Reflection at Normal Incidence

When an electromagnetic wave travelling through one medium encounters a planar boundary with a second medium, part of the wave energy is reflected back and part is transmitted forward. At normal incidence, the wave strikes the boundary perpendicularly, making this the simplest and most analytically clean case of wave reflection. Understanding this scenario is foundational for GATE problems on standing waves, impedance matching, and power transfer across boundaries.

Core Concept Explanation

Consider a uniform plane wave propagating in the +z direction in Medium 1, with intrinsic impedance η₁. At the boundary (z = 0), the wave encounters Medium 2 with intrinsic impedance η₂. The mismatch in impedance is the root cause of reflection. If η₁ = η₂, no reflection occurs and the wave continues unaffected. The greater the impedance mismatch, the stronger the reflection.

The reflection coefficient Γ (Gamma) quantifies the ratio of the reflected electric field amplitude to the incident electric field amplitude at the boundary. Similarly, the transmission coefficient τ (tau) gives the ratio of the transmitted electric field amplitude to the incident amplitude. These coefficients are derived entirely from the boundary conditions: tangential E and tangential H must both be continuous across the interface.

Applying boundary conditions at z = 0, the incident, reflected, and transmitted fields satisfy Eᵢ + Eᵣ = Eₜ for the electric field, and Hᵢ - Hᵣ = Hₜ for the magnetic field. Using the relationship H = E/η for plane waves in each respective medium, these two equations together yield the Fresnel coefficients for normal incidence.

The standing wave pattern that forms in Medium 1 is a direct consequence of superposition of the forward-travelling incident wave and the backward-travelling reflected wave. The standing wave ratio (SWR) characterises the severity of this interference and is deeply linked to how much reflection is occurring at the boundary.

Mathematical Expressions

The intrinsic impedance of a medium is given by η = sqrt(μ/ε), measured in ohms. For free space, η₀ = 377 Ω. The reflection and transmission coefficients at normal incidence are:

Reflection coefficient: Γ = (η₂ - η₁)/(η₂ + η₁). Transmission coefficient: τ = 2η₂/(η₂ + η₁). These satisfy the identity 1 + Γ = τ at all times.

The power reflectance R and power transmittance T are: R = |Γ|² and T = 1 - R = (η₁/η₂)|τ|². Note that T is not simply |τ|² because the transmitted wave carries power in a medium of different impedance.

The standing wave ratio (SWR) in Medium 1 is defined as SWR = (1 + |Γ|)/(1 - |Γ|). An SWR of 1 means perfect matching (no reflection). An SWR approaching infinity means total reflection, as occurs when Medium 2 is a perfect conductor (η₂ = 0).

Practical Understanding

In transmission line engineering, the same reflection coefficient formula applies when load impedance ZL replaces η₂ and source impedance Z₀ replaces η₁. This analogy makes normal incidence EM theory directly applicable to practical RF circuit design, antenna feed networks, and coaxial cable connections.

At a perfect electric conductor (PEC) boundary, η₂ = 0, giving Γ = -1. This means the reflected electric field is equal in magnitude but opposite in sign to the incident field. The total electric field at the boundary is zero, and a perfect standing wave forms in Medium 1 with nulls at the boundary and every half-wavelength behind it.

In fibre optic connectors, glass-to-glass interfaces are designed to minimise the refractive index step between joined fibres, thereby minimising Γ and reducing return loss. Even a small air gap at a connector (n_air = 1 vs n_glass ≈ 1.5) introduces a significant reflection that degrades signal quality.

Given:
Medium 1: free space, η₁ = 377 Ω
Medium 2: dielectric with εᵣ = 4, μᵣ = 1
  → η₂ = η₀/sqrt(εᵣ) = 377/2 = 188.5 Ω

Why this formula applies:
Normal incidence → use Γ = (η₂ - η₁)/(η₂ + η₁)

Formula:
Γ = (η₂ - η₁)/(η₂ + η₁)
τ = 2η₂/(η₂ + η₁)
SWR = (1 + |Γ|)/(1 - |Γ|)

Substitution:
Γ = (188.5 - 377)/(188.5 + 377) = (-188.5)/(565.5)
τ = 2 × 188.5 / 565.5

Calculation:
Γ = -0.333
τ = 377/565.5 = 0.667
R = |Γ|² = 0.111 → 11.1% power reflected
T = 1 - 0.111 = 0.889 → 88.9% power transmitted
SWR = (1 + 0.333)/(1 - 0.333) = 1.333/0.667

Final Answer:
Γ = -0.333, τ = 0.667, SWR = 2.0
11.1% of incident power is reflected; SWR in Medium 1 = 2.

Exam Tip: GATE frequently tests the sign of Γ. If η₂ < η₁ (wave going from rarer to denser medium), Γ is negative, meaning the reflected E-field undergoes a 180° phase reversal. The transmitted field always has the same sign as the incident field — τ is always positive for real impedances.

Mechanism — Standing Wave Formation

• In Medium 1, the total field is a superposition of forward and backward waves, creating a standing wave pattern with fixed maxima and minima in space.
• Electric field minima (nulls) occur at multiples of λ/2 from the boundary when Γ = -1 (PEC case). For partial reflection, the minima are not true zeros.
• In Medium 2, only the transmitted wave exists — it is a pure travelling wave with no standing wave component.
• Power continuity must hold: power carried by incident wave = power reflected + power transmitted. This is expressed as 1 = R + T = |Γ|² + (η₁/η₂)|τ|².
• SWR = 1 indicates no mismatch; SWR = infinity indicates total reflection (short or open circuit analogy in transmission lines).

Quick Revision

• Γ = (η₂ - η₁)/(η₂ + η₁) and τ = 2η₂/(η₂ + η₁); always 1 + Γ = τ.
• η = sqrt(μ/ε); for free space η₀ = 377 Ω; for a dielectric η = η₀/sqrt(εᵣ) when μᵣ = 1.
• Γ negative → reflected E undergoes 180° phase flip; occurs when wave goes from lower to higher impedance medium.
• Power reflectance R = |Γ|²; power transmittance T = 1 - |Γ|² (not |τ|²).
• SWR = (1 + |Γ|)/(1 - |Γ|); SWR = 1 for perfect match, SWR → ∞ for PEC boundary.
• Standing wave exists only in Medium 1; Medium 2 has a pure travelling wave.
• Common trap: τ > 1 is possible (e.g., when η₂ >> η₁) but T < 1 always — no energy violation because power depends on impedance ratio.