|*Note: The information below can be referenced to: Iskander, M., Electromagnetic Fields and Waves, Waveland Press, Prospect Heights, IL, 1992, ISBN: 1-57766-115-X. Edminister, J., Electromagnetics (Schaum’s Outline), McGraw-Hill, New York, NY, 1993, ISBN: 0-07-018993-5. Hecht, E., Optics (Schaum’s Outlines), McGraw-Hill, New York, NY, ISBN: 0-07-027730-3
Normal Incidence Plane Wave Reflection and Transmission at Plane at Plane Boundaries
Why is there a need to study the reflection and transmission properties of plane waves when incident on boundaries between regions of different electric properties? Perhaps you had no idea that we experience this topic daily in our lives. For instance, when you try and make a call on your cell phone and you are downtown amongst all those tall buildings. Will you always have great reception? When the hot sun penetrates your window it can quickly heat up your room, but maybe you have blinds, curtains, or tinted material to prevent some of that intense heat. For those of you that wear glasses, you know what happens when you get your picture taken; that annoying glare from those glasses. What about a light ray on the surface of a mirror? A reflection can be seen and some of that ray will penetrate the glass.
This chapter focuses on the reflection and transmission properties related to one-dimensional problems that have normal-incident plane waves converging on infinite plane interfaces that will separate two or more different media.
The Figure 5.1 is the fundamental concept. It illustrates the geometry of the positive z propagating plane wave that is normally incident on a plane interface between regions 1 & 2.
5.2 Normal Incidence Plane Wave Reflection and Transmissions at Plane
Boundary Between Two Conductive Media
The electric and magnetic fields related to the incident wave are given by the following:
* Note: (i) incident, (m1) medium 1, (γ1) propagation constant in region 1, (η1) wave impedance in region 1, (z) direction of propagating wave
The complex propagation constant in region 1 is. *Note: α and β are the real and imaginary parts respectively. The propagation constant γ is that square root of γ2 whose real and imaginary parts are positive:
γ = α + jβ
The wave impedance as defined in chapter 2 as the ratio between the electric and magnetic fields is
The wave impedance η in a conductive medium is a complex number meaning that the electric and magnetic fields are not in phase. The phase velocity will be less than the velocity of light vp < c. The wavelength λ in the conductive medium will be shorter than the wavelength λo in free space at the same frequency, λ = 2π/β < λo. The factor will attenuate the magnitudes of both E and H as they propagate in the +z direction.
Figure 5.2 The electric field associated with a plane wave propagating along the positive z direction.
What happens when this wave hits the boundary?
Some of the energy related to the incident wave will transmit across the boundary surface at z = 0 in region 2, therefore providing a transmitted wave in the +z direction in medium 2. The following are the electric and magnetic fields related to the transmitted wave:
* Note: (t) transmitted wave
Recall Maxwell’s equations:
· E = 0
· H = 0
The Wave equation for H:
For now, let’s look at the simplest system, that consisting of a plane wave of coordinate z.
Therefore, according to the wave equation as noted above, equations (5.1) & (5.2) satisfy Maxwell’s equations.
If the amplitude of the transmitted wave is unknown then boundary conditions at the interface z = 0 separating the two media must be satisfied.
Good conductors are often treated as if they were perfect conductors. Metallic conductors such as copper have a high conductivity σ = 6 * 107 S/m, however, only superconductors have infinite conductivity and are truly perfect conductors.
Static (time independent)
n · D1 = ρs
n · (B1 – B2) = 0
* Subscripts denote the conducting medium.
Characteristics of static cases:
Electrostatic field inside a good conducting medium is zero. Free charge can exist on the surface of a conductor, thus making the normal component of D discontinuous being zero inside the conductor and nonzero outside. The tangential component of E just inside the conductor must be zero even if the surface is charged.
The electric and magnetic fields in the static case are independent. A static magnetic field can therefore exist inside a metallic body, even though an E field cannot. The normal component of B and the tangential components of H are therefore continuous across the interface.
For time-varying fields, the boundary conditions for good (perfect) conductors are:
Time –varying fields (time dependent)
n ·D = 0
n · B = 0
The subscripts have been deleted because in this case the only nonvanishing fields are those outside the conducting body.
& are tangential to the interface therefore the boundary conditions will require these fields be equal at z = 0. Equate &, and set z = 0. *Note: (t) is transmitted wave, (i) incident wave. The result will be
*Note: (m) medium, (+) transmitted wave (5.3)
& are also tangential to the interface, so by applying the same procedure as above you will notice that it is impossible to satisfy the magnetic field boundary conditions if.
We can then, include a reflected wave in region 1 traveling away from the interface, or in other words in the –z direction. Only part of the energy related to the incident wave will be transmitted to region 2 because of the process the incident fields must encounter prior to crossing the boundary. The fields left behind during this process will in fact be the reflected wave.
The electric and magnetic fields related to the reflected wave are
Note: (r) reflected wave, (-) wave traveling in the –z direction
Equation (5.4) is related by because the reflected wave is traveling in the –z direction and the Poynting vector will be in the -az direction. To satisfy the boundary conditions for the tangential electric field, z = 0. This is important because the basic model assumes three waves incident, reflected and transmitted:
This can be simplified, by adding the E field transmitted to the E field reflected of medium 1 with a result equal to the E field transmitted wave of medium 2:
*Note: We can model the system as three waves incident, reflected and transmitted. Boundary conditions must be met for the E field as well as the H field. Waves have both E & H fields - .
Similarly, enforcing the continuity of the tangential magnetic field at z = 0,
To solve for, multiply equation (5.6) by and add the result to equation (5.5). The result is:
The transmission coefficient is the ratio of the amplitudes of the transmitted to the incident fields:
The amplitude of the reflected wave can be solved for by multiplying equation (5.6) byand subtracting the result from equation (5.5) for a result of:
The reflection coefficient is the ratio of the amplitudes of the reflected and incident electric fields given by:
From equations (5.8) & (5.10), note that the reflection and transmission coefficients are
An H field travels in the direction in free space with a phaseshift constant (β) of
30.0 rad/m and an amplitude of A/m. If the field has the direction –ay when t = 0 and z = 0, write suitable expressions for E and H. Determine the frequency and wavelength.
In a medium of conductivity σ, the intrinsic impedance η, which relates E and H, would be complex, and so the phase of E and H would have to be written in complex form. In free space the restriction is unnecessary. Using cosines, then
H (z, t) =
For propagation on –z,
Determine the propagation constant γ for a material having and, if the wave frequency is 1.6MHz.
In this case,
α = 0
And. The material behaves like a perfect dielectric at the given frequency. Conductivity of the order indicates that the material is more like an insulator than a conductor.
Normal Incidence Plane-Wave Reflection at Perfectly conducting Plane
Special case (analysis of material presented in section 5.2)
Assumptions-(region 2) perfect conductor → ∞, wave impedance
, as σ2→ ∞, (5.11)
To simplify the standing wave analysis, assume that region 1 is a perfect dielectric σ1 = 0.
Using substitution: take equation (5.11) in the reflection and transmission coefficient expressions in equations (5.8) and (5.10) in order to obtain
The zero value of the transmission coefficient simply means that the amplitude of the transmitted field in region 2 is = 0. This can be explained in terms of the following:
The depth of penetration parameter is zero in a perfectly conducting region,
(Chapter 3, p. 241). Therefore, there will be no transmitted wave in a perfectly conducting region, because of the inability of time-varying fields to penetrate media with conductivities converging toward infinity.
Only the incident and reflected fields will be present in region 1.
The amplitude of the reflected wave is. The reflected wave is therefore equal in amplitude and is opposite in phase to the incident wave. This simply means that the entire incident energy wave is reflected back by the perfect conductor.
The combination of the two fields meets the boundary condition at the surface of the perfect conductor.
This can be illustrated by examining the expression for the total electric field in region 1, which is assumed to be a perfect dielectric (i.e., α1 = 0)
Substituting, for a result of:
= -2j (5.12)
Note: The total electric field is zero at the perfectly conducting surface (z = 0) meeting the boundary condition.
To study the propagation characteristics of the compound wave in front of the perfect conductor, we must obtain the real-time form of the electric field.
Step 1: Multiply the complex form of the field in equation (5.12) by ejωt
Step 2: Take the real part of the resulting expression
In equation (5.13) the amplitude of the electric field was assumed real. Our objective is then to complete the following step:
Step 3: Show that the total field in region 1 is not a traveling wave, although it was obtained by combining two traveling waves of the same frequency and equal amplitudes of which are propagating in the opposite direction.
Figure 5.2 shows a variation of the total electric field in equation (5.13) as a function of z at various time intervals.
Figure 5.2b The variation of the total electric field in front of the perfect conductor as a function of z and at various time intervals ωt.
From figure 5.2 you can make the following observations:
α = 0, indicating that the total field meets the boundary condition at all times.
The total electric field has maximum amplitude twice that of the incident wave. The maximum amplitude occurs at z = λ/4, at z = 3λ/4, etc., when ωt = π/2, ωt = 3π/2, etc. happening when both the incident and reflected waves constructively interfere.
When z = λ/2, z = λ, z = 3λ/2, etc., in front of the perfect conductor the total electric field is always zero. This is happening when the two fields are going through destructive interference process for all values of ωt, also known as null locations.
The occurrence of the null and constructive interference locations do not change with time. The only thing that changes with time is the amplitude of the total field at nonnull locations. Therefore, the wave resulting from the interference of the two waves is called “standing waves”.
It should also be emphasized that the difference between the electric field expressions for the traveling and standing waves. For a traveling wave, the electric field is given by:
The term emphasizes the coupling between the location as a function of time of a specific point (constant phase) propagating along the wave.
It also indicates with an increase in t, z should also increase in order to maintain a constant value of (t –z/v1), and it characterizes a specific point on the wave. This means that a wave with an electric field expression which includes cosis a propagating wave in the positive z direction. The time t and location z variables are uncoupled in equation (5.13), or in other words, the electric field distribution as a function of z in front of the perfect conductor follows a sinvariation, with the locations of the field nulls being those values of z at which sin = 0.
The sin (ωt) term modifies the amplitude of the field allowing a variation of a function of time located at the nonzero field locations as illustrated in Figure 5.2.
By finding the values of the permanent locations of the electric field nulls can be determined, thus making the value of the field zero. So, from equation (5.12) we can see that
This simply shows that is zero at the boundary z = 0, and at every half wavelength distance away from the boundary in region 1 which is illustrated in Figure (5.2).
Total Magnetic Field Expression:
The minus sign in the reflected magnetic field expression is simply because for a –z propagating wave the amplitude of the reflected magnetic field is related to that of the reflected electric field by (). Substituting yields:
The time-domain magnetic field expression is obtained from equation (5.15) as: (5.16)
ωt = π/2