Questions: Evanescent Waves and Total Internal Reflection

Question 1

Two glass prisms are arranged hypotenuse-to-hypotenuse with an air gap. Total internal reflection occurs at the first hypotenuse. When the gap is reduced to a fraction of a wavelength, light is observed in the second prism. A student says: 'This is impossible — TIR means total reflection, so no field exists in the air gap.' What is the correct explanation?

• A. The student is correct: reducing the gap mechanically disrupts the TIR condition, causing partial transmission through a different mechanism
• B. At very small gaps, quantum tunneling allows photons to jump across the gap, bypassing the classical wave equations
• C. TIR reflects the traveling wave but an evanescent field — non-propagating, exponentially decaying — still exists in the air gap; when the second prism is close enough, this field couples into propagating modes there, transferring power (frustrated TIR)
• D. The student is correct that no field exists, but diffraction at the edge of the prism redirects light into the second prism

Answer: (C)

TIR means the time-averaged energy flux (Poynting vector) into the second medium is zero — no net power is transmitted. But fields are not zero: Maxwell's boundary conditions require a field in the second medium, which takes the form of an exponentially decaying evanescent wave. When a second medium is brought within this decay length, the evanescent field can couple into propagating modes there. This is frustrated TIR. The student's error is conflating 'total reflection of propagating energy' with 'no fields at all.'

Question 2

A wave vector k = iκ (with κ real and positive) in the wave factor e^{ikx} describes which physical situation?

• A. A traveling wave with a 90° phase shift relative to a standard plane wave
• B. A standing wave formed by two counterpropagating waves with equal amplitude
• C. An exponentially decaying field (e^{−κx}) — an evanescent wave that carries no net power in the decay direction
• D. A wave with reduced phase velocity due to dispersion in a dense medium

Answer: (C)

Substituting k = iκ into e^{ikx} gives e^{i(iκ)x} = e^{−κx}: pure exponential decay with no oscillation in x. This is an evanescent wave. It occurs whenever k² < 0, which happens in total internal reflection (angle exceeds critical angle), below plasma cutoff frequency, or in a waveguide below its cutoff. The field amplitude decays on a length scale ~1/κ, typically of order one wavelength.

Question 3

In total internal reflection, the electromagnetic field in the second (rarer) medium is exactly zero — TIR produces complete exclusion of the field from that medium.

• T. True
• F. False

Answer: False

TIR requires that Maxwell's boundary conditions be satisfied at the interface, which forces the existence of a field in the second medium. That field is evanescent: it decays exponentially with distance from the interface and carries zero net power. The fields are real and measurable — frustrated TIR, near-field microscopy, and attenuated total reflectance spectroscopy all exploit this non-zero evanescent field. 'Total reflection' refers to the power balance, not the field amplitude.

Question 4

Because evanescent waves carry no net power, they have no physical consequences and can seldom be detected or exploited in technology.

• T. True
• F. False

Answer: False

Evanescent fields are physically significant despite carrying no net power in the decay direction. Near-field optical microscopy uses evanescent waves to achieve resolution beyond the classical diffraction limit. Attenuated total reflectance spectroscopy places a sample in contact with the evanescent field to measure its absorption spectrum. Optical fiber couplers operate by overlapping evanescent tails of two fibers. Frustrated TIR itself is the basis for some optical switches and sensors. Zero net power flux is not the same as zero physical effect.

Question 5

Why is frustrated total internal reflection considered the optical analogue of quantum-mechanical tunneling?

Model answer: In quantum tunneling, a particle's wavefunction decays exponentially through a classically forbidden potential barrier but re-emerges as a propagating wave on the other side, allowing transmission with finite probability. In frustrated TIR, the evanescent electromagnetic field decays exponentially through the air gap (the 'barrier') and re-emerges as a propagating wave in the second prism. Both phenomena are described by the same mathematical equation — exponential decay of the wave amplitude through a region where the wave vector is imaginary — and in both cases, the 'barrier' does not need to be traversed by a classical traveling wave for transmission to occur.

The mathematical identity is not coincidental: the Schrödinger equation and Maxwell's wave equations are both second-order linear equations, and imaginary wave vector solutions play the same structural role in both. This connection is part of why wave mechanics and electromagnetic theory share so much formal structure, and why insights from optics (TIR, evanescent waves) historically contributed to early quantum theory.