of 10
Subwavelength-thick Lenses with High Numerical Apertures and Large Efficiency Based on High
Contrast Transmitarrays
Amir Arbabi,
1
Yu Horie,
1
Alexander J. Ball,
1
Mahmood Bagheri,
2
and Andrei Faraon
1,
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, 1200 E California Blvd., Pasadena, CA 91125, USA
2
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
We report subwavelength-thick, polarization insensitive micro-lenses operating at telecom wavelength with
focal spots as small as 0.57 wavelengths and measured focusing efficiency up to 82%. The lens design is based
on high contrast transmitarrays that enable control of optical phase fronts with subwavelength spatial resolution.
A rigorous method for ultra-thin lens design, and the trade-off between high efficiency and small spot size (or
large numerical aperture) are discussed. The transmitarrays, composed of silicon nano-posts on glass, could be
fabricated by high-throughput photo or nanoimprint lithography, thus enabling widespread adoption.
Flat optical devices thinner than a wavelength promise to
replace conventional free-space components for wavefront
and polarization control [1, 2]. Transmissive flat lenses are
particularly interesting for applications in imaging and on-
chip optoelectronic integration. Several designs based on
metasurfaces [3–6], high contrast transmitarrays (HCTA) and
gratings [7], have been recently implemented but did not pro-
vide a performance comparable to conventional curved lenses.
Here, for the first time, we report polarization insensitive,
micron-thick, HCTA micro-lenses with high numerical aper-
tures and large efficiency.
Flat lenses are most commonly realized with Fresnel struc-
tures. However, geometrical complexity of Fresnel lenses and
the low efficiency of Fresnel zone plates make them not well
suited for integration with wafer-scale processing. Effective
medium structures have been proposed as an alternative [8–
12], but their deep subwavelength structures and high aspect
ratios make their fabrication challenging. Recently, diffrac-
tive elements based on plasmonic metasurfaces composed of
2D arrays of ultrathin scatterers have attracted significant at-
tention [1–5, 13–15], but their efficiency is limited to 25% by
fundamental limitations [16] and also suffer from material ab-
sorption [4].
A novel approach is to use high contrast gratings (HCGs),
fabricated from semiconductors or high index dielectrics [6,
17–19], that can be designed with large reflection [20] or
transmission [21] efficiencies. Wavefront control was origi-
nally achieved by rendering one dimensional gratings aperi-
odic by gradually modifying the local period and duty cycle
of the grating [21–23]. Reflecting focusing mirrors were real-
ized using this approach [21, 23, 24].
A promising class of aperiodic HCGs can be realized by
positioning high index dielectric scatterers on a periodic sub-
wavelength 2D lattice, as shown schematically in Fig. 1a for a
hexagonal lattice of circular posts. Due to subwavelength di-
mensions, the structure acts as a zero order grating completely
described by its transmission and reflection coefficients. The
high index results in negligible interaction among scatterers,
so the light scattered at each lattice site is dominated by the
scatterer proprieties rather than by the collective behavior of
multiple coupled scatterers in the lattice. We refer to high con-
trast zero order gratings composed of disconnected scatterers,
which operate in this local scattering regime, as high contrast
arrays. The term high contrast transmitarrays (HCTAs) is used
when they are designed for large transmission. Low numeri-
cal aperture (NA) lenses based on HCTA have been recently
reported [7, 25].
Here, we explain the HCTA concept and discuss its unique
features resulting from the localized scattering phenomena,
which enable implementation of diffractive elements with
rapidly varying phase profiles. To demonstrate the HCTA ver-
satility, we present design, simulation, fabrication and char-
acterization results of polarization insensitive high NA micro-
lenses with high focusing efficacy. Large NA lenses are re-
quired in microscopy, high-density data recording, focal plane
arrays, and coupling between on-chip photonic components
and free-space beams. Current techniques to fabricate on-chip
devices that impose rapid phase variation require gray-scale
lithography [26, 27], a process that is difficult to control. On
the other hand, the HCTA devices provide a more reliable al-
ternative with fabrication techniques that lend themselves to
wafer-scale processing.
Figure 1b shows the simulated transmission and phase of
the transmission coefficient for two gratings composed of cir-
cular amorphous silicon posts arranged in hexagonal and rect-
angular lattices, as functions of lattice constant and post diam-
eter (See Fig. 1 legend for dimensions). The gratings are de-
signed to operate at
λ
=1550 nm, but the concept is scalable to
any wavelength. In each case, the amplitude and phase depend
primarily on the post diameter, which is an indication that, in
this parameter regime, the scattering is a local effect and is
not affected significantly by coupling among scatterers. This
is also confirmed by the almost identical transmission proper-
ties of the hexagonal and square lattice. Examining the near
field distribution of the grating reveals the underlying physi-
cal mechanism behind the local scattering effect. As shown
in Fig. 1c, light is concentrated inside the posts that behave as
weakly coupled low quality factor resonators. This behavior is
fundamentally different from the low contrast gratings operat-
ing in the effective medium regime whose diffractive charac-
teristics are mainly determined by the duty cycle and the fill-
ing factor. We also note that the structures created by chang-
ing the bar width and period of 1D HCGs [21, 23, 24] are not
considered HCTA since there is a strong coupling along the
arXiv:1410.8261v1 [physics.optics] 30 Oct 2014
2
b
d
c
200
300
400
500
0
0.2
0.4
0.6
0.8
1
Post diameter
(
nm
)
Transmission
Phase/(2
)
10
20
30
10
20
30
D=500 nm
1
m
1
m
1
2
3
4
5
1
2
3
D=200 nm
1
m
1
m
Post diameter (nm)
Lattice constant (nm)
200
300
400
500
600
650
700
750
800
850
900
0.2
0.4
0.6
0.8
Post diameter (nm)
Lattice constant (nm)
200
300
400
500
600
650
700
750
800
850
900
0
0.5
1
1.5
Phase /(
2
)
Transmission (%
)
Post diameter (nm)
Lattice constant (nm)
200
300
400
500
600
650
700
750
800
850
900
0.2
0.4
0.6
0.8
Transmission (%
)
Post diameter (nm)
Lattice constant (nm)
200
300
400
500
600
650
700
750
800
850
900
0
0.2
0.4
0.6
0.8
1
1.2
Phase /(
2
)
20
40
60
80
100
20
40
60
80
D=350 nm
1
m
1
m
r
t
a
FIG. 1.
Simulation results of periodic HCTAs
.
a
A schematic representation of a periodic HCTA composed of high index posts in a
hexagonal lattice (side view on left, top view on right) with transmission coefficient
t
and reflection coefficient
r
. The posts rest on a low
index substrate.
b
, Simulated transmission and phase of the transmission coefficient for hexagonal and square periodic HCTAs composed of
circular amorphous silicon posts on a fused silica substrate as a function of the lattice constant and the post diameter. The insets show the
corresponding lattices.
c
, Top view (on the left) and side view (on the right) of the color coded magnetic energy density in a periodic HCTA
for different post diameters
D
. The dashed black lines depict the boundary of the silicon posts. A plane wave with magnetic energy of 1 is
normally incident on the silicon posts from the top.
d
, Simulated transmission and phase of the transmission coefficient for a family of periodic
hexagonal HCTAs with lattice constant of 800 nm, and varying post diameters. The shaded part of the graph is excluded when using this graph
to map transmission phase to post diameter. In all these simulations, the posts are made of amorphous silicon (
n
= 3
.
43
), are 940 nm tall, and
the wavelength is
λ
=1550 nm.
bar direction and a rapid variation of the local transmission or
reflection properties of the structure is not achievable along
that direction.
To design an HCTA that implements a transmissive phase
mask, we find a family of periodic HCTAs with the same lat-
tice but with different scatterers that provide large transmis-
sion amplitude while the phase spans the entire 0 to
2
π
range.
Such a family is shown in Figure 1d where the post diame-
ter is varied from 200 nm to 550 nm in a hexagonal lattice
with 800 nm period while transmission is greater than 92%.
To create the phase mask we start from an empty lattice and
at each lattice site we place a scatterer from the periodic HC-
TAs that most closely imparts the desired phase change onto
the transmitted light. Any arbitrary transmissive phase masks
can be realized using this method. An example phase mask
for a transmissive micro-lens is shown in Fig. 3a. To mini-
mize scattering from aperiodic HCTAs into non-zero orders,
a gradual change in the scatterer size is preferable.
The unprecedented possibility to realize any transmissive
masks using HCTAs enables the implementation of micro-
lenses with exotic phase profiles, optimized for specific tasks,
such as large NA lenses. To design these components, the
conventional ray tracing technique is not applicable. A gen-
eral rigorous technique for determining the optimum trans-
missive mask to shape an incident optical wavefront to a de-
sired form is given in Supplementary Information, S.1. Using
this technique and the HCTAs in Fig. 1d, we found the opti-
mum phase mask for micro-lenses that focus
λ
=1550 nm light
3
b
20
40
60
80
100
120
0.
5
1.
0
1.
5
2.
0
2.
5
FWHM spot size
Focusing effici
enc
y
Tr
ansmission
d(
μ
m)
FWHM spot
size
(
wavel
engt
h
)
50
60
70
80
90
100
(%
)
Excitation
30
m
30
m
a
d
c
30
20
10
0
40
dB
20
m
2
m
0
1
Substrate
Lens
Fiber
d
0
1
1
FIG. 2.
Simulation results of high NA HCTA micro-lenses
.
a
, Illustration of high NA focusing of the light from a cleaved optical fiber using
an HCTA micro-lens.
b
, Electric field distribution at the
xz
cross section, in the excitation plane (inset), and immediately after passing through
the micro-lens (inset).
c
, Logarithmic scale electric energy density in the
xz
cross section. the inset shows the real part of the z component of
the Poynting vector at the plane of focus.
d
, Simulated plane of focus FWHM spot size, transmission, and focusing efficiency of the high NA
HCTA micro-lenses for devices with varying focusing distances. The simulated points are shown by the symbols and the solid lines are eye
guides. All the devices simulated in this figure are a factor of four smaller than the devices fabricated and measured in Fig. 3 and Fig. 4
.
from a single mode fiber to the smallest spot. We designed a
set of 400
μ
m diameter high NA micro-lenses that focus the
light from a single mode fiber located 600
μ
m away from the
backside of the substrate (500
μ
m thick fused silica) to points
located at distances ranging from
d
=50
μ
m to
d
=500
μ
m away
from the micro-lenses (Fig. 2a). We refer to
d
as the focusing
distance.
The lens performance was evaluated by full 3D finite dif-
ference time domain (FDTD) simulations [28]. To reduce the
simulation size, micro-lenses with the same NA but with a
factor of four smaller dimensions (100
μ
m diameter, 150
μ
m
spacing between the illuminating fiber and the back side of
the 125
μ
m thick substrate) were simulated. Figures 2b and
c show the results for a micro-lens that focuses at
d
=25
μ
m
away from the lens. The full width at half maximum (FWHM)
of the focal spot is 1.06
μ
m or 0.68
λ
(
λ
=1550 nm). The sim-
ulation indicates that 85% of the incident light is transmitted
by the lens and 72% is concentrated in the focus (See Methods
for details).
Simulated values of the transmission, focusing efficiency
and FWHM spot size for several micro-lenses with
d
ranging
from 12.5
μ
m to 125
μ
m are presented in Fig. 2d. Higher
NAs and smaller spot sizes correlate with decreased transmis-
sion and lower focusing efficiencies. This is due to the rapidly
varying phase profile close to the circumference of high NA
lenses, which is under-sampled by the HCTA lattice (shrink-
ing the lattice constant of the HCTA reduces this trade-off,
see Supplementary Information S.2). The micro-lens with
d
=12.5
μ
m has a simulated FWHM spot size of 0.51
λ
which is
close to the smallest possible diffraction limited value of 0.5
λ
.
The relatively high efficiency (more than 50%) and the diffrac-
tion limited focusing of this micro-lens confirms the validity
of our technique for determining the optimum phase profile,
and demonstrates an example of the high performance that can
be achieved by HCTA flat diffractive elements.
The high NA lenses were fabricated in a hydrogenated
amorphous silicon film deposited on a 500
μ
m thick fused
silica substrate as described in the Methods section. Images
of the fabricated devices are shown in Fig. 3a-c. The char-
acterization was performed in a setup (Fig. 4a) consisting of
a custom built microscope that images the plane of focus of
the micro-lens (See Methods). The micro-lenses were illumi-
nated with 1550 nm light emitted from a cleaved single mode
fiber positioned 600
μ
m away from the substrate backside.
The normalized measured intensity profile at the plane of fo-
cus for a micro-lens with the focusing distance of
d
=50
μ
m is
shown in Fig. 4b. The intensity profiles for micro-lenses with
different focusing distance are plotted in Fig. 4c.
Figure 4d shows the measured FWHM spot size, transmis-
sion, and focusing efficiency for devices with different fo-
cusing distances. The micro-lens designed for
d
=50
μ
m fo-
cuses light to a 0.57
λ
FWHM spot size, and the micro-lens
4
a
1
μ
m
100
μ
m
b
c
1
μ
m
d
FIG. 3.
Schematic illustration and images of fabricated HCTA lenses
.
a
, Schematic of the aperiodic HCTA used in the high NA micro-lens.
b
, Optical microscope image of a fabricated HCTA lens with large NA.
c,d
, Scanning electron microscope images of the silicon posts forming
the HCTA micro-lens.
designed for
d
=500
μ
m shows more than 82% focusing effi-
ciency. These results agree well with the simulation results
presented in Fig. 2d, although the measured focusing efficien-
cies are 10% smaller (3% is attributed to reflection from sub-
strate backside interface and 7% to scattering by the random
roughness of the etched silicon posts). As expected from sim-
ulations (Fig. 2d), the measured focusing efficiency decreases
as the NA increases.
The wavelength dependence of the FWHM spot size and
focusing efficiency of a micro-lens with
d
=175
μ
m are pre-
sented in Fig. 4e. The FWHM spot size increases slightly
at shorter wavelength, and the focusing efficiency reduces by
5 % at 50 nm away from the design wavelength. Also, by
changing the laser wavelength from 1550 nm to 1450 nm, the
focusing distance changed from
175
μ
m to
195
μ
m.
In conclusion, the HCTAs enable shaping of the wavefront
of light at will, efficiently, and with sub-wavelength resolu-
tion. The exceptional freedom provided in the implementa-
tion of any desired masks allows for achieving the best per-
formance for any particular functionality. The HCTA micro-
lenses with focusing efficiencies up to 82%, and FWHM spot
sizes down to 0.57
λ
, to the best of our knowledge, represent
the best performance among any types of flat high NA micro-
lens experimentally reported so far. Combined with their pla-
nar form factor, these structures will enable on-chip optical
systems created by cascading multiple diffractive elements.
One recent demonstration is a planar retroreflector integrat-
ing a HCTA lens and a reflectarray focusing mirror [29]. We
envision near future application of HCTA based devices in re-
alization of more complex optical systems with new function-
alities.
5
Camer
a
Photodetector
Tube len
s
Objective lens
HCTA microlens
Cleaved fiber
Flip mirror
Iris
a
b
c
d
0
100
200
300
400
500
0.5
1.0
1.5
2.0
2.5
FWHM spot size
Focusing effici
enc
y
Transmission
d(
μ
m)
FWHM spot size
(wavelengt
h)
40
50
60
70
80
90
(%
)
d
1440
1480
1520
1560
1600
0.9
1.1
1.2
1.3
Wavele
n
g
th (nm)
FWHM spot size
(w avelength
)
60
64
68
72
76
80
Focusing efficiency (%
)
e
-4
-2
024
0.0
0.5
1.0
No
rmaliz
ed intensity (a.u.)
x(
μ
m)
d
500
μ
m
250
μ
m
175
μ
m
100
μ
m
50
μ
m
x
0
1
x
y
1
μ
m
0.5
FIG. 4.
Measurement results of high NA HCTA micro-lenses
.
a
, Measurement setup for imaging the focal spot size and measuring the
efficiency of the high NA HCTA micro-lenses. The flip mirror was inserted into the setup only during efficiency measurements.
b
, Measured
2D intensity profile at the plane of focus for a micro-lens with
d
=50
μ
m.
c
, Normalized measured intensity profiles of high NA micro-lenses
with different focal lengths at their plane of focus.
d
, Measured plane of focus FWHM spot size, transmission, and focusing efficiency of the
HCTA micro-lenses as a function of their focusing distance. The measurement data are represented by the symbols and the solid lines are eye
guides.
e
, Wavelength dependence of the FWHM spot size, transmission, and focusing efficiency for the micro-lens with
d
=175
μ
m.
6
METHODS
Sample Fabrication
. The HCTA pattern was defined in
ZEP520A positive resist using a Vistec EBPG5000+ electron
beam lithography system. After developing the resist, the pat-
tern was transferred into a 70 nm thick aluminum oxide layer
deposited by electron beam evaporation using the lift-off tech-
nique. The patterned aluminum oxide served as hard mask for
the subsequent dry etching of the 940 nm thick silicon layer
in a mixture of C
4
F
8
and SF
6
plasma.
Measurement procedure
. The microscope uses a 100X
objective (Olympus UMPlanFl) with the NA of 0.95, and a
tube lens (Thorlabs LB1945-C) with focal distance of 20 cm
which is anti-reflection coated for the 1050-1620 nm wave-
length range. The magnification of the microscope was found
by imaging a calibration sample with known feature dimen-
sions.
The transmission and focusing efficiency of the micro-
lenses were measured by inserting a flip mirror (as shown in
Fig. 4a) in front of the camera. To measure the optical power
focused by the micro-lens, the active area of the photodetec-
tor (Newport 818-IR) was reduced using an iris. The radius of
the iris aperture was adjusted to three times of the measured
FWHM spot size of the micro-lens on the camera. The to-
tal transmitted power was measured by opening the iris aper-
ture completely. The total power incident on the microlens
was measured by removing the micro-lens from the setup and
bringing the fiber tip into the focus of the microscope. The
non-uniformities seen in the intensity profile in Fig. 4 are due
to the local variations in the sensitivity of the camera (Digital
CamIR 1550 by Applied Scintillation Technologies) and are
observed even when directly imaging the light from an optical
fiber.
Simulations
. We found the electric and magnetic fields
of the light from the fiber on a plane close to the lens using
the plane wave expansion technique. Then, these fields were
used to determine the equivalent electric and magnetic sur-
face current densities, which were used as excitation sources
in the FDTD simulation. This allowed us to reduce the size
of the simulation domain to a smaller volume surrounding the
micro-lens.
In Fig. 2 , the FWHM spot size is found by fitting a 2D
Gaussian function to the z component of the Poynting vec-
tor at the plane of focus (shown in Fig. 2c). The focusing
efficiency is defined as the fraction of the incident light that
passes through a circular aperture in the plane of focus with a
radius equal to three times the FWHM spot size
ACKNOWLEDGEMENT
This work was supported by Caltech/JPL president and
director fund (PDF) . Amir Arbabi was also supported by
DARPA. Yu Horie was supported by JASSO fellowship and
the Light-Material Interactions in Energy Conversion Energy
Frontier Research Center funded by the US Department of En-
ergy, Office of Science, Office of Basic Energy Sciences un-
der Award No. de-sc0001293. Alexander Ball was supported
by the Summer Undergraduate Research Fellowship (SURF)
at Caltech. The device nanofabrication was performed in the
Kavli Nanoscience Institute at Caltech.
The authors thank to David Fattal and Sonny Vo for useful
discussion.
Corresponding authors:
A.F: faraon@caltech.edu,
A.A:
amir@caltech.edu
[1] N. Yu and F. Capasso, Nature materials
13
, 139 (2014).
[2] A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, Science (New
York, N.Y.)
339
, 1232009 (2013).
[3] N. Yu, P. Genevet, M. a. Kats, F. Aieta, J.-P. Tetienne, F. Capasso,
and Z. Gaburro, Science (New York, N.Y.)
334
, 333 (2011).
[4] F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard,
Z. Gaburro, and F. Capasso, Nano letters
12
, 4932 (2012).
[5] P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard,
M. O. Scully, Z. Gaburro, and F. Capasso, Applied Physics Let-
ters
100
, 013101 (2012).
[6] D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, Science
345
,
298 (2014).
[7] S. Vo, D. Fattal, W. V. Sorin, Z. Peng, T. Tran, R. G. Beau-
soleil, and M. Fiorentino, IEEE Photonics Technology Letters
26
, 1 (2014).
[8] W. Stork, N. Streibl, H. Haidner, and P. Kipfer, Optics letters
16
,
1921 (1991).
[9] F. T. Chen and H. G. Craighead, Optics Letters
20
, 121 (1995).
[10] M. E. Warren, R. E. Smith, G. A. Vawter, and J. R. Wendt,
Optics Letters
20
, 1441 (1995).
[11] F. T. Chen and H. G. Craighead, Optics Letters
21
, 177 (1996).
[12] P. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois,
Optics Letters
23
, 1081 (1998).
[13] F. M. Huang, T. S. Kao, V. a. Fedotov, Y. Chen, and N. I.
Zheludev, Nano letters
8
, 2469 (2008).
[14] X. Ni, S. Ishii, A. V. Kildishev, and V. M. Shalaev, Light: Sci-
ence & Applications
2
, e72 (2013).
[15] E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and
R. W. Boyd, Light: Science & Applications
3
, e167 (2014).
[16] F. Monticone, N. M. Estakhri, and A. Al
`
u, Physical Review
Letters
110
, 203903 (2013).
[17] L. Chen, M. C. Y. Huang, C. F. R. Mateus, C. J. Chang-Hasnain,
and Y. Suzuki, Applied Physics Letters
88
, 031102 (2006).
[18] U. Kemiktarak, M. Metcalfe, M. Durand, and J. Lawall, Ap-
plied Physics Letters
100
, 061124 (2012).
[19] T. T. Wu, Y. C. Syu, S. H. Wu, W. T. Chen, T. C. Lu, S. C.
Wang, H. P. Chiang, and D. P. Tsai, Optics express
20
, 20551
(2012).
[20] C. Mateus, M. Huang, Y. Deng, A. Neureuther, and C. Chang-
Hasnain, IEEE Photonics Technology Letters
16
, 518 (2004).
[21] F. Lu, F. G. Sedgwick, V. Karagodsky, C. Chase, and C. J.
Chang-Hasnain, Optics express
18
, 12606 (2010).
[22] S. Astilean, P. Lalanne, P. Chavel, E. Cambril, and H. Launois,
Optics Letters
23
, 552 (1998).
[23] D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil,
Nature Photonics
4
, 466 (2010).
[24] A. B. Klemm, D. Stellinga, E. R. Martins, L. Lewis, G. Huyet,
L. O. Faolain, T. F. Krauss, and L. O’Faolain, Optics letters
38
,
3410 (2013).
7
[25] A. Arbabi, M. Bagheri, A. J. Ball, Y. Horie, D. Fattal, and
A. Faraon, in
CLEO: 2014 (Optical Society of America, San
Jose, California, 2014) p. STu3M.4.
[26] T. Shiono, M. Kitagawa, K. Setsune, and T. Mitsuyu, Applied
optics
28
, 3434 (1989).
[27] M. Haruna, M. Takahashi, K. Wakahayashi, and H. Nishihara,
Applied optics
29
, 5120 (1990).
[28] A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel,
J. Joannopoulos, and S. G. Johnson, Computer Physics Com-
munications
181
, 687 (2010).
[29] A. Arbabi, Y. Horie, and A. Faraon, in
CLEO: 2014 (Optical
Society of America, San Jose, California, 2014) p. STu3M.5.
8
SUPPLEMENTARY INFORMATION FOR “SUBWAVELENGTH-THICK LENSES WITH HIGH NUMERICAL APERTURES
AND LARGE EFFICIENCY BASED ON HIGH CONTRAST TRANSMITARRAYS”
S.1 Optimum Transmissive Mask Design
J
M
(1)
(2)
a
(1)
(2)
b
(1)
(2)
c
d
FIG. S1.
Optimum transmissive mask design for shaping an incident light to the desired tangential form
.
a
, The light from the sources
and scatterers in half space (1) passes through the transmission mask
t
(
x,y
)
, and its tangential electric field on the target plane is represented
by
E
tan
.
b
, The equivalent magnetic surface current density
M
s
emits the same tangential electric field
E
tan
on the target plane.
c
, The
magnetic field
H
(2)
d
is emitted by an electric surface current density
J
s
which is located on the target plane.
We consider the general case of using a transmissive mask to modify the optical wavefront emitted by given sources to a
desired form. The light is generated by sources located in half space (1) as shown schematically in Fig. S1a. The tangential
component of the electric field of the incident light just before propagating through the transmissive mask is represented by
E
(1)
i
. The incident field might be, for example, a diverging beam from a semiconductor laser or a collimated Gaussian beam.
The desired output wavefront can be chosen arbitrarily (examples include a beam that is matched to a mode of an optical fiber, a
Bessel beam, or a tightly focused beam). Since the propagation is governed by Maxwell’s equations, the desired output beam is
fully described by the tangential components of its electric field on a target plane parallel to the transmissive mask. We represent
this desired tangential component by
E
d
(as shown in Fig. S1a)).
The output beam formed by the transmissive mask is in general different from the desired beam, and its tangential electric
field on the target plane
E
tan
is an approximation for the desired tangential electric field. Our main objective is to determine the
transmissive mask
t
(
x,y
)
such that
E
tan
is the best possible approximation for the
E
d
. A useful measure for quantifying the
accuracy of the approximation is the norm of the projection integral defined as
|
<
E
tan
,
E
d
>
|
=
E
tan
·
E
d
d
s
,
(1)
where
represents the complex conjugate operation; and the surface integral is evaluated over the target plane. For the best
approximation, this norm should be maximized.
To relate the projection integral to the transmission mask we use the equivalence principle and the reciprocity theorem.
According to the definition of a transmissive mask, the tangential electric field of the light just after it passes through the
transmissive mask is given by
t
E
i
. Using the equivalence principle, a magnetic surface current density
M
s
= 2
t
E
i
×
ˆ
z
located
at the output plane of the transmissive mask and emitting in vacuum, will generate the same beam in the region (2) as the original
sources and the transmissive mask (see Fig S1b). Now we consider an electric surface current density with
J
s
=
E
d
on the
target plane which is emitting in vacuum. We refer to the magnetic field emited by
J
s
at the output plane of the transmissive
mask by
H
(2)
d
. Using the reciprocity theorem [1], we can write
E
tan
·
J
s
d
s
=
H
(2)
d
·
M
s
d
s.
(2)
9
From (1) and (2) we obtain
|
<
E
tan
,
E
d
>
|
= 2
t
H
(2)
d
·
(
E
(1)
i
×
ˆ
z
)d
s
.
(3)
Using (3), we see that the best approximation to a desired output beam is achieved when
|
t
|
= 1
and
t
=
(
H
(2)
d
·
(
E
(1)
i
×
ˆ
z
)
)
.
(4)
In other words, the best transmissive mask is a phase mask, and we can determine its phase profile as follows. First, we find the
tangential component of the incident light at the location of the transmissive mask (
E
(1)
i
); this can be done either analytically or
numerically depending on the type of the excitation. Next, we consider an electric surface current density with the same spatial
distribution as the desired tangential electric field at the target plane which is emitting in the free space. We find the magnetic
field emitted by this current at the location of the transmission mask (
H
(2)
d
). Since
J
s
is planar and is radiating in free space,
its fields can be obtained using a simple method such as the plane wave expansion technique [2]. Finally, we obtain the phase
profile of the transmission mask using (4). To achieve a lens with tight focus, the desired field should be set to a uniform electric
field confined inside a circle with deep subwavelength radius located at the plane of focus.
S.2 Under-sampling of the phase profile
Considering the micro-lens in the geometrical optics picture offers an intuitive understanding of how the under-sampling of
the phase profile affects the lens performance. According to the geometrical optics, a ray that is propagating parallel to the
lens’s optical axis is deflected by the lens toward its focal point. The rays that are propagating farther away from the optical
axis are deflected by larger angles, and the lens’s NA represents the sine of the largest deflection angle. Therefore, a lens can
be considered as a deflector whose local deflection angle gradually increases from zero at the center of the lens to its maximum
(which is given by
sin
1
(NA)
) at the perimeter of the lens. We show that due to the under sampling of the phase profile by
the HCTA lattice, the deflection efficiency of the HCTA deflectors decreases by increasing their deflection angle. The lower
deflection efficiency at larger deflection angles leads to the lower focusing efficiency of the micro-lenses with higher NA.
b
TE
TM
a
c
02
04
06
08
0
0
20
40
60
80
TE
TM
Defl
ection efficiency (%
)
Deflection an
g
le (d
e
g
rees
)
FIG. S2.
Measurement results of HCTA beam deflectors. a
, Schematic illustration of an ideal beam deflector.
b
, The HCTA pattern that
implements a uniform phase ramp along the horizontal direction. The polarization direction for the TE and TM polarizations are also shown.
c
, Measured deflection efficiency of a set of beam deflectors for the transverse electric and magnetic polarized incident beam as a function of
deflection angle.
A uniform deflector functions similar to a blazed grating and deflects a monochromatic normally incident light by a fixed
angle (as shown in Fig. S2a). The uniform deflector has a linearly varying phase profile whose slope is proportional to the sine
of its deflection angle. A schematic illustration of an HCTA uniform deflector, with phase profile varying linearly along the
horizontal direction, is depicted in Fig. S2b. Nine 400
μ
m diameter uniform deflectors with different deflection angles were
fabricated using the same family of periodic HCTAs used for the high NA micro-lenses and the same fabrication process.
The deflectors were illuminated with a linearly polarized collimated laser beam with beam radius of approximately 100
μ
m.
The deflected power was measured using a photodetector located 10 cm away from the deflector along the expected deflection
10
direction. The deflection efficiency was obtained by dividing the measured deflected power by the incident power. The measured
deflection efficiencies for two linear orthogonal polarizations are depicted in Fig. S2c. The directions of polarization for the TE
and TM polarized lights are shown in Fig. S2b. The TE polarization corresponds to the transverse electric polarized deflected
light, while the deflected light is transverse magnetic polarized for the TM polarization.
As Fig. S2c shows, the deflection efficiency of an HCTA deflector decreases as its deflection angle increases. The deflection
efficiency drop is faster for the TM polarized incident light compared to the TE one. We attribute the efficiency reduction to
the under sampling of the phase profile of the deflectors with large deflection angles. The desired phase profile of a deflector
with deflection angle of
θ
is sampled by
n
=
λ/
(
a
sin(
θ
))
unit cells over
2
π
phase variation, where
a
is the lattice constant of
the HCTA. For the HCTA used in this study, the lattice constant is roughly equal to a half of a wavelength; therefore, for a 40
deflector, the full phase range of
2
π
is sampled by approximately three unit cells. Similar diffraction efficiency reduction due to
phase sampling and quantization error is encountered in the design and implementation of Fresnel lenses with a limited number
of levels [3]. One approach to increase the efficiency of a high NA micro-lens is to use an HCTA with a smaller lattice constant.
This leads to a finer sampling of the desired micro-lens profile. For example, as Fig.1b,d in the main manuscript show, we could
use the lattice constant of 650 nm instead of 800 nm and still achieve the full
2
π
transmission phase range by changing the post
diameters.
Corresponding authors: A.F: faraon@caltech.edu, A.A: amir@caltech.edu
[1] R. F. Harrington,
Time Harmonic Electromagnetic Fields (Wiley-IEEE Press, 2001).
[2] M. Born and E. Wolf,
Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999).
[3] F. Wyrowski, Optics Communications
92
, 119 (1992).