CN104428771A - System and methods of calculating diffuse reflection of an optical stack with a nanowire - Google Patents

Abstract

The present disclosure relates to a method for improving optical qualities of transparent conductive films including a multilayer optical stack and conductive nanowires embedded therein.

Description

System and method for calculating optical stack diffuse reflectance with nanowires

Cross Reference to Related Applications

This application is based on the claims of 35u.s.c. § 119(e), claiming rights to U.S. provisional application No. 61/621,359 filed 4/6/2012 and U.S. provisional application No. 61/678,886 filed 8/2/2012, and U.S. patent application No. 13/667,556 filed 11/2/2012 and U.S. patent application No. 13/831,351 filed 3/14/2013, which are incorporated herein by reference in their entirety.

Background

Transparent conductive films include a conductive material coated on a highly light-transmitting surface or substrate, and are widely used in flat panel displays such as Liquid Crystal Displays (LCDs), touch panels or touch sensors, electroluminescent devices (e.g., light emitting diodes), thin film photovoltaic cells, or as antistatic layers and electromagnetic wave shielding layers.

Currently, vacuum deposited metal oxides such as Indium Tin Oxide (ITO) are industry standard materials for providing optical transparency and electrical conductivity to dielectric surfaces such as glass and polymer films. However, metal oxide films are brittle and prone to damage under bending or other physical stress. The metal oxide film also requires elevated deposition temperatures and/or high annealing temperatures to achieve high conductivity levels. For certain substrates that are prone to moisture absorption, such as plastics and organic substrates (e.g., polycarbonate), proper adhesion of the metal oxide film becomes a problem. Therefore, the application of the metal oxide film to the flexible substrate is severely limited. In addition, vacuum deposition is a costly process and requires specialized equipment.

In recent years, there has been a trend to replace the current industry standard transparent conductive ITO films in flat panel displays with composite metal nanomaterials (e.g., silver nanowires). Generally, a transparent conductive film is formed by first coating an ink composition including silver nanowires and a binder on a substrate. A transparent UV or thermally curable polymeric material may then be applied to form the protective layer. Nanowire-based coating techniques are particularly useful for printed electronics. By using solution-based formats, printed electronics technology makes it possible to fabricate robust electronic devices on large area flexible substrates.

The presence of nanowires in transparent conductive films can lead to certain optical challenges that are not typically encountered in continuous ITO films. For example, when an ITO touch sensor is off, the ITO touch sensor displays black under ambient light; whereas touch sensors made from transparent films based on silver nanowires may have a "milky" or "cloudy" appearance. The milky appearance may affect the image quality (when the LCD module is turned on), presenting a lower contrast or other image problems. Accordingly, there is a need to address the optical challenges specific to nanowire-based transparent conductors.

Disclosure of Invention

Various embodiments are provided herein that relate to methods of addressing the milky appearance of nanowire displays by reducing or minimizing diffuse reflection in an optical stack that includes at least one nanowire-based conductive film.

One embodiment is a method, comprising: selecting optical stack parameters for an optical stack having nanowires and calculating a plurality of diffuse reflectance values from the optical stack parameters, wherein each diffuse reflectance value is for a respective configuration of a plurality of optical stack configurations. The method also includes selecting one of the optical stack configurations based at least on the comparison of the diffuse reflectance values and forming the optical stack according to the selected optical stack configuration.

In one embodiment, a method includes calculating a plurality of specular reflectance values, where each specular reflectance value is for a respective configuration of an optical stack configuration.

In one embodiment, calculating the diffuse reflectance value comprises calculating a scattering cross section of the nanowire.

In one embodiment, calculating the diffuse reflectance values includes, for each optical stack configuration, separately calculating an electromagnetic field of incident light at the nanowire location within the optical stack and calculating a transfer matrix of light scattered from the nanowires within the optical stack.

In one embodiment, calculating the diffuse reflection includes calculating an amount of light scattered from the nanowire based on the scattering cross-section and a field of incident light at the nanowire location.

In one embodiment, calculating the field of incident light includes calculating an electromagnetic field of diffusely scattered light at the nanowire location.

In one embodiment, the plurality of optical stack parameters includes a number of layers of the optical stack. In one embodiment, the plurality of optical stack parameters includes a range of thicknesses of layers of the optical stack. In one embodiment, the plurality of optical stack parameters includes a range of refractive indices of layers of the optical stack.

In one embodiment, forming the layers of the optical stack includes forming a first layer on the substrate and forming a second layer on the first layer, the nanowires being disposed within either the first layer or the second layer.

In one embodiment, a method includes calculating a plurality of specular reflectance values from optical stack parameters, where each specular reflectance value is for a respective one of a plurality of optical stack configurations.

In one embodiment, calculating the plurality of specular reflectance values includes calculating a transfer matrix for light incident to each optical stack configuration.

In one embodiment, selecting one of the optical stack configurations is based in part on a comparison of specular reflectance values.

In one embodiment, selecting one of the optical stack configurations includes selecting the optical stack configuration that corresponds to the smallest diffuse reflectance value.

In one embodiment, a method includes inputting input optical stack parameters for an optical stack having nanowires to a processor, and storing the input optical stack parameters in a memory circuit coupled to the processor. The method also includes calculating, in the processor, a plurality of diffuse reflectance values for a plurality of optical stacks, wherein each optical stack has a respective configuration according to optical stack parameters. Calculating the diffuse reflectance values includes, for each configuration, separately calculating an electromagnetic field value of incident light at a location within the optical stack corresponding to the optical stack nanowire location, and calculating a transfer matrix based in part on the electromagnetic field values to provide diffuse reflectance values at the optical stack surface.

In one embodiment, the method includes comparing the diffuse reflectance values to each other and selecting one of the diffuse reflectance values.

In one embodiment, the method includes outputting from the processor a selected optical stack configuration corresponding to a selected value of diffuse reflectance.

In one embodiment, the input optical stack parameters include a refractive index range of at least one layer of the optical stack. In one embodiment, the selected optical stack configuration includes a refractive index within a range of refractive indices. In one embodiment, the input optical stack parameter includes a range of thicknesses of layers of the optical stack.

In one embodiment, the selected optical stack configuration includes a thickness within a range of thicknesses of the layers of the optical stack.

In one embodiment, a method includes forming an optical stack according to a selected optical stack configuration.

In one embodiment, calculating the diffuse reflectance value comprises calculating a scattering cross section of the nanowire.

One embodiment is a system comprising: a processor; a memory coupled to the processor; an input coupled to the processor is configured to receive a first parameter of the optical stack. The processor is configured to calculate a set of incident optical electromagnetic field values corresponding to nanowire positions within the optical stack, calculate a light scattering distribution of the nanowires, calculate a set of surface diffuse reflectance values for the optical stack, and estimate a second set of parameters for the optical stack. The second parameter corresponds to a preferred value of the set of diffuse reflection values. The output is coupled to the processor and configured to receive the second parameter from the processor.

In one embodiment, the system includes a display coupled to the output, the display configured to display the second parameter.

In one embodiment, the system includes a deposition apparatus coupled to the output, the deposition apparatus configured to receive the second parameter and deposit the first optical layer of the optical stack according to the second parameter.

One embodiment is a method comprising inputting parameters of an optical stack to a processor, estimating, in the processor, a set of electromagnetic field values of incident light corresponding to a position of a nanowire of the optical stack, and estimating, in the processor, a light scattering distribution of the nanowire. The method further includes, in the processor, estimating a set of diffuse reflection values at the surface of the optical stack based on the electromagnetic field values and the scattering cross-section, and outputting from the processor an optical stack configuration corresponding to the selected diffuse reflection values.

In one embodiment, estimating the set of electromagnetic fields includes calculating a first transfer matrix from the optical stack parameters.

In one embodiment, estimating the set of diffuse reflection values includes calculating a second transfer matrix from the optical stack parameters.

Drawings

In the drawings, like reference numerals designate like elements or acts. The sizes and relative positions of elements in the drawings are not necessarily shown to scale. For example, the shapes of various elements and angles are not shown to scale, and some elements are arbitrarily enlarged and positioned to improve drawing legibility. Further, the particular shapes of the elements as shown are not intended to convey any information regarding the actual shape of the particular elements, and have been solely selected for ease of recognition in the drawings.

FIG. 1 is a cross-sectional view of an optical stack containing nanowires in accordance with this embodiment.

FIG. 2A illustrates specular reflection of an optical stack in accordance with one embodiment.

FIG. 2B illustrates the diffuse reflection of an optical stack in accordance with one embodiment.

FIG. 3A is a graph of the diffuse reflectance within an optical stack.

FIG. 3B is a plot of specular reflection within an optical stack.

Fig. 4A-4C show diffuse reflectance curves for several wavelengths in different media and at different thicknesses.

Fig. 5A-5C show the specular reflection of several wavelengths in different media and at different thicknesses.

FIG. 6 is a cross-section of an optical stack in accordance with one embodiment.

Fig. 7 illustrates total internal reflection in an optical stack.

FIG. 8 is a cross-sectional view of an optical stack including three layers in accordance with one embodiment.

FIG. 9A is a cross-section of an optical stack illustrating top and bottom mode propagation in the optical stack, in accordance with one embodiment.

FIG. 9B is a cross-section of an optical stack illustrating a top mode and a bottom mode of diffusely scattered light propagation in the optical stack, in accordance with one embodiment.

FIG. 10A illustrates a GUI in accordance with one embodiment.

FIG. 10B illustrates a GUI in accordance with another embodiment.

FIG. 10C illustrates specular and diffuse reflectance curves optimized in accordance with one embodiment.

FIG. 10D illustrates a GUI in accordance with one embodiment.

FIG. 10E illustrates a GUI in accordance with another embodiment.

FIG. 11 is a block diagram of a system in accordance with one embodiment.

FIG. 12 is a method for reducing diffuse reflection in an optical stack, in accordance with one embodiment.

FIG. 13 is a method for reducing diffuse reflection in an optical stack, in accordance with another embodiment.

FIG. 14 shows a flat panel device incorporating an optical stack, in accordance with one embodiment.

Detailed Description

The description herein includes potential causes of a "milky" appearance of the nanowire display, methods of addressing the "milky" appearance, and optical stacks with lower or no milky appearance. As used herein, an "optical stack" refers to a multi-layer stack of thin films through which light from an external light source or an internal light source passes, one or more layers having an effect on the optical properties of the light. The thin films within the optical stack are typically functional films such as transparent conductive films, polarizers, color filters, antiglare films, or antireflective films as well as protective coatings and transparent adhesives. The film may be flexible (e.g., a polymer substrate) or rigid (e.g., a glass substrate). The optical stack is typically coupled to another functional unit such as a display. In addition to films, the voids between films or between a film and a display also affect the optical properties of light and are considered part of the optical stack.

Further, in the context of film orientation, a film that is "on" another film is configured to be closer to an external light source (or viewer) than the other film. For example, an over-coating located over the nanowire layer is always disposed between an external light source (or viewer) and the nanowire layer. A film that is "under" another film is configured to be farther from an external light source (or viewer) than the other film. For example, in an optical stack that employs an undercoat layer located below the nanolayers, the nanowire layer is always disposed between an external light source (or viewer) and the undercoat layer.

Fig. 1 shows an optical stack 30 of transparent conductive films. In basic optical stacks (30), as in more complex optical stacks (e.g., in an entire touch panel), diffuse reflection may be caused to some extent by multiple or all layers or structural elements. Various embodiments described herein are methods of attenuating diffuse reflection by manipulating and modifying various layers or structural elements. However, it is to be understood that any one or more of the embodiments may be combined to provide additional benefits of further reducing diffuse reflection. Accordingly, various embodiments are directed to an optical stack comprising at least one nanowire layer; and at least one substrate adjacent to the nanowire layer, wherein the nanowire layer comprises a plurality of electrically conductive nanowires, and the diffuse reflection of the incident light is a percentage of the incident light when viewed from the same side of the optical stack as the incident light. As used herein, "adjacent" refers to the relative position of the substrate and the nanowire layer. The substrate and nanowire layers may be in direct contact or in close proximity to each other with one or more intervening layers interposed therebetween.

The optical stack 30 includes conductive nanowires 32 embedded within a transparent insulating layer 34. The transparent insulating layer 34 and the nanowires 32 are located on a substrate 36.

The optical stack 30 is of the type that may be used in a flat panel display. Accordingly, attributes that have greatly enhanced visual characteristics of the optical stack are desirable for the optical stack 30. As described above, the optical stack 30 containing the nanowires 32 may suffer from "milky" or hazy quality. This milky quality can detract from the visual characteristics of the optical stack 30. In particular, when it is desired to display a dark color such as black, the optical stack 30 may conversely display a milky color, which negatively impacts the quality of the displayed image.

One source of these undesirable features is diffuse reflection from the nanowires 32. Typically, when light encounters a surface or object, the angle of reflection is equal to the angle of incidence, which is referred to as specular reflection. Specular reflection is shown in fig. 2A. In FIG. 2A, the light rays are at an angle of incidence Φ_iIs incident on surface 37 of optical stack 30. The light ray is at an angle phi_rIs reflected from surface 37 of optical stack 30, where Φ_rIs equal to phi_i。

However, as shown in FIG. 2B, some of the light impinging on surface 37, or indeed any surface, of optical stack 30 is also at a plurality of angles θ_rIs diffusely reflected. This diffuse reflection is now that the light is scattered in a number of directions different from the intended angle of reflection at the mirror. Although only one angle theta is labeled in fig. 2B_rBut diffusely reflects light at a plurality of angles theta_rIs reflected. In fig. 2B, light incident on surface 37 is scattered in many directions. While typically a relatively small portion of the light is diffusely reflected from any surface, the optical stack 30 in fig. 2B is more diffusely reflected due to the presence of the nanowires 32.

When light is incident on an object or structure having a size smaller than the wavelength of the light, the light is diffusely scattered from the object. Typically, the nanowires 32 and the optical stack 30 have a radius of less than 100nm, for example a radius between 5nm and 100 nm. 100nm is much smaller than the minimum wavelength of visible light. Thus, when any visible light encounters the nanowire 32, it is diffusely reflected from the nanowire 32. In a transparent film, most of the light incident on the surface 37 is transmitted through the surface 37 and into the layer 34 in which the nanowires 32 are embedded. Only a small portion of the light is reflected at the surface. However, some portion of the light interacting with the nanowires 32 is diffusely reflected. This diffuse reflection is a major cause of the milky quality that can sometimes reduce the appearance of the optical stack 30 containing the nanowires 32. It has been demonstrated that when the nanowires 32 are embedded in the optical stack 30, the diffuse reflection of the nanowires 32 can be reduced in several ways using calculations.

One such method is to reduce the refractive index of the layer 34 in which the nanowires 32 are embedded. Fig. 3A shows a plot of diffuse reflection versus wavelength of light incident on the nanowire 32. Three curves are shown, each corresponding to a layer having a refractive index of 1.43, 1.33 and 1.23, respectively. The peak of the refractive index 1.43 curve is much higher than the n-1.33 and n-1.23 curves. For a layer with a refractive index equal to 1.43, the peak of the diffuse reflection occurs at a light wavelength of about 400 nm. 400nm is the boundary of the visible spectrum and corresponds to violet light. The wavelength of less than 380nm, which corresponds to ultraviolet light, is generally not visible to humans.

When the refractive index is reduced to n-1.33, not only does the peak of the diffuse reflection decrease, but it also shifts to smaller wavelengths. For a material with a refractive index n of 1.33, the peak is reduced to about 6 × 10^-4And a peak wavelength of about 370 nm. Thus, not only is less light diffusely reflected back out of the surface 37 of the optical stack 30, but a greater portion of the reflected light is moved out of the visible spectrum and into the ultraviolet spectrum. It should be noted here that the value of the diffuse reflection in this graph is arbitrary, but is helpful in understanding the relative effect on the diffuse reflection of varying the parameters of the optical stack 30.

The diffuse reflection of a material with a refractive index n ═ 1.23 is the smallest of the three curves. For n-1.23, the peak diffuse reflection is about 4.5 × 10^-4. And it is also important that the peak wavelength is shifted even further into the ultraviolet range which is not visible to the human eye. Thus, placing the nanowires 32 in the layer 34 having a lower refractive index can both reduce the diffuse reflection and shift the peak diffuse reflection away from the visible spectrum.

It is also desirable to reduce specular reflection as much as possible. Fig. 3B shows three curves of specular reflection versus wavelength of light for the same three indices of refraction n as in fig. 3A. As can be seen from fig. 3B, the specular reflection is highest for the layer 34 having a refractive index n of 1.43. For n-1.43, the peak specular reflection is about 0.04. However, the peak at about 300nm is outside the visible range. For layer 34 with an index of refraction n 1.33, there is a small drop in peak specular reflection. However, for most of the visible spectrum, the corresponding wavelength is about 400nm to 700nm, and the specular reflection at n-1.33 is much lower than that at n-1.43. Thus, although the present disclosure focuses primarily on reducing diffuse reflection, specular reflection is not ignored. Reducing both specular and diffuse reflections can greatly enhance the visual characteristics of the optical stack 30.

For a layer 34 with a refractive index of n 1.23, the specular reflection is the lowest. Not only is the peak specular reflection reduced, but the specular reflection is nearly 0 across most of the visible spectrum, with the low points occurring at about 500 nm. Therefore, reducing the refractive index of the layer in which the nanowires 32 are embedded is very beneficial for both diffuse reflection and specular reflection.

Another parameter of the optical stack 30 that can affect specular as well as diffuse reflection is the thickness of the layer 34 in which the nanowires 32 are embedded. Fig. 4A shows a plot of diffuse reflection versus thickness of the layer 34 in which the nanowire 32 is embedded, for several wavelengths and a refractive index of n ═ 1.23. It can be seen that the diffuse reflection of light with a wavelength of 400nm is slightly higher than the diffuse reflection of light with a wavelength of 450nm, 500nm or 650 nm. Perhaps most significantly, the diffuse reflection remains largely constant for any given wavelength throughout the thickness range of layer 34, which is about 20nm to 400nm thick. The diffuse reflection of 400nm light is both greater in magnitude and varies more than the diffuse reflection of the other wavelengths in fig. 4A. In other optical stacks this is not the case. In fact, the thickness of the layer may be very important in certain configurations.

Fig. 4B plots the diffuse reflection of the nanowire 32 in a layer 34 with a refractive index n of 1.33. A slight increase in the refractive index results in an increase in the magnitude of the diffuse reflection. Specifically, the diffuse reflection of light having a wavelength of 400nm is increased more than the diffuse reflection of light having a wavelength of 450nm, 500nm, or 650 nm. Thus, fig. 3A and 3B and fig. 4A-4C show that the diffuse reflectance fluctuates most strongly near the violet end of the visible spectrum.

In fig. 4C, the refractive index n is 1.43. With this increase in refractive index, a large increase in the diffuse reflection of 400nm light occurs. Smaller increases in the diffuse reflection of light at wavelengths of 450nm, 500nm and 650nm also occur, but to a much lesser extent.

However, the specular reflection fluctuates widely with changes in the thickness of the layer 34 in which the nanowires 32 are embedded. For each of the four wavelengths of light plotted in fig. 5A, the specular reflection follows a sine wave plot. As the thickness of layer 34 increases, all wavelengths of light experience peaks and valleys in their specular reflectance values. When the thickness of the layer is close to 0, the specular reflection is close to a peak of about 4% for each of the four wavelengths of light.

All four wavelengths plotted in fig. 5A experience a minimum of specular reflection as the thickness increases to about 100 nm. As the thickness of layer 34 increases toward 200nm, all four wavelengths of light again approach a peak. Depending on the thickness of the layers in the stack, light will experience constructive and destructive interference at locations throughout the optical stack. In addition, light reflected from surface 37 may be 180 degrees out of phase with light reflected from below. Thus, depending on the thickness and material of layers 34 and 38, light reflected from below may interfere destructively with light reflected from surface 37 and thus reduce specular reflection.

In fig. 5B, the specular reflection of four wavelengths of light is plotted when the refractive index of layer 34 is 1.33. The peaks and valleys occur at approximately the same locations that occur when the index of refraction is 1.23. However, the present minimum value is higher than when n is 1.23. Specifically, the minimum value drops only to about 1% of specular reflection, and when n is 1.23, the minimum value drops to about 0.

In fig. 5C, the layer 34 in which the nanowire 32 is embedded has a refractive index n of 1.43. As in fig. 5B and 5A, the peak here remains at about 4%. However, the minimum value of the percentage of specular reflection has increased to about 2.5% relative to 1% for n 1.33 and 0% for n 1.23. Therefore, to reduce specular reflection, it is desirable to have a lower index of refraction in some optical stacks.

FIG. 6 shows an optical stack 30 in accordance with one embodiment. In accordance with one embodiment, the optical stack 30 includes nanowires 32 within an insulating layer 34. Layer 34 is disposed on layer 38 and layer 38 is a high index of refraction layer. Layer 38 is also optically transparent. The layer 38 may enhance forward scattering of diffuse light from the nanowires 32. When the nanowires 32 are disposed within a layer 34 having a lower refractive index relative to the layer 38, more forward scattering of the diffuse light is promoted. In other words, as light diffusely reflects from the nanowires 32, more light will be forward scattered toward the layer 38. Thus, less light will be diffusely reflected back toward the surface 37 of the optical stack 30. This is due, in part, to the increased density of states for forward scattering relative to backward scattering when there is a higher index layer next to a lower index layer. As previously mentioned, the increased density of states promotes forward scattering.

Another advantage of having the high index of refraction layer 38 under the nanowires 32, as shown in fig. 7, is that total internal reflection of diffusely reflected light can occur within the high index of refraction layer 38. Incident angle greater than critical angle theta_cWhen this occurs, total internal reflection occurs. The angle of incidence is measured relative to the normal at the refractive interface. As light propagates from the high index layer 38 to the low index layer 34, light reaching the interface of layer 38 and layer 34 is refracted toward the high index layer 38. When the angle of incidence is sufficiently large, the transmission angle within the low refractive index layer 34 reaches 90 degrees with respect to the normal. At this point, light is no longer transmitted into the low refractive index layer 34. This interaction follows snell's law and is expressed as:

n₁sin(θ)₁＝n₂sin(θ)₂

by simple operation, the critical angle theta at which total internal reflection will occur can be calculated_cThe following are:

θ_c＝arcsin(n₂/n₁)

therefore, the larger the difference between the low refractive index layer 34 and the high refractive index layer 38, the smaller the critical angle will be. As the critical angle becomes smaller, more light will undergo total internal reflection upon reaching the interface of high index layer 38 and low index layer 34. Therefore, selecting a layer 38 with a sufficiently high index of refraction can further reduce the amount of diffusely reflected light reaching the surface 37 of the optical stack 30. Thus, as described with reference to FIG. 6, promoting total internal reflection is associated with enhanced forward scattering. In particular, the more light that is forward scattered from the nanowires 32 into the high refractive index layer 38, the more light will be totally internally reflected within the high refractive index layer 38 and will not reach the surface and thus cause an increase in opalescence.

In accordance with the principles discussed with reference to fig. 6 and 7, fig. 8 discloses an optical stack 30 in accordance with one embodiment in which a high index layer 38 is disposed below a low index layer 34 and above a substrate 36, as previously described. Having an optical stack 30 that includes a low index layer 34, nanowires 32 embedded within the low index layer 34, and a high index layer adjacent below the low index layer provides the enhancements described with reference to fig. 6 and 7. The substrate 36, having an index of refraction generally between that of the low index layer 34 and the high index layer 38, provides additional structural support and enables its attachment to the flat panel device.

Although the foregoing embodiments of the optical stack 30 provide advantages, optimization of the optical stack may still be very difficult in order to minimize diffuse as well as specular reflection. In order to provide an optical stack with minimal diffuse reflection, it is beneficial to utilize an efficient method to calculate or estimate the diffuse reflection of the optical stack 30 for a given layer and nanowire configuration. The diffuse reflection of the optical stack 30 can be calculated by solving the system of maxwell equations of the optical stack 30. The differential form of the Maisfiel equation describes the properties of the electric field E and the magnetic field B as follows:

<math> <mrow> <mo>&dtri;</mo> <mo>&CenterDot;</mo> <mi>E</mi> <mo>=</mo> <mfrac> <mi>&rho;</mi> <msub> <mi>&epsiv;</mi> <mn>0</mn> </msub> </mfrac> <mo>,</mo> </mrow> </math>

<math> <mrow> <mo>&dtri;</mo> <mo>&CenterDot;</mo> <mi>B</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </math>

<math> <mrow> <mo>&dtri;</mo> <mo>&times;</mo> <mi>E</mi> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mo>&PartialD;</mo> <mi>B</mi> </mrow> <mrow> <mo>&PartialD;</mo> <mi>t</mi> </mrow> </mfrac> <mo>,</mo> </mrow> </math>

<math> <mrow> <mo>&dtri;</mo> <mo>&times;</mo> <mi>B</mi> <mo>=</mo> <msub> <mi>&mu;</mi> <mn>0</mn> </msub> <mi>J</mi> <mo>+</mo> <msub> <mi>&mu;</mi> <mn>0</mn> </msub> <msub> <mi>&epsiv;</mi> <mn>0</mn> </msub> <mfrac> <mrow> <mo>&PartialD;</mo> <mi>E</mi> </mrow> <mrow> <mo>&PartialD;</mo> <mi>t</mi> </mrow> </mfrac> <mo>,</mo> </mrow> </math>

where ρ is the charge density due to free and polarized charges, J is the current density,₀is the dielectric constant of free space, and₀is the permeability of free space. When calculating the diffuse reflectance of many optical stacks 30, it is relatively difficult to utilize the system of maxwell's equations in a method of calculating the diffuse reflectance, and a large amount of time and processing resources may be required. The complexity of the system of maxwell equations makes it very difficult to solve and calculate the preferred parameters of the optical stack 30.

In addition, it is not easy to manipulate the maxwell system of equations each time a new and different layer is added to the optical stack 30 to again provide for fast optimization of the optical stack 30 including additional parameters. In some cases, it is possible to add many layers both below and above the nanowire 32. Some optical stacks may be subject to specific constraints. The system of mausweirs equations will be solved anew each time the parameters or constraints of the optical stack 30 change, and therefore more time and processing resources will be used.

A less resource-intensive method for calculating the specular and diffuse reflections of an optical stack will be described by means of a transfer matrix with reference to fig. 9A and 9B. Because the system of maxwell equations are second-order partial differential equations, the full set of these equation solutions utilizes a set of at least two linearly independent solution families (modes). One embodiment defines the two families as top and bottom modes. The first solution, the top mode, corresponds to the field distribution that can be produced by light incident on the system from the top, as is the case with specular reflection. The second solution, bottom mode, describes the field distribution throughout the system that can be produced by a light source located on the substrate side of the structure (below layer 36 in fig. 9 a). These solutions are also present in the process of diffuse reflection.

The specular reflection process is now considered in more detail with reference to the arrow on the right side of FIG. 9A. In this process, a certain amount of incident light is transmitted. FIG. 9A shows an optical stack 30 in accordance with one embodiment. The nanowires 32 are not shown in fig. 9A because the associated calculations for the optical stack in fig. 9A are performed assuming that the nanowires 32 are not present. The position in the optical stack at which y-0 corresponds to the position that the nanowire 32 would occupy within the optical stack. As previously described, the optical stack 30 includes a low index layer 34. Low index layer 34 is located on high index layer 38 and high index layer 38 is located on substrate 36. The light source illuminates the optical stack 30. Light is incident on the surface 37 of the low refractive index layer 34.

The distribution of the EM field throughout the multilayer structure was calculated assuming no nanowires. These calculations are performed using a method of transfer matrices, where the field for each layer is represented as a series of plane waves (reflected/transmitted waves) moving up and down through the system, and the amplitudes of these waves are correlated at adjacent layers by the transfer matrices.

Light from the light source is incident on the surface 37 of the optical stack 30. The arrows on the right side of the optical stack correspond to the top modes because they carry energy from the top of the system. A certain amount of incident light from a light source above the optical stack is transmitted through surface 37 into low index layer 34 as indicated by the arrow in fig. 9A propagating down the right side of the optical stack into layer 34. A proportion of the light incident on the optical stack 30 is reflected from the surface 37 as indicated by the angled off-going arrow from the arrow propagating down into the layer 34. The angle of this arrow is not intended to indicate the angle at which light is reflected from the surface, but merely indicates that some light returns upwards and some light passes through. This is true for all arrows in fig. 9A. These arrows which appear angled are merely angled to distinguish them from arrows passing through the interface. In fact, the direction of light propagation depends on the illumination source and is described by the solution of the system of maxwell equations.

Some light from the air passing through interface 37 continues to propagate within layer 34 until it reaches interface 44 between layers 34 and 38. At interface 44, some light passes through and some light is reflected, as indicated by the arrows on the right side down through interface 44 and the arrows back into layer 34, indicating the reflection at interface 44. This reflected light, in turn, will return to the interface 37, partly contributing to the initial specular reflection (upward arrow) and partly contributing to the initial transmission (downward arrow). In the context of fig. 9a, these subsequent re-reflections and re-transmissions are combined and represented by a single combination of transmission (downward) and reflection (upward) arrows. This description is consistent with a transfer matrix approach according to one embodiment that can be used to automatically calculate the total reflection/transmission coefficient.

Again, at the interface 42, some portion of the light passing through the layer 38 to the interface 42 passes through the interface 42 into the layer 36. Likewise, a portion of the light incident on interface 42 is reflected back into layer 38. An amount of light passes through interface 40 and into any layers located below layer 36.

Below the layer 36 of the optical stack 30, a hypothetical light source is shown. The dashed arrows on the left side of the optical stack originate from this light source and correspond to the "bottom mode" because they carry energy from the bottom of the system upwards. Light that passes through the interface 40 is transmitted into the layers 36 of the optical stack towards the interface 42, and some portion of the light will be reflected. At interface 42, a portion of the amount of light passes through interface 42 from layer 36 to layer 38. At the same time, at the interface 42, part of the light is reflected back towards the interface 40. Again, at interface 44, some light passes through to layer 34, while some light is reflected back into layer 38 at interface 44.

Finally, at the surface 37 of the optical stack 30, some light enters the air surrounding the optical stack 30 from passing through the layer 34.

Using the transfer matrix, the magnitude of the up-down propagating field in each layer can be calculated. In particular, for the top mode, the calculation of the magnitude of the light reflected upward from the interface 37 can be used to calculate the total specular reflection very accurately. In addition, the amplitude of other waves containing the top mode can be used to calculate the electromagnetic field for any given vertical position within the stack 30. In this way, the field at the position of the nanowire 32 can be calculated.

In one embodiment, the size of the stack in the z-axis direction, i.e., the direction into the page, is assumed to be infinite. Thus, the total field of the optical stack 30 can be represented as a linear combination of two fields having different polarizations. The first type of field, known as a TE wave, has an electric field component along the z-axis such that the magnetic field of the wave has only x and y components. Similarly, the second type of wave, the TM wave, has a magnetic field aligned with the z-axis and the electric field of the wave is in the xy-plane.

At the interface of two arbitrarily selected adjacent layers (j and j +1) within the optical stack 30, the component of incident light, assuming it isWith wave vectors. The relationship between the amplitudes of plane waves in adjacent layers can be determined by considering the boundary conditions of the electric and magnetic fields. Specifically, for the interface between layers j and j +1 (e.g., corresponding to layers 34 and 38), the relationship is expressed as:

$\left(\begin{array}{c}{a}_{j+1}^{-}\\ a_{j+1}^{+}\end{array}\right)=\frac{1}{2}\left(\begin{array}{cc}1+K_j& 1-K_j\\ 1-K_j& 1+K_J\end{array}\right)\left(\begin{array}{c}{a}_j^{-}\\ a_j^{+}\end{array}\right)$

wherein, a^-And a⁺Amplitude of wave propagating in the negative and positive y-directions, respectively, polarization dependent constant K_jFor TE polarized waveAnd for TM polarized waves is represented byAnd (4) showing. The matrix that interconnects the magnitudes of the fields in adjacent layers is called the transfer matrix. This transfer matrix is only one type of transfer matrix that can be used to calculate the specular reflection, diffuse reflection, or amplitude of the light waves of the optical stack 30. Many other types of transfer matrices may be used. In addition, other methods for calculating diffuse reflectance that do not use a transfer matrix may also be employed in accordance with the principles of the present disclosure.

Fig. 9B shows the optical stack 30 of fig. 9A, wherein the nanowires 32 have scattered light incident on the optical stack 30 of fig. 9A. The light sources appearing in fig. 9A are not presented in fig. 9B to emphasize that the present focal point is diffusely reflecting rather than specularly reflecting. Thus, the nanowire 32 has light scattered in multiple directions.

The diffuse reflection corresponds to the amount of light scattered by the nanowires 32 that exits the optical stack 30 through the surface 37. Thus, according to one embodiment, a method for calculating diffuse reflectance includes calculating the amount of light scattered from the nanowires 32 in all directions. As previously described, when calculating the transfer matrix to determine specular reflection, the field at any location in the optical stack may also be calculated. One step in calculating the light scattered by the nanowires 32 is to calculate the field at the location of the nanowires 32.

Once the field at the nanowire location has been calculated or estimated, the amount of light scattered by the nanowire can be obtained by calculating or estimating the scattering cross-section of the nanowire 32. The scattering cross section of the nanowire can be obtained by solving the system of maxwell's equations for a long cylindrical wire of a given shape. For a line with a circular cross-section, the scattering cross-section can be calculated without putting a great burden on the processing resources. The scattering cross-section may also be calculated for other shapes of lines, such as lines having a polygonal or other cross-section. In one example of such a calculation, the solution to the system of maxwell's equations is represented as a set of cylindrical waves, and boundary conditions along the circumference of the line are used to correlate the amplitudes of these waves. In the article (Viktor a. podolsbiy, Evgenii nairimanov, Wei Fang, and HuiCao, "Chaotic ceramics based on dynamic positioning," proc. nat. acad. sci. v.101(29) pp.10498-10500(2004) and its references), an implementation of this formula is described using an example from dielectric resonator light emission. This article is incorporated by reference herein in its entirety. Once this correlation is found, it is simple to correlate the energy flux scattered by the line with the energy flux incident on the line, and to use this correlation to calculate the scattering cross-section of the line. The scattering cross-section describes the proportion of light incident to the nanowire 32 that will be scattered by the nanowire 32.

By multiplying the field of light incident on the nanowire 32 by the scattering cross section of the nanowire, the amount of light scattered by the nanowire 32 can be calculated. By again calculating the transfer matrix of light scattered by the nanowires 32 within the optical stack 30, the total diffuse reflection of the optical stack 30 can be calculated or estimated. Diffuse reflection is the amount of light scattered by the nanowires 32 that exit the optical stack 30 from the surface 37. In one embodiment, the nanowires are considered to scatter light equally in all directions. Mathematically, the spectrum A (k) of the diffusely reflected light_x) X component k independent of wave vector_x。

Similar to the specularly reflected light of FIG. 9A, in FIG. 9B, the light diffusely reflected from the nanowires is also transmitted and reflected at each interface within the optical stack. As described above, the transfer matrix for calculating the diffuse reflection is calculated for the top mode and the bottom mode.

As described above, light forward scattered from the nanowires 32 will be incident on the interface 44 between the layers 34 and 38. A portion of the light will be reflected back towards the surface 37. A portion of the light forward scattered from the nanowires 32 will be transmitted through the interface 44 into the layer 38. The light will again propagate to the interface 44 between the layers 38 and 36 where some light will be transmitted and some light will be reflected back up towards the interface 44. Some of the light transmitted through the interface 36 will be reflected at the interface 40 and some will pass through the interface 40. All light passing through the interface 40 will represent diffuse transmitted light. The light reflected by each interface 44, 4240 will contribute to diffuse reflection. However, the main contribution to diffuse reflection comes from the light impinging into the bottom mode of the system (shown above the nanowires in fig. 9B). The amount of light transmitted through interface 37 (which will represent all light scattered by the lines into the bottom mode and the portion of light initially impinging into the top mode and subsequently reflected by interfaces 44, 42 and 40) represents the total diffuse reflection in the system.

The diffuse reflection can be calculated by a method similar to the specular reflection described with reference to fig. 9A. That is, the scattering cross-section is obtained and the transfer matrix calculation is performed for the diffuse reflected light transmitted and reflected at all interfaces within the optical stack 30. In this way, the total diffuse reflection can be very close, however, relatively few processing resources are used.

In one embodiment, the field at the nanowire location may include both the field from the incident light and the field from the previously scattered light. In other words, some of the light scattered by the nanowires 32 will be reflected within the optical stack 30 and will be scattered again by the nanowires 32. By taking into account the field from the diffusely reflected light at the nanowire location, the accuracy of the diffuse reflection calculation can be improved.

The calculation of the light scattering is generalized to take into account the phase of the scattered light. To achieve this, it is assumed that the radius of the nanowire is very small, so that the scattering of the nanowire is dominated by the lowest possible cylindrical harmonic (empirical calculations indicate that TE scattering is dominated by cylindrical modes with m 0[ independent of polar angle ], while TM scattering is dominated by cylindrical modes with m 1[ dipole-like ]. Thus, the scattering spectrum is proportional to:

<math> <mrow> <msub> <mi>A</mi> <mi>TE</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mi>x</mi> </msub> <mo>)</mo> </mrow> <mo>&Proportional;</mo> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mi>y</mi> </msub> </mfrac> </mrow> </math>

<math> <mrow> <msub> <mi>A</mi> <mi>TM</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mi>x</mi> </msub> <mo>)</mo> </mrow> <mo>&Proportional;</mo> <mfrac> <mi>c</mi> <mi>n&omega;</mi> </mfrac> </mrow> </math>

where n is the refractive index of the material around the line, k is the wave vector, and ω is the angular frequency. Note that: when the radius of the nanowire is sufficiently small, both cases are reduced to the above-mentioned k_xIndependent spectrum of (a).

The scattered light is represented as the sum of the "illuminated" waves (bottom mode is y >0, top mode is y <0) plus the sum of the top and bottom mode reflected components, respectively. The amplitude of the top and bottom modes illuminated by the light source is the same for the TE polarization and opposite for the dipole TM polarization. When considering the interference of the top and bottom modes, the effective amplitude of the illumination light becomes:

for TE waves:

$a_b^{+}=a\left(k_x\right)\frac{1+r_t}{1-r_t{r}_b};$ $a_t^{-}=a\left(k_x\right)\frac{1+r_b}{1-r_b{r}_t}$

for TM waves:

$a_b^{+}=a\left(k_x\right)\frac{1-r_t}{1-r_t{r}_b};$ $a_t^{-}=-a\left(k_x\right)\frac{1-r_b}{1-r_b{r}_t}$

wherein, a (k)_x) Is the amplitude of the irradiated light, and r_t、r_bThe reflection coefficients of the top mode and bottom mode components.

To calculate the field that is fed back, i.e. the diffusely scattered light is again incident on the nanowire 32, we multiply the emitted fields with their respective reflection coefficients and add the two. Thus, the total amplitude of the field at the line location becomes:

for TE waves, it is:

for TM waves, is

Factor dk_xRepresenting the step size of the wave vector spectrum used for the mathematical calculation. The self-consistent calculation of the field at the nanowire location includes incident light from an external light source as well as diffusely reflected light. In a self-consistent approach, the field may be described as:

resulting in a matrix relationship describing the illumination spectrum:

<math> <mrow> <mi>a</mi> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mi>x</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>dk</mi> <mi>x</mi> </msub> <mover> <mi>R</mi> <mo>~</mo> </mover> <mi>A</mi> <mo>]</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>Aa</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <msubsup> <mi>k</mi> <mi>x</mi> <mo>&prime;</mo> </msubsup> <mo>)</mo> </mrow> <mo>.</mo> </mrow> </math>

wherein, A (k)_x,k_x') describe a wave from a wave vector k_x' the plane wave enters a wave vector k_xScattering of plane waves, and (diagonal) matrixHaving a value corresponding to the foregoing a_totThe composition of the coefficients of (a).

As described above, these calculations may be simplified where the percentage of diffusely reflected light that returns to the nanowire location is small. In this case, the energy flux of the spectral components of the diffuse reflected light enhances:

${\left|\frac{1+r_t}{1-r_b{r}_t}\right|}^2$

in some applications, partial diffuse scattering is calculated or estimatedDiffuse reflection of light, rather than all, may be advantageous. For example, it may be important to estimate the amount of diffusely reflected light towards the observer or only the amount of light scattered away from the observer, rather than the total amount of diffusely scattered light in all directions. In these cases, the formula developed above can be used to calculate the energy flux due to diffuse reflection, which is the angle of incidence Φ_iAnd angle of reflection theta_rAs a function of (c). The above equation for the transfer matrix used to calculate/estimate the total diffuse reflection can be used to calculate or estimate the angular distribution of this diffuse reflection. In these cases, both the angle of incidence and the angle of reflection may pass through the wave vector k_xIs parameterized and then based on the amplitude a (k)_x) To calculate an angular distribution of the energy flux representing the diffuse reflection.

To improve the estimation accuracy of the angular dependent diffuse reflection, the scattering probabilities a (k) of the different modes may be used_x,k_x') is incorporated into the calculation. Specifically, k may be used_xUncorrelated spectra, dipole type directional spectra, combinations thereof, or other directional spectra. In one embodiment, the scattering probability A depends on polarization, for TE polarized waves, a directionally uncorrelated energy flux is generated, and for TM polarized waves, a dipole type radiation pattern is generated (with an energy flux inversely proportional to cos)²(φ_i+θ_r)). In other embodiments, the energy flux of the TE polarized wave is 1/cos²θ_rProportional, and energy flux of TM polarized wave and cos²(φ_i+θ_r)/cos²θ_rIn inverse proportion.

There may be a number of different models of the scattering probability a, all of which fall within the scope of the present disclosure. When developing these models, it is helpful to remember that the transfer matrix model represents an estimate of the diffuse scattering process. Thus, the coefficients can be fine-tuned by comparing the prediction of the transfer matrix code with a strict (but more time-consuming) solution of the system of maxwell's equations using finite element methods, time-domain finite difference methods, strict coupled wave approximation methods, or other methods.

Using the simple method described above to calculate or estimate the diffuse reflectance of an optical stack, the diffuse reflectance of many optical stacks 30 with different parameters can be calculated using an optimization program to find the optical stack 30 that produces the lowest diffuse reflectance. Commercially available optimization programs, such as those available in Matlab, may be used to optimize the diffuse reflectance for many optical stack configurations in accordance with the principles of the present disclosure. In conjunction with the above-described method of calculating diffuse reflectance, these optimization routines may assist in finding an optical stack with relatively low diffuse reflectance.

The specific optimization goals of this process depend on the final application. For example, the total diffuse reflection of the system can be optimized for a given wavelength. The weighted averages corresponding to diffuse reflections in one or more particular directions may also be combined to weight the total diffuse reflection, subject to the constraint that the diffuse reflection in a particular direction remains below a certain value. It is also possible to estimate the diffuse reflection of light at different wavelengths and aggregate these estimates by some method (e.g., averaging, weighted averaging, etc.) to arrive at the final target figure of merit to be optimized. All such combinations may be implemented by one of ordinary skill in the art in light of this disclosure.

While diffuse and specular reflectance calculations are described above in terms of a transfer matrix, other methods other than a transfer matrix may be used to obtain diffuse reflectance values in accordance with the principles of the present disclosure. These other methods are also within the scope of the present disclosure.

One example of these methods includes an extension of the proposed method to optimize specular and diffuse reflection for an optical stack that incorporates inside the optical stack a set of fixed thick layers that may include a thick base layer (e.g., optical glue) or a thick protective layer (e.g., cover glass layer). By "optically thick" is meant here that the thickness of the layer is greater than or equal to the coherence length of the radiation present in the stack.

Light propagation through the optically thick layer is somewhat similar to the process described above for generating the top and bottom mode contributions. For example, consider the specular reflection of the top mode shown in FIG. 9A. As described above, light entering the optical stack through the interface 37 will be partially reflected and partially transmitted through this interface. The transmitted portion will enter layer 34 and reach interface 44 where part of the light will be transmitted into layer 38 and part of the light will be reflected back into layer 34. This reflected light will reach the interface 37 where it will be partially transmitted out of the stack (contributing to specular reflection) and partially reflected back into the stack. When layer 34 is optically thick, the second (and subsequent) contribution to specular reflection will not interfere with light initially reflected by interface 37. Instead, the corresponding energy fluxes will be added together. Based on the (energy flux based) reflectivity (R) and the transmissivity (T) of the inter-layer interface, it is simple to calculate the specular reflection of a stack incorporating several optically thick layers.

For example, the following recursive method may be used. The layers in fig. 9A are assumed to be optically thick. Then the reflectivity of the interface 40 can be calculated using the square of the absolute value of the corresponding fresnel coefficient. The refractive index of the light entering the interface 42 can then be calculated as follows:

${\tilde{R}}_i=R_i^{+}+\frac{T_i^2{\tilde{R}}_{i-1}}{1-R_i^{-}{\tilde{R}}_{i-1}}$

wherein,for the (total) reflectivity of light entering the system from layer 38 and from 42,is the single-phase interface reflectivity of the interface 42,is the reflectance of the same interface to light propagating into layer 38 (typically,) And is andis the total reflectance of light entering the interface 40. The same equation can then be used to calculate the total reflectance of light entering interface 44 and finally the reflectance of interface 37.

If the system includes a mixture of optically thick layers and optically thin layers, the optical properties (reflectivity and transmissivity) of the optically thin layers can be calculated using the transfer matrix equation, which can then be approximated as individual interfaces (with known reflectances/transmittances) in the optically thick stack.

A similar technique can be used to calculate the diffuse reflection for the presence of an optically thick layer.

FIG. 10A shows a Graphical User Interface (GUI)48 of an optical stack optimization software program stored in a computer readable medium. The GUI may be displayed on a display coupled to the processor to allow a technician to implement an optimization procedure for finding an optical stack 30 having parameters that will produce a preferred diffuse reflectance value. The processor reads the software instructions from a memory circuit coupled to the processor. Accordingly, software instructions for running an optimization program to find preferred parameters of the optical stack 30 are stored in a memory coupled to the processor. Thus, the processor causes the display to display a GUI, and the technician may enter the parameter ranges for the optical stack via a mouse, keyboard, or any other suitable input device. The parameters may include the number of layers in the stack, the index of refraction of the substrate 36, and the environment in which the optical stack 30 is to be placed.

Thus, in the exemplary GUI 48 of fig. 10A, the refractive index of the protective layer is 1 because it is air. The refractive index of the substrate is listed as 1.5 and corresponds to the substrate 36 of the optical stack 30. The user can input the refractive index of any substrate or protective layer as desired. The radius of the nanowire 32 is also input for calculating the scattering cross section. According to one embodiment, the line radius entered in the GUI 48 is 50 nm. However, the radius of the wire may be any other suitable radius depending on the particular nanowire 32 or other nanostructure used in the optical stack 30. The user may also select which layer of the optical stack 30 the nanowires 32 are located on. In the GUI 48 illustrated in FIG. 10A, the line layer has been selected as layer 2, which corresponds to layer 34 of the optical stack 30. In the fields labeled as active layer parameters, the user may enter minimum and maximum thicknesses for layers 34 and 36, corresponding to layers 1 and 2 of GUI 48. In the example of fig. 10A, both layer 34 and layer 36 have a thickness ranging from 50nm to 200 nm. The refractive index of each of layers 34 and 36 ranges from 1.2 to 2.2. These ranges are the limits within which the optimization program will select parameters for the optical stack 30 to calculate which parameters produce the best diffuse reflection. When the program is executed, diffuse and specular reflections are calculated for several optical stacks having parameters within the input ranges of layer thicknesses, refractive indices, and light wavelengths. The optimization program calculates diffuse and specular reflections according to the methods described above or using other suitable methods in accordance with the principles of the present disclosure.

In one embodiment, the optimization program calculates the diffuse reflectance for a first set of optical stacks having different parameters within a given range, rather than iteratively calculating the diffuse reflectance for each possible iteration within the input range. The optimization program then selects a second set of optical stacks having parameters slightly different from those in the first set that produce the lowest diffuse reflectance. In this manner, the optimization program continues to calculate the diffuse reflectance of the optical stack until a preferred diffuse reflectance is found. Without computing each possible iteration, the optimization program can efficiently find the parameters that produce the preferred diffuse reflection. In this way, a particular configuration of the optical stack 30 that produces relatively low diffuse reflectance may be selected. This is possible due to the more simplified method of calculating or estimating the diffuse reflectance of the optical stack 30 described above.

Possibly with low diffuse reflection and at the same time with unacceptably high specular reflection. For this reason, below the active layer parameter field is a field labeled maximum reflection. In this field, the skilled person can specify the maximum tolerable specular reflection. In this example, the maximum specular reflection has been chosen to be 1.5%. This means that when running the transfer matrix for both specular and diffuse reflection, the preferred stack configuration will be chosen to be the lowest diffuse reflectance where the specular reflection is no more than 1.5%.

In the right area, the optical stack is shown. The optical stack 30 includes a lower index of refraction layer 34 comprising nanowires 32 overlying a higher index of refraction layer 38. Layer 38 is located on substrate 36 having a refractive index of 1.5. The refractive index of the air above the optical stack is 1. In layers 34 and 38, on the left side of each layer, a thickness range and a refractive index range are given. On the left side of the layer 34, this is denoted w₂50nm to 200nm and n₂1.2 to 2.2. These are the thickness range of layer 34 and the refractive index range within which iterations are performed to find specular and diffuse reflections when calculating the transfer matrix. Also, the range w is specified on the left side of the layer 38₁50nm to 200nm and n₁1.2 to 2.2. On the right side of layer 34, preferred thicknesses are listed, as well as preferred refractive indices. In particular, a preferred thickness of layer 34 is given as 118.2 nm. The preferred refractive index of layer 34 is 1.2. The preferred thickness of the high refractive index layer 38 is 50nm and the preferred refractive index is 1.7779. Under the optical stack, the specular reflection is listed as R₀0.0144 or about 1.4%. Diffuse reflection R_{Free game}Listed as 5.469X 10^-5。

Thus, the GUI 48, which is capable of operating the optimization method of the optical stack 30, allows a user to enter a first optical stack parameter or input parameters, and run the program, perform calculations, list preferred specular and diffuse reflections, and list the layer thicknesses and refractive indices that produce these preferred results. Those skilled in the art will appreciate, in light of the present disclosure, that many changes can be made to the methods described, as well as to the specific GUIs and inputs and outputs provided thereby.

FIG. 10B illustrates a GUI 50 according to one embodiment. The GUI 50 relates to a method by which a detailed plot of specular and diffuse reflections for a variety of wavelengths can be calculated based on the preferred parameters output from the GUI 48 in fig. 10A. Specifically, the user may input the number of layers (2 in this example), the substrate index of refraction (1.5) of the substrate layer 36, and the index of refraction (1) of the protective layer from the preferred output. Then, an active layer may be selected, in this example, active layer 2 is highlighted, which means that the parameters of layer 34 may be entered in the fixed parameters field. The preferred features as determined by the GUI 48 of fig. 10A are a thickness of 118.2nm for the layer 34 and a refractive index of 1.2. The active layer may then be highlighted and the preferred characteristics of the layer 38 calculated with reference to the GUI 48 of FIG. 10A may be entered. In this example, the preferred feature is a thickness of 50nm and a refractive index of 1.7779. The range of wavelengths of light for which the plot is to be generated may be entered in a field (labeled as wavelength nm) below the fixed layer parameters. In this example, the minimum wavelength to be iterated is 300nm and the maximum wavelength is 800nm, step size 10 nm.

FIG. 10C shows the drawing generated by the GUI 50 of FIG. 10B. In particular, fig. 10C is a plot of specular and diffuse reflection over the wavelength range specified in fig. 10B. Both specular and diffuse reflection experience peaks in the ultraviolet range slightly less than 400 nm. The specular reflection drops and reaches a minimum of about 1% at a wavelength of 500nm and then gradually increases to about 2.5% at 800 nm. The diffuse reflection drops and also reaches low values at about 500nm, but remains relatively flat up to 800nm, only very muchA gentle upward slope. In this example, the diffuse reflectance remains at about 5 × 10 over most of the visible spectrum^-5. For most of the visible spectrum, the specular reflection remains between 1% and 2%.

Certain wavelengths of light may be weighted more heavily than other wavelengths of light in the software instructions stored in the memory. When calculating the transfer matrices, each transfer matrix is performed for a range of wavelengths, except for a range of thicknesses of the layers and a range of refractive indices of the layers. In one embodiment, when calculating the preferred diffuse reflectance, the reflectance at certain wavelengths may be weighted more heavily than the reflectance at other wavelengths. The human eye is more sensitive to certain wavelengths than others. Thus, for some optical stacks, the diffuse reflection may be slightly higher at less prominent wavelengths and close to a minimum at more prominent wavelengths. In such an example, diffuse reflection may be the preferred diffuse reflection, although at some wavelengths it is not near the minimum diffuse reflection. For this reason, it may be desirable to give greater weight to diffuse reflection at certain wavelengths. In one example, the visible spectrum is separated in 50nm increments between 400nm and 700 nm. The software stored for calculating the diffuse reflectance can be modified to give higher or lower relative weights to different wavelengths. For example, in one embodiment, wavelengths between 450nm and 600nm are weighted more heavily than others. Of course, the weighting may be a technician choice that changes the code stored in memory. For the calculation of the specular reflection, weighting may also be implemented.

FIG. 10D illustrates a GUI of a software program configured to find an optical stack with relatively low diffuse reflectance, in accordance with one embodiment. The GUI of fig. 10D allows the user to select the number of layers of the optical stack 30 and which layer the nanowires 32 are to be placed on. In the example of fig. 10D, the number of layers is three and the nanowire layer is layer 2. After the nanowire layer is selected, the user may enter the thickness ranges and refractive index ranges of the other layers in the optical stack 30. However, in the embodiment of fig. 10D, the thickness and refractive index of the nanowire layer cannot be changed by using the GUI; these parameters are fixed in the embodiment of fig. 10D. The parameters of layer 1 can be entered by selecting layer 1 as the active layer and then entering the thickness range and the refractive index range in the field of the mark. In the same way, the parameters of the layer 3 can be entered. In FIG. 10D, the user has selected that the thickness of both layer 1 and layer 3 range from 30nm to 300 nm. The refractive index range of layers 1 and 3 has been selected to be 1.2-2.2. These ranges are the ranges from which the optimization routine will select the thickness values and refractive index values of the layers during the optimization process.

The refractive indices of the protective layer and the substrate layer may also be selected by entering a value in the field of the mark. In the example of fig. 10D, these have been chosen to be 1 and 1.5, respectively. Once these parameters have been selected, a basic graphic of the layers of the optical stack 30 is displayed on the right side of the GUI indicating the position of the layers, the position of the nanowire layer, the range of refractive indices and thicknesses, and the refractive indices of the protective layer and the substrate.

The user may also select whether the optimization routine will optimize diffuse or specular reflection by checking the appropriate options in the optimization field. If optimized diffuse reflection is chosen, the maximum specular reflection can also be chosen by entering a value in the field of maximum specular reflection. The program will select an optical stack that has low diffuse reflection and specular reflection equal to or less than a selected maximum. Alternatively, if the user selects optimal specular reflection, the user may enter a maximum diffuse reflection value for the optical stack.

Finally, the user can click the start button to run the optimization program. The optimization program then calculates the diffuse and specular reflections for a number of possible optical stacks and selects an optical stack that has a relatively low diffuse reflection and a specular reflection that is less than a selected maximum. The parameters of the selected optical stack will then be output. The user may also save the optimal optical stack parameters or load a previously saved optical stack by clicking on the appropriate button.

FIG. 10E illustrates a GUI for calculating and rendering diffuse and specular reflectance in accordance with an alternative embodiment. The GUI of fig. 10E allows the user to select the number of layers of the optical stack 30 and which layer the nanowires 32 are to be placed on. In the example of fig. 10E, the number of layers is three and the nanowire layer is layer 2. Thickness and refractive index cannot be changed by using GUI; these parameters are fixed in the embodiment of fig. 10E. After selecting the nanowire layer, the user may input the thickness and refractive index of the other layers in the optical stack 30. By selecting layer 3 as the fixed layer, the parameters of layer 3 are entered, and then the thickness and the refractive index are entered under the field of the mark. In the same way, the parameters of layer 1 can be entered. The refractive indices of the protective layer and the substrate may also be selected; these parameters have been chosen to be 1 and 1.5, respectively. Once these parameters are selected, the basic graphic of the layers of the optical stack 30 will be displayed on the right side of the GUI. In the example of fig. 10E, the nanowire layer is layer 2.

The wavelength range and step size for calculation and mapping may also be input. In the example of fig. 10E, the selected wavelength range is from 300nm to 800nm, step size 10 nm. When all fields have been filled, the user may click on the start button to start the calculation routine. For all wavelengths, specular as well as diffuse reflection were calculated. A graph showing the specular reflection as well as the diffuse reflection for each wavelength can be output. A table may also be output showing numerical values for diffuse reflection and specular reflection for each step wavelength in the wavelength range. Many other configurations of the GUI are possible as will become apparent in light of this disclosure. All such other configurations are also within the scope of the present disclosure.

As described above, the diffuse reflection may be calculated for a selected angle of diffuse reflection or range of angles of diffuse reflection relative to the surface of the optical stack 30. In some applications, it is beneficial to know how much light is diffusely reflected at a particular angle or angles relative to the surface of the optical stack 30. Accordingly, in one embodiment, a user of the optimization software may select a plurality of angles for which diffuse reflectance is estimated for each optical stack configuration.

In one embodiment, a set of diffuse reflection values is calculated or estimated for each iteration of the optical stack parameters. Each set of diffuse reflectance values includes a plurality of diffuse reflectance values at selected angles relative to the optical stack 30. The optimization routine may be configured to select an optical stack configuration based on the set of diffuse reflectance values. In particular, the sets of diffuse reflection values are compared to each other, and based on the comparison, the optimization routine may select an optical stack configuration. The optimization routine may also be configured to compare the diffuse reflectance values of the various corners to a threshold. The optimization routine then selects a set of diffuse reflection values based in part on the comparison to the threshold.

In one example, the skilled person may select eleven different reflection angles for which to calculate the diffuse reflection. The eleven angles with respect to the normal may include: 75 °, 60 °, 45 °, 30 °, 15 °,0 ° (i.e. normal), -15 °, -30 °, -45 °, -60 °, and-75 °. Each set of diffuse reflection values will include a diffuse reflection value for each selected angle. In this example, each set of diffuse reflection values will include eleven diffuse reflection values. Of course, more or fewer angles may be selected. The specific angles and number of angles described above are given by way of example only.

In one example, a large reflection angle with respect to the normal has a higher threshold than an angle closer to the normal. In other words, higher diffuse reflection can be tolerated at large angles relative to the normal. This is because, in some embodiments, the optical stack 30 may be included in a display screen where display quality at angles very close to normal is more important. Large angles relative to the normal correspond to peripheral viewing angles of a display screen incorporating the optical stack 30, and maintaining high optical quality at these angles may be less important. Thus, the threshold for diffuse reflection at angles close to normal may be much smaller relative to the threshold for angles far from normal. This is because the display is more often viewed from angles close to normal relative to the display screen.

In one embodiment, if any one of the values of diffuse reflectance in a particular set exceeds a corresponding threshold value of diffuse reflectance, the optical stack configuration associated with that particular set will not be selected.

Optionally, each set of diffuse reflectance values may be compared to a single diffuse reflectance value threshold. If any one of the diffuse reflectance values in a particular set exceeds the diffuse reflectance value threshold, the optical stack configuration associated with that particular set will not be selected.

In one embodiment, for each set of diffuse reflection values, a total diffuse reflection value may be calculated. The optimization routine may select the optical stack configuration that corresponds to the lowest total diffuse reflectance value. Calculating the total diffuse reflectance value may include summing the diffuse reflectance values. Alternatively, calculating the total diffuse reflectance value may comprise assigning a respective weighting factor to each reflection angle.

In one embodiment, an average diffuse reflectance value for each set may be calculated. The average diffuse reflection of a collection corresponds to the average of the diffuse reflection values calculated in the collection. Based on the average diffuse reflectance for each set, the optimization routine may select an optical stack configuration.

The optimization routine may be configured to assign greater weight to the diffuse reflectance values of some corners and lesser weight to the diffuse reflectance values of other corners. The optimization routine may also assign greater weight to certain wavelengths of light at multiple angles. The optimization routine may select the optical stack configuration by considering the diffuse reflection of many wavelengths per reflection angle.

Many other optimization routines, software programs and methods of calculating or estimating diffuse reflections at different angles may be implemented. All such other routines and methods are included within the scope of the present disclosure.

FIG. 11 illustrates a system 60 in accordance with one embodiment. The system 60 includes a processor 62 configured to execute software instructions stored in a memory circuit 64. The memory circuit 64 stores data that the processor reads to perform the optimization method described above. An input module 66 is also coupled to the processor 62. At input block 66, a technician operating system 60 may input parameters for optical stack 30, and processor 62 will then optimize these parameters and output parameters in response to the optimization. A display 68 is coupled to the processor 62. The processor 62 may cause the GUI 48 or the GUI 50 to be displayed on the display 60. The technician operating the input module 66 may then enter the appropriate fields by visually observing the GUI 48 or the GUI 50 on the display 68. Likewise, the optimization parameters may be displayed on the display 68.

In one embodiment, the system 60 includes a manufacturing device 70 coupled to the processor 62. In this embodiment, processor 62 outputs the output parameters directly to the fabrication facility, which then deposits the appropriate layers and thicknesses as described in the optimization output. For example, for an optical stack 30 (the optical stack 30 includes a low index layer 34 with nanowires 32 embedded therein, a high index layer 38 below the low index layer 34, and a substrate 36 below the high index layer 38), the optimized output may be provided to a fabrication device 70, which then deposits the layer 38 on the substrate 36 and the layer 34 on the layer 38. The aforementioned system 60 is given by way of example. But may also include many other components and software instructions not described herein. When a user operates the input module 66 to input parameters, the input parameters are stored in a memory 64 coupled to the processor 62.

In one embodiment, the memory 64 may comprise EEPROM, ROM, SRAM, DRAM, or any other suitable memory. Software instructions for performing the optimization process are stored in memory 64. The input instructions may be temporarily stored in the memory 64 or in a separate cache coupled to the processor. Any suitable component may be used for storing the input parameters and software instructions so that they can be read by the processor 62 used. Alternatively, the optical stack may be fabricated using the output of the process of selecting parameters for optics without physically coupling the fabrication equipment to the circuitry used in selecting the optical stack parameters.

FIG. 12 is a flow chart illustrating a method for optimizing parameters of the optical stack 30. At 80, the technician inputs the layer parameters into the processor. The input parameters are then stored in a memory coupled to the processor. The input parameters may include the number of layers of the optical stack 30, the thickness range of the layers in the optical stack 30, the refractive index range of the layers in the optical stack 30, the wavelength ranges of the diffuse and specular reflections to be calculated, and the relative weight values to be assigned to different wavelengths in the wavelength range. At 82, the processor calculates the field at the location of the nanowire 32 within the optical stack 30. The calculation of the field at the nanowire locations may be performed by using a transfer matrix or any other suitable calculation that provides the field at the nanowire 32 locations. In an alternative embodiment, the calculation of the field at the nanowire location may include a field from previously scattered light, as described above.

At 84, the scattering cross section of the nanowire 32 is calculated. The scattering cross section of the nanowire 32 gives an indication of the direction and magnitude of the scattering of the diffusely reflected light from the nanowire 32. The nanowires 32 may diffusely reflect light in any direction. At 86, the processor calculates the diffuse reflection based on the calculated field at the nanowire location and the scattering cross section. In one embodiment, the diffuse reflection is estimated by calculating a transfer matrix for the transmission and reflection of the diffusely reflected light at each layer interface and through each layer in the optical stack 30.

At 88, the calculation of the field at the location of the nanowire 32, the scattering cross section of the nanowire 32, and the diffuse reflected light reaching the surface is performed iteratively for a number of optical stacks 30 within the input parameters. In one embodiment, the diffuse reflectance calculations are performed for a first set of optical stacks. The first set of optical stacks may have layer thickness values, refractive indices of the layers, etc., selected to provide a wide first sampling of the optical stacks over a range of possible inputs. For example, the first set of optical stacks may include optical stacks having first layers with respective minimum and maximum thicknesses and some thicknesses dispersed therebetween. For the first group, the diffuse reflections were calculated and compared to each other.

The diffuse reflectance of the second set of optical stacks is then calculated. In one embodiment, the parameters of the second set of optical stacks are selected based in part on the diffuse reflectance of the first set. For example, the second set of optical stacks includes optical stacks having one or more parameters that are close to the one or more parameters that produce the lowest diffuse reflectance values in the first set of optical stacks. This allows the processor to find the preferred diffuse reflectance value without calculating every possible optical stack within the range. Instead, the processor may analyze the optical stack whose parameters are most likely to have low diffuse reflectance. This process can be continued as long as necessary to obtain a sufficiently deep optimization process, where time and computing power allow. Finally, the processor can select the optical stack parameters that produce the best diffuse reflectance values. At 92, the optical stack 30 is formed by depositing layers having characteristics corresponding to the optimal output parameters.

Materials that can be used to fabricate the layers of an optical stack according to the present invention are known in the art. Examples of such materials include, for example, TiO₂(R_D1.8), polyimide (R)_D1.7) and embedded with e.g. ZnO, ZrO₂And TiO₂A transparent polymer of high refractive index particles.

Table 1 shows some relatively low index optical materials that may be used for the layers of an optical stack produced in accordance with the present invention.

TABLE 1

Table 2 shows some relatively high index optical stack materials that may be used for the layers of an optical stack produced in accordance with the present invention.

TABLE 2

It will be appreciated in the art that coating, printing, sputtering or other techniques may be used to deposit the optical stack with the desired thickness. Coatings having the desired wet film thickness are discussed in terms of coating Technology, particularly in "Modern coating and Drying Technology" by Edwards Cohen and Edgar Gutoff (John Wiley & Sons,1992, see pp.11 and 25-28), which is incorporated herein by reference. The dry film thickness resulting from a given wet film thickness depends on the composition of the coating solution used and is understood by those skilled in the art. Methods of coating and printing nanowire conductive layers are disclosed, for example, in U.S. patent No. 8,094,247 and U.S. patent application nos. 12/380,293 and 12/380,294, each of which is incorporated herein by reference.

FIG. 13 illustrates a method for optimizing parameters of an optical stack 30 in accordance with one embodiment. At 94, input parameters for the optical stack are input into a processor, which stores the input parameters in a memory circuit. The memory executes software instructions stored in the memory to begin a method of optimizing optical stack parameters. At 96, the processor calculates a transfer matrix for light incident on the optical stack having values within the parameters input to the processor at step 94. By calculating the transfer matrix, the specular reflection from the surface 37 of the optical stack 30 can be obtained. Also by calculating the transfer matrix, the field at the location of the nanowires 32 within the stack 30 can be calculated at 98.

At 99, the scattering cross section of the nanowire 32 is calculated. The scattering cross section of the nanowires 32 is indicative of the amount of diffusely reflected light scattered in each direction within the optical stack 30. At 100, a transfer matrix is calculated for diffusely reflected light scattered in all directions from the nanowires 32 within the optical stack 30. The transfer matrix gives the portion of the diffusely reflected light that reaches the surface 37 of the optical stack 30.

At 102, the processor checks whether more iterations of the input parameters are required. In one embodiment, for the first set of optical stacks, the processor will perform the diffuse reflectance calculations. For example, if the possible thickness of the first layer ranges between 50nm and 200nm, the processor may calculate the minimum and maximum thicknesses and the diffuse reflectance values for some thicknesses in between while keeping other parameters constant. The diffuse reflectance values are compared and the processor selects a next iteration value based on the comparison of the diffuse reflectance of the first set of optical stacks. At 104, the processor selects the parameters for the next iteration, and the processor performs the calculations of specular reflection, field at the nanowire location, and diffuse reflection for the new set of parameters. At 106, the processor selects a preferred diffuse reflectance from the set of diffuse reflectances (which has been calculated for the range of input parameters) and outputs specific preferred parameters of the optical stack 30 that produce the preferred diffuse reflectance.

FIG. 14 illustrates a flat panel device 120 incorporating an optical stack 30 according to one embodiment in a touch display screen. The optical stack 30 has been fabricated with the layer parameters obtained from the optimization process described above. The display of the tablet device 120 does not suffer from the above-described milky or hazy problem.

Although specific layers, thicknesses, and properties of the optical stack 30 are described herein, many other suitable configurations of optical stacks are possible, including more or fewer layers, multi-layer nanostructures, or any other suitable features. All such stacks are within the scope of the present disclosure.

Also, although the present disclosure discloses specific methods for optimizing the optical characteristics of the optical stack 30, many other suitable variations in the present method are possible. For example, fields, specular reflections, and diffuse reflections may be approximated by other means, but still fall within the scope of the present disclosure. More, fewer, or different parameters may be input into the processor to optimize the heap. Also, optimization may be performed for other parameters besides specular reflection and diffuse reflection. The word "best" should not be construed to mean the best possible configuration, but rather one value or configuration is preferred over another. Also, the optimum reflectance does not necessarily mean the lowest reflectance, but is a desired reflectance among possible reflectances.

The various embodiments described above can be combined to provide further embodiments. All U.S. patents, U.S. patent application publications, U.S. patent applications, foreign patents, foreign patent applications, and non-patent material referred to in this specification and/or listed in the application data sheet, are incorporated herein by reference, in their entirety. Aspects of the embodiments can be modified, if necessary to employ concepts of the various patents, applications and publications to provide yet further embodiments.

These and other changes can be made to the embodiments in light of the above detailed description. In general, in the following claims, the terms used should not be construed to limit the claims to the specific embodiments disclosed in the specification and the claims, but should be construed to include all possible embodiments along with the full scope of equivalents to which such claims are entitled. Accordingly, the claims are not limited by this disclosure.

Claims (31)

1. A method, comprising:

selecting optical stack parameters for an optical stack having nanowires;

calculating, for each of a plurality of optical stack configurations, a plurality of sets of diffuse reflectance values in accordance with the optical stack parameters, each set of diffuse reflectance values comprising a plurality of diffuse reflectance values for a respective angle of reflection from the optical stack;

selecting one of the optical stack configurations based at least in part on a comparison of the sets of diffuse reflectance values for the plurality of angles; and

the layers of the optical stack are formed according to a selected optical stack configuration.

2. The method of claim 1, comprising:

for each of the optical stack configurations of one of the optical stack configurations, a plurality of specular reflection values for the plurality of reflection angles is calculated.

3. The method of claim 1, comprising:

calculating a plurality of specular reflectance values, wherein each specular reflectance value is for a respective optical stack configuration;

comparing the specular reflection value to a predetermined specular reflection value; and

selecting one of the optical stack configurations based on a comparison with the predetermined specular reflectance values.

4. The method of claim 2, wherein selecting one of the optical stack configurations comprises: selecting an optical stack configuration having a specular reflectance value that is lower than the predetermined specular reflectance value.

5. The method of claim 1, comprising:

comparing each set of diffuse reflectance values to at least one predetermined diffuse reflectance value; and

selecting one of the optical stack configurations based at least in part on the comparison to the at least one predetermined diffuse reflectance value.

6. The method of claim 5, wherein selecting one of the optical stack configurations comprises: an optical stack configuration is selected in which the diffuse reflectance value for each reflection angle is below at least one predetermined diffuse reflectance threshold.

7. The method of claim 1, wherein selecting one of the optical stack configurations comprises: an optical stack configuration corresponding to a minimum diffuse reflectance value is selected.

8. The method of claim 1, comprising:

calculating a respective total diffuse reflectance value for each set; and

an optical stack configuration corresponding to a minimum total diffuse reflectance value is selected.

9. The method of claim 8, wherein calculating the respective total diffuse reflectance value for each set comprises: summing the diffuse reflectance values of the set.

10. The method of claim 8, wherein calculating the respective total diffuse reflectance values comprises: and distributing respective weight factors for the diffuse reflection according to the reflection angles.

11. The method of claim 1, comprising:

calculating a plurality of respective diffuse reflectance averages, each diffuse reflectance average corresponding to a respective set of averages of the diffuse reflectance values; and

selecting the optical stack configuration based at least in part on the plurality of diffuse reflectance averages.

12. The method of claim 1, wherein calculating the diffuse reflectance value comprises: the scattering cross section of the nanowire is calculated.

13. The method of claim 1, wherein calculating the diffuse reflectance value comprises: for each optical stack configuration, separately:

calculating an electromagnetic field of incident light at a location of the nanowire within the optical stack; and

a transfer matrix of light scattered from the nanowires within the optical stack is calculated.

14. The method of claim 13, wherein calculating the diffuse reflectance value comprises: calculating an amount of light scattered from the nanowire based on the scattering cross-section and a field of incident light at the location of the nanowire.

15. The method of claim 14, wherein calculating the field of incident light comprises: calculating an electromagnetic field of diffusely scattered light at the location of the nanowire.

16. The method of claim 1, wherein the plurality of optical stack parameters comprises a number of layers of the optical stack.

17. The method of claim 1, wherein forming layers of the optical stack comprises:

forming a first layer on a substrate; and

forming a second layer on the first layer, the nanowire being disposed on the first layer or the second layer.

18. A method, comprising:

storing the input optical stack parameters in a memory circuit coupled to the processor;

in the processor, for a plurality of optical stack configurations, calculating a plurality of sets of diffuse reflection values, wherein each optical stack has a respective configuration according to the optical stack parameters, each set of diffuse reflection values comprising a plurality of diffuse reflection values for respective angles of reflection from surfaces of the optical stack, calculating the sets of diffuse reflections comprising, for each configuration, respectively:

calculating an electromagnetic field value of incident light at a location within the optical stack corresponding to a location of a nanowire within the optical stack; and

calculating a transfer matrix to provide the plurality of diffuse reflectance values for a plurality of reflection angles from a surface of the optical stack based in part on the electromagnetic field values,

an optical stack configuration is selected based on the diffuse reflectance value.

19. The method of claim 18, wherein selecting the optical stack configuration comprises: an optical stack configuration corresponding to a minimum diffuse reflectance value is selected.

20. The method of claim 18, comprising:

for each set, calculating a respective total diffuse reflectance value; and

an optical stack configuration corresponding to a minimum total diffuse reflectance value is selected.

21. The method of claim 18, wherein the input optical stack parameter comprises a refractive index range of at least one layer of the optical stack.

22. The method of claim 21, wherein the selected optical stack configuration includes an index of refraction from within the range of indices of refraction.

23. The method of claim 18, wherein the input optical stack parameter comprises a thickness range of layers of the optical stack.

24. The method of claim 23, wherein the selected optical stack configuration comprises a thickness from within a range of thicknesses of layers of the optical stack

25. The method of claim 17, wherein calculating the set of diffuse reflection values comprises: the scattering cross section of the nanowire is calculated.

26. A system, comprising:

a processor;

a memory coupled to the processor;

an input coupled to the processor, configured to receive a first parameter of an optical stack, the processor configured to calculate a set of incident optical electromagnetic field values for a location corresponding to a nanowire within the optical stack, calculate a light scattering distribution of the nanowire, calculate a set of a plurality of diffuse reflectance values at a surface of the optical stack, and estimate a set of a second parameter of the optical stack, the second parameter corresponding to a set of preferred diffuse reflectance values, each set of diffuse reflectance values comprising a plurality of diffuse reflectance values for a respective angle of reflectance from the surface of the optical stack; and

an output coupled to the processor configured to receive the second parameter from the processor.

27. The system of claim 26, comprising a display coupled to the output, the display configured to display the second parameter.

28. The system of claim 26, comprising a deposition device coupled to the output, the deposition device configured to receive the second parameter and deposit a first optical layer of the optical stack according to the second parameter.

29. A method, comprising:

inputting parameters of the optical stack to a processor;

estimating, in the processor, a set of electromagnetic field values of incident light at locations corresponding to nanowires within an optical stack;

estimating, in the processor, a light scattering distribution of the nanowires;

estimating, in the processor, a plurality of sets of diffuse reflection values at a surface of the optical stack based on electromagnetic field values and the scattering cross-section, each set of diffuse reflection values comprising a plurality of diffuse reflection values for a respective reflection angle from the surface of the optical stack; and

outputting from the processor an optical stack configuration corresponding to the selected set of diffuse reflectance values.

30. The method of claim 29, wherein estimating the set of electromagnetic field values comprises: a first transfer matrix is calculated from the parameters of the optical stack.

31. The method of claim 30, wherein estimating the set of diffuse reflection values comprises: a second transfer matrix is calculated from the parameters of the optical stack.