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Off the Beaten Path with Total Internal Reflection by csgirla

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									          Off the Beaten Path with Total Internal Reflection

                             Lorne A. Whitehead and Michele A. Mossman
                Department of Physics and Astronomy, University of British Columbia
                       6224 Agricultural Rd, Vancouver, Canada V6T 1Z1

Abstract
This paper begins by “re-introducing” the phenomenon of total internal reflection and the associated critical angle,
including a careful discussion of the extent to which the “total” in TIR is truly total, and the “critical” in critical
angle is truly critical. Although in one sense these points are largely of theoretical interest, they also have an applied
aspect in relation to controlling TIR. From this perspective, two practical applications of TIR are discussed. The
first involves illuminating engineering applications with prism light guides, and the second concerns electronic
image displays employing frustrated TIR. In both cases the unique attributes of TIR play a key role in the efficiency
and practicality of these applications.



Introduction
Total internal reflection (TIR) can occur when a light ray traveling in a transparent material encounters an interface
with another transparent, but less optically dense material. This effect is well know and has a wide variety of uses.
The physics of TIR is fascinating, as the reflection itself can be very close to perfect, and because the effect has a
critical dependence on the angle of incidence. The phenomenon of TIR itself is well known, but in this paper, there
will be a discussion of some subtle aspects of these special characteristics of TIR, and a discussion of its uses in two
less well known applications with which the authors have considerable experience. The first example is a device
known as a prism light guide – a hollow dielectric structure which can transport a large luminous flux for
illuminating engineering applications. The second application is a form of electronic paper based on electrically
controlled frustration of TIR. In both of these applications, it is the unique characteristics of TIR that make these
practical devices possible.


Exactly how “total” is TIR and how “critical” is the critical angle?
It is commonly stated that TIR occurs when the angle of incidence exceeds the critical angle, and in this case the
reflection is total. This, however, is an unrealizable idealization, for fundamental physical reasons. First, this
common description is based upon the inherent approximation of the ray model of light. In any system of finite
extent, the idea of light rays, representing the propagation direction of plane waves, is physically incorrect. In a real
system, the finite wavefront is equivalent to as a superposition of a distribution of plane waves, which in turn have a
range of propagation directions.
In other words, the idea of angle of incidence is somewhat unclear, and in effect there is always a distribution of
effective angle of incidence which has a range, in radians, of order the ratio of the wavelength to the physical extent
of the system. Therefore, for all angles of incidence, there will be a fraction of the incident energy that is,
effectively, within the critical angle and hence does not entirely reflect. Another fundamental shortcoming is that in
any real system there is some degree of absorption loss in the materials.
Thus, in principle, TIR is never truly total and in this sense it bears a similarity to its “distant cousin”, metallic
reflection – both phenomena are imperfect in real systems using real materials. But, there is nevertheless a profound
quantitative difference between TIR and metallic reflection. For example, in the visible wavelength band 95%
reflectance is very good reflectance for a metal, but even small transparent plastic prisms employing TIR can reflect
99.9% of the light. So, TIR is important in that it is able to realize, in a practical manner, higher reflectance than is
possible with metals. Another important difference is that in the case of TIR, the reflection can be modified by
changing the external medium, which allows interesting possible applications.




                                                                                                                        1
It is interesting to consider these ideas a bit more carefully and quantitatively. Figure 1 shows the reflectance as a
function of angle of incidence for each polarization, for light rays traveling in polymethyl methacrylate (PMMA)
resin (n1=1.5) and encountering an interface with air (n2=1.0). This well-known plot shows that light is transmitted
for small angles of incidence, although the reflectance grows rapidly as a function of angle as the angle of incidence
approaches the well known critical angle given by


                                                 θ c = arcsin(n2 n1 )                                                (1)



                                1.0

                                0.8
                  Reflectance




                                0.6                                            parallel polarization

                                                                               perpendicular polarization
                                0.4

                                0.2

                                0.0
                                      0     20                 40             60               80
                                                   Angle of incidence (degrees)
         Figure 1. Reflectance versus angle of incidence for light passing from PMMA to air


It is well-known that the reflectance as a function of the angle of incidence varies in a continuous manner for all
angles, but nevertheless there is a remarkable transition at the critical angle. In fact the derivative of the reflectance
value with respect to angle does undergo a discontinuity at the critical angle, which seems both puzzling and perhaps
unphysical. We shall see shortly that it is both.
Let us consider now the reflectance values for angles of incidence that are very close to the critical angle, as
depicted in Figure 2. (Note that for all these graphs the results are slightly different for the two polarizations, but the
behaviours are very similar. For simplicity, only one polarization is shown in Figure 2.)




                                                                                                                           2
                           1.000


             Reflectance   0.998


                           0.996                                                             AR coated TIR
                                                                                             interface
                                                                                             simple interface
                           0.994
                                                                                             50 microdegree
                                                                                             beamspread
                           0.992
                                                                                             slight external
                                                                                             absorption
                           0.990
                              41.8101       41.8102            41.8103             41.8104            41.8105

                                                   Angle of incidence (degrees)
         Figure 2. Reflectance versus angle of incidence very close to the critical angle


It is interesting to note how tiny the angular range is in this plot. Let us first consider the dotted line. This line
depicts the fact that there truly is a fundamental difference in principle between the situation for light rays below the
critical angle than for above. Simply stated, for angles of incidence below the critical angle, an anti-reflective
coating can be designed for the interface to completely eliminate reflection. (The design of the coating depends on
angle, and there is no general anti-reflective dielectric coating that will work for all angles of incidence.) Above the
critical angle, that situation is fundamentally different – in that case the reflection will be total, and a dielectric
coating cannot change that situation at all.
It is remarkable that such a radically different situation could exist across an infinitely thin mathematical boundary,
and intuition might suggest that this would be unphysical. Indeed it is, as depicted by the three additional curves in
the figure. The second curve shows that for an uncoated surface the actual surface reflectance does rise to approach
1 as the angle of incidence approaches the critical angle. This is true even if there is a fixed AR coating present, but
the shape of the approach will be somewhat different. At any rate, the point is that the reflectance is continuous
across the TIR boundary, but clearly the derivative is not. The next curve shows the result of building in a small
amount of angular spread (in this case about 1 microradian, equivalent to the diffraction spread of a 0.5 m
telescope!). Clearly, this angular spread smoothes out all of the derivatives at the critical angle. Even without the
angular spread, another real effect – absorption in the external medium – also has a similar smoothing effect, as
shown in the last curve. Thus while the effect of TIR is critical and has a surprising onset, it is clear that in reality it
does not represent a fundamental non-physical discontinuity. It is, rather, a physical effect with surprising strength
and usefulness.
As will be discussed shortly, there are some situations in which it is desirable to actively modify the external
medium in order to introduce, at a selected time, a high degree of absorption in that medium. Not surprisingly, such
absorption can reduce the amount of reflected light, as shown in Figure 3 which is a graph of reflectance versus
absorption coefficient, k, in the external medium. It is interesting to note in Figure 3 that there is a limit to this
effect. If the absorption becomes too great, the external material functions like a metallic reflector. Thus, there is a
value of absorption coefficient which minimizes the reflected intensity.




                                                                                                                           3
                                     1.0

                                     0.8


                       Reflectance
                                     0.6

                                     0.4

                                     0.2

                                     0.0
                                           0        0.5                   1                  1.5
                                                                              -1
                                                Absorption coefficient k (m )
         Figure 3. Reflectance versus absorption coefficient in the second medium


These interesting behaviours have led to TIR being used in a wide variety of practical devices, including optical
fibers, fingerprint sensors, TIR microscopy, light deflection films, and many others. We consider here two less well
known applications based on the authors’ research focus.


Illuminating Engineering with Hollow Light Guides using TIR
In the field of illumination engineering, it is sometimes desirable to transport light from a bright source and to
distribute it where needed. Optical fibers effectively guide light for small-scale applications, but they are unsuitable
for large-scale applications since the cost of a large solid-core fiber would be prohibitive and, moreover, the weight
of such a structure would be impractical. For these reasons, hollow light guides are preferred in such cases. Prism
light guides are hollow structures which pipe light by means of TIR, and can achieve high efficiency and uniform
illumination [1].




         Figure 4. Prism light guide cross section and isometric view


As shown in Figure 4, the wall of the hollow guide has molded in its external surface a series of isosceles right angle
prisms, running parallel to the axis of the guide. These prisms reflect with near perfect efficiency light rays that
travel within a range of directions such that their angular deviation from the guide axial direction, defined as θ, is




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less than a value θplg given by the following equation, which is about 30° for the refractive index of n=1.59 for the
polycarbonate material which is often used for such microreplicated structures [2]:


                                     θ ≤ θ p lg = sin −1   (3 − 2 2 )(n   2
                                                                              − 1)                               (2)

This represents is a sufficiently large solid angle for transmission of a large luminous flux from readily available
light sources. Furthermore, it is possible to controllably extract light by gradually deflecting it to larger angles
which are not trapped and therefore escape from the guide. The escaping light is often used for illumination in
remote hazardous locations. Such emission of light can be engineered using specifically shaped patches of diffuse
reflectance, so that the illumination levels can be controlled. The extractor, usually a piece of white diffusive film,
increases in size as a function of distance from the source. A typical prism light guide of arbitrary cross sectional
shape is depicted schematically in Figure 5. (For simplicity in the drawing, the extractor is not shown, but is located
inside the guide, immediately opposite the emitting area.)
                                                                              end mirror




                                  prism light
                                     guide                  emitting
                                                             area




                                                                                     reflective cover




                                        light injector

         Figure 5. Schematic depiction of a typical prism light guide


Prism light guides are currently widely used, primarily in applications where it is desirable to separate the source
from the region being illuminated. Common installations include tunnel lighting, industrial spaces such as
warehouses, high ceiling areas, and architectural highlights on buildings. A photo of an installation of prism light
guide in the Callahan Tunnel in Boston, MA, is shown in Figure 6.




                                                                                                                        5
         Figure 6. Prism Light Guide installation in the Callahan Tunnel in Boston, MA


Most recently, research is underway with the goal of cost-effectively illuminating office buildings with sunlight, in
order to reduce the required electrical lighting load and increase the quality of illumination [3]. In typical buildings,
the penetration of daylight is limited to the zones very close to the windows, and it is impractical to daylight the core
of the building. Over the past few decades, there have been a number of demonstrations of various guided daylight
systems, which have shown that it is possible to bring substantial amounts of solar flux into the building interior,
enabling the electrical lighting load to be reduced [4,5]. However, the capital costs of those demonstrated systems
have been prohibitively high compared to their energy savings. The goal of the current research program is to
provide a new optical and mechanical system for core daylighting, for which the associated lifecycle cost is
sufficiently low that the technology can be adopted in standard building construction.
The system is largely housed within a glazed canopy that runs horizontally along the length of the south-facing wall
of the building, above the windows and adjacent to the plenum space on each floor of the building. Sunlight is
directed into the system using an array of thin, approximately square mirrors as shown schematically in Figure 7(a).
Each mirror is supported by three fibers, one in each of two corners of the mirror and the third in the center of the
opposite edge. By fixing the central fiber and moving the two fibers on the opposite corners using two simple,
inexpensive linear actuators, the mirrors can be reoriented in unison. This low-cost design has been demonstrated to
be sufficiently accurate and very robust. A photo of the preliminary experimental set-up is shown in Figure 7(b).




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                                (a)                                                      (b)

    Figure 7. (a) Schematic drawing and (b) photograph of new mirror array used to direct sunlight into
                buildings


Once directed inside the building, the sunlight is distributed throughout the workspace using a modified prism light
guide which we refer to as the hybrid prism light guide [6]. This structure not only guides sunlight, releasing it in a
controlled fashion into the space below, but it also operates as an efficient luminaire. In this new hybrid light guide
design, the fixture has a rectangular cross-section. The emitting surface is lined with the prismatic reflective film,
but the side and top inner surfaces are instead lined with a highly reflective multi-layer optical film which has a
luminous reflectance of greater than 98%. Because of the highly reflective surfaces and the angular constraints of
the concentrated light, sunlight is efficiently guided along the length of the guide and is coupled out into the room
below by an appropriately tapered extraction panel. The light from the fluorescent lamps, on the other hand, strikes
the prismatic film primarily in “non-trapped” directions, and as a result it transmits more or less directly through the
film to illuminate the room below. The preliminary hybrid light guide, illuminated using only sunlight, is shown in
the photo in Figure 8.




                                      (a)                                               (b)

         Figure 8. (a) Experimental hybrid prism light guide and (b) room illuminated by guided sunlight




                                                                                                                       7
The illumination levels provided by the new solar canopy system are well above the recommended standards, so
under direct sunlight there is no need for additional electrical lighting under such conditions. In addition, the color
rendering properties are high, so the quality of illumination is very good. The next step will be the installation and
ongoing monitoring of a fully-operational system in several building sites. Although it is premature to precisely
quantify the energy savings that will result from this system, we are optimistic that this system will, for the first
time, yield truly economical daylight-based energy savings in the core regions of standard office buildings.


Electronic paper employing frustrated TIR
Another interesting use of TIR is in the emerging field of electronic paper – the creation of an electronic information
display in which contrast is produced by changing the reflectance of a surface via controlled application of electric
field. There is considerable interest in the display industry in developing such an electronic display that has the
visual appearance of ink on paper. One of the most important optical characteristics of paper is the substantial
difference in reflectance between the approximately 85% reflective white regions and the approximately 5%
reflective black regions. This large difference in reflectance allows the displayed information to be interpreted
quickly and easily under typical lighting conditions.
Liquid crystal displays (LCD), currently the leading reflective display technology, require polarized light to
appropriately reflect or absorb the incident rays. To accomplish this, a polarizer must absorb at least half of the
incident light, reducing the theoretical maximum reflectance to 50%. In practice, typical reflectance values are
lower than 40%. There is substantial work underway in a variety of emerging technologies to develop alternate
approaches that improve the overall reflectance [7,8,9]. While all of these approaches have the potential to yield a
higher maximum reflectance than conventional liquid crystal displays, the TIR-based approach discussed here
shares this capability and has additional unique advantages.
In addition to the need for high reflectance, a useful electronic paper display must have a wide viewing angle. In
order to maintain its maximum brightness, the typical viewing range of a reflective LCD is quite limited, typically
less than a half-angle of 30º. The ability to produce a bright, high contrast image even at wide viewing angles is
critical to the development of a suitable paper-like electronic image display. The TIR-based approach, which is
referred to as the “CLEAR” display (which stands for Charged Liquid Electro-Active Response) can achieve this
goal.
The CLEAR technology relies on TIR in a microstructured polymer sheet to generate a high surface reflectance
[10]. The reflectance of this surface can be controllably adjusted by moving an absorptive material into optical
contact with the reflective surface, thus preventing the reflection by a mechanism known as “frustration” of TIR,
depicted schematically in Figure 9. The challenge with this technology lies in designing optical structures which
allow TIR to produce an effectively diffuse reflection, and mechanisms which allow a control voltage to variably
“frustrate” the TIR.




         Figure 9. (a) Reflection by TIR at an interface and (b) frustration by absorbing evanescent wave
                    energy




                                                                                                                          8
A primary advantage of this technique is that switching from the reflective state to the absorptive state requires only
a microscopic movement (< 1 micron) of the absorbing material into and out of the evanescent region associated
with TIR, so the reflectance level can be modulated quickly and efficiently. This enables the CLEAR display to
have both low power requirements and fast response times.
In the CLEAR system, the underside of a polymer sheet is textured with tiny hemispheres [11], in which the TIR
takes place as depicted in Figure 10. Although light rays incident on the central region will pass through the
hemisphere and thus not reflect, the majority strike closer to the outer region of the hemisphere and undergo a series
of reflections by TIR until they exit from the flat surface. As a result, the hemisphere displays a bright ring with a
dark central spot when it is viewed through the flat side.

                                 reflected light
                                      rays
                                                                                  bright
                                                                                 annulus
                                      n1
                                 n2                                          dark
                                                                            center
                                           (a)                        (b)
         Figure 10. Reflection of light by TIR in a hemisphere


The average reflectance R of the flat surface of a hemisphere is determined by the ratio of the bright area to the total
area. Based on simple geometric optics, this area, and hence R, increases with the ratio of the refractive index of the
sphere, n1, to that of its surroundings, n2.
                                                                  2
                                                        ⎛n    ⎞                                                     (3)
                                                 R = 1− ⎜ 2
                                                        ⎜n    ⎟
                                                              ⎟
                                                        ⎝ 1   ⎠
The high reflectance of the hemisphere occurs even when it is viewed at an oblique angle, since a substantial
fraction of the incident light rays still encounter the curved surface at a sufficiently large angle to undergo TIR. This
means that the reflection is apparent even at extremely wide viewing angles, as depicted in Figure 11. Although the
bright area somewhat decreases at glancing angles, the reflectance remains high.




         Figure 11.    A hemisphere appears bright from virtually all viewing angles


An additional benefit of the micro-hemisphere array is the tendency to preferentially reflect incident light rays
approximately toward their source. We call this behavior “semi-retro-reflection”, in comparison to retro-reflection,
in which light reflects almost exactly back to the source. In a reflective display application, semi-retro-reflection is
important because under typical reading conditions, where the light source is located above and behind the viewer's
head, the reflected light will return predominantly toward the viewer. This can result in a factor of two increase in
brightness, relative to a diffuse white standard, under common lighting circumstances, so that the image appears
almost as white as paper even when using an red-green-blue color filter layer. In addition, this reflection is




                                                                                                                       9
somewhat diffuse so that the surface appears white from virtually all angles, and the surface can have an appearance
that is comparable to white paper.
There are a number of possible methods for absorbing the ambient light. Currently, our work focuses on the
movement of light-absorbing particles that are stably suspended in a low refractive index fluid and can be moved
electrostatically toward the TIR interface, by the well-known process of electrophoresis. The initial focus was
placed on the electrophoresis of pigment particles [12], as depicted in Figure 12.




                              Micro-hemisphere array                 Pigment suspension

         Figure 12. TIR-based reflective display geometry using microhemispheres


More recently, we have become interested instead in using a solution of dye ions that can similarly be moved in and
out of the evanescent wave region associated with TIR by applying the appropriate electric field. While the above
description focused on the use of micro-hemispheres, it is important to note that the approach can work equivalently
well with any TIR-based reflective structure.
The optical performance of the device can be improved by substantially increasing the effective surface area of the
electrodes, thereby increasing the capacitance of the device, and correspondingly increasing the resultant charge
transfer in response to the electric field. This can be achieved by using a nanostructured or nanoporous electrode, as
depicted in Figure 13. The effective surface area of such an electrode can be easily several hundred times larger
than that of an equally-sized flat electrode, but since the structures are on the nanometer scale and are therefore
much smaller than the wavelength of light, the ability for TIR to occur at the electrode is not affected. One possible
such electrode is a carbon nanotube film [13], which has been shown to have high conductivity due to the
overlapping fiber structure and also relatively high transmission.




         Figure 13. Carbon nanotube film used in (a) reflective and (b) absorptive states in a TIR-based
                    reflectance modulation device


Initial experiments using flat, transparent layers of indium tin oxide (ITO) as the electrodes, and several solutions of
common dye molecules in water, have demonstrated a reflectance modulation of about 1%. This level of
modulation is consistent with the expected value, taking into consideration the ionic charge, the ionic absorption
cross section and electrode surface area. Using instead a conductive carbon nanotube film, absorption levels of
greater than 30% have been observed. The spectral dependence of the reflected light, when frustrated by the dye
ions, was measured using as spectrophotometer, and the results are plotted in Figure 14(a) as the negative logarithm
of the spectral transmission, which corresponds to the absorption coefficient.




                                                                                                                     10
                                                                                                                        pH 7.0
                            3.0
                                                                                            3.0                         pH 12.4
                            2.5                                                                                         pH 13.4
                                                                                            2.5
      -ln(Transmission)




                                                                                                                        pH 14.0




                                                                        -ln(Transmission)
                            2.0
                                                                                            2.0

                            1.5                                                             1.5

                            1.0                                                             1.0

                            0.5                                                             0.5

                            0.0                                                             0.0
                               400       500      600         700                              400   500        600    700
                                        Wavelength (nm)                                              Wavelength (nm)

                                                  (a)                                           (b)
                          Figure 14. Spectral transmission of (a) frustrated TIR system and (b) dye solutions at different pH


The spectral transmission of the dye solution was also measured as a function of pH value; as illustrated by the
graph in Figure 14(b), the transmission changes dramatically as the pH increases. Comparing and interpolating the
graphs in Figure 14(a) and (b), it is apparent that the absorption spectrum observed in the frustrated TIR case
corresponds to a relatively high pH value of approximately 13.7. This is not surprising, since in addition to the
negatively-charged dye ions, dissociated OH- ions in the solution would migrate toward the positively-charged TIR
interface, generating a thin layer of high pH solution near the interface. This effect necessitates careful selection of
the dye in terms of its absorptive properties at different pH levels.
In a practical device using this principle, a microstructured surface such as the hemisphere array would be coated
with the nanoporous layer. It has previously been shown that the hemispheres can be as small as 2 microns in
diameter and still very efficiently reflect the light by TIR [14], and the coating need not be highly conformal to
substantially coat the surface; the demonstrated flexibility of a carbon nanotube film may be sufficient in this regard.
As well, the coating itself need not have high lateral conductivity since the hemispheres could first be coated by a
thin continuous layer of a transparent conductive material such as ITO, in order to generate the desired lateral
conductivity. Since the average light ray reflects several times as it passes through the hemisphere structure, even a
modest level of absorption of 30% at the interface will result in an extremely low net reflectance, yielding a very
high contrast display.
The dye molecules have several advantages over pigment particles, since the chemical composition, electrostatic
charge and size of the molecules can be very accurately known which enables a good understanding of the
interfacial physics in the system. As well, importantly, the molecules can respond very quickly to the applied
electric field, and because of thermodynamics the dye molecules will not agglomerate over time, as tends to happen
with light-absorbing pigment particles [15]. Finally, we believe that the dye molecules will exhibit a completely
non-hysteretic response, which means that a known reflectance can reliably and reproducibly be achieved by simply
applying a predetermined voltage level. These advantages make the dye-electrophoresis system uniquely useful for
controlling TIR in various microstructured sheets for high-reflectance, low power image displays. This approach
has the potential to create a very high-brightness, video-rate image display with paper-like appearance.




                                                                                                                                  11
Conclusions
Total internal reflection is a well-known and well-studied phenomenon in optics, but nonetheless there are some
subtle and interesting characteristics relating the nature of the critical angle that are generally not well known.
Investigations in this area have led to the development of several new applications for TIR. The prism light guide, a
hollow light guidance structure that uses prismatic microstructures, is now used in many installations for large-scale
illumination. The structure uses TIR to guide light within a well-defined range of angles with a very low loss rate
while controllably allowing light rays outside the key angular range to escape. The result is the uniform illumination
in areas where a remote source is desirable.
A more recent application is an electronic paper display using hemispherical microstructures to efficiently redirect
ambient light toward the viewer. This TIR-based approach can yield a full-colour video-rate image display that has
a high-brightness, paper-like appearance. Optical and electrical characteristics of early prototype devices have been
encouraging, suggesting that the TIR-based approach may be useful in the field of electronic paper displays.



Acknowledgements
The authors thank the Natural Sciences and Engineering Research Council of Canada and 3M Company for their
support of this project. The authors also thank Dr. A. Rinzler for providing the carbon nanotube film samples that
were used in these preliminary investigations.




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