Reconfigurable Imaging Systems Using Elliptical Nanowires

Reconfigurable Imaging Systems Using Elliptical Nanowires
LETTER
pubs.acs.org/NanoLett
Reconfigurable Imaging Systems Using Elliptical Nanowires
Ethan Schonbrun,*,† Kwanyong Seo,‡ and Kenneth B. Crozier‡
†
‡
Rowland Institute for Science, Harvard University, 100 Edwin H. Land Boulevard, Cambridge, Massachusetts 02142, United States
School of Engineering and Applied Sciences, Harvard University, 29 Oxford Street, Cambridge, Massachusetts 02138, United States
bS Supporting Information
ABSTRACT: Materials that have subwavelength structure can
add degrees of freedom to optical system design that are not
possible with bulk materials. We demonstrate two lenses that
are composed out of lithographically patterned arrays of elliptical cross-section silicon nanowires, which can dynamically
reconfigure their imaging properties in response to the polarization of the illumination. In each element, two different
focusing functions are polarization encoded into a single lens.
The first nanowire lens has a different focal length for each linear
polarization state, thereby realizing the front end of a nonmechanical zoom imaging system. The second nanowire lens has a different optical axis for each linear polarization state,
demonstrating stereoscopic image capture from a single physical aperture.
KEYWORDS: Artificial dielectrics, nanowires, diffractive optics, birefringence
T
he miniaturization and integration of optical components
has enabled imaging systems to appear in many new areas.
For example, medicine has seen several advances in endoscopy
and other small form factor microscopes for in situ inspection. In
addition, extremely compact cameras threaten to redefine consumer photography and machine vision. Miniaturized optical
imaging systems can often obtain performance competitive with
larger bulk systems in terms of resolution and aberration
correction.1 What is frequently lost in miniaturized imaging
systems, however, is the flexibility and functionality of larger
systems. Traditionally, much of an imaging system’s functionality
is based on dynamic positioning of the lens with respect to the
image sensor. Both focusing and changing the magnification, for
example, require mechanical motion or swapping of lenses and
are consequently frequently abandoned in miniaturized systems.
One possibility for adding functionality to compact imaging
systems is using active elements based on liquid crystals,2 micromirror arrays,3 or deformable mirrors.4 These systems have an
extremely large number of degrees of freedom, owing to their
large pixel count, and consequently can perform an enormous
number of different optical functions. Their drawback, however,
is that they require electrical connections to address each of the
individual pixels which adds complexity and bulk to the miniaturized optical system. Other promising directions are the
implementation of liquid lenses that use the physical boundary
between two immiscible fluids for refraction, where the shape of
the boundary can be controlled electronically.5 For applications
that require a continuous change of focal length, such as
autofocus, liquid lenses are effective. Yet there are many surface
profiles that are inaccessible due to the physics of surface tension,
for example, high numerical aperture lenses and other aspherical
surfaces.
r 2011 American Chemical Society
Holography offers another method for adding functionality to
an optical system. Instead of changing their optical properties as a
function of time, holograms can be designed to have properties
that vary as a function of wavelength or incident angle. This
feature of volume holograms, called Bragg selectivity, has primarily been taken advantage of in optical data storage schemes6 and
the storage of images for projection.7 Recently there has also been
interest in holographic imaging systems that use Bragg selectivity
to encode multiple filters into the same volume element.8,9 While
volume holography is effective at multiplexing filters, it does not
make for ideal lenses because Bragg selectivity reduces the ability
of the element to collect light over a broad angular range, which is
crucial for high-numerical aperture imaging.
Instead of using Bragg selectivity, information can be holographically encoded into optical elements using the polarization
dependence of subwavelength dielectric structures. This effect is
called form birefringence and has previously been realized using
linear subwavelength gratings.10 By locally changing the orientation or the fill factor of the grating, the birefringence of the
hologram can be spatially controlled. The ability to control the
birefringence of a material has enabled new forms of polarization
beam splitters,10 the generation of nonuniformly polarized light
beams,11 polarization multiplexed Fourier holograms,12,13 and a
high-efficiency multilevel phase lens.14
In this Letter, we demonstrate a holographic encoding scheme
that utilizes single elliptical nanowires to locally control the form
birefringence. Our scheme uses a nanowire pitch of one-quarter
of the free space wavelength of the incident light, enabling the
Received: July 8, 2011
Revised:
September 14, 2011
Published: September 16, 2011
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LETTER
Figure 1. Elliptical nanowire form birefringence. (a) Geometry of an elliptical nanowire. The nanowire has a cross-section defined by the two radii,
rx and ry, and a height h. (b) Finite element simulations of a periodic array of elliptical nanowires of different cross sections for Ex. The nanowires are
composed of amorphous silicon, which has a a bulk refractive index of 3.95 at a wavelength of 980 nm. The contour lines join points with the same mode
effective index. (c) Contour lines for Ey. (dg) Field plots of the distribution of Ex for the four different nanowire geometries: (d) and (g) have the same
mode effective index and (e) and (f) have the same mode effective index for Ex. (h) Scanning electron micrograph of fabricated elliptical nanowires of
type (f) having rx = 100 nm and ry = 50 nm.
control of birefringence with extremely high resolution. Instead
of holographically encoding data or images, we use this scheme to
encode two lens functions into a single element. Using this
design freedom, we implement two reconfigurable imaging
systems. The first is the front end of a nonmechanical zoom
system, capable of projecting images with two different magnifications. The second is a single aperture stereoscopic lens,
capable of projecting images from two different perspectives.
Form birefringent materials in the optical frequency range are
challenging to fabricate. Previous efforts have focused on fabricating deep subwavelength gratings, where a large aspect ratio is
needed to produce the required polarization-dependent phase
shift. Instead of using subwavelength gratings, we demonstrate
that elliptical nanowires are also capable of large form birefringence. Many reports have demonstrated that silicon nanowires
can be patterned with an extraordinarily large aspect ratio,1517
making them an ideal candidate for form birefringence. Using
electron beam lithography and reactive ion etching, it is possible
to control the cross-sectional shape of each nanowire in the array.
Each nanowire can be considered to be a waveguide that has a
corresponding effective refractive index (neff) for each of the two
different incident linear polarization states, Ex and Ey. Nanowires
that have a circular cross-section have the same neff for both
Ex and Ey polarized incident light. However, nanowires that have
an elliptical cross-section have a different neff for Ex and Ey, which
is equivalent to form birefringence.
Our goal is to realize a pattern acting as a two-level phase
grating for one polarization and a second completely independent two-level phase grating for the other polarization.18,12 For
each of the two phase functions, light must experience a phase
delay that can take one of two values, depending on which of two
polarization states it comprises. Four different nanowire structures are therefore needed. The difference between these phase
delays should be π, in order to achieve the maximum possible
diffraction efficiency for a two level grating of 40.5%. A similar
encoding scheme was implemented by Yu et al.12 to encode
holographic images using linear gratings and square arrays of
posts and holes. Figure 1 shows the nanowire geometry, which
has three geometric degrees of freedom, rx, ry, and h. We restrict h
to a value of 1 μm because in previous work17 we developed a
reliable fabrication process for silicon nanowires with this height,
with radii down to 40 nm. Further details of the nanowire
fabrication are included in the Supporting Information.
To find the required dimensions rx and ry, we use a finite
element method (FEM) mode solver (COMSOL) to evaluate
the effect of cross-sectional shape on the form birefringence. The
simulated structure is illustrated in Figure 1a. Figure 1b and c
shows the results for the effective index of elliptical nanowires
with radii rx and ry for both polarizations Ex and Ey. The
nanowires are composed of amorphous silicon that has a bulk
refractive index of 3.95 at a wavelength of 980 nm,19 and they are
placed in a square unit cell with a length 250 nm. Further details
of the numerical simulation are included in the Supporting
Information. We restrict the parameter search to nanowire radii
that support only two modes, corresponding to the lowest order
modes that are primarily polarized in X and Y. The contour plots
in Figure 1b show that as rx of the nanowire increases, the neff for
the Ex mode increases dramatically. While, as ry of the nanowire
increases, the effective index of the Ex mode increases much
slower. If the effective index of Ex were completely uncoupled
from the nanowire radius in Y, then the contour plot would be an
array of vertical lines. Instead, the two are coupled slightly, which
results in the diagonally curved contours that are shown. The
equivalent contour plot of neff for Ey is also shown (Figure 1c)
and is a rotated version of the case of Ex.
We first choose two circular cross-section nanowires, corresponding to points along the dotted diagonal line of Figure 1b. As
discussed, the difference in phase delay of a wave traveling
through these two structures should ideally be π and will operate
equally on both Ex and Ey. This condition is satisfied when Δneff
is equal to λ/2h = 0.49. We find that nanowires with radii of 62
and 82 nm satisfy this condition and have neff values of 1.3 and
1.8, which we call nlow and nhigh, respectively. These are denoted
as points (d) and (e) in Figure 1b. The field distributions of the
Ex modes for these two nanowires are shown in Figure 1d and e.
By periodically arranging these two nanowires, it is possible to
realize a polarization-insensitive dielectric grating.2022
Next, we consider the implementation of form birefringence. In
this case, an elliptical nanowire with nlow for Ex and nhigh for Ey is
needed. The same nanowire will have nhigh for Ex and nlow for Ey,
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when rotated by 90°. From inspection of the contours of Figure 1b,
we can see that point (f) with rx and ry of 100 and 50 nm,
respectively, approximately satisfies this criteria and has a neff of 1.82
and 1.3 for Ex and Ey, respectively. The opposite criteria holds for
point (g), with neff taking values of 1.3 and 1.82 for Ex and Ey,
respectively. The field distributions of the Ex modes for the two
elliptical nanowires are shown in Figure 1f and g. Using these four
nanowire geometries, patterns can be built for which each linear
polarization encounters a completely different phase profile. With
the nanowire geometries chosen, we next encode them into
computer-generated holograms. Two separate desired phase functions for Ex and Ey are first analytically derived, based on the desired
application. Each phase function is sampled in two dimensions on a
pitch of 250 nm. We then discretize the phase for each incident
polarization into one of two values, and the appropriate nanowire
geometry for that point is chosen from the four possibilities. The
result, a pattern of nanowires on a 250 nm pitch, achieves the goal of
a two-level phase grating for one polarization, with a second,
independent, two-level phase grating for the other.
To demonstrate and characterize nanowire form birefringence, we first encode test images into arrays of elliptical
nanowires. Figure 2a shows a bright field image, obtained with
Figure 2. Polarization image multiplexing. (a) Bright field microscope
image of nanowire array taken in reflection with unpolarized light
(visible wavelengths). (b) Scanning electron micrograph of center of
nanowire array, where the two arrows overlap. All four geometries of
nanowires can be seen in the image. (c and d) Microscope image (50
objective lens), obtained in trasmission mode, of nanowire array
illuminated with an infrared LED (λ0 = 940 nm). (c) Image when illumination is polarized in Ex and (d) when illumination is polarized in Ey.
The scale bars are 10 μm in (a), (c), and (d) and 500 nm in (b).
LETTER
unpolarized light, of a nanowire array where two perpendicular
arrows have been polarization multiplexed. The array is designed
so that, when the near-field is imaged, only the arrow that is
directed along the incident polarization is visible. Figure 2b
shows the center of the nanowire array. In regions where the
arrows overlap, the nanowires are circular, while in the nonoverlapping regions, the nanowires are elliptical and birefringent.
The nanowire array is illuminated with polarized infrared light
from a near-infrared LED (wavelength λ0 = 940 nm), with the
transmitted light being imaged by an objective lens (50
magnification) onto a CCD camera (Figure 2c and d). Each
arrow is only four nanowires wide, slightly larger than the free
space wavelength of the illumination, demonstrating the high
resolution possible with nanowire form birefringence. The
images in Figure 2c and d show good contrast for the arrow
pointed along the polarization direction as well as a small amount
of contrast for the perpendicular arrow. The presence of the
perpendicular arrow is due to polarization cross-talk and is
discussed further in the following case.
In addition to images, grating patterns can be encoded to
measure the diffraction efficiency of the fabricated nanowire arrays
as a function of incident polarization. Figure 3 shows a grating
structure designed to diffract Ex vertically and Ey horizontally. The
effective period of the grating is 4 μm for both polarizations. By
imaging the far field diffraction pattern, we can quantify both the
diffraction efficiency and the polarization contrast of the nanowire
grating. In this case, the diffraction efficiency is defined as the ratio
of the power in the desired diffraction order divided by the power
in the total far field, excluding power that is absorbed and reflected.
The polarization contrast is defined to be the power in a desired
diffraction order relative to the power in an undesired diffraction
order. After accounting for the size of the incident beam relative to
the nanowire array, we measure diffraction efficiencies of 28.3 and
27.5% for the two horizontal diffraction orders in Figure 3c and
25.1 and 26.5% efficiencies for the two vertical diffraction order
shown in Figure 3d. We measure polarization contrasts of 15.3:1
and 26.8:1 for Ey and Ex, respectively. The diffraction efficiencies
are somewhat less than the ideal value of 40.5%. The reduction in
efficiency is due to different reflection coefficients23 of the
different nanowire geometries and from the difficulty in fabricating
the exact nanowires dimensions suggested by FEM simulation.
Despite the tight tolerances, the fabricated elliptical nanowire
gratings are still more than twice as efficient as amplitude gratings,
and additionally, they show high polarization contrast ratios.
Encoding lenses into the nanowire array is similar to encoding
images or simple gratings. Phase functions for each multiplexed
lens can be defined analytically based on the desired lens
properties. Two monochromatic phase functions (j1, j2) for
Figure 3. Elliptical nanowire gratings. (a) Bright field microscope image of the two-dimensional elliptical nanowire grating taken in reflection with
unpolarized light (visible wavelength). (b) Scanning electron micrograph showing one period of the grating. Repetition of this in both the horizontal and
vertical directions yields the grating. (c and d) Images of the far field of the grating when illuminated with a laser beam (λ0 = 976 nm) that is linearly
polarized in (c) Ey and (d) Ex.
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LETTER
Figure 4. Nanowire multiplexed imaging systems. (a) Microscope image (reflection mode) of nanowire lens with two polarization-encoded focal
lengths. (b and c) Images of lens obtained under x- and y-polarized illumination, respectively. (d and e) Image planes of the lens for x- and y-polarized
illumination, respectively, demonstrating magnifications of 1.12 and 0.59, respectively. (f) Microscope image (reflection mode) of stereoscopic nanowire
lens. (g and h) Images of the lens obtained under x- and y-polarized illumination, respectively. (i and j) Image planes of the lens for x- and y-polarized
illumination, respectively, exhibiting parallax.
nonparaxial lenses defined on a plane can be written as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ϕ1ðx, yÞ ¼ expðjk ðx x2 Þ2 þ ðy y2 Þ2 þ ðf1 Þ2 Þ
ð1aÞ
ϕ2ðx, yÞ ¼ expðjk
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðx x2 Þ2 þ ðy y2 Þ2 þ ðf2 Þ2 Þ
ð1bÞ
where k is the wave vector (2π/λ), xn and yn define the position
of the optical axis of the lens in the transverse plane, and fn is the
focal length.24,25 We encode these two phase functions using an
array of nanowires, each of which is one of the four elliptical
nanowires described previously.
We first investigate a nanowire lens whose focal length
depends on the incident polarization. Figure 4a shows a microscope image taken in reflection of this lens. The two encoded lens
functions have the same optical axis but have different focal
lengths for different polarizations (180 and 120 μm). The focal
lengths are chosen so that, at an object to nanowire lens distance
of 360 μm, the nanowire lens to image distances for the two
polarizations is 360 and 180 μm, respectively. The resulting
magnifications are predicted to be 1.0 and 0.5. Figure 4b and
c shows images of the lens when illuminated with x- and
y-polarized light, respectively. These images reveal the two
diffractive lenses encoded into the nanowire array via polarization. It can be seen that the lens in Figure 4c has more rings and
consequently a shorter focal length than the lens in Figure 4b.
Figure 4d and e shows the image planes of the nanowire lens. By
changing the polarization and then adjusting the position of the
relay lens (see Supporting Information Figure 1), we find the
optimal focusing conditions which result in magnifications of
1.12 and 0.59. With a single tunable focal length lens, it is not
possible to make a nonmechanical zoom system. However, as
was recently shown, two tunable focal length lenses can be
arranged in a Galilean telescope to realize this goal.26 The
nanowire imaging system demonstrated here can therefore be
considered the front end of a nonmechanical zoom system.
The second multiplexed nanowire imaging system is designed
for recording stereoscopic image pairs. Stereoscopic imaging
enables three-dimensional imaging and traditionally requires two
imaging paths collected at a small angular or spatial separation.
This is most frequently done using two spatially offset lenses but
can also be performed by spatially dividing the aperture of a
single lens. Our multiplexed imaging method enables the capture
of two stereoscopic images from the same aperture of a single
lens without any subdivision. Figure 4f shows the stereoscopic
lens, where the focal length is 150 μm for both polarizations, but
the optical axis of each lens is offset from the center by 3 μm
vertically. Figure 4gj shows images of the stereoscopic lens and
its image plane for both illumination polarizations. The images
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show parallax, which is a function of the optical axis offset
between the two multiplexed nanowire lenses and the nanowire
lens to object distance. Using the measured parallax and the
known lens properties, we calculate the distance from the lens to
the object to be 430 μm (see Supporting Information). This
compares reasonably well to the distance of 400 μm determined
from the image size and the known object size. The parallax
method for distance determination, however, does not require
the object size to be known. Unlike other microscopes demonstrated recently that use parallax,27,28 this method requires no
registration between the image pairs, and the only shift observed
on the image sensor is from the parallax. In addition, the quality
of the image is not degraded by subdividing the back aperture.
In this paper, we have demonstrated a method to encode
phase patterns into arrays of elliptical nanowires. Using the
differential effective mode index of elliptical nanowires, we have
shown that two phase patterns can be polarization encoded into a
single array of elliptical cross-section nanowires. While we have
demonstrated only two level phase gratings, it is possible to
obtain higher diffraction efficiency with additional phase levels.
A three level phase grating having a possible diffraction efficiency
of 68% could be made in the same way. It would require, however,
nine different nanowire geometries instead of four.
This method of polarization multiplexing is particularly useful
in microoptical imaging systems, where it is traditionally difficult
to dynamically adjust imaging properties. We have demonstrated
two imaging systems that benefit from polarization multiplexing,
the front half of a nonmechanical zoom and a stereoscopic lens.
Further advances in nanofabrication techniques and phase encoding methods will continue to expand the capabilities of microoptical imaging systems.
’ ASSOCIATED CONTENT
bS
Supporting Information. Additional information and
figures. This material is available free of charge via the Internet
at http://pubs.acs.org.
LETTER
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’ AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected] Telephone: 617-497-4704.
’ ACKNOWLEDGMENT
The study was supported by the Rowland Junior Fellows
program. Fabrication work was performed at the Harvard Center
for Nanoscale Systems (CNS), which is supported by the
National Science Foundation. K.S. acknowledges support from
Zena Technologies.
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