Normal incidence, reflection and transmission

Phenomenon

When a uniform plane wave meets the boundary between two different media, part of its energy is reflected back into the first medium and the rest is transmitted into the second. For normal (perpendicular) incidence, the incident, reflected and transmitted waves all travel along the normal to the interface, with E and H tangential to the boundary. The reflection produces a standing wave in medium 1 (a mixture of the incident and reflected waves), while medium 2 carries only a travelling wave.

  Medium 1 (ε1, μ1, η1) | Medium 2 (ε2, μ2, η2)
                        |
   Ei  ───────────►     |   Et ───────────►
   Er  ◄───────────     |
                        |
                       z = 0

Derivation

Take the boundary at z = 0, media 1 (z < 0) and 2 (z > 0), perfect dielectrics, and E along a_x (H along a_y).

Incident:     E_i = E_i0 e^{−jβ₁z} a_x,        H_i = (E_i0/η₁) e^{−jβ₁z} a_y
Reflected:    E_r = E_r0 e^{+jβ₁z} a_x,        H_r = −(E_r0/η₁) e^{+jβ₁z} a_y
Transmitted:  E_t = E_t0 e^{−jβ₂z} a_x,        H_t = (E_t0/η₂) e^{−jβ₂z} a_y

The reflected H has a negative sign because the wave travels in −z (E × H must point in −z).

Boundary conditions at z = 0 (tangential E and H continuous, no surface current):

E: E_i0 + E_r0 = E_t0                              ... (1)
H: (E_i0 − E_r0)/η₁ = E_t0/η₂                      ... (2)

Substitute (1) into (2): (E_i0 − E_r0) η₂ = (E_i0 + E_r0) η₁, so

E_r0 (η₁ + η₂) = E_i0 (η₂ − η₁)

Reflection coefficient

Γ = E_r0/E_i0 = (η₂ − η₁)/(η₂ + η₁)

Transmission coefficient (from (1): τ = 1 + Γ)

τ = E_t0/E_i0 = 2η₂/(η₂ + η₁)

Remarks

• Relation: 1 + Γ = τ.
• For non-magnetic dielectrics η = η₀/√ε_r, so Γ = (√ε_r1 − √ε_r2)/(√ε_r1 + √ε_r2).
• If η₂ = η₁, Γ = 0 (matched, no reflection). Perfect conductor (η₂ = 0): Γ = −1, τ = 0, total reflection.
• Power: reflected fraction = |Γ|², transmitted fraction = 1 − |Γ|² = (η₁/η₂)|τ|².
• SWR in medium 1: s = (1 + |Γ|)/(1 − |Γ|).