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1. Basic Concepts of Optoelectronic Devices

1.1 Maxwell's Equations in Linear Media

Optoelectronic devices are structures in which the electromagnetic field described

by Maxwell's equations and the quantum mechanicallaws for the wavefunction of

the charge carriers cooperate in order to generate, propagate or detect optical

fields.

The laws of quantum mechanics appear explicitly in the description of

optoelectronic devices based on mesoscopic structures and therefore will be

presented briefly at the beginning of Chap. 2 in connection with the definition and

characteristics of structures with mesoscopic dimensions. However, Maxwell's

equations and their solution in dielectric waveguides form the basis of the

description of any optoelectronic devices including quantum ones. Therefore, the

present chapter is entirely dedicated to Maxwell's equations and their solutions for

coupled, periodic, linear and nonlinear dielectric media. We will focus first on

linear media, i.e. media in which the dielectric constants are independent of the

propagating electromagnetic fields, while nonlinear media will be treated in Sect.

1.5.

1.1.1 Maxwell's Equations in Linear and Inhomogeneous Media

Maxwell's equations are:

V xE= -oB I 0(,

V xH = oDI ot+J,

V·B=O,

V·D=p,

(1.1)

where E and H are the electric and magnetic fields, D and Bare the electric

displacement and magnetic induction field, respectively, p is the density of

charge and J is the current density vector. The four field vectors which appear in

Maxwell's equations are related through two constitutive equations that describe

D. Dragoman et al., Advanced Optoelectronic Devices© Springer-Verlag Berlin Heidelberg 1999

2 1. Basie Coneepts of Optoeleetronie Deviees

the response of the medium to the eleetromagnetie field exeitation. These are

B = jJll and D = eE with jJ and e the magnetie permeability and eleetrie

permittivity tensors, respeetively. Sinee throughout the book we will be mainly

eoneemed with eleetromagnetie propagation problems we ean safely ass urne

throughout that p = 0 and J = O. Moreover, for isotropie media, whieh represent

the most eommon situations in optoeleetronie deviees, the e and jJ tensors

beeome sealar quantities. The time domain derivative 8/ 8t appears explieitly

only in problems of optieal pulse propagation. However, any pulse ean be

deeomposed into aseries of harmonie waves, eaeh eharaeterized by a fixed

frequeney w. In this ease all field veetors have an exponential time dependenee

given by the faetor exp(iwt), so that the operator 8/ 8t reduees to a simple

multiplieation with iw. Maxwell's equations for harmonie wave fields in an

isotropie medium then beeome

v xE = -iwf..iH,

V xH = iw&E,

V(f..iH) = 0,

V(&E) = O.

(1.2)

The system of equations (1.2) whieh relates now only the E and H fields ean be

transformed by using simple operator transformations in two independent

equations for E and H whieh deseribe the propagation of the eleetrie and magnetie

field veetors in inhomogeneous and isotropie media:

V2 E + (V f..i I f..i) x V x E + V( EV & 1&) + w2&f..iE = 0,

V2H +(V&I &) xV x H + V(HVf..i1 f..i) + w2&f..iH = O. (1.3)

In a homogenous medium for whieh e(r) = f..i(r) = const. the above equations

reduee to the equations of a harmonie oseillator (Heimholtz equation):

V2F + w2&f..iF = 0, F = E, H. These resemble the equation deseribing the

propagation of the eleetromagnetie field in the vaeuum: V2 F + k1; F = 0 with the

only differenee that the waveveetor ko = sw I c should be replaeed by nko where

the refraetive index n is defmed as n2 = &f..i I &of..io = C2&f..i. S denotes the unit

veetor along the direction of propagation and &0 = 107 I 4Jrc2 Firn, f..io =

4JrlO-7 Hirn are the electric permittivity and magnetic permeability in vacuum,

respectively, with c = 2.998· 108 m/s the velocity of light.

The system of equations (1.3) is generally difficult to solve. However, in most

dielectric waveguides the parameters & and f..i (and implieitly n) have eonstant

values along a direction defined by the unit vector Z which may or may not

1.1 MaxwelI's Equations in Linear Media 3

coincide with the direction of light propagation. When I; and J1. are constant along

the direction of light propagation, the solution ofthe system of equations (1.3) is

found by decomposing the total electric and magnetic fields into modes; otherwise

one can use the matrix method to find the solution of (1.3). These methods will be

briefly discussed in Sects. 1.3 and 1.2, respectively. In both cases, due to the

translational invariance of waveguide properties along the Z direction, the electric

and magnetic fields can be separated as

E(r) = e(Ii )exp(ißz),

H(r) = h(Ii)exp(ißz), (1.4)

where ß is the propagation constant along the Z direction and Ii is the coordinate

vector transverse to the Z axis. Although in a waveguide the complete solutions of

Maxwell's equations are E(r,t) = E(r)exp(iwt) and H{r,t}=H{r}exp{iwt}, with

the spatially varying parts given by (1.4), only their real parts, Re( E( r, t)) and

Re(H(r,t)), have physical meaning. With the exception of the cases where

temporal averages of products of harmonie functions are involved, the form (1.4)

is used to describe the spatially varying parts of the fields. These temporal

averages can however be expressed directly in terms of the spatially varying

components ofthe fields by

F(r, t)G(r, t) = Re(F(r) G*(r))12. (1.5)

In orthogonal coordinates, employed in studying planar or channel waveguides,

r = (x,y,z), Ii = (x,y) while in polar cylindrical coordinates, used for optical

fibers, r = (r, q), z), Ii = (r, q)). The constant of propagation ß can be positive or negative. In this book we

adopt the convention that a positive ß describes an electromagnetic wave

propagating along the positive z direction while a negative value of ß indicates

that the electromagnetic field propagates in the opposite, negative z direction.

Moreover, e and h which are in general complex quantities, can be

decomposed into transverse and longitudinal parts with respect to z:

e(Ii) = et(Ii) + zez(Ii), h(Ii) = ~(Ii) + zhAIi)·

(1.6)

According to the symmetry properties of the electric and magnetic fields at the

inversion ofthe propagation direction, these fields transforms as

4 I. Basic Concepts of OptoeJectronic Devices

e('i) = et('i) - zez('i), h('i) = -Ilt('i) + zhz(Ii),

when z is replaced with -z.

1.1.2 Relations Between the Transverse

and Longitudinal Components of the Electromagnetic Field

in a Translationa"y Invariant Medium

(1.7)

It can be shown that the trans verse and longitudinal components of the

electromagnetic field are not independent, but can be related through a set of

formulae derived by introducing (1.6) into Maxwell's equations (1.2). The result

is:

Ilt = -z x (ßet - iVtez) / mJi,

et = z x (ßIlt - iV thz) / mB,

hz = iZ . V t X et / mJi = - i( V tllt + Ilt . V t In Ji) / ß, ez = iz· Vt x Ilt / mB = -i(Vtet + et . Vt InB) / ß,

(1.8)

where V t is the transverse part of the gradient operator V = V t + ZO / &. Equations (1.8) hold for both orthogonal and polar transverse coordinates. In

nonabsorbing waveguides (with no propagating loss) Band Ji, as weIl as n, have

real values, while for absorbing waveguides they are complex quantities. For

nonabsorbing waveguides (1.7) is consistent with a particular choice of the e and

h components, such that the transverse components are real and the longitudinal

ones imaginary or vice versa. Such a choice, as for example et and Ilt real and

ez and hz imaginary, gives more insight into the physical interpretation.

By eliminating Ilt and et from the first equations in (1.8), the following

relations can be obtained between the transverse and longitudinal components of

the electromagnetic field in a transiationally invariant medium:

et = -i(ßV tez + mJ1i x V thz) / (m 2BJi - ß2),

Ilt = -i(ßVthz - ma x Vthz) /(m2BJi- ß2). (1.9)

The relations derived between the transverse and longitudinal field components

are important because they point out that it is not necessary to solve for the

components ofboth e and h in order to obtain the complete solution of MaxweIl's

1.1 MaxweU's Equations in Linear Media 5

equations. The computation of only one field is sufficient for the calculation of the

remaining three field vectors (ifuse is made also ofthe constitutive equations).

The last two equations in (1.8) have been derived directly from Maxwell's

equations (1.2). They could have been derived also from the system of equations

(1.3) since it is equivalent to (1.2). However, (1.3) gives us additional information

regarding the coupling between the transverse and longitudinal field vectors.

Namely by introducing (1.6) into (1.3) it follows that:

(v; + m2&J.l- ß2)h = -(Vt~ + ißhz )' Vt InJ.l-(V t x h+ ißZ x~) x Vt ln&,

(V; + m2&J.l- ß2)e = -(Vtet + ißez)' Vt ln&-(V t xe + ißZ x et) x Vt InJ.l.

(1.10)

The above result expresses the fact that the terms containing V t ln & and V t In J.l

couple the different field components. In their absence all field components would

independently be solutions of the harmonie oscillator type equations in the left

hand side of (1.10). The terms containing Vt ln& and Vt InJ.l describe the

polarization phenomena due to the waveguide structure. Even in a waveguide with

a step refractive index profile these terms do not vanish everywhere; they are

different from zero at the interfaces between regions with different refractive

indices.

1.2 Matrix Method for Electromagnetic Field Propagation in a Stratified Medium

The simplest method of finding the solution of the electromagnetic field which

propagates in a stratified medium is the matrix method. As mentioned above, it is

applicable when the direction of the field propagation does not coincide with the

direction along which the parameters & and J.l are constant.

Let us ass urne that our stratified medium is formed from aseries of N

altemating dielectrie layers with different & and J.l values. As before, the direction

along wh ich the dielectric constants are invariant is denoted by z, x is the direction

perpendicular to the dielectric layers (stratification direction) and y is chosen such

that the directions x, y, z form a right handed orthogonal system of coordinates

(see Fig. l.l). Ifthe electromagnetic field, which we will suppose for the moment

to be transverse electric (TE) (with e perpendicularly polarized with respect to the

propagation direction), is obliquely incident - in the xz plane - on the stratified

medium, Maxwell's equations with E(x,y,z) = yEy(x,y,z) are:

6 1. Basic Concepts of Optoelectronic Devices

z

~.----t~--r---~-r_X

Fig. 1.1. Schematic diagram of a stratified medium and the coordinate axes

imJ1l!y = 0,

oEy / ox + imJ1l!z = 0,

oE y / oz - imJ1l!x = 0,

OHx / oz - oHz / ox - imeEy = 0,

OHY /ox - oHx /oy = 0,

OHz / oy - OHy /oz = O.

(1.11)

From these it follows that Hy = 0 and Ey, Hz, Hx are functions only ofx and z. From symmetry considerations they can be written asEAx,z)=eAx)exp(ißz),

HAx,z) = hAx)exp(ißz), HAx,z) = hAx) exp(ißz). ey, hx and hz are further related by Maxwell's equations (1.11). One can easily show that eyand hz satisfy

a system of first order coupled equations:

dey / dx = -imflhz ,

dhz / dx = i(ß / mfl- me)ey, (1.12)

with hx = ßey / mfl. So, eyand hz can be expressed as a linear combination of

two particular solutions, ey', hz ' and ey", hz", which satisfy the system (1.12)

and for which d( ey" hz '-ey' hz ' ,) / dx = 0, i.e. the determinant of the matrix

is constant. By choosing these particular solutions such that ey' (0) = hz " (0) = 0,

ey " (0) = hz ' (0) = 1, the general solution for the electric and magnetic field at an

arbitrary x plane can be written in the matrix form

1.2 Matrix Method in a Stratified Medium 7

where the matrix S has unit determinant. Usually the inverse M of the S matrix is

used to relate the input fields to the output fields. For a homogeneous medium

with refractive index n on which the electromagnetic field is incident at an angle ()

with respect to the x axis, ß = kon sin () and the matrix M is

_ ( ) _ ( co~ konxcos()) isin( konxcos())! p) M - M n, Lopt - ( ) ( )' ipsin konxcos() cos konxcos()

(1.14)

with p = ~ c! f1 cos (). F or the particular optical wavelengths for which the optical

thickness Lopt = nxcos() is an integral number of half-wavelengths, the layer is

optically transparent, i.e. M( n, Lopt ) is proportional to the 2 x 2 unit matrix. Since

the tangential components ofthe electric and magnetic fields are continuous at the

interface, the total matrix of a stratified medium consisting of a succession of N

homogeneous layers, each labeled by i, is equal to the product of the matrices

corresponding to each layer:

N

M = DM(n;,Lopt;). 1=1

If p in (1.14) is replaced by q = ~ f1! c cos(), the matrix for a homogeneous

medium is obtained under a transverse magnetic (TM) electromagnetic wave

(with h polarized perpendicular to the propagation direction) excitation; the

general matrix relation (1.13) holds also for this case.

An important case in applications is that of a periodic stratified medium with

M(A) - the matrix corresponding to one period with width A. Then, the matrix

of the total stratified medium formed of N periods is

m12 CN_I(m) )

m22CN-I(m) - CN- 2(m) ,

(1.15)

where m = Tr( M(A)) ! 2 = (mI I + m22 ) ! 2 with mij (i,) = 1,2) the elements of the

M( A) matrix, and


the Chebyshev polynomials. For optical wavelengths for which Im! > 1, the

periodic layer acts as a perfect reflector, i.e. stopbands for incident light

wavelengths appear where the argument of the cos-I function in CN(m) takes

forbidden values. For the remaining spectrum regions the radiation passes through

- these are passband regions for the incident wavelength.

The matrix method works equaIly weIl for passive and active dielectric layers.

In the first case n is real while in the second it becomes complex, its imaginary

part describing the gain (or loss).

The most important applications of passive periodic media are as wavelength

filters. In this case one is concemed with the dependence of the reflectance R with

the wavelength A of the electromagnetic field. If the stratified medium

characterized by a total matrix Mwith elements mij (i,j = 1,2) is placed between

a medium with parameters Bin and !-lin from which the light is incident at an

angle Bin and an output medium characterized by Bollt ' !-lout and Bollt , the

intensity reflection coefficient is:

(1.16)

where Pin = ~Bin / !-lin cosBin , Pout = ~Bout / !-lOllt cosBollt ' So, a stratified medium acts as a mirror with an intensity reflection coefficient R. If the period of the

stratified medium consists of two layers with refractive indices nl and n2, widths

LI and L2, and opticallengths nl LI cos BI and n2 L2 cos B2 equal to multiples of

quarter wavelengths, the filter is

- a band-pass filter for:

M = (M(n\>A / 4)M(n2,A /4))N (M(n2,A / 4)M(n\>A / 4)t,

- a short-wave pass for:

M = ( M( n2 ' A /4) M( nl , A /4)) N M( n2 ,A /4) , - a long-wave pass for:

M = (M(nl,A / 4)M(n2,A / 4))N M(nl,A / 4). These filters are extremely useful in a large number of optoelectronic devices. For

example, in semiconductor lasers the usual dielectric mirrors can be replaced by

such filters in order to assure the coherence of the laser radiation. They are used

also as resonant cavity filters in the photodetection process in some p-i-n

structures, to select the wavelength at which the quantum efficiency is a maximum

(Tung and Lee 1997). The most important application for active periodic layers is

the very recently proposed photonic band-edge laser. It consists of altemating

1.2 Matrix Method in a Stratified Medium 9

active and passive layers designed such that the structure works near the stop band

edge where the group velocity has a very low value. This implies that the optical

path length is enlarged and the laser gain is therefore increased by a factor

between 4 and 5 (Dowling et al. 1994).

1.3 Modal Method for Electromagnetic Field Propagation in a Dielectric Waveguide

This method for studying the electromagnetic field propagation is generally used

if the direction along which c and f..l are constant coincides with the propagation

direction of the light wave. In this case the total electric and magnetic fields are

decomposed into a sum over the modal fields, indexed by):

(1.17)

The modal fields are solutions of the source-free Maxwell's equations and

propagate along the z direction with propagation constants ßj determined from

the boundary conditions at the interface between the different layers. All the

relations derived in Sect. 1.1.2 hold for the modes; an important characteristic is

that since ej , h j depend only on Ti, the form of the modes is invariant at

propagation. The possible solutions for ßj can have a discrete spectrum, the case

in which the modes are guided, or a continuous spectrum for the radiation modes.

In the latter case the sum in (1.17) must be replaced by an integral expression.

Radiation modes have an oscillatory behavior and dissipate the energy in a

direction transverse to that of propagation. Therefore, they are undesired modes

and can be suppressed by an appropriate design of the waveguide; we will not

refer to them further. Guided modes are possible only if the refractive index of the

propagating region (core) is higher than that of the surrounding layers (claddings).

Namely,

(1.18)

where ncl and neo are the maximum refractive index of the claddings and core,

respectively. The coefficients aj characterize the coupling between the source

and the modal field). In order to calculate aj we use the orthogonality relations

between any two guided modes:


f ej x hk * idA = f ek * x hjidA = N k0j,k, (1.19) Aoo Aoo

where Aoo is the cross-section of the waveguide at infinity and * stands for the

complex conjugate operation. Actually, only the transverse components of the

electric and magnetic fields contribute to the integrals in (1.19). The relation

(1.19) is valid for guided modes in nonabsorbing dielectric waveguides. For

absorbing ones a relation similar to (1.19) can be written but without the complex

conjugate operation. From (1.17) and (1.19) it follows that:

ak = fEx H/idAl Nb (1.20a) Aoo

for nonabsorbing waveguides and

ak = JE x HkidA I Nk (1.20b) Aoo

for absorbing waveguides.

The power carried by the jth mode is obtained by integrating the power density

along the z direction,

S· = (la'l2 12) Re(E . x H* . i) ././ J./

(the component ofthe Poynting vector along i), over A",,:

(1.21a)

It is important to point out that if Pi is positive then P_ j , which represents the

power carried by the jth mode when propagating in the -z direction, is negative.

Its expression can be written directiy as:

P = ~ I a .12 f e . x h .*. z~ dA = - ~ I a ·1 N -./ 2 - } - } -./ 2 - } ./' Aoo

(1.21 b)

and yields a negative result because h_ jt = -h jt

(the relation f e_ j x h_ j ' . idA = f e_ jl x h_ j,* . idA was used). Aoo Aoo

1.3 Modal Method in a Dielectric Waveguide 11

1.3.1 Modes in Slab Waveguides

To exemplify the calculation of modes, we consider first slab waveguides with

constant refractive indices in the core and cladding layers - step refractive index

profiles (see Fig. 1.2).

x n

nell ncl2 d ncl1

z neo -d d x y

-d ncl2

Fig. 1.2. Dielectric slab waveguide

Let us suppose for generality that neo> ncl2 > nell (asymmetrie waveguides).

In the case of symmetrie waveguides ncl2 = ncl1 = nct. For step refractive index

waveguides V t In n 2 = ° (V t In 6" = 0, V t In,l1 = 0) in each layer, but differs from

zero at the interfaces. However, even if Vt ln6",tO, from (l.9) it follows that

there are two exceptional cases in wh ich the different components of the electric

and magnetic field do not couple: the case when ez = ° -transverse electric (TE)

modes, and hz = ° -transverse magnetic (TM) modes.

For a TE mode (ez = 0) by choosing also ex = 0, ey satisfies the Helmholtz

like equations:

(d 2 / dx2 + kJn~o - ß2 )ey = ° (d2 / dx2 + kJn~ll - ß2 )ey = ° (d2 / dx 2 + kJn~12 - ß2 )ey = °

o:::::lxl<d,

x>d,

x<-d.

(l.22)

As mentioned above, for guided modes kOncl2 < ß < konco . lndeed, this is the

requirement to have a sinusoidal behavior of the field in the core and an

exponential field decay in the claddings. Thus, the energy is concentrated in the

core region. The concentration of the field in the core is more effective if the

difference between the core and cladding refractive indices increases.

For symmetrie waveguides, the solution for TE modes is:

12 I. Basic Concepts of Optoelectronic Devices

{

cos Ux 0 :::; I xl < d cosUd

e y = exp( _ WI xl) even modes Ixl~d

exp(- Wd)

(1.23a)

{

sin Ux 0 :::; I xl < d sinUd

e y = x exp( - WI xl) odd mo des

R exp( - W d) I xl ~ d

(1.23b)

where U = ~k~n;o - ß2, W = ~ß2 - k~n;I. From the continuity conditions for ey and its derivative with respect to x at

x = ± d, one obtains the transcendental equations to be solved in order to get ß:

W=UtanUd even modes, (1.24)

W = -U cotanUd - odd modes.

The values of ß obtained from these equations (and the corresponding modes)

are numbered in decreasing order of their value: ßo > ßI > ... > ß N. The index of

the TE mode also indicates the number of zeros ofthe guided electric field: TEo has no zero, TE 1 has only one, and so on. The number of guided TE modes for a

slab waveguide with constant core and cladding refractive indices is M TE = Int[2V / lr], where V is the most important characteristic of the waveguide - the

normalized frequency ofthe waveguide - defined as:

(1.25)

The expression for MTE is derived from the condition that a mode appears only if

ß takes its minimum value kOncl - see also Fig. 1.3. So, the number ofTE modes

is a function of the wavenumber (frequency) of the incident light as weIl as a

function of the geometrical dimensions of the waveguide. The minimum

frequency for which a mode can propagate is called the cut-off frequency of the

mode. Modes can be cutoff not only by decreasing the frequency but also by

reducing the width of the waveguide, since it has the same effect on V. For

waveguides in which the refractive indices have an x variation inside the core

and/or cladding layers, the number of TE modes depends in general on the

variation law of the refractive index.


Ud

V=Ud ..

n ............................ .r= , 0- :

n/2 nnuA= .,' :

TMo

TEo

11 v

Fig. 1.3. Schematic diagram for the number of solutions of ß as a function of

V in a slab waveguide with constant refractive index profiles

The TM modes for a symmetric waveguide satisfy the same equation as (I.22)

for the y component of the magnetic field (hz = 0, hx = 0), with the same solution

as in (1.23). The only difference with respect to TE modes appears in the

boundary conditions, a fact which modifies the tran sc enden tal equations for ß:

n~oW = n~ptanUd even modes,

n~oW = -n~IU cotanUd - odd modes. (1.26)

So, the propagation constants of the TM and TE modes are different, as one

can observe also from Fig. 1.3. However, the indexing of modes is performed in

the same way and the number of zeros is similarly related to the mode index.

Moreover, M TE = M TM . For 0< V < Jr / 2 only one mode (actually two if the

polarization is also considered) is propagating through the waveguide - the

waveguide is called single-moded. If V;::: Jr / 2 several modes can propagate

simultaneously - the waveguide is multimoded. Apart from V, the numerical

aperture NA, defined as

is an important parameter of optical waveguides especially in coupling problems.

For waveguides with large V values the propagation of the electromagnetic

field can be described in the approximation of geometrical optics. In this situation

the diameter of the waveguide is much larger than the wavelength of the light and

the concept of waves can be safely replaced by that of rays. The light field can


then be described as a bunch of rays, each incident into the waveguide under a

slightly different angle (J, with the z axis, the individual rays undergoing total

internal reflection at the interface between the core and claddings. Between

successive reflections the ray path is straight in waveguides with a constant

refractive index in the core and cladding; otherwise the path r(s) is determined

by solving the eikonal equation

! (n(r) :) = Vn(r), (1.27)

with s the distance along the ray path. The ray path in a slab waveguide is periodic

if the refractive index of the waveguide is constant along the z direction. In this

case a propagation constant can be defined, as in the wave-like treatment. Namely,

jJ = n( li) cos B( li) = n( 0) cos B( 0) is constant, where (J(!i) is the local angle

between the ray and the z axis at a transverse position li, and 0 = (0,0) is the

trans verse coordinate vector of points situated on the z axis. ß is called the

longitudinal ray invariant. In planar waveguides the ray path equation can be

separately and independently solved in the xz and yz planes. We can focus on the

ray propagation in only one of these planes, say xz, as in the rest of this section.

If total reflection does not take place, i.e. if the ray is incident on the interface

at an angle with the z axis greater than the critical angle at the interface between

the core and cladding Ber{x = ±d) = cos- I (ncl{x = ±d)) / neo{x = ±d), it penetrates

into the cladding region and is lost. These rays are called refracting rays and

correspond to the radiation modes in the wave treatment. Alternatively, a ray at

the trans verse position x = 0 is totally reflected at the interface if its angle with the

z axis is sm aller than the critical angle Ber (0) = cos-1 (ncl (x = ±d)) / neo (x = 0), at

the interface. (Jer = cos-1 (ncl / neo) in waveguides with con.:tant core and cladding

refractive indices. Bound rays are those for which ncl < ß :s; neo and refracting

rays are those for which 0 :s; ß < ncl . Not only bound and refracting rays can be

defined in the geometrical approximation, as for the wave-like treatment, but also

a number of müdes M can be defined by the condition that the phase accumulated

by the ray after propagating one period, including the phase changes at the

reflectiün at the core/claddings interfaces, is a multiple M of 21l.


1.3.2 Modes in a Step Index Fiber

Cireularly symmetrie fields in a step-index fiber (Fig. 1.4) ean be, as for the planar

waveguide, either TE or TM modes. However, (1.10) in polar coordinates also

allows other types of modes, denoted by EH, HE, or LP modes, for which ez and

hz are different from zero. These are called hybrid modes.

x

r

z

y

Fig. 1.4. Circu1ar step fiber

Let us consider such a hybrid mode in a fiber with constant core refractive

index neo and constant cladding refractive index ncl. The z components of the

electric and magnetic fields satisfy the following Helmholtz-like equations in

polar coordinates:

o ~ r < d,

(1.28)

r ;:C:d,

where, as before U = ~ k~ n~o - ß2, W = ~ ß2 - k~ n~l. The solutions separable

in r and r/J of (I .28) are:

ez = !AJ/(Ur)exP(ilr/J), BK/(Wr)exp(ilr/J),

h. = !CJ/(ur)exP(ilr/J), " DK,(Wr)exp(ilr/J),

O~r~d

r?d

O~r~d

r?d

(1.29a)

(1.29b)

with arbitrary constants A, B, C, D and integer I values. The other components of

the electric and magnetic fields can be obtained from (1.8), i.e.


er = ~ [ AUJ/(Ur) - iaJ;ol CJ,(Ur) }XP(il~),

elP = i~ [~A UJ, (Ur) + aJJicol CUJ/ (Ur )]exp( il ~), U r ß

hr = ~ [ CUJ/(Ur) + iaJ~ol AJ,(Ur) }XP(il~), O~r~d

elP = i~ [~CJ,(ur) _ aJ&col AUJ"(Ur)]eXP(il~), U r ß

(1.30)

r?d

By writing the eontinuity eonditions for elP , ez , hIP' hz at r = d one obtains a

homogeneous system of four equations with four unknowns: A, B, C, D. This ean

be solved only if its determinant is set equal to zero, a eondition whieh yields a

transeendental equation for ß:

( J/(Ud) + K/(Wd) )(n;oJ,'(Ud) + n;IK/(Wd)) UdJ,(Ud) WdK,(Wd) UdJ,(Ud) WdK,(Wd)

= lz[(~dr +(~dr)(~r (1.31)

For 1= 0, t3 / t3~ = 0 , and all field eomponents are radially symmetrie. There

are two families of solutions for ß in this ease. In the first one ß is obtained from:

(1.32)

( Jo I = -J1, Ko I = - K1); this ease eorresponds to TE modes sinee the only

nonvanishing field eomponents are hr , hz and elP . These are designated as TEOm

where the first subseript stands for 1=0 and the seeond subseript m = 1,2, ...

denotes the number of the solution for ß .


The other class corresponds to the equation:

J1(Ud) _ nz,K1(Wd)

U Jo(Ud) - nZoWKo(Wd) (1.33)

and describes TMOm modes with nonvanishing field components er' ez ' htP .•

For 1,* 0, (1.31) is a second order equation for J/(Ud) / UJ/(Ud). The two

classes of solutions for J/'(Ud) / UJ/(Ud) yield the two hybrid types of modes:

EH/rn and HE/rn, characterized by two indices. The first is equal to land the

second indicates the number of the solution for ß. The fundamental mode, i.e. the

mode with the highest value of the propagation constant is HEll (see also Fig.

1.5).

Ud v= Ud EH2!

5.134···········································:.~

... / : EH 11 ... ~

3.832 ............................... :.~ : ./ :: HE2 !

2.9 ..................... :.~ TMO!

TEO!

2.405 ............... ::-:

..

.. . . . .

~--------~~----~----~----------+v o 2.405 2.9 3.832 5.134

Fig. 1.5. Schematic diagram for the number of solutions of ß as a function of V

in an optical fiber with constant refractive index profiles

The total number of modes for a step-index circular fiber is M = Int[V2 /2];

for fibers with variable core refractive index (the refractive index of the cladding

is generally constant) M depends on the variation law of neo. A schematic

diagram of the number of solutions for ß as a fimction of V in a step-index fiber

is given in Fig. 1.5. For V< 2.405 the fiber is single-moded whereas for greater

V values it is multimoded.

The solutions for the electric and magnetic fields for the EH/rn and HE/rn

modes, even in a step profile fiber, are quite complicated. They can be simplified


in weakly guiding fibers, i.e. in fibers where the refractive index of the core

differs only slightly from that ofthe cladding: neo = nc1 = ß / ko and U, W« ß. In this case the continuity condition at the core-cladding interface for the

tangential component of h becomes identical to that for the tangential component

of e; cartesian coordinates can therefore be introduced, simplifying the solutions

ofthe wave equation. For example, erp can be decomposed in terms ofthese new

cartesian components, ex and ey , as erp = -ex sin r/J + ey cos r/J. In such weakly

guiding fibers we can defme a new set of modes - linearly polarized modes - for

wh ich ex or ey vanish. The y-polarized solution of the MaxweII's equations has

the form (Yariv 1985)

ex =0,

e = {AJ,(Ur) exp(ilr/J) , r<d y BK, (Wr) exp(ilr/J), r > d

Iß uey ß 2 • ::J {U ~(J'+I(Ur)exp(i(1 + 1)r/J) + J'_I(Ur)exp(i(I-I)r/J)),r < d

ez = - OJ2&f-l oy = ~ ~ (K'+I(Wr)exp(i{I + 1)r/J)- K'_I(Wr)exp(i{I -I)r/J)),r > d

ß hx =--ey , OJj.1.

hy =0,

h =_i oey ={-~ ~(J'+I(ur)exP(i(t+l)r/J)-J/-I(ur)exP(i(l-l)r/J)),r<d z OJj.1. oX _ lW B (K'+l (Wr) exp(i{I + 1)r/J) + J/-I (Wr) exp(i(I-I)r/J)),r > d

OJj.1. 2

(1.34)

from which it follows that the dominant field components are ey and hx ' i.e. the

electromagnetic field is alm ost transverse (TEM field). The two constants A and B

are not independent; from the continuity condition of ey (and hx ) at r = d it

follows that B = AJ,(Ud) / K,(Wd). Analogously, the x-polarized solution of the

MaxweII's equations in a weakly guiding fiber has a similar form as that given

above for the y-polarized solution, but with its ex , ey, ez ' hx ' hy' hz

components identical with the ey, ex ' OJj.1.hz / ß, hy' -hx , ßez / OJj.1. components

of the y-polarized field. The electromagnetic field is nearly transverse in this case

also since the dominant field components are ex and hy- The B constant is

determined, as above, from the continuity condition at the core-cladding interface

imposed upon ex '

I.3 Modal Method in a Dielectric Waveguide 19

The propagation constants for the new set of approximate solutions of

Maxwell's equations in weakly guiding fibers are determined from the continuity

conditions of ez and hz at the boundary between core and cladding. For both

cases, of the x- and y-polarized modes, the continuity conditions, which must be

valid for all azimuthai angles rjJ, impose the conditions

(1.35)

These two conditions are mathematically equivalent since the recurrence relations

for the Bessel functions are J,(x)/X=(J'_I(X)+J'+I(X))/2 and K,(x)/x= (K'+I(X)-K'_I(X))12. The transcendental equations for the propagation

constants of the modes in weakly guiding fibers are much simpler than the exact

equation (1.31). Since both orthogonally linearly polarized modes have the same

set of propagation constants, they are degenerate; this corresponds to the fact that

the propagation constants ofthe HE'+I,m and EH,_I,m modes become degenerate

in weakly guiding fibers. So, the linearly polarized modes in fibers with

approximately the same refractive index in core and cladding regions, denoted by

LP'm' are a superposition of HE'+I,m and EH,_I,m modes. The set ofEH and HE

modes is replaced by the set of x- and y-polarized LP modes. In particular LP01 is

the new designation for the HEll mode in weakly guiding fibers, the LP02 mode

now denotes the HE 12 mode, the LPll modes include four possible field

distributions denoted formerly as HE 21 , TM o1 and TE o1 modes, and so on. For

the transversely polarized LP modes there is one dominant electric field

component and one dominant magnetic field component, orthogonal of the

electric one. For an LP mode with e and h chosen along two orthogonal radial

directions there is always another LP mode with e, h fields orthogonal to the first

pair.

Unlike the case of planar waveguides, in step profile fibers with V» 1 the

eikonal equation has two invariant quantities for longitudinally independent

refractive index profiles. One is identical to the ß parameter in: the planar

waveguides and the other is [ = neo sinBcosrjJ where rjJ is the angle in the core

cross-section between the tangent to the interface and the projection of the ray

path; rjJ has the same value at any reflection. The azimuthai invariant [ appears

as a consequence of the axisymmetric refractive index profile. The rays for which

[ = 0 are called meridional rays since between reflections they pass through the

center of the fiber. The properties of the meridional rays are identical to those of

bound rays in a planar waveguide; they are also characterized by only one

invariant parameter. The rays which do not pass through the fiber center are called


skew rays; they have a helicoidal path determined by both invariants, i.e. by both

angles () and rp. For step index fibers with large V values, as for planar

waveguides, it is possible to define bound rays and refracting rays as those for

which nc1 < jJ ~ neo and 0 ~ jJ2 + P < n;l' respectively. However, the conditions

for the bound and refracting rays do not cover all the possibilities for incident

rays. The remaining part, namely the rays for which n;1 < jJ2 + P ~ n;o and

o ~ ß < nc1 are called tunneling rays and have the interesting property of

tunneling a finite distance into the cladding. At the core-cladding interface the ray

path actually separates into a core path and a cladding path, spatially separated by

a region in which the propagation cannot take place. As a result, the power of the

tunneling rays is lost at propagation due to the cladding part of the solution of the

eikonal equation, but this loss is much sm aller than that associated with the

refracting rays, which have no core path after reflection.

1.3.3 Modes in a Periodic Waveguide

Let us consider a nonmagnetic stratified medium whose period, of length A,

consists of two layers with refractive indices and widths nl' LI and n2 , ~

respectively, such that LI + ~ = A. The propagation ofthe electromagnetic field

through this structure can be described by the matrix method with

M(A) = M(nl,ßtLl)M(n2'~~) where ßl> ~ are the light wavenumbers in the two media along the direction of propagation. Due to the periodicity, the electric

and magnetic fields of a TE mode, for example, are identical after aperiod A to

the incident ones up to a phase constant exp( ißA):

(1.36)

where ß is the propagation constant of the mode. It is determined by equating to

zero the determinant of M( A) exp( ißA) - I, with I the 2 x 2 unit matrix. The

resulting equation is

where X = 1 for the TE modes and X = ni / nr for TM modes. This equation can

have a number ofsolutions ßo,ßt,/h, ... ,ßN ifthe modulus ofthe right-hand side

of (1.37) is less than or equal to 1; otherwise no solution is found, so no modes


propagate through the structure. The frequency ranges for which Icos(ßA)I> 1 are

called stopbands.

Now, (1.36) determines ß from the condition imposed by the periodic

behavior after a stratification period. If the whole periodic structure is considered

(Ma 1994), the propagation constant of the modes are the same as those derived

from (1.36) if A is sufficiently large (A» ,1,). With decreasing A, the coupling

between waveguide periods become important and each mode splits into a number

of submodes equal to the number of periodic sequences and changes itself into a

band. This behavior is to be expected from the results of solid state physics, since

a periodic dielectric structure is similar to a periodic sequence of elementary cells

in a crystal. In Sect. 1.3.1 we have shown that modes can be cutoff (they are not

allowed to propagate ) if the width of the waveguide is decreased, maintaining the

same frequency of electromagnetic radiation. The same happens in the stratified

medium. As A decreases, the modes are cutoff successively until only the TEo mode (band) is allowed to propagate. By further decreasing A (A « ,1,) only one

subband propagates. The periodic structure is now equivalent to a single mode

dielectric waveguide. The stratified medium in these excitation conditions is

called a superlattice. In a superlattice the coupling between different periods

wh ich have widths much smaller than the incident light wavelength is so strong

that the properties of the structure are equivalent to the properties of a

homogeneous medium through which only one mode can propagate. These

properties can sometimes differ a lot from the properties of the materials wh ich

form the periodic waveguide. In particular, an equivalent refractive index

distribution can be defined for the periodic structure, which is different from and

can be determined in terms of the refractive indices of the individual materials of

the stratified medium. The equivalent refractive index for a TE mode is usually

approximated by

I.n2 L . .I .I

n2 -_.I_TE -I. Lj .I

(1.38)

where the sum is taken over all the layers which form the superlattice, whereas for

a TM mode

1 I. 2 Lj j nj

I. Lj .I

(1.39)

22 1. Basic Concepts of Optoe1ectronic Devices

In particular, for an asymmetrie structure the number of layers with refractive

indices nl and n2 are equal, and the sums can be restricted to one period, while in

the symmetrie structure the number of layers differs by one.

Another example of a superlattice is a multiple baITier structure in which in

one material ofthe periodic structure (the barrier) the wavefunction is not allowed

to propagate (it is an evanescent wave). Mathematically, this implies that the

effective wavenumber in that medium, say 1, is imaginary, i.e. ßI must be

replaced by ißI where flJ. is the wavenumber along the propagation direction of

the wave. The other material, in which the propagation of the wavefunction is

aIlowed is called a weIl. In such a structure Saini and Sharma (1996a) have shown

that a better approximation of the equivalent refractive index can be defined, for a

TE mode, as

(1.40)

nm in the formula above is the equivalent refractive index for a given mode

determined from a transcendental equation for the propagation constant of the

structure, as discussed in Sect. 1.3.1, and CN are the Chebyshev polynomials w

with N w the number of weIls in the structure. The quantum analog of a multiple

barrier dielectric structure is called a multiple quantum weIl (MQW); in these

structures the electron wavefunction is not allowed to propagate in the baITier

regions, but only in the weIls.

MQW structures are generally made from altemating layers of semiconductor

elements or semiconductor compounds. The compound semiconductors can be

binary as in III-V or lI-VI semiconductors or temary, or even composed from four

elements. These MQWs have different properties compared to the materials

forming the different layers. For example, an MQW structure formed from

altemating layers of GaAs and AIAs has an unusually high coefficient of

nonlinearity as compared to that ofthe GaAs and AIAs constituents.

1.3.4 Modes in a Bent Step Profile Planar Waveguide

Under the assumption that the refractive index profile is not affected by the bend,

the propagation of the rays in a bent planar waveguide can be reduced to the

propagation of skew rays in a circular fiber of radius R + d where R is the bend

radius (see Fig. 1.6). These rays can therefore be characterized by the azimuthai

invariant land by a longitudinal ray invariant ß = 0 (the ray propagates in a

cross-sectional plane of the equivalent fiber). Since ß = 0, every ray in a bent


waveguide is leaky, i.e. it is either a refracting ray or a tunneling ray. The power

attenuation in bent waveguides is described by the formula

p( 9) = p( 0) exp( - T 9 / 9 p ), (lAI)

where p( 0) is the initial power in the straight section, T is the power

transmission coefficient at ray reflection, given by

(l.42)

whispering-gallery rays 2d

9

Fig. 1.6. Ray paths in a bent waveguide

and 9p is the angular separation between two successive reflections (Snyder and

Love 1983). For waveguides with large V values the transmitted power decreases

as the normalized radius of curvature R / d is reduced. The refracting rays and

the tunneling rays with the largest power loss coefficient are first lost at

propagation through the bent waveguide, the remaining tunneling rays being then

lost with a much smailer power loss coefficient. Actually, the radiation emitted

from bent waveguides is not continuous but has the form of discrete, divergent

rays, the number of radiated beams per unit length increasing as the bend radius

increases (Gambling et al. 1976).

In bent waveguides with a large V value the guided light rays can foilow two

paths between successive reflections at the interfaces: (i) they can undergo

reflections altematively from the inner and outer interfaces or (ii) they can reflect

only from the outer interface. In the latter case they are cailed whispering-gailery

rays. By writing the azimuthai ray invariant as


- R-d I == neo cos~ == --neo cos~',

R+d (l.43)

where ~' is the angle between the ray path and the inner interface of the bent

waveguide, it follows that the whispering-gallery rays must satisfy the condition

cos ~ > (R - d) / (R + d). Since in practice R > > d, only a sm all number of rays

satisfy this condition. They propagate almost tangentially to the outer interface of

the bent waveguides, with ~ = o. For whispering-gallery rays ~ is usually much

smaller than the critical angle at the interface between the core and cladding,

0er == cos-1(nc1 / neo), so that they are tunneling rays. Otherwise the rays are

refracting. Although leaky, the power lost by whispering-gallery rays can be low,

especially when the difference between the refractive indices of the core and

cladding is high. The whispering-gallery rays are used in some optoelectronic

devices as for example, in microdisk lasers; their repeated reflections at the bent

interface is equivalent to the repeated reflections of a ray in aresonant cavity.

From a wave-like point of view, it is possible to obtain the whispering-gallery

modes in a bent waveguide (or in a disk-like waveguide) by using the effective

index method. The TE and TM modes are found to be of the form rpM(r) =

J M (r) exp( iM.9) where J M is the Bessel function of order M and r is the radial

coordinate. F or high M values the asymptotic behavior of the Bessel function can

be used to simplify the expression of the modes to rpM(r hA r M, where A is a

constant. If the boundary condition is such that rpM vanishes at the outer

interface, no optical energy would theoretically escape outside the bent

waveguide, in the radial direction. For waveguide disks the radiation can however

escape above the disk; it is emitted within a small divergence angle centered on

the plane of the disk (Li and Liu, 1996).

1.4 Coupled Mode Method for Electromagnetic Field Propagation in Perturbed Waveguides

The coupled mode method is used to calculate the electromagnetic field when an

extemal perturbation acts on a waveguide. This extemal perturbation can be the

presence of another waveguide, as for example in waveguide couplers, or a

perturbation of the shape and/or refractive index profile. We would like to point

out that the periodic perturbation of the reftactive index profile for example could

also be treated by the matrix method applied to periodic structures. Actually the

two methods give results which agree with one another as long as the fractional

change of the refractive index from one layer to the other is relatively small.

1.4 Coupled Mode Method in Perturbed Waveguides 25

In the coupled mode method the guided electric and magnetic modal fields

E k , Hk and their constants of propagation ßk in the unperturbed waveguide are

supposed to be known. The transverse electric and magnetic field components in

the perturbed waveguide, Ept and H pt are written as a sum over the transverse

components ofthe guided fields in the unperturbed waveguide:

(1.44)

where ak and a_k refer to the same kth mode, propagating in the forward and

backward direction, respectively; use has been made of the transformation

relations of the electric and magnetic field components at the inversion of

propagation direction, as given in (1.6) and (1.7).

Supposing now a nonmagnetic medium (f.L = J10 and & = n2 &0) and using the

last equation in (1.8) as weIl as (1.6) and (1.7), one can also express the

longitudinal component of the guided field in the perturbed waveguide in terms of

those of guided modes in the unperturbed one:

(1.45)

In the above equations n and np denote respectively the refractive indices of the

unperturbed and perturbed media.

Our aim is to find the as-yet unknown coefficients bk and b_k (ak and a_k).

To perform this task we use the divergence theorem

J VF.dA=~ JF·zdA+t Fndl A", i}z A", I

(1.46)

where I is the contour of the cross-section at infmity. We apply this theorem for

F = Ep x HZ + EZ x Hp. Due to the property that at great distances from the

considered waveguide, the guided fields become negligible, the last term on the

right-hand side of (1.46) can be neglected. By introducing (1.44) and (1.45) into

the expression for F, using the orthogonality condition for guided modes in the

unperturbed waveguide and after performing simple algebraic operations on the


remaining terms in (1.46) one fmaBy obtains the differential equations satisfied by

bk and b_k . For bk it looks like:

(1.47)

where Ck,l are the coupling coefficients defined as:

(1.48)

Ck,_1 is given by an analogous expression with elz replaced by -elz- The

coefficient b_k of the kth mode propagating in the backward direction is

determined from equations similar to (1.47), (1.48) in wh ich bk is replaced with

b_k , ßk with ß-k = -ßb ko with -ko and eZz with - eZz· The coupled mode theory describes the electromagnetic field propagation and

so, the performances of the most passive optoelectronic devices. However, there

are two basic examples of its utility: the directional coupler and the Bragg

reflector. In these examples only two modes are supposed to be coupled so that we

have to solve for bk only a set of two coupled first order differential equations of

the type given in (1.47). We will study in detail these two examples since they

have a lot of applications of their own or as parts of more sophisticated

optoelectronic devices and/or circuits. Solutions for coupled mode equations in

other structures, as for example the coupling due to Dirac-like singularities in &,

can be found as weB (Boucher 1997).

1.4.1 The Directional Waveguide Coupler

The directional waveguide coupler consists of two waveguides, the distance

between which is sufficiently small such that the guided field in one of them is

perturbed by the presence of the other. Let us suppose that the waveguides are

parallel and singlemoded, the transverse guided fields and propagation constants

in the isolated waveguides being denoted by elt,ßI and e2t,ß2 respectively. The

trans verse component of the total field propagating in this structure can then be

written as:

(1.49)


where the coefficients b1, b2 satisfy a system of coupled equations which can be

put in matrix form:

(1.50)

According to (1.48) and to OUf supposition that the perturbation is due to the

presence of the other waveguide, we arrive at the conclusion that Ci,i are

negligible for guided modes since n2 - n~ (we have supposed real refraction

indices) is equal to zero over the region occupied by the waveguide i. The only

contribution to Ci,i comes from the integral over the waveguide j *' i, where the

field components of the waveguide i are very small (the guided fields are

concentrated in the region of the respective waveguide). Therefore, we neglect the

terms Ci';' i = 1,2 in (l.50). Moreover, for a lossless waveguide, i.e. a waveguide

for which the total optical power

is preserved at propagation, the additional relation CI 2 = C; I = C holds with

the assumption that leH I = le2t I = l. The system of equations (l.50) can be solved

easily by replacing the rapidly varying coefficients bi with slowly varying ones

b;(z) = bi ( z) exp( - i(ßI + ~)z /2); with these considerations (l.50) reduces to

where I'!.ß = (ßI - ß2) ! 2. The solution of (1.51) is:

(b;(z)) = ~ ()(b;(0)) ~() M12 z ~() b2 z b2 0

= [cos(p) + iI'!.ßsin(p)! r iC' (sin(p) ! r)

iC(sin(p)!r) )(b;(0)) cos(p)-ißsin(p)!r b;(o) ,

(1.51 )

(l.52)

where r = ~ I'!.ß2 + Ic12 . The original unknowns bl, b2 are related by a matrix

MJ2(z) = exp(i(ßl + ß2)Z !2)Mn(z). One should observe that detMn(z) = 1 and

that for a directional coupler there are no stopbands since ITrMJ2 I::; 2. MJ2 is

periodic, i.e. Mn( z + 2 Lb ) = MJ2 (z) where Lb = Jr ! r is the beat length of the


coupler, for wh ich the power distribution in the coupler is identical to that at its

input ( Ml2 = -I with I the 2 x 2 unit matrix).

In most applications the optical power, normalized to unity, is launched in one

waveguide, say waveguide I, i.e. Fj(O) = I, P2(0) = O. Then after a propagation

length L the distribution ofpower in the two waveguides is:

Fj( L) = Ib1( L)12 = Ih;( L)I = I-Ie / rl 2 sin2(rL),

P2(L) = Ib2(L)1 2 = 1b;(L)1 = le / rl 2 sin2(rL). (I.53)

The power transfer in the initially unexcited waveguide after a length L is

characterized by the splitting ratio SR= P2(L)/ Fj(O). As seen from the above

formula, the power transfer depends on the waveguide characteristics and the

coupling length. If the two waveguides are identical, Ir / CI equals I, otherwise it

is greater than I. The t.ß value, and thus the power transfer, for a coupler with

given (identical or nonidentical) waveguides, can be controlled externally, after

the fabrication ofthe coupler by bending the coupling region or by applying along

it an extern al electric field. The last solution is applicable for waveguides made

from electro-optic materials in which the refractive index can be modified by an

applied electric field but has the disadvantage of requiring voltage sourees. The

first solution can be applied to any coupler, without the need of an external

voltage, and it was shown to be fuHy equivalent to the electro-optic tuning of the

power transfer (Feuerstein et a1. 1996). It is based on the modification of the

refractive index profile in bent waveguides due to the stress. In order not to

modify the direction of emerging light from a bent coupler, sine-bends of

controlled length and curvature can be used to tune the power transfer. In these

structures, the coupled waveguides are bent in the coupling plane, the lateral

deviation being described by the expression x(z) = h(z / L - (1I27Z) sin(2m / L)) where (x,z) =(0,0) at the inflection point ofthe bend, L is the length on which

the bend is applied and +h is the maximum transverse deviation from the initial

direction of the waveguides.

Equation (I.53) which expresses the conservation of the total power in the

directional coupler (11 ( L) + P2 ( L) = const. = I) allows us to introduce another

characteristic length of the coupler: the coupling length Lc = Lb /2. Lc is the

length after which the power in the initially unexcited waveguide (the coupled

power) is maximum. This maximum value can at most equal the initial launched

power Fj(O) ifthe two waveguides are identical (if t.ß = 0); it decreases rapidly

with increasing t.ß. Therefore, a directional coupler couples modes with alm ost

the same propagation constant; this implies that they have also the same


propagation direction. For this reason the directional coupler is an example of a

device which couples copropagating modes.

In directional couplers made from non linear media, i.e. media in which the

refractive index depends on the optical light intensity which propagates through

the coupled waveguides, the coupling length becomes a function of the incident

light intensity (Davis and Digonnet 1996) and even bistability in the power

transfer characteristics can occur (Thirstrup 1995b).

1.4.2 Bragg Reflectors

Efficient mode coupling between modes propagating in opposite directions

(counterpropagating modes) can be realized with a periodic perturbation in the

waveguide shape and/or refractive index profile. A periodic refractive index

perturbation can always be written as a Fourier series in the period A:

cf)

~n2('i,z) = I aA'i)exp(i2;rqz / A). (1.54) q=-cf)

If two guided modes EI (r, t) = e l (Ii) exp( ißlz + imt) and E2 (r,t) = e2 (Ii) exp(iß2z + imt) propagate simultaneously in the waveguide, the perturbation

polarization Pp(r,t) = &o~nEI(r,t) due to EI excites E2 if the time mediated

power density which excites E2

(1.55)

(see (1.5)) is different from zero. Conservation of total power requires that the

perturbation polarization due to E2 excites E1 with a time mediated power

density P21 = - ~2·

An overall power transfer between these two guided modes occurs if the

excitation power, obtained by integrating the power density, is different from

zero. This requirement imposes two conditions: - a phase condition, expressed as

f exp(i(ß2 - ßt + 2;rqz / A)z)dz * 0, L»A

(1.56a)

or

ßI - ß2 = 2;rq / A, (l.56b)

30 1. Basic Concepts ofOptoelectronic Devices

- and an amplitude condition:

(l.57a)

or

J aAIi)el*(Ii)e2(Ii)dA =1= O. (1.57b) Aoo

So, the coupling between the two modes is realized through the qth Fourier

harmonics of the perturbation, and its strength is determined by aq .

If one decomposes the total transverse electric field into a sum over eIl and

e21 , the coefficients bj (i = 1,2) of this decomposition satisfy the same set of

coupled equations as (1.47). The solution ofthis set can again be put into a matrix

form as in Sect. 1.4.1 with the difference that I1ß wh ich describes the coupling

efficiency is now given by the expression I1ß = (ßI - ß2 - 2trq I A) 12. I1ß = 0

indicates, as in Sect. 1.4.1, the condition of maximum coupling efficiency. The

wavelength for which I1ß = 0 is called the re sonant wavelength of the Bragg

reflector or simply the Bragg wavelength of the structure. The strength of the

coupling efficiency, given by CI,2 and C2,1, is consistent with the amplitude

condition, since for a TE wave for example, they now become proportional to

These considerations are valid for any counterpropagating coupled modes.

The Bragg reflector in semiconductor structures is formed from a succession of

alternating dielectric layers with the stratification direction parallel with the

propagation direction of the electromagnetic field; the perturbation is in this case

induced by the periodicity of I1n. When the waveguide width is periodically

perturbed, the structure is usually called a Bragg grating. In a Bragg reflector (or

grating) the coupling between the forward and backward propagating modes is

generally due to the first-order Fourier term in the decomposition of I1n2. In this

case ßI = -ß2 = ß, q = land I1ß becomes equal to I1ß = ß - TC I A. The Bragg

wavelength of the structure can simply be calculated from (1.56b) as AB =

2neffA where neff is the effective refractive index of the guided mode in the

structure (=ßlko). The set ofcoupled equations (1.51) in Sect. 1.4.1 can be

directly rewritten for a Bragg reflector as


where CI2 = C; I = C and the indices 1 and 2 of b were replaced by + and - for a better link to their physical interpretation as the coefficients of forward

and backward propagating modes. The relation between CI,2 and C2,l is again

derived from the conservation condition of optical power

(the power in a backward propagating mode is negative).

The solution of (1.58) can be put in a matrix form, analogous to (1.52) in Sect.

1.4.1, but with the difference that the elements of the M + _ matrix depend now on

the value of

Unlike the case of the directional coupler, for a Bragg reflector r can take also

imaginary values. If f:..ß2::?: Iq2 (oscillatory region - permitted for

electromagnetic field propagation) r is real and

~ (cOS(rz)-if:..ßsin(rz)/r iCsin(rz)exp(i2ßz) Ir) M+_ = _ iC' sin(rz) exp( - i2ßz) I r co~rz) + if:..ßsin(rz) Ir'

(1.59a)

whereas if f:..ß2 < ICI2 (stopbands), ris imaginary and

~ _ (cosh(lrl z) - if:..ßsinh(lrlz)/lrl iCsinh(lrlz)/lrl J M+ - • ( ) () ()' (l.59b) - - iC sinh Irlz Ilrl cosh Irlz + if:..ßsinh Irlz Ilr

The parameter which characterizes the performance of a Bragg reflector is the

power reflection coefficient R = ~_ (0) I b+ (0)12 . If an electromagnetic field is

incident on the z = 0 plane on a Bragg reflector of length L, b_ (L) must be zero at

z = Land so we find that

(1.60a)

in the oscillatory regions and

32 1. Basic Concepts of OptoeIectronic Devices

(1.60b)

inside the stopbands. The reflection coefficient, which is a symmetric function of

Aß, i.e. R( Aß) = R( - Aß), is higher in stopbands than in the oscillatory regions

and its maximum value

(1.61 )

is obtained at the center of the stopband for which Aß = O. Rmax depends only on

ICI and L. For ICiL < 1 we have a poor reflector, a Bragg reflector with IClL> 10

is an excellent one over the whole stopband of width 2Aß = 21C1, whereas

lelL "" 3 is an optimum choice for a good reflector obtained with a quite short

Bragg reflector length. We would like to point out that a periodic stratified

medium is not actually equivalent to a mirror with the reflectivity given by

(1.60b) but with a free space propagation length followed by the mirror; the

length of this free space propagation region is given by the length the field

penetrates inside the reflector and depends on the difference in refractive index

between the layers in the periodic structure.

The power transmission coefficient T follows from the power conservation law

as T = 1- R. Since the transmitted power is always less than the incident power

(in media with no net gain) and can be controlled by modifying the frequency

(Aß) ofthe incident electromagnetic field, a Bragg reflector acts as a transmission

amplitude modulator. A periodic stratified medium can also act as a phase

modulator if, unlike the case of the Bragg reflector, the direction of stratification

is perpendicular to that of light propagation (Stegeman and Hall 1990). These

applications ofthe Bragg reflector will be discussed in more detail in Chap. 3.

From formula (1.60b) it follows that a uniform Bragg reflector (grating) has a

reflection spectrum with large sidelobes. In the applications where it is used as a

pass-band filter, it is often desirable to suppress the sidelobes. One way of doing

this is to modulate the coupling efficiency by modulating the grating depth or,

simpler, by using nonperiodic sampled gratings (Shibata et al. 1994). A sampled

grating consists of two sections: the grating region and the uniform, non

modulated region, the ratio between their lengths varying along the waveguide. In

this case the change in the refractive index profile for a ID grating can be

expressed as

An2(x,z) = N(x,z) cos(2JlZ / A)


the grating period A determining the Bragg wavelength and the Fourier

component N determining the shape of the reflection spectrum. More precisely,

the equivalent coupling efficiency between the forward and backward propagating

modes depends on z as

where Lo is the unit length.

To obtain the desired shape of the reflection spectrum N(x,z) must be

computed and then the grating must be fabricated accordingly. To suppress the

sidelobes from 3 dB as in a uniform grating to 10 dB, for example, a sampled

grating is needed with a ratio of the grating to uniform region lengths decreasing

symmetrically in an alm ost parabolical shape towards the edges of the reflecting

structure. Such a structure can also be employed for controlling the field

distribution inside a laser cavity in order to avoid the spatial hole buming effect.

The bandwidth L1 v of a Bragg filter, i.e. the frequency interval in which the

reflection coefficient varies up to a given fraction of its maximum value (usually

to 0.707 of its maximum value) is proportional to the variation of the refractive

index in the periodic structure: L1 v! v::::< L1n! n in the L1n« n approximation.

Filters with broad bandwidths are used in some linear applications as rejection

filters for ultrashort pulses, of the ps and fs scale, or in nonlinear applications for

all-optical switching devices. In waveguides microfabricated in AIGaAs,

broadband Bragg reflectors can be realized by fabricating deep gratings in the

guiding layer (etching the guiding layer) so that a large difference in the refractive

index between AlGaAs and air, of L1n! n = 0.6%, is created. For a depth of the

etching of 300 nm and A = 250 nm, the bandwidth of the filter is 15 nm, centered

around A = 1.6 flm (Espindola et al. 1996). This filter was used for the rejection of

150 fs pulses and had a minimum transmission of 2-5% due to leakage in higher

order modes, not rejected by the Bragg filter wh ich was designed to reject only the

lowest order mode ofthe waveguide.

The expression for the reflection coefficient of a Bragg reflector obtained

above is valid for TE modes; for TM modes the reflection coefficient is different,

as can be inferred for example from the matrix treatment of mode propagation in a

stratified medium (see Sect. 1.2). So, Bragg reflectors are polarization sensitive. In

applications where polarization in sensitive Bragg reflectors are needed, one can

use the structure proposed by Huang et al. (1997). It consists of two periodic

structures separated by a uniform, phase-shift section of length Lp . Each of the

periodic sections has its period A formed from a grating section of length Lg and


period Ag and a uniform, phase-shift section of length Ls ' These periodic

inserted phase-shift sections modulate the grating and generate a transmission

spectrum with multiple peaks corresponding to the different spatial harmonics of

the Fourier series of the index perturbation along the structure. Supposing that

only two spatial harmonics are dominant, m = Lg / Ag and m + I, a polarization

independent reflector can be fabricated provided that the resonant condition is

simultaneously satisfied for the TE and TM modes, i.e. provided that

2nTE / A-rn = (m+ 1) / (m+ s)Ag and2nTM / A..m = m / (m+ s)Ag where s = Ls / Ag

and nTE' nTM are the effective refractive indices of the TE and TM modes at the

wavelengths A-rn, A-rM' For a common stopband of the Bragg reflector for both

TE and TM modes with A-rn = hrM = A,B the above conditions give the design

parameters:

where B is the normalized modal birefringence B = (nTE - nTM) / (nTE + nTM)' A

true polarization-independent Bragg reflector is obtained if m is an integer;

otherwise the centers of the stopbands for the two polarizations are off-set.

However, the polarization independent reflector cannot be operated at the center

of the common stopband but at a shifted wavelength. At this operating wavelength

the transmission spectra for the two polarizations are the same, although the

stopbands of the TE and TM modes are shifted in opposite directions. By a proper

design the operating wavelength can be tuned in a range limited by the common

stopbands of the TE and TM modes.

A conventional Bragg reflector used as a mirror usually consists of altemating

layers with high and low refractive indices, the optical lengths of the layers being

equal to A, /4. The CL parameter which determines the reflection coefficient is in

this case a function of the number of pairs and the difference between the

refractive indices in the altemating layers. The phase difference between the rays

reflected from the interfaces of a layer with refractive index n and length I, for a

normal incidence upon the structure, must be

- exp( - 2ikon/) = - exp( - 2ikon( A, /4n)) = - exp( - in)

(the light reflected within the high index layer does not suffer any phase shift at

reflection, whereas that reflected within the low index layer undergoes a phase

shift of 180°). Thus, the reflected rays from the interfaces of one layer are in

phase (Murtaza et al. 1995a). The reflectivity is a maximum for those n and I

values for which the real part of the phase factor is maximized, i.e. for

- co~ 2konl) = 1, from which one recovers the condition I = A, /4n. In these


symmetrie Bragg refleetors the thiekness of eaeh layer must be preeisely given

and, in some materials, espeeially in strained ones, the thiekness of the layer ean

be larger than the eritieal thiekness for whieh no disloeations appear. In sueh

materials it is possible to fabricate asymmetrie Bragg refleetors in whieh the

refleetions from every seeond interface are in phase, the intermediate refleetions

being slightly out of phase, with a phase deviation dependent on the optieal

thiekness deviation from the required ..1., / 4 value. In these refleeting struetures

the peak refleetivity and the bandwidth are sm aller than for symmetrie Bragg

refleetors with the same number of periods.

Asymmetrie or asymmetrie Bragg refleetor has peak refleetivity at a eertain

wavelength; Murtaza et al. (1995a) showed that it is possible to have Bragg

refleetors with peak refleetivities at 2N wavelengths for modulated Bragg

struetures. For example, to obtain a dual-wavelength symmetrie Bragg mirror with

peak refleetivities at two wavelengths, Al and ..1.,2' the expression

- (eos( 2k1l) + eos( 2k21)) = -2 eos( 2kl) eos( 2L'lkl)

must be maximized at the wavelengths ..1.,] and ~; here k], k2 are the

eorresponding waveveetors inside the medium with refraetive index n and

k = (k] + k2 ) /2, L'lk = (k2 - k1) /2. From this eondition it follows that the phase

faetor for a Bragg mirror should be modulated by another faetor with a smaller

spatial frequeney L'lk; if this modulation is done with a square instead of a eosine

funetion, ..1., /4 additional layers should be inserted at appropriate positions to

shift the refleeted phase by Jr. The refleeting strueture is arepetition of abasie unit

formed from a Bragg reflector with a Bragg wavelength ..1., = 2..1.,1 ~ / (..1.,1 + ~) and

L'lA / ..1., - 1 periods with an additional ..1., / 4 layer added sueh that the overall

periodieity is (L'lA/A)..1.,/4, with L'lA=2Al~/(~-Al)' In a similar manner

asymmetrie dual-wavelength Bragg refleetors and higher order Bragg refleetors

ean be designed. In partieular, for a four-wavelength peak mirror, a phase faetor

of the form eos( 2kl) eos( 2L'lk] I) eos( 2L'lk21) must be maximized, with L'lk], L'lk2

the spatial frequeneies of the modulating terms. In the hypothesis of

k > L'lk] > L'lk2 , the four wavelengths for whieh the refleetion eoeffieient has a

peak are now k + L'lk] + L'lk2 , k + L'lk] - L'lk2 , k - MI + L'lk2 , and k - M] - L'lk2 •

If the differenee between the refraetive indices of the alternating Iayers whieh

form the Bragg refleetor is suffieiently high so that L'ln ean no longer be

eonsidered to eause only a perturbation, an exaet analysis of the field propagation

must replaee the approximation derived here for low L'ln. A treatment of this

situation ean be found in the work of Matusehek et al. (1997).


1.4.3 Distributed Feedback Lasers

and Lasers with Distributed Bragg Reflectors

The theory in Sect. 1.4.2 can be extended to the case where the perturbed

dielectric medium has a net gain for the propagating electromagnetic field, i.e.

when the medium is amplifying the incident radiation. If we denote the gain

coefficient by g then (1.56) has to be replaced by

(1.62)

The solution of this system of coupled equations can be expressed in a matrix

form where the matrix is identical to that in (1.59) if Aß is replaced by Aß + ig.

Supposing again an electromagnetic field incident on the z == 0 plane with an

amplitude b+(O) = bo, and using the condition that at the output ofthe medium of

length L, [( L) == 0, one obtains inside the stopbands that

- . C sinh(rL )bo b_(O) == -I rcosh(rL) + i(Aß+ ig)sinh(rL) , (1.63a)

b (L) == rbo + r cosh(rL) + i(Aß + ig) sinh(rL) ,

(1.63b)

where r == ~ - (Aß + ig) 2 + Icf. Because r is now complex, a quite different

situation appears if

r cosh(rL) = (g - iLlß) sinh(rL). (1.64)

In this case the reflected field h:. ( 0) and the transmitted field b+ ( L) can be

different from zero even if bo == 0 - the device is working as an oscillator and

(1.64) describes the threshold condition for laser operation. The light wavelengths

for wh ich the condition (1.64) is satisfied are called the oscillating or lasing

modes of the structure. The periodic perturbation couples the modes in an

amplifying medium (creates a feedback between them) so that they become

coherent if condition (1.64) is met. In this way the periodic perturbation replaces

the mirrors in usual lasers. If the periodic perturbation is imposed over the entire

length of the amplifying medium, the device is called a distributed feedback

(DFB) laser while if it is imposed only on regions surrounding the active medium

the device is called a distributed Bragg reflector (DBR) laser (see Fig. 1.7). In the

last case the periodic perturbed regions are not amplifying (are passive or even


absorbing regions with a negative g) and act as mirrors which surround the active

region with a positive gain. The lasing condition for DBR lasers is not given by

(1.64) but by the condition that the field intensity after a round-trip passing

through the structure is identical. This condition will be derived in Sect. 2.3. The

lasing modes are those for which standing wave light patterns are formed in the

structure. A more complete description of DFB and DBR lasers can be found in

Chap. 13 ofthe excellent book ofYariv (1985).

active region

a b

active region

Fig. 1.7. Schematic diagram of(a) the DBR and (b) the DFB laser

In all the expressions above the gain coefficient g should be understood as the

net gain coefficient, i.e. the difference between the gain and los ses in the media.

Moreover, in DFB lasers the above coupled-mode theory holds only for perfectly

antireflection coated structures, with no additional feedback than that provided by

the counterpropagating modes themselves.

DFB lasers are compact, have low threshold currents, narrow linewidth due to

the spectral filtering of the Bragg structure and reduced sensitivity to optical

feedback. Therefore, they are widely used in many applications; quantum weil

(QW) DFB lasers have been fabricated recently (see Sect. 2.3.5). For this reason

we will discuss in detail in this section the coupling mechanisms and characteristic

features of DFB lasers, leaving for Sect. 2.3.5 only the presentation of the DFB

lasers with QWs as active materials. There are two coupling mechanisms in DFB

lasers, both described by the coupled mode theory presented above: the gain

coupling and index coupling mechanisms.

In gain-coupled DFB lasers the net optical gain is periodically modulated (!ln

above refers to the complex part ofthe refractive index), the frequency-dependent

feedback between the forward and backward traveling waves fixing the lasing

wavelength at the modulation period. These lasers are basically single-mode light

sources with good side mode suppression between the lasing mode and other

possible modes; they are more sensitive to external feedback than strong index

coupled lasers (with CL > 3) but less sensitive than weakly index coupled lasers

(with CL < 0.5). For non-antireflection coated, gain-coupled DFB lasers, the yield

38 l. Basic Concepts ofOptoelectronic Devices

is better than that for antireflection coated devices with the same gain coupling

coefficient; moreover, the feedback sensitivity and spatial hole burning effects are

decreased in non-antireflection coated devices with a low reflection coefficient

(as-cleaved for example).

In index-coupled DFB lasers there is no modulation of the optical gain and

therefore the laser tends to emit in two degenerate modes located at the edges of

the Bragg reflection band; An is in this case the real part of the refractive index.

To avoid the emission degeneracy, a quarter-wave shift, a width-modulated stripe

or a taper is introduced in the central region of the active medium. These solutions

have the disadvantage of spatial hole burning which cause in its turn multimode

operation at injection currents not far from threshold. However, the spatial hole

burning is beneficial in that it lowers the threshold current in strongly coupled

structures, due to the induced nonlinearities. The spatial hole burning is due to the

nonuniform photon density, carrier distribution (and thus the gain coefficient, the

refractive index profile of the active medium and propagation coefficient) along

the cavity length caused by the periodic coupling structure.

To increase the modulation bandwidth (the frequency excitation range for a

-3 dB (0.707 from the maximum value) response - see Sect. 2.3) well beyond

that of usual DFB lasers, which is limited by the electron-photon resonance, a

two-contact, push-pull modulated DFB laser can be used. The modulation is done

by changing the currents in the two electrodes such that their sum remains

constant. In this way the optical power within the laser is longitudinally displaced

towards the highly pumped part while the carrier density changes during

switching are smaller than in devices with a single contact. At low powers the

modulation bandwidth is limited by a cut-off frequency which increases linearly

with the power and can attain values between 20 GHz and 100 GHz. This cut-off

frequency is related to the time needed for the carrier and photons to reach a new

equilibrium state when the excitation currents switch in the two contacts; it is

smaller the larger the change in the gain coefficient due to the change in carrier

concentration (the larger the differential gain - see Sect. 2.3) and is inversely

proportional to the optical power. This behavior can be explained by a small

signal model as that developed for example by Marcenac et al. (1994). The cut-off

frequency is upper limited by the resonance frequency corresponding to the

spacing between the lasing mode and its closest side mode. To increase this

resonance frequency shorter lasers with lower coupling coefficients should be

used; a trade-off of the coupling efficiency between low power threshold and

large modulation bandwidth is however desired. For a second order grating, anti

reflection coated structure the modulation bandwidth can be more than doubled


(for example from 21 GHz for a single contact laser to 49 GHz for one with two

contacts for an injection current of200 mA).

1.4.3.1 Index-Coupled Distributed Feedback Lasers

with Reduced Spatial Hole Burning

In order to reduce spatial hole burning the electric field distribution along the

cavity length has to be flattened, without disturbing the periodic mode coupling.

Theoretical analysis has established that for index-coupled DFB lasers there is an

optimum coupling factor (CL == 1.25) which assures a trade-off between hole

buming and stable, single-mode operation even at high output powers. Some

solutions for spatial hole buming decrease in DFB lasers are: quarter wavelength

shifted (or Ir / 2 phase shifted) lasers (Fig. 1.8a), DFB lasers with nonuniform

stripe width as wide (Fig. 1.8b) or narrow (Fig. 1.8c) stripe geometries and

double-tapered DFB lasers (Fig. 1.8d).

"'I a b c

Fig. 1.8. (a) Quarter wavelength shifted, (b) wide stripe, (c) narrow stripe and

(d) double tapered DFB lasers

A comparison between the first three solutions for hole buming decrease can be

found in Correc (1994). He has shown that the phase-shifted lasers have a high

yield in the single-mode operation regime. In strong phase-shifted coupled

devices (CL == 3) the photon density is higher in the center of the phase-shifted

structure, the number of carriers consumed there is higher at currents higher than

the threshold and the output power-injection current characteristic is superlinear.

As a consequence of the fact that the threshold gain increases with the injection

current, the spectral position of the stopband moves towards shorter wavelengths

and the two modes become nondegenerate. The one which lases first is that

situated at the longer wavelength side of the stopband, the other lasing also at


higher injection currents; the net result is a slight decrease ofthe lasing mode with

an increase in the injection current. In low coupled phase-shifted structures (CL ;:::;

0.5) the situation is opposite: the photon density is higher at the facets than in the

center, due to insufficient grating confinement, and the first lasing mode is that at

longer wavelengths inside the stopband.

To flatten the photon density variation along the cavity due to the localized

phase shift of the grating, this region can be spread over a larger distance Ls by

widening or narrowing the waveguide width by modulating the stripe width. To

obtain a phase shift of tr / 2 the propagation constant of the phase shifter region

must differ from that in the uniform region by I1ß = 2trl1neff / A =(2m + l)tr / 2Ls

where I1neff is the difference between the effective refractive indices of the phase

shifter and uniform regions and m is an integer. In wide phase shifter structures

(Fig. I.8b) with I1neff > 0 the lasing mode at threshold is located on the left side

of the stopband (I1ßL > 0); with increasing current densities the stopband moves

towards lower wavelengths (the carrier density increases), the lasing mode moves

towards its right side and the other mode begins lasing also at higher injection

levels. In narrower phase shifters with I1neff < 0 the lasing mode is located at

threshold on the right side of the stopband and moves to the right, away from the

Bragg wavelength as the injection current increases. Comparing the intensity of

the lasing mode, it was found that the most sensitive structure is the wide phase

shifter whereas the less sensitive one is the narrow phase shifter; the wide phase

shifter is also more sensitive for hole burning than the narrow phase shifter, for

the same photon distribution at threshold.

Double-tapered DFBs (Fig. I.8d) can also be used to reduce the spatial hole

burning effects and to maintain a stable, single-mode operation in structures with

CL > 2. Due to their special geometry the optical power density at the facets is

minimized since the optical mode has a greater spatial extent at the facets

compared to the central, phase adjusting region, and so the double-tapered DFB

laser can be used at higher output powers than the other configurations. Yu (1997)

have shown that double-tapered DFBs also have a better dynamic response under

electrical modulation. In these devices the index and gain profiles, the propagation

constant and its deviation from the Bragg constant and any other characteristics

are dependent on the waveguide dimensions. Numerical simulations have shown

that the relaxation frequency (the frequency of maximum small-signal response to

a modulated excitation - see Sect. 2.3) generally increases with the stripe width,

the CL product and the lateral carrier diffusion constant, up to a saturation value.

So, high relaxation frequencies and single-lateral mode operation can be obtained

in double-tapered DFBs by widening the tapered regions whereas in other DFB


configurations the increase of the relaxation frequency would imply the increase

ofthe stripe width. This will however also excite the higher order modes.

The introduction of a modulated grating pitch (period) can also introduce a

A /4 phase shift over a distributed length Lc if the period of the center section

Ac (over which the corrugation pitch is modulated) differs from that of the outer

sections, A, such that Ac = A( I ± A / 2 LJ However, unlike in the localized A /4

phase shifter where the lasing wavelength coincides with the Bragg wavelength,

here, as for the modulated stripe lasers, the emission wavelength does not coincide

with AB. The detuning of the lasing wavenumber ß = 2nneff / A from the Bragg

wavenumber ßB = TC / A, !':..ß = ß - ßB' is negative if A > A and positive if the

inverse inequality occurs. A small negative detuning causes laser oscillations at

the upper frequency of the band edge, the lasing mode approaching the Bragg

wavelength as the detuning increases. The transition from positive to negative

values of the detuning corresponds to a mode jump from the high frequency to the

low frequency edge of the stopband. Zero Bragg detuning can be obtained for a

given corrugation pitch at a fixed Lc value, but this does not correspond to the

condition of a distributed A /4 phase shift. However, for a fixed length of the

DFB laser and of the detuning region, an optimum Bragg detuning can be found

to maximize the output power for given sidemode suppression. As a rule of

thumb, for stable single-mode operation up to high powers the inequality

1 :s; TC( I/Ac - I / A) / C :s; 2 should be satisfied. When the above defined quantity

is less than one and approach es zero, an unstable region associated with

bifurcations appears at high optical powers. A modulated corrugation pitch can

also be produced by S-bending a waveguide DFB laser (see Olesen et al. (1995)

and references therein). The tilt of the stripe with respect to the grating in the

center of an S-bent structure creates a distributed phase shift region. A more

complex version of the modulated grating pitch laser is the distributed coupling

structure, consisting of a four-section laser with identical inner and outer sections,

of lengths Li and Lo respectively and grating periods Ai and Ao . At the center

of the structure a A / 4 phase shifter is introduced. The single mode operation

range is greatly enhanced and the laser behavior under transient conditions is

much improved since the carrier density longitudinal profile remains clamped

(Fessant 1997).

1.4.3.2 Mixed Coupled Distributed Feedback Lasers

For better performances the index- and gain-coupling mechanisms can be

combined in the so-called mixed coupled lasers. Lowery and Novak (1994) have

studied the performances of perfectly antireflection coated, mixed coupled DFB


lasers as a function of r - the ratio between the index coupling to the gain

coupling coefficients; r is positive for in-phase index-to-gain coupling and can be

negative for antiphase index-to-gain coupling. In these structures the maximum

modulation bandwidth for example can increase up to 11 GHz for an antiphase

coupled laser with r = -153 from 7 GHz for a pure index-coupled laser or 6.5

GHz for a pure gain-coupled laser; in-phase coupling decreases the modulation

bandwidth by up to 3 GHz for r = 1.73. It was shown that the threshold current

(minimum injection current needed to obtain a positive g value - see Sect. 2.3)

decreases with increasing modulus of the r parameter. The lasing mode has the

same side mode suppression ratio as in pure gain-coupled structures but its

wavelength can be shifted from the center of the stopband by varying r: to the

blue end of the stopband for anti-phase coupling and to the red end for in-phase

coupling. The resonant frequency ofthe anti-phased coupled lasers (7.3 GHz) was

found to be higher than for in-phase coupled structures (2.1 GHz) and higher than

that of index-coupled lasers (= 4 GHz) or gain coupled lasers (5.6 GHz). In-phase

coupled lasers have a larger chirp (nonlinear phase response at modulating

excitations) due to spatial hole burning than pure gain-coupled lasers, whereas

antiphase coupled structures have smaller chirps. The latter structures also have

smaller noises than pure gain coupled devices and are more tolerant to external

feedback than index-coupled structures.

In even more complex DFB laser structures the index- and gain-coupling

periodic structures can be dephased with an arbitrary phase !':!.rjJ, i.e. the index and

gain variation (supposedly sinusoidal) along the z axis are described by

n(z)=n+!':!.nco~2JlZ/A+rjJ)and g(z) = g+!':!.gcos(2JlZ/A+rjJ+!':!.rjJ). !':!.rjJ=O or

lf correspond to the in-phase and antiphase coupled grating structures,

respectively. By varying !':!.rjJ the overlap between the light standing wave pattern

formed due to the index coupling and the gain coupling structure can be modified

such that the depth of the standing pattern is decreased and the optical field

uniformity is improved, thus decreasing spatial hole burning effects. The output

light is spectrally narrower and its possible frequency range is larger than in in

phased coupled structures. Numerical simulations (Kwon 1996) have shown that

increasing !':!.rjJ from 0 to lf results in an increase of the effective coupling

coefficient between the counterpropagating modes when !':!.ßL < 0 and to a

decrease of the effective coupling coefficient if !':!.ßL > O. The single mode yield

dependence on !':!.rjJ is determined by the coatings. For example, for antireflection

coated structures it is a maximum for !':!.rjJ = 0 or lf, decreases for !':!.rjJ = lf / 4 or

3lf / 4 and is even less for !':!.rjJ = TC / 2. On the contrary, for laser structures with

one high-reflection facet and one antireflection coated (HR-AR lasers), the single

mode yield is a maximum for !':!.rjJ = TC / 4 and 3TC / 4 and decreases in the order


I':!.fjJ = 7r /2, I':!.fjJ = 0 or 7r, I':!.fjJ = 57r / 4 or 77r / 4 and I':!.fjJ = 37r / 2. Moreover, the

yield curves for I':!.fjJ = 0 are more sensitive than those for I':!.fjJ = 7r / 4 at an optical

field nonuniformity inside the structure. This better field uniformity in HR-AR

lasers with I':!.fjJ = 7r / 4 explains the 2.4 times higher yield than for I':!.fjJ = 0 lasers.

As the reflectivity of the anti-reflection coating increases, however, the yield of

the I':!.fjJ = 7r / 4 laser decreases whereas that for I':!.fjJ = 0 increases up to a point

where they are equal. The phase difference between the index and the gain

gratings also influences the reflection coefficient of the periodic structure: it is a

minimum for I':!.fjJ = 7r / 2 where the feedback from the index and gain gratings are

out-of-phase and cancel each other, and increases with approximately an order of

magnitude if moving away, on either side, from the I':!.fjJ = 7r / 2 condition.

Actually, by modifying I':!.fjJ while keeping constant the overall coupling

coefficient it is possible to move the peak of the reflection coefficient at different

positions relative to the Bragg frequency (Cardimona et al. 1995). The lasing

frequency of such a dephased mixed coupled DFB laser could also be tuned if I':!.fjJ

could be varied in real time by applying an extern al electric field.

In DFB lasers with periodic modulation of either the refractive index profile or

the gain, and in which the phase condition I':!.ß = 0 is not realized for the first

order (q = I) Fourier term, the emitted wavelength can be derived from (1.56b)

for ß2 = -ßI = koneff as A = 2Aneff / q where neff is the effective refractive

index of the structure. So, the emission wavelength can be tuned by varying the

period of the grating. However, when the grating period is sm all the wavelength

variation can be too coarse. For example, for a ZnSe based grating with A = 90

nm, used to obtain blue laser radiation, a variation of the grating period of 1 nm

results in a change ofthe energy ofthe emitted light of25 meV. In order to obtain

a much smoother variation of the output laser radiation wavelength periodic

modulated gratings should be used (Eisert et al. 1996). Let us suppose for

example that the grating consists of periodic modules with an average period

Aav = (qIAI + q2A2) / (ql + q2) formed by a succession of one or more periods of

lengths AI and A2 with ql' q2 integer numbers. Since the effect of this

composed grating on A averages over Aav , by varying ql and q2 it should be

possible to tune the emission wavelength for both refractive index profile and gain

modulated structures. In particular, for gain modulated DFB lasers such a

composed grating can be realized by a selective implantation with a Ga focused

ion beam. For AI = 87.7 nm, A2 = 91.2 nm, and ql and q2 related by the

condition that ql + q2 = 20, the wavelength of the emitted light can be tuned in

steps ofO.7 nm, corresponding to a step ofthe average period of0.14 nm; or, with

AI = 92 nm, A2 = 96 nm, with ql' q2 independently varying between 1 and 3, a


fine tuning of A between 506.8 nm and 482.8 nm has been obtained with the same

variation step of the gratings of I nm.

DFB lasers with longitudinal variation of the coupling constant along the

propagation axis have recently shown remarkable properties (Boucher et a1.

1997). In these mixed-coupled tapered gratings both the index- and gain-coupling

constants depend on Z as q(z)exp(iqJJ and Cg(z)exp(iqJg), respectively, giving

rise to qualitatively new effects. For example, for a pure index-coupled DFB

structure with Cj (z) = Co exp( ajz) the exponential dependent coupling coefficient

introduces an asymmetry between the forward and backward propagating waves.

The first experiences an average gain of g - aj /2 whereas the average gain for

the second wave is g + aj /2, g being the gain of the active medium. If the

gradient of the coupling coefficient is adequately changed, the forward

propagating wave is left unchanged whereas the backward propagating wave can

be enhanced or attenuated. In this case the device behaves in an apparently

puzzling manner: it has equal transmission coefficients for left-to-right and right

to-left propagating waves but different reflectivities. An explanation can be given

by taking into account that close to the Bragg condition the wave incident on the

structure by the weaker coupling side will penetrate deeper than a wave incident

from the opposite direction. The emitted intensities from the two facets of a DFB

laser with such a tapered grating are different and dependent on the mode,

although the mode location is the same as in periodic DFB structures. For

complex-coupled gratings with qJj-qJg=Jr/2, Cj(z)=Cocosh(az), Cg(z)= Co sinh( az), the device is symmetric with respect to the two propagating waves

and the average gain is shifted by a. However, the reflection coefficients for the

two directions are different, due to the dephasing of the index and gain grating.

For mixed coupled gratings with qJj-qJg=O, Cj(z)=Coco~az), Cg(Z) =

Co sin( az) the structure is again asymmetric, but a is equivalent to a shift of the

detuning. The reflectivity curve is nonsymmetrically shifted for the two propaga

ting directions; it is possible to use it as a one-way-only dichroic reflective filter.

1.4.3.3 Tunable Distributed Feedback

and Distributed Bragg Reflector Lasers

In many applications tunable light sources are needed. One of the simplest ways

to tune the wavelength of a DFB laser is to vary the ridge width of the laser,

modifying in this way, in a controllable manner, the modal index of the

waveguide structure. Altematively, one can tilt the stripe with respect to the

grating at the expense of deteriorating the coupling coefficient with increasing tilt

angle; the threshold current as weil as the stop bandwidth are strongly varying


functions of the tilt angle. The coupling coefficient reduces by a factor

co~](x sin 0) where K is the grating wavevector, 0 the tilt angle and x the

transverse direction. For example, it was found to decrease from 120 cm-] to

20 cm-1 as the ridge was tilted from 0° to 10°. At the same time the stopband is

red-shifted, narrowed and the power in the lasing mode decreases. The threshold

current density also increases with the tilt angle as J th = J 0 exp( gth ( 0) ! go) where gth is the threshold gain coefficient and go a constant. Oue to the increase

of the threshold current with tilt angle, the operating temperature of the laser

increases, and the ga in peak moves toward the longer wavelength side. Practical

wavelength tuning ranges with this method do not however exceed 10 nm - a

quite modest result (Sarangan et al. 1996).

Another simple method of tuning OFB lasers is to use multielectrode

structures. Usually, the upper electrode is divided into three parts, the outer ones

being electrically connected. In phase-shifted OFB lasers the central electrode is

grown upon the phase-shifting region: by varying the current in the central

electrode the propagation constant, the phase-shift and the emission wavelength

can be modified. An increase of the current in the phase-shifting region leads to a

decrease in the lasing wavelength, without observable degradation in the side

mode suppression ratio. The tuning range for optimized structures is alm ost as

large as the stopband. Even for uniform OFB lasers with electrodes of equal

length, the injection currents select the emission wavelength. If the current in the

central electrode 1 c is larger than that in the lateral electrodes, 1], the right mode

in the stopband lases, whereas the left mode emits for the opposite case.

Moreover, when 1 c > 1] the emission wavelength can be tuned: it increases with

the current in the central electrode, decreases with the current in the lateral

electrodes and the rate of change decreases with 1 c. The tunability range

decreases with laser length and increases when the light confinement factor in the

active region increases. However, the tunability range is extremely smalI: about

6 Ä. The multielectrode solution is thus not suitable for wide tunable sources - it

is more useful for reducing the hole burning effects when operated in the 1 c > 1]

regime since then the current injection profile is similar to that of the photons in

the cavity (Correc 1996).

Tunable distributed Bragg reflector lasers can be realized in a three-region

configuration. The laser consists of the active region, the passive phase control

region and the OBR, the light being emitted through the cleaved or coated facet of

the active region. The three regions are separately electrically connected, the

wavelength being switched by modifying the currents in the phase control and

OBR regions. The wavelength tuning is due to a modification of the effective

refractive index when the injection current is modified inside the different regions

46 1. Basic Concepts of Optoe1ectronic Devices

of the cavity. The tuning can be discrete (different longitudinal modes of the laser

cavity are selected) when only the DBR current I d is changed, or continuous

when the lasing wavelength is tuned within the interval between different

longitudinal cavity modes. In the last case both the phase control current I p and

the DBR current are changed according to the relation 1p = kId + 10 where the

offset current 10 adjusts the phase gap between the lasing wavelength and the

Bragg wavelength. In this case the lasing wavelength changes continuously with

the Bragg wavelength, both decreasing as the DBR current increases; however,

the cavity loss under continuous tuning is larger due to the additional loss

introduced by the phase control current. The tuning range is limited by the

generated heat which increases the refractive index and counteracts the refractive

index change produced by the injected currents. The tuning range is limited to a

few nm, but can be switched very fast. Dynamic continuous wavelength tuning is

realized if the wavelength change of the cavity mode equals the change of the

Bragg wavelength. Under fast tuning conditions power oscillations occur when

the current is tumed on and mode hopping occurs, the wavelength rise-up and

fall-down times being different due to the different carrier lifetimes in the two

conditions. This behavior was confirmed by the theoretical model of Teshima

(1995). The switching delay varies between land 2 ns within a mode and

increases to 7.5 ns when a mode jump occurs; within a mode, it decreases as the

Bragg current increases (Zhang and Cartledge 1995).

A continuously tuning three electrode DBR laser with the variation of only one

current is possible when the DBR eIectrode is interdigitated (Ishii et al. 1995);

except for this, the structure of the device is similar to that described above.

However, there is now no need to tune synchronously the Bragg and the

longitudinal cavity modes. One part of the interdigital electrode, of Iength Ld! is

connected to the phase controI electrode, driven by a current I P' The other part,

with interdigital electrode length Ld2 is separately connected to a current source

I d2' WaveIength tuning is achieved by varying the current in the phase shift

region. The wavelength shift of the Bragg mode is given by

~AB / AB = (Ld ! / Ld! + Ld2 )~n / n where ~n is the refractive index change in the

Bragg region when the injection current is modified. By denoting by La' Lp ' LB the lengths of the active region, phase control region and effective length of the

Bragg region, the cavity mode shift can be approximated by

~A / A = (~n / n)[( Lp + LBLd! / (Ld! + Ld2 )) / (La + Lp + LB )]. A mode matching condition between the cavity and Bragg mode can be achieved and maintained if

the interdigitated electrode is designed such that Lp / La = Ld ! / Ld2 . Moreover,

for continuous tuning it is necessary that the spacing between supermodes, created

1.5 Solutions ofMaxwell's Equations in Nonlinear Media 47

by the periodic refractive index variation in the DBR region, be larger than the

tuning range. This imposes another condition for the interdigitated electrode, since

the supermode spacing is given by ~Ilsm = Il~ ! 2n( Ld1 + Ld2 ). For a supermode

spacing of 5.6 nm, a wavelength tuning range of 4.6 nm was obtained with an

output power variation of ll dB during tuning caused by an increased loss; the

side mode suppression ratio was better than 30 dB over the whole tuning range.

1.5 Solutions of Maxwell's Equations in Nonlinear Media

Throughout this book we will refer to nonlinear media and nonlinear effects only

in connection to the media in which the refractive index depends on the

propagating light intensity. We will not refer to the many applications, also

generally referred to as nonlinear phenomena, in which the light frequency is

modified, as in second harmonic generation or parametric amplification. Many

dielectric media become nonlinear at optical frequencies when excited at

moderate power levels, i.e. the refractive index n depends on the intensity of the

optical radiation field Iopt = ( & ! 2 )IEI2 . This is due to the dependence of the

electrical polarization vector P on the electric field E; the explicit form of this

dependence is determined by the symmetry properties of the dielectric crystal.

The most common variation is

n=no+~nlopt, (1.65a)

which is known as the Kerr-like variation of the refractive index. f...n can be

positive or negative, for focusing or defocusing media, respectively. Other

possible types of variation inc1ude

(1.65b)

for saturable nonlinear media and

(1.65c)

no can eventually be complex to inc1ude the effect of gain or loss. The dependence

n = f( Iopt ) changes qualitatively the solutions of Maxwell's equations. For

example, the optical modes are no longer plane waves even for a translationally

invariant waveguide, the stopbands (Little 1994) and modes of periodic structures

(Saini and Sharma 1996b) depend on the light intensity, the transfer characteristic

of directional couplers changes, and optical bistability in directional couplers and


lasers appears. However, these effects can only be accounted for in a numerical

analysis, no analytic formulae being in general available.

To study the effect of nonlinearities on the electromagnetic field propagation

we will consider a Kerr-like dependence ofthe refractive index in a waveguide. In

this case some analytical results can be obtained, which can help the

understanding ofthe nonlinear effects. We will also make some assumptions:

i) the nonlinear term of the refractive index is considered as aperturbation of

the linear one I~nl« no, ii) the field polarization is maintained during the propagation, i.e. the tensorial

character of ~n is neglected,

iii) the electromagnetic field is quasi-monochromatic, i.e. its frequency

spectrum centered around lüo has a width ~lü such that ~lü« lüo. So, it is no

longer an harmonic field but an optical pulse.

To avoid unnecessary complications we will solve the equation satisfied by the

electric field (the first equation in (1.3)) in a nonlinear step refractive index

profile, i.e. we are searching for solutions ofthe following equation:

V2E + k~n2(lü)E = O. (1.66)

We look for separable solutions of (1.66) for each component of the electric

field; for example the solution for Ez is supposed to have the form

(1.67)

with G a slow dependent function of z. ßo is the value of the propagation

constant for lü = lüo. With this decomposition (1.66) is equivalent to:

{V; F + (n 2 k5 -~ )F = 0

. IJG 2 2 21ßO Tz + (ßNL - ßo )G = 0, (1.68)

where PNL is introduced as a constant for now. The first equation of (1.68) is

identical to the equation satisfied by the electrical field in a linear waveguide with

a step-like refraction index profile. Therefore, PNL has the meaning of a

propagation constant in such a waveguide and is determined from the boundary

conditions imposed on the electric field. Since n2 is only aperturbation of the

index of the linear waveguide, n2 = (no + ~nlopt)2 == n~ + 2n~nlopt, we will

suppose that it does not affect F, but it pro duces only a slight modification of the

propagation constant from its value ß if no nonlinearity is present. So

1.5 Solutions of Maxwell's Equations in Nonlinear Media 49

where

(1.70)

Thus, finally, the last equation in (1.68) can be rewritten as

If ß( w) is replaced by its Taylor series, and we neglect the terms of order

higher than two in this series, we arrive at

(1.72)

where

is the inverse of the group velocity of the electromagnetic pulse (ßI = I/v g) and

is responsable for the spatial dispersion of the pulse and is therefore called the

dispersion coefficient.

Equation (1.72) can be transformed into a differential equation in time for the

Fourier transform A(t) of G, defined as:

A( z, t) = f G(z, w - wo) exp(i( w - wo)t )dw.

The result is

oA -ß oA +~ß ~ A = i IAI2 A oz I ot 2 2 ot2 r ,

where

(1.73)

(1.74)


y = (Linko&! 2) JIFI4 dA / JIFI2 dA A." A",

is the non linear coefficient. The dispersion and nonlinear coefficients which

appear in (1.74) depend on the width and the peak power of the electromagnetic

pulse, respectively. In order to compare their effects on the electromagnetic pulse

propagation, normalized time and amplitude coordinates are introduced, and

moreover, a system of reference is chosen which moves together with the pulse

such that t is replaced by T = t - z / v g = t - ß 1 z. The normalized form of (1.74) is

(1.75)

where r = T /1'0 with To the width of the pulse and U(z, r) = A(z, r) / Fo with

Po the peak power. The discussion of the effects of nonlinearity and dispersion on

the propagation of an electromagnetic pulse on a distance L can now be done in

terms of two characteristic lengths: LNL = 1/ yPo, the nonlinear length and

L D = 1(} /Iß21, the dispersion length. If

i) L« LNL , L« Lo , neither dispersion nor nonlinearity plays a role in the

propagation and the solution of (1.54) is U( L, r) = U( 0, r). The amplitude of the

pulse remains constant and the waveguide has only a passive role in the

transmission ofthe pulse. For y= 20 W-1km-1, IßzI = 20 pS2 / km, /l = l.55 11m

this regime occurs for large pulses 10 ~ 100 ps with small peak power Po :5:

0.1 mW

ii) L« LNL , L ~ Lo , the evolution of the pulse is govemed by dispersion.

Generally, the width of the pulse increases during propagation and its shape

changes. The only pulse wh ich does not change its shape during propagation

through a dispersive medium (at least when only the second term in the dispersion

relation is considered) is the Gaussian one. However, the width 10 of a chirped

Gaussian input pulse

U(O,T) = exp(-(l + iK)T2 /102 )

where K is a chirp constant, becomes after a propagation distance z equal to

For chirped Gaussian pulses it is possible that for a propagation length z < zmin = IKILo / 2( 1 + K 2 ) the width of the pulse decreases if ß2 K < O. For pure Gaussian


pulses with K = 0, the propagation in a dispersive medium always implies an

increase of its width. This situation is undesirable in applications, in particular in

signal transmission applications, since the pulse dispersion limits the information

speed. F or the same values of y, 1ß21 and A, as above and for Ta ~ 1 ps, this

dispersive propagation regime occurs for small optical powers Po «1 W.

iii) L« LD , L:::: LNL , the nonlinear effect dominates and the phenomenon of

self-phase modulation occurs. The solution of the nonlinear propagation equation

has in this case the form

u( z, T) = U( 0, T) exp( ilU( 0, TW Zeff ! LNL )

where Zeff is the effective propagation length. Zeff = Z for a nonabsorbing

medium and Zeff = (1 - exp( - az)) ! a if the absorption is taken into account,

where a is the absorption coefficient. This expression indicates that the shape of

the pulse, determined by IU(z, T)I, is constant during propagation, but the

spectrum of the pulse, defined as

S(m)=IU(z,m)12 = 4~2Ifu(z,T)exP(-imT)dTI2 (1.76)

where U(z,m) is the Fourier transform ofthe pulse, is drastically modified. For

example, the spectrum of an initial Gaussian pulse, itself a well-behaved, one

peaked Gaussian function, becomes multi-peaked at propagation, the number M of

peaks increasing linearly with zeff! LNL . This behavior is motivated by the fact

that at propagation new frequency components are continuously generated by the

nonlinear phase; these frequency components

are increasingly far away from the central frequency of the incident pulse. The

peaks in the pulse spectrum appear as a result of the constructive or destruetive

interference of the same frequency components whieh appear at different T values

in the pulse. In a Gaussian pulse for example, it is easy to show that the same

frequency component m is generated at two different, symmetrie T values in the

pulse, the number of peaks in the spectrum being given in this case by zeff! LNL =

(M - I! 2)lT. This self-phase modulation regime is eharacteristie for large pulses

(To > 100 ps) with large peak powers (Po:::: 1 W).

iv) L:::: LD , L:::: LNL , the dispersion and nonlinearity aet simultaneously and

they can eompensate one another under certain conditions. In this regime


stationary solutions of (1. 7 5) can be found for the electromagnetic field

propagation. These solutions appear for weIl defmed light excitation conditions

and are called solitons. Their form is unchanged during propagation through the

non linear and dispersive waveguide, as optical modes in linear and nondispersive

waveguides (see Sect. 1.3). Therefore, solitons are modes for the nonlinear and

dispersive waveguide.

1.5.1 Optical Solitons

The stationary solution of (1.75), i.e. the solutions which depend on z at most

through a phase factor, are separable as U( r,z) = f( r)exp(i~(z)) with realfand ~ functions. By supposing that, analogously to the modes in linear, nondispersive

waveguides, ~ is proportional to z, i.e. ~ = Ta + ~o, the equation satisfied by f can

be put, after some algebraic calculations, in the following form:

(1.77)

where K} = 2Lo / sgn(~)LNL' K2 = 2KLo / sgn(ß2) and Ko is a constant

parameter.

If Ko = 0, fhas solutions only for negative values of ß2 (sgn(ß2) = -1), in

the anomalous dispersion regime. The solution is

(1.78)

and, correspondingly

u( r, z) = J2KLNL sech( J2KLo r) exp( iK z + i~o). (1.79)

Since from its definition the width ro = 1/ J2KLo ofthe pulse and the maximum

value of IU( r ,z )12 must be equal to 1, it follows that K = 1/ 2LNL , Lo / LNL = 1

and the propagation constant of the pulse is determined by LNL . From the three

pulse parameters, i.e. the amplitude, width and propagation constant, only one is

independent the others being determined by the parameters Lo and LNL . So,

u( r,z) = sech( r) exp(iz / 2LNL + i~o). (1.80)

1.5 Solutions ofMaxwell's Equations in Nonlinear Media 53

This solution of the electromagnetic field is called a bright soliton due to its form

(see Fig. 1.9) and has a phase independent of r, i.e. sgn( U) is constant with

respect to r(see Fig. 1.9).

IUI s (U)

0.5

t t

·4 ·2 2 4 ·4 ·2 o 2 4

Fig. 1.9. The variation ofthe pulse amplitude (left) and phase (right) of a bright

soliton as a function ofthe normalized time coordinate r

IUI gn(U)

-3 ·u 3

t

-3 -1.5 15 3

Fig. 1.10. The variation of the pulse amplitude (left) and phase (right) of a dark

soliton as a function ofthe normalized time coordinate r

If Ko *- 0, a solution for f can be found only if sgn(ß2) = +1 (normal

dispersion region) and has the form

f( r) = ~KLNL tanh( ~KLo r), (1.81 )

with the normalized complete solution far U

U( T,Z) = tanh( T) exp( iz / LNL + i~o), (1.82)

where K = 1/ LNL from the amplitude normalization condition and Lo / LNL = 1

from the pulse width normalization condition. The form of the pulse is given in


Fig. 1.10; its phase has a jump at r = ° since, unlike the solution for the bright

soliton, U( r,z) for a dark one is an odd function of r. Dark solitons are harder to

realize experimentally since it is difficult to realize the constant infmite

"background" .

The bright and dark solitons are fundamental solitons, since for them the

dispersion and nonlinear effects exactly compensate each other, i.e. for them

Lo / Lm. = 1. For Lo / LNL *" 1 the dispersion and nonlinearity are alternatively

dominant and the evolution of an incident soliton has a periodic behavior with z -

these are the higher order soliton solutions. The parameter N 2 = Lo / LNL

determines the order ofthe high er soliton. For example, for a bright N = 2 soliton,

an input pulse of the form U( 0, r) = Nsech( r) has a periodic behavior with

respect to z in a nonlinear and dispersive medium, given by (Agrawal 1989):

cosh( 3r) + 3 exp( 4i~) cosh( r) . U(~, r) = 4 exp(l~ 12),

cosh( 4r) + 4 cosh(2r) + 3 cos( 4~) (1.83)

where ~ = z / Lo . The evolution of the amplitude of a bright N = 2 order soliton

along a distance equal to ~ = 7r is shown in Fig. 1.11.

4

Fig. 1.11. The evolution of a bright N = 2 higher-order soliton

Although the higher-order solitons compress and broaden succesively due to

the alternate predominance of the non linear and dispersion effects, the initial form

of the pulse is recovered after aperiod of zp = nLo /2, for integer N values (see

Fig. 1.11). If N is not integer or if the initial pulse is not a soliton, only apart of

the incident power is recovered in the form of a soliton (fundamental or of higher


order) after the propagation through a distance of a few zp' The remaining part is

radiated.

N 2 = 1 is the condition for the creation of a fundamental soliton. This

condition implies that

(l.84)

i.e. there is a threshold power POlh = I~ 1/ rTr} for soliton generation which

depends on both the waveguide parameters, rand 1ß21, and on the pulse width 10. For an N-order soliton this threshold power increases N2 times.

1.5.2 Optical Solitons in Real Transmission Systems

The above derived stationary solutions of the Maxwell's equations in dispersive

and nonlinear media propagate, ideally, over theoretically infinite distances

without changing their shape. In real communication transmission systems

incident soliton pulses change their form and/or energy due to the effect of losses,

higher-order dispersion, Raman scattering, deviations from the ideal sech2 form of

the input soliton, nonuniformities (periodic or not) of the propagating medium, the

soliton self-frequency shift and so on. As a result, the maximum propagation

distance of real solitons is limited to about 50 km without repeater stations. The

exact contribution of the above mentioned effects is hard to estimate since this can

be done generally only by numerical calculations. One notable exception is the

effect of loss: the evolution of an incident fundamental bright soliton with

u( r,O) = sech( r) through a medium with an energy absorption coefficient a can

be described by:

u( r,z) = exp( - az)sech( exp( - az)r)exp( i(l- exp( - 2az)) I 4a). (l.85)

The soliton broadens due to the loss, without changing its form; this can result in

collisions/interactions between successively sent solitons. Two interacting solitons

can give rise to soliton collapse or not, depending on their relative phases and

amplitudes; information errors occur. The other mentioned effects contribute also

to a change of the pulse shape or even its wavelength. The self-frequency shift

effect for example, observed in long fibers in which pulses with an energy greater

than the threshold for the fundamental soliton are launched, consists of the split of

the pulse into a fundamental soliton and a nonsoliton part. The latter has its central

frequency approximately identical to that of the original pulse, but the soliton part


has the central frequency shifted to lower values. This effect can be explained by

the Raman effect in optical tibers, the higher frequency components of the pulse

acting as a pump to provide Raman gain for the lower frequency components. In

long optical transmission systems where periodic amplitications are needed to

prevent soliton degradation, transmission instability may occur due to the

resonance of the soliton phase with the amplitication period. Such instabilities

occur when the ratio between the amplitication period and the soliton period is

approximately equal to 8.

1.5.3 Stretched Pulses

Stretched pulses have been recently shown to be more suitable than optical

solitons for propagation over long distances (> IOD km) in tibers with the same

amount of averaged dispersion and in conditions of weak tixed frequency

filtering. Stretched pulses are formed in nonlinear tibers with alternating normal

and anomalous dispersion regions; stable pulses can be formed with higher

energies than the solitons due to a reduced averaged self-phase modulation. The

pulses are called stretched since during the propagation through the positive and

negative dispersion tibers the pulse widths have alternate maximum and minimum

values. The stretching factor, detined as the ratio between the maximum and

minimum pulse widths, can have values greater than 20. The stretching and

compressing effects are linear and reversible. Unlike solitons, stretched pulses that

are not in a steady state condition suffer oscillations in the pulse parameters with a

period determined by the average dispersion; these oscillations can be damped by

tiltering. The higher energy of the stretched pulses make the effect of noise less

important and the presence of both dispersion types leads to no pulse collapse

during the interaction of two stretched pulses if the dispersion has sm all deviations

with respect to its average value (Matsumoto and Haus 1997). The form of the

stretched pulses can be well approximated with a Gaussian function:

K (T2 1) f(T) = ~exp 21;} 1- i{b (1.86)

with three parameters: height K, width 10 and chirp {b. When the pulse is

adiabatically perturbed such that the chirp parameter is approximately constant,

the height and width change so as to leave invariant the product of the peak

intensity and the pulsewidth to the fourth power (Yu et al. 1997).

1.6 Materials 57

1.6 Materials

There are two categories of materials used in advanced optoelectronic devices:

glasses for fiber devices and semiconductor compounds for integrated opto

electronic devices.

1.6.1 Glasses

The glass technologies (chemical vapor or flame hydro lysis deposition on a

substrate) refer mainly to Si02 pure or doped materials. Si02 is doped with P or

Ge in thc core region in order to increase its retractive index and is doped with B

in the cladding region to decrease the Si02 refractive index.

The quantity L1 = (n;o - n;,) /2n;o reflecting the difference between ncl and

neo ranges from 0.003 < L1 < 0.01 for single mode fibers to 0.01 < L1 < 0.03 for

multimode fibers (neo = 1.461). The corresponding core radius ranges between

2 11m < d< 5 11m for single mode fibers and between 12.5 11m < d< 100 11m for

multimode fibers.

The fibers usually operate with low losses « 0.4 dBIKm) between 0.8 11m < A

< l.6 11m. At 1.3 11m the los ses have a quite low value « 0.2 dB/Km) and the

dispersion coefficient is approximately zero making the fiber useful for

information transmission. The minimum loss value is attained at 1.55 11m - the

other wavelength of interest in applications. But at A = 1.55 f..lm the dispersion

coefficient is not zero and an equalizer is necessary to compensate the dispersion

effects. This can be done by realizing a sinusoidal refractive index profile inside

the core (a fiber Bragg grating). In practice, this Bragg grating is made by

illuminating the fiber core with a stationary UV laser beam.

Active fibers are obtained by doping the core with different rare earth

materials, depending on the operating wavelength. At A = 1.3 11m the Nd3+ ions

are mostly used to fabricate fibers with net gain coefficients while at A = 1.55 f..lm

the Er 3+ ions realize the same task. These dopants produce a gain of up to

O. I dB/cm. In glasses, the index of refraction has a non linear behavior with the

light intensity due to a nonharmonie motion ofthe bound electrons.

Since Si02 has no symmetry inversion this nonlinearity can be described by a

Kerr-like dependence with I1n = 3.2.10-16 cm2 / w.


1.6.2 Semiconductors

In advanced optoelectronics, semiconductor heterostructures such as GaAsl

Ga1_xAlxAs and Inl_xGaxAsyPl_y IInP playa central role. These are band

engineered structures because their parameters such as Eg , n, the effective masses

of electron and holes me and mh' etc. are controlled through the alloy

composition x and y. In this way, the operating wavelength A [J..Lm] = 1.241 Eg (e V)

as weil as other characteristics such as the absorption coefficient can be tuned by

modifying the composition.

There are two types of heterostructures depending on the band structure of the

semiconductors, as displayed in Fig. 1.12 (Esaki 1986).

type I a

type II staggered b

type II misaligned e

Fig. 1.12. Band energy diagrams for different types ofheterostructures (a) type

I, (b) type II staggered, and (e) type II misaligned

In type I heterostructures the motion of the electrons in the conduction band

and of the holes in the valence band can be considered as independent. The

semiconductor with the smaller Eg plays the role of a weil for both electrons and

holes, as can be observed from Fig. 1.12a; the semiconductor with the higher Eg

1.6 Materials 59

is then a baITier for both electron and hole motions. In type 11 heterostructures the

motions of electrons and holes are coupled. In Figs. 1.12b and 1.12c the

semiconductor with the energy gap Eg2 plays the role of a weil for electrons,

while that with energy gap Eg1 plays the role of a weil for holes. Examples of

type I heterostructures are GaAsl AIAs, GaSbl AISb, GaAs/GaP systems; type 11

heterostructures are InAs/GaSb, InAs1_xGaAsx I GaSb1_yGaAsy etc.

Some important parameters of the semiconductor materials used in advanced

optoelectronic devices are given in Table 1.1 (Adachi 1985; Silver and O'Reilly

1995; Yeh 1988; Yang and Xu 1992a). These parameters are very sensitive to the

temperature and in general depend on the experimental conditions under which

they are determined; in particular the refractive index n is also sensitive to the

wavelength of the incident light. The values given in Table 1.1 must therefore be

taken as values given for illustrative purposes only. Slightly different values can

also be found in the literature. Both the electron and hole masses in Table 1.1 are

normalized to the mass of the free electron, mo. The effective mass of holes in

Table 1.1 is actually a weighted sum ofthe light and heavy hole effective masses.

Table 1.1. Parameters of semiconductor materials

Material Eg(eV) A(llm) me /mo mh /mo n

GaAs 1.424 0.8 0.067 0.087 3.6 AIAs 2.168 0.8 0.15 0.15 2.97

AlxGa1_xAs 1.424 + l.247x 0.8 0.067 + 0.087 + 3.6 -(x< 0.45) 0.083x 0.063x 0.63x 1.9 + 0.125x + 0.143x2

(x> 0.45)

AISb 2.22 0.3 0.146 3.31 InAs 0.36 0.0185 0.0255 3.63 InP 1.35 0.08 0.12 3.327 InSb 0.16 1 0.013 4.27 GaP 2.74 0.8 0.17 0.04 3.27

GaSb 0.68 3.87

The semiconductors used in advanced optoelectronic devices also have

nonlinear properties. These properties are due to several physical mechanisms.

Among them we mention the following:

i) the nonharmonic motion ofbound electrons,

ii) confinement effects in MQW which enhance the exciton absorption,

iii) the Stark effect due to energy level shifts of quantum confined electrons

and holes in the presence of an applied field,


iv) the photorefractive effect in which the refractive index is spatially varied

due to nonunifonnities in the electric tield detennined by charge separation in the

carrier transport,

v) band tilling due to the excitation of carriers near the conduction band edge.

Some of these mechanisms can be simultaneously present in a device. A

comparison of these mechanisms for GaAs/GaAlAs heterostructures is given in

Table 1.2 (Gannire et al. 1989). By careful design the nonlinearities due to carrier

transport (iv) can be made much larger than those due to other nonlinear

mechanisms. This is the case for example of a nipi structure.

Table 1.2. Comparison ofthe strength of different nonlinear mechanisms

Mechanism

Exciton (75 A)

Bandtilling

nipi MQWnipi

Iln (cm2/W) 4.10 5

1.10-5

0.0125

13

The nipi structure is fonned by altemating p and n-doped regions separated by

isolating layers. In a nipi structure at each interface charge separation occurs in

the depletion region and therefore, in it, the nonlinear mechanism (iv) dominates.

MQW nipi are altemating p and n regions separated by isolating regions made by

MQWs. In this last case the nonlinear mechanisms (iii) and (v) concur.

From Table 1.2 it is obvious that the nonlinear coefficients of semiconductor

materials are several orders ofmagnitude higher than in Si02 (3.2 .10-16 cm2/W).

However, since the diameter of the optical tibers is generally smaller than that of

integrated semiconductor waveguides and since the propagation length of the

electromagnetic tield in tibers is generally several orders of magnitude larger, the

overall nonlinear effect in optical tibers is comparable with and sometimes larger

than in integrated semiconductor waveguides.

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