Controlling the sign of chromatic dispersion in diffractive optics
Ehsan Arbabi,
1
Amir Arbabi,
1
Seyedeh Mahsa Kamali,
1
Yu Horie,
1
and Andrei Faraon
1,
∗
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology,
1200 E. California Blvd., Pasadena, CA 91125, USA
Abstract
Diffraction gratings disperse light in a rainbow of colors with the opposite order than refractive prisms,
a phenomenon known as negative dispersion [1, 2]. While refractive dispersion can be controlled via ma-
terial refractive index, diffractive dispersion is fundamentally an interference effect dictated by geometry.
Here we show that this fundamental property can be altered using dielectric metasurfaces [3–5], and we ex-
perimentally demonstrate diffractive gratings and focusing mirrors with positive, zero, and hyper negative
dispersion. These optical elements are implemented using a reflective metasurface composed of dielectric
nano-posts that provide simultaneous control over phase and its wavelength derivative. In addition, as a
first practical application, we demonstrate a focusing mirror that exhibits a five fold reduction in chromatic
dispersion, and thus an almost three times increase in operation bandwidth compared to a regular diffractive
element. This concept challenges the generally accepted dispersive properties of diffractive optical devices
and extends their applications and functionalities.
∗
faraon@caltech.edu
1
arXiv:1701.07178v1 [physics.optics] 25 Jan 2017
Most optical materials have positive (normal) dispersion, which means that the refractive index
decreases at longer wavelengths. As a consequence, blue light is deflected more than red light by
dielectric prisms (Fig. 1a). The reason why diffraction gratings are said to have negative dispersion
is because they disperse light similar to hypothetical refractive prisms made of a material with neg-
ative (anomalous) dispersion (Fig. 1b). For diffractive devices, dispersion is not related to material
properties, and it refers to the derivative of a certain device parameter with respect to wavelength.
For example, the angular dispersion of a grating that deflects normally incident light by a positive
angle
θ
is given by
d
θ/
d
λ
= tan(
θ
)
/λ
(see [1] and Supplementary Section S1). Similarly, the
wavelength dependence of the focal length (
f
) of a diffractive lens is given by
d
f/
d
λ
=
−
f/λ
[1, 2]. Here we refer to diffractive devices that follow these fundamental chromatic dispersion
relations as “
regular
”. Achieving new regimes of dispersion control in diffractive optics is im-
portant both at the fundamental level and for numerous practical applications. Several distinct
regimes can be differentiated as follows. Diffractive devices are dispersionless when the deriva-
tive is zero (i.e.
d
θ/
d
λ
= 0
,
d
f/
d
λ
= 0
shown schematically in Fig. 1c), have positive dispersion
when the derivative has opposite sign compared to a regular diffractive device of the same kind
(i.e.
d
θ/
d
λ <
0
,
d
f/
d
λ >
0
) as shown in Fig. 1d, and are hyper-dispersive when the derivative
has a larger absolute value than a regular device (i.e.
|
d
θ/
d
λ
|
>
|
tan(
θ
)
/λ
|
,
|
d
f/
d
λ
|
>
|−
f/λ
|
,
Fig. 1e). Here we show that these regimes can be achieved in diffractive devices based on optical
metasurfaces.
Metasurfaces have attracted great interest in the recent years [3–12] because they enable precise
control of optical wavefronts and are easy to fabricate with conventional microfabrication technol-
ogy in a flat, thin, and light weight form factor. Various conventional devices such as gratings and
lenses [7–9, 13–24] as well as novel devices [25, 26] have been demonstrated using metasurfaces.
These optical elements are composed of large numbers of scatterers, or meta-atoms placed on a
two-dimensional lattice to locally shape optical wavefronts. Similar to other diffractive devices,
metasurfaces that locally change the propagation direction (e.g. lenses, beam deflectors, holo-
grams) have negative chromatic dispersion [1, 2, 27, 28]. This is because most of these devices
are divided in Fresnel zones whose boundaries are designed for a specific wavelength [28, 29].
This chromatic dispersion is an important limiting factor in many applications and its control is
of great interest. Metasurfaces with zero and positive dispersion would be useful for making
achromatic singlet and doublet lenses, and the larger-than-regular dispersion of hyper-dispersive
metasurface gratings would enable high resolution spectrometers. We emphasize that the devices
2
with zero chromatic dispersion discussed here are fundamentally different from the multiwave-
length metasurface gratings and lenses recently reported [28–38]. Multiwavelength devices have
several diffraction orders, which result in lenses (gratings) with the same focal length (deflection
angle) at a few discrete wavelengths. However, at each of these focal distances (deflection angles),
the multi-wavelength lenses (gratings) exhibit the regular negative diffractive chromatic dispersion
(see [28, 29], Supplementary Section S2 and Supplementary Extended Data Fig. 1).
Here we argue that simultaneously controlling the phase imparted by the meta-atoms compos-
ing the metasurface (
φ
) and its derivative with respect to frequency (
φ
′
=
∂φ/∂ω
which we refer
to as chromatic phase dispersion or dispersion for brevity) makes it possible to dramatically al-
ter the fundamental chromatic dispersion of diffractive components. This, in effect, is equivalent
to simultaneously controlling the “effective refractive index” and “chromatic dispersion” of the
meta-atoms. Recently we used this concept to demonstrate metasurface focusing mirrors with
zero dispersion [39]. Using this concept, here we experimentally show metasurface gratings and
focusing mirrors that have positive, zero, and hyper chromatic dispersions. We also demonstrate an
achromatic focusing mirror with a highly diminished focal length chromatic dispersion, resulting
in an almost three times increase in its operation bandwidth.
First, we consider the case of devices with zero chromatic dispersion. For frequency inde-
pendent operation, a device should impart a constant delay for different frequencies, similar to
a refractive device made of a non-dispersive material [1]. Therefore, the phase profile will be
proportional to the frequency:
φ
(
x,y
;
ω
) =
ωT
(
x,y
)
,
(1)
where
ω
= 2
πc/λ
is the angular frequency (
λ
: wavelength,
c
: speed of light) and
T
(
x,y
)
deter-
mines the function of the device (for instance
T
(
x,y
) =
−
x
sin
θ
0
/c
for a grating that deflects
light by angle
θ
0
;
T
(
x,y
) =
−
√
x
2
+
y
2
+
f
2
/c
for a spherical-aberration-free lens with a focal
distance
f
). Since the phase profile is a linear function of
ω
, it can be realized using a metasur-
face composed of meta-atoms that control the phase
φ
(
x,y
;
ω
0
) =
T
(
x,y
)
ω
0
and its dispersion
φ
′
=
∂φ
(
x,y
;
ω
)
/∂ω
=
T
(
x,y
)
. The bandwidth of dispersionless operation corresponds to the
frequency interval over which the phase locally imposed by the meta-atoms is linear with fre-
quency
ω
. For gratings or lenses, a large device size results in a large
|
T
(
x,y
)
|
, which means
that the meta-atoms should impart a large phase dispersion. Since the phase values at the center
wavelength
λ
0
= 2
πc/ω
0
can be wrapped into the 0 to 2
π
interval, the meta-atoms only need to
3
cover a rectangular region in the
phase-dispersion
plane bounded by
φ
= 0
and 2
π
lines, and
φ
′
= 0
and
φ
′
max
lines, where
φ
′
max
is the maximum required dispersion which is related to the de-
vice size (see Supplementary Section S4 and Supplementary Extended Data Fig. 2). The required
phase-dispersion coverage means that, to implement devices with various phase profiles, for each
specific value of the phase we need various meta-atoms providing that specific phase, but with
different dispersion values.
To realize metasurface devices with non-zero dispersion of a certain parameter
ξ
(
ω
)
, phase
profiles of the following form are needed:
φ
(
x,y
;
ω
) =
ωT
(
x,y,ξ
(
ω
))
.
(2)
For instance, the parameter
ξ
(
ω
)
can be the deflection angle of a diffraction grating
θ
(
ω
)
or the
focal length of a diffractive lens
f
(
ω
)
. As we show in the Supplementary Section S3, to inde-
pendently control the parameter
ξ
(
ω
)
and its chromatic dispersion
∂ξ/∂ω
at
ω
=
ω
0
, we need to
control the phase dispersion at this frequency in addition to the phase. The required dispersion for
a certain parameter value
ξ
0
=
ξ
(
ω
0
)
, and a certain dispersion
∂ξ/∂ω
|
ω
=
ω
0
is given by:
∂φ
(
x,y
;
ω
)
∂ω
|
ω
=
ω
0
=
T
(
x,y,ξ
0
) +
∂ξ/∂ω
|
ω
=
ω
0
ω
0
∂T
(
x,y,ξ
)
∂ξ
|
ξ
=
ξ
0
.
(3)
This dispersion relation is valid over a bandwidth where a linear approximation of
ξ
(
ω
)
is valid.
Assuming hypothetical meta-atoms that provide independent control of phase and dispersion
up to a dispersion of
−
150
Rad
/μ
m (to adhere to the commonly used convention, we report
the dispersion in terms of wavelength) at the center wavelength of 1520 nm, we have designed
and simulated four gratings with different chromatic dispersions (see Methods for details). The
simulated deflection angles as functions of wavelength are plotted in Fig. 2a. All gratings are
150
μ
m wide, and have a deflection angle of 10 degrees at their center wavelength of 1520 nm. The
positive dispersion grating exhibits a dispersion equal in absolute value to the negative dispersion
of a regular grating with the same deflection angle, but with an opposite sign. The hyper-dispersive
design is three times more dispersive than the regular grating, and the dispersionless beam deflector
shows almost no change in its deflection angle. Besides gratings, we have also designed focusing
mirrors exhibiting regular, zero, positive, and hyper dispersions. The focusing mirrors have a
diameter of
500
μ
m and a focal distance of
850
μ
m at 1520 nm. Hypothetical meta-atoms with a
4
maximum dispersion of
−
200
Rad
/μ
m are required to implement these focusing mirror designs.
The simulated focal distances of the four designs are plotted in Fig. 2b. The axial plane intensity
distributions at three wavelengths are plotted in Figs. 2c-2f (for intensity plots at other wavelengths
see Supplementary Extended Data Fig. 3).
An example of meta-atoms capable of providing 0 to 2
π
phase coverage and different disper-
sions is shown in Fig. 3a. The meta-atoms are composed of a square cross-section amorphous
silicon (
α
-Si) nano-post on a low refractive index silicon dioxide (SiO
2
) spacer layer on an alu-
minum reflector. They are located on a periodic square lattice (Fig. 3a, middle). The simulated
dispersion versus phase plot for the meta-atoms at the wavelength of
λ
0
= 1520
nm is depicted
in Fig. 3b, and shows a partial coverage up to the dispersion value of
∼ −
100
Rad
/μ
m (the
meta-atoms are 725 nm tall, the SiO
2
layer is 325 nm thick, the lattice constant is 740 nm, and the
nano-post side length is varied from 74 to 666 nm at 1.5 nm steps). Simulated reflection amplitude
and phase for the periodic lattice are plotted in Figs. 3c and 3d, respectively. The reflection am-
plitude over the bandwidth of interest is close to 1 for all nano-post side lengths. The operation of
the nano-post meta-atoms is best intuitively understood as truncated multi-mode waveguides with
many resonances in the bandwidth of interest [26, 40]. By going through the nano-post twice,
light can obtain larger phase shifts compared to the transmissive operation mode of the metasur-
face (i.e. without the metallic reflector). The metallic reflector keeps the reflection amplitude high
for all sizes, which makes the use of high quality factor resonances possible. High quality fac-
tor resonances are necessary for achieving large dispersion values, because, as we have shown in
Supplementary Section S5, dispersion is given by
φ
′
≈ −
Q/λ
0
, where
Q
is the quality factor of
the resonance. In an alternative view, resonances with large quality factors are necessary as they
correspond to large group delays in the meta-atoms. For example, in the special case of a lens with
zero dispersion, light passing through the middle of the lens should experience a delay with respect
to light passing close to the lens circumference, so the mata-atoms in the middle must compensate
for this delay. Therefore, the largest achievable meta-atom quality factors directly limit the device
size.
Using the dispersion-phase parameters provided by this metasurface, we designed four gratings
operating in various dispersion regimes. The gratings are
∼
90
μ
m wide and have a 10-degree de-
flection angle at 1520 nm. They are designed to operate in the 1450 to 1590 nm wavelength range,
and have regular negative, zero, positive, and hyper (three-times-larger negative) dispersion. Since
the phase of the meta-atoms does not follow a linear frequency dependence over this wavelength
5
interval (Fig. 3d, top right), we calculate the desired phase profile of the devices at 8 wavelengths
in the range (1450 to 1590 nm at 20 nm steps), and form an 8
×
1 complex reflection coefficient
vector at each point on the metasurface. Using Figs. 3c and 3d, a similar complex reflection
coefficient vector is calculated for each meta-atom. Then, at each lattice site of the metasurface,
we place a meta-atom whose reflection vector has the shortest weighted Euclidean distance to
the desired reflection vector at that site. The weights allow for emphasizing different parts of the
operation bandwidth, and can be chosen based on the optical spectrum of interest or other consid-
erations. Here, we used an inverted Gaussian weight (
exp((
λ
−
λ
0
)
2
/
2
σ
2
)
,
σ
= 300
nm), which
values wavelengths farther away from the center wavelength of
λ
0
= 1520
nm. The same de-
sign method is used for the other devices discussed in the manuscript. The designed devices were
fabricated using standard semiconductor fabrication techniques as described in Methods. Figures
3e-3g show scanning electron micrographs of the nano-posts, and some of the devices fabricated
using the proposed reflective meta-atoms.
Figures 4a and 4b show the simulated and measured deflection angles for gratings, respectively.
The measured values are calculated by finding the center of mass of the deflected beam 3 mm away
from the grating surface (see Methods and Supplementary Extended Data Fig. 5 for more details).
As expected, the zero dispersion grating shows an apochromatic behavior resulting in a reduced
dispersion, the positive grating shows positive dispersion in the
∼
1490-1550 nm bandwidth, and
the hyper-dispersive one shows an enhanced dispersion in the measurement bandwidth. This can
also be viewed from the grating momentum point of view: a regular grating has a constant momen-
tum set by its period, resulting in a constant transverse wave-vector. In contrary, the momentum of
the hyper-dispersive grating increases with wavelength, while that of the zero and positive gratings
decreases with it. This means that the effective period of the non-regular gratings changes with
wavelength, resulting in the desired chromatic dispersion. Figures 4e-4h show good agreement
between simulated intensities of these gratings versus wavelength and transverse wave-vector (see
Methods for details) and the measured beam deflection (black stars). The green line is the the-
oretical expectation of the maximum intensity trajectory. Measured deflection efficiencies of the
gratings, defined as the power deflected by the gratings to the desired order, divided by the power
reflected from a plain aluminum reflector (see Methods and Supplementary Extended Data Fig. 5
for details) are plotted in Figs. 4c and 4d for TE and TM illuminations, respectively. A similar
difference in the efficiency of the gratings for TE and TM illuminations has also been observed in
previous works [16, 26].
6
As another example for diffractive devices with controlled chromatic dispersion, four spherical-
aberration-free focusing mirrors with different chromatic dispersions were designed, fabricated
and measured using the same reflective dielectric meta-atoms. The mirrors are 240
μ
m in diameter
and are designed to have a focal distance of 650
μ
m at 1520 nm. Figures 5a and 5b show simulated
and measured focal distances for the four focusing mirrors (see Extended Data Figs. 6, 7, and
8 for detailed simulation and measurement results). The positive dispersion mirror is designed
with dispersion twice as large as a regular mirror with the same focal distance, and the hyper-
dispersive mirror has a negative dispersion three and a half times larger than a regular one. The
zero dispersion mirror shows a significantly reduced dispersion, while the hyper-dispersive one
shows a highly enhanced dispersion. The positive mirror shows the expected dispersion in the
∼
1470 to 1560 nm range.
As an application of diffractive devices with dispersion control, we demonstrate a spherical-
aberration-free focusing mirror with increased operation bandwidth. For brevity, we call this de-
vice dispersionless mirror. Since the absolute focal distance change is proportional to the focal
distance itself, a relatively long focal distance is helpful for unambiguously observing the change
in the device dispersion. Also, a higher NA value is preferred because it results in a shorter depth
of focus, thus making the measurements easier. Having these considerations in mind, we have
chosen a diameter of 500
μ
m and a focal distance of 850
μ
m (NA
≈
0.28) for the mirror, requiring
a maximum dispersion of
φ
′
max
≈ −
98
Rad/
μ
m which is achievable with the proposed reflective
meta-atoms. We designed two dispersionless mirrors with two
σ
values of 300 and 50 nm. For
comparison, we also designed a regular metasurface mirror for operation at
λ
0
= 1520
nm and
with the same diameter and focal distance as the dispersionless mirrors. The simulated focal dis-
tance deviations (from the designed 850
μ
m) for the regular and dispersionless (
σ
= 300
nm)
mirrors are plotted in Fig. 5c, showing a considerable reduction in chromatic dispersion for the
dispersionless mirror. Detailed simulation results for these mirrors are plotted in Extended Data
Fig. 9.
Figures 5d-5g summarize the measurement results for the dispersionless and regular mirrors
(see Methods and Extended Data Fig. 5 for measurement details and setup). As Figs. 5d and
5g show, the focal distance of the regular mirror changes almost linearly with wavelength. The
dispersionless mirror, however, shows a highly diminished chromatic dispersion. Besides, as seen
from the focal plane intensity measurements, while the dispersionless mirrors are in focus in the
850
μ
m plane throughout the measured bandwidth, the regular mirror is in focus only from 1500
7
to 1550 nm (see Extended Data Figs. 10 and 11 for complete measurement results, and the Strehl
ratios). Focusing efficiencies, defined as the ratio of the optical power focused by the mirrors to the
power incident on them, were measured at different wavelengths for the regular and dispersionless
mirrors (see Methods for details). The measured efficiencies were normalized to the efficiency
of the regular metasurface mirror at its center wavelength of 1520 nm (which is estimated to
be
∼
80
%
–90
%
based on Fig. 3, measured grating efficiencies, and our previous works [16]).
The normalized efficiency of the dispersionless mirror is between 50
%
and 60
%
in the whole
wavelength range and shows no significant reduction in contrast to the regular metasurface mirror.
The reduction in efficiency compared to a mirror designed only for the center wavelength (i.e.
the regular mirror) is caused by two main factors. First, the required region of the phase-dispersion
plane is not completely covered by the reflective nano-post meta-atoms. Second, the meta-atom
phase does not change linearly with respect to frequency in the relatively large bandwidth of
140 nm as would be ideal for a dispersionless metasurface. Both of these factors result in deviation
of the phase profiles of the demonstrated dispersionless mirrors from the ideal ones. Furthermore,
dispersionless metasurfaces use meta-atoms supporting resonances with high quality factors, thus
leading to higher sensitivity of these devices to fabrication errors compared to the regular meta-
surfaces.
In conclusion, we demonstrated that independent control over phase and dispersion of meta-
atoms can be used to engineer the chromatic dispersion of diffractive metasurface devices over
continuous wavelength regions. This is in effect similar to controlling the “material dispersion”
of meta-atoms to compensate, over-compensate, or increase the structural dispersion of diffractive
devices. In addition, we developed a reflective dielectric metasurface platform that provides this
independent control. Using this platform, we experimentally demonstrated gratings and focusing
mirrors exhibiting positive, negative, zero, and enhanced dispersions. We also corrected the chro-
matic aberrations of a focusing mirror resulting in a
∼
3 times bandwidth increase (based on an
Strehl ratio
>
0
.
6
, see Extended Data Fig. 11). In addition, the introduced concept of metasurface
design based on dispersion-phase parameters of the meta-atoms is general and can also be used
for developing transmissive dispersion engineered metasurface devices.
8
METHODS
Simulation and design.
The gratings with different dispersions discussed in Fig. 2a were de-
signed using hypothetical meta-atoms that completely cover the required region of the phase-
dispersion plane. We assumed that the meta-atoms provide 100 different phase steps from 0 to
2
π
, and that for each phase, 10 different dispersion values are possible, linearly spanning the 0 to
−
150
Rad
/μ
m range. We assumed that all the meta-atoms have a transmission amplitude of 1.
The design began with constructing the ideal phase masks at eight wavelengths equally spaced in
the 1450 to 1590 nm range. This results in a vector of eight complex numbers for the ideal trans-
mission at each point on the metasurface grating. The meta-atoms were assumed to form a two
dimensional square lattice with a lattice constant of 740 nm, and one vector was generated for each
lattice site. The optimum meta-atom for each site was then found by minimizing the Euclidean
distance between the transmission vector of the meta-atoms and the ideal transmission vector for
that site. The resulting phase mask of the grating was then found through a two-dimensional inter-
polation of the complex valued transmission coefficients of the chosen meta-atoms. The grating
area was assumed to be illuminated uniformly, and the deflection angle of the grating was found
by taking the Fourier transform of the field after passing through the phase mask, and finding the
angle with maximum intensity. A similar method was used to design and simulate the focusing
mirrors discussed in Figs. 2b-2f. In this case, the meta-atoms are assumed to cover dispersion
values up to
−
200
Rad
/μ
m. The meta-atoms provide 21 different dispersion values distributed
uniformly in the 0 to
−
200
Rad
/μ
m range. The focusing mirrors were designed and the corre-
sponding phase masks were found in a similar manner to the gratings. A uniform illumination
was used as the source, and the resulting field after reflection from the mirror was propagated in
free space using a plane wave expansion method to find the intensity in the axial plane. The focal
distances plotted in Fig. 2b show the distance of the maximum intensity point from the mirrors at
each wavelength. The gratings and focusing mirrors discussed in Figs. 4a, 5a, and 5c are designed
and simulated in exactly the same manner, except for using actual dielectric meta-atom reflection
amplitudes and phases instead of the hypothetical ones.
Reflection amplitude and phase of the meta-atoms were found using rigorous coupled wave
analysis technique [41]. For each meta-atom size, a uniform array on a subwavelength lattice was
simulated using a normally incident plane wave. The subwavelength lattice ensures the existence
of only one propagating mode which justifies the use of only one amplitude and phase for describ-
9
ing the optical behavior at each wavelength. In the simulations, the amorphous silicon layer was
assumed to be 725 nm thick, the SiO
2
layer was 325 nm, and the aluminum layer was 100 nm
thick. A 30-nm-thick Al
2
O
3
layer was added between the Al and the oxide layer (this layer served
as an etch stop layer to avoid exposing the aluminum layer during the etch process). Refractive in-
dices were set as follows in the simulations: SiO
2
: 1.444, Al
2
O
3
: 1.6217, and Al: 1.3139-
i
13.858.
The refractive index of amorphous silicon used in the simulations is plotted in Extended Data Fig.
10.
The FDTD simulations of the gratings (Figs. 4e-4h) were performed using a normally incident
plane-wave illumination with a Gaussian amplitude in time (and thus a Gaussian spectrum) in
MEEP [42]. The reflected electric field was saved in a plane placed one wavelength above the
input plane at time steps of 0.05 of the temporal period. The results in Figs. 4e-4h are obtained via
Fourier transforming the fields in time and space resulting in the reflection intensities as a function
of frequency and transverse wave-vector.
Sample fabrication.
A 100-nm aluminum layer and a 30-nm Al
2
O
3
layer were deposited on a
silicon wafer using electron beam evaporation. This was followed by deposition of 325 nm of
SiO
2
and 725 nm of amorphous silicon using the plasma enhanced chemical vapor deposition
(PECVD) technique at
200
◦
C
. A
∼
300 nm thick layer of ZEP-520A positive electron-beam resist
was spun on the sample at 5000 rpm for 1 min, and was baked at
180
◦
C
for 3 min. The pattern
was generated using a Vistec EBPG5000+ electron beam lithography system, and was developed
for 3 minutes in the ZED-N50 developer (from Zeon Chemicals). A
∼
70-nm Al
2
O
3
layer was
subsequently evaporated on the sample, and the pattern was reversed with a lift off process. The
Al
2
O
3
hard mask was then used to etch the amorphous silicon layer in a 3:1 mixture of
SF
6
and
C
4
F
8
plasma. The mask was later removed using a 1:1 solution of ammonium hydroxide and
hydrogen peroxide at
80
◦
C.
Measurement procedure.
The measurement setup is shown in Extended Data Fig. 5a. Light emit-
ted from a tunable laser source (Photonetics TUNICS-Plus) was collimated using a fiber collima-
tion package (Thorlabs F240APC-1550), passed through a 50/50 beamsplitter (Thorlabs BSW06),
and illuminated the device. For grating measurements a lens with a 50 mm focal distance was
also placed before the grating at a distance of
∼
45 mm to partially focus the beam and reduce the
beam divergence after being deflected by the grating in order to decrease the measurement error
(similar to Extended Data Fig. 5b). The light reflected from the device was redirected using the
same beamsplitter, and imaged using a custom built microscope. The microscope consists of a
10
50X objective (Olympus LMPlanFL N, NA=0.5), a tube lens with a 20 cm focal distance (Thor-
labs AC254-200-C-ML), and an InGaAs camera (Sensors Unlimited 320HX-1.7RT). The grating
deflection angle was found by calculating the center of mass for the deflected beam imaged 3 mm
away from the gratings surface. For efficiency measurements of the focusing mirrors, a flip mirror
was used to send light towards an iris (2 mm diameter, corresponding to an approximately 40
μ
m
iris in the object plane) and a photodetector (Thorlabs PM100D with a Thorlabs S122C head).
The efficiencies were normalized to the efficiency of the regular mirror at its center wavelength
by dividing the detected power through the iris by the power measured for the regular mirror at
its center wavelength. The measured intensities were up-sampled using their Fourier transforms
in order to achieve smooth intensity profiles in the focal and axial planes. To measure the grat-
ing efficiencies, the setup shown in Extended Data Fig. 5b was used, and the photodetector was
placed
∼
50 mm away from the grating, such that the other diffraction orders fall outside its active
area. The efficiency was found by calculating the ratio of the power deflected by the grating to the
power normally reflected by the aluminum reflector in areas of the sample with no grating. The
beam-diameter on the grating was calculated using the setup parameters, and it was found that
∼
84
%
of the power was incident on the 90
μ
m wide gratings. This number was used to correct for
the lost power due to the larger size of the beam compared to the grating.
ACKNOWLEDGEMENTS
This work was supported by Samsung Electronics. E.A. and A.A. were also supported by
National Science Foundation award 1512266. A.A. and Y.H. were also supported by DARPA, and
S.M.K. was supported as part of the Department of Energy (DOE) “Light-Material Interactions in
Energy Conversion” Energy Frontier Research Center under grant no. de-sc0001293. The device
nanofabrication was performed at the Kavli Nanoscience Institute at Caltech.
Author contributions
E.A., A.A., and A.F. conceived the experiment. E.A., S.M.K., and Y.H.
fabricated the samples. E.A., S.M.K., A.A., and Y.H. performed the simulations, measurements,
and analyzed the data. E.A., A.F., and A.A. co-wrote the manuscript. All authors discussed the
results and commented on the manuscript.
Competing financial interests
The authors declare no competing financial interests.
11
Hyper-dispersive
Positive
Zero
Refractive
(Positive)
abc
e
d
O
O
O
O
O
O
Regular
(Negative)
Figure 1
|
Schematic illustrations of different dispersion regimes. a
, Positive chromatic dispersion in
refractive prisms and lenses made of materials with normal dispersion.
b
, Regular (negative) dispersion in
typical diffractive and metasurface gratings and lenses.
c
, Schematic illustration of zero,
d
, positive, and
e
,
hyper dispersion in dispersion-controlled metasurfaces. Only three wavelengths are shown here, but the
dispersions are valid for any other wavelength in the bandwidth. The diffractive devices are shown in
transmission mode for ease of illustration, while the actual devices fabricated in this paper are designed to
operate in reflection mode.
12
a
Wavelength [
P
m]
Deflectionangle [degrees]
1.46
1.52
1.58
9
10
11
Positive
Hyper-dispersive
Dispersionless
Regular negative
b
Wavelength [
P
m]
Positive
Hyper-dispersive
Dispersionless
Regular negative
1.46
1.52
1.58
f [
P
m]
750
850
950
c
30 [
P
m]
O
=1450nm
O
=1520nm
O
=1590nm
850
1000
700
z [
P
m]
1
0
Intensity [a.u.]
f
850
1000
700
30 [
P
m]
z [
P
m]
O
=1450nm
O
=1520nm
O
=1590nm
e
30 [
P
m]
850
1000
700
z [
P
m]
O
=1450nm
O
=1520nm
O
=1590nm
d
30 [
P
m]
850
1000
700
z [
P
m]
O
=1450nm
O
=1520nm
O
=1590nm
Figure 2
|
Simulation results of dispersion-engineered metasurfaces based on hypothetical
meta-atoms. a
, Simulated deflection angles for gratings with regular, zero, positive, and hyper dispersions.
The gratings are 150
μ
m wide and have a 10-degree deflection angle at 1520 nm.
b
, Simulated focal
distances for metasurface focusing mirrors with different types of dispersion. The mirrors are 500
μ
m in
diameter and have a focal distance of 850
μ
m at 1520 nm. All gratings and focusing mirrors are designed
using hypothetical meta-atoms that provide independent control over phase and dispersion (see Methods
for details).
c
, Intensity in the axial plane for the focusing mirrors with regular negative,
d
zero,
e
positive,
and
f
hyper dispersions plotted at three wavelengths (see Extended Data Fig. 3 for other wavelengths).
13
200
400
600
1.44
Post side [nm]
1.52
1.6
Wavelength [
P
m]
Post side [nm]
200
400
600
1
0.8
c
1
0.8
Reflection amplitude
Reflection amplitude (|r|)
Reflection amplitude
1.6
Wavelength [
P
m]
1.52
1.44
1
0.8
I
[Rad]
Reflection phase (
I
)
Post side [nm]
Wavelength [
P
m]
200
400
600
1.44
1.52
1.6
d
0
S
S
Phase [Rad]
Phase/2
S
200
400
600
Post side [nm]
0
2
4
1.6
Wavelength [
P
m]
1.52
1.44
0
-0.8
b
Top view
a
Metal
D
-Si
SiO
2
|r|Ee
i
I
E
0
2
S
S
-200
-100
0
I
'
[Rad
/
O
]
Side view
e
500 nm
f
5
P
m
g
2
P
m
Figure 3
|
High dispersion silicon meta-atoms. a
, A meta-atom composed of a square cross-section
amorphous silicon nano-post on a silicon dioxide layer on a metallic reflector. Top and side views of the
meta-atoms arranged on a square lattice are also shown.
b
, Simulated dispersion versus phase plot for the
meta-atom shown in (a) at
λ
0
=
1520 nm.
c
, Simulated reflection amplitude, and
d
phase as a function of
the nano-post side length and wavelength. The reflection amplitude and phase along the dashed lines are
plotted on the right.
e
-
g
, Scanning electron micrographs of the fabricated nano-posts and devices.
14