Throughout this set of lectures, I will make extensive use of a wonderful set of animations which can be found at Iowa State University's QuickTime Movies of Electromagnetic Waves.Characteristics of Plane Wave Solutions:
For the record, let us restate the frequency domain, macroscopic Maxwell's equations which are valid in the high frequency or optical regime for a linear, local, isotropic medium  viz.
[ VIII 1a ] 
[ VIII 1b ] 
[ VIII 1c ] 
[ VIII 1d ] 
Further, in regions free of explicit sources of current and charge we may write
[ VIII 2a ] 
[ VIII 2b ] 
[ VIII 2c ] 
[ VIII 2d ] 
where . In this set of lectures it is our intention to explore in some depth plane wave propagation within a uniform medium  i.e. . To that end we consider a plane wave solution in the form
[ VIII 3 ] 
which may pictorially represented as follows:
Therefore
[ VIII 4a ] 
[ VIII 4b ] 
and the Maxwell's equations formulated in Equation [VIII2 ] become
[ VIII 5a ] 
[ VIII 5b ] 
[ VIII 5c ] 
[ VIII 5d ] 
Operate through on both sides of Equation [ VIII 5a ] with the operator "^{"} we obtain
[ VIII 6a ] 
Using the "baccab" rule[1] and Equation [ VIII 5b ] this becomes
[ VIII 6b ] 
or finally
[ VIII 6c ] 
Substituting these results into Equation [ VIII 5a ] we obtain
[ VIII 7 ] 
so that the so called wave impedance is given by
[ VIII 11 ] 
Thus, the complete expression for an electromagnetic plane wave propagating in a direction in a uniform medium is given by
[ VIII 9a ] 
[ VIII 9b ] 
Electromagnetic Interfacial Continuity (Saltus) Conditions:
The previous section gives a complete plane wave solution within a particular uniform, linear, isotropic medium. The key remaining problem is to find how that solution may be extended into a second uniform, linear, isotropic medium. The conditions for extending the solution across an interface between two materials are give by consideration of the appropriate integral forms of Maxwell's equations  viz.
[ VIII 10a ] 
[ VIII 10b ] 
Applying these equations to the small thoughtloop that spans the interfacial surface, as illustrated below
it is seen that Equation [ VIII 10a ] yields
[ VIII 11 ] 
unless is pathologically large over the loop. Similarly, it is seen that Equation [ VIII 10b ] yields
[ VIII 12 ] 
unless and/or are pathologically large over the loop.In words and in general, the tangential component of the electric field strength and the magnetic field strength are continuous across an interfacial surface between two materials unless the electric current density , the magnetic flux density , or the electric flux density are pathologically large near that interfacial surface.
The Interfacial Fresnel Equation:
Consider then a plane wave incident on a planar interfacial surface.
The Mathematical Representation of Fields:In abstract vector form, the incident field is given by [3]
[ VIII 13a ] 
the reflected field is given by
[ VIII 14a ] 
and the transmitted field is given by
[ VIII 15a ] 
In coordinate form these equations become:
[ VIII 13b ] 
[ VIII 14b ] 
[ VIII 15b ] 
Or expanding out the crossproducts:
[ VIII 13c ] 
[ VIII 14c ] 
[ VIII 15c ] 
Applying any kind of continuity condition at the interface requires that
Law of Sinus
[ VIII 16a ] 
Law of Snell
[ VIII 16b ] 
Movies: See an animation of transmission through a planar interface and an animation of reflection from a planar interface.
Applying, in particular, the continuity conditions discussed in the previous section  viz.
and  [ VIII 17 ] 
which requires at the interface that
[ VIII 18 ] 
and that
[ VIII 19 ] 
These two sets of equations yield the Fresnel Reflection Equations  viz.
[ VIII 20a ] 
and
[ VIII 21a ] 
Since
[ VIII 20b ] 
and
[ VIII 21b ] 
These equations taken together with the first equations from Equations [ VIII 18 ] and [ VIII 19 ] yield the Fresnel Transmission Equations  viz.
[ VIII 22 ] 
and
[ VIII 23 ] 
FAMOUS FRESNEL REFLECTION CURVES ( )
The minimum (zero) in occurs at the Brewster angle where
[ VIII 24a ] 
or
[ VIII 24b ] 
or (from Snell's equation)
[ VIII 24c ] 
Total Internal Reflection
Reconsider Equation [ VIII 15c ] and use Snell's law to write the exponential factors in the form
[ VIII 25 ] 
When , , the solution in medium 2, is attenuated!
Reconsideration of Equation [ VIII 20a ] and [ VIII 21a ] shows that the magnitude of the reflection coefficients are one when  viz.
[ VIII 26a ] 
and
[ VIII 26b ] 
Movie: See an animation field configuration due to total internal reflection. Note how evanecient field develops at the critical angle of incidence.
Parallel Plate Waveguide:
To establish the language of guided wave propagation, consider the propagation of a plane wave between two parallel perfectly conducting planes.
Perspective view Side view
Combining Equations [ VIII 13c ] and [ VIII 14c ], the electric field strength of the TE wave in the region between the plates may be written
[ VIII27 ]
At the field parallel to the surface of a perfect conductor must be zero so that and, therefore,
[ VIII28 ]
where . At the upper surface  i.e.  the field parallel to the surface of a perfect conductor must also be zero so that
[ VIII29 ]
and, therefore,
[ VIII30 ] 
which is the dispersion relationship for TE waves in a parallel plate waveguide with "cutoff" frequencies at
[ VIII31 ] 
Movie: See an animation of TE_{1} mode propagation, an animation of TE_{1} mode propagation as a function of frequency and an animation of TE_{2} mode propagation.
Again combining Equations [ VIII 13c ] and [ VIII 14c ], the electric field strength of the TM wave in the region between the plates may be written
[ VIII32 ] 
At the field parallel to the surface of a perfect conductor must be zero so that and, therefore,
[ VIII33 ] 
where . At the upper surface  i.e.  again the field parallel to the surface of a perfect conductor must also be zero so that
[ VIII34 ] 
and, therefore,
[ VIII35 ] 
which is the dispersion relationship for TM waves in a parallel plate waveguide.
Note that the TM0 mode is a bona fide mode of propagation which does not have a "cutoff" frequency!
Dielectric Slab Waveguides  The Basis of Integrated Optics
Consider the propagation of waves "trap in" or "guided by" a dielectric slab of thickness d.
In its full generality this is moderately complicated problem, but a rather simple ray optics model of the propagation is sufficient to yield dispersion relationships for the various possible modes of propagation. To obtain such relationships, consider the total internal reflection of a sequence of plane waves as illustrated below.
In order for the multiple reflected wave to be selfconsistence or coherent the following, relatively obvious, phase condition must hold:[4]
[ VIII 36 ] 
where and are, respectively, the phase shifts associated with the reflections at the upper and lower dielectric boundaries.
For TEmodes of propagation, Equation [ VIII 26a ] gives the phase shift at the boundary (called in the trade the TE GoosHänchen shift) and Equation [ VIII 36 ] becomes
[ VIII 37a ] 
where (n is the effective index of the propagation mode).
[ VIII 37b ] For the symmetric case (i.e.,), the selfconsistence relationship for the TE modesis given by
[ VIII 38 ] 
where . This is a transcendental equation in the single variable . Its solutions yield the allowed bounce angles, , of possible modes and, hence, the allowed propagation constants since The left and right sides of this equation may be plot as a function of with and as a parameters. The intersections of such curves yield the allowed bounce angles as illustrated below
for and 
for and 


for and 
for and 
Movie: See an animation of TE_{1} mode propagation and an animation of TE_{2 }propagation as a funtion of frequency.
Modeling tools for integrated and fiber optical devicesFor reference in lecture discussion:1974, Kogelnik and Ramaswamy developed a convenient formalism for treating slabwaveguide problems. First they introduced three new waveguide parameter  viz.
The normalized frequency/slab thickness parameter [ VIII39a ]The normalized waveguide index parameter [ VIII39b ]The normalized waveguide asymmetry parameter [ VIII37c ]
They then showed that Equation [ VIII37b ] could be written
[ VIII39c ]which can be used to generate the following family of curves:
It is also useful to differentiate Equation [ VIII40 ] to obtain
[ VIII40 ]which, in turn, can be used to generate the following family of curves:
Notes on absorption in fiber optic glassNotes on dispersion in fiber optic glass
[2] Note: In this figure we have taken the plane of reflection to be identical to the plane of incidence. While assumed here for simplicity, this important identity is establish in the analysis below.
[3] A note on notation: The subscripts and refer to the polariztion of the electric field taken with respect to the plane of incidence. The field components are also called transverse electricor TE components and the field components are called transverse magnetic or TM components.
[4] This equation is a direct generalization of
Equations
[ V29 ] and [ V34 ] which figured in our analysis of parallel plane
waveguides.