Total Internal Reflection

Total internal reflection (TIR) occurs when a wave travelling in a denser medium strikes a less dense medium at an angle greater than a specific critical angle, causing the wave to be completely reflected back with no transmitted energy. This is not merely a special case of oblique incidence — it is a fundamentally important phenomenon that makes optical fibre communication possible and is a recurring topic in GATE electromagnetics.

Core Concept Explanation

When a wave travels from a medium of higher refractive index n₁ to lower refractive index n₂ (n₁ > n₂), Snell's law gives sinθₜ = (n₁/n₂)sinθᵢ. Since n₁/n₂ > 1, there exists a finite angle θᵢ at which sinθₜ = 1, meaning θₜ = 90°. The transmitted ray skims along the boundary. This angle is the critical angle θc = arcsin(n₂/n₁). For any θᵢ > θc, sinθₜ would exceed 1, which has no physical solution for a real transmitted angle — the wave cannot penetrate Medium 2 and is totally reflected.

Importantly, TIR is not accompanied by any power loss. The reflection coefficient magnitude |Γ| = 1 for both TE and TM polarisations when θᵢ > θc, though both pick up a phase shift that is polarisation-dependent. The reflected wave carries exactly the same power as the incident wave. This makes TIR ideal for lossless wave guiding, which is the physical principle behind optical fibres.

The evanescent wave is a critical consequence of TIR that is frequently misunderstood. Despite zero net power being transmitted into Medium 2, the electromagnetic field does not abruptly vanish at the boundary. Instead, a rapidly decaying field penetrates a short distance into Medium 2 (roughly one wavelength). This evanescent field decays exponentially in the direction normal to the boundary but oscillates along the boundary. It carries no net power flow in the normal direction, consistent with TIR, but it is physically real and can be coupled out by placing another medium very close to the boundary — a phenomenon called frustrated TIR, used in devices like optical couplers and ATR spectroscopy.

Mathematical Expressions

The critical angle is obtained by setting θₜ = 90° in Snell's law: sinθc = n₂/n₁. For θᵢ > θc, the transmitted wave vector component kz in Medium 2 becomes imaginary: kz = k₂cosθₜ = j·α, where α = k₀sqrt(n₁²sin²θᵢ - n₂²) is the attenuation constant of the evanescent wave.

The evanescent field in Medium 2 varies as exp(-αz), where z is the depth into Medium 2 measured from the boundary. The penetration depth δ = 1/α gives the characteristic depth at which the field falls to 1/e of its boundary value. The phase shifts for TE and TM polarisations upon TIR are different and are given by the Fresnel equations evaluated with complex cosθₜ.

Practical Understanding

Optical fibre is the most important application of TIR. The fibre core has a higher refractive index than the cladding. Light injected within the acceptance cone of the fibre undergoes repeated TIR at the core-cladding boundary and propagates with negligible loss over long distances. The numerical aperture (NA) of a fibre is NA = sqrt(n₁² - n₂²), characterising the range of angles over which the fibre can accept and guide light.

Prism-based periscopes, binoculars, and retroreflectors exploit TIR at glass surfaces to redirect light without silvered mirrors. TIR gives 100% reflection efficiency, unlike metallic mirrors that absorb a few percent of light.

In microwave and millimeter-wave engineering, TIR at dielectric slab boundaries is used to design dielectric waveguides, including the substrate-integrated waveguide (SIW) structures used in modern RF chips. The evanescent field outside the guide is intentionally used in sensors and couplers.

Given:
Core refractive index: n₁ = 1.48 (silica glass)
Cladding refractive index: n₂ = 1.46
Frequency: optical (λ = 1550 nm)

Why this formula applies:
TIR condition at core-cladding interface of optical fibre
sinθc = n₂/n₁, NA = sqrt(n₁² - n₂²)

Formula:
sinθc = n₂/n₁
NA = sqrt(n₁² - n₂²)
Acceptance half-angle in air: θa = arcsin(NA)

Substitution:
sinθc = 1.46/1.48 = 0.9865
NA = sqrt(1.48² - 1.46²)
   = sqrt(2.1904 - 2.1316)
   = sqrt(0.0588)

Calculation:
θc = arcsin(0.9865) = 80.57°
NA = sqrt(0.0588) = 0.2425
θa = arcsin(0.2425) = 14.03°

Final Answer:
Critical angle at core-cladding boundary = 80.57°
Numerical aperture NA = 0.242
Fibre accepts light within ±14° cone from its axis.

Exam Tip: TIR requires the wave to travel from denser to rarer medium (n₁ > n₂). The critical angle formula sinθc = n₂/n₁ is valid only in this direction. Reversing the direction (n₂ > n₁) means TIR cannot occur regardless of angle. Also, the evanescent wave in Medium 2 does NOT carry power in the normal direction — but it does carry power along the interface, which is sometimes asked as a trap question.

Mechanism — Evanescent Wave and Fibre Guiding

• TIR occurs only when travelling from denser (n₁) to rarer (n₂) medium, and only when θᵢ exceeds the critical angle θc = arcsin(n₂/n₁).
• The evanescent wave penetrates Medium 2 exponentially with characteristic depth δ = 1/α. It carries zero net power in the direction normal to the boundary.
• In optical fibres, the guided ray undergoes TIR at every core-cladding interface, propagating along the fibre axis with nearly zero radiation loss.
• The numerical aperture NA = sqrt(n₁² - n₂²) defines the acceptance cone — light injected within angle θa = arcsin(NA) from the fibre axis will be guided.
• Frustrated TIR: If a third medium is brought within a wavelength of the TIR boundary, the evanescent field can couple energy into it, allowing selective energy transfer — used in optical couplers and ATR spectroscopy.

Quick Revision

• Critical angle: sinθc = n₂/n₁ — valid only when n₁ > n₂ (denser to rarer).
• TIR condition: θᵢ ≥ θc; reflected power = incident power (|Γ| = 1), but phase shifts.
• Evanescent wave exists in Medium 2 even under TIR — decays as exp(-αz) with α = k₀sqrt(n₁²sin²θᵢ - n₂²).
• Optical fibre NA = sqrt(n₁² - n₂²); guides light within acceptance cone ±arcsin(NA).
• Evanescent wave carries no net normal power — but it IS a real field in Medium 2.
• Common trap: TIR cannot happen if wave goes from rarer to denser medium, regardless of angle.
• Frustrated TIR enables energy coupling through evanescent field — used in optical couplers.