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1872 J. Opt. Soc. Am. B/Vol. 23, No. 9 /September 2006 Vladimir G. Avramenko

Generalized optical transfer-matrix technique:application to the nonlinear response of multiple

quantum wells

Vladimir G. Avramenko

Department of Physics, Moscow State University, 119992 Moscow, Russia

Received October 20, 2005; revised April 24, 2006; accepted April 25, 2006; posted May 4, 2006 (Doc. ID 65450)

A theoretical study of the electromagnetic field inside a multiple-quantum-well (MQW) structure possessing aquadratic optical nonlinearity is undertaken. The nonlocal response of each quantum well of the structure isanalyzed on the basis of the integral equation for the local field inside the well. The light propagation insidethe structure at the fundamental and sum frequencies is described within the generalized transfer-matrixtechnique. An application of the developed technique to calculation of the optical response of a model MQWstructure is demonstrated. © 2006 Optical Society of America

OCIS codes: 160.6000, 190.4160.

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. INTRODUCTIONhere is increasing interest in the investigation of bothhe linear and nonlinear optical responses of multipleuantum wells (MQW).1–4 A MQW is a periodic structurehat mediates between superlattices5,6 and photonicrystals.7 On the one hand, there is no tunneling of elec-rons between adjacent quantum wells (QWs) of the struc-ure; therefore the quantum-size effects inherent to indi-idual QWs are pronounced in MQW structures too.8–10

n the other hand, the period of the structure is too smalln comparison with optical wavelengths to provide photo-ic bandgap formation in the visible range. The charac-eristic feature of MQW structures is that the componentf the local electric field perpendicular to well boundariesssentially varies within each QW layer; therefore theonlocality of the electromagnetic response inside the

ayer has to be taken into account, e.g., to describe the re-ponse of the structure to p-polarized radiation. Linearptical properties of such systems can be characterized inerms of integral quantities such as d parameters11 orheet-conductivity tensors.12–14

Generally, theoretical investigations of the optical re-ponse of systems containing regions with the nonlocallectromagnetic response are based on the solution of thentegral equation for the local field inside the regions.15–17

ost theoretical works concern the optical response (bothhe linear and nonlinear) of systems containing a singleW. MQW systems can be analyzed in a similar manner,ut instead of a single integral equation, a system ofoupled integral equations has to be solved.18 Dimensionsf the system increase proportionally to the number ofWs in the MQW structure; therefore the approach leads

o extremely increasing numerical efforts in the case ofQW structures.An effective procedure of calculation of the linear opti-

al response of multilayer structures is provided by theptical transfer-matrix technique.19 This technique can bedapted to the case of multilayer structures possessing

0740-3224/06/091872-10/$15.00 © 2

onlinearities.20,21 Unfortunately, the standard transfer-atrix technique is applicable to MQW structures onlyhen the nonlocality of the electromagnetic response in

he direction perpendicular to the boundaries of QWs cane neglected.22 The goal of this work is the development ofmethod of calculation of both the linear and nonlinear

ptical response of MQW structures that, on the oneand, takes into, account the nonlocality of the electro-agnetic response inside QW layers and, on the other,

rovides an effective calculation procedure in a waynalogous to the standard optical transfer-matrix tech-ique.The paper is organized as follows. In Section 2 I

resent a review of the approach to calculation of the localeld inside MQW structures based on the solution of the

ntegral equation. In Section 3 I show that the linear op-ical response of a QW can be described in terms of a gen-ralized transfer matrix. The procedure of calculation ofhe linear response of the MQW structure is described inection 4. In Section 5 I solve the integral equation for the

ocal field produced by the nonlinear currents inside theW. I derive a relation between the amplitudes of

orward- and backward-propagating fields on the mixedrequency at the boundaries of the QW and the ampli-udes of fields at the mixing frequencies. In Section 6, arocedure of calculation of the nonlinear response of theQW structure is described. In Section 7 an application

f the developed formalism to calculation of linear re-ponse of a model MQW structure is demonstrated. Theomparison of the developed technique with the current-heet formalism and the standard transfer-matrix tech-ique is presented. The paper ends in Section 8 with aiscussion of the main features of the developed general-zed transfer-matrix technique.

. INTEGRAL EQUATION FOR THEOCAL FIELDonsider a typical MQW structure consisting of N QWseposited on a semi-infinite substrate [see Fig. 1(a)]. In

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Vladimir G. Avramenko Vol. 23, No. 9 /September 2006 /J. Opt. Soc. Am. B 1873

he MQW structure, layers of thickness d forming QWslternate with layers of the host material of thickness D.he electromagnetic response of the substrate and theost layers is supposed to be a local one; therefore it cane characterized by dielectric constants, �s and �h, respec-ively. The confinement of an electron motion across theW layers results in the appearance of a nonlocal part of

he electromagnetic response of the QW layers that isharacterized by the linear and quadratic conductivityensors, ��z ,z�� and �2��z ,z� ,z��, respectively. The localomponent of the linear-optical response of each QW layers described by the background dielectric constant �q. Ashe QWs have a finite depth, there is a “spill-out” of thelectronic density into the adjacent host layers; thereforehe nonlocal part of the response has to be taken into ac-ount in the QW layers and in the regions or thickness �ear the boundaries between the QW and the host layerssee Fig. 1(b)]. We will use the notation Dqw

�n� for the regionhere the nonlocal component of the electromagnetic re-

ponse of the nth QW is nonzero. It must be noted that inhe MQW structure the tails of the electronic density ofhe neighboring QWs do not overlap, i.e., 2��D; there-ore the regions with the nonlocal response are separatedy the regions of the host material with the local re-ponse.

To analyze the optical response of the MWQ structuree direct the z axis of a Cartesian xyz frame perpendicu-

arly to the layers of the structure. The xz plane is choseno coincide with the incidence plane of the electromag-etic field with frequency � and angle of incidence �counted from the z direction). We suppose that the MQWtructure possesses the translational invariance againstrbitrary displacements parallel to the xy plane. There-

ig. 1. Schematics of (a) a typical MQW structure consisting ofQWs and (b) an individual QW. Due to “spill-out” of the elec-

ronic density p�z�, the nonlocal component of the optical re-ponse is nonzero in the crosshatched region that includes theW layer and the regions of thickness � in the host layers.

ore a generic form of the electric field of the mth har-onic of the fundamental radiation (written in a complex

orm) is given by

Em�r,t� = Em��z�exp�im��x sin �/c − t��, m = 1,2. �1�

n consideration of the electric field at frequency m� weill omit the factor exp�im��x sin � /c− t�� as it is the same

or all quantities determined by the electromagnetic re-ponse of the system at frequency m�.

The local field E��zn� inside the region Dqw�n� satisfies the

ntegral equation18

E��zn� = E�b�zn� − i��

m=1

N

G��n� · �� · E��zm�, �2�

ith �=4� /c2 and notations

G��n� · f�zm� =�

Dqw�m�

G��zn,zm� · f�zm�dzm, �3�

�� · E��zm� =�Dqw

�m��zm,zm� � · E��zm� �dzm� , �4�

here f�z� is an arbitrary vector function and zm�Dqw�m�. In

q. (2) E�b�zn� is the background field; that is, the electro-

agnetic field in the absence of the nonlocal currents in-ide all QWs [as if �� ,zn ,zn�� is equal to zero for n1, . . . ,N], which satisfies the inhomogeneous wave equa-

ion

Lz · G��z,z�� = U�z − z��, �5�

here U is a unit tensor, �z−z�� is the Dirac’s function,nd the operator Lz in a dyadic form is given by23

Lz = U��z��2

c2 − qx2 +

�2

�z2� − iexqx + ez

�

�ziexqx + ez

�

�z .

�6�

ere qx=� sin � /c and ��z� is the piecewise-constant func-ion describing the local part of the linear electromagneticesponse of the MQW structure: ��z� takes the values 1,h, �q, and �s within the regions corresponding to vacuum,ost-medium layers, quantum wells, and substrate, re-pectively.

The local field at frequency 2� inside the region Dqw�n�

atisfies the integral equation [Eq. (2)] with � substitutedith 2�, � with 2� and the background field given by23

E2�b �z� = − i2��

m=1

N

G2��n� · Jnl�zm�, �7�

here

Jnl�zn� =�Dqw

�n��

Dqw�n�

�2��zn,z�,z��:E��n��z��E�

�n��z��dz�dz�.

�8�

It follows from Eq. (2) that the procedure of calculationf the linear response of the MQW structure with N QWseads to solving a system of N coupled integral equations.his procedure requires great numerical efforts that in-

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lude a large number (proportional to N2) of operations ofouble integration [see Eq. (2)]. As a result, the durationf calculations of the response of the MQW structure is N2

imes greater than that in the case of the single QW.The essence of the problem described above is that in

q. (2) the value of the local field inside the QW is ex-ressed through the background field. Alternatively, weill express the local field inside the QW through thealue of the local field taken outside the QW just at itsoundaries. I will show that the amplitudes of the localeld at opposite boundaries of the QW can be related withach other by means of a matrix, which will allow us toalculate the optical response of the whole MQW struc-ure by multiplying the corresponding matrices.

. LINEAR RESPONSE OF A SINGLEUANTUM WELL

o calculate the local field inside the nth QW of the struc-ure, we place the origin of the Cartesian frame at theenter of the nth well; therefore Dqw

�n�= �zl ,zr�, where zl=0.5d−� and zr=0.5d+� are coordinates of the left andight boundaries of the region Dqw

�n�, respectively.Outside regions Dqw

�m��m=1, . . . ,N� the electromagneticesponse of the medium is local and the electric field�� ,z� can be represented as a superposition of the

orward- and backward-propagating fields20:

E��,z� = ��=s,p

�e+��E�

��z�exp�iqzz� + e−��E�

��z�exp�− iqzz��,

�9�

here the superscript �=s ,p denotes the polarizationtate of the field, e±

�s�= �0,1,0� , e±�p�= �qx

2+qz2�−1/2

�qz ,0 , qx�, and qz= ��h�2 /c2−qx2�1/2. The functions E�

��

�z� and E��z� remain constant values in a homogeneous

egion with the local response.This notation,

E�,��m,�� = E�

��zm,��exp�iqzzm,��, �10�

ill be used for the forward-propagating field, and

E�,��m,�� = E�

��zm,��exp�− iqzzm,�� 11�

ill be used for the backward-propagating field at theoint zm,�, where �= l ,r and zm,l, and zm,r are coordinatesf the left (index l) and right (index r) boundaries of theegion Dqw

�m�, respectively.To simplify the structure of the integral equation [Eq.

2)] we represent G��z ,z�� in the following form:

G��z,z�� = G��z,z�� + g��z,z��, �12�

here G��z ,z�� is the Green’s function of a three-layeredtructure with the dielectric function

�qw�z� = �h, �z� � d/2

�q, �z� � d/2�13�

hat describes the local component of the response of anndividual QW [see Fig. 1(b)]. The Green’s function��z ,z�� satisfies the inhomogeneous wave equation in theegion D�n�:
qw
Lz · G��z,z�� = U�z − z��, �14�

here z ,z��Dqw�n�.

Another term in Eq. (12), g��z ,z��, satisfies the homo-eneous wave equation

Lz · g��z,z�� = 0, �15�

here z ,z��Dqw�n�.

With the expansion (12), the integral equation (2) forhe local field inside the region Dqw

�n� can be rewritten inhe following form:

E��z� = E�ext�z� − i�G� · � · E��z�, �16�

ith

G� · f�z� =�zl

zr

G��z,z�� · f�z��dz�. �17�

he external field

Eext�z� = − i� �m�n

G��nm� · �� · E��zm� − i�g� · � · E��z�

�18�

atisfies the homogeneous wave equation

Lz · E�ext�z� = 0 �19�

nside the region Dqw�n� owing to Eq. (15). In Eq. (18) the

ontraction of the tensor g� with the vector function f�z�� ·E��z� is defined by Eq. (17) where G is replaced by g.he external field E�

ext�z� can be considered as a field in-uced by effective current sheets placed outside the re-ion Dqw

�n� just at its boundaries. The current sheet placedt the left (right) boundary represents contributions fromurrents generated in semi-infinite regions at the leftright) of the region Dqw

�n�. The left region includes theacuum and QWs with numbers m=1, . . . , �n−1�. Theight region includes the QWs with numbers m= �n1� , . . . ,N and the substrate.It follows from Eq. (19) that E�

ext�z� can be expressed inerms of the amplitudes of forward- and backward-ropagating fields at the boundaries of the region Dqw

�n�:

E�ext�z� = T�

ext�z� · E��n�, �20�

here

E��n� = �

E�,s�n,l�

E�,s�n,r�

E�,p�n,l�

E�,p�n,r�� . �21�

t is worth noting that E�ext�z� is a three-component vector

escribing the Cartesian components of the external field,nd E�

�n� is a four-component vector describing the ampli-udes of forward- and backward-propagating fields at theoundaries of the layer; therefore the propagator Text�z� is3�4 matrix. The explicit expression for Text�z� is pre-

ented in Appendix A.

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Instead of the amplitudes of the fields E�,��n,l� and E�,�

�n,r�

�=s ,p� incident on the QW, any other two pairs of am-litudes can be used, for example, E�,�

�n,l� and E�,��n,l�, or E�,�

�n,l�

nd E�,��n,r� and so on, but it is convenient to use E�,�

�n,l� and¯

�,��n,r� because the last term in Eq. (16) does not contribute

hem.The solution of Eq. (16) can be expressed through theatrix Q��z ,z��, which satisfies the integral equation

Q��z,z�� + i�T��z,z�� = U�z − z��, �22�

here

T��z,z�� =�zl

zr�zl

zr

G��z,z�� · ��z�,z�� · Q��z�,z��dz�dz�.

�23�

he integral equation [Eq. (22)] can be solved analyticallyn the case of a factorable linear conductivity tensor of theW.Once the matrix Q��z ,z�� is found, the solution of Eq.

16) can be straightforwardly obtained:

E��z� = P��z� · E��n�, �24�

here

P��z� =�zl

zr

Q��z,z�� · T�ext�z��dz�. �25�

It follows from Eq. (24) that the amplitudes of forward-nd backward-propagating fields at the opposite bound-ries of the region Dqw

�n� can be related with each other byeans of a generalized propagation matrix:

E��n,r� = M�

qw · E��n,l�, �26�

here

E��n,�� = �

E�,s�n,��

E�,s�n,��

E�,p�n,��

E�,p�n,��

� �27�

nd �= l ,r. If the components xy, yx, zy, and yz of the lin-ar conductivity tensor ��z ,z�� of the QW are equal toero, then M�

qw is a 4�4 block-diagonal matrix:

M�qw = �M�

qw 0

0 M�,pqw� , �28�

here the matrices M�,sqw and M�,p

qw are defined by

M�,�qw =

1

�4��1�4 + �2�3 �2

�3 1 � , �29�

here �1= �P��zr��21, �2= �P��zr��22−1, �3=1− �P�zl��21, �4

�P��zl��22 for �=s, and �1= �P��zr��13/cos �h, �2=−1�P��zr��14/cos �h, �3=1− �P�zl��13/cos �h, �4

�P��zl��14/cos �h for �=p. Here �h is the angle betweenhe wave vector of the forward-propagating field in theost medium and the z axis, cos � = �1−sin2� /� �1/2. We
h h
se the notation �M�ij for an ij component of a matrix M.n general case of the anisotropic electromagnetic re-ponse of the QW, Eq. (26) remains valid, but the compo-ents of the matrix M�

qw are defined by more complex ex-ressions.The matrix M�

qw is a generalization of the standard op-ical transfer matrix for the case of regions with nonlocalesponse. It is determined from solving the integral equa-ion (16) rather than the system of coupled integral equa-ions (2).

In order to calculate the distribution of electric field in-ide the MQW structure, it is convenient to express thelectric field inside the QW through the amplitudes oforward- and backward-propagating fields at the leftoundary of the QW:

E��z� = P��z� · E��n,l�, �30�

here the components of the propagator P��z� are giveny

�P��z��ij = �P��z��ij + �P��z��i2�M�qw�2j + �P��z��i4�M�

qw�4j

�31�

or i=1,2,3 and j=1,3, and

�P��z��ij = �P��z��i2�M�qw�2j + �P��z��i4�M�

qw�4j �32�

or i=1,2,3 and j=2,4.

. LINEAR RESPONSE OF MULTIPLEUANTUM WELL STRUCTUREquation (26) allows the calculation of the linear opticalesponse of the whole MQW structure within the transfer-atrix technique.20,21 Consider a structure consisting ofQWs with values of thicknesses of the layers equal to

hose shown in Fig. 1(a). In what follows throughout thisection, we suppose that the incident field of (given) am-litude E�,s

�v� is s polarized and that the xy, yx, yz, and zyomponents of the linear conductivity tensor are equal toero.

To find the linear response of the MQW structure, weeed to calculate the propagator

T�,smqw = M�,s

�s,h� · T��e� · �M�,s

qw · T�,�b��N−1 · M�,s

qw · T��e� · M�,s

�h,v�,

�33�

here M�,sqw is the linear transfer matrix of the s-polarized

eld through the QW defined by Eq. (29) with �=s. Theatrix M�,s

�i,j� with i=s,h and j=h,v (the index v stands forhe vacuum, h for the host material, and s for the sub-trate) is the transfer matrix that describes the propaga-ion of the s-polarized field at frequency � through the in-erface between media i and j. The transfer matrices T�

�e�

nd T��b� are defined by the following expressions:

T��e� = T�

�h��D − ��, �34�

T��b� = T�

�h��D − 2��, �35�

here T��h��z� is the propagation matrix that relates the

mplitudes of forward- and backward-propagating fields

aep

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wd

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olo

a

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t frequency � in the host layer at points z and z+z. Thexplicit expressions for the matrices M�,s

�i,j� and T��h��z� are

resented in Appendix A [see Eqs. (A2) and (A4)].The linear reflection coefficient r�,s is given by the well-

nown expression20

r�,s = −�T�,s

mqw�21

�T�,smqw�22

. �36�

he amplitudes of forward- and backward-propagatingelds at the left boundary of the jth QW are given by

E�,s�j,l�

E�,s�j,l� = L�,s

�j� · 1

r�,sE�,s

�v� , �37�

here

L�,s�j� = �T�

�b� · M�,s�qw��j−1 · T�

�e� · M�,s�h,v�. �38�

aving calculated these amplitudes, we can determinehe local field inside the jth QW with the use of Eq. (30).

. NONLINEAR RESPONSE OF A SINGLEUANTUM WELL

he electromagnetic field at the double frequency inducedy the nonlinear currents [Eq. (8)] in the region Dqw

�n� sat-sfies the integral equation (2) with � substituted with�, � with 2�, and background field given by Eq. (7). Mak-ng use of Eq. (30) and the Green’s function expansionEq. (12)] we derive the modified integral equation

E2��z� = T2�nl �z�:E�

�n,l�E��n,l� + T2�

ext�z� · E2��n�

− 2i�G2� · �2� · E2��z�, �39�

here T�ext�z� and E2�

n are defined by Eqs. (20) and (21) athe double frequency, respectively, and the tensor T2�

nl �z� isiven by

T2�nl �z� = − 2i�G2� · �2��z�:P�P�, �40�

ith

�2��z�:P�P� =�zl

zr�zl

zr

�2��z,z�,z��:P��z��P��z��dz�dz�.

�41�

solution of Eq. (39) can be expressed through the matrix2q��z ,z��, defined by Eq. (22) at the double frequency

E2��z� = P2��nl��z�:E�

�n,l�E��n,l� + P2��z� · E2�

�n�, �42�

here the tensor P2��z� is defined by Eq. (25) at theouble frequency, and P2�

nl �z� is given by

P2�nl �z� =�

zl

zr

Q2��z,z�� · T2�nl �z��dz�. �43�

inally, we derive the relation between the field ampli-udes at the boundaries

E2��n,r� = M2�

qw · E2��n,l� + S2�

qw:E��n,l�E�

�n,l�, �44�

here M2�qw is the generalized transfer matrix defined by

q. (28) at the double frequency and the components ofhe 4�4�4 tensor S2�

qw are given by

�S2�qw�1jk = �P2�

nl �zr��2jk −�1 − 1

�2�P2�

nl �zl��2jk, �45�

�S2�qw�2jk = −

�P2�nl �zr��2jk

�2, �46�

�S2�qw�3jk =


cos �h−

�3 − 1

�4�P2�

nl �zl��1jk, �47�

�S2�qw�4jk =


�4, �48�

here �1= �P2��zr��22, �2= �P2��zl��22, �3= �P2��zr�14

/cos �h, and �4= �P2��zl��14. Equations (45)–(48) are validf the components xy, yx, zy, and yz of the linear conduc-ivity tensor �2��z ,z�� are equal to zero. In the generalase of the anisotropic electromagnetic response of theW, Eq. (44) remains valid, but the components of S2�

qw areefined by more complex expressions.

. NONLINEAR RESPONSE OF THEULTIPLE QUANTUM WELL STRUCTUREaking use of the quantity

S�n� = S2�qw:E�

�n�E��n�, �49�

e can calculate the nonlinear response of the MQWtructure in a way analogous to the optical transfer-atrix technique [compare S�n� with the quantity �jMkj ·�j ·Sj, defined by Eq. 11(a) in Ref. 20].Consider, for example, the s-polarized field generated

y the MQW structure at frequency 2�. The total nonlin-ar response of the structure is a superposition of contri-utions from all layers. To calculate the contribution fromhe jth layer, let us suppose that the nonlinear currentsre equal to zero in all layers except the jth one.Solution of the linear problem provides the amplitudes

f the electric field at the fundamental frequency at theeft boundary of the jth layer, which allows the calculationf the vector

S�j� = �S�n��1

�S�n��2 �50�

nd the following propagation matrices:

L2�,s�j� = M2�,s

qw · �T2��b� · M2�,s

qw �j−1 · T2��e� · M2�,s

�h,v�, �51�

R2�,s�j� = �M2�,s

s,h · T2��e� · �M2�,s

qw · T2��b��N−j�−1. �52�

he amplitude E2�,s�s,j� of the forward-propagating field in

he substrate and the amplitude E2�,s�v,j� of the backward-

ropagating field in vacuum are given by

w

o

lta

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w

wlTtTcdQ

r

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pstsftt

aQ

aS

w

at


E2�,s�s,j�

E2�,s�v,j� = K2�,s�L2�,s

�j� �22 − �L2�,s�j� �12

�R2�,s�j� �21 − �R2�,s

�j� �11 · S�j�, �53�

here

K2�,s =1

�R2�,s�j� �11�L2�,s

�j� �22 − �R2�,s�j� �21�L2�,s

�j� �12

. �54�

The amplitude of the reflected field is given by the sumf contributions from all layers,

E2�,s�v� = �

j=1

N

E2�,s�v,j� . �55�

It must be noted that the described procedure of calcu-ation of the nonlinear response neglects the depletion ofhe energy of the field at the fundamental frequency. Fortypical MQW structure this assumption is valid.

. COMPARISON WITHXISTING TECHNIQUEShe integral equation for the local field [Eq. (16)] can beolved analytically in the practically important case whenhe linear conductivity tensor of the QW can beactorized.14,18,23 Consider a QW with only two size-uantized energy levels �1 and �2. The value of �1 is lo-ated below the Fermi level, �F, while �2 lies above �F.lectrons of the QW exhibit freelike motion (with effec-

ive mass m*) in directions parallel to the boundaries ofhe QW. Within the framework of the random-phase ap-roximation, and in the long-wavelength �qx→0� and low-emperature �T→0� limits, the nonlocal linear conductiv-ty tensor ��z ,z�� of the QW takes a diagonal form withhe following elements23:

��z,z��xx = ��z,z��yy = ��z��z��, �56�

��z,z��zz = ��z��z��, �57�

ith

�� =

ie2

�2�

��2 − �1��F − �1�2

�� + i/��2 − ��2 − �1�2 , �58�

�� =

ie2

2m*�

��2 − �1��F − �1�

�� + i/��2 − ��2 − �1�2 , �59�

��z� = �1�z��2�z�, �60�

��z� = �1�z��2�z�

�z− �2�z�

��1�z�

�z, �61�

here �j�z� is the wave function of the electron at the jthevel �j=1,2� and � is the relaxation time of the electrons.he QW also possesses the nonresonant local responsehat can be characterized by the dielectric constant �q.his contribution is due to bound electrons of the materialonstituting the QW. This system provides a satisfactoryescription of typical QWs, for example GaAs/AlxGa1−xAsW, when the optical response of the QW in the near-

esonance frequency range is considered.By inserting Eqs. (56) and (57) into integral equation

22), we can transform Eq. (22) into the system of the al-ebraic equations, which allows us to derive analytical ex-ressions for the components of tensor Q��z ,z��. Finally,e can derive analytical expressions for the componentsf the generalized transfer matrix M�

qw and tensor S2�qw.

or brevity, I will not give here explicit expressions for theomponents of M�

qw and S2�qw (the details of solving the in-

egral analogous to Eq. (22) can be found in Ref. 23).In the case of the nonfactorable linear conductivity ten-

or, the integral equation (22) can be solved either com-letely numerically or with the use of the self-field andlave-model approximations that are considered in Ref.4.It is worth numerically comparing the results provided

y the developed generalized transfer-matrix techniqueGTMT) with that given by the current-sheetormalism13,14 (CSF) and the standard transfer-matrixechnique20,21 (STMT).

First, consider the relation between the GTMT andSF. Within the framework of CSF the QW is considereds a homogeneous layer of thickness d with a currentheet with surface current

Jsh�� =�−d/2

d/2

��z,z�� · E��z��dzdz� = Ssh��E��− d/2�,

�62�

laced at the center of the layer. Matrix Ssh is called theheet-conductivity tensor. In applying CSF to MQW struc-ures it is assumed12 the terms of the local field E��z� in-ide QWs of the second order in qxd are neglected. There-ore, to calculate the components of sheet-conductivityensor Ssh we can use the self-field approximation14 forhe Green’s function,

G��z,z�� =c2

�2�q�z − z��ezez, �63�

nd neglect a variation of the external field inside theW,

E�ext�z� = E��− d/2�. �64�

Performing the calculations described in Section 3, werrive at the following expressions for the components ofsh:

Sxxsh�� = Syy

sh�� = ��

−d/2

d/2

��z�dz�2

, �65�

Szzsh�� =

��

−d/2

d/2

��z�dz�2

1 + i�4/�q��

−d/2

d/2

�2�z�dz

, �66�

ith all other components Sij equal to zero.The physical meaning of the parameters S�

sh=Sxxsh=Syy

sh

nd S�sh=Szz

sh is the same as that of the d parameters in-roduced by Feibelman,11 namely, parameters Ssh and Ssh

� �

rwp

cs

w

eed

m=itQmet=t

rptqMasrwsQirtsT

aa

sNso

wt

Tico

cictwnitcfipue

FctSrpl


elate the integral current inside a thin nonlocal layerith the component of the electric field parallel and per-endicular to the layer boundaries, respectively.The components of the generalized transfer matrices

an be readily expressed through the components of theheet-conductivity tensor:

M�,sqw = U + iq�dU +

2�Syysh

q�c2 �− 1 − 1

1 1 � , �67�

M�,pqw = U + iq�dU +

2q�Sxxsh

��− 1 − 1

1 1 �+

2qx2Szz

sh

�q��− 1 1

− 1 1� , �68�

here q�=��q�2 /c2−qx2, U is a 2�2 units matrix and

U = �1 0

0 − 1� · �69�

Second, to apply STMT to the MQW structure we treatach QW as a homogeneous layer of thickness d with locallectromagnetic response characterized by the effectiveielectric tensor:

�xx�� = �yy�� = �q + i4

�d��

��−d/2

d/2

��z��dz��2

, �70�

�zz�� = �q + i4

�d��

��−d/2

d/2

��z��dz��2

· �71�

Consider the p-polarized linear optical response of aodel MQW structure: GaAs/ALxGa1−xAs MQW with �q1. We use the infinite-barrier wave functions of electrons

nside each QW in the structure. The structure is charac-erized by the following parameters: effective thickness ofW d=12 nm, period of the structure �=20 nm, effectiveass of the electron m*=0.0665me, relaxation time of the

lectron system �=0.2 ps. The energy separation betweenhe lowest levels in conduction band is ��=�2−�1118 meV. The Fermi energy �f=�1+54 meV corresponds

o the surface concentration of electrons 1.5�1012 cm−2.The results of calculations of the intensity linear-

eflection spectrum in the near-resonance range for-polarized light incident at angle �= /4 on the struc-ure with N=50 QWs are shown in Fig. 2. CSF provides auite precise description of the linear response of theQW structure: The deviation from the exact spectrum

t resonance is about 2%. In contrast, STMT results in es-ential error: The position of the resonance of the linearesponse is shifted toward the red region of the spectrumith respect to the exact position ��0− �0� ·��4.8�. The

hift exists for structure with an arbitrary number ofWs, as shown in the inset in Fig. 2. The electromagnetic

nteraction between the QWs of the structure shifts theesonance position toward the blue region of the spec-rum, but the essential discrepancy between the peak po-itions of the reflectance spectra exists in all range of N.he peak position calculated with use of the current-sheet

pproximation coincides with the exact result with highccuracy (the relative deviation is less than 0.1%).The difference in values of �0 and �0 stems from con-

idering a QW as a thin layer with a local response.amely, at small N, in the case of QW with a nonlocal re-

ponse, the peak position is determined by the singularityf the component:

1 + i4

�0��0

��−d/2

d/2

�2�z�dz = 0, �72�

hereas for the layers with local response it correspondso zero of the effective dielectric constant �zz��:

1 + i4

�0d��0

��−d/2

d/2

��z�dz�2

= 0. �73�

hus, to calculate the linear response of the QW correctlyt is necessary to take into account the variation of the lo-al field inside the QW, which can be accomplished onlyn the basis of solution of the integral equation (16).

Although the CSF is an effective technique for the cal-ulation of the linear optical response of MQW structures,t is unusable when we deal with nonlinear response. Toalculate the nonlinear currents it is necessary to knowhe distribution of the local field inside the QW; therefore,e cannot consider the QW as an object without an inter-al structure. Moreover, in the case of symmetrical QWs

t is necessary to take into account high-order terms (inhe qxd expansion) of the local field inside a QW. In fact, itan be readily demonstrated that if the uniform externaleld inside the QW [see Eq. (64)] and the self-field ap-roximation for the Green’s function [see Eq. (63)] aresed, then the integral quadratic current inside the QW isqual to zero,

�−d/2

d/2

Jnl�z�dz =�−d/2

d/2 �−d/2

d/2 �−d/2

d/2

�2��z,z�,z��:E��z��E��z��

�dzdz dz = 0, �74�

ig. 2. Intensity reflectance spectra of the p-polarized field Rpalculated within the frameworks of a few techniques: developedechnique (exact solution, solid curve), CSF (dotted curve) andTMT (dashed curve). Dependencies of the peak position of theeflectance spectra �0 on the number of QW in the structure arelotted in the inset: the solid curve corresponds to the exact so-ution, and the dashed curve is calculated with a STMT.

� �

dncnnsf

8Toispait(tttb=

stmtotttooaratctsbf(mpn

ht�

o

wa

h

Moictw

ATTmeqmbt

wt�

=

wmd

Tbz

wa

wmiIat1

Tct[


ue to the symmetry properties of the nonzero compo-ents of the zero-order (in qxd) terms of �2��z ,z� ,z��. Fororrect calculation of nonlinear response of the QW it isecessary to take into account the variation of the exter-al field E�

ext�z� inside the QW and to use the full expres-ion for Green’s function G��z ,z�� instead of its reducedorm [Eq. (63)].

. DISCUSSION AND CONCLUSIONShe analogy between the STMT and the method devel-ped in the present paper consists in expressing the fieldnside a region possessing either local or nonlocal re-ponse through the amplitudes of forward- and backward-ropagating fields at a boundary of the region by means ofpropagation matrix [see Eq. (30)]. The difference is that

n the standard technique the propagator is expressed inerms of matrices M�,�

�i,j� and T��j�, defined by Eqs. (A2) and

A4), respectively, whereas in the generalized techniquehe propagator is determined from the solution of the in-egral equation (16). The nonlinear nonlocal response ofhe QW is determined by the vector S�j�, which is definedy Eq. (49) and is an analog of the quantity �jMkj ·�j ·Sj in the STMT [see Eq. (11a) in Ref. 20].The linear optical response of thin layers with an es-

entially nonlocal linear conductivity tensor can be effec-ively described within the CSF.13,14 The nonlocal electro-agnetic response of a thin layer can be described in

erms of a sheet-conductivity tensor that relates the valuef the integral current inside the layer with the values ofhe (Cartesian) components of the electric field outsidehe layer, just at its boundary. The CSF and the developedechnique are almost equivalent in the case of calculationf the linear optical response of MQW structures: The usef both quantities—sheet-conductivity tensor and gener-lized transfer matrix—provides the jump conditions thatelate the amplitude of the local field at opposite bound-ries of the region. Moreover, the CSF provides a satisfac-ory accuracy of calculations of the linear response of typi-al MQW structures as is demonstrated in Section 7. Onhe other hand, calculation of the nonlinear nonlocal re-ponse of each QW requires information about the distri-ution of the local field inside the QW at the fundamentalrequency. In this case the propagation matrix [see Eq.30)] has to be used instead of the generalized transferatrix (or, equivalently, sheet-conductivity tensor) that

rovides information only about the integral electromag-etic response of the QW.In Section 5, the process of second-harmonic generation

as been considered but the approach can be readily ex-ended to description of any three-wave mixing process�3=�1+�2�:

E�3

�n,r� = M�3

qw · E�3

�n,l� + S�3,�1,�2

qw :E�1

�n,l�E�2

�n,l�, �75�

r mth-harmonic generation:

Em��n,r� = Mm�

qw · Em��n,l� + Sm�

qw :�E��n,l��m, �76�

here tensors S�3,�1,�2

qw and Sm�qw are calculated in a way

nalogous to that described in Section 5.To summarize, the optical transfer-matrix technique

as been generalized to calculation of light propagation in

QW structures, in which both the linear- and nonlinear-ptical response of each QW possesses strong nonlocalityn the direction normal to the well boundaries. In prin-iple, the developed technique can be applied to an arbi-rary multilayer object that consists of alternate layersith nonlocal and local electromagnetic response.

PPENDIX A: STANDARDRANSFER-MATRIX TECHNIQUEhe STMT provides an effective description of the electro-agnetic field inside multilayer structures with local

lectromagnetic response of each layer. One of the basicuantities in the transfer-matrix technique is the transferatrix M�,�

�i,j� that relates the amplitudes of forward- andackward-propagating fields adjacent to the interface be-ween media i and j,

E�,��i�

E�,��i� = M�,�

�i,j� · E�,��i�

E�,��j� , �A1�

here the notations E�,�� or E�,�

�� means that the ampli-ude of, respectively, forward- or backward-propagating-polarized field �a=s ,p� at frequency � in medium � ��i , j� is considered. The components of M�,�

�i,j� are given by

M�,��i,j� =

1

t�,��i,j�� 1 r�,�

�i,j�

r�,��i,j� 1 � , �A2�

here r�,��i,j� and t�,�

�i,j� are, respectively, reflection and trans-ission coefficients for the �-polarized field ��=s ,p� inci-

ent upon the i–j interface from medium i.Another important quantity is the propagation matrix

��j��z�, which relates the amplitudes of forward- andackward-propagating fields in medium j at points z and+z:

E�,��j� �z + z�

E�,��j� �z + z�

= T��j��z� · E�,�

�j� �z�

E�,��j� �z�

, �A3�

here �=s ,p and an argument of E�,��j� �z� is a coordinate of

point at which the field is evaluated.The components of T�

�j��z� are given by

T��j��z� = �exp�iq�

�j�z� 0

0 exp�− iq��j�z�� , �A4�

here q��j�= ��j�

2 /c2−qx2�1/2, �j is a dielectric constant of the

edium j and qx is the component of the wave vector ofncident radiation parallel to the boundary of the layer j.n the case of MQW structures qx=sin �� /c�, where � isn angle of incidence of the fundamental radiation uponhe MQW structure „counted from the z direction [see Fig.(b)].Consider the explicit expression for the propagator

�ext�z� [see Eq. (20)] of the external field in a point with aoordinate z (the origin of the Cartesian frame is placed athe center of the QW) for a QW with a typical structuresee Fig. 1(b)].

Let us introduce the following matrices:

w

w

Iasl

bEo

w

wb

z

w=

AINtSip

R

1

1

1

1

1

1


L��z� = �

�T��h��z + 0.5d + ��−1, z � − 0.5d

�T��q��z + 0.5d� · M�,�

�1,1��−1, �z� � 0.5d

�T��h��z − 0.5d� · M�,�

�1,2��−1, z � 0.5d� ,

�A5�

here

M�,��1,1� = M�,�

�q,h� · T�,��h� ��, �A6�

M�,��1,2� = M�,�

�h,q� · T��q��d� · M�,�

�q,h� · T��h��, �A7�

R��z� = �

M�,��2,1� · T�

�h��− 0.5d − z�, z � − 0.5d

M�,��2,2� · T�

�h��0.5d − z�, �z� � 0.5d

T��h��0.5d + � − z�, z � 0.5d

� ,

�A8�

here

M�,��2,1� = T�

�h�� · M�,��h,q� · T�

�h��d� · M�,��q,h�, �A9�

M�,��2,2� = T�

�h�� · M�,��h,q�· �A10�

n Eqs. (A5)–(A10) �=s ,p, and matrices T��j��z� and M�,�

�i,j�

re defined by Eqs. (A2) and (A4), respectively. The super-cript h stands for the host material and q for the QWayer material.

Now we can relate the amplitudes of forward- andackward-propagating fields at the point z�E�

��z� and¯

��z�, respectively] with the amplitudes of fields incidentn the well �E�,�

�n,l� and E�,��n,r� at Fig. 1(b)]:

E��z� = �E�,�

�n,l�R22��,��z� − E�,�

�n,r�L12��,��z��D�

��z�, �A11�

E��z� = �E�,�

�n,r�L11��,��z� − E�,�

�n,l�R21��,��z��D�

��z�, �A12�

ith

D��z� =

1

L11��,��z�R22

��,��z� − L12��,��z�R21

��,��z�, �A13�

here we use the notation Mij��,��= �M�

��ij �M=L ,R� forrevity.Finally, we derive, the following expressions for non-

ero components of the propagator:

�T�ext�z��13 = cos �h�R22

��,p��z� − R21��,p��z��D�

�p��z�, �A14�

�T�ext�z��14 = cos �h�L11

��,p��z� − L12��,p��z��D�

�p��z�, �A15�

�T�ext�z��33 = − sin �h�R21

��,p��z� − R22��,p��z��D�

�p��z�,

�A16�

�T�ext�z��34 = sin �h�L11

��,p��z� + L12��,p��z��D�

�p��z�, �A17�

�T�ext�z��21 = �R22

��,s��z� − R21��,s��z��D�

�s��z�, �A18�

�T�ext�z��22 = �L11

��,s��z� − L12��,s��z��D�

�s��z�, �A19�

here cos �j= �1−sin2 � /�j�1/2, sin �j= ��j�−1/2 sin � and jh,q. All other components of T�

ext�z� are equal to zero.

CKNOWLEDGMENTSthank the staff of the laboratory of Non-linear Optics ofanostructures and Photonic Crystals (Quantum Elec-

ronics Division of Department of Physics of Moscowtate University) for fruitful discussions and assistance

n accomplishing the work. This work was partly sup-orted by RFBR grant 04-02-16847.The author’s e-mail address is [email protected].

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