# On Classical Electromagnetic Fields (cont.)

## VIII. Guided Waves in Planar Structures (pdf copy)

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 [VIII-2 ] 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 "bac-cab" 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 Spatial Configuration:[2]
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 cross-products:
 [ 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

 [ VIII-27 ]
At  the field parallel to the surface of a perfect conductor must be zero so that  and, therefore,

 [ VIII-28 ]
where .  At the upper surface -- i.e.  -- the field parallel to the surface of a perfect conductor must also be zero so that

 [ VIII-29 ]
and, therefore,
 [ VIII-30 ]
which is the dispersion relationship for TE waves in a parallel plate waveguide with "cutoff" frequencies at
 [ VIII-31 ]

Movie: See an animation of TE1 mode propagation, an animation of TE1 mode propagation as a function of frequency and an animation of TE2 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
 [ VIII-32 ]
At  the field parallel to the surface of a perfect conductor must be zero so that  and, therefore,
 [ VIII-33 ]
where .  At the upper surface -- i.e.  -- again the field parallel to the surface of a perfect conductor must also be zero so that
 [ VIII-34 ]
and, therefore,
 [ VIII-35 ]
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 self-consistence 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 TE-modes of propagation, Equation [ VIII- 26a ] gives the phase shift at the boundary (called in the trade the TE Goos-Hänchen 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 self-consistence 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
 LHS and RHS of Equation [ VIII- 38 ] for  and LHS and RHS of Equation [ VIII- 38 ] for  and

 LHS and RHS of Equation [ VIII- 38 ] for  and LHS and RHS of Equation [ VIII- 38 ] for  and
Movie: See an animation of TE1 mode propagation and an animation of TE2 propagation as a funtion of frequency.
Modeling tools for integrated and fiber optical devices

1974, Kogelnik and Ramaswamy    developed a convenient formalism for treating slab-waveguide problems.  First they introduced three new waveguide parameter -- viz.

 The normalized frequency/slab thickness parameter [ VIII-39a ] The normalized waveguide index parameter [ VIII-39b ] The normalized waveguide asymmetry parameter [ VIII-37c ]

They then showed that Equation [ VIII-37b ] could be written

 [ VIII-39c ]

which can be used to generate the following family of curves:

It is also useful to differentiate Equation [ VIII-40 ] to obtain

 [ VIII-40 ]

which, in turn, can be used to generate the following family of curves:

For reference in lecture discussion:

Notes on absorption in fiber optic glass

[1] That is

[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 [ V-29 ] and [ V-34 ] which figured in our analysis of parallel plane waveguides.

This page was prepared and is maintained by R. Victor Jones, jones@deas.harvard.edu
Last updated February 29, 2000