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Appendix 4
From slab waveguide to optical fibre
1. By exploiting a pair of reflector to form a slab waveguide, we manage to confine the
wave in one direction (x) by enforcing a resonance along that direction. As a result,
the wave becomes a standing wave (in a sinusoidal form) inside the core region of the
slab waveguide along x, and evanescent wave (in an exponentially decaying form)
inside the top and bottom claddings.
2. The standing wave in a sinusoidal form comes from the summation of contrapropagating plane (traveling) waves, since e jkx x  e jkx x  cos(k x x),sin(k x x) .
3. The evanescent wave in an exponentially decaying form comes from the fact that the
propagation constant of the plan (traveling) wave turns from real to imaginary, hence
e jkx x  e j ( j|kx |) x  e|kx | x .
4. The wave propagation along z is set free. Hence we still have the free-space solution
for the wave in z: e j ( kz z t )  e j (  z t ) (plane traveling wave).
5. The field must take a homogenous form as E ( x, z, t )   ( x)e j (  z t ) , with  ( x) given
in a form of either standing wave or evanescent wave, depending on the region where
the field stays, as described in above item 2 and 3, simply because the aforementioned
resonance must happen everywhere along z, otherwise the wave will be leaking at the
specific location where the wave is not resonating.
6. Therefore, the field of the guided-wave by a general waveguide can always be
factorized as: E ( x, y, z, t )   ( x, y ) E0 ( z, t )e j (  z t ) , with  ( x, y ) representing the
resonant field in the cross-sectional plane (y is restored to cover the general 3D case),
e j (  z t ) the traveling wave factor along z (the direction the wave is sent), E0 a slowvarying function of z and time (t) as the wave will eventually be modulated by the
base-band signal [ E0 ( z, t ) must be a slow-varying function in comparing with
e j (  z t ) , namely
|  2 E0 (t ) / z 2 | / | E0 (t ) ||  2e j (  z t ) / z 2 | / | e j (  z t ) |  2 ,
|  2 E0 (t ) / t 2 | / | E0 (t ) ||  2e j (  z t ) / t 2 | / | e j (  z t ) |  2
as otherwise E0 ( z, t )e j (  z t ) cannot be put into a form of f (  z  t ) and the wave
is no longer a traveling wave along z.]
7. The unknown field to be solved in the wave equation can then be substituted by the
above factorized form to obtain:
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2 E
 E    E  0 
 E    E  0 
 E  2   2 E  0
z
E ( z , t )
3

 E0 ( z , t )e j (  z t )T2   2 j  e j (  z t ) 0
 ( 2   2 ) E0 ( z , t )e j (  z t )  0
z
,

E
(
z
,
t
)
 E0 ( z , t )[T2   ( 2   02 ) ]  E0 ( z, t )(  02   2 )  2 j  0
0
z
E ( z , t )
4

 E0 ( z , t )(  02   2 )  2 j  0
0
z
2
2
1
2
2
2
2
T
where at above step 1, only a single polarization direction is put under consideration
without losing generality by knowing the fact that the orthogonal polarization
components are decoupled in the wave equation and the wave equation itself is linear
(so that it is sufficient to consider one polarization component at a time and make a
summation if multiple polarization components exist); at step 2, the 3D Laplacian
operator (  2 ) is decomposed into a 2D (in the cross-section x, y plane) Laplacian
operator (  T2 ) plus a longitudinal (in z) one (  2 / z 2 ); at step 3, the 2nd order
derivative of the field slow-varying amplitude ( E0 ) is neglected due to the fact shown
in above item 6; and finally at step 4, as a resonant solution in the cross-section,
 ( x, y ) must satisfy (the eigen-value equation) T2   ( 2   02 )  0 , as otherwise,
in the special case of E0 as a constant (i.e., without modulation) and    0 , the
above equation cannot hold.
8. Hence we have the eigen-value equation:
T2  ( x, y )  ( 2   02 ) ( x, y )  0 ,
and the propagation equation:
E0 ( z, t )  2  02

E0 ( z, t )  0
z
2 j
held for the resonant field factor (  ) and the wave amplitude ( E0 ). By solving these
two equations subject to the given waveguide structure (with a specific set of
boundary conditions extracted from the waveguide for  ) and the given initial
condition E0 (0, t ) at the source (i.e., the base-band modulation signal at the
transmitter), we will be able to obtain the general field E at any place and at any time.
(We certainly will be able to find the field at the receiver end.)
9. For a slab waveguide, the eigen-value equation in above item 8 will be reduced to:
2
2
 ntop

 cladding


2
2
2
2
2
2
d  ( x) / dx  (    0 ) ( x)  0 , with   0 and    0 n ( x)   0 
ncore
,
2
 nbottom

 cladding 

which can then be readily solved analytically as shown in the slides – the result is
given as the sinusoidal (co-sine or sine) function in the core region, and as
exponentially decaying functions in the top and bottom cladding. (We didn’t
consider the modulation case when we firstly introduced the waveguide concept so
that the field amplitude was assumed as a constant then.)
10. Unfortunately, for a general 2D dielectric waveguide, there is no closed-form
analytical solution for the eigen-value problem. We have to rely on the numerical
approach to find the solution. An optical fiber can be viewed as the result of winding
of a slab waveguide along its centre symmetric line in its left uniform direction (y) to
form a cylinder. By doing so, the eigen-value equation in above item 8 will take the
form of:
T2  ( x, y )  ( 2   02 ) ( x, y )  0

,
 2 (r ,  ) 1  (r ,  ) 1  2 (r ,  )


 2
 [( ) 2 n 2 (r )   02 ] (r ,  )  0
2
2
r
r r
r

c
where we have transformed the equation from the Cartesian system to the polar
system in order to accommodate the boundary condition in cylindrical symmetry.
With the rotational symmetry, the solution stands as a combined azimuthal resonance
given in the form of a standing wave e jm [i.e., cos(m ) and sin( m ) ] and a radial
resonance given by a family of the Bessel functions. The latter is obviously not in a
closed-form.
11. The wave propagation equation as given in above item 8 takes the same form as the
one shown in Lecture Note #3, although they are derived by different approaches.
For further discussions of optical fibers, please refer to Lecture Note #3.
12. A summary of the knowledge we have learnt so far:
Coulomb’s law that describes the static electronic interaction → implicit expression of
the static electric field by specifying its divergence (given by the source charge
distribution) and curl (free) → Biot-Savert’s law that describes the static magnetic
interaction in a symmetric fashion to the Coulomb’s law → implicit expression of the
static magnetic field by specifying its divergence (free) and curl (given by the current
density, also known as the Ampere’s law) → Faraday’s law showing the fact that the total
electric and magnetic field is conserved in a combined space and time domain, hence the
time-varying magnetic field supports the curl of the electric field (be careful with the
negative sign!) → modified Ampere’s law to remove the inconsistency lies in between
the original Ampere’s law and the charge conservation law, by adding on a displacement
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current term, hence the time-varying electric field supports the curl of the magnetic field
→ Maxwell’s equations are obtained as a general and full description of the
electromagnetic field → Faraday’s law + modified Ampere’s law leads to a sustainable
oscillation between the electric and magnetic fields in both space and time domain, the
fact that the oscillation walks away from the source as time goes (a nature related to the
curl operation) implies the existence of the traveling wave (E and M field oscillation) that
can propagate away from its excitation source → to quantify this effect, we extract the
electric and magnetic wave equations from Maxwell’s equations by eliminating the
magnetic field and electric field in Faraday’s and modified Ampere’s equations,
respectively → the solution to the wave equation in free space (sourceless and
homogenous) is a plane traveling wave having its wave vector with the value of
   ( / c)n  (2 /  )n and with its direction pointing right at the direction of wave
propagation, having the electric and magnetic field both polarized in a plane
perpendicular to the wave propagation direction, having the electric and magnetic field
polarized orthogonally [i.e., the wave vector or the wave propagation direction, the
electric field, and the magnetic field must stay all orthogonally, from which we
immediately come to the conclusion that the traveling electromagnetic wave can only
exist in a 3D or higher-dimensioned space], and having the ratio between the electric and
magnetic field amplitude given as the free-space impedance characterized by  /  →
reflection and refraction (transmission) of the plane wave at the dielectric interface
(boundary between two different dielectric media) [please remember the Snell’s law and
the formula for reflectivity under normal incidence!] → the idea to build a slab
waveguide → extraction of the general waveguide concept, i.e., the guided wave must
take a factorized form with one factor given as a stationary distribution
(resonant/standing wave inside the core region and evanescent/decaying wave in
claddings) in the cross-sectional (x, y) plane (conventionally termed as “mode”) and the
other as a traveling wave term f ( z  t ) ~ A( z, t )e j (  z t ) ; its amplitude (A) can either
be a constant (for DC wave propagation in applications such as power transmission) or a
slow-varying function in terms of z and t compared to the traveling wave factor e j (  z t )
(when the wave is modulated by base-band signals) → the field in the wave equation for
a waveguide problem can then be substituted by its factorized form, which lead to an
eigen-value equation and an wave propagation equation; the former governs the crosssectional stationary field (mode) distribution, whereas the latter governs the wave
amplitude → solve the eigen-value problem subject to the boundary condition defined by
the waveguide cross-sectional structure to obtain the mode; solve the wave propagation
equation subject to the initial condition to obtain the wave amplitude; their product then
gives the total field and the entire waveguide problem is solved
Please prepare your cheat sheet for the midterm by following this summary and record
equations/expressions/formulas along with the path shown above once you feel it is
necessary.
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