Research Article
Vol. 2, No. 7 / July 2015 / Optica
589
Continuous color reflective displays using
interferometric absorption
JOHN HONG,* EDWARD CHAN, TALLIS CHANG, TZE-CHING FUNG, BRANDON HONG, CHEONHONG KIM,
JIAN MA, YAOLING PAN, ROB VAN LIER, SHEN-GE WANG, BING WEN, AND LIXIA ZHOU
Qualcomm MEMS Technologies, Inc., San Jose, California 95134, USA
*Corresponding author: johnhong@qmt.qualcomm.com
Received 30 October 2014; revised 7 April 2015; accepted 12 April 2015 (Doc. ID 225521); published 23 June 2015
Reflective displays that rely on ambient light as opposed to an internal light source have been making inroads for a
variety of important applications, especially those involving mobility where power usage must be aggressively controlled. The underlying color rendering strategies for both reflective and emissive displays have largely been the same,
combining three fixed, primary color subpixels to compose the rich gamut that users expect. The result, for reflective
color displays, is unfavorable brightness/gamut performance since each of the color subpixels absorbs roughly 2/3 of
the incident white light. We demonstrate a new technology that we call the single mirror interferometric display that
overcomes such limitations with pixels whose reflectance properties can tune through a continuum of colors, including
high contrast black and white reflectance states. We use an effect that we call interferometric absorption in which a
thin absorbing metal layer in front of a highly reflective mirror surface selectively absorbs different colors, depending
on the gap that separates the two. The gap is controlled by electrostatic actuation in a relatively simple microelectro-mechanical-system structure. We describe this elegant and powerful color rendering principle and present
experimental results for the basic pixel device as well as early system demonstrations. © 2015 Optical Society of America
OCIS codes: (120.2040) Displays; (120.3180) Interferometry.
http://dx.doi.org/10.1364/OPTICA.2.000589
1. INTRODUCTION
A display system consists of a spatial array of pixels whose optical
properties (reflection or emission) can be controlled to render
two-dimensional color images. A system is needed to realize this
in practice, as illustrated by the active matrix backplane system of
Fig. 1, where each pixel element can be addressed via a thin film
transistor switch, allowing row-by-row updating of the visual
information. The optical state of each pixel is set by a voltage or
current, and a given row of pixels is selected by a corresponding
row of pixel switches and energized by the column drivers. Most
display devices assign the primary color components of an image
pixel to separate but adjacent device pixels so that three physical
pixels are needed to render a single image pixel, as illustrated.
Reflective display technologies have used this architecture,
including the first generation micro-electro-mechanical-system
(MEMS) display [1], color E-ink [2], and reflective LCD [3].
Variations on this theme for emissive technologies include
RGBW subpixels [4], where the addition of a white pixel allows
for added flexibility in the image rendering and, in some cases,
power savings. In the case of organic light emitting diodes
(OLEDs), PenTile and related arrangements use a lower density
of blue and red subpixels compared to the green subpixels
(RGBG) with the claim that the higher-density green subpixels
can be used to specify the resolution of the display [5].
2334-2536/15/070589-09$15/0$15.00 © 2015 Optical Society of America
The single mirror interferometric (SMI) display takes a completely different approach in rendering an image. As opposed to
having a collection of primary color subpixels to represent each
image pixel, all of the device pixels on the SMI panel are identical.
Each device pixel can tune its reflective color continuously across
the visible spectrum and also provide high contrast black and
white states. Instead of limiting the primaries to three or four
choices, the SMI palette has potentially a large number of primaries to choose from.
RGB-based reflective displays exhibit less than ideal brightness
and gamut performance. The brightness problem is straightforward and illustrated in Fig. 2. In order to represent a particular
primary color, the other two primaries must be black, which
means that only ∼1∕3 of the incident light can at best be reflected
back toward the viewer (assuming that the incident light has a flat
spectrum and each color pixel reflects ∼1∕3 of the total spectrum). This generalizes to the rendering of any color or white.
The latter will be at best a neutral gray with maximum reflectivity
of ∼33% (in most reflective LCDs, there is an additional 50%
loss due to the polarizer that is needed). The lowered gamut is
less obvious and has to do with leakage from the subpixels that
are turned off and supposed to be black. The black state leakage,
along with scattered light from imperfections and other surface
reflections, conspire to degrade the saturation of colors. In the
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Buffer
Pixel
Render
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Video
Source
Row Selects
Data (Column) Lines
Fig. 3. IMOD pixel—optical components to harness interferometric
absorption.
Fig. 1. Display system architecture.
SMI display, however, the 2/3 brightness penalty is avoided since
every pixel can render the correct primary color. The gamut,
likewise, is not degraded by unwanted black state leakage.
There is one difficulty, however, in the inability of the SMI
pixel to realize grayscale modulation. The rendering of full color
with grayscale modulation requires spatial and/or temporal
dithering where dithering artifacts must be carefully controlled.
This is discussed in some detail in a later section.
In what follows, we explain how the reflective colors are
realized in the SMI display system using a novel combination
of interferometric absorption and MEMS actuation devices.
Developmental system demonstrations are given to prove the
basic principles of operation.
to allow partial absorption and partial transmission of the light. In
Appendix A, we present a simple theory that specifies the essential
characteristics of the absorbing layer. The incident and reflected
light interfere with one another, and the constituent spectral components in the light set up standing-wave interference patterns
that differ in periodicity, following the component’s wavelength.
The IMOD is essentially a programmable reflection filter that
achieves its color (wavelength) selectivity by moving the mirror
to align the node (null) in the interference pattern with the
absorber layer. The movement of the mirror is achieved through
MEMS technology, as is explained later.
Referring to the illustration in Fig. 4(a), incident light at wavelength λ will interfere with its own reflection from the mirror to
create a standing wave with local peaks and nulls. The first null is
λ∕2 from the mirror, and subsequent nulls are placed at λ∕2
intervals. For that wavelength, a very thin absorber placed at
one of the null positions will absorb very little energy. When
the absorber is placed at the null of the red interference pattern
[Fig. 4(b)], other colors are absorbed, reflecting the red with high
efficiency. As the absorber moves closer to the mirror, the reflection color turns green, then blue, then black, where the colors
2. COLOR BY INTERFEROMETRIC ABSORPTION
Interferometric absorption takes place in an optical configuration
that requires a mirror, a thin metallic absorbing layer, and an air
gap that separates the two. Such a system, which we call the interferometric modulator (IMOD), is illustrated in Fig. 3 where
the incident light first passes through the thin layer of metal
before reflecting off a mirror and passing back through the same
metal to the observer. The metallic layer is sufficiently thin so as
Fig. 2. RGB versus SMI color rendering.
Fig. 4. Standing-wave electric field diagram.
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across the visible spectrum are absorbed nearly uniformly, and
finally white, when the absorber is collapsed onto the mirror,
where no part of the visible spectrum is absorbed.
3. SOME OPTICAL DETAILS
The illustration shown in Fig. 4 indicates that the absorber is in
contact with the mirror for the white state. In practice, the mirror
is not a perfect reflector, and even for the most reflective metals,
the light partially penetrates into the surface so that the first-order
node is inside the metal near the surface (the skin depth effect).
Additionally, some lubricating surface treatment is often needed
to prevent the contacting surfaces from sticking to one another (or
an air gap imposed by mechanical stops). Furthermore, a passivation layer is required to protect the metal absorber from contacting the metal mirror for electrostatic actuation. As such, the
first-order node cannot be made coincident with the absorber for
the white state. For example, when a 9 nm Al2 O3 passivation and
10 nm air gap is applied, the white state reflectance is very low
(∼45%) because the standing-wave field intensity is quite high at
the absorber. Therefore, the second-order node should be used as
the white state absorber location. To reduce the spatial separation
of the nodes of different wavelengths, we can use a high index/low
index pair of layers with appropriate thicknesses as illustrated in
Fig. 5, where the secondary null can be accessible for the white
state with approximate alignment across an appreciable fraction of
the visible spectrum. At the black state, minimum reflection
across the entire visible spectrum is desired. Again, the simple
absorber layer on its own does not achieve the best performance
because of the effects of dispersion (in the absorber and the mirror
materials), and impedance matching layers of high/low dispersion
dielectric materials can be incorporated into the design.
By using well-established thin film optimization procedures
[6], we arrive at a prescription for dielectric layers for the mirror
as well as the impedance matching layers for the absorbing layer,
as shown in Fig. 5. The corresponding reflectivities for various gap
values are plotted in Fig. 6. The reflective spectra can also be characterized by a corresponding plot on a colorimetric diagram—this
time with the air gap as a parameter as illustrated in Fig. 7. The
spiral of colors straddles the sRGB color periphery (the blue triangle), as the air gap varies from 10 to 640 nm. As the air gap is
further increased, the colors become more and more saturated to a
point until either the interference pattern becomes washed out
Fig. 5. Dielectric coatings to enhance the interferometric absorptive
colors. The red/green/blue curves indicate the electric field intensities
for 630, 530, and 440 nm light. Light enters from the right, and the
mirror reflective surface (AlCu) is on the left side of the figure.
Fig. 6. Reflectivity at various gap values.
due to coherence limits or multiple higher-order colors start
to mix.
These colors thus characterized correspond to light incident
normal to the mirror surface. The SMI pixel by itself behaves
as a colored mirror and would have significant viewing issues
without an adequate diffuser that would enable the rendered color
to be viewed across a range of viewing angles. In addition, a diffuser with sufficient haze is needed to partially compensate for a
shift in the reflective spectrum that is common to interferometric
optical devices.
If a diffuser is placed on the display, between the reflective
pixel and the viewer, incident light is scattered before passing
through the pixel optics and the reflected light is scattered once
more before reaching the observer, thus widening the viewing
angle range. An optical diffuser can be characterized by its haze
value, which represents the percentage of incident light power
scattered more than 2.5° away from a collimated incident beam
[7]. As the haze is increased, the viewing angle range increases at
the cost of reduced color saturation. Figure 8 shows the desaturating characteristic (calculated) of using a diffuser with a haze of
78% for the viewing condition where the viewing direction is 8°
from surface normal and the illumination is a 50/50 mixture of
diffuse and collimated (directed 20° off normal of surface) light.
Fig. 7. Interferometric colors plotted on u 0 –v 0 as a function of air gap
(the trajectory starts near the X-marked location and spirals clockwise as
the air gap is increased from 10 to 640 nm with a 5 nm step).
Research Article
Vol. 2, No. 7 / July 2015 / Optica
Fig. 8. Reflective color spiral with the application of a diffuser in a
typical indoor illumination condition (50% diffused and 50% collimated
light).
The diffuser haze of 78% does not materially affect the stated
343 ppi resolution, with the display substrate glass thickness
of 0.5 mm.
The materials used in fabricating the IMOD pixels are widely
available, and processes for their deposition and etch do not
present significant challenges for manufacturing, with one exception. The in-plane stresses and vertical stress gradients must be
properly controlled so as to keep the mirror relatively flat.
This is discussed in the next section.
4. ELECTROMECHANICS OF THE PIXEL
The requirements for the electromechanics of the pixel are to
(1) allow accurate control of the mirror movement by balancing
the applied electrostatic force against mechanical restoration
forces (i.e., hinges) and (2) maintain mirror flatness so as to
not adversely impact the color gamut. The electrostatic control
of a two-plate capacitive structure is well known and was used
in the first interferometric absorptive color display [1].
The SMI shown in Fig. 9 is configured as a three-terminal device to minimize the mirror curvature for a given actuation range
and to linearize the displacement-voltage actuation curve. If the
mirror is not sufficiently thick, the structure will bend during actuation, which will result in desaturation of the reflective colors
since different parts of the bent mirror will have different gaps
to the absorber, giving rise to different colors. The mirror, which
is the center terminal, is driven by TFT electronics to position it at
the desired location somewhere in between the two voltage-biased
outer terminals, which have fixed positions. Equilibrium between
the electrostatic and mechanical forces exists when
F −K x − x 0 −
V d − V b 2 ϵA V 2d ϵA
0;
2
2 d − x2
x2
Fig. 9. Three-terminal configuration of SMI pixel. The mirror is in
between the top (common) and bottom (bias) electrical terminals and
driven through the TFT switch. Although not shown, the top terminal
and absorber are fixed to a thick dielectric layer on the top and the substrate, respectively, and do not move, while the mirror is suspended by
mechanical hinges to a quiescent position.
thus limiting the range across which the mirror can be stably
controlled.
The mirror is suspended by hinges on all fours sides and is
rotationally symmetric (four-fold) to resist tilting. The center
portion of the mirror is thicker than the hinges due to the addition of a stiffener to decouple the requirement of maintaining a
very flat and stiff mirror and the requirement of having adequately
compliant hinges to maintain low-power operation. Since the
mirrors have multiple optical coatings, the stresses of the thin
films must be precisely controlled to maintain flatness throughout
all operating conditions.
To drive the mirror to a specific position, a voltage is applied to
the mirror through the TFT that is controlled by a row select
pulse of duration less than F r N r −1 , where F r is the refresh rate
of the display and N r is the number of rows in the display. The
mirror movement is comparatively slower than the row select
pulse due to the significant squeezed-film damping at such small
gaps. Since the mirror has not yet settled by the time the TFT
isolates the mirror from the data lines, the voltage on the mirror
varies as the mirror is moving since the three-terminal device
(1)
where the voltages and displacements are defined in Fig. 9. K is the
restoring force on the mirror provided by the hinges shown in
Fig. 10, A is the area of the mirror, and ϵ is the dielectric
permittivity of air. This equilibrium state is stable only if
ϵAV 2d
∂F
ϵAV d − V b 2
0>
−K ;
3
x
d − x3
∂x
592
(2)
Fig. 10. Three-dimensional exploded view of the SMI MEMS. The
mirror is suspended by four hinges and has a stiffener in the middle
to maintain flatness. The top terminal consists of a metal layer covered
by a thick dielectric to form a mechanical platform for TFT electronics.
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of SMI pixels, is used to infer the position of the mirror as a function of voltage. Small regions of the display panel are scanned
systematically to determine the uniformity and aging of the panel
over time and environmental conditions.
5. IGZO TFT/MEMS INTEGRATION
Fig. 11.
Mirror position as a function of voltage applied.
functions as a capacitive divider. This provides a form of negative
feedback that increases the stable travel range of this electromechanical actuator by reducing the quadratic increase in electrostatic forces as electrical terminals are brought together. Figure 11
shows the mirror position as a function of applied voltage for the
cases in which (1) the mirror voltage is held constant until the
mirror settles and (2) the mirror voltage is pulsed briefly and then
isolated and allowed to settle to the voltage level determined by
the capacitive divider.
The kink in the response curve at the 140 nm mirror position
is due to the compound spring effect when a specifically designed
feature of the mirror hinge contacts the absorber terminal first
before the mirror makes full contact with the absorber, thus increasing the mechanical restoring force to aid mirror peel back.
In order to monitor the characteristics of the pixel to account
for variations in the manufacturing process and aging, a circuit
measures the current flowing into a group of SMI pixels that
is actuated by a voltage ramp. The circuit shown in Fig. 12 uses
a high-precision voltage-controlled-current-source that is driven
in a closed loop to minimize noise. The output of the opamp, which indicates the amount of current going into the group
Fig. 12. Circuit to measure current flowing into SMI pixels as it is
actuated. A ramp is applied to the positive terminal of the op-amp to
drive the mirror through its entire range. The output of the analogto-digital converter is the current drawn by the group of SMI pixels.
The SMI display panel architecture is described in Fig. 13, showing the active matrix backplane as an array of transistor switches
that allows individual mirror connections to be made to the corresponding column electrode that is driven by a silicon CMOS
circuit. Also shown is an integrated row driver circuit that is made
directly on the glass using the same TFT process as for the backplane switches. This is common practice in the display industry
using both low temperature poly silicon (LTPS) [8] and indium
gallium zinc oxide (IGZO) [9,10] technologies. What is unique
in our approach is the integration of the IGZO TFT and MEMS
processes in a way that does not compromise the performance of
either. A detailed discussion of the process integration techniques
that were developed is outside the scope of this paper, but we
describe the general fabrication process sequence, pointing out
the key process steps that were optimized to yield the desired outcome. We also highlight the TFT performance characteristics that
were achieved using our integrated process.
The pixel architecture is shown schematically in Fig. 14, where
the MEMS device layers are shown to be fabricated first, followed
by the TFT sections that are built on top of the superstructure
that houses the MEMS portions. Light enters through the transparent glass substrate and is reflected back through the absorber
toward the viewer. The TFT is shielded from the light by the
mirror and black matrix patches that block the light from entering
the panel between the mirrors. All of the air gaps are defined during the process by a sacrificial material that is etched away as the
last step before packaging by a vapor phase process. The sacrificial
material that we used is sputtered Mo, and the etchant is XeF2 .
Fig. 13. SMI panel system architecture. IGZO TFT technology was
developed to provide both the pixel switches as well as to implement an
integrated row driver.
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Vol. 2, No. 7 / July 2015 / Optica
Fig. 14. Exploded view of the MEMS + IGZO TFT pixel built up
starting from the viewing side.
The top dielectric roof also serves as the platform on which the
TFTs are built. The sequence of fabrication is important as the
TFT device characteristics are known to be very sensitive to process temperature and ambient gas conditions. Building the MEMS
first allows for minimal perturbation of the TFT, but care still
needs to be taken since the release etch and subsequent encapsulation processes can adversely affect the device characteristics.
The TFT technology chosen is IGZO, which is a metal oxide
semiconductor, typically prepared by sputter deposition using a
compound indium-oxide–gallium-oxide–zinc-oxide target [11].
The device geometry used in our development is a bottom-gate
configuration with etch stop layer as illustrated in Fig. 15. The
transistor is an n-type field-effect transistor in which the application of a gate bias controls the current flow between the source
and drain contacts.
While many have reported on IGZO and related oxide transistors in the literature, we are the first, to the best of our knowledge, to report on a MEMS/TFT integrated process to implement
both an active matrix backplane and integrated row drivers. The
gate insulator is plasma enhanced chemical vapor deposition
(PECVD) SiO2 , and the metal contacts are made with sputtered
molybdenum. The IGZO active channel is also sputtered.
Typical device transfer characteristics (I d s versus V gs at V d s 0.1 and 10.1 V) of the fabricated IGZO TFTs with W∕L 4 μm∕8 μm are shown in Fig. 16. The field-effect mobility
(μsat ) in the saturation regime is higher than 10 cm2 ∕V- sec ,
and the devices exhibit sharp switching characteristics (exceeding
the display application requirements) as we can see the
Passivation
ESL (SiO2)
IGZO
Gate Insulator
594
Fig. 16. Typical device transfer curves. The device dimensions are
W∕L 4 μm∕8 μm, and nine devices in 6 0 0 glass wafer are plotted
together. The inset table summarizes the TFT parameters. The curves
taken from different parts of the wafer are overlaid together.
subthreshold slope (SS) of ∼150 mV∕dec. The inset table
summarizes μsat , SS, and threshold voltage (V th ).
6. SMI IMAGE RENDERING
To display images in full color, almost all displays, emissive or
reflective, combine multiple display elements (more than one
per image pixel to be rendered) to achieve the desired color
appearance. These display elements can be subpixels for spatial
mixing or pixels that are temporally modulated within each video
frame to mix colors in the time domain. The human visual system
has limited resolution, in terms of both spatial and temporal
acuity. For instance, normal eyes cannot resolve pixels that are
smaller than 50 cycles∕ deg , which corresponds roughly to a pixel
width of 43 μm, at a viewing distance of 0.25 m, nor track temporal fluctuations in excess of 30 Hz [12]. Therefore, when a displayed image is observed under normal viewing conditions, the
human visual system blends the colors of those display elements,
both spatially and temporally.
Each pixel in the SMI display is capable of changing its color
across the visible spectrum, including the important black and
white states. Because it does not have a gray level control capability, however, general images must be rendered through a spatial or
temporal dithering (or a combination of the two) approach. Each
pixel in a given source image is mapped onto the gamut volume
that is accessible by the SMI system. Each SMI pixel is then assigned to one of the many available primaries (five is the minimum, consisting of R, G, B, black, and white, but can be as many
as 16 in our current system as limited by the driver electronics) by
halftone techniques such as error diffusion and vector screening
[13] in which the difference between the source and human visual
perception of rendered images is minimized following welldefined algorithms. Figure 17 shows a computer generated image
rendering with a palette of 16 primary colors available in the SMI
system, using spatial dithering.
Gate Metal
Buffer SiO2
Fig. 15. Bottom-gate configuration with etch stop layer (ESL). The
ESL protects the IGZO surface (back-channel) from process damage
during source and drain (S/D) metal etching.
7. DEMONSTRATIONS
Square panels of 384 × 384 pixels with a diagonal of 1.58 0 0 and
74 μm pixel pitch were made, implementing the full threeterminal MEMS mirror array integrated with an IGZO active
matrix backplane and integrated row driver circuits, using both
Research Article
Vol. 2, No. 7 / July 2015 / Optica
Fig. 20.
Fig. 17. Computer simulation of a spatially dithered image using 16
SMI primary colors. This simulation used an error diffusion algorithm
that included the splitting of the image into four subframes with equal
duration to implement a spatio-temporal dither [14].
a 6 0 0 glass wafer process and a Gen 4.5 glass sheet process, with
similar results. Figure 18 shows the reflected colors from a patch
of mirrors driven with varying voltages, showing microscope images to demonstrate the relative flatness of the mirror structures.
Figure 19 shows a close-up of the mirrors in an array. The flatness
of each mirror is typically within 10 nm of maximum sag.
The panel without diffusers was characterized with an optical
spectrum analyzer to compare the actual device with the theoretical predictions (see Fig. 7). The measurement result for 15 colors
that were selected is shown in Fig. 20, where the encircled gamut
is approximately 108% of sRGB. With the application of a diffuser (haze value 78), measured with a completely diffuse ambient
Fig. 18. Mirrors driven to several colors demonstrating the flatness
and uniformity that is achievable.
595
Measured color gamut of 15 SMI colors.
light source (with the observer viewing 8° off surface normal), the
color gamut decreases due to a number of factors, including the
diffusive mixing of angle-dependent colors and additional reflective and scatter losses arising from front of screen optical components including the cover glass, frontlight, touch screen, and
adhesive layers.
Figure 21 shows the complete panel driven with a silicon
driver chip that was mounted onto the glass using well-established
chip-on-glass techniques. The panel was driven with 60 Hz video
content with no ghosting artifacts, proving the adequate response
speed of both the MEMS structures and the associated TFT circuits. The photos were taken with a digital camera in an office
lighting environment with a simple plastic diffuser that was laminated onto the display panel. The extremely low leakage backplane enabled by the IGZO transistors was also demonstrated
by holding an image for 10 s without refreshing, during which
time no degradation of the image was observed. This aspect of
operation is a key enabler for mobile and wearable applications
where the conservation of precious battery power is an important
consideration.
The off-axis viewing properties are illustrated in Fig. 22, for
the same demonstration panel, viewed at angles of 10°, 20°,
and 40° with respect to the surface normal. The lighting conditions are the same, an average office environment with office
room light at near normal of the panel. The drop in brightness
is due to the particular haze value that was used, and a shift in
some of the colors can be seen at the extreme 40° angle.
8. SUMMARY
A completely new approach to realizing color displays, breaking
away from the RGB construct, was presented. The potential for a
Fig. 19. Close-up image of mirrors exhibiting a maximum mirror sag
of less than 10 nm across the 74 μm pitch.
Fig. 21. Initial SMI panel demonstrations. Both images and video
were rendered at 60 Hz. The illumination used was the available light
in an average, fluorescent lamp lit office environment.
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The characteristic impedance of the substrate is given by
Zs Fig. 22. SMI off-axis viewing examples: (a) camera directed 10° off
surface normal, (b) 20°, and (c) 40°.
bright, high gamut display in a reflective mode using available
ambient light for viewing was demonstrated. The physics of
both the optical and electromechanical aspects of operation were
discussed in detail, introducing the concept of interferometric
absorption to provide a rich palette of colors at each mirror-pixel
location.
APPENDIX A: SELECTING THE ABSORBER
MATERIAL AND THICKNESS
Although the IMOD effect is typically analyzed by thin film modeling techniques wherein the layers are treated as discrete interfaces and a numerical procedure is used to track the multiple
reflected and transmitted beam components, the transmission line
analysis [15] approach offers important insights into how the
absorber layer should be designed.
The IMOD is modeled using a transmission line construct as
illustrated in Fig. 23. The mirror is modeled as an electrical
short, the air gap is a section of transmission line of characteristic
impedance Z 0 377Ω, the absorber is a very short transmission
line of characteristic impedance Z M , and the substrate glass, on
which the absorber is placed, is a transmission line of characteristic impedance Z S . In terms of material parameters, the refractive
index of the substrate glass is ns , and the complex refractive
index of the absorber metal is nc n − jk, where n is the real part
of the index, k is the so-called extinction coefficient (defined so
that both n and k are real and positive), and j is the square root
of −1.
The characteristic impedance of the absorber metal is given by
Z0
:
(A1)
ZM n − jk
Fig. 23.
Transmission line model.
Z0
:
ns
(A2)
The absorbing metal layer can be crudely modeled as a resistive
load, but to actually derive the conditions for an optimum
material choice and thickness prescription, the key function of
the absorbing layer must be described. Intuitively, the condition
for high reflectivity is a simple short termination behind the
absorber, easily met by making the air gap zero to achieve a white
state (for all wavelengths) or one half of the wavelength of the
color desired for high reflectivity. The other constraint on the
function of the absorber is to efficiently absorb the colors not desired in the reflection. To formulate a simple condition, we examine one wavelength at a time, and set the air gap between the
absorber and the mirror to one quarter of the wavelength of the
light component. Of course, all color components not matching
the one quarter wavelength condition will also be absorbed to
some degree. For the one quarter wavelength air gap, the short
termination presented by the mirror is transformed to an open
circuit condition for the absorber, and the effective impedance
looking into the absorbing layer is given by
Z0
Z MQ
ZM
Z0
n−jk
≈
2πn−jk2 d a
j tanβd a j tan 2πn−jkd a
j
λ
λ
Z 0 2nk − jn2 − k2 ;
2πd a
n2 k 2 2
(A3)
λ
where d a is the thickness of the absorbing layer and β is the
complex propagation constant in the absorber. Maximizing
absorption is tantamount to minimizing the reflection for light
entering the absorber. Since the light enters from the substrate
glass into the absorber, the effective impedance Z M Q must
match that of the substrate glass, which is purely real. This yields
two conditions. First, we seek metals for which the real and
imaginary components of the complex refractive index are the
same: n k. This would set the reactive part of the load of
the above equation to zero. Second, the remaining real part
Fig. 24. n − k for various metals. Vanadium (V) and the alloy
molybdenum chromium (MoCr) exhibit the closest match between
the real and imaginary parts of the refractive index across the visible
spectrum.
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Vol. 2, No. 7 / July 2015 / Optica
of the effective impedance should be set equal to the substrate
impedance resulting in
Z0
2πd a
λ
2nk
Z
d
nkns
0→ a
:
n2 k2 2
ns
λ
πn2 k2 2
(A4)
This yields a wavelength-dependent thickness to ensure a
matching condition for all wavelengths, which is not possible.
We can instead choose a wavelength for which the matching
is perfect and accept a degree of mismatch for others.
To compare some metals, the difference between the real and
imaginary parts of the refractive index is plotted in Fig. 24. The
more absorptive metals have the two components more closely
matched than the highly reflective Ag. The best absorbing metals
turn out to be MoCr and V among readily available lists, and the
optimum thickness predicted by this theory is 6–7 nm, verified by
our experimental experience.
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