Lecture 15

October 4, 2004

8-8 Normal Incidence at a Plane Dielectric Boundary-continued

If medium 2 is not a perfect conductor, partial reflection will result. The total electric field in medium 1 can be written as

or, in view of ,

E₁(z) is composed of two parts: a traveling wave with an amplitude τE_i0 and a standing wave with an amplitude 2ΓE_i0.

The locations of maximum and minimum |E₁(z)| are conveniently found by rewriting E₁(z) as

For dissipationless media, η₁ and η₂ are real, making both Γ and τ real. Consider the following cases.

For Γ > 0 (η₂ > η₁) the maximum value of |E₁(z)| is E_i0(1 + Γ), which occurs when 2β₁z_max= -2nπ, or at

The minimum value of |E₁(z)| is E_i0(1 – Γ), which occurs when 2β₁z_min= -(2n + 1)π, or at

For Γ < 0 (η₂ < η₁) the maximum value of |E₁(z)| is E_i0(1 – Γ), which occurs at z_min and the minimum value of |E₁(z)| is E_i0(1 + Γ) which occurs at z_max.

The ratio of the maximum value to the minimum value of the electric field intensity of a standing wave is called the standing-wave ratio (SWR), S.

An inverse relation is

The magnetic field intensity in medium 1 is obtained by combining H_i(z) and H_r(z):

This should be compared with E₁(z).

In medium 2, (E_t , H_t) constitute the transmitted wave propagating in +z-direction.

Example 8-11 A uniform plane wave in lossless media with η₁ and η₂. Obtain the expressions for the time-average power densities.

Solution The time-average Poynting vector: P_av= ½Re(E × H*).

In medium 1

where Γ is a real number.

In medium 2

Since we are dealing with lossless media, the power flow in medium 1 must equal that in medium 2:

8-9 Normal Incidence at Multiple Dielectric Interfaces

A uniform plane wave traveling in the +z-direction in medium 1(ε₁, µ₁) impinges normally at a plane boundary with medium 2(ε₂, µ₂), at z = 0. Medium 2 has a finite thickness and interfaces with medium 3(ε₃, µ₃) at z = d.

The H₁ in region 1 that corresponds to the E₁ is

The electric and magnetic fields in region 2

In region 3

There are four unknown amplitudes: They can be determined by solving the four boundary-condition equations.

At z = 0:

At z = d:

8-9.1 Wave Impedance of the Total Field

We define the wave impedance of the total field at any plane parallel to the plane boundary as the ratio of the total electric field intensity to the total magnetic field intensity.

In the case of a uniform plane wave incident from medium 1 normally on a plane boundary with an infinite medium 2, the magnitudes of the total electric and magnetic field intensities in medium 1 are,

The wave impedance of the total field in medium 1 at a distance z from the boundary plane:

which is a function of z.

At distance

Using the definition of , we obtain

which reduces to .

If the plane boundary is conducting (ε₂ = ∞), η₂ = 0 and Γ = -1,

8-9.2 Impedance Transformation with Multiple Dielectrics

The wave impedance of the total field in medium 2 at the left-hand interface z = 0 can be found from

The effective reflection coefficient at z = 0 for the incident wave in medium 1 is