# Phase Front Design with Metallic Pillar Arrays

**Abstract**

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: shanhui@stanford.edu

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

photonic components.

© 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

[8]. A limitation of phase control with slit arrays,

however, is that slits provide control in one dimen-

sion only. Moreover, these structures are essentially

polarizing devices.

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 [11]. 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 [12]. 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-

spective.

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

m

=−10.78+0.79i [13]. 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 [9]. The array geometry

also eliminates any effects that might occur owing to

the small aperture size of individual microlenses

[8,14].

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

=

R

+i

I

of

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 [15]. 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

R

d,

where the distance d corresponds to the thickness of

the nanopatterned metallic ﬁlm, i.e., the height of the

pillars. The complex propagation constant

depends

strongly on the cross-sectional size of the pillar [Fig.

2(a)]. The real part of the effective index is deﬁned as

n

R

=

R

/k

0

, with k

0

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

R

Ⰷn

I

共=

I

/k

0

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:

共x,y兲 =

2

f

−

2

冑

f

2

+ x

2

+ y

2

, 共1兲

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

R

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

y

兩

2

in

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

E

y

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 [18]. 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% [19], 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-

havior.

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 [9], 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 [22].

The authors acknowledge N. Engheta for suggest-

ing the study of this structure. This research was

supported by the MARCO Interconnect Focus Center.

References and Notes

1. N. Engheta, Science 317, 1698 (2007).

2. V. M. Shalaev, Nat. Photonics 1, 41 (2007).

3. S. Lal, S. Link, and N. J. Halas, Nat. Photonics 1, 641

(2007).

4. Z. Sun and H. K. Kim, Appl. Phys. Lett. 85, 642 (2004).

5. H. Shi, C. Wang, C. Du, X. Luo, X. Dong, and H. Gao,

Opt. Express 13, 6815 (2005).

6. T. Xu, C. Wang, C. Du, and X. Luo, Opt. Express 16,

4753 (2008).

7. C. Min, P. Wang, X. Jiao, Y. Deng, and H. Ming, Opt.

Express 15, 9541 (2007).

8. L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. Bar-

nard, M. L. Brongersma, and S. Fan, Nano Lett. 9, 235

(2009).

9. L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Appl.

Phys. Lett. 95, 071112 (2009).

10. Z. Sun, Appl. Phys. Lett. 89, 261119 (2006).

11. S. Yin, C. Zhou, X. Luo, and C. Du, Opt. Express 16,

2578 (2008).

12. Y. Fu, W. Zhou, L. E. N. Lim, C. L. Du, and X. G. Luo,

Appl. Phys. Lett. 91, 061124 (2007).

13. D. R. Lide, ed., CRC Handbook of Chemistry and Phys-

ics, 88th ed. (CRC, 2007).

14. P. Rufﬁeux, T. Scharf, H. P. Herzig, R. Völkel, and K. J.

Weible, Opt. Express 14, 4687 (2006).

15. G. Veronis and S. Fan, Opt. Lett. 30, 3359 (2005).

16. G. Veronis and S. Fan, in Surface Plasmon Nanophoto-

nics, M. L. Brongersma and P. G. Kik, eds. (Springer,

2007), p. 169.

17. A detailed description of the three-dimensional FDFD

method will be published elsewhere.

18. B. Richards and E. Wolf, Proc. R. Soc. London Ser. A

253, 358 (1959).

19. C. C. Fesenmaier, Y. Huo, and P. B. Catrysse, Opt. Ex-

press 16, 20457 (2008).

20. R. Yan, D. Gargas, and P. Yang, Nat. Photonics 3, 569

(2009).

21. A. V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren,

G. A. Wurtz, R. Atkinson, R. Pollard, V. A. Podolskiy,

and A. V. Zayats, Nat. Mater. 8, 867 (2009).

22. L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Phys.

Rev. Lett. 103, 033902 (2009).

Fig. 4. (Color online) Two-dimensional focusing in a micro-

lens application. (a) Electric ﬁeld intensity 兩E

y

兩

2

for an xy

cross section through the center of the lens structure (from

Fig. 1). Real part of the E

y

ﬁeld for an (b) xy and (c) xz cross

section near the center of the lens structure. (d)–(f) Electric

ﬁeld intensity 兩E

y

兩

2

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

- CitationsCitations26
- ReferencesReferences34

- "It is reported for dielectric gratings [27], and acoustic waves [28,29] . It should be noted that nonresonant EOT has miscellaneous applications, e.g., in broadband absorber [19,21], energy concentration [19,26] , selective emit- ters [21], directive antennas [30], microlens applications [30,31], energy harvesting, and solar cells [32]. Although many accurate analytical models have been proposed for 1D metallic slit arrays [11,33343536, no accurate enough analytical model has ever been proposed to obtain the Brewster angle in 2D metallic pillar arrays. "

[Show abstract] [Hide abstract]**ABSTRACT:**The physics behind the broadband Brewster transmission through square arrays made of rectangular metallic pillars is explored by appealing to the effective medium theory. First, an analytical solution is given for the principal electromagnetic mode propagating between the pillars, and then, the mode matching technique is invoked to extract the parameters of the effective medium model. In this fashion, the pillars are homogenized via a diagonal anisotropic tensor and the effects of higher diffracted orders are included in the effective medium theory by attributing a surface conductivity to the surface boundary of the array. It is shown that the former effectuates the wideband Brewster effect while the latter causes the narrowband spoof surface plasmon resonances. The accuracy of the proposed model is verified by full-wave numerical simulations.- "Phase modulation has been adopted in micro-lenses, where spatial phase variation is realized by controlling the effective refractive index through subwavelength structures. Metallic pillars [5], waveguides [6], [7], and slits [8]–[10] have been reported for phase control by varying their geometrical size, and phase delay is proportional to the thickness of these structures. However, large phase delay is a great challenge in phase-based micro-lens for subwavelength focusing, especially for short wavelength, due to the difficulty in fabricating sub-wavelength structure with high depth-to-width ratio. "

[Show abstract] [Hide abstract]**ABSTRACT:**A super-oscillation far-field focusing micro-lens based on continuous amplitude modulation is experimentally demonstrated with 40-nm thick width-varied sub-wavelength metallic slit array. The $228times 200~mu text{m}^{mathrm {mathbf {2}}}$ micro-lens is designed and fabricated with numerical aperture 0.976 and focal length $40.1lambda $ for wavelength $lambda =632.8$ nm. Experimental results show that the focal length is $sim 26.5pm 1~mu text{m}$ , and the focal line full width at half maximum is $0.379lambda $ , which is smaller than the corresponding diffraction limit $0.512lambda $ and the super-oscillation criterion $0.389lambda $ . A great suppression of sidelobes was observed in the measured focal plane area, and the largest sidelobe intensity was found only 10.6% of the central lobe intensity, leading to a wide field of view.- "Due to this, a lot of current research studies the possibility of achieving a plasmonic lens based on arrays of nanoapertures in metallic films123. Different structures with different configurations and under various polarized illuminations have been simulated and experimented with toward the goal of introducing a phase delay distribution by a modulation of the geometrical parameters (shape, size, positions, and distance between the nanoobjects)4567891011. However, such systems induce a focusing process in the micrometer range, but they have a low transmittance due to the subwavelength width of their diffracting objects. "

[Show abstract] [Hide abstract]**ABSTRACT:**We report on the optimization of ultrasmall microlenses based on the diffraction of two parallel metallic nanowires. The Rayleigh-Sommerfeld integral is used in the visible range to simulate the near field diffraction patterns induced by single and pairs of planar silver wires. We demonstrate that the wire width w affects only the diffraction efficiency and the contrast of the diffraction pattern. The wire inter-distance D control the focal length and the depth of focus, that are equal and vary in the 0.1 to 10 µm range when D/λ increases from 1 to 8. The transversal full width half maximum FWHM increases from 200 to 700 nm and a normalized intensity greater than 2.2 is obtained at the focal point when w is about 300 nm and D/λ = 3. There is an excellent agreement between these calculated properties, and experimental results obtained for single and pairs of parallel silver nanowires. We show that in our micro-sized geometry, the plasmon contribution is negligible with respect to pure diffraction effect. In addition, these nanowire microlenses have focusing properties similar than those of ideal refractive lenses limited by diffraction.

Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.

This publication is from a journal that may support self archiving.

Learn more