SYSTEM AND METHODS OF REDUCING DIFFUSE REFLECTION  OF AN OPTICAL STACK

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.

DETAILED DESCRIPTION

Described herein include the underlying cause for the “milky” appearance of a nanowire display, methods for addressing the same, and optical stacks that have lower or no milky appearance. As used herein, “optical stack” refers to a multi-layer stack of thin films through which light from either an external or an internal source travels, one or more layers having an impact on the optical behavior of the light. The thin films within the optical stack are typically functional films such as transparent conductive films, polarizers, color filters, anti-glare films, or anti-reflective films, as well as protective coatings and clear adhesives. The thin films can be flexible (e.g., polymer substrate) or rigid (e.g., glass substrate). The optical stack is typically coupled to another functional unit such as a display. In addition to the films, air space between films or between the films and the display also contribute to the optical behavior of the light, and is considered a part of the optical stack.

Also in the context of film orientations, a film that “overlies” another film is configured to be more proximate to the external light source (or the viewer) than the other film. For instance, an overcoat that overlies the nanowire layer is always disposed between the external light source (or the viewer) and the nanowire layer. A film that “underlies” another film is configured to be less proximate to the external light (or the viewer) than the other film. For instance, in an optical stack that employs an undercoat that underlies the nanowire layer, the nanowire layer is always disposed between the external light source (or the viewer) and the undercoat.

FIG. 1illustrates an optical stack30of a conductive transparent thin film. In the basic optical stack (30), as in more complex ones (e.g., in an entire touch panel), many or all of the layers or structural elements may contribute to the diffuse reflection to certain degrees. Various embodiments described here are approaches to lessen the diffusion reflection through manipulations and modification of individual layers or structural elements. However, it should be understood that any one or more individual embodiments may be combined to provide additive benefit in further reducing diffuse reflection. Thus, various embodiments are directed to optical stacks that comprises at least one nanowire layer; and at least one substrate adjacent to the nanowire layer, wherein the nanowire layer includes a plurality of conductive nanowires, and wherein a diffuse reflection of an incident light, as viewed from the same side of the optical stack as the incident light, is some percentage of the incident light. As used herein, “adjacent” refers to the relative locations of the substrate and the nanowire layer. They may be in immediate contact, or are near each other with one or more intermediate layers interposed between.

The optical stack30includes conductive nanowires32embedded in a transparent insulating layer34. The transparent insulating layer34and the nanowires32are on a substrate36.

The optical stack30is of a type which can be used in a flat panel display. As such, it is desirable for the optical stack30to have properties which most enhance the visual characteristics of the optical stack. As described previously, an optical stack30including nanowires32can suffer from a milky or hazy quality. This milky quality can detract from the visual characteristics of the optical stack30. In particular, when it is desirable to display dark colors such as black, the optical stack30can instead display a milky color which adversely affects quality of the displayed images.

One source of these undesirable characteristics is diffuse reflection from the nanowires32. Typically, when light encounters a surface or an object, the angle of reflection is identical to the angle of incidence. This is called specular reflection. Specular reflection is illustrated inFIG. 2A. InFIG. 2A, a ray of light is incident on the surface37of the optical stack30with an angle of incidence φi. The ray is reflected from the surface37of the optical stack30at an angle φr, which is equal to φi.

However, as illustrated inFIG. 2B, some of the light which strikes the surface37of the optical stack30, or indeed any surface, is also reflected diffusely at a plurality of angles θr. This diffuse reflection manifests itself in light being scattered in many directions other than the expected angle of reflection for a specular reflection. While only one angle is labeled θrinFIG. 2B, the diffusely reflected light is reflected at many angles θr. The light incident on the surface37is scattered in many directions inFIG. 2B. While a very small portion of light is typically reflected diffusely from any surface, the optical stack30ofFIG. 2Bsuffers from further diffuse reflection due to the presence of the nanowires32.

When light is incident on an object or a structure which has dimensions smaller than the wavelength of the light, the light is scattered diffusely from the object. The nanowires32and the optical stack30are, in general, smaller than 100 nm in radius, for example between 5 and 100 nm in radius. 100 nm is much smaller than the smallest wavelength of visible light. Thus, when any visible light encounters the nanowire32, it is diffusely reflected from the nanowire32. In a transparent film, the great majority of light which is incident on the surface37, is transmitted through the surface37and into the layer34in which the nanowires32are embedded. It is only a small percentage of the light that is reflected at the surface. However, some fraction of the light which interacts with the nanowires32is reflected diffusely. This diffuse reflection is the major cause of the milky quality which can sometimes diminish the appearance an optical stack30including nanowires32. It has been shown using calculations that the diffuse reflection from the nanowires32can be reduced in several ways when the nanowires32are incorporated in an optical stack30.

One such method is to reduce the index of refraction of the layer34in which the nanowires32are embedded.FIG. 3Aillustrates curves of diffuse reflection versus the wavelength of light which is incident on the nanowires32. Three curves are shown, one each for layers having indices of refraction of 1.43, 1.33, and 1.23 respectively. The peak of the curve for the index of refraction of 1.43 is significantly higher than the curve for n equals 1.33 and n equals 1.23. For the layer whose index of refraction equals 1.43, the peak of diffuse reflection occurs when the wavelength of light is about 400 nm. 400 nm is on the edge of the visible spectrum, and corresponds to violet light. Humans typically are unable to see wavelengths smaller than 380 nm, which corresponds to ultraviolet light.

When the index of refraction is reduced to n=1.33, not only is the peak diffuse reflection reduced, but it is also shifted to a smaller wavelength. For the material whose index of refraction is n=1.33, the peak is reduced to about 6×10−4and the peak wavelength is about 370 nm. Thus, not only is less light diffusely reflected back out of the surface37of the optical stack30, but a larger portion of the light that is reflected is shifted out of the visible spectrum and into the ultraviolet spectrum. It should be noted here that diffuse reflection values are in arbitrary units in this graph but are nevertheless useful to understand the relative effect that changing parameters of an optical stack30has on the diffuse reflection.

The diffuse reflection for the material whose index of refraction is n=1.23 is smallest of the three curves. The peak diffuse reflection for n=1.23 is about 4.5×10−4. and, just as importantly, the peak wavelength is shifted even further into the ultraviolet range, which is not visible to the human eye. Thus, placing the nanowires32in a layer34whose index of refraction is smaller can both reduce the diffuse reflection and shift the peak diffuse reflection away from the visible spectrum.

It is also desirable to reduce the specular reflection as much as possible.FIG. 3Billustrates three curves of specular reflection versus the wavelength of light for the same three indices of refraction n as inFIG. 3A. As can be seen fromFIG. 3B, specular reflection is highest for the layer34whose index of refraction is n=1.43. The peak specular reflection is about 0.04 for n=1.43. However, the peak is outside of the visible range at about 300 nm. For a layer34which has an index of refraction n=1.33, the peak specular reflection is decreased by a small amount. However, for most of the visible spectrum, which corresponds to about 400 nm to 700 nm in wavelength, the specular reflection for n=1.33 is far lower than the specular reflection for n=1.43. Thus, while the primary concern of the present disclosure is to reduce diffuse reflection, specular reflection is not to be neglected. Reducing both the specular reflection and the diffuse reflection can most enhance the visual characteristics of the optical stack30.

For a layer34whose index of refraction is n=1.23, the specular reflection is lowest of all. Not only is the peak specular reflection reduced, but the specular reflection in a large portion of the visible spectrum is very close to 0, with a low point coming around 500 nm. Thus, it is highly beneficial for both the diffuse reflection and the specular reflection to reduce the index of refraction of the layer in which the nanowires32are embedded.

Another parameter of the optical stack30which can affect the specular and the diffuse reflection is the thickness of the layer34in which the nanowires32are embedded.FIG. 4Aillustrates a curve of diffuse reflection versus the thickness of the layer34in which the nanowires32are embedded for several wavelengths and for an index of refraction of n=1.23. As can be seen, the diffuse reflection for light whose wavelength is 400 nm is slightly higher than for light whose wavelength is 450, 500, or 650 nm. Perhaps most notably, the diffuse reflection for any given wavelength remains mostly constant throughout the entire range of thickness from about 20 to 400 nm in thickness of the layer34. Diffuse reflection for 400 nm light is both larger in magnitude and more variable than the diffuse reflection for the other wavelengths inFIG. 4A. In other optical stacks this is not the case. In fact, the thickness of the layer can be very important in some configurations.

FIG. 4Bplots the diffuse reflection from nanowires32in a layer34whose index of refraction is n=1.33. The slight increase in the index of refraction results in an increase in the magnitude of the diffuse reflection. In particular, the diffuse reflection for light whose wavelength is 400 nm has increased more than the diffuse reflection for light whose wavelength is 450, 500, or 650 nm. Thus,FIGS. 3A and 3BandFIGS. 4A-4Cillustrate that the diffuse reflection fluctuates most heavily near the violet end of the visible spectrum.

InFIG. 4C, the in index of refraction is n=1.43. With this increase in the index of refraction, a large increase in the diffuse reflection of 400 nm light has occurred. Smaller increases in the diffuse reflection for light whose wavelength is 450 nm, 500 nm, and 650 nm has also occurred, but to a much smaller extent.

However, specular reflection fluctuates greatly with a change in the thickness of the layer34in which the nanowires32are embedded. The specular reflection follows a sinusoidal wave pattern for each of the four wavelengths of light that are plotted inFIG. 5A. All of the wavelengths of light experience peaks and valleys in the magnitude of their specular reflection as the thickness of the layer34increases. As the thickness of the layer approaches 0, the specular reflection approaches a peak of about 4% for each of the four wavelengths of light.

As the thickness increases to about 100 nm, all four of the wavelengths plotted inFIG. 5Aexperience a minimum in the specular reflection. As the thickness of the layer34increases toward 200 nm, all four of the wavelengths of light again approach a peak. Light will experience constructive and destructive interference at positions throughout the optical stack according to the thicknesses of the layers in the stack. Additionally, light reflected from the surface37may be 180 degrees out of phase with the light reflected from below. Thus, depending on the thicknesses and materials of the layers34,38, light reflected from below can destructively interfere with light reflected from the surface37and thereby reduce the specular reflection.

InFIG. 5B, the specular reflection is plotted for the four wavelengths of light when the layer34has an index of refraction 1.33. The peaks and valleys occur at roughly the same places as they did when the index of refraction was 1.23. However, the minimums are now higher than for n=1.23. In particular, the minimums only dropped to about 1% specular reflection, whereas for n=1.23, the minimums drop to about 0.

InFIG. 5C, the index of refraction n=1.43 for the layer34in which the nanowires32are embedded. Here the peaks remain at about 4%, as inFIGS. 5B and 5A. However, the minimums in the percent specular reflection have increased to about 2.5% as opposed to 1% for n=1.33 and 0% for n=1.23. Thus, for reducing specular reflection, it is desirable to have a lower index of refraction in some optical stacks.

FIG. 6illustrates an optical stack30according to one embodiment. The optical stack30includes nanowires32in an insulating layer34, according to one embodiment. The layer34is placed on a layer38which is a high index of refraction layer. The layer38is also optically transparent. The layer38can enhance forward scattering of diffuse light from the nanowire32. When nanowire32is placed in a layer34having a relatively low index of refraction compared to the layer38more forward scattering of diffuse light is promoted. In other words, when light is diffusely reflected from a nanowire32, more light will be scattered forward toward the layer38. Thus, less light will diffusely reflect back toward the surface37of the optical stack30. This is partially because there is an increased density of states for forward scattering relative to backward scattering when there is a higher index of refraction layer abutting the lower index refraction layer. The increased density of states promotes forward scattering as described previously.

A further advantage of having a high index of refraction layer38below the nanowire32is that total internal reflection of diffusely reflected light can occur within the high index layer38as illustrated inFIG. 7. The critical angle θcis the angle of incidence above which total internal reflection occurs. The angle of incidence is measured with respect to normal at the refractive boundary. When light is passing from a layer38of high index of refraction to a layer34of low index of refraction, the light which strikes the interface between the layers34and38is bent toward the high index of refraction layer38. When the incident angle is large enough, the transmitted angle in the low index of refraction layer34reaches 90° relative to normal. At this point, the light is no longer transmitted into the low index of refraction layer34. This interaction is governed by Snell's law which states:

By simple mathematics, the critical angle θcat which total internal reflection will occur can be calculated as follows:

Therefore, the greater the difference of the low index of fraction layer34and the high index of refraction layer38, the smaller the critical angle will be. As the critical angle becomes smaller, more light will undergo total internal reflection upon reaching the boundary of the high index of refraction layer38and the low index of refraction layer34. Therefore, selecting a layer38having a sufficiently high index of refraction can further decrease the amount of diffusely reflected light that reaches the surface37of the optical stack30. Thus, promoting total internal reflection is related to the enhanced forward scattering as described in relation toFIG. 6. In particular, the more light that is scattered forward from the nanowire32into the high index of refraction layer38, the more light that will be totally internally reflected within the high index of refraction layer38and will not reach the surface and thus result in an increase in milkiness.

In accordance with the principles discussed in relation toFIGS. 6 and 7,FIG. 8discloses an optical stack30according to one embodiment in which a high index of refraction layer38is placed below the low index of refraction layer34and above the substrate36as described previously. Having an optical stack30which includes a low index of refraction layer34, nanowires32embedded in the low index of refraction layer34and adjacent a high index of refraction layer below the low index of refraction layer provides the enhancements described in relation toFIGS. 6 and 7. Having the substrate36, which commonly will have an index of refraction that is between that of the low index of refraction layer34and the high index of refraction layer38, provides additional structural support as well as enabling attachment to a flat panel device.

In spite of the benefits which are provided by the aforedescribed embodiments of an optical stack30, optimization of the optical stack in order to minimize the diffuse reflection as well as the specular reflection, can still be very difficult. In order to provide an optical stack with minimal diffuse reflection, it is beneficial to utilize an efficient method for calculating or estimating the diffuse reflection of an optical stack30for a given configuration of layers and nanowires. The diffuse reflection for an optical stack30can be calculated by solving Maxwell's equations for the optical stack30. The differential forms of Maxwell's equations which describe properties of electric fields E and magnetic fields B are as follows:

where ρ is the charge density due to free and polarization charges, J is the current density, ∈ois the permittivity of free space, and μ0is the permeability of free space. Utilizing Maxwell's equations in a method of calculating the diffuse reflection is relatively difficult and can require large amounts of time and processing resources when diffuse reflection is to be calculated for many optical stacks30. The complexity of Maxwell's equations make it very difficult to solve and compute preferred parameters of an optical stack30.

Furthermore, each time new and different layers are added to an optical stack30, Maxwell's equations are not easily manipulated to again provide a quick optimization of an optical stack30including additional parameters. In some cases, many layers both below and above the nanowires32may be added. Some optical stacks may be subject to particular constraints. Each time the parameters or constraints of the optical stack30are changed, Maxwell's equations would be solved anew, thereby using more time and processor resources.

A less resource intensive method for calculating the specular and diffuse reflections of an optical stack will be described in relation toFIGS. 9A and 9Bvia transfer matrices. Because Maxwell's equations are second-order partial differential equations, the complete set of solutions of these equations utilizes a set of at least two linearly independent families of solutions (modes). One embodiment defines these two families as “top” and “bottom” modes. The first class of solutions, top modes, corresponds to field distributions that could be initiated by the light incident on the system from the top, as in the case of specular reflection. The second class of solutions, bottom modes, describes the field distributions across the system that could be initiated by a light source on the substrate side of the structure (below layer36inFIG. 9a). These solutions also exist in the process of diffuse reflection.

Consider now the process of specular reflection in more detail in relation to the arrows on the right side ofFIG. 9A. In this process, some amount of incident light is transmitted.FIG. 9Aillustrates an optical stack30according to one embodiment. The nanowire32is not shown inFIG. 9Abecause calculations relating to the optical stack ofFIG. 9Aare performed as though the nanowire32is not present. The position y=0 in the optical stack corresponds to the position that the nanowire32would occupy in the optical stack. The optical stack30includes a low index of refraction layer34as described previously. The low index of refraction layer34is on a high index of refraction layer38and the high index of refraction layer38is on a substrate36. A light source is irradiating the optical stack30. Light is incident on the surface37of the low index of refraction layer34.

The distribution of the EM field throughout the multilayered structure is calculated as if there were no nanowire. These calculations are performed with the transfer-matrices approach, where the field in each layer is represented as a series of plane waves moving up and down through the system (reflected/transmitted waves), and the amplitudes of these waves in the neighboring layers are related via transfer matrices.

The light from the light source is incident on the surface37of the optical stack30. 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. Some amount of the incident light from the light source above the optical stack is transmitted through the surface37into the low index of refraction layer34as indicated by the arrow passing down into the layer34on the right side of the optical stack inFIG. 9A. Some percentage of the light incident on the optical stack30is reflected from the surface37as indicated by the arrow coming off at an angle from the arrow passing down into the layer34. The angle of this arrow is not intended to be representative of the angle at which light is reflected from the surface, only to indicate that some of the light goes back up while some passes through. This is true with all of the arrows inFIG. 9A. Those arrows that seem to be at an angle are only at an angle to distinguish them from the arrows that pass through a boundary. In fact, the direction of light propagation depends on the source of illumination, and is described by the solutions of Maxwell equations.

Some of the light that passes through the boundary37from air continues in layer34until it reaches a boundary44between layers34and38. At the boundary44some of the light passes through and some is reflected as indicated on the right side by the arrows passing through the boundary44downward and the arrow coming back into the layer34that represents the reflection at the boundary44. This reflected light, in turn will come back to the boundary37, partially contributing to the initial specular reflection (upward arrow) and partially to the initial transmission (downward arrow). In the context of theFIG. 9a, these subsequent re-reflections and re-transmissions are combined together and are represented by the single combination of transmitted (downward) and reflected (upward) arrows. Such a description is consistent with transfer the matrix technique, according to one embodiment, that can be used to automatically calculate the overall reflection/transmission coefficients.

Again at the boundary42, some portion of the light that passes through layer38to the boundary42is passed through the boundary42into the layer36. Likewise, a portion of the light that is incident on the boundary42is reflected back into the layer38. Some amount of light is passed through the boundary40and into any layers that are below the layer36.

A hypothetical light source is illustrated below layer36of the optical stack30. The dashed arrows on the left side of the optical stack originate from this light source and correspond to the “bottom modes” because they carry energy from the bottom of the system upward. Some amount of the light, passing through the boundary40propagates into layer36of the optical stack toward the boundary42while some portion of this light will be reflected. At the boundary42, some amount of the light is passed through the boundary42from layer36to layer38. At the same time, some light is reflected at the boundary42back toward the boundary40. Again, at the boundary44, some light is passed through to the layer34while some is reflected at the boundary44back into the layer38.

Finally, at the surface37of the optical stack30some light passes from layer34into the air surrounding the optical stack30.

By utilizing transfer matrices, the amplitudes of the fields propagating up and down in each layer can be calculated. In particular, calculation of the amplitude of the light reflected up from the interface37for the top mode can be used to calculate the total specular reflection very accurately. Furthermore, the amplitudes of other waves composing the top modes can be used to calculate the electromagnetic field at any given vertical position within the stack30. In this manner the field at the position of the nanowire32can be calculated.

In one embodiment, the dimensions of the optical stack in the z direction, ie., the direction into the page, are assumed to be infinite. Therefore, the total field in the optical stack30can be represented as a linear combination of two fields with different polarization. The first class of fields, known as TE waves, has its electric field component along the z axis so that its magnetic field has only x and y components. Similarly, the second class of waves, TM waves, have its magnetic field aligned with the z axis, and its electric field in the xy plane.

At an the interface between two arbitrarily chosen adjacent layers (j and j+1) within the optical stack30it is assumed that the incident light is a plane wave with the wave vector having components {Kx1, kj1}. The relationship between the amplitudes of the plane waves in the neighboring layers can be determined by considering boundary conditions on electric and magnetic fields. Explicitly, for the interface between layers j and j+1 (corresponding to layers34and38for example) this relationship is given by

(?)=12(1+?1-?1-?1+?)(?),?indicates text missing or illegible when filed

where a−and a+are the amplitudes of the waves propagating in the negative and positive y direction respectively, the polarization-dependent constant Kjis given by

???indicates text missing or illegible when filed

for TE-polarized waves and by

???indicates text missing or illegible when filed

for TM-polarized ones. The matrix connecting the amplitudes of the fields in the neighboring layers to each other is called a transfer matrix. Such a transfer matrix is only one type of transfer matrix which can be used in calculating specular reflection, diffuse reflection, or the amplitude of light waves for an optical stack30. Many other types of transfer matrices may be used. Additionally, other methods which do not use transfer matrices can be used in calculating diffuse reflection according to principles of the present disclosure.

FIG. 9Billustrates the optical stack30ofFIG. 9Ain which the nanowire32has scattered light that was incident on the optical stack30onFIG. 9A. The light source that was present inFIG. 9Ais not present inFIG. 9Bto emphasize the focus now on diffuse reflection as opposed to specular reflection. Thus, the nanowire32has light scattering off it in a plurality of directions.

The diffuse reflection corresponds to the amount of light scattered from the nanowire32that exits the optical stack30through the surface37. A method for calculating the diffuse reflection according to one embodiment therefore includes calculating the amount of light scattered from the nanowire32in all directions. As described previously, when calculating the transfer matrices to determine the specular reflection, the field at any position in the optical stack can also be calculated. One step in calculating the light scattered by the nanowire32is to calculate the field at the position of the nanowire32.

Once the field at the position of the nanowire 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 nanowire32. The scattering cross-section of the nanowire can be obtained by solving Maxwell's equations for a long cylindrical wire of the given shape. For a wire 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 can also be calculated for other shapes of wires such as wires with polygonal or other cross-sections. In one example of such calculations, the solutions of Maxwell equations is represented as a set of cylindrical waves, and the boundary conditions along the wire circumference are used to relate the amplitude of these waves. One realization of such a formalism is described, using an example of light emission from dielectric resonators, in the article (Viktor A. Podolskiy, Evgenii Narimanov, Wei Fang, and Hui Cao, Chaotic microlasers based on dynamical localization, Proc. Nat. Acad. Sci. v. 101 (29) pp. 10498-10500 (2004) and in references therein). This article is incorporated by reference herein in its entirety. Once such a relation is found, it is straightforward to relate the energy flux scattered by the wire to the energy flux incident on the wire, and use this relationship to calculate the scattering cross-section of the wire. The scattering cross-section describes what proportion of the light that is incident on the nanowire32will be scattered by the nanowire32.

By multiplying the field from the light incident on the nanowire32by the scattering cross-section of the nanowire, the amount of light scattered by the nanowire32can be calculated. The total diffuse reflection from the optical stack30can be calculated or estimated by again calculating transfer matrices for the light scattered by the nanowire32within the optical stack30. The diffuse reflection is the amount of light scattered by the nanowire32that exits the optical stack30from the surface37. In one embodiment the nanowire is treated as though it scatters light in all directions equally. Mathematically, the spectrum of the diffusely scattered light A(kx) does not depend on the x component of the wavevector kx.

Similar to the specularly reflected light inFIG. 9A, the diffusely reflected light from the nanowire inFIG. 9Balso both transmits through and reflects at each boundary within the optical stack. The transfer matrices for calculating the diffuse reflection are computed for top and bottom modes as described previously.

Light that is forward scattered from the nanowire32as described previously will be incident on the boundary44between the layers34and38. A portion of this light will be reflected back toward the surface37. A portion of the forward scattered light from the nanowire32will transmit through the boundary44into the layer38. Light will again propagate to the boundary44between the layers38and36where some of it will be transmitted and some will be reflected back up toward the boundary44. Some of the light that is transmitted through the boundary36will reflect at the boundary40and some will pass through the boundary40. The total light that passes through the boundary40will represent diffusely transmitted light. The light reflected by each of the interfaces44,42,40, will contribute to the diffuse reflection. However, the main contribution to diffuse reflection comes from the light emitted into the bottom modes of the system (shown above the nanowire inFIG. 9B). The amount of light transmitted through the interface37(that will represent the total of the light scattered by the wire into the bottom modes and the portion of the light originally emitted into the top modes that is subsequently reflected by the interfaces44,42,40) represents the total diffuse reflection in the system.

The diffuse reflection can be calculated in a manner similar to the specular reflection as described in relation toFIG. 9A. Namely, the scattering cross section is acquired and transfer matrix calculations are performed for the diffusely scattered light for transmission and reflection at all boundaries within the stack30. In this manner, the total diffuse reflection can be closely approximated while using relatively little processing resources.

In one embodiment, the field at the position of the nanowire can include both the field from the incident light and the field from previously scattered light. In other words, some of the light scattered by the nanowire32will reflect within the optical stack30and again be scattered by the nanowire32. The accuracy of the calculation of diffuse reflection can be improved by taking into account the field from diffusely scattered light at the position of the nanowire.

Calculation of light scattering is generalized to take into account the phase of the scattered light. To achieve this, the radius of the nanowire is assumed to be extremely small, so that its scattering is dominated by the lowest-possible cylindrical harmonics (empirical calculations indicate that TE scattering is dominated by m=Q [polar-angle-independent] cylindrical mode, while TM scattering is dominated by m=1 [dipole-like] cylindrical mode. As such, the spectrum of the scattered waves is proportional to:

ATE(?)∝1?ATM(?)∝??,?indicates text missing or illegible when filed

where n is refractive index of the material surrounding the wire, k is the wave vector and w is the angular frequency. Note that when the radius of the nanowire is sufficiently small, both cases reduce to the previously described kxindependent spectrum.

The scattered light is represented as a sum of the “emitted” waves (bottom modes for y>0, top modes for y<0) plus the sum of the reflected components of the top and bottom modes respectively. The amplitude of the top and bottom modes emitted by the source is the same for the TE polarization and is opposite for the “dipole” TM polarization. When the interference of the top and bottom modes is taken into account, the effective amplitude of the emitted light becomes:

ab+=a(?)??;?=a(?)???indicates text missing or illegible when filed

for the TE waves and

ab+=a(?)??;?=-a(?)???indicates text missing or illegible when filed

for TM waves, with a(kx) being the amplitude of the emitted light and ra, rbbeing the reflection coefficients of the components of top and bottom modes.

To calculate the feedback field, ie the diffusely scattered light again incident on the nanowire32, we multiply the emitted fields by their respective reflection coefficients and add the two together. Therefore, the total amplitude of the field at the location of the wire becomes:

atot=a0+?[???+???]a(?)?,?indicates text missing or illegible when filed

for TE waves, and

atot=a0+?[???-???]a(?)?,?indicates text missing or illegible when filed

for TM waves.

The factor dkxrepresents the step in wavevector spectrum utilized in numerical calculations. A self consistent calculation of the field at a position of the nanowire can include incident light from the external light source and diffusely reflected light. In a self consistent solution, the field can be described as

leading to the matrix relation describing the emission spectrum:

Where A(kx,k′x) describes the scattering from the plane wave with wavevector k′xinto the plane wave with the wavevector kx, and the (diagonal) matrix {tilde over (R)} has components corresponding to the coefficients for atotdescribed previously.

As mentioned above, these calculations can be simplified in the case when the percentage of the diffusely reflected light coming back to the wire position of the nanowire is small. In this case, the energy flux of the spectral component of the diffusely reflected light is enhanced by

In some applications, it may be advantageous to calculate or estimate a portion of diffusely scattered light rather than the total diffuse reflection. It may, for example, be important to estimate the amount of light diffusely scattered towards the observer or only the amount of light scattered away from observer rather than the total amount of light diffusely scattered in all directions. In these situations, the formalism developed above can be used to calculate the energy flux due to diffuse reflection as a function of the angle of incidence φiand the angle of reflection θr. The transfer-matrix formalism, described above for calculating/estimating the total diffuse reflection, can be used to calculate or estimate the angular distribution of such diffuse reflection. In these instances, the angles of incidence and reflection may be both parameterized by the longitudinal component of the wavevector kxand then the angular distribution of the energy flux representing the diffuse reflection is calculated based on the angular spectrum of amplitudes a(kx).

To improve the accuracy of estimations of angle-dependent diffuse reflection, different models for scattering probability A(kx, k′x) may be incorporated into the calculations. In particular, one may use kx-independent spectra, dipole-type directionality spectra, a combination of thereof, or other directionality spectra. In one embodiment, scattering probability A depends on the polarization, producing direction-independent energy flux for TE-polarized waves, and dipole-type radiation pattern [with energy flux∝cos2(φi+θr)] for TM-polarized waves. In other embodiment, the energy flux of TE-polarized waves is proportional to 1/cos2(θr), while that of TM-polarized waves is∝cos2(φi+θr)/cos2θr.

There may exist many different models of scattering probability A, which fall within the scope of this disclosure. When developing these models, it is helpful to keep in mind that the transfer-matrix model represents an estimate of the diffuse scattering process. Thus, the coefficients can be fine-tuned by comparing the predictions of transfer-matrix codes to rigorous (but more time-consuming) solutions of Maxwell's equations with a Finite-elements method, finite-difference time-domain method, rigorous coupled-wave approximation method, or other methods.

Using the convenient methods for calculating or estimating the diffuse reflection of an optical stack as described above, an optimization program can be utilized to calculate the diffuse reflection of many optical stacks30having different parameters in order to find an optical stack30which gives the lowest diffuse reflection. Commercially available optimization programs, such as those available in Matlab, can be used to optimize diffuse reflection, according to principles of the present disclosure, for many optical stack configurations. Such optimization programs can assist in finding an optical stack having a relatively low diffuse reflection in conjunction with the methods for calculating diffuse reflections described above.

The particular optimization goals of such a process depend on the final application. For example, one may optimize total diffuse reflection of the system for a given wavelength. One may also a weighted average corresponding to diffuse reflection in a particular direction or directions, or a combination of weighed total diffuse reflection with constraints that the diffuse reflection in particular direction remains below a certain value. One may also estimate diffuse reflection for different wavelength of light, and aggregate these estimations in some manner (averaging, weighted averaging, etc.) to arrive at the final goal figure of merit that will be optimized. All such combinations can be implemented by those of skill in the art in light of this disclosure.

While the calculations of diffuse and specular reflection have been described above in terms of transfer matrices, other methods besides transfer matrices can be used to obtain a value of diffuse reflection according to principles of the present disclosure. Such other methods also fall within the scope of the present disclosure.

One example of such methods include the extension of the presented approach to optimize specular and diffuse reflection from the optical stack that is incorporated inside a fixed set of thick layers, which may include thick underlayer (for example, optical adhesive) or thick overlayer (for example, protective glass layer). Here “optically thick” means that the thickness of the layer is greater or comparable to the coherence length of the radiation present it the stack.

Propagation of light through optically-thick layers is somewhat similar to the process that yields the formation of top and bottom modes described above. Consider, for example, specular reflection of the top mode shown inFIG. 9A. As described above, light that enters the optical stack through the interface37will partially reflect and partially transmit through this interface. The transmitted portion will enter the layer34and reach the interface44, where part of the light will be transmitted into the layer38, and part of the light will be reflected back into the layer34. This reflected light will reach the interface37, where it will be partially transmitted outside the optical stack (contributing to specular reflection), and partially reflected back into optical stack. When the layer34is optically thick, the second (and subsequent) contributions to specular reflection will not interfere with the light originally reflected by the interface37. Rather, the corresponding energy fluxes will be added together. It is straightforward to calculate specular reflection of the stack incorporating several optically-thick layers based on (energy-flux-based) reflectivities (R), and transmittivities (T) of the inter-layer interfaces.

For example, the following recursive technique can be used. Assume that layers inFIG. 9Aare optically-thick. Then reflectivity of the interface40can be calculated using absolute value square of the corresponding Fresnel coefficient. Then the reflectivity of light entering interface42can be calculated as

Here {tilde over (R)}iis the (overall) reflectivity of light entering the system from42from layer38, Ri+is the single-interface reflectivity of the interface42when light travels from the layer38, Ri−is the reflectivity of the same interface for the light travelling into layer38(often Ri−=Ri+), and {tilde over (R)}i−1is the overall reflectivity of light entering on the interface40. The same equation can be then used to calculate overall reflectivity of light entering interface44, and finally, interface37.

If the system contains a mix of optically-thick and optically-thin layers, the transfer matrix formalism can be used to calculate the optical properties (reflectivity and transmittivity) of optically-thin layers, which can be then be approximated as single interfaces (with known reflectance/transmittance) in optically-thick stacks.

Similar techniques can be utilized to calculate diffuse reflection in the presence of optically-thick layers.

FIG. 10Arepresents a graphical user interface (GUI)48of an optical stack optimization software program stored in a computer readable medium. The GUI can be displayed on a display coupled to a processor to allow a technician to implement the optimization program for finding an optical stack30having parameters that will yield a preferred value of diffuse reflection. The processor reads software instructions from a memory circuit coupled to the processor. The software instructions for running an optimization program to find preferred parameters of the optical stack30are therefore stored in the memory coupled to the processor. The processor therefore causes the display to display the GUI and a technician can input via a mouse and keyboard or any other suitable input device the ranges of parameters for an optical stack. The parameters can include the number of layers in the stack, the indices of refraction of the substrate36, as well as the environment in which the optical stack30will be placed.

Therefore, in the exemplary GUI48inFIG. 10Athe superstrate index of refraction is 1 because it is air. The substrate index of refraction is listed as 1.5 and corresponds to the substrate36of the optical stack30. The user can input any substrate or superstrate index as desired. The radius of the nanowire32is also entered in order to calculate the scattering cross section. The wire radius entered in the GUI48according to one embodiment is 50 nm. However, the wire radius can be any other suitable radius according to the particular nanowires32or other nanostructure that are used in the optical stack30. The user can likewise select which layer of the optical stack30the nanowires32are located in. In the exemplary GUI48ofFIG. 10A, the wire layer has been selected as layer two which corresponds to layer34of the optical stack30. In the field labeled active layer parameters, the user can enter minimum and maximum thicknesses for the layers34and36corresponding to layers one and two of the GUI48. In the example ofFIG. 10A, both the layer34and the layer36have ranges from 50 nm to 200 nm in thickness. The index of refraction for each of the layers34and36ranges from 1.2 to 2.2. These ranges are limits within which the optimization program will select parameters for optical stacks30in order to calculate which parameters yield the best diffuse reflection. When the program is executed, the diffuse and specular reflections are calculated for several optical stacks having parameters within the input ranges of layer thicknesses, indices of refraction, and wavelengths of light. The optimization program calculates the diffuse and specular reflection according to the methods previously described or using other suitable methods according to principles of the present disclosure.

In one embodiment, rather than calculating the diffuse reflection for every possible iteration within the input ranges, the optimization program calculates diffuse reflection for a first group of optical stacks having a variety of parameters within the given ranges. The optimization program then selects a second set of optical stacks having parameters varying somewhat from those which yielded the lowest diffuse reflections in the first group. The optimization program continues to calculate the diffuse reflection of optical stacks in this manner until a preferred diffuse reflection has been found. The optimization program can efficiently find the parameters which yield a preferred diffuse reflection without computing every possible iteration. In this way, the particular configuration of the optical stack30which yields a relatively low diffuse reflection can be selected. This is possible because of the aforedescribed simpler method for calculating or estimating the diffuse reflection of an optical stack30.

It is possible to have a low diffuse reflection while having an unacceptably high specular reflection. For this reason, below the active layers parameters field is a field labeled as max reflection. In this field a technician can specify the maximum tolerable specular reflection. In this case the maximum specular reflection has been selected as 1.5%. This means that when the transfer matrices are run for both the specular reflection and the diffuse reflection, the preferred stack configuration will be chosen for the lowest diffuse reflection yield in which the specular reflection not greater than 1.5%.

In the field to the right, is illustrated an optical stack. The optical stack30includes the layer34of a lower index of refraction, including the nanowire32, on top of layer38of higher index of refraction. Layer38is on the substrate36which has an index of refraction of 1.5. The index of refraction for the air above the optical stack is 1. In the layers34and38on the left side of each layer, the ranges of thickness and the ranges of the indices of refraction are given. This is noted by w2=50 nm to 200 nm on the left side of layer34and n2=1.2 to 2.2. These are the ranges for the thickness and the index of refraction of layer34for which the iterations will be performed in calculating the transfer matrices to find the specular and diffuse reflection. The layer38on the left side likewise specifies the range w1=50 nm to 200 nm and n1=1.2 to 2.2. On the right side of the layer34, the preferred thickness and the preferred index of refraction are listed. In particular, the preferred thickness of the layer34is given as 118.2 nm. The preferred index of refraction of the layer34is 1.2. The preferred thickness of the high index of refraction layer38is 50 nm and the preferred index of refraction is 1.7779. Below the optical stack, the specular reflection is listed as R0=0.0144 or about 1.4%. The diffuse reflection Rdiffuseis listed as 5.469×10−5.

Thus, the GUI48which enables operation of the method for optimizing an optical stack30allows a user to input first parameters for the optical stack or input parameters and the program is run, the calculations are made, and the preferred specular and diffuse reflections are listed as well as the layer thicknesses and indices of refraction which yield those preferred results. It will be understood by those of skill in the art in light of the present disclosure that many modifications can be made to the method which has been described as well as the particular GUI and the inputs and outputs provided thereby.

FIG. 10Billustrates a GUI50according to one embodiment. The GUI50relates to a method by which a detailed plot of the specular and diffuse reflections for a variety of wavelengths can be computed based on the preferred parameters output in the GUI48fromFIG. 10A. In particular, a user can enter the number of layers from the preferred output which is two in this case, the substrate index for the substrate layer36which is 1.5, and the superstrate index of refraction which is 1. Thereafter, an active layer can be selected, in this case the active layer two is highlighted which means that the parameters of the layer34can be entered in the fixed parameters field. The preferred characteristics as determined by the GUI48fromFIG. 10Awere 118.2 nm for the thickness of the layer34with an index of refraction of 1.2. The active layer can then be highlighted and the preferred characteristics calculated in relation to GUI48from FIG.10A for the layer38can be entered. In this case, the preferred characteristics are 50 nm in thickness and an index of refraction of 1.7779. The range of the wavelength of light for which the plot will be generated can be entered in the field below to fix layer parameters labeled as wavelength in nm. In this case, the minimum wavelength is 300 nm and the maximum wavelength is 800 nm to be iterated in steps of 10 nm.

FIG. 10Cillustrates the plot generated by the GUI50fromFIG. 10B. In particular,FIG. 10Cis a plot of the specular and the diffuse reflection for a range of wavelengths as specified inFIG. 10B. Both the specular and the diffuse reflection experience a peak just short of 400 nm in the ultraviolet range. The specular reflection dips down and hits a minimum of about 1% at a wavelength of 500 nm and then increases gradually to about 2.5% at800nm. The diffuse reflection dips down and hits a low around 500 nm as well, but stays relatively flat all the way through 800 nm, just sloping up very gradually. The diffuse reflection, in this case, has been kept to about 5×10−5through the majority of the visible spectrum. The specular reflection has been kept between 1% and 2% for the majority of the visible spectrum.

Within the software instructions stored in memory, certain wavelengths of light can be weighted more heavily than other wavelengths of light. When the transfer matrices are calculated, each transfer matrix is performed for a range of wave lengths in addition to the range of thicknesses of the layers and indices of refraction of the layers. When calculating the preferred diffuse reflection, the reflection at some wavelengths can be weighted more heavily than others in one embodiment. The human eye is more sensitive to certain wavelengths than to others. Thus for some optical stacks, the diffuse reflection may be somewhat higher at less prominent wavelengths, while more prominent wavelengths are near a minimum. In such a case the diffuse reflection may be a preferred diffuse reflection despite some wavelengths not being near a minimum of diffuse reflection. For this reason it may be desirable to give a stronger weight to the diffuse reflection of some wavelengths. In one example the visible spectrum is divided in increments of 50 nm between 400 and 700 nm. The software which stores the program for calculating diffuse reflection can be modified to give higher or lower relative weights to the various wavelengths. For example, in one embodiment wavelengths between 450 nm and 600 nm are weighted more heavily than other wavelengths. The weighting, of course, can be selected by a technician who alters the code stored in the memory. The weighting can also be implemented for calculations of the specular reflection.

FIG. 10Dillustrates a GUI for software program configured to find an optical stack having a relatively low diffuse reflection according to one embodiment. The GUI ofFIG. 10Dallows a user to select a number of layers of the optical stack30and in which layer the nanowire32will be located. The number of layers in the example ofFIG. 10Dis three and the nanowire layer is layer2. After the nanowire layer has been selected, the user can enter the ranges of thickness and index of refraction of the other layers in the optical stack30. However, in the embodiment ofFIG. 10Dthe thickness and index of refraction of the nanowire layer cannot be altered through use of the GUI; these parameters are fixed in the embodiment ofFIG. 10D. The parameters of layer1can be entered by selecting layer1as the active layer, then entering the ranges of thickness and index of refraction in the labeled fields. The parameters of layer3can be entered in the same way. InFIG. 10Dthe user has selected a range of 30-300 nm for thicknesses of both layer1and layer3. A range of 1.2-2.2 for indices of refraction for layers1and3has been selected. These ranges are the ranges from which the optimization program will select values of thickness and index of refraction for the layers during the optimization routine.

The index of refraction of the superstrate and substrate can also be selected by entering values in the labeled fields. These have been selected as 1 and 1.5 respectively in the example ofFIG. 10D. Once these parameters have been selected a basic diagram of the layers of the optical stack30is displayed on the right side of the GUI indicating the position of the layers, the position of the nanowire layer, the ranges of index of refraction and thickness, and the indices of refraction for the superstrate and the substrate.

The user can also select whether the optimization routine will optimize diffuse or specular reflection by checking the appropriate selection in the optimize field. If diffuse reflection is selected for optimization, then a maximum specular reflection can also be selected by entering a value in the max specular reflection field. The program will select an optical stack having a low diffuse reflection and a specular reflection equal to or less than the selected maximum value. Alternatively, if the user selects the specular reflection for optimization, then the user can enter a maximum diffuse reflection value for the optical stack.

Finally the user can click on the start button to run the optimization program. The optimization program will then calculate the diffuse reflection and specular reflection for a number of possible optical stacks and select an optical stack having a relatively low diffuse reflection and a specular reflection less than the selected maximum value. The parameters of the selected optical stack will then be output. The user can also save the optimum optical stack parameters or load a previously saved optical stack by clicking on the appropriate buttons.

FIG. 10Eillustrates a GUI for calculating and plotting both the diffuse and specular reflection from a according to an alternative embodiment. The GUI ofFIG. 10Eallows a user to select a number of layers of the optical stack30and in which layer the nanowire32will be located. The number of layers in the example ofFIG. 10Eis three and the nanowire layer is layer2. The thickness and index of refraction cannot be altered through use of the GUI; these parameters are fixed in the embodiment ofFIG. 10E. After the nanowire layer has been selected, the user can enter the thickness and index of refraction of the other layers in the optical stack30. The parameters of layer3are entered by selecting layer3as the fixed layer, then entering the thickness and index of refraction in the labeled fields below. The parameters of layer1can be entered in the same way. The index of refraction of the superstrate and substrate can also be selected; these have been selected as 1 and 1.5 respectively. Once these parameters have been selected a basic diagram of the layers of the optical stack30is shown on the right side of the GUI. The nanowire layer is layer2in the example ofFIG. 10E.

The range of wavelengths and the step size for the calculations and plots can also be entered. In the example ofFIG. 10E, the range of wavelengths selected is between 300 and 800 nm in stems of 10 nm. After all fields have been populated the user can click the start button to begin the calculation routine. The specular and diffuse reflections are calculated for all wavelengths. A graph can be output showing the specular and diffuse reflection for each wavelength. A table can also be output showing the numerical values of the diffuse and specular reflection for each wavelength step in the range of wavelengths. Many other configurations of a GUI are possible as will be apparent in light of the present disclosure. All such other configurations fall within the scope of the present disclosure.

As described previously, the diffuse reflection can be calculated for selected angles of diffuse reflection or ranges of angles of diffuse reflection with respect to the surface of the optical stack30. In some applications it is useful to know how much light is diffusely reflected at a particular angle or angles with respect to the surface of the optical stack30. Accordingly, in one embodiment, a user of the optimization software can select a plurality of angles for which diffuse reflection will be estimated for each optical stack configuration.

In one embodiment, for each iteration of optical stack parameters, a set of values of diffuse reflection is calculated or estimated. Each set of values of diffuse reflection includes a plurality of values of diffuse reflection for the selected angles with respect to the optical stack30. The optimization routine can be configured to select an optical stack configuration based on the sets of values of diffuse reflection. In particular, the sets of values of diffuse reflection can be compared to each other and the optimization routine can select an optical stack configuration based on the comparison. The optimization routine can also be configured to compare the values of diffuse reflection for the respective angles with threshold values. The optimization routine can then select one of the sets of diffuse reflection values based in part on the comparison with the threshold values.

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

In one example large angles of reflection with respect to normal have higher threshold values than angles closer to normal. In other words, a higher diffuse reflection may be tolerated at large angles with respect to normal. This is because in some embodiments the optical stack30may be included in a display screen in which it is more important that the display quality be high at angles very close to normal. Angles which are large with respect to normal correspond to peripheral viewing angles of a display screen including the optical stack30and maintaining high optical quality may be less important at these angles. Thus, the threshold values of diffuse reflection at angles close to normal may be much smaller than the threshold values for angles further from normal. This is because displays are more commonly viewed from angles close to normal with respect to the display screen.

In one embodiment, if any of the values of diffuse reflection in a particular set exceeds the respective threshold value of diffuse reflection, the optical stack configuration associated with that particular set is not selected.

Alternatively, the values of diffuse reflection of each set can be compared to a single threshold value of diffuse reflection. If any of the values of diffuse reflection in a particular set exceeds the threshold value of diffuse reflection, the optical stack configuration associated with that particular set is not selected.

In one embodiment, an aggregate diffuse reflection value can be calculated for each set of diffuse reflection values. The optimization routine can select the optical stack configuration corresponding to the lowest aggregate diffuse reflection value. Calculating the aggregate diffuse reflection value can include summing the diffuse reflection values. Alternatively, calculating the aggregate diffuse reflection value can include assigning relative weight factors to each angle of reflection.

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

The optimization routine can be configured to give greater weight to the diffuse reflection values at some angles while giving lower weight to the diffuse reflection values at other angles. The optimization routine can also give greater weight to certain wavelengths of light at the various angles. The optimization routine can select an optical stack configuration by taking into account the diffuse reflection for a large number of wavelengths at each of the angles of reflection.

Many other optimization routines, software programs, and methods of calculating or estimating the diffuse reflection at various angles can be implemented. All such other routines programs and methods involved in the scope of the present disclosure.

FIG. 11illustrates a system60according to one embodiment. The system60includes a processor62configured to execute software instructions stored in the memory circuit64. The memory circuit64stores data which is read by the processor to execute the optimization methods described previously. An input module66is also coupled to the processor62. A technician operating the system60can input, at the input module66, the input parameters of the optical stack30, which parameters the processor62will then optimize and output parameters reflecting the optimization. A display68is coupled to the processor62. The processor62can cause the GUI48or50to be displayed on the display60. The technician then operating the input module66can input the appropriate field by visually viewing the GUI48or50on the display68. The optimization parameters can be displayed on the display68as well.

In one embodiment, the system60includes manufacturing equipment70coupled to the processor62. In such an embodiment, the processor62outputs the output parameters directly to the manufacturing equipment which then deposits the appropriate layers and thicknesses as described in the optimization output. For example, for an optical stack30including a low index of refraction layer34in which is embedded nanowires32and a high index of refraction layer38below the low index of refraction layer34, as well as a substrate36below the high index of refraction layer38, the optimization outputs can be given to the manufacturing equipment70which can then deposit the layer38on the substrate36and the layer34on the layer38. The foregoing system60is given by way of example. Many other components and software instructions can be included but which have not been described herein. When a user operates the input module66to input the input parameters, the input parameters are stored in the memory64coupled to the processor62.

In one embodiment, the memory64can include an EEPROM, ROM, SRAM, DRAM, or any other suitable memory. The software instructions for performing the optimization process can be stored in the memory64. The input instructions can be temporarily stored in the memory64or in a separate cache memory coupled to the processor. Any suitable components for storing the input parameters and the software instructions such that they can be read by the processor62can be used. Alternatively, the output from a process for selecting parameters for an optical can be used to manufacture the optical stack without manufacturing equipment physically coupled to circuitry used in selecting the optical stack parameters.

FIG. 12is a flow diagram illustrating a method for optimizing parameters of an optical stack30. At80, the technician inputs layer parameters to a processor. The input parameters are then stored in a memory coupled to the processor. The input parameters can include numbers of layers in an optical stack30, ranges of thicknesses of the layers in the optical stack30, ranges of indices of refraction of the layers in the optical stack30, a range of wavelengths for which diffuse and specular reflection are to be calculated, and relative weight values to be given to the various wavelengths in the range of wavelengths. At82, the processor calculates the field at the position of the nanowire32in the optical stack30. The calculation of field at the position of the nanowire can be performed by using transfer matrices or any other suitable calculation that can provide the field at the position of the nanowire32. In an alternative embodiment, as described previously, the calculation of field at the position of the nanowire can include the field from previously scattered light.

At84, the scattering cross section of the nanowire32is calculated. The scattering cross section of the nanowire32gives an indication of the directions and magnitudes of scattering of diffusely reflected light from the nanowire32. The nanowire32can diffusely reflect the light in any direction. At86, the processor calculates the diffuse reflection based on the calculated field at the nanowire position and the scattering cross-section. In one embodiment, the diffuse reflection is estimated by calculating transfer matrices for the transmission and reflections of diffusely reflected light at each of the layer boundaries and through each of the layers in the optical stack30.

At88, the calculations of field at the position of the nanowire32, the scattering cross section of the nanowire32and the diffusely reflected light that reaches the surface are repeatedly performed for a large number of optical stacks30across the range of input parameters. In one embodiment the diffuse reflection calculations are performed for a first group of optical stacks. The optical stacks of the first group can have values for layer thicknesses, indices of refraction of the layers, etc. selected to give a broad first sampling of optical stacks across the possible input ranges. For example, the first group of optical stacks can include optical stacks whose first layers respectively have a minimal thickness, a maximal thickness, and a few thicknesses spread out between. The diffuse reflections are calculated for the first group and compared to each other.

Diffuse reflection is then calculated for a second group of optical stacks. In one embodiment the parameters of the optical stacks of the second group are chosen, based in part on the diffuse reflections of the first group. For example, the second group of optical stacks includes optical stacks having one or more parameters close to one or more of the parameters of the optical stacks of the first group which yielded the lowest values of diffuse reflection. This allows the processor to find an preferred value of diffuse reflection without computing every possible optical stack within the ranges. But rather, the processor can analyze optical stacks whose parameters are most likely to have a low diffuse reflection. This process can be continued as long as desired to obtain as thorough an optimization process as time and computing power allow. In the end the processor can select the optical stack whose parameters yielded the best value of diffuse reflection. At92, the optical stack30is formed by depositing layers having the characteristics corresponding to the optimum output parameters.

Materials that may be used in the layers of an optical stack fabricated in accordance with the present invention are known in the art. Examples of such materials include, for example, TiO2(RD=1.8), polyimides (RD=1.7), as well as clear polymers embedded with high refractive index particles such as ZnO, ZrO2, and TiO2.

Table 1 shows a number of relatively low refractive index optical materials that may be used in the layers of an optical stack fabricated in accordance with the present invention.

Table 2 shows a number of relatively high refractive index optical materials that may be used in the layers of an optical stack fabricated in accordance with the present invention.

Methods of depositing optical layers having desired thicknesses using coating, printing, sputtering or other techniques are understood in the art. Regarding coating techniques in particular, Edward Cohen and Edgar Gutoff, “Modern coating and Drying Technology” (John Wiley & Sons, 1992, see pp. 11 and 25-28), which is incorporated herein by reference, discusses coating layers having desired wet film thicknesses. 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 of ordinary skill. Methods of coating and printing nanowire conductive layers are disclosed, for example, in U.S. Pat. No. 8,094,247 and U.S. patent application Ser. Nos. 12/380,293 and 12/380,294, each of which is incorporated by reference herein.

FIG. 13illustrates a method for optimization of the parameters of an optical stack30according to one embodiment. At94, optical stack input parameters are input to a processor which stores the input parameters in a memory circuit. The processor executes software instructions stored in the memory to begin a method for optimizing the parameters of the optical stack. At96, the processor calculates transfer matrices for light incident on an optical stack having values in the range of parameters that were input to the processor at step94. By calculating the transfer matrices, the specular reflection from the surface37of an optical stack30can be obtained. Also calculating the transfer matrices, the field at the position of the nanowire32within the stack30can be calculated at98.

At99, the scattering cross section of the nanowire32is calculated. The scattering cross section of the nanowire32is an indication of the magnitude of diffusely reflected light scattered in each direction within the optical stack30. At100, transfer matrices are calculated for diffusely reflected light scattered in all directions from the nanowire32within the optical stack30. The transfer matrices give the portion of the diffusely reflected light that reaches the surface37of the optical stack30.

At102, the processor checks to see if more iterations of the input parameters are needed. In one embodiment the processor will perform the diffuse reflection calculations for a first group of optical stacks. For example, if the range of possible thicknesses of the first layer is between 50 nm and 200 nm, the processor can compute values of diffuse reflection for the minimum and maximum thicknesses, as well as for a few thicknesses in between while holding other parameters constant. The values of diffuse reflection are compared and the processor selects the next iteration of values based on the comparisons of the diffuse reflections of the first group of optical stacks. The processor chooses the parameters for the next iteration at104and the processor performs the calculations for specular reflection, field at the nanowire position, and diffuse reflection for the new set of parameters. At106, the processor selects a preferred diffuse reflection from the set of diffuse reflections that have been calculated for the range of input parameters and outputs the particular preferred parameters of the optical stack30that produce the preferred diffuse reflection.

FIG. 14illustrates a flat panel device120including in a touch screen display an optical stack30according to one embodiment. The optical stack30has been produced having layer parameters obtained from the optimization process as described previously. The display of the flat panel device120does not suffer from the problems of milkiness or haziness as described previously.

While particular layers, thicknesses, and properties of an optical stack30have been described herein, many other suitable configurations of optical stacks are possible, including more or fewer layers, multiple layers of nanostructures, or any other suitable characteristics. All such stacks fall within the scope of the present disclosure.

Likewise, while the present disclosure has disclosed particular methods for optimizing optical characteristics of an optical stack30, many other suitable variations in the method are possible. For example, the field, specular reflection, and diffuse reflection can be approximated in other ways while still falling within the scope of the present disclosure. More, fewer, or different parameters can be input to a processor to optimize the stack. Likewise, optimization can be performed in regards to other parameters aside from the specular and diffuse reflection. The word optimum should not be understood to mean the best possible configuration, but rather one value or configuration preferred above other values or configurations. Likewise, an optimum reflectance does not necessarily mean the lowest reflectance, but rather a desired reflectance among the possible reflectances.

The various embodiments described above can be combined to provide further embodiments. All of the U.S. patents, U.S. patent application publications, U.S. patent application, foreign patents, foreign patent application and non-patent publications 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, application and publications to provide yet further embodiments.