CVI Melles Griot Optics Guide - Single-Layer Antireflection Coatings

Coating Theory

Single-Layer Antireflection Coatings

The simple principles of single-layer antireflection coatings should now be clear. The substrate (glass, quartz, etc.) is coated with a thin layer of material so that reflections from the outer surface of the film and the outer surface of the substrate cancel each other by destructive interference. The intensity of the transmitted beam is correspondingly increased so that, ignoring scattering and absorption, incident energy = reflected energy + transmitted energy. Two requirements create an exact cancellation of reflected beams with a single-layer coating: The reflections are exactly 180 degrees (p radians) out of phase, and they have the same intensity.
Film Thickness
The thickness of a single-layer antireflection film must be an odd number of quarter wavelengths in order to achieve the correct phase for cancellation. This requirement is shown in the figure below, which illustrates the mechanism of a hypothetically perfect single-layer antireflection coating. There is a p/2 phase shift for reflections at both interfaces because they are low to high index medium interfaces. These identical phase shifts cancel each other out. The net phase shift between the two reflections is therefore determined solely by the optical path difference 2_t x n_c, where t is the physical thickness of the coating layer and n_c is the refractive index of the coating material.The phase shift is therefore 2t n/l.

Schematic representation of a single-layer antireflection coating
Single-layer antireflection coatings are generally deposited with a thickness of l/4, where l is the desired wavelength for peak performance. The phase shift is 180 degrees (p radians), and the reflections are in a condition of exact destructive interference.
Refractive Index
The intensity of a reflected beam from a single surface, at normal incidence, is given by

where p is the ratio of the refractive indices of the two materials at the interface.
For the two reflected beams to be equal in intensity, it is necessary that p, the refractive index ratio, be the same at both the interfaces

(i.e., the three refractive indices must form a geometric progression). Since the refractive index of air is 1.0, the thin antireflection film ideally should have a refractive index of

Optical glasses typically have refractive indices of between 1.5 and 1.75. Unfortunately, there is no ideal material that can be deposited in durable thin layers with a low enough refractive index to satisfy this requirement exactly (n = 1.23 for an antireflection coating on (crown glass). However, magnesium fluoride (MgF₂) is a good compromise because it forms high-quality, stable films and has a reasonably low refractive index, 1.38 at a wavelength of 550 nm. Magnesium fluoride is probably the most widely used thin-film material for optical coatings. Although its performance is not outstanding, it represents a significant improvement over an uncoated surface. Typical crown glass surfaces reflect from 4% to 5% of visible (light at normal incidence. A high-quality MgF₂ coating can reduce this value to 1.5%. For many applications this improvement is sufficient, and sophisticated multilayer coatings are not necessary. Such coatings work extremely well over a wide range of wavelengths and angles of incidence, despite the fact that the theoretical target of 0% reflectance is achieved by a film of quarter wavelength optical thickness only for normal incidence, and only if the refractive index of the coating material is exactly the geometric mean of the substrate and air. In fact, the single layer of quarter-wave-thickness MgF₂ coating designed for normal incidence makes its most)significant contribution to the transmission of steep surfaces, where most rays are incident at large angles (see the figure below).

Performance of a nominal incidence coating design for 550 nm working at 45 degrees compared with a 45-degree incidence coating working at 45 degrees.
Wavelength Dependence
As with any thin film, performance depends on the incident light wavelength for two reasons. First, at other than the design wavelength, film thickness is no longer the ideal l/4. This is taken into account by all thin-film design programs. A more subtle effect, which can be quite important, is caused by the change in refractive index of the coating and substrate with wavelength (i.e., dispersion). Only the most up-to-date computer design packages, such as those used by CVI Melles Griot, include this higher level of sophistication for multilayer coatings. For single-layer antireflection coatings, wavelength dependence of the coating performance can be evaluated from analytical expressions.
Angle of Incidence
The irradiance reflectance of any thin-film coating varies with the angle of incidence. Two main effects lead to a complicated dependence of reflectance (hence transmission) on the angle of incidence. First, the path difference of the front and rear surface reflection from any layer is a function of angle. As the angle of incidence increases from zero (normal incidence), the optical path difference is decreased. The change in path difference results in a change of phase difference between the two interfering reflections in an identical manner to the phase change resulting from tilting a Fabry-Perot interferometer. The reflectance of any optical interface varies according to the angle of incidence (click here for illustration). Thin-film performance evaluation at arbitrary angles of incidence is therefore quite complex, even for a simple one-layer antireflection coating. In short, the phase difference between the two pertinent reflections changes together with their relative amplitude.
Coating Formulas
Because of the practical importance and wide usage of single-layer coatings, especially at oblique incidence, it is valuable to have formulas from which coating reflectance curves, as functions of wavelength, angle of incidence, and polarization, can be calculated.
Coating Dispersion Formula
The first step in evaluating performance of a single-layer anti-reflection coating is to calculate the refractive index of the film and substrate at the wavelength of interest. For optical purposes, a thin film may be considered to be perfectly homogeneous. The refractive index of MgF₂, whether amorphous or crystalline, is connected to)density with the Lorentz-Lorenz formula. The crystalline ordinary and extraordinary indices of refraction may be averaged for the amorphous phase. The formulas for crystalline MgF₂ are, respectively,

and

for the ordinary and extraordinary rays, where l is the wavelength in microns. For the average of the ordinary and extraordinary indices of refraction,

The value 1.38 is the universally accepted amorphous film index for MgF₂ at a wavelength of 550 nanometers, which assumes a packing density of 100%. Real films, however, tend to be slightly porous. The refractive index of a real magnesium fluoride film is usually slightly lower than 1.38 because the packing density is rarely 100% in practice. Because it is a complex function of the manufacturing process, packing density varies slightly from batch to batch. Air and water vapor can also settle in the film and affect its refractive index. For CVI Melles Griot magnesium fluoride coatings, this will usually correspond to an effective refractive index between 97% and 100% of the 1.38 theoretical value.
Coated Surface Reflectance at Normal Incidence
Suppose that the coating is of quarterwave optical thickness for some wavelength l. Let n_a denote the refractive index of the external medium at this wavelength (1.0 for air or vacuum), and let n_f and n_s, respectively, denote the film and substrate indices, as shown in the figure below.

Reflectance at normal incidence
For normal incidence at this wavelength, the single-pass irradiance reflectance of the coated surface can be shown to be

regardless of the polarization state of the incident radiation. The function is plotted in the figure below.

Reflectance at surface of substrate with index n_g when coated with a quarter wavelength of magnesium fluoride (index n = 1.38)
Coating Surface Reflectance at Oblique Incidence
At oblique incidence, the situation is more complex. Let n₁, n₂, and n₃, respectively, represent the wavelength-dependent refractive indices)of the external medium (air or vacuum), coating film, and substrate as shown in the figure below.

Reflectance at oblique incidence
Assume that the coating exhibits a reflectance extremum of the first order for some wavelength l_d and angle of incidence q₁_d in the external medium. The coating is completely specified when q₁_d and l_d are known. One may then identify n₂ with the film index n_f (1.38 for MgF₂ at 550 nm). The extremum is a minimum if n₂ is less than n₃ and a maximum if n₂ exceeds n₃. The same formulas apply in either case. Corresponding to the angle of incidence q_1d is an angle of refraction in film:

As q₁is reduced from q_1d to zero, the reflectance extremum shifts in wavelength from l_d to l_n, where the subscript n denotes normal incidence. The wavelength is given by the equation

Corresponding to the arbitrary angle of incidence q₁ and arbitrary wavelength l₁ are angles of refraction in the coating and substrate, given by

and

Following are formulas for the single-interface amplitude reflectances for both the p- and s-polarizations:




The subscript "12p," for example, means that the formula gives the amplitude reflectance for the p-polarization at the interface between the first and second media. The corresponding irradiance reflectances for the coated surface, accounting for both interferences and the phase differences between the reflected waves, are given by

and

where b is the phase difference (in the external medium) between waves reflected from the first and second surfaces of the coating.

The cosines must be in radians. The average reflectance is given by

With these formulas, reflectance curves can be calculated as functions of either wavelength l₁or angle of incidence q₁.

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