Anti-Reflection Coatings: How a Quarter-Wave of MgF₂ Makes Glass Vanish

The problem: Fresnel reflection at every interface

Whenever light crosses a boundary between two transparent media with different refractive indices, part of it reflects. At normal incidence, the fraction of intensity reflected is given by the Fresnel equation:

• R = ((n₂ − n₁)/(n₂ + n₁))²

Here n₁ and n₂ are the refractive indices on either side of the interface. For air (n₁ = 1.00) meeting crown glass (n₂ = 1.52):

• R = ((1.52 − 1.00)/(1.52 + 1.00))² = (0.52/2.52)² = (0.2063)² ≈ 0.043, i.e. 4.3%.

That 4.3% happens twice per lens — once entering, once leaving — so a single air-spaced element transmits only about 0.957² ≈ 91.6% of the light. In a compound lens with, say, 14 air/glass surfaces, the surviving fraction is 0.957¹⁴ ≈ 0.54. Nearly half the light is gone, and the stray reflections don't just vanish — they bounce around inside the barrel and reappear as ghost images and veiling flare that crush contrast. Cutting each surface's reflectance from 4% to 0.2% raises that 14-surface transmission from 54% to about 97% and kills the ghosts. That is the whole motivation for AR coatings.

The quarter-wave trick: two waves, exactly out of step

Put a thin transparent film of index n₁ (intermediate between air n₀ and glass n₂) on the surface. Now light reflects from two interfaces: the top (air/film) and the bottom (film/glass). The AR idea is to make these two reflected waves cancel.

Cancellation needs two things at once — the right phase and the right amplitude.

• Phase (thickness) condition. The bottom reflection travels an extra distance of twice the film thickness, 2d, inside the film. For the two waves to be a half-wavelength out of phase — a path difference of λ/2 — we need 2·n₁·d = λ₀/2, which gives the famous quarter-wave rule: n₁·d = λ₀/4. The film's optical thickness is one quarter of the design wavelength.
• Phase from reflection. Both reflections here occur going from a lower to a higher index (air→film and film→glass, since n₀ < n₁ < n₂), so each picks up the same π phase shift. Those cancel out, and only the 2n₁d path term matters — cleanly delivering the π (half-wave) offset we want.

For MgF₂ (n₁ = 1.38) designed at λ₀ = 550 nm, the physical thickness is d = λ₀/(4n₁) = 550/(4·1.38) ≈ 99.6 nm. A hundred nanometers of fluoride is all it takes.

The amplitude condition: why n₁ = √(n₀·n₂)

Getting the phase right only guarantees the two reflected waves point in opposite directions. To reach true zero they must also be equal in magnitude. The amplitude reflection coefficients (small-r Fresnel coefficients) at the two interfaces are:

• r₁ = (n₀ − n₁)/(n₀ + n₁) at the air/film boundary
• r₂ = (n₁ − n₂)/(n₁ + n₂) at the film/glass boundary

Setting |r₁| = |r₂| and solving gives the elegant index-matching condition:

• n₁ = √(n₀·n₂) — the coating index should be the geometric mean of the media it sits between.

For air and crown glass: n₁ = √(1.00 × 1.52) = √1.52 ≈ 1.23. When both conditions are met, the exact minimum reflectance is R = ((n₀n₂ − n₁²)/(n₀n₂ + n₁²))², which goes to zero at n₁² = n₀n₂. The catch: no durable solid material has an index as low as 1.23. The lowest practical coating is magnesium fluoride at n₁ = 1.38. Plugging that in, the residual reflectance at the design wavelength is R = ((1.52 − 1.38²)/(1.52 + 1.38²))² = ((1.52 − 1.904)/(1.52 + 1.904))² ≈ (0.112)² ≈ 1.3% — a threefold improvement over bare glass, but not zero. Chasing that last percent is why single layers give way to multilayers.

Controlling variables: wavelength, angle, and polarization

A quarter-wave film is tuned to one wavelength, but light is broadband and rarely hits dead-on. Three variables shift the performance:

• Wavelength. Because the phase term is 2π·(2n₁d)/λ, a film that is exactly λ/4 at 550 nm is only ~0.20λ at 700 nm and ~0.31λ at 450 nm. The reflectance minimum is a shallow V-shape in wavelength — hence designers place the minimum near 510–550 nm, where the eye is most sensitive, leaving a faint purple/magenta residual glow (leftover blue and red) that is the signature of coated lenses.
• Angle of incidence. At oblique angle θ the light travels a longer path through the film, so the effective optical thickness scales with cos of the internal angle. The reflectance minimum blue-shifts as θ increases, which is why a coated surface looks more colored and more reflective when tilted toward grazing.
• Polarization. Off-axis, the s- and p-polarizations reflect differently (this is the same physics as Brewster's angle), so a coating optimized for unpolarized normal incidence degrades unevenly at steep angles.

Real systems also must respect the substrate index: a high-index flint glass (n ≈ 1.9) or a silicon solar cell (n ≈ 3.9 in the visible) needs a completely different coating index than crown glass — for silicon, an ideal single layer wants n₁ = √(1×3.9) ≈ 1.97, close to silicon nitride (Si₃N₄, n ≈ 2.0), which is exactly why solar cells wear that characteristic dark-blue Si₃N₄ coat.

Beyond one layer: V-coats, broadband stacks, and moth eyes

To beat the single-layer floor, engineers stack multiple dielectric films and treat the whole thing as an interference problem solved by the transfer-matrix method (a 2×2 characteristic matrix per layer, multiplied together to get the total reflectance). Three families dominate:

• V-coat (two layers). A high-index layer under a low-index layer relaxes the √(n₀n₂) constraint and can drive R below 0.25% — but only at one wavelength, giving a narrow V-shaped dip. Ideal for lasers, where the wavelength is fixed (e.g. 1064 nm).
• Broadband multilayer (3–7+ layers). Alternating high/low index films (e.g. TiO₂/SiO₂, Ta₂O₅/SiO₂) flatten the reflectance below 0.2% across the whole 400–700 nm visible band. This is the standard on premium camera lenses and eyeglasses.
• Moth-eye / gradient-index. Instead of discrete films, a forest of sub-wavelength bumps (spacing < λ) makes the effective index rise gradually from air to glass. With no sharp interface, there is nothing to reflect — reflectance drops below 0.1% over a huge angular and spectral range. Nature invented it: the nocturnal moth's cornea is nanostructured this way so predators can't see the glint of its eyes.

Where AR coatings show up — and the numbers that matter

The energy that a coating stops reflecting is transmitted, so AR coatings quietly boost the performance of almost every optical product:

• Camera and telescope optics. Multicoating (Zeiss T*, Nikon Nano-Crystal, Canon SubWavelength Structure) lifts a complex lens from ~55% to ~97% transmission and suppresses the ghost images that ruin backlit shots. The residual reflection is what gives coated front elements their green-violet sheen.
• Solar cells. A bare silicon wafer reflects >35% of sunlight; a ~75 nm Si₃N₄ AR layer cuts that to under 3%, adding several absolute percentage points to module efficiency — a huge lever on levelized cost.
• Eyeglasses and displays. AR coatings remove the distracting reflections others see in your lenses and stop the ~4%-per-surface haze that washes out phone and monitor screens in bright light.
• Lasers and fiber optics. V-coated windows and lens ends hold reflection to < 0.1% at the operating wavelength, preventing feedback that can destabilize the cavity or damage the source.

A rough figure of merit: each 4% reflection you eliminate is roughly 0.18 dB of insertion loss recovered, and in a many-surface system those decibels add up fast.

Subtleties and misconceptions

It doesn't absorb light — it redirects the energy. An AR coating is a lossless interference device; the light that isn't reflected is transmitted, not soaked up. This is the opposite of a neutral-density filter or a dark tint.

Reflected energy really does go into transmission. Conservation of energy still holds. The two reflected wavelets carry near-zero net energy backward; that missing reflected power reappears as extra transmitted power. There is no violation — destructive interference in one direction is always paired with constructive interference in the complementary channel.

You cannot get zero over all wavelengths with one film. The quarter-wave condition n₁d = λ₀/4 is wavelength-specific; a single layer is a narrowband trick, and its residual color (that magenta bloom) is unavoidable. Only multilayers approximate broadband suppression.

A single dielectric layer never gives an exact zero on glass unless the index matches. Because MgF₂'s 1.38 exceeds the ideal 1.23, even a perfect quarter-wave leaves ~1.3%. People sometimes assume MgF₂ makes reflection vanish — it merely quarters it.

The coating must be tuned to the substrate. The same MgF₂ film that is excellent on crown glass is mediocre on high-index flint and useless on silicon; the optimal n₁ = √(n₀n₂) shifts with the substrate, which is why solar cells, camera glass, and IR windows all wear different coatings.

Reflection loss per surface for common coating strategies at 550 nm (crown glass, n₂ = 1.52, normal incidence)

Surface treatment	Coating index n₁	Thickness	Reflectance R	Note
Bare glass	—	none	~4.3%	Fresnel loss at air/glass
MgF₂ single layer	1.38	~100 nm	~1.3%	n₁ > √n₂, so imperfect
Ideal single layer	1.23	~112 nm	~0%	No solid is this low
V-coat (2 layers)	1.38 / 2.1	λ/4 each	<0.25%	Narrowband, one wavelength
Broadband multilayer	3–7 layers	mixed	<0.2%	Flat across visible
Moth-eye nanostructure	gradient	~250 nm	<0.1%	Index gradient, no discrete film