Frustrated total internal reflection

Frustrated total internal reflection (FTIR) occurs when light that is totally reflected at a high-to-low index boundary finds a second high-index medium a short distance beyond it. The evanescent wave that extends into the gap couples into a propagating wave in the second medium, and part of the power crosses the gap while the rest is reflected. The transmitted fraction falls roughly exponentially with gap width on a scale of the evanescent penetration depth, typically 100–300 nm for glass and air at visible wavelengths. Two 45° prisms of index 1.52 at 633 nm transmit 72 % of s-polarized light across a 100 nm air gap, 16 % across 300 nm and 0.07 % across 1 µm.

Transmission through the gap

Above the critical angle, 41.1° for glass of index 1.52 against air, the field in the gap decays as $e^{-\kappa z}$ with

$$\kappa = k_0\sqrt{n_1^2\sin^2\theta - n_2^2},$$

as described under total internal reflection. For a gap of width $g$ between two identical media and s-polarized light, the transmittance is

$$\frac{1}{T_s} = 1 + \left(\frac{k_z^2+\kappa^2}{2k_z\kappa}\right)^{2} \sinh^2(\kappa g),$$

where $k_z = k_0 n_1\cos\theta$ is the normal wavevector component in the prisms. This expression has the same form as the transmission of a quantum particle through a rectangular potential barrier. For gaps much larger than $1/\kappa$ it reduces to an exponential with a prefactor,

$$T_s \approx 4\left(\frac{2k_z\kappa}{k_z^2+\kappa^2}\right)^{2} e^{-2\kappa g}.$$

For $n_1$ = 1.52, $\theta$ = 45° and $\lambda_0$ = 633 nm, the field decay length $1/\kappa$ is 256 nm and the prefactor is 1.67. The full expression gives:

The p-polarized wave, obtained by weighting $k_z$ and $\kappa$ by $1/n^2$ in the same formula, tunnels more easily, so a gap that transmits half the light is 156 nm for s and 223 nm for p. The difference makes FTIR devices polarization dependent (see s- and p-polarization). Near the critical angle the decay length grows without limit, and the transmission at a fixed gap rises steeply.

Observing it in the lab

The classic demonstration presses two right-angle prisms together with a slight curvature or a thin spacer on one face. Where the surfaces touch, light passes straight through; around the contact the transmitted spot fades over a ring a few hundred nanometers in gap height. Because only skin within a fraction of a wavelength of the glass frustrates the reflection, a fingerprint pressed on a prism face appears as dark ridges in the reflected image where the skin contacts the glass, which is the basis of optical fingerprint readers and some multitouch screens.

Where it is used

• Variable beam splitters. Two prisms separated by a piezo-controlled gap act as a beam splitter whose ratio is set by the gap.
• Prism coupling. A prism coupler brings a high-index prism within about 0.1 µm of a thin film so that light tunnels into a guided mode; the coupling angle gives the mode's effective index.
• Surface plasmon excitation. In the Otto configuration, the evanescent field crosses a gap to excite a surface plasmon on a metal surface.
• Waveguide coupling. The power transfer in a directional coupler is the guided-wave version: evanescent tails of two waveguides overlap across a gap of a few hundred nanometers.

Pitfalls

Gap widths of 100–300 nm are smaller than many dust particles, so contamination can hold the surfaces apart and stop the effect entirely. Surfaces must be flat and parallel to a small fraction of the decay length across the beam, or the splitting ratio varies across the aperture. Unintended FTIR also occurs: two optical surfaces in near-contact, a fiber end close to a component, or a finger on a TIR prism can leak light from a path that was designed to be totally reflecting. An index-matching gel in the gap removes TIR altogether, which is the reverse case.

Common questions

Is frustrated total internal reflection the same as quantum tunneling?

The mathematics is the same, with the gap playing the role of the potential barrier and the evanescent field the role of the decaying wavefunction. FTIR is a classical wave effect fully described by Maxwell's equations.

How small must the gap be?

A gap comparable to the field decay length, typically 100–300 nm for glass and air in the visible, transmits a large fraction; at two decay lengths, about 512 nm in the example above, s-polarized transmission falls to about 3 %, and at four to under 0.1 %.

Does any light cross the gap in ordinary total internal reflection?

No time-averaged power crosses into a semi-infinite low-index medium. Power crosses only when a second high-index medium, or an absorber, is within reach of the evanescent field.

References: M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999); E. Hecht, Optics, 5th ed. (Pearson, 2017); S. Zhu, A. W. Yu, D. Hawley and R. Roy, "Frustrated total internal reflection: A demonstration and review," American Journal of Physics 54, 601 (1986).