Reflection

Reflection is the return of light from a boundary into the medium it came from. It happens wherever the refractive index changes: at an air–glass surface, at a metal, at the facet of a laser chip. The fraction of power returned, the reflectance $R$, depends on the index contrast, the angle of incidence and the polarization. For light arriving perpendicular to a surface between two transparent media,

$$R = \left(\frac{n_1 - n_2}{n_1 + n_2}\right)^2 .$$

An air–glass surface with $n = 1.5$ reflects 4.0%; N-BK7 at 587.6 nm ($n = 1.5168$) reflects 4.2%; water ($n = 1.333$) reflects 2.0%; silicon at 1550 nm ($n \approx 3.48$) reflects 30.6%. Metals reflect far more because their free electrons respond strongly to the optical field: aluminum around 90% across the visible, silver and gold above 95% in the near infrared.

Law of reflection and specular versus diffuse

At a smooth surface the reflected ray lies in the plane of incidence and leaves at the same angle to the normal as the incident ray, $\theta_r = \theta_i$. This is specular reflection, and it preserves the wavefront, so a flat mirror forms an image. When the surface roughness is comparable to or larger than the wavelength, each patch reflects in a different direction and the light is scattered into a broad range of angles: diffuse reflection. White paper, ground glass and matte paint are diffuse reflectors; a polished mirror substrate with roughness of a nanometer or less is specular. Most real surfaces show both components, and an optical scatter measurement separates them by angle.

Angle and polarization dependence

Away from normal incidence the reflectance differs for the two polarizations and is given by the Fresnel equations. For p-polarized light (electric field in the plane of incidence) the reflectance falls to zero at Brewster's angle, $\tan\theta_B = n_2/n_1$, which is 56.3° for $n = 1.5$. The s-polarized reflectance rises monotonically with angle. Both approach 100% at grazing incidence. When light travels from the denser medium toward a lower-index one beyond the critical angle (41.8° for glass to air at $n = 1.5$), no power is transmitted at all: total internal reflection, the mechanism of prisms used as mirrors and of optical fiber.

Worked number: a glass window

An uncoated window has two surfaces. Each reflects 4.0% at $n = 1.5$, so a quick estimate gives $1 - (1 - 0.04)^2 = 7.8\%$ total reflected. Summing the incoherent multiple reflections between the two faces gives a transmittance of

$$T = \frac{1 - R}{1 + R} = \frac{2n}{n^2 + 1} = 92.3\%,$$

so the window reflects 7.7%, assuming no absorption. With a coherent source and a thin, parallel window the two reflections interfere and the result oscillates between about 0% and 15% with thickness and wavelength: the window is then a weak etalon.

Where it matters

Every uncoated surface in a multi-element lens loses a few percent and sends that light back toward the sensor as ghosts and flare, which is why anti-reflection coatings are standard. In fiber systems, a reflection returning to a laser destabilizes it; the relevant specification is optical return loss, and angled connectors exist to keep it high. In the other direction, laser cavities need controlled high reflectance: cleaved semiconductor facets give about 30% without any coating, and dielectric stacks such as the distributed Bragg reflector reach 99.9% and above by adding many weak reflections in phase.

Measurement and pitfalls

Reflectance is measured by comparing the reflected power with the incident power on the same detector, or against a calibrated reference mirror in a spectrophotometer reflectance accessory. For low-reflectance coatings, the back-surface reflection of the sample often exceeds the front-surface value being measured; a wedged or index-matched, blackened back surface removes it. Metal mirror data sheets usually quote averages over a band and at a stated angle, typically 45° for fold mirrors, and the values for s and p polarization can differ by several percent at that angle.

Common questions

What is the law of reflection?

The angle of reflection equals the angle of incidence, both measured from the surface normal, and the incident ray, the reflected ray and the normal lie in one plane. It follows from requiring the phase of the wave to match along the surface on both sides of the reflection.

Why does a window act as a mirror at night?

The glass reflects about 8% of the light striking it from either side. In daylight the transmitted outdoor scene is far brighter than the reflected room; at night the outdoor scene is dark, so the 8% reflection of the lit room dominates what the eye sees.

How is reflection different from refraction?

Both occur at the same boundary. Reflection is the part returned into the first medium at an equal angle; refraction is the part transmitted into the second medium and bent according to Snell's law. The Fresnel equations give how the power divides between them, and the two fractions sum to one in the absence of absorption, the complement of reflectance being the transmittance.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 4; M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 1; B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019), Ch. 6.