[Radiance-general] BRTDfunc for dichroic film revisited

[John An]

Hi all,

I was wondering if I could enlist the assistance of some of the experts 
here on the list.  I have been trying to figure out how do define 
dichroic films for use in Radiance, but really have not made much 
headway.

I have found a paper (copied below) which gives a description of the 
behavior of dichroics.  I can make a pdf with all the figures and 
images available.

I realize that this may be an extremely open-ended question, but I 
would really appreciate any guidance and/or assistance.

Thanks.


John An


Title: GIANT BIREFRINGENT OPTICS IN MULTILAYER POLYMER MIRRORS ,  By: 
Weber, Michael F., Stover, Carl A., Gilbert, Larry R., Nevitt, Timothy 
J., Ouderkirk, Andrew J., Science, 0036-8075, March 31, 2000, Vol. 287, 
Issue 5462
Database: Academic Search Premier


Section: RESEARCH ARTICLE

GIANT BIREFRINGENT OPTICS IN MULTILAYER POLYMER MIRRORS





  Multilayer mirrors that maintain or increase their reflectivity with 
increasing incidence angle can be constructed using polymers that 
exhibit large birefringence in their indices of refraction. The most 
important feature of these multilayer interference stacks is the index 
difference in the thickness direction (z axis) relative to the in-plane 
directions of the film. This z-axis refractive index difference 
provides a variable that determines the existence and value of the 
Brewster's angle at layer interfaces, and it controls both the 
interfacial Fresnel reflection coefficient and the phase relations that 
determine the optics of multilayer stacks. These films can yield 
optical results that are difficult or impossible to achieve with 
conventional multilayer optical designs. The materials and processes 
necessary to fabricate such films are amenable to large-scale 
manufacturing.

  There are two conventional ways to create a mirror: using the surface 
of a layer of metal, or using a tuned interference stack composed of 
multiple layers of transparent dielectric materials. Metal mirrors are 
inexpensive and perform robustly across a broad range of angles, 
wavelengths, and polarizations, but they exhibit limited reflectivity. 
Multilayer interference mirrors are routinely used for optical 
applications requiring high reflectivity and wavelength selectivity. 
Although they can be designed to achieve a wide range of optical 
characteristics, each design typically performs across a limited range 
of incidence angles, wavelengths, and polarizations. A key limitation 
of multilayer mirrors stems from Brewster's law, a nearly 200-year-old 
maxim of optics, which predicts the decrease of reflection for 
p-polarized light at material interfaces with increasing incidence 
angle. Specifically, Brewster's law states that there is an angle of 
incidence (Brewster's angle) for which the reflectivity for p-polarized 
light vanishes at a material interface. As a result, a multilayer 
interference mirror that is designed to have a 1% loss for reflection 
of p-polarized light (99% reflectivity) at normal incidence can have 
many times that loss at high incidence angles.

  Using highly birefringent polymers, we have found that multilayer 
mirrors can be constructed that maintain or increase their reflectivity 
with increasing incidence angle. The reflective characteristics of 
these mirrors require a generalization of Brewster's law. This 
generalization has enabled the development of a new class of multilayer 
interference optics with design freedoms that can result in 
unprecedented means for transporting, filtering, and reflecting light.

  Optical birefringence describes the difference of a material's 
refractive index with direction. When birefringence is on the order of 
the change of the in-plane refractive index between adjacent material 
layers, surprising and useful optical effects occur. We refer to these 
effects as giant birefringent optics (GBO). A central feature of GBO is 
improved control of the reflectivity of p-polarized light. With the 
additional design freedom allowed by GBO, Brewster's angle can be 
controlled to any angle from 0 Degree (normal incidence) to 90 Degrees 
(grazing incidence), to imaginary values for light incident from media 
of any index of refraction. For imaginary values of Brewster's angle, 
the reflectivity at material interfaces (referred to as Fresnel or 
interfacial reflectivity) for p-polarized light increases with angle of 
incidence in a similar or identical form to that for s-polarized light. 
By comparison, isotropic materials have no substantial optical 
birefringence; that is, their refractive index values are equal for all 
directions. Interfaces of these conventional isotropic materials 
exhibit a limited range of Brewster's angles.

  Because the optical effects presented are based on the fundamental 
physics of interracial reflection and phase thickness and not on a 
particular multilayer interference stack design, new design freedoms 
are possible. For example, designs for wide-angle, broadband 
applications are simplified if optical elements with no Brewster's 
angle are used, particularly if immersed in a high-index medium such as 
a glass prism. Color filters can be designed that provide high color 
saturation at all incidence angles and polarizations. Alternatively, a 
mirror or reflecting polarizer can be designed to have a Brewster's 
angle that is accessible in air.

  Conventional polymer film-making processes have been enhanced to 
fabricate a wide array of GBO films from commercially available 
polymers and monomers for use in a range of applications. These 
applications include high-efficiency mirrors for piping visible light 
over long distances or uniformly lighting small optical displays. GBO 
multilayer films have been used to create reflective polarizers that 
make liquid crystal displays brighter and easier to view. Other 
applications include decorative products, cosmetics, security films, 
optoelectronic components, and infrared solar control reflectors for 
architectural and automotive glazing. After a review of birefringent 
optics, we discuss the relations describing GBO and show the 
implications of GBO on optical film performance and applications.

  Background.  Multilayer interference optics can generally be described 
as the use of the amplitudes and phases of light reflected at planar 
material boundaries to produce constructive and destructive 
interference effects. Pairs or groupings of adjacent layers (termed 
unit cells) can produce constructive interference effects when their 
thicknesses are properly scaled to the wavelengths of interest. These 
interference effects in multilayered structures result in the 
development of wavelength regions of high reflectivity (reflection 
bands) with adjacent wavelength regions of high transmission (pass 
bands) (1).

  Much of the design effort in multilayer interference optics is devoted 
to controlling the angular dependence of reflection bands, which is 
complicated by polarization effects. These effects have long been 
known, with publications dating to before the mm of the century [see, 
e.g., Drude (2,3)]. Sir David Brewster empirically deduced the law 
named for him by observing that light reflected from an air-glass 
interface is highly polarized at a specific angle (4). The same 
phenomenon occurs for all interfaces between isotropic materials. Aside 
from the well-known MacNeille polarizing beamsplitters (5) and 
magneto-optic materials (6), such polarization effects are typically 
undesirable, as they limit the angular performance of multilayer 
interference stacks. Various researchers (7-10) have developed a 
variety of limited solutions to the problem. In addition, modern 
computer optimization codes have dealt admirably with the problem. 
However, the basic phenomenon associated with Brewster's angle still 
continues to constrain ire angular and wavelength performance of 
multilayer interference stacks fabricated from materials having 
isotropic indices of refraction.

  Multilayer polymeric interference mirrors were pioneered in the late 
1960s (11), and even though the large birefringence of oriented 
polyethylene terephthalate (PET) was known at the time (12), the use of 
materials with large optical birefringence in a multilayer mirror 
(polymeric or otherwise) has not been reported. Numerous other works 
have been published on birefringent optical materials (13-18), but none 
of these discuss the use of birefringence to control (or eliminate) 
Brewster's angle effects and phase thickness relations among interfaces 
in multilayer interference stacks.

  Giant birefringent optics.  The coordinate system used to reference 
the material axes and the incident electric field for different linear 
polarization states is shown in Fig. 1. For GBO, each birefringent 
layer is either uniaxial, with its z-direction index different from the 
equal in-plane indices (equal x-y direction indices), or biaxial, with 
the x-, y-, and z-direction indices all being unequal.

  Part of the optical behavior of a multilayer interference stack 
originates in the angular dependence of the Fresnel interface 
reflection coefficients, including the nature of the Brewster's angle 
ThetaB. Figure 2 compares the magnitudes of Fresnel reflection for 
various internal interfaces (that is, between materials 1 and 2 in Fig. 
1) as a function of angle of incidence (from the external medium). For 
convenient comparison of material pairs having a range of index 
differences, all of the reflectance values plotted for a given material 
interface have been normalized to their value at normal incidence. An 
external medium with a refractive index n0 = 1.60 (e.g., a glass prism) 
is chosen so that a wide range of propagation angles can be explored. 
Snell's law requires that the larger the external medium index, the 
greater the range of propagation angles that can be achieved within the 
films. For most isotropic material pairs, ThetaB is not accessible for 
light incident from air.

  Curve c in Fig. 2 shows the interfacial reflectivity for a common 
material pair used in the multilayer interference film industry, 
SiO2-TiO2, which in this case has ThetaB = 52 Degrees. The range of 
ThetaB for other commonly used isotropic material pairs that are 
transparent in the visible portion of the spectrum is indicated by the 
shaded portion of the plot (about 40 Degrees to 70 Degrees in a n0 = 
1.60 medium); the lower bound of 40 Degrees occurs for a material pair 
with indices 1.35 and 1.50 and the upper bound for a pair with indices 
1.95 and 2.4. Tellurium-polystyrene, an interesting material system 
that is transparent only at mid-infrared wavelengths, was recently 
reported by Fink et al. (19) and is represented by curve d; in this 
case it has ThetaB = 71 Degrees (similar to ZrO2-TiO2).

  These examples illustrate behavior that is indeed a "law" for 
interfaces between two isotropic materials, regardless of the incident 
medium index. From its value for normally incident light, the 
interfacial reflection for p-polarized light decreases monotonically 
with increasing incidence angle up to ThetaB (20). Whether ThetaB is 
observed depends on the range of propagation angles that are accessible 
in the materials, as determined by Snell's law of refraction and the 
incident medium index.

  Curves a, b, e, and f in Fig, 2 represent interfacial reflection of 
various birefringent material pairs from which we have fabricated 
multilayer interference stacks. Curve e is for the special case of 
matched z-direction indices where reflectivity is constant with angle 
of incidence. When the interface materials have a z-direction index 
difference Delta nz of opposite sign relative to the in-plane index 
difference Delta ny, the interfacial reflection behavior for 
p-polarized light is similar to that for s-polarized light (curve f). 
The material pairs used for curves a and b demonstrate that ThetaB can 
be reduced to any value, including 0 Degree, by the appropriate choice 
of z-direction index values relative to the in-plane indices.

  The quantitative relations that provide the basis for GBO offer 
physical insight into the optical effects that are achievable with 
birefringent multilayer stacks. These are discussed below.

  Fresnel coefficients and phase relations for GBO.  At the boundary 
between two birefringent materials 1 and 2 that have their orthogonal 
optic axes, coincident with the film axes (see Fig. 1), the Fresnel 
reflection coefficient for p-polarized light propagating from material 
layer 1 into material layer 2 can be found in textbooks (21) and is 
given by


(1) rp = (n2zn2y square root of n2, sub 1z -n2, sub 0 sin2 Theta0 - 
n1zn1y square root of n2, sub 2z - n2, sub 0 sin2 Theta0)/ (n2zn2y 
square root of n2, sub 1z - n2, sub 0 sin2 Theta0 + n1zn1y square root 
of n2, sub 2z -n2, sub 0 sin2 Theta0)

  where n0; and Theta0 refer to the index and angle in the external 
isotropic medium, respectively. In the limit of isotropic indices, Eq. 
1 reduces to that given by Born and Wolf (22). For such a material 
system, s-polarized light interacts only with the in-plane indices and 
the Fresnel coefficient is the same as for isotropic materials:


(2) rs = square root of n2, sub 1x - n2, sub 0 sin2 Theta0 - square 
root of n2, sub 2x - n2, sub 0 sin2 Theta0 / square root of n2, sub 1x 
- n2, sub 0 sin2 Theta0 - square root of n2, sub 2x - n2, sub 0 sin2 
Theta0

In Eqs. 1 and 2, the plane of incidence (see Fig. 1) is taken to be 
along they axis. If the plane of incidence were along the x axis, the 
values of nx and ny, would be exchanged in Eqs. 1 and 2. For uniaxial 
material systems, nx = ny.

  By inspection, we can arrive at the effective interfacial indices for 
the ith layer of a birefringent material:


(3) nint, sub is = square root of n2, sub ix - n2, sub 0 sin2 Theta0 / 
cos Theta0

for s-polarized light and


(4) nint, sub ip = niyniz cos Theta0 / square root of n2, sub iz - n2, 
sub 0 sin2 Theta0

for p-polarized light. Effective indices are useful in that they 
combine angle and polarization effects into a simple expression with 
the form of a refractive index. Equation 4 leads to a generalized 
version of Brewster's law that can be used to solve for ThetaB, the 
incidence angle for which


(5) nint, sub 1p = nint, sub 2p

There are some interesting limits to Eq. 1. For the case of materials 1 
and 2 having equal z-direction indices n1z = n2z (Delta nz = 0), Eq. 1 
reduces to


(6) rp = n2y - n1y / n2y - n1y

which is independent of angle (shown by curve e in Fig. 2).

  For a broader class of materials, when the z-direction index 
difference (n1z - n2z) is nonzero and has the opposite sign from the 
in-plane index difference (n[sub 1y - n2y), the fractional bandwidth of 
a multilayer stack reflection band and its reflectivity can actually 
increase with angle of incidence. Also, consider the special case where 
the two sets of index differences in materials 1 and 2 are equal with 
opposite sign, and n1y = n2z, or n2y = n1z. Equation 1 then reduces to


(7) rp = -rs

for all angles of incidence. A quarter-wave multilayer interference 
reflector constructed with this material combination has identical 
s-and p-polarization reflection bands at all angles.

  Multilayer interference optics depend not only on the interfacial 
reflections but also on the phase thickness relations that govern 
coherent interference. For example, reflection bands centered about a 
given wavelength Lambda0 develop from a multilayer stack composed of 
alternating materials of high and low index, where the phase thickness 
of each of the layers in the structure is Lambda0/4. The center 
wavelength Lambda0 for a reflection band follows from a simple 
relation:


(8) Lambda0 = 2(nphz, sub 1d1 + nphz, sub 2d2)

  where d1 and d2 are the physical thicknesses and nphz, sub 1 and nphz, 
sub 2 are the effective phase thickness indices of each material. The 
effective indices that are used to determine the phase relations of 
birefringent materials are


(9) nphz, sub is = square root of n2, sub ix - n2, sub 0 sin2 Theta0

for s-polarized light and


(10) nphz, sub ip = niy / niz square root of n2, sub iz - n2, sub 0 
sin2 Theta0

for p-polarized light (21). Equations 1 to 4 and 8 to 10 are sufficient 
to describe the optical behavior of reflection bands developed from 
multilayer interference stacks, whether they are composed of 
conventional isotropic materials or from materials exhibiting large 
optical birefringence.

  Reflection band examples.  Reflection bands have characteristic 
features that describe their optical behavior. A reflection band is 
positioned about a particular wavelength, the center wavelength (Eq. 
8), and the bandwidth, which refers to the span of wavelengths of high 
reflectivity. These characteristics are determined by the interfacial 
reflectivity and phase thickness of the layers constituting the 
multilayer stack. Each of these has its own dependence on the incidence 
angle and polarization. The details of a multilayer stack structure 
(the sequence of unit cells) also affect reflection band 
characteristics. Generally, the greater the number of unit cells in a 
stack and the larger the index difference between adjacent layers, the 
greater the reflectivity at and around the center wavelength. The 
simplest reflection band designs use many repeats of identical unit 
cells. Other designs may use a sequence of unit cells that have a 
gradation of thicknesses so as to increase the overall bandwidth of the 
reflection band (23).

  The reflection band characteristics of a pair of isotropic materials 
are compared to those for a hypothetical pair of birefringent materials 
in Fig. 3. The inset in Fig. 3A shows the material configuration, with 
the length of the arrows along the x, y, and z directions representing 
the magnitude of each material's indices along the respective 
directions. The magnitude of the reflection at the interface between 
materials 1 and 2 versus incidence angle for p-polarized and 
s-polarized incident light (Fig. 3A) for this pair of isotropic 
materials shows the typical behavior of interfacial reflection for 
p-polarized light. At 55 Degrees, the value of the Fresnel reflection 
drops to zero (ThetaB) for light incident from an external medium with 
n0 = 1.4. Figure 3B shows reflection bandwidth versus incidence angle 
for a tuned (quarter-wave at normal-angle incidence) interference stack 
composed of alternating layers of these isotropic materials 1 and 2. As 
incidence angle increases, the centers of the s-polarized and 
p-polarized reflection bands move to shorter wavelengths as the 
effective phase thickness of the layers decreases. The reflection band 
behavior is calculated using the four effective indices with the 
characteristic matrix method (24) and locates the band edges for a 
design with a large number of unit cells. In this instance, the 
reflection band edges are plotted. Note that the reflection band 
"disappears" for p-polarized light at the ThetaB values for these 
isotropic material interfaces.

  In Fig. 3, C and D, both materials 1 and 2 are birefringent. Material 
1, with the higher in-plane index, is negatively birefringent; its 
z-direction index is lower and matched to the in-plane index of 
material 2. Material 2 is positively birefringent, with its z-direction 
index higher than its in-plane index and nearly matched to the in-plane 
index of material 1. As before, the external medium has an index of 
1.4. In this instance, the Fresnel reflection of p-polarized light at 
the interface between materials 1 and 2 actually increases with 
incidence angle (Fig. 3C), much the way it does for s-polarized light. 
Figure 3D shows how the p-polarized light reflection band of a 
multilayer quarter-wave stack of these materials has an increasing 
fractional bandwidth with increasing incidence angle, in a manner 
nearly identical to the s-polarized light reflection band.

  Another important parameter affecting the behavior of multilayer stack 
reflection bands is the relative phase thicknesses of the material 
components in a unit cell. A measure of relative phase thicknesses, 
termed the f-ratio, is the ratio of the phase thickness for each layer 
relative to the aggregate phase thickness of the repeating unit cell. 
It determines how the Fresnel reflections of each layer interface are 
coherently summed across the unit cells in the optical stack, which in 
turn determines reflection band behavior with changing incident angle. 
In many optical stack designs, suppression of higher order reflection 
bands (harmonics of the primary, first-order reflection band) is an 
important consideration (25). For p-polarized light, GBO provides an 
increased level of control of f-ratio with changing incidence angles. 
By using effective phase indices (Eq. 10), it can be shown that the 
f-ratio for a z-direction index-matched unit cell for p-polarized light 
is unchanged with angle. This control of the f-ratio can lead to 
pass-band designs (pass filters) that are very robust with incidence 
angle.

  The reflection bandwidth of a multilayer interference mirror made from 
a sequence of repeated, identical unit cells is determined by the 
effective interfacial indices (Eqs. 3 and 4) of the materials and their 
f-ratios. To create a wider reflective band, a standard technique is to 
use a graded unit cell thickness profile. A 60-unit cell interference 
mirror with a 25% thickness gradient was fabricated from a birefringent 
polyester and polymethylmethacrylate (PMMA) [see, e.g., (12,26-28) for 
optical properties]. A cross section of this multilayer interference 
structure was characterized with an atomic force microscope (AFM) (Fig. 
4).

  The layer thickness distribution developed from the AFM 
characterization was used in conjunction with measured dispersive 
values n1x, n1y, n1z, n2x, n2y, and n2z as input for a multilayer 
interference optical film model. These refractive index values were 
measured for thick monolithic films of both PMMA and birefringent 
polyester that had undergone the same film fabrication process as the 
multilayer mirror. These measurements indicate that the PMMA and 
birefringent polyester constituting the unit cells are matched in their 
z-axis refractive indices, with a substantial mismatch for their 
in-plane indices (GBO). Curve e in Fig. 2 shows the expected behavior 
of the interfacial reflectivity (p-polarized light) versus incidence 
angle for these GBO material interfaces.

  The measured and calculated spectra for this mirror sample are 
compared in Fig. 5. For ease of comparison, optical density 
[essentially -log(1-reflectance) for these low-loss, low-scatter 
polymers] is plotted. With the AFM-measured layer profile and 
dispersive refractive index values, the measured transmission spectra 
at normal incidence are well matched by spectra modeled using either 
the GBO refractive indices or isotropic refractive index values (set to 
the measured in-plane values). Using the same measured indices and 
layer thickness profile, the GBO calculations agree very well with 60 
Degrees incidence (from air) p-polarization measurements, but the 
agreement is very poor for the isotropic refractive index calculation. 
Both the details of the reflection band and the band edge positions are 
faithfully reproduced with the model calculation incorporating the GBO 
refractive indices. Indeed, as expected for a GBO system with matched 
z-axis refractive indices, all of the characteristics of the normal 
incidence reflection band are maintained for high-incidence p-polarized 
light. Calculations using the isotropic refractive index values in the 
model result in a substantially weaker reflection band that is easily 
differentiated from the mirror using z-axis index-matched materials.

  The refractive index values measured for the thick monolithic films of 
both PMMA and birefringent polyester are consistent with the GBO 
effects shown in Fig. 5 for the mirror sample with individual layer 
thicknesses ranging from 90 to 120 nm. This result supports recent work 
on the optical and physical properties of thin polymer layers (29).

  Applications.  The enhanced control of p-polarized light enabled by 
GBO allows different multilayer interference stack designs to be 
developed for numerous optical applications. GBO multilayer 
interference stacks can be fabricated with a variety of manufacturing 
methods. One economical method is polymer coextrusion (11). With this 
technique, we have produced a variety of optical films having between 
100 and 1000 layers. With no rigid substrate, they are thin and 
flexible.

  GBO broadband mirrors. As shown above (curve e in Fig. 2), a 
z-direction index-matched multilayer interference stack exhibits 
reflection band behavior for p-polarized light that enables previously 
unavailable performance. One optical application that can take 
advantage of this characteristic is a broadband mirror, intended to be 
an efficient transporter of visible light. The measured spectra in Fig. 
6A illustrate the angular behavior of such a GBO broadband mirror 
containing 448 layers of birefringent polyester and PMMA. The normal 
incidence reflection spectrum is compared with the reflectance spectra 
measured for p-polarized light incident at 45 Degrees from air and from 
a glass prism. Note how the p-polarized light reflection stays very 
high at high incidence angle, particularly that demonstrated by the 
spectra for the mirror "immersed" in a glass medium.

  Broadband mirror applications that use multibounce reflections are 
very sensitive to reflectivity levels and color changes upon 
reflection. Figure 7A shows a set of three circular cylinders, each of 
which is lined with a high reflectivity mirror. A broadband, "white" 
light source is obliquely illuminating each tube's entrance aperture. 
Tube a is lined with a multilayer GBO film with matched z-direction 
indices. Tube b is lined with a high-quality, second-surface aluminum 
mirror, and tube c is lined with a high-quality second-surface silver 
mirror. The light exiting each tube has undergone a large number of 
reflections across a range of high incidence angles. The resulting 
light intensity and color fidelity of the exiting light provide a 
measure of the level of omnidirectional reflection quality. As can be 
seen in the photograph, the light exiting the GBO broadband mirror tube 
has both high intensity and good color fidelity. The silver mirror tube 
shows an obvious "yellowing" of the exit light, and the aluminum tube 
has markedly lower exit-light intensity.

  GBO color mirrors. A nonpolarizing color mirror that operates over a 
range of incidence angles and wavelengths is a difficult task for a 
designer using conventional optical materials (30). For non-normal 
incidence, polarization effects limit band edge sharpness, which can 
greatly affect color purity. GBO techniques can be used to construct a 
color mirror that has a matched band edge at all angles for both p- and 
s-polarized light, eliminating these difficulties.

  The importance of the use of GBO for color mirrors is illustrated in 
the following example. Transmission spectra for a GBO stack with all 
layers having matched z-direction indices near 1.5 are shown in Fig. 
6B. Measurements for normal incidence and 60 Degrees angle of incidence 
for s- and p-polarized light are shown. Note that the small midband 
leak at normal incidence is reproduced with its intensity unchanged in 
the 60 Degrees p-polarization measurement. Because the air-polymer 
interface does not meet GBO criteria, typical Brewster's law behavior 
is observed for wavelengths outside the reflective band (transmission 
levels of 60% for s-polarization and 98% for p-polarization). Although 
the long-wavelength band edges are substantially different, the 
short-wavelength band edges for s- and p-polarization are nearly 
identical.

  The range and intensity of colors that are created in a film cavity 
made of these materials is shown in Fig. 7B. In this photograph, the 
cavity is externally illuminated with a "white" light. The multiple 
bounces produced in a cavity with high reflectivity over a portion of 
the visible spectrum accentuate the reflected intensity variation at 
different wavelengths, creating intense color. The highly saturated 
colors seen at all observation angles are a result of the matched s- 
and p-polarization band edges at all angles, combining light 
transmitted through and reflected from the cavity surfaces.

  GBO reflective polarizers. GBO multilayer interference stacks can be 
fabricated with a high refractive index difference developed along only 
one in-plane axis, creating a reflecting polarizer. A schematic of a 
unit cell with appropriate indices is shown in Fig. 8A, indicating a 
biaxial refractive index for at least one of the layers. Figure 8B 
shows reflection measurements along the two principal axes (see Fig. 
1). With the use of GBO techniques, the ultimate omnidirectional 
reflective polarizer can be made where the index differences between 
layers are zero along both the x and z axes. In such a system, light 
polarized along the reflective axis (y direction) behaves according to 
curve e in Fig. 2 for p-polarized light. For light polarized along the 
pass axis (x direction), neither s- nor p-polarized light is reflected 
by the multilayer stack for any angle of incidence, as the relevant 
index differences are zero.

  Although more complex than a mirror with uniaxial symmetry, GBO design 
concepts for reflective polarizers can be applied separately for light 
polarized along each axis. The Fresnel and phase thickness equations 
given above hold for light incident with its polarization direction 
parallel to either the x axis or y axis. Reflective polarizers 
constructed with the polymers discussed above have a demonstrated 
extinction ratio of 300:1 averaged across all visible wavelengths at 
all angles of incidence.

  3M Film/Light Management Technology Center, 3M Center, St. Paul, MN 
55144, USA.

  DIAGRAM: Fig. 1. The normal conventions for polarization are followed 
here, with p-polarized light having its electric field in the plane of 
incidence and s-polarized light with its electric field perpendicular 
to the plane of incidence. They and z directions in the layer are 
shown. Note that only p-polarization interacts with the indices along 
the z axis of the layer. For clarity, only the resultant reflected 
waves are indicated in the right-side diagram.

  GRAPH: Fig. 2. In order of increasing ThetaB, curves a through f 
illustrate p-polarized interfacial reflectivity for the following sets 
of indices: (a) GBO n1y = 1.63, n1z = 1.5, n2y = 1.63, n2z = 1.63 
(birefringent polyesterisotropic polyester), ThetaB = 0 Degree; (b) GBO 
n1y = 1.54, n1z = 1.63, n2y = 1.5, n2z = 1.5 (syndiotactic 
polystyrene-PMMA), ThetaB = 30 Degrees; (c) Isotropic n1y = 2.4, n1z = 
2.4, n2y = 1.46, n2z = 1.46 (TiO2-SiO2), ThetaB = 52 Degrees; (d) 
Isotropic n1y = 5.0, n1z = 5.0, n2y = 1.58, n2z = 1.58 
(telluriumpolystyrene), ThetaB = 71 Degrees; (e) GBO n1y = 1.8, n1z = 
1.5, n2y = 1.5 n2z = 1.5 (birefringent polyester-PMMA), ThetaB is 
imaginary; and (f) GBO n1y = 1.8 n1z = 1.5 n2y = 1.56, n2z = 1.56 
(birefringent polyester-isotropic polyester), ThetaB is imaginary. The 
shaded portion indicates the range of ThetaB for isotropic material 
pairs that are transparent in the visible portion of the spectrum.

  GRAPH: Fig. 3. Angular dependence of (A and C) interfacial reflection 
and (B and D) the long- and short-wavelength band edges for an 
isotropic layer pair and a GBO layer pair, respectively, of low and 
high in-plane indices of refraction. In (A), the p-polarization ThetaB 
is near 55 Degrees and the reflection band disappears at that angle in 
(B). For (A) and (B), n1x = n1y = 1.8, n1z = 1.8, n2x = n2y = 1.5, n2z 
= 1.5, and n0 = 1.4 For the GBO material pair, the low and high indices 
of refraction in the x-y plane have the opposite sign index difference 
compared with that along the z axis. In (C) and (D), the p-polarization 
reflection is higher than the s-polarization reflection with angle. For 
this GBO example, n1x = n1y = 1.8, n1z = 1.5, n2x = n2y = 1.5, n2z = 
1.9, and n0 = 1.4.

  GRAPH: Fig. 4. An AFM image of a GBO stack (31); the dark-colored 
layers are PMMA and the light-colored layers are birefringent polyester 
(polyethylene naphthalate). Layers on the left side of the image are 
about 25% thicker than those on the right.

  GRAPH: Fig. 5. Comparison of measured and modeled results for light 
transmission (T) at normal incidence and for 60 Degrees incidence of 
p-polarized light. The plot shows good agreement between measurements 
and high-incidence p-polarized GBO model calculations, and poor 
agreement for an isotropic materials calculation at high incidence 
angles with p-polarized light.

  GRAPH: Fig. 6. (A) Measured broadband visible mirror reflection for 
various incidence angles. The p-polarized light measurements show no 
loss of reflection, only an increased band shift upon immersion in a 
glass incidence medium (n0 = 1.52). (B) Measured spectrum for a GBO 
color mirror at normal and 60 Degrees angle of incidence, for s- and 
p-polarizations. In (A) and (B), the sequence of unit cells has a 
gradient in thickness to increase the reflection bandwidth.

  PHOTO (COLOR): Fig. 7. (A) Light transport tubes using (a) GBO 
broadband mirror, (b) commercial aluminum mirror, and (c) commercial 
silver mirror. The ratio of length to diameter of the tubes is 17, and 
white light is used to illuminate the open aperture. (B) A GBO film 
cavity that is illuminated from the front aperture with white light. 
Note the change of highly saturated color with observing angle.

  GRAPH: Fig. 8. Measured reflectance for a GBO multilayer reflecting 
polarizer whose indices consist of alternating layers that are matched 
along both the x and z axes and mismatched along they axis. For this 
example (A), n1x = 1.57, n1y = 1.86, n1z = 1.57, n2x = 1.57, n2y = 
1.57, n2z = 1.57, and n0 = 1.0. (B) When measured in the y direction, 
reflection shows a strong band at near 100% intensity. Along the x 
direction, there are only air interface reflections.


References and Notes

  (1.)  P. Baumeister and G. Pineus, Sci. Am. 223, 58 (December 1970).


(2.)  P. Drude, Wied. Ann. 43, 146 (1891).


(3.)  -----, Ann. Phys. Chem. 38, 865 (1891).


(4.)  D. Brewster, A Treatise on Optics (Lea & Blanchard, London, 1839).


(5.)  M. Banning, J. Opt. Soc. Am. 37, 792 (1947).


(6.)  P. Lissberger, Rep. Prog. Phys. 33, 197 (1970).


(7.)  V. R. Costrich, Appl. Opt. 9, 866 (1970).


(8.)  A. Thelen, Appl. Opt. 15, 2983 (1976).


(9.)  -----, J. Opt. Soc. Am. 70, 118 (1980).


(10.)  T. Ito, U.S. Patent 5,579,159 (26 November 1996).


(11.)  T. Alfrey, E. F. Gurnee, W. J. Schrenk, Polym. Eng. Sci. 9, 400 
(1969).


(12.)  C. J. Heffelfinger and K. L. Knox, The Science & Technology of 
Polymer Films (Interscience, New York, 1971), p. 587.


(13.)  D. A. Holmes and D. L. Feucht, J. Opt. Soc. Am. 56, 1763 (1966).


(14.)  R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized 
Light (Elsevier, Amsterdam, 1987), p. 119.


(15.)  D. W. Berreman, J. Opt. Soc. Am. 62, 502 (1972).


(16.)  P. Yeh, J. Opt. Soc. Am. 69, 742 (1979).


(17.)  I. Hodgkinson and Q. H. Wu, J. Opt. Soc. Am. A 10, 2065 (1993).


(18.)  J. Lekner, J. Opt. Soc. Am. A 10, 2059 (1993).


(19.)  Y. Fink et al., Science 282, 1679 (1998).


(20.)  Optical stack designs with isotropic materials often compensate 
for this reduction of p-polarization reflectivity with incidence angle 
by increasing the number of unit cells in the multilayer stack to such 
a level that the reflectivity falloff is minimized. This approach can 
be quite successful for isotropic multilayer interference layers with 
air as the external medium, but has limited utility when the external 
medium allows for larger propagation angles to be present in the stack.


(21.)  R. M. A. Azzam and N. M. Bashara, in (14), p. 357 (corrected 
equation).


(22.)  M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 
ed. 5, 1975), p. 40.


(23.)  O. S. Heavens and H. M. Liddel, Appl. Opt. 5, 373 (1966).


(24.)  M. Born and E. Wolf, in (22), p. 66.


(25.)  A. Thelen, J. Opt. Soc. Am. 53, 1266 (1963).


(26.)  J. C. Kim, M. Cakmak, X. Zhou, Polymers 39, 4225 (1998).


(27.)  J. C. Seferis, Polymer Handbook (Wiley, New York, ed. 3, 1989), 
p. 45.


(28.)  M. Cakmak and J. L. White, Polym. Eng. Sci. 29, 1534 (1989).


(29.)  R. L. Jones, S. K. Kumar, D. L. Ho, R. M. Briber, T. P. Russel, 
Nature 400, 146 (1999).


(30.)  H. A. Macleod, Thin-Film Optical Filters (Macmillan, New York, 
ed. 2, 1986), p. 334.


(31.)  AFM image and layer thickness analyses were provided by V. W. 
Jones of 3M Corporate Analytical Technology Center.

  25 October 1999; accepted 15 February 2000

  ~~~~~~~~

By Michael F. Weber; Carl A. Stover; Larry R. Gilbert; Timothy J. 
Nevitt and Andrew J. Ouderkirk

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Source: Science, 03/31/2000, Vol. 287 Issue 5462, p2451, 6p, 1 diagram, 
6 graphs, 1c.
Item Number: 3010032	
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