Controlling Reflection with Thin-Film Interference

An optical coating changes how a surface transmits, reflects, and absorbs light. Its response depends on the material at each interface and the phase accumulated between interfaces. A film that suppresses reflection at one wavelength can enhance it elsewhere, so a useful coating specification always includes the operating spectrum and illumination geometry.

Match amplitudes and phases

At normal incidence, the power reflectance of a bare interface between transparent media is:

R_{\mathrm{bare}}=\left(\frac{n_0-n_s}{n_0+n_s}\right)^2

Air against glass with $n_s=1.52$ reflects about 4.26% at that one surface. Adding a transparent film creates two interfaces and multiple internally reflected waves. An intermediate index helps set their amplitudes; the film’s thickness sets their relative phase. Both determine the total reflection.

A dielectric film introduces reflected waves at two interfaces; their combined field depends on amplitude and propagation phase.

View diagram at full size

For a film of index $n_1$ and physical thickness $d$ , the normal-incidence optical thickness is $n_1d$ . The one-way phase thickness is $\delta=2\pi n_1d/\lambda$ , using vacuum wavelength $\lambda$ . Reflection from the lower interface adds a round trip through the film, as well as the phase changes associated with reflection.

Start with a quarter-wave AR layer

An ideal single-layer antireflection coating at wavelength $\lambda_0$ satisfies:

n_1=\sqrt{n_0n_s},\qquad n_1d=\frac{\lambda_0}{4}

Quarter-wave optical thickness, or QWOT, gives a half-wave propagation difference between successive returned beams. The matching-index condition supplies the required amplitude balance. Together they yield zero reflectance in this lossless, planar, normal-incidence model.

Quarter-wave optical thickness produces a reflection minimum when the layer index also matches the entrance and substrate media.

View diagram at full size

For air and $n_s=1.52$ , the ideal index is approximately 1.233. A practical material may have a different index and leave residual reflection. Absorption, dispersion, oblique incidence, and deposition variation also change the result. The substrate’s rear surface needs separate treatment.

Calculate the complete single-film reflection

For real, positive refractive indices at normal incidence, define the electric-field reflection amplitudes $r_{01}=(n_0-n_1)/(n_0+n_1)$ and $r_{1s}=(n_1-n_s)/(n_1+n_s)$ . Summing the successive coherent reflections gives:

r=\frac{r_{01}+r_{1s}e^{2i\delta}}{1+r_{01}r_{1s}e^{2i\delta}},\qquad R=|r|^2

At $\delta=\pi/2$ , this becomes:

R_{\mathrm{QW}}=\left(\frac{n_0n_s-n_1^2}{n_0n_s+n_1^2}\right)^2

Thus quarter-wave thickness alone does not guarantee complete cancellation. A zero-thickness film or an integer half-wave film returns the bare-interface reflectance at that wavelength. This expression assumes a semi-infinite substrate and excludes a second coated or uncoated rear face.

Build a multilayer response

Alternating higher- and lower-index dielectrics, such as tantalum pentoxide and silica, provide a common design framework. A sequence of quarter-wave H/L pairs forms a Bragg reflector: the reflected contributions reinforce over a stopband. Adding pairs strengthens rejection; increasing index contrast generally broadens that band.

A resonant spacer between reflecting stacks introduces a transmission band. Coupled cavities and adjusted layer thicknesses then shape its width, ripple, and transitions. A simple quarter-wave reflector is therefore a building block, rather than a finished narrowband telecom filter.

Alternating quarter-wave layers form a reflector; a resonant spacer between reflecting stacks creates a transmission band.

View diagram at full size

Connect the model to deposited material

Evaporation and sputtering are physical vapor deposition processes. Chemical vapor deposition and atomic layer deposition provide other growth mechanisms. Process conditions affect the deposited index, absorption, density, and stress; material names alone do not define those properties. Metals and graded-index coatings extend the design options beyond alternating transparent layers.

Iris Optics can connect coating requirements with illumination geometry, material assumptions, and practical verification so that the specified response supports the complete optical system.

Controlling Reflection with Thin-Film Interference

Match amplitudes and phases

Start with a quarter-wave AR layer

Build a multilayer response

Connect the model to deposited material

A quarter-wave antireflection layer

Build a quarter-wave Bragg reflector

Match the coating to the optical interface.