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Chapter 8
Integrated Optical Waveguides
7.1 Dielectric Slab Waveguides
7.1.1 Introduction: A variety of different integrated optical waveguides are used to confine and guide light on a chip. The most basic optical waveguide is a slab waveguides shown below. The structure is uniform in the y- direction. Light is guided inside the core region by total internal reflection at the core-cladding interfaces.
Most actual waveguides are not uniform and infinite in the y-direction but can be approximated as slab waveguides if their width W is much larger than the core thickness d , as shown below.
A better description of the guided light is in terms of the optical modes. The slab waveguide supports two different kinds of propagating modes: i. TE (transverse electric) mode: In this mode, the electric field has no component in the direction of propagation. ii. TM (transverse magnetic) modes: In this mode, the magnetic field has no component in the direction of propagation.
Cladding
Cladding
Core
z
x n 1
n 2
n 1
d
To study these modes we start from Maxwell’s equations. The complex form of Maxwell’s equations is,
n x E c
E
H i n xE
E i H
o
o
2 2
2
2
Since E E E
( ). In general E 0
. Rather [ n^2 ( x ) E ] 0
. But if index
is piecewise uniform in different regions then inside each region one may assume E 0
. So we have in each region,
n x E
c
E
2
2
^2 ^
Similarly, with the assumption of piecewise uniform index we can write for the H
field,
n x H
c
H
2
2
^2 ^
7.1.2 TE Modes: For TE modes, the electric field can be written as, i z Exz yEo xe
In each region of piecewise uniform index (core and cladding), ( x )satisfies,
2 2 (^ ) ( )^2 (^ )
2 2
2 n x x x x c
Given a value for the frequency , we can find all solutions of the above equation which is an
eigenvalue equation with eigenfunction ( x )and eigenvalue ^2. Once we have the electric field,
the magnetic field H
can be found as follows,
i z o
o o
o
o
x e
E
x x
x i
zE
i
i z Exz x
x
i
E
Hxz
^
0
The boundary conditions needed to solve the eigenvalue equation above are as follows:
i) y-component of the electric field is continuous at the core-cladding interfaces ii) z-component of the magnetic field is continuous at the core-cladding interfaces iii) x-component of the magnetic field is continuous at the core-cladding interfaces (this is automatically satisfied when (i) above is satisfied)
The solutions are labeled with the integer index m ( m 0 , 1 , 2 , 3 ,).So the field for the TE (^) m mode is,
i z o m
E ( x , z ) y ˆ E ( x ,) e m (^ )
where the dependence of the eigenfunctions and the propagation vector on the frequency is
explicitly indicated. For the TE modes we assume the solution,
sin( )
cos( )
( / 2 ) 1
2
( / 2 ) 1
Ce x d
x d kx
kx C
Ce x d
x
x d
x d
The propagation vectors (^) m ( ) behave as n 1 c
near then cut-off frequencies, and asymptotically
approach the n 2 c
curve at high frequencies. In most applications, waveguide dimensions are chosen
such that it supports only one mode. Unless necessary we will usually drop the mode index “ m “.
7.1.3 TM Modes: Similar analysis can be done for the TM modes. Assume,
2 2 2
i z o
o o
o o
iz o
x e n x
H
x i n x x
H
z i n x
H
Exz
Hxz yH xe
For piecewise uniform index n ( x ), ( x ),satisfies,
2
2 2
(^2) n x x x x c
Assume solution of the form,
2 2 1
2 2 2
2 2 2
2 2 2
( / 2 ) 1
2
( / 2 ) 1
sin( )
cos( )
n c
n c
k
Ce x d
x d kx
kx C
Ce x d
x
x d
x d
The boundary conditions needed to solve the eigenvalue equation above are as follows:
i) y-component of the magnetic field is continuous at the core-cladding interfaces ii) z-component of the electric field is continuous at the core-cladding interfaces iii) x-component of the electric field weighted by the square of the index is continuous at the core-cladding interfaces (this is automatically satisfied when (i) above is satisfied)
Using the boundary conditions give,
(sinesolutions oddTMmodes ) 2
cot
(cosinesolutions evenTMmodes) 2
tan
2 1
2 2
2 1
2 2
^
n^ k
kd n
n k
kd n
The above relations are similar to those obtained for the TE modes other than the factors containing the squares of the core and cladding refractive indices. The general behavior of the TM modes is similar to the TE modes.
Cladding
Cladding
Core
z
x n 1
n 2
n 1
d E y E (^) y
TE 0 TE 1
7.1.4 Effective Index and Group Index: The effective index (^) n (^) eff ( )of a mode is defined by the relation,
neff
c
The effective index of each mode equals n (^) 1 at their cut-off frequencies and approaches n (^) 2 at high
frequencies. The group velocity of a mode is,
v g v (^) g is written as,
g
g n
c v
where ng ( )is called the group index of the mode.
7.2 Two Dimensional Waveguides
7.2.1 Introduction: In slab waveguides, light is confined in only one dimension (the x-dimension in our notation) as it travels in the z-direction. In most actual waveguides light is confined in two dimensions (x and y- dimensions) and travels in the z-direction. For example, the cross-section of a rectangular waveguide is shown below.
Unfortunately, the exact modes of such 2D waveguides are not as easy to complete as those of slab
waveguide. The modes are neither TE nor TM. All modes have a small component of E
and H
fields in the direction of propagation. The E
and the H
fields can be written as, E ( x , y , z ) x ˆ Ex ( x , y ) y ˆ Ey ( x , y ) z ˆ Ez ( x , y ) ei z^ Et x , y z ˆ Ez ( x , y ) ei^ z
H ( x , y , z ) x ˆ Hx ( x , y ) y ˆ Hy ( x , y ) z ˆ Hz ( x , y ) ei z^ Ht x , y z ˆ Hz ( x , y ) ei^ z
We have defined the transverse components of the fields as follows,
H xy xH xy yH x y
E xy xE xy yE xy
t x y
t x y
i z t
iz t t
iz Et e Ee Ee ^2 ^ ^2 2 ^
Using the results above, Equation (3) becomes,
t t t t t t t n Et E^ t
c
nE
n
E E
2
2 2 2
2.^1 .( ) ^
The above eigenvalue equation is what one needs to solve to get the exact solution. This equation can be put in the form,
ˆ ˆ (^4 )
2
y
x y
x yx yy
xx xy E
E
E
E
P P
P P
where the differential operators are.
x y
E
y
n E x n
P E
n E y c
E
x
n E x n
P E
y y xy y
x
x x xx x
2 2
2
2 2
2 2
2 2 2
ˆ^1 ( )
ˆ^1 ( )^
y x
E
x
n E y n
P E
n E y c
n E
x y n
E
P E
x x yx x
y
y y yy y
2 2 2
2 2
(^22)
2 2
2
ˆ^1 ( )
ˆ^
The above equation is an eigenvalue equation and its solution gives the transverse components of the
electric field E
for the mode and the corresponding propagation constant (^) ( ). E (^) z ( x , y )can be
obtained from E (^) x ( x , y )and E (^) y ( x , y )as already explained, and H
field can be obtained from the
relation,
0
i
E
H
For piecewise uniform indices, all derivatives of the index n ( x , y ) can be dropped provided
appropriate boundary conditions are used at all the interfaces.
7.2.3 The Semi-Vectorial Approximation: In many cases of practical interest one transverse component of the electric field (either E (^) x or Ey )
dominates over the other component. In such cases, we may assume that the other transverse field component is zero. For example, if we know a priori that E (^) x dominates then we may assume that
E (^) y is zero and solve the much simpler eigenvalue equation,
2 2 2
2 2
2 2 2
2
x x x x
xx x x
n xyE E y c
E
x
n E x n
P E E
The above equation is called the semi-vectorial approximation. For piecewise uniform dielectrics we can also write it as,
t x x^ x
x x x x
n xyE E c
E
n xyE E c
E
y
E
x
2 2 2
2 2
2 2 2
2 2
2 2
2
provided we take care to impose the boundary conditions on E (^) x ( x , y )at all the dielectric interfaces
as appropriate for the x-component of the electric field. Once the dominant (^) E (^) x ( x , y )component has
been found, the remaining field components can be found as follows,
y
E
i
H
H
x
n E y n
H
x
nE x n
H E
i
E
H
x
nE n
i E
n E
x o
z
x o
x
x o
x o
y
o
x z
2 2
2 2
2 2
2
When the horizontal electric field component dominates the modes are sometimes called HE pq modes or TE pq modes (with a slight abuse of terminology). When the vertical electric field component dominates the modes are called EH pq modes or TM pq modes (again with a slight abuse of terminology). The two subscripts p and q indicate the number of nodes the dominant electric field component has in the horizontal and vertical directions, respectively.
7.2.4 The Scalar Field Approximation: If one component of the transverse electric field dominates over the other transverse component and the index differences among different regions in the structure are also relatively small, then one may use the semi-vectorial approximation and also do away with the boundary conditions on the normal component of the electric field at all dielectric interfaces. For example, if we know a priori that E (^) x ( x , y )dominates then we may solve the equation,
W z (^) on ngME. E * dxdy 2
The effective index n (^) eff ( )of a mode is,
neff
c
The group velocity of a mode is,
v g v (^) g is frequently expressed in terms of the group index of the mode,
g
g (^) n
c v
where ng ( ) is called the group index. One can prove the following relation between the power
P ( z )and the energy per unit length W ( z ), P ( z ) vgW ( z )
which can also be written as,
^
E H zdxdy
nn EE dxdy Pz
Wz c
n
t t
M g o g Re *.^ ˆ
In the slab-waveguide approximation, assuming TE modes with the transverse component of the electric field given by x , y , one obtains the following expressions for various quantities of interest,
W z (^) onngMEE dxdy onnMg dxdy 2 2
Pz E H E H z dxdy dxdy o
(^) (^) t t t t 2 2
ˆ^1
[ * * ].
dxdy
nn dxdy n n
M g g eff 2
2
7.2.7 Properties of Waveguide Modes and Orthogonality of Modes: Frequently, solutions in various cases involve expansions in terms of all the waveguide modes. In such cases, knowledge of the orthogonality of the modes is useful. The electric and the magnetic fields for the m -th mode can be written as,
Em^ Etm x , y z ˆ Ezm ( x , y ) ei m ^ z
m i ^ z z
m t
H m^ H x , y z ˆ H ( x , y ) e m^
Some useful properties of the waveguides modes are listed below:
i) When the indices are real, the propagation vectors are also real, and the transverse field components can be chosen to be real as well. Equations (1) and (2) show that in this case the z-components of the fields are purely imaginary.
ii) When the indices are real, the complex conjugate of the electric field mode gives the field for the mode propagating in the opposite (time-reversed) direction. For example, if,
Em^ Etm x , y z ˆ Ezm ( x , y ) ei m ^ z
represents the field for the forward propagating mode then,
^ m i ^ z z
m t
E m^ * E x , y z ˆ E ( x , y ) e m^
represents the field for the backward propagating mode.
iii) When the indices are real, the negative of the complex conjugate of the magnetic field mode gives the field for the mode propagating in the opposite (time-reversed) direction. For example, if,
Hm^ Htm x , y z ˆ Hzm ( x , y ) ei m ^ z
represents the field for the forward propagating mode then,
H m^ * Htm x , y z ˆ Hmz ( x , y ) e i m ^ ^ z
represents the field for the backward propagating mode.
iv) Consider two different modes, “ m “ and “ p “ with the same propagation vector but different frequencies then the orthogonality between the mode fields is expressed as,
dxdy^ n^2 ^ x , y ^ Em^ . E^ p *^ ^0 for m p
dxdy^ Hm^.^ H^ p *^ ^0 for m p
v) Consider two different modes, “ m “ and “ p “ with the same frequency but different propagation vectors then the orthogonality between the mode fields is expressed as,
dxdy^ ^ Etm ^ H^ tp *^. z ˆ^0 for m p
vi) The most general way of expanding a time harmonic field of a particular frequency inside a waveguide is in terms of the waveguide modes,
(^) ^ ^ ^ m
m i z z
m m t m
m i z z
m m (^) t
Ex , y , z a E x , y z ˆ E ( x , y ) e m^^ b E x , y z ˆ E ( x , y ) e m^
(^) ^ ^ ^ m
m i z z
m m t m
m i z z
m m (^) t
Hx , y , z a H x , y z ˆ H ( x , y ) e m^^ b H x , y z ˆ H ( x , y ) e m^
7.2.8 Longitudinal vs Transverse Modes of a Waveguide: Consider the model dispersion relations shown below for the first three modes of a waveguide. The HE and EH modes are called the transverse modes of the waveguide since these modes describe the field profile in dimensions x and y that are transverse to the direction of propagation.
The field profile in the direction of propagation is described by the propagation vector . The
dispersion for the lowest HE 00 mode is plotted below in a slightly different way that also shows the dispersion for negative values of the propagation vector (propagation in the –z-direction). Different
values of correspond to different longitudinal modes of the waveguide.
HE 00 mode
HE 01 mode
EH 00 mode
7.2.8 Mode Confinement Factors: Consider the waveguide shown below. The refractive index of the core is n 2 and that of the cladding
is n 1.
Suppose the refractive index of the core changes by n 2. The change in the propagation vector can
be found as follows,
2 2
2
core (^22)
2
2
core (^22)
(^22)
2 2
core 2 2 (^22)
2 2
core 2 2 core
Re *.ˆ
Re *.ˆ
Re *.ˆ
Re *.ˆ
Re *.ˆ
n n
n c
c
n nn EE dxdy
n n EE dxdy
n
n
E H zdxdy
nn EE dxdy
nn EE dxdy
n n EE dxdy
n n
n n
E H zdxdy
n n EE dxdy
n n
n n
E H zdxdy
EE dxdy n n E H zdxdy
n nEE dxdy
E H zdxdy
n nEE dxdy
M g
g
g M o g
M o g M g
t t
M o g M o g
M o g M g
t t
M o g M g
t t
o t t
o
t t
o
^
Note that nM 2 g is the material group index of the waveguide core, and ng is the group index of the
waveguide mode. The overlap integral 2 , defined as,
^ ^
nn EE dxdy
n n EE dxdy M o g
M o g
. *
core 2 2. * 2
represents the fraction of the mode energy density confined in the waveguide core. 2 is called the
transverse mode confinement factor for the waveguide core. The change in the propagation vector is, as expected, proportional to n 2 and also to the fraction of the modal energy inside the core.
Suppose now the refractive index of the cladding also changes by n 1 then the total change in the
propagation vector can be written as a simple sum,
n 2 → n 2 + n 2
2 2
1 2 1
1 n n
n c
n n
n c M g
g M g
g
In the slab waveguide approximation, assuming a TE mode and the transverse component of the electric field given by x , y , one obtains the following expression for the transverse mode
confinement factor for the core,
nn dxdy
n n dxdy M g
M g 2
core
2 (^22) 2
The waveguide perturbation theory can be used to calculate the change in the propagation vector in the presence of material loss (or gain). Suppose the core of the waveguide becomes lossy and the imaginary part of the core refractive index acquires a non-zero value given by,
2 2 2
c n n i
In this case, we can take the index perturbation n 2 to be,
2 2
c n i
The change in the waveguide propagation constant becomes,
2 2 2 2
2 2 2
2 2 2
2
i i
n
n i
c i n
n c
n n
n c M g
g M g
g M g
g
where,
2 2 2 2
2
M g
g n
n
The propagation vector acquires a small imaginary part because of optical loss in the waveguide
core. The imaginary part of the propagation vector will cause the wave energy to decay with distance as it propagates in the waveguide.
