Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

In medium 1 the incident and the reflected waves are present together, and they add:

E_{1} (x) H_{1} (x) = E_{0}^{i} (e^{- j β_{1} x} + Γ e^{j β_{1} x}) a_{y}, = \frac{E _{0}^{i}}{η _{1}} (e^{- j β_{1} x} - Γ e^{j β_{1} x}) a_{z} . (6.3.18)

Write $Γ = ∣Γ∣ e^{j θ_{Γ}}$ and take the factor $e^{- j β_{1} x}$ out of the bracket. Its modulus is $1$ , so the amplitude of the total field at the point $x$ is

∣ E_{1} (x) ∣ = ∣ E_{0}^{i} ∣ 1 + ∣Γ∣ e^{j (2 β_{1} x + θ_{Γ})} . (6.3.19)

The second term in the bracket is a phasor of fixed length $∣Γ∣$ that turns as $x$ changes. It is aligned with the $1$ when its angle is a multiple of $2 π$ and opposite to it when the angle is an odd multiple of $π$ . Hence the field amplitude has

maxima $∣ E_{0}^{i} ∣ (1 + ∣Γ∣)$ where $2 β_{1} x + θ_{Γ} = - 2 mπ$ ,
minima $∣ E_{0}^{i} ∣ (1 - ∣Γ∣)$ where $2 β_{1} x + θ_{Γ} = - (2 m + 1) π$ ,

with $m$ an integer chosen so that the point lies in medium 1, $x \leq 0$ . Since $2 β_{1} x$ grows by $2 π$ when $x$ grows by $λ_{1} /2$ , the maxima repeat every $λ_{1} /2$ , the minima likewise, and each minimum lies $λ_{1} /4$ from a maximum. The ratio of the largest to the smallest amplitude is the standing-wave ratio.

Standing-wave ratio

S ∣Γ∣ = \frac{∣ E _{1} ∣ _{m a x}}{∣ E _{1} ∣ _{m i n}} = \frac{1 + ∣Γ∣}{1 - ∣Γ∣}, = \frac{S - 1}{S + 1} . (6.3.20)

$S = 1$ for no reflection, $S \to \infty$ for total reflection ( $∣Γ∣ = 1$ ).

In plain words

A reflected wave beats against the incident one and turns the uniform travelling wave into a rippled pattern, fixed in space, with peaks and dips half a wavelength apart. The depth of the ripples measures $∣Γ∣$ , their position measures the phase of $Γ$ .

For a real $Γ$ (lossless media) the pattern has a simple reading. Splitting the incident wave as $(1 - Γ) + Γ$ ,

E_{0}^{i} (e^{- j β_{1} x} + Γ e^{j β_{1} x}) = travelling wave (1 - Γ) E_{0}^{i} e^{- j β_{1} x} + standing wave 2Γ E_{0}^{i} cos β_{1} x,

the total field in medium 1 is a travelling wave carrying the net power towards the interface plus a pure standing wave, which carries none. This is why the minima of a partial standing wave never reach zero.

Standing-wave patterns in medium 1 for \Gamma=-\frac12 (medium 2 denser, solid) and \Gamma=+\frac12 (dashed). Both have S=3. A negative \Gamma puts a *minimum* of |E_1| on the interface, a positive \Gamma a maximum. In medium 2 there is only a travelling wave and |E_2|=|\tau|\,|E_0^i| is flat. — **Figure 6.3.4.** Standing-wave patterns in medium 1 for $Γ = - \frac{1}{2}$ (medium 2 denser, solid) and $Γ = + \frac{1}{2}$ (dashed). Both have $S = 3$ . A negative $Γ$ puts a *minimum* of $∣ E_{1} ∣$ on the interface, a positive $Γ$ a maximum. In medium 2 there is only a travelling wave and $∣ E_{2} ∣ = ∣ τ ∣ ∣ E_{0}^{i} ∣$ is flat.

Read the pattern

$Γ < 0$ (denser medium 2): a minimum of $∣ E_{1} ∣$ at the interface. $Γ > 0$ : a maximum. The magnetic field Eq. (6.3.18) has $- Γ$ in place of $Γ$ , so the maxima of $∣ H_{1} ∣$ sit at the minima of $∣ E_{1} ∣$ and vice versa. This is how a field probe moved along a slotted line or in front of a sample measures a reflection coefficient: $S$ gives $∣Γ∣$ , the position of the first minimum gives $θ_{Γ}$ (Home exercise 6.3.2).

Quick check

A probe in front of a dielectric half space measures $∣ E_{1} ∣_{m a x} = 15$ V/m and $∣ E_{1} ∣_{m i n} = 5$ V/m, with a maximum exactly on the surface. Find $Γ$ .

$S = 3$ , so $∣Γ∣ = \frac{3 - 1}{3 + 1} = \frac{1}{2}$ . A maximum on the surface means $∣1 + Γ∣ > 1$ , hence $Γ = + \frac{1}{2}$ : medium 2 has three times the impedance of medium 1, $η_{2} / η_{1} = (1 + Γ) / (1 - Γ) = 3$ .

Below, air meets $ε_{r} = 9$ (Problem 6.3.1), so $Γ = - \frac{1}{2}$ and a minimum of $∣ E_{1} ∣$ sits on the interface. Swap the media, $ε_{r 1} = 9$ and $ε_{r 2} = 1$ , and a maximum settles there instead, with $Γ = + \frac{1}{2}$ as in the quick check. Back in Problem 6.3.1, switch the layer on for a quarter-wave match.

InteractiveLoading simulation…

6.3.3 Standing waves and the standing-wave ratio