How High-Reflector Mirrors Work: From Metal Coatings to Quarter-Wave Stacks

Every laser cavity needs a mirror. Every telescope primary needs a reflective coating. But a bare glass surface only reflects about 4% of light. How do you get to 99%? Or 99.99%?

Metal Mirrors: The Simple Approach

A metal reflects light because its free electrons oscillate in response to the incoming electric field and re-radiate most of the energy backward. The reflectance of a bare metal surface at normal incidence follows from the Fresnel equation for complex refractive indices:

where ${\tilde{n}}_{\text{metal}}=n+ik$ is the complex refractive index. For metals, the extinction coefficient k is large (meaning strong absorption), and the real part n is often less than 1. The power reflectance is $R=\mathrm{\mid }r{\mathrm{\mid }}^2$.

Three metals dominate optics:

Silver has the highest visible reflectance of any metal. At 550 nm, silver has n = 0.059 and k = 3.59, giving R = 98.3% from the Fresnel formula. A 100 nm silver film on glass achieves R = 94.9% at 550 nm with a thin SiO₂ protective overcoat. Without the overcoat, silver would tarnish within days in a normal atmosphere, forming dark silver sulfide.

Aluminum is the workhorse mirror material. At 550 nm, aluminum has n = 0.79 and k = 5.85, giving a theoretical R = 91.6%. Protected aluminum shows R = 90.5% at 550 nm with a 25 nm SiO₂ overcoat. Aluminum's great advantage is its flat spectral response: R stays between 88% and 91% across the entire visible spectrum, with no dip at short wavelengths. This is why nearly every telescope primary mirror uses aluminum.

Gold is the mirror of choice for infrared optics. In the red and near-IR (above 600 nm), gold reaches R = 94–97%. But below 520 nm, gold's reflectance collapses. At 500 nm, R drops to 51%, and at 400 nm, it falls to 39%. This happens because gold's electrons can absorb blue and violet photons via an electronic transition — the same physics that gives gold its color.

The key limitation of metal mirrors: absorption. That 1–2% of light that silver absorbs at each reflection doesn't seem like much, but in a laser cavity where light bounces back and forth hundreds of times, every percent matters. At R = 98.3%, after 100 round trips only 3.3% of the original intensity survives. For a high-finesse cavity, you need R > 99.99%, which no metal can achieve.

Quarter-Wave Stacks: The Dielectric Approach

A quarter-wave stack replaces the metal with alternating layers of two transparent materials: one with a high refractive index (H) and one with a low refractive index (L). Each layer is exactly one quarter-wavelength thick at the design wavelength ${\lambda }_0$:

The standard material pair for visible-light mirrors is TiO₂ (n = 2.65 at 550 nm) and SiO₂ (n = 1.46 at 550 nm). At 550 nm, the quarter-wave thicknesses are:

A stack of N pairs is written as (HL)^N. A 7-pair stack means Air | (TiO₂ SiO₂)⁷ | BK7 — 14 layers total, starting with TiO₂ (high index) on the air side and ending with SiO₂ touching the BK7 substrate.

Why Does It Work?

At the design wavelength, each layer is exactly λ/4 thick optically. The quarter-wave thickness is chosen so that all the reflected partial waves add up constructively at the front surface. Here's why, step by step:

1. Light hits the first interface (air to H layer) and partially reflects. The transmitted part enters the H layer.
2. It travels through the H layer (one quarter-wave, λ/4 optical path), reaches the H–L interface, and partially reflects back.
3. The reflected part travels another λ/4 back through the H layer. Total round trip: λ/2, which shifts the phase by π (half a cycle).
4. But there's a second phase shift. Reflection at an interface where light goes from high index to low index (H → L) has no extra phase shift. Reflection where light goes from low to high (L → H) adds another π.
5. π from the round trip + π from the interface = 2π = back in phase with the original reflection.
6. This happens at every interface in the stack. All reflected partial waves arrive at the front surface in phase, adding constructively.

The same interference that maximizes reflection also minimizes transmission: the forward-traveling partial waves interfere destructively inside the stack. At the design wavelength, the stack acts like a mirror.

Peak Reflectance vs. Number of Pairs

The peak reflectance at the design wavelength has an analytical formula:

where N is the number of HL pairs and $n_{\text{sub}}$ is the substrate index. The important part: every additional pair multiplies the transmittance by $(n_L\mathrm{/}{n}_H{)}^2$. For TiO₂/SiO₂, that factor is (1.460/2.648)² = 0.304, so each new pair blocks 70% of the remaining transmitted light.

For TiO₂/SiO₂ on BK7 at 550 nm:

Seven pairs of TiO₂/SiO₂ already give R = 99.94% at the design wavelength. That's an order of magnitude better than silver. And unlike silver, there's zero absorption: R + T = 100% (to within the material's transparency). For laser mirrors that need R = 99.99%, you just add more pairs.

The Stop Band: Where the Mirror Works

A quarter-wave stack doesn't just reflect at a single wavelength. It creates a high-reflectance zone called the stop band centered at the design wavelength. Within the stop band, R is close to the peak value. Outside it, R drops rapidly and oscillates (the familiar sideband ripples).

The width of the stop band depends on the refractive index contrast between H and L. The half-width in normalized frequency units is:

For TiO₂/SiO₂ at 550 nm, $(n_H-n_L)\mathrm{/}(n_H+n_L)=0.289$, giving Δg = 0.187. In wavelength terms, the stop band extends from about 463 nm to 676 nm — a width of 213 nm or 39% of the design wavelength.

That's a wide stop band. The 7-pair TiO₂/SiO₂ mirror maintains R > 99% from 475 nm to 630 nm, covering most of the visible spectrum. The index ratio $n_H\mathrm{/}{n}_L=1.81$ is one of the highest available from common coating materials, which is exactly why TiO₂/SiO₂ is the standard pair for dielectric mirrors.

This matters in practice. If you design a mirror for 550 nm and need R > 99% at 632.8 nm (HeNe laser), you can check: at 625 nm, R = 99.48%, and at 650 nm it drops to 97.62%. So 632.8 nm is right at the edge of the useful range.

Comparing the Two Approaches

Here is where it all comes together. When should you use a metal mirror, and when a dielectric stack?

The rule of thumb: use metal mirrors when you need broadband coverage and can tolerate a few percent absorption. Use dielectric mirrors when you need R > 99.9% or zero absorption, and your wavelength range is limited.

The Enhanced Aluminum Mirror: When More Coating Isn't Better

Consider an aluminum mirror with a quarter-wave MgF₂ overcoat instead of the thin protective SiO₂. You might expect the dielectric overcoat to enhance reflectance.

At 550 nm, it does the opposite: R drops to 86.0%, compared to 90.5% for plain protected aluminum. At 700 nm it's even worse, falling to 81.9%. The non-quarter-wave MgF₂ lets more light penetrate into the aluminum, where it's absorbed. The enhancement only works near 400 nm, where the MgF₂ thickness is closer to a quarter-wave, and R does improve slightly to 91.0%.

The lesson: thin-film interference can work against you if the layer thicknesses aren't matched to the wavelength. This is why coating design is an optimization problem, not just a matter of adding more layers.

Real-World Applications

Laser cavity mirrorsare the highest-performance dielectric mirrors. A typical Nd:YAG laser at 1064 nm uses a high-reflector rear mirror with R > 99.95% (achieved with 15–20 pairs of Ta₂O₅/SiO₂) and a partially-reflective output coupler with R = 80–95%.

Telescope mirrors almost universally use protected aluminum. The Hubble Space Telescope primary mirror is aluminum-coated with a MgF₂ protective layer, optimized for UV performance. Ground-based telescopes like Keck and VLT use bare aluminum or protected silver, recoating every few years as the metal degrades.

The same quarter-wave stack principle also underlies interference filters for fluorescence microscopy, telecom, and astronomy: bandpass filters, edge filters, notch filters, and more.

Going Further

The quarter-wave stack is the starting point for dielectric mirror design, not the end. Real coatings use modified layer thicknesses to widen the stop band, suppress sideband ripples, or create specific spectral profiles. Advanced techniques include chirped mirrors (layers with gradually varying thickness), rugate filters (continuous index gradients), and hybrid metal-dielectric designs that combine broadband coverage with high peak reflectance.

Try loading a mirror preset in the Photizon Thin-Film Simulator and experimenting: change the number of pairs, swap TiO₂ for Ta₂O₅ (n = 2.14, lower contrast), or shift the design wavelength. You'll quickly develop intuition for how quarter-wave stacks behave.

Designing HR mirrors for a high-Q cavity? See the Q-factor tutorial →

Further Reading

Textbook references:

• H. A. Macleod, Thin-Film Optical Filters, 5th ed. (CRC Press, 2017), Ch. 4–5. The definitive treatment of quarter-wave stacks and high-reflectance coatings.
• P. W. Baumeister, Optical Coating Technology (SPIE Press, 2004), Ch. 4. Clear coverage of dielectric mirror design with worked examples.
• E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 9. Undergraduate-level treatment of multilayer interference and reflective coatings.

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