Theory Anti-reflective coating

1 theory

1.1 reflection
1.2 rayleigh s film
1.3 interference coatings
1.4 textured coatings

theory

an anti-reflection coated window, shown @ 45° , 0° angle of incidence

there 2 separate causes of optical effects due coatings, called thick-film , thin-film effects. thick-film effects arise because of difference in index of refraction between layers above , below coating (or film); in simplest case, these 3 layers air, coating, , glass. thick-film coatings not depend on how thick coating is, long coating thicker wavelength of light. thin-film effects arise when thickness of coating approximately same quarter or half wavelength of light. in case, reflections of steady source of light can made add destructively , hence reduce reflections separate mechanism. in addition depending on thickness of film , wavelength of light, thin-film coatings depend on angle @ light strikes coated surface.

reflection

whenever ray of light moves 1 medium (for example, when light enters sheet of glass after travelling through air), portion of light reflected surface (known interface) between 2 media. can observed when looking through window, instance, (weak) reflection front , surfaces of window glass can seen. strength of reflection depends on ratio of refractive indices of 2 media, angle of surface beam of light. exact value can calculated using fresnel equations.

when light meets interface @ normal incidence (perpendicularly surface), intensity of light reflected given reflection coefficient, or reflectance, r:

r
=


(




n

0


−

n

s





n

0


+

n

s





)


2


,


{\displaystyle r=\left({\frac {n_{0}-n_{s}}{n_{0}+n_{s}}}\right)^{2},}

where n0 , ns refractive indices of first , second media respectively. value of r varies 0 (no reflection) 1 (all light reflected) , quoted percentage. complementary r transmission coefficient, or transmittance, t. if absorption , scattering neglected, value t 1 − r. if beam of light intensity incident on surface, beam of intensity ri reflected, , beam intensity ti transmitted medium.

for simplified scenario of visible light travelling air (n0 ≈ 1.0) common glass (ns ≈ 1.5), value of r 0.04, or 4%, on single reflection. @ 96% of light (t = 1 − r = 0.96) enters glass, , rest reflected surface. amount of light reflected known reflection loss.

in more complicated scenario of multiple reflections, light travelling through window, light reflected both when going air glass , @ other side of window when going glass air. size of loss same in both cases. light may bounce 1 surface multiple times, being partially reflected , partially transmitted each time so. in all, combined reflection coefficient given 2r/(1 + r). glass in air, 7.7%.

rayleigh s film

as observed lord rayleigh, thin film (such tarnish) on surface of glass can reduce reflectivity. effect can explained envisioning thin layer of material refractive index n1 between air (index n0) , glass (index ns). light ray reflects twice: once surface between air , thin layer, , once layer-to-glass interface.

from equation above , known refractive indices, reflectivities both interfaces can calculated, denoted r01 , r1s respectively. transmission @ each interface therefore t01 = 1 − r01 , t1s = 1 − r1s. total transmittance glass t1st01. calculating value various values of n1, can found @ 1 particular value of optimal refractive index of layer, transmittance of both interfaces equal, , corresponds maximal total transmittance glass.

this optimal value given geometric mean of 2 surrounding indices:

n

1


=



n

0



n

s




.


{\displaystyle n_{1}={\sqrt {n_{0}n_{s}}}.}

for example of glass (ns ≈ 1.5) in air (n0 ≈ 1.0), optimal refractive index n1 ≈ 1.225.

the reflection loss of each interface approximately 1.0% (with combined loss of 2.0%), , overall transmission t1st01 of approximately 98%. therefore, intermediate coating between air , glass can halve reflection loss.

interference coatings

the use of intermediate layer form anti-reflection coating can thought of analoguous technique of impedance matching of electrical signals. (a similar method used in fibre optic research, index-matching oil used temporarily defeat total internal reflection light may coupled or out of fiber.) further reduced reflection in theory made extending process several layers of material, gradually blending refractive index of each layer between index of air , index of substrate.

practical anti-reflection coatings, however, rely on intermediate layer not direct reduction of reflection coefficient, use interference effect of thin layer. assume layer s thickness controlled precisely, such 1 quarter of wavelength of light in layer (λ/4 = λ0/(4n1), λ0 vacuum wavelength). layer called quarter-wave coating. type of coating incident beam i, when reflected second interface, travel half own wavelength further beam reflected first surface, leading destructive interference. true thicker coating layers (3λ/4, 5λ/4, etc.), anti-reflective performance worse in case due stronger dependence of reflectance on wavelength , angle of incidence.

if intensities of 2 beams r1 , r2 equal, destructively interfere , cancel each other, since out of phase. therefore, there no reflection surface, , energy of beam must in transmitted ray, t. in calculation of reflection stack of layers, transfer-matrix method can used.

real coatings not reach perfect performance, though capable of reducing surface reflection coefficient less 0.1%. also, layer have ideal thickness 1 distinct wavelength of light. other difficulties include finding suitable materials use on ordinary glass, since few useful substances have required refractive index (n ≈ 1.23) make both reflected rays equal in intensity. magnesium fluoride (mgf2) used, since hard-wearing , can applied substrates using physical vapor deposition, though index higher desirable (n = 1.38).

further reduction possible using multiple coating layers, designed such reflections surfaces undergo maximal destructive interference. 1 way add second quarter-wave thick higher-index layer between low-index layer , substrate. reflection 3 interfaces produces destructive interference , anti-reflection. other techniques use varying thicknesses of coatings. using 2 or more layers, each of material chosen give best possible match of desired refractive index , dispersion, broadband anti-reflection coatings covering visible range (400–700 nm) maximal reflectivities of less 0.5% commonly achievable.

the exact nature of coating determines appearance of coated optic; common ar coatings on eyeglasses , photographic lenses bluish (since reflect more blue light other visible wavelengths), though green , pink-tinged coatings used.

if coated optic used @ non-normal incidence (that is, light rays not perpendicular surface), anti-reflection capabilities degraded somewhat. occurs because phase accumulated in layer relative phase of light reflected decreases angle increases normal. counterintuitive, since ray experiences greater total phase shift in layer normal incidence. paradox resolved noting ray exit layer spatially offset entered , interfere reflections incoming rays had travel further (thus accumulating more phase of own) arrive @ interface. net effect relative phase reduced, shifting coating, such anti-reflection band of coating tends move shorter wavelengths optic tilted. non-normal incidence angles cause reflection polarization-dependent.

textured coatings

reflection can reduced texturing surface 3d pyramids or 2d grooves (gratings).

if wavelength greater texture size, texture behaves gradient-index film reduced reflection. calculate reflection in case, effective medium approximations can used. minimize reflection, various profiles of pyramids have been proposed, such cubic, quintic or integral exponential profiles.

if wavelength smaller textured size, reflection reduction can explained of geometric optics approximation: rays should reflected many times before sent toward source. in case reflection can calculated using ray tracing.

using texture reduces reflection wavelengths comparable feature size well. in case no approximation valid, , reflection can calculated solving maxwell equations numerically.

antireflective properties of textured surfaces discussed in literature wide range of size-to-wavelength ratios (including long- , short-wave limits) find optimal texture size.