Phase Front Design with Metallic Pillar Arrays
We demonstrate numerically, using a three-dimensional finite-difference frequency-domain method, the ability to design a phase front using an array of metallic pillars. We show that in such structures, the local phase delay upon transmission can be tuned by local geometry. We apply this knowledge to demonstrate a metallic microlens. The presented design principles apply to a wider range of wavelength-size integrated photonic components.
Phase front design with metallic pillar arrays
Lieven Verslegers, Peter B. Catrysse, Zongfu Yu, Wonseok Shin, Zhichao Ruan, and Shanhui Fan
E. L. Ginzton Laboratory and Department of Electrical Engineering,
Stanford University, Stanford, California 94305, USA
Corresponding author: email@example.com
Received November 18, 2009; revised January 14, 2010; accepted February 3, 2010;
posted February 16, 2010 (Doc. ID 120122); published March 12, 2010
We demonstrate numerically, using a three-dimensional ﬁnite-difference frequency-domain method, the
ability to design a phase front using an array of metallic pillars. We show that in such structures, the local
phase delay upon transmission can be tuned by local geometry. We apply this knowledge to demonstrate a
metallic microlens. The presented design principles apply to a wider range of wavelength-size integrated
© 2010 Optical Society of America
OCIS codes: 310.6628, 250.5403, 230.0230, 310.6805.
A variety of compact photonic devices based on the
manipulation of light by nanostructured metals has
been suggested and demonstrated recently [1–3]. Slit
arrays in an optically thick metallic ﬁlm, for ex-
ample, were proposed as a means to achieve phase
front control for the focusing and redirecting of light
[4–10]. The use of nanostructured metal ﬁlms to con-
trol the phase front provides several advantages over
conventional shaped dielectrics, including a larger in-
dex contrast, greater design freedom, and a planar
geometry that facilitates integration while being
compatible with existing manufacturing techniques
. A limitation of phase control with slit arrays,
however, is that slits provide control in one dimen-
sion only. Moreover, these structures are essentially
Three-dimensional variations on the slit array
structure have been suggested to address such limi-
tations, but the published designs also forego some of
the advantages of slit arrays. Arrays of square holes
with different sizes in a metallic ﬁlm have been
shown to act as lenses with focusing capabilities in
two dimensions . Such structures, however, rely
on the presence of a propagating mode in each hole,
which dictates the size of the hole to be at least half a
wavelength, and, hence, prevents scaling of the en-
tire device to sizes on the order of a wavelength. In a
different approach, a microzone plate, based on the
principle of diffraction, was shown to focus an inci-
dent plane wave into a subwavelength spot . This
microzone plate, however, is strongly polarization de-
pendent, fairly large, and has a circular shape that is
undesirable in the context of periodic arrays of such
elements on a square lattice, e.g., square lenses are
preferable from a light collection and real-estate per-
In this Letter, we numerically demonstrate the
ability to perform phase front design in two dimen-
sions by varying the local geometry in metallic nano-
scale pillar arrays. Our approach maintains the ad-
vantages of nanoscale slit arrays while overcoming
its limitations. The proposed structure is planar, can
be made polarization independent, and is well suited
to miniaturization because its building blocks can be
made much smaller than a wavelength. We demon-
strate the design freedom that is offered by metallic
pillar arrays through the simulation of a wavelength-
size metallic microlens. The design principles are
more general though and could be applied to a larger
range of compact photonic devices for the beneﬁt of
controlling the incoupling and outcoupling of light.
Figure 1 shows an example structure that is based
on a two-dimensional array of square gold pillars
with varying base size depending on location within
the array. The pillars are 400 nm tall and have base
sizes ranging from 25 to 75 nm. They are centered in
100 nm by 100 nm unit cells that are organized on an
11 by 11 grid. The structure operates on light inci-
dent along the z direction. The calculations and simu-
lations are performed for a y-polarized plane wave
with a wavelength of 632.8 nm. The permittivity of
gold at this wavelength is
=−10.78+0.79i . For
simulation purposes, we opt for this structure to re-
peat itself in the x and y directions. The simulation
results then correspond to the case of a periodic array
of microlenses. Microlens arrays are very common in
practical applications, where microlenses are often
placed on top of a photodetector array to improve the
individual detector efﬁciency . The array geometry
also eliminates any effects that might occur owing to
the small aperture size of individual microlenses
The purpose of the metallic pillar array is to create
a position-dependent phase delay as light propagates
Fig. 1. (Color online) Metallic nanoscale pillar array for
two-dimensional phase front design. The entire structure is
on the order of the wavelength size and consists of an array
of 11 by 11 unit cells with 400 nm tall gold pillars of vari-
able base width (between 25 and 75 nm) in unit cells of 100
nm by 100 nm.
844 OPTICS LETTERS / Vol. 35, No. 6 / March 15, 2010
0146-9592/10/060844-3/$15.00 © 2010 Optical Society of America
through the structure. To calculate the phase delay
associated with light propagation through a local re-
gion consisting of a single nanoscale pillar, we calcu-
late the complex propagation constant
the symmetric eigenmode of a periodic array of nano-
scale pillars of inﬁnite length using a full-vector
ﬁnite-difference frequency-domain (FDFD) mode
solver . The symmetric eigenmode is relevant be-
cause this is the mode that a plane wave incident
along the normal direction couples to. The phase de-
lay through the local region is then estimated as
where the distance d corresponds to the thickness of
the nanopatterned metallic ﬁlm, i.e., the height of the
pillars. The complex propagation constant
strongly on the cross-sectional size of the pillar [Fig.
2(a)]. The real part of the effective index is deﬁned as
, with k
the free space propagation con-
stant, and ranges from one for vacuum to more than
three for s =0.9a =90 nm, where a = 100 nm is the size
of the unit cell. In all cases considered, n
兲, and material losses do not obscure the
phase delay effect over the thickness of the structure.
Figure 2(b) shows a vector plot of the electric ﬁeld
for the symmetric eigenmode in the array that corre-
sponds to a pillar with s = 65 nm. The electric ﬁeld
concentrates in the gap between the metal pillars.
For a y-polarized (x-polarized) plane wave, the corre-
sponding eigenmodes have their electric ﬁeld concen-
trate at the top/bottom (left/right) sides of the unit
cell. These two modes are degenerate, having the
same effective index.
Based on the properties of the eigenmodes of an in-
dividual element as outlined above, we now design a
wavelength-size microlens based on an array of such
elements. To achieve lensing, for an incident plane-
wave front, the metallic pillar array needs to gener-
ate an outgoing phase front that is curved. The re-
quired phase delay ideally has the following form:
given as a function of position x and y, as measured
from the center of the lens, where is the wave-
length, and f is the geometrical focal length. This
analytical phase delay is shown, up to an arbitrary
constant, in Fig. 3(a) for f=0.5
m. It can be approxi-
mated by picking the correct pillar size, which results
in a local phase delay of
d, for each unit cell. This
approximate phase delay, corresponding to the struc-
ture from Fig. 1, is shown in Fig. 3(b).
The microlens design is validated with full three-
dimensional FDFD simulations of the entire struc-
ture [16,17]. This method allows us to model materi-
als using the measured, tabulated permittivity for
every wavelength, thus directly taking into account
both exact material dispersion as well as loss. For
this simulation, we set the grid size to 5 nm in the
transverse x and y directions and 20 nm in the longi-
tudinal z direction, resulting in a manageable simu-
lation domain size. This allows us to ﬁnd the phase
past the exit surface [Fig. 3(c)]. This phase is in rea-
sonably good agreement with the design value.
Figure 4(a) shows the electric ﬁeld intensity 兩E
an xy cross section through the center of the lens.
The ﬁeld distribution around each individual pillar
element in the array is quite similar to the symmet-
ric eigenmode calculated above for a pillar of the
same size, conﬁrming the unit cell approximation we
made. Figures 4(b) and 4(c) show the real part of the
ﬁeld component for yz and xz cross sections near
the center of the structure and allow us to observe
the curvature introduced by the structure. An xy
cross section of the clearly deﬁned focal spot is shown
in Fig. 4(d). Figures 4(e) and 4(f) conﬁrm that the fo-
cusing behavior is very similar for yz and xz cross
sections. The actual focal length turns out to be
somewhat larger than the analytical value, because
the phase delay introduced is somewhat smaller than
for the analytical case. This small deviation can be
attributed to the assumption of a single pass phase
delay and a symmetric eigenmode.
The focal spot has a near-circular shape [Fig. 4(d)].
It can be seen, however, that the intensity distribu-
tion in the focal spot does not exhibit exact 90° rota-
tion symmetry. This is expected: for linearly polar-
ized incident light, with the use of a lens with a large
numerical aperture, there is in fact no reason a priori
that the focal spot has to be circular . The struc-
ture, however, is inherently polarization indepen-
dent: the two polarizations behave in exactly the
same way by rotational symmetry.
The ﬂux directly underneath the lens divided by
the incident ﬂux of a reference measurement is 77%.
As a point of comparison, a wavelength-size solid-
state image sensor pixel with a dielectric shaped
Fig. 2. (Color online) Basic building block for two-
dimensional phase front design. (a) Real and imaginary
parts of the effective index as a function of the base width s
of the metal pillar. The inset shows the metal pillar, cen-
tered in the 100 nm by 100 nm unit cell. (b) Vector plot of
the electric ﬁeld in a unit cell with s=65 nm.
Fig. 3. (Color online) Phase front design for wavelength-
size metallic microlens. (a) Phase delay calculated analyti-
cally from Eq. (1). (b) Phase delay approximated by an ar-
ray of unit cells. (c) Phase delay measured underneath the
lens in a three-dimensional FDFD simulation.
March 15, 2010 / Vol. 35, No. 6 / OPTICS LETTERS 845
microlens has an optical efﬁciency of 67% , show-
ing that the losses of the metallic microlens design
are quite reasonable. Reﬂection of the structure is
calculated at 9% and can likely be improved by opti-
mization. Propagation losses inside the metallic
structure are estimated at about 14%. Since the
structure consists of 37% metal, funneling of light oc-
curs in the gaps and transmission is slightly extraor-
dinary. The intensity is locally modulated as well.
This can be expected from a structure that is on reso-
nance at certain locations, and off resonance in other
places, but it is not detrimental to the structure’s be-
Fabricating these structures using top-down ap-
proaches, including lithography or focused ion beam
milling, might prove challenging owing to the high
aspect ratios that are required. Fabrication require-
ments could be eased by increasing structure size
and focal length. A bottom-up approach by growing
the pillars could provide an alternative implementa-
tion method. In fact, the nanoscale pillar dimensions
correspond well with realistic nanowire diameters
[20,21]. In such a setup, local phase delay could be
tuned by either pillar size or pillar density.
In conclusion, the presented design principles al-
low a nanophotonic component designer to tailor
phase delay in wavelength-size devices by varying lo-
cal geometry. Whereas we focused on a lens struc-
ture, the principles should be extendible to more com-
plex designs, for instance to compensate for angles
of incidence , or to intentionally introduce polar-
ization dependence by using rectangular base instead
of square base pillars. Similar structures, exten-
ded in the z direction, should also enable deep-
subwavelength focusing within the structure, as we
demonstrated before in a slit array .
The authors acknowledge N. Engheta for suggest-
ing the study of this structure. This research was
supported by the MARCO Interconnect Focus Center.
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Fig. 4. (Color online) Two-dimensional focusing in a micro-
lens application. (a) Electric ﬁeld intensity 兩E
for an xy
cross section through the center of the lens structure (from
Fig. 1). Real part of the E
ﬁeld for an (b) xy and (c) xz cross
section near the center of the lens structure. (d)–(f) Electric
ﬁeld intensity 兩E
for (d) xy, (e) yz, and (f) xz cross sections
of the focal spot. Where necessary, the ﬁeld intensities in-
side the gaps are saturated.
846 OPTICS LETTERS / Vol. 35, No. 6 / March 15, 2010