

II.M AXWELL–B LOCH E QUATIONS

We consider a one-dimensional(1-D)Bragg grating formed in a host medium with Kerr nonlinearity.The periodic variations of the refractive index inside the grating region is assumed to be

[17]

(2.1)

where is the frequency-dependent refractive

index,is the Kerr nonlinear-index co-

1077-260X/02$17.00?2002IEEE

efficient,is the magnitude of the periodic index variations,

and

(2.2)

where

is the electric induced polarization including

the linear and nonlinear contributions of the host medium,

and

,

and

propagating along

the

c.c.

(2.4)

where c.c.stands for complex conjugate,is the polarization unit vector of the light assumed to be linearly polarized along

the

is the transverse modal

distribution,

c.c.

(2.5)

where

and are assumed to be much

smaller than the refractive

index

of the host medium,so that they can be treated as perturbations for expanding

.Consequently,the time-domain coupled-mode equations de-scribing pulse propagation in a uniformly doped PBG structure are written

as

(2.6)

where

via

terms to time domain [18].In the literature,

the wave equation for pulse propagation is usually derived by neglecting the second derivative of electromagnetic field with respect

to

terms can be comparable to the other terms in (2.6).

This effect will be justified in the following derivation.There-fore,we keep

the

terms in our equation.We now consider the atomic Bloch equations.If the relax-ation times of the polarization and population difference are long compared with the pulsewidth,the relaxtion effects of the two-level system can be ignored.Therefore,under the rotating wave approximation,the electric field and the macroscopic po-larization satisfy the Bloch

equations

is the

macroscopic population difference multiplied by the transition matrix

element

(2.9)

where

is the normalized inhomogeneous-broadening

line-shape function.To keep a closed set of Bloch equations,we assume

that

(2.10a)

TSENG AND CHI:DISTORTIONLESS PULSE-TRAIN PROPAGA TION683 After we substitute(2.4),(2.5),and(2.8)–(2.10)into(2.7),the

Bloch equations are expressed

as

(2.11a)

(2.11b)

(2.11c)

to get a closed set of equations.Strictly speaking,

we have ignored all higher-order spatial terms of material

polarization oscillating with multiples of the light wavenumber.

In a periodic structure,such higher-order terms might be formed

by beating of two counterpropagating and phase-modulated

(chirped)waves.This process in general produce many spatial

harmonics of material polarization which are converging very

slowly.Nevertheless,in this paper we is devoted to finding an

exact solution propagating along one-direction and exciting no

spatial harmonic of material polarization[16].

III.E FFECTIVE NLCME S FOR M AXWELL–B LOCH E QUATIONS

In this section,we show how to reduce the Maxwell–Bloch

equations to effective NLCMEs for pulse propagating in a non-

linear PBG structure doped uniformly with inhomogeneously

broadening two-level atoms.In order to obtain the analytic

solution we

assume

(3.1)

where is known as the dipole spectra-response function

and is normalized

as.Integrating(2.11a),we

have

(3.2)

where

(3.3b)

Similarly,by integrating(2.11c)and(2.11d),we

obtain

(3.4a)

,

where

and are defined

as

(3.5a)

and(3.5b)

Substituting(3.1)–(3.5)into(2.11b),we

have

(3.7)

and

(3.9a)

(3.10a)