Reflection at Oblique Incidence

When an electromagnetic wave strikes a boundary at an angle other than 90 degrees, the analysis becomes richer and more physically meaningful. Oblique incidence introduces the concepts of polarisation-dependent reflection, Snell's law for EM waves, and the Brewster angle at which one polarisation is transmitted without any reflection. These ideas are central to GATE problems and directly explain real-world optical phenomena such as polarised sunglasses and anti-reflection coatings.

Core Concept Explanation

At oblique incidence, the incoming wave makes an angle θᵢ with the normal to the boundary. The law of reflection states that the angle of reflection θᵣ equals the angle of incidence θᵢ. The transmitted wave refracts into Medium 2 at angle θₜ governed by Snell's law: n₁ sinθᵢ = n₂ sinθₜ, where n₁ and n₂ are the refractive indices of the two media. This is identical to the optical form of Snell's law and emerges naturally from the phase matching condition at the boundary.

The behaviour of the wave depends critically on the polarisation of the incident electric field relative to the plane of incidence. The plane of incidence is the plane containing the incident ray and the normal to the boundary. For TE polarisation (also called s-polarisation or perpendicular polarisation), the electric field is perpendicular to the plane of incidence. For TM polarisation (also called p-polarisation or parallel polarisation), the electric field lies within the plane of incidence.

The Fresnel equations provide the reflection and transmission coefficients separately for TE and TM polarisations. These are not the same as in normal incidence — they depend on angle. For TE polarisation, the reflection coefficient is ΓTE = (η₂cosθᵢ - η₁cosθₜ)/(η₂cosθᵢ + η₁cosθₜ). For TM polarisation, ΓTM = (η₂cosθₜ - η₁cosθᵢ)/(η₂cosθₜ + η₁cosθᵢ).

The Brewster angle θB is the special angle of incidence at which ΓTM = 0, meaning TM-polarised light is completely transmitted with no reflection. Setting ΓTM = 0 and using Snell's law gives tanθB = n₂/n₁. At this angle, only TE-polarised light is reflected. This principle is used in polarising filters and laser cavity windows (Brewster windows).

Mathematical Expressions

Starting from the phase matching condition at the boundary and applying boundary conditions for each polarisation, the Fresnel reflection coefficients are derived. For non-magnetic media (μ₁ = μ₂ = μ₀), refractive index n = sqrt(εᵣ), and the coefficients simplify as follows.

TE: ΓTE = (n₁cosθᵢ - n₂cosθₜ)/(n₁cosθᵢ + n₂cosθₜ). TM: ΓTM = (n₂cosθᵢ - n₁cosθₜ)/(n₂cosθᵢ + n₁cosθₜ). Transmission coefficients: τTE = 2n₁cosθᵢ/(n₁cosθᵢ + n₂cosθₜ) and τTM = 2n₁cosθᵢ/(n₂cosθᵢ + n₁cosθₜ). Brewster angle for TM: tanθB = n₂/n₁. Note: TE polarisation has no Brewster angle for non-magnetic dielectrics.

Practical Understanding

Polarised sunglasses exploit the Brewster angle effect. Light reflected off horizontal surfaces (roads, water) is predominantly TE-polarised after the TM component is transmitted through at the Brewster angle. The sunglasses absorb this TE-polarised glare, reducing eye strain significantly.

In laser systems, Brewster windows are flat glass plates tilted at the Brewster angle inside the laser cavity. TM-polarised laser light passes through without reflection loss, while any TE component gets progressively attenuated by repeated reflection losses. Over many cavity round trips, the output becomes highly linearly polarised.

Anti-reflection coatings on camera lenses and optical instruments use the principle of destructive interference between reflections from the top and bottom surfaces of a thin film. The design is angle-sensitive, and understanding oblique incidence Fresnel coefficients is necessary to optimise performance off-normal angles.

Given:
Medium 1: air, n₁ = 1.0
Medium 2: glass, n₂ = 1.5
Angle of incidence: θᵢ = 40°

Why this formula applies:
Oblique incidence, both polarisations → use Fresnel equations
First apply Snell's law to find θₜ

Formula:
Snell: n₁ sinθᵢ = n₂ sinθₜ
ΓTE = (n₁cosθᵢ - n₂cosθₜ)/(n₁cosθᵢ + n₂cosθₜ)
ΓTM = (n₂cosθᵢ - n₁cosθₜ)/(n₂cosθᵢ + n₁cosθₜ)
tanθB = n₂/n₁

Substitution:
sinθₜ = (1.0 × sin40°)/1.5 = 0.6428/1.5 = 0.4285
θₜ = arcsin(0.4285) = 25.37°
cosθᵢ = cos40° = 0.766, cosθₜ = cos25.37° = 0.902

Calculation:
ΓTE = (1×0.766 - 1.5×0.902)/(1×0.766 + 1.5×0.902)
     = (0.766 - 1.353)/(0.766 + 1.353)
     = -0.587/2.119 = -0.277
ΓTM = (1.5×0.766 - 1×0.902)/(1.5×0.766 + 1×0.902)
     = (1.149 - 0.902)/(1.149 + 0.902)
     = 0.247/2.051 = +0.120
θB = arctan(1.5/1.0) = arctan(1.5) = 56.3°

Final Answer:
ΓTE = -0.277 (11.3% TE power reflected at 40°)
ΓTM = +0.120 (1.44% TM power reflected at 40°)
Brewster angle = 56.3° (TM reflection = 0 at this angle)

Exam Tip: GATE often asks to identify polarisation type or find Brewster angle. Remember tanθB = n₂/n₁ applies only for TM polarisation in non-magnetic media. TE polarisation has no Brewster angle. Also, at normal incidence (θᵢ = 0), both TE and TM Fresnel formulas reduce to the single normal-incidence formula Γ = (η₂ - η₁)/(η₂ + η₁).

Mechanism — Polarisation Dependence and Brewster Angle

• TE reflectance increases monotonically from its value at normal incidence to 1.0 at grazing incidence (90°).
• TM reflectance first decreases as θᵢ increases, reaches zero at the Brewster angle θB, then rises back to 1.0 at grazing incidence.
• At normal incidence (θᵢ = 0°), both polarisations give the same reflectance, consistent with normal incidence theory.
• At grazing incidence (θᵢ → 90°), all surfaces become perfect reflectors for both polarisations — this is why the sky near the horizon appears brighter.
• Brewster angle exists only for TM polarisation in non-magnetic media; for magnetic media or TM at a magnetic boundary, a Brewster angle may or may not exist.

Quick Revision

• Snell's law: n₁ sinθᵢ = n₂ sinθₜ — derived from phase matching at boundary.
• TE (s-pol): E perpendicular to plane of incidence. TM (p-pol): E parallel to plane of incidence.
• Brewster angle (TM only): tanθB = n₂/n₁ — zero TM reflection at this angle.
• At θᵢ = 0°, both Fresnel equations reduce to normal incidence formula.
• TE reflectance always increases with angle; TM reflectance dips to zero at θB before rising.
• Common trap: Brewster angle does NOT exist for TE polarisation in non-magnetic media.
• At grazing incidence, reflectance → 1 for all polarisations — perfect reflection.