Theory of slow light excitation in 1D photonic crystals

Theory of slow light excitation in1D photonic

crystals

D.Yudistira1,D.A.I.Marpaung1,H.P.Handoyo1,H.J.W.M.Hoekstra1,M.Hammer1,

M.O.Tjia2,A.A.Iskandar2

1Mesa+Institute,University of Twente,The Netherlands

2Photonics group,Department of Physics,Institute Teknologi Bandung,Indonesia

Slow light(SL)states corresponding to wavelength regions near the bandgap edge of grated structures are known to show strong?eld enhancement.Such states may be excited ef?ciently by well-optimised adiabatic transitions in grating structures,e.g.,by slowly turning on the modulation depth.To study adiabatic excitations,a detailed investigation in1D is performed to obtain insight into the relation between the device parameters and properties like?eld enhancement and modal re?ection.The results enable the design of an adiabatic device for ef?cient excitation of SL states in1D,and may be the basis for further research on2D and3D photonic crystals.

Introduction

Recently,periodic dielectric structures(i.e.photonic crystals(PCs))have attracted much interest.The main reason for that is that materials with a photonic band gap can be realised by means of a proper choice of both lattice structure and index contrast.This leads to a variety of(possible)applications such as the inhibition of spontaneous emission[1],low loss waveguides with sharp bends[2],narrow-band?lters,and strong?eld enhancement related to low group velocity,i.e.slow light(SL),modes propagating at frequencies near the band edge[3].

Due to the mismatch of both modal pro?les and phase velocities between the incoming propagating wave and the modes in SL devices(e.g.gratings),direct excitation of SL modes will cause high losses[4].One promising technique,that has been introduced in several papers to overcome this problem,is the so-called adiabatic excitation[4].By means of tapering either by gradually changing index or geometry,it is possible to change the pro?le of an incoming wave gradually into that of the SL mode.Thus,the effects of pro?le mismatch and so of losses can be minimised.In this paper,we will present a theory for SL excitation in1D.In particular,we discuss the relation between device parameters,like the modulation depth and,modal properties like?eld enhancement and modal re?ection.

Basic theory

We consider a1D model structure as depicted in Fig.1,assuming a plane wave at normal incidence coming in from the left.The Helmholtz equation is to be solved is(?zz+ n2(z)k2)E y=0with vacuum wavenumber k and index distribution n(z)(see Fig.1).In each unit cell the?eld solution can be written as a sum of right and left traveling modal ?elds,corresponding to that unit cell.These solutions are of the form of plane waves as follows:

Figure1:Refractive index as a function of z of a typical adiabatic1D gratings. Indices2and1distinguish the local high and low index regions.

n

E±(z)=u±(z)e?iβ±z≡v1(2),±(z)+w1(2),±(z)(1) where the+(-)sign labels the right(left)traveling Bloch mode,βis the Bloch-wave

number and v1(2),±and w1(2),±are the corresponding right and left traveling plane wave ?elds in the low and high index of each grating period.We limit ourself to wavelengths outside the band gap leading to realβandβ?=?β+,with its value chosen in the1st Brillouin zone.From the modal?eld solution the power enhancement,η,de?ned by

η=

|v2|2+|w2|2

dz b=κbb b+κba ae?2i

R z

0β(z

)dz ,i d

real quantities.The coupling coef?cients can be determined by comparing Eq.(3)with

the results of modal re?ection in the presence of changes in the modulation depth. Relation between modal and structure parameters

We now present the relation between the power enhancement,η,the coupling between

left and right traveling modes,described byκ≡|κab|,and the structure parameters will be investigated.For each wavelength with considered region(0.89544μm λ 0.9959μm),

there is a modulation depth n g de?ning the band edge of the uniform grating(0.01< n g<0.3).It is found that the structure dependence ofηandκcan be described more conveniently in terms of the latter quantity.By careful?tting of the numerical results we arrive at

κ n g C1?z n m

h1/2

+

C2n m

?Cn m,max

?Cn max～κLn max

L=