Interference filter design using flip-flop optimization

An antireflective optical film is placed between an incident media and a substrate to effect minimal reflectivity from the incident media-substrate interface over a broad spectral band. The film is designed by selecting a first material with a low refractive index and a second material with a high refractive index. A theoretical film is defined with a plurality of thin layers of equal thickness. The low refractive index or the high refractive index material is specified for each layer in the film. The reflectivity of the theoretical film is evaluated. The refractive index of one of the layers is changed to the alternate index and the reflectivity of the defined film is reevaluated. If the reflectivity of the film is improved, the alternate refractive index is assigned to the changed layer. The steps of changing, reevaluating, and assigning are repeated for each of the layers in the film until no further improvement in the reflectivity of the film is obtained. The thin layers may be defined with equal physical or optical thickness and should be substantially thinner than wavelengths within the spectral band to be reflected by the film. The initial refractive index profile may be defined with all of the layers having the low refractive index, all of the layers having the high refractive index, the layers having alternately low and high refractive indices, or such that the initial refractive index profile for the film approximates a predetermined gradient index profile.

BACKGROUND OF THE INVENTION 
This invention relates to the design of optical coatings for reducing the 
amount of light which is reflected from an optical surface. 
Modification in the intensity of light which may occur when two or more 
beams of light are superposed is known as interference. The principle of 
superposition states that the resultant amplitude is the sum of the 
amplitudes of the individual beams. The brilliant colors, for example, 
which may be seen when light is reflected from a soap bubble or from a 
thin layer of oil floating on water are produced by interference effects 
between two trains of light waves. The light waves are reflected at 
opposite surfaces of the thin film of soap solution or of oil. 
One important practical application for the principles of interference in 
thin films involves the production of coated optical surfaces. If a film 
of a transparent substance is deposited on glass, for example, with a 
thickness which is one quarter of a particular wavelength of light in the 
film, the reflection of that light from the glass surface can be almost 
completely suppressed. The light which would otherwise be reflected is not 
absorbed by a nonreflecting film; rather, the energy in the incident light 
is redistributed so that a decrease in reflection is accompanied by a 
concomitant increase in the intensity of the light which is transmitted. 
Nonreflecting films are of practical importance because they can be used to 
greatly reduce the loss of light by reflection at the various surfaces of 
an optical system. Stray light, which could otherwise reach the image 
because of these reflections, can also be substantially eliminated, with a 
resulting increase in contrast. Such improvements are particularly useful 
where an image is formed by a highly corrected lens system which employs a 
large number of air-glass surfaces. Consequently, almost all optical 
components of high quality are coated to reduce reflection. These coatings 
were first made by depositing several monomolecular layers of an organic 
substance on glass plates. More durable coatings may be fabricated by 
evaporating calcium or magnesium fluoride on the surface in a vacuum, or 
by chemical treatment with acids which leave a thin layer of silica on the 
surface of the glass. 
Considerable improvements have been achieved in the antireflective 
performance of such films by using a composite film having two or more 
superimposed layers. The use of gradient index layers, in which the index 
of refraction within the layer is made to vary continuously as a function 
of depth in the layer, further increases the degrees of freedom available 
in the design of such films. Modern applications of optical technology, 
however, require antireflective films with even lower levels of reflection 
than have previously been attainable in the art. One of the ways in which 
higher performance antireflective layers have been obtained is through 
utilizing digital equivalents of continuous gradient index layers. The 
spectral performance of an arbitrary generalized gradient index 
interference coating may be closely approximated by some digital 
configuration which is a sequence of thin layers each having a high or a 
low refractive index. It is the goal of this invention to further advance 
the performance of antireflective coatings by providing a new technique 
for defining an optimized digital index profile for a film with a given 
thickness and spectral performance. 
SUMMARY OF THE INVENTION 
The invention provides an antireflective optical film which, when placed 
between an incident media and a substrate, effects minimal reflectivity 
from the incident-substrate interface over a broad spectral band. 
The film is designed by the steps of selecting a first material with a low 
refractive index and a second material with a high refractive index, then 
defining a theoretical film with a plurality of thin layers of equal 
thickness. The low refractive index or the high refractive index is 
specified for each layer in the film and the reflectivity of the 
theoretical film is evaluated. The refractive index of one of the layers 
is then changed to the alternate index and the reflectivity of the defined 
film is reevaluated. If the reflectivity of the film is improved, then the 
alternate refractive index is assigned to the changed layer. The steps of 
changing, reevaluating, and assigning are repeated for each of the layers 
in the film until no further improvement in the reflectivity of the film 
is obtained. 
In a more particular embodiment, each layer in the film is made 
substantially thinner than wavelengths within the spectral band to be 
reflected by the film. The step of defining the theoretical film may 
further define the thin layers to be of equal physical thickness or of 
equal optical thickness. 
In another embodiment, the step of specifying the low index or the high 
index for each layer may specify the low index for all of the layers or 
specify the high index for all layers. The layers may be specified to 
initially have alternately low and high refractive indices. The refractive 
indices may be defined such that the initial refractive index profile for 
the film approximates a predetermined gradient index profile.

DESCRIPTION OF THE INVENTION 
This invention addresses the problem of designing a high performance 
digital antireflective coating from materials with two discrete values of 
refractive index so that the digital coating closely approximates a 
gradient index coating. 
FIG. 1 is a cross sectional side view illustrating an optical interface 
with an antireflective film. As those skilled in the art will appreciate, 
some of the dimensions in this drawing are greatly exaggerated in order to 
effectively illustrate the optical interference phenomena which are 
involved. An antireflective film 10 is placed between an incident media 12 
and a substrate media 14, with the incident media having a lower index of 
refraction than the substrate media. When a ray of light 16 traverses this 
system, a portion of the light is refracted and a portion is reflected at 
each interface between the different media. Thus the incident ray 16 is 
divided into a reflected ray 22r and a refracted ray 18 at the boundary 
between the incident media 12 and the film 10. Similarly, the refracted 
ray 18 divides into a reflected ray 20 and a reflected ray 22t at the 
boundary between the film 10 and the substrate media 14. This division 
continues at the incident media-film and film-substrate media boundaries 
for each ray which is internally reflected within the film, resulting in a 
series of reflected rays 22r, 24r, 26r . . . and a series of transmitted 
rays 22t, 24t, 26t . . . . The antireflective film 10 is designed to have 
a refractive index profile and a thickness which are calculated to 
minimize the total intensity of the reflected rays 22r, 24r, 26r . . . . 
This minimal reflection is to be accomplished for those light rays with 
wavelengths within a predetermined range of the optical spectrum. 
It is difficult, however, if not impossible, to actually fabricate a true 
gradient index profile film, because the actual materials which can be 
used to make such films are available with only discrete values of 
refractive index. A gradient index profile can be approximated, however, 
by dividing a theoretical fixed thickness gradient index film into many 
incremental layers each having a discrete refractive index value. The 
number of layers is chosen to be sufficiently large that each layer is 
substantially thinner than wavelengths within the spectral band which is 
to be antireflected, thereby causing the index variation to appear 
substantially continuous. 
According to the Herpin equivalent, any symmetrical combination of thin 
films is equivalent at any given wavelength to a simple layer. See 
Epstein, The Design of Optical Filters, Journal of the Optical Society of 
America, Volume 42, Page 806 (1952). This equivalence has proven to be a 
useful tool in the design of interference coatings. The Herpin equivalent 
is also commonly used in reverse, that is, to find a symmetrical 
three-layer combination of high and low index layers to substitute for a 
single layer of some unattainable index. 
The use of Herpin equivalent layers in the design of coatings has been 
described by Epstein. Two important features of Herpin equivalent layers 
are that they are equivalent only at one wavelength and that they apply 
only to symmetrical combinations (3, 5, or some odd number of layers). 
Berning has shown that when Herpin equivalents are applied to very thin 
layers (i.e., layers with optical thicknesses much less than the 
wavelengths of interest) the resulting configuration does not suffer from 
dispersion. Berning, Use of Equivalent Films in the Design of Infrared 
Multilayer Antireflection Coatings, Journal of the Optical Society of 
America, Volume 52, Page 431 (1962). This means that the symmetrical 
three-layer combination has the same spectral response as the single thin 
layer. This non-dispersive feature could be deduced from Epstein's paper 
by examining his plots of equivalent index versus phase thickness (which 
is proportional to 1/.lambda.). In the limit of thin layers the equivalent 
index is independent of wavelength. But Berning went on to point out that 
this dispersion-free property can be extended to layers of any thickness. 
This may be accomplished by simply dividing the layer into many thin 
layers and then replacing each thin layer by its symmetrical three-layer 
equivalent. 
Thin layers, however, are not only equivalent at all wavelengths but they 
also need not be symmetrical. That is, a simple high-index/low-index layer 
pair of appropriate thicknesses will be equivalent to a given single layer 
whose index is bracketed by the high-low pair. Furthermore, any arbitrary 
interference coating consisting of homogeneous or inhomogeneous 
gradient-index layers is equivalent to a sequence of high and low index 
layers. This generalized equivalence principle may be used to formulate a 
synthesis algorithm which exhibits rapid convergence for broadband 
antireflection coatings. 
In designing such a film, the assumption is made that a thin layer (i.e., 
one having an optical thickness much smaller that the wavelengths of 
interest) with an arbitrary index of refraction may be approximated by a 
pair of high and low index layers having the same total physical and 
optical thickness. This assumption leads to a simple algorithm for 
converting an arbitrary inhomogeneous index profile to one consisting of 
discrete layers of high and low index materials. 
For a given thin homogeneous layer of index N and physical (or geometrical) 
thickness T, the characteristic matrix M (see Epstein, supra) is, at 
normal incidence: 
##EQU1## 
where the phase thickness .phi. is given by: 
EQU .phi.=(2.pi./.lambda.)NT (2) 
For thin layers satisfying: 
EQU NT&lt;&lt;.lambda. (3) 
the trigonometric functions in Equation (1) may be replaced by their small 
argument approximations, so that M becomes: 
##EQU2## 
For a combination of two thin layers, one having a high index (n.sub.H, 
t.sub.H) and the other having a low index (n.sub.L, t.sub.L), the 
characteristic matrix is found by matrix multiplication of the single 
layer characteristic matrices to obtain: 
##EQU3## 
By comparing Equations (4) and (5), a single layer equivalent to the 
high-low pair can be identified. The conditions are: 
EQU T=t.sub.H +t.sub.L (6) 
and 
EQU N.sup.2 =(n.sub.H.sup.2 t.sub.H +n.sub.L.sup.2 t.sub.L)/(t.sub.H +t.sub.L) 
(7) 
Given a single layer with index N and thickness T, and two materials with 
indices n.sub.H and n.sub.L, Equations (6) and (7) can be solved for the 
thicknesses of the high-low pair which is equivalent to the single layer: 
EQU t.sub.H =T(n.sup.2 -n.sub.L.sup.2)/(n.sub.H.sup.2 -n.sub.L.sup.2) (8) 
EQU t.sub.L =T-t.sub.H (9) 
The inhomogeneous layer is divided into many thin, discrete layers, the 
number being sufficient to maintain performance. Each one of these 
sublayers is then replaced by a two layer approximation whose component 
thicknesses are specified by equations (8) and (9). 
Note from Equation (6) that the equivalent layer has the same physical 
thickness as the sum of the component layers. This is a surprising result, 
since one might have thought that, in the thin layer limit, equal optical 
thicknesses would be necessary to keep things equivalent. 
Another feature of Equations (4) and (5) is that thin layer characteristic 
matrices commute. That means that it does not matter in which order the 
matrices appear. In fact, it can be shown that in the thin layer 
approximation the symmetric three-layer Herpin combinations reduce to 
Equations (6) and (7) when two of the layers are interchanged (commuted) 
to produce a high-low pair. 
A third important feature of the thin layer high-low equivalent result 
proceeds from Equation (7), which indicates that the dielectric constant 
of the mixture .epsilon.=N.sup.2 is linear with respect to the dielectric 
constants of the component layers. 
The technique for replacing a generalized gradient-index coating 
(consisting of either homogeneous or inhomogeneous layers) by a sequence 
of high and low index layers is as follows. Divide the coating into thin 
homogeneous layers, even though adjacent layers may have the same index. 
Let the thickness of these sublayers satisfy Equation (3) and set the 
index of each layer to the average index across its thickness. Next, 
replace each of these layers by a thin high-low pair whose thicknesses are 
determined according to Equations (8) and (9). The resulting configuration 
is a digital equivalent and has the same total physical thickness and the 
same spectral characteristics. 
An example is shown in FIG. 2. A quintic (fifth-order polynomial) index 
profile is known to represent an effective broadband antireflection 
coating. FIG. 2a illustrates a quintic refractive index profile and its 
digital equivalent. FIG. 2b depicts the reflectivity which is obtained for 
the gradient and digital profiles. Agreement between the two reflectivity 
curves becomes perfect as the sublayers used are made thinner. 
A second example is provided in FIG. 3, which illustrates the use of thin 
high-low equivalents to represent a layer of an intermediate index. Here, 
the antireflection coating is designed to reduce reflections from a glass 
substrate. The original coating consists of a quarter-wave of low index 
material, a half-wave of high index material, and a quarter-wave of an 
intermediate index layer whose value is determined to optimize the 
broadband performance. In the index profile of FIG. 3a, this third layer 
is replaced by a digital equivalent. The use of the Herpin symmetrical 
three-layer combination could also replace the third layer, but broadband 
performance would be slightly different due to the dispersion of the 
Herpin equivalent layers. The calculated reflectivity for the coating 
shown in FIG. 3a is depicted in FIG. 3b. 
The coating synthesis approach seeks in general to find an index 
distribution which produces a given spectral reflectance (or 
transmittance). This is basically an inverse scattering problem and its 
most general solution is a generalized gradient-index distribution. By 
generalized is meant a distribution which allows index discontinuities, 
homogeneous index regions, and inhomogeneous or gradient-index regions. 
Current design practice, however, seeks approximate solutions consisting 
of homogeneous layers of nominally quarter-wave optical thickness. Since 
generalized gradient-index distributions have thin layer high-low 
equivalents, searching for thin layer digital solutions can yield improved 
solutions. 
The performance of such a multilayered gradient index film can be further 
improved by applying a synthesis technique, such as that described by 
Snedaker, "New Numerical Thin-Film Synthesis Technique", Journal of the 
Optical Society of America, Volume 72, Page 1732 (1982). In this approach, 
the reflectivity of the film at several wavelengths over the spectrum is 
evaluated using conventional matrix theory for homogeneous layers. The 
index of each layer is then separately adjusted, by varying the 
thicknesses of the sublayers within the appropriate sublayer pair, to 
minimize the broadband reflectivity. Snedaker's method uses fixed layer 
thicknesses, but adjusts the index of each sublayer in a continuous 
fashion to improve the merit function. This sublayer index optimization 
requires several merit function evaluations including those required to 
form derivatives for a nonlinear search technique. Furthermore, 
convergence is slow, requiring several dozen or even hundreds of passes. 
Then, when a solution is found, intermediate index sublayers must be dealt 
with (these sublayers could be replaced by high-low thin layer 
equivalents). The approach of the present invention, however, directly 
obtains a high-low digital equivalent to a generalized gradient-index 
solution. 
It is an outstanding feature of this invention to derive digital solutions 
for a given desired spectral response profile by using the following 
synthesis algorithm: 
1. Specify a total physical thickness for the coating. Divide this 
thickness into a series of thin layers, each of equal thickness. 
2. Assign one of two indices, either high or low, to each layer in the 
series. The convergence of an iterative solution will usually depend on 
the starting values, so this step can be important. Four possible initial 
schemes are: 
a. Start with all high index layers. 
b. Start with all low index layers. 
c. Start with alternating high and low index layers. 
d. Start from some known approximate solution. 
The first three approaches require no knowledge of thin-film theory, while 
the fourth attempts to utilize such experience. 
3. Evaluate a merit function based on the desired spectral response. One 
example is the least squares sum, in which the difference between the 
calculated reflectivity and the desired reflectivity at various 
wavelengths across the band of interest is squared and the squared 
differences are summed. The characteristic matrix theory, supra, is used 
to evaluate the calculated response. 
4. Change the state of each layer (from low to high index or from high to 
low) one at a time and reevaluate the merit function. If the performance 
is better with that layer having the "flipped" index state, then retain 
the change; otherwise, restore that layer to its previous index. 
5. If, after testing all the layers (a single pass), the merit function 
shows improvement, then repeat step 4. If no further improvement is 
indicated, the optimum filter design has been obtained. Layers of the 
specified thickness and index value are then deposited on the optical 
surface to be antireflected. This process can be accomplished by any of 
the deposition techniques known to those in the optical coating art. 
One advantage of this digital search technique is that there are only two 
index values. This eliminates the infinity of arbitrary index values to 
sample. Furthermore, by using a fixed thin-layer thickness, the infinity 
of thicknesses to sample is eliminated. Of course, by using fixed 
thin-layer thicknesses some of the generality in being able to duplicate 
an arbitrary thin-layer index is lost. This loss of generality may be 
restored, however, by simply employing thinner layers. 
Various refinements could be added to the above algorithm. One could, for 
example, make the high and low index sublayers have the same optical, 
rather than physical, thickness. 
To demonstrate the advantages of this invention, a broadband antireflection 
coating similar to that shown in FIG. 3 was designed for a glass surface 
with a total coating thickness of 0.279 .mu.m (to have the same total 
thickness as the more classical design shown in FIG. 3). The substrate had 
a refractive index of 1.52 and layer materials with incices of n.sub.L 
=1.388 and n.sub.H =2.027 were used. The film was divided into 100 
sublayers. All sublayers were initially assigned the high index material. 
A least-squares merit function based on zero reflectivity targets at eight 
wavelengths equally spaced in wave number from 0.4 to 0.7 .mu.m was used 
to evaluate the reflectivity of the film. After four passes in which the 
index state of each sublayer was flipped and the coating reevaluated, no 
further improvement in the reflectivity was noted. The resulting index 
profile is shown in FIG. 4a, with the corresponding reflectivity depicted 
in FIG. 4b. 
A second film was designed for the same performance parameters, starting 
this time for alternating high and low index sublayers. Convergence was 
achieved in only two passes through all the layers. The results of this 
design are shown in FIG. 5. 
Experiments with this design technique using low and high index substrates 
and a number of different total thicknesses and starting points yielded 
solutions which seldom required over 10 passes and always exhibited good 
coating performance. 
A comparison of FIGS. 4 and 5 shows that the two coating designs are 
basically similar. There is an approximately quarter-wave low-index layer 
in each, followed by a half-wave region of predominantly high-index. There 
is then a third, approximately quarter-wave, region at some intermediate 
index. These are not identical solutions, but both are acceptable. 
Generally the small details of the solutions depend on the initial 
refractive indices specified for the layers, while the gross features tend 
to be independent of the starting point. 
In conclusion, although several particular embodiments of the invention 
have been described, modifications and additional embodiments will 
undoubtedly be apparent to those skilled in the art. Consequently, the 
exemplary embodiments should be considered as illustrative, rather than 
inclusive, and the appended claims are intended to define the full scope 
of the invention.