Goos-Hänchen Shift: The Lateral Beam Displacement at Total Internal Reflection

What the Goos-Hänchen Shift Is

Point a collimated light beam at the inside face of a glass block at an angle steeper than the critical angle θc = arcsin(1/n). Geometric optics says the beam reflects perfectly, its virtual point of reflection sitting exactly where the incident ray strikes the interface. Experiment says otherwise: the reflected beam emerges laterally displaced by a distance D, sliding forward along the surface in the plane of incidence.

This is the Goos-Hänchen (GH) shift. Crucially, it appears only for a bounded beam — a real beam is a superposition of many plane-wave components spanning a small range of angles. It is not a change in the beam's direction; the reflected beam travels parallel to the geometric prediction but is offset sideways, as if reflecting off a mirror buried a fraction of a wavelength inside the rarer medium.

• Occurs under total internal reflection (θ > θc), where |reflection coefficient| = 1.
• Magnitude is on the order of the wavelength λ.
• Depends on polarization: TE (s) and TM (p) shifts differ.

The Mechanism: Evanescent Waves and the Reflection Phase

Under total internal reflection the transmitted field does not vanish — it becomes an evanescent wave that penetrates the rarer medium, decaying exponentially with depth over a scale d ≈ λ/(2π√(n²sin²θ − 1)), typically 100-300 nm. This wave carries energy parallel to the surface for a short distance before the light re-couples into the reflected beam, physically offsetting it.

Quantitatively, the reflection coefficient has unit magnitude but a nonzero phase δ(θ) that varies with angle. Karl Artmann showed in 1948 by a stationary-phase argument that each plane-wave component of the beam picks up a slightly different phase, and their superposition shifts the beam by

• D = -(λ/2π)·(dδ/dθ) = -(1/k)·(dδ/dθ), with k = 2π/λ the wavenumber.

The steeper the phase's angular slope, the larger the shift. Because the Fresnel phases for TE and TM light differ, so do their shifts — the hallmark of the GH effect.

Key Quantities and a Worked Example

For light going from a dense medium (index n) into vacuum/air, the total-internal-reflection phases are

• TE (s): δ_TE = -2·arctan(√(n²sin²θ − 1) / (n cosθ))
• TM (p): δ_TM = -2·arctan(n√(n²sin²θ − 1) / cosθ)

Feeding these into Artmann's formula gives the compact result

• D_TE = (λ/π) · tanθ / √(n²sin²θ − 1)
• D_TM = D_TE / (n²sin²θ − cos²θ)

Example. Take n = 1.5 (BK7 glass), λ = 633 nm (HeNe laser), critical angle θc ≈ 41.8°. At θ = 45°, n²sin²θ − 1 = 1.5²·0.5 − 1 = 0.125, so √(0.125) ≈ 0.354, tanθ = 1. Then D_TE = (633 nm/π)·(1/0.354) ≈ 569 nm — just under one wavelength. The TM shift is smaller: dividing by (n²sin²θ − cos²θ) = 1.125 − 0.5 = 0.625 gives D_TM ≈ 569/0.625 ≈ 910 nm here, because near the critical angle that denominator drops below 1 and boosts TM. As θ → θc the formula diverges, a known artifact resolved by finite-beam theories.

How It Is Observed and Applied

A single shift of ~500 nm is nearly impossible to see directly, so Goos and Hänchen used a clever trick: a beam that undergoes many successive total internal reflections inside a slab accumulates the shift N times. By comparing a totally reflecting region against a silver-coated strip (ordinary metallic reflection, no GH shift) on the same prism, they amplified a sub-micron effect into a measurable displacement of the exit spot, confirming Picht's 1929 prediction.

Modern detection is far more sensitive:

• Weak-value amplification uses nearly-crossed polarizers to magnify the polarization-dependent shift by factors of 10³-10⁴, resolving displacements of ångström scale.
• Position-sensitive detectors and interferometry track the beam centroid directly.

Applications exploit the shift's exquisite sensitivity to surface conditions: optical sensing and biosensing (a monolayer of adsorbed molecules changes the local index and thus D), surface-plasmon-resonance readouts, temperature and refractive-index metrology, and beam-steering in integrated photonics and waveguide couplers.

Comparison to Related Beam-Shift Phenomena

The Goos-Hänchen shift is one member of a family of non-specular reflection effects that arise because a real beam is not a single plane wave. Distinguishing them clarifies what GH is not:

• Imbert-Fedorov (IF) shift: a transverse displacement, perpendicular to the plane of incidence, driven by the spin-orbit interaction of light. It flips sign with the handedness of circular polarization, whereas GH lives entirely in the plane of incidence.
• Angular GH deviation: a slight change in beam direction (not position), governed by the gradient of the reflection amplitude rather than its phase; it becomes important right at the critical angle where partial transmission returns.
• Wigner / Hartman time delay: the temporal analogue, D_time = dδ/dω, the dwell time of the pulse in the evanescent region — the same phase-gradient idea applied to frequency instead of angle.

All share the stationary-phase mathematics; they differ in which variable (in-plane angle, polarization, or frequency) the reflection phase is differentiated against.

Significance, Open Questions, and Famous Cases

The Goos-Hänchen shift matters because it is a rare, tabletop manifestation of the evanescent wave — the same clinging near-field that underlies frustrated total internal reflection, optical tunneling, near-field microscopy, and the quantum-tunneling analogy. Isaac Newton anticipated a version of it in his 1704 Opticks, describing light appearing to "turn" before leaving a surface; it took 243 years for Goos and Hänchen to catch it.

The effect is genuinely universal. It has been observed not only for light but for microwaves, X-rays, acoustic and seismic waves, matter waves (neutrons, 2010), and electrons in graphene, where the GH shift at a p-n junction can reach tens of nanometers and is being explored for valley-tronic beam control.

• Negative and giant shifts: near surface-plasmon resonances or in metamaterials and photonic crystals, D can reverse sign or reach tens to hundreds of wavelengths.
• Open issue: the Artmann formula's divergence exactly at θc; finite-beam and Bessel-function treatments keep D finite, but the precise value at the critical angle remains subtle and depends on beam width.

The Goos-Hänchen shift compared with related non-specular beam effects at an interface

Effect	Displacement plane	Origin	Polarization behavior
Goos-Hänchen (GH)	In plane of incidence (along surface)	dφ/dθ — angular gradient of reflection phase	D_TM = D_TE / (n²sin²θ − cos²θ)
Imbert-Fedorov (IF)	Perpendicular to plane of incidence (transverse)	Spin-orbit coupling of light; dφ/d(polarization)	Opposite sign for left/right circular polarization
Angular GH deviation	In-plane change of beam direction	Amplitude (not phase) gradient across beam	Vanishes deep inside TIR; grows near θc
Fresnel filtering	In-plane, curved interfaces	Angular reflectivity selecting sub-beams	Strong near critical angle on curved surfaces
Wigner time delay (temporal cousin)	Time, not space	dφ/dω — frequency gradient of phase	Both polarizations delayed on reflection