of 21
Controlling the sign of chromatic dispersion in
diffractive optics with dielectric metasurfaces:
supplementary materials
E
HSAN
A
RB
ABI
1
,
A
MIR
A
RB
ABI
1,2
,
S
EYEDEH
M
AHSA
K
AMALI
1
,
Y
U
H
ORIE
1
,
AND
A
NDREI
F
ARA
ON
1,*
1
T.
J.
Watson
Laboratory
of
Applied
Physics,
California
Institute
of
Technology,
1200
E.
California
Blvd.,
Pasadena,
CA
91125,
USA
2
Department
of
Electrical
and
Computer
Engineering,
University
of
Massachusetts
Amherst,
151
Holdsworth
Wa y,
Amherst,
MA
01003,
USA
*
Corresponding
author:
A.F.:
faraon@caltech.edu
Published 7 June 2017
This
document
contains
supplementary
methods
and
materials
for
“Controlling
the
sign
of
chro-matic
dispersion
in
diffractive
optics
with
dielectric
metasurfaces
,
"
http
s
://
doi.org/10.1364/optica.4.000625
.
©
2017
Optical
Society
of
America
http
s
://doi.org/10.1364/optica.
4
.
000625
.s
00
1
S1. MATERIALS AND METHODS
Simulation and design.
The gratings with different dispersions discussed in Fig. 2(d)
were designed 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 transmission at each point on the metasurface grat-
ing. 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 interpolation
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. 2(e-i). 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 corresponding 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 ex-
pansion method to find the intensity in the axial plane. The focal
distances plotted in Fig. 2(e) show the distance of the maximum
intensity point from the mirrors at each wavelength. The grat-
ings and focusing mirrors discussed in Figs. 4(a), 5(a), and 5(c)
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.
If the actual meta-atoms provided an exactly linear disper-
sion (i.e. if their phase was exactly linear with frequency over
the operation bandwidth), one could use the required values
of the phase and dispersion at each lattice site to choose the
best meta-atom (knowing the coordinates of one point on a line
and its slope would suffice to determine the line exactly). The
phases of the actual meta-atoms, however, do not follow an ex-
actly linear curve [Fig. 3(d)]. Therefore, to minimize the error
between the required phases, and the actual ones provided by
the meta-atoms, we have used a minimum weighted Euclidean
distance method to design the devices fabricated and tested in
the manuscript: at each point on the metasurface, we calculate
the required complex reflection at eight wavelengths (1450 nm
to 1590 nm, at 20 nm distances). We also calculate the complex
reflection provided by each nano-post at the same wavelengths.
To find the best meta-atom for each position, we calculate the
weighted Euclidean distance between the required reflection vec-
tor, and the reflection vectors provided by the actual nano-posts.
The nano-post with the minimum distance is chosen at each
point. As a result, the chromatic dispersion is indirectly taken
into account, not directly. The weight function can be used to
increase or decrease the importance of each part of the spectrum
depending on the specific application. In this work, we have cho-
sen an inverted Gaussian weight function (
exp
((
λ
λ
0
)
2
/
2
σ
2
)
,
λ
0
=
1520 nm,
σ
=
300 nm) for all the devices to slightly em-
phasize the importance of wavelengths farther from the center.
In addition, we have also designed a dispersionless lens with
σ
=
50 nm (the measurement results of which are provided in
Figs.
S13
and
S14
) for comparison. The choice of 8 wavelengths
to form and compare the reflection vectors is relatively arbi-
trary; however, the phases of the nano-posts versus wavelength
are smooth enough, such that they can be well approximated
by line segments in 20 nm intervals. In addition, performing
the simulations at 8 wavelengths is computationally not very
expensive. Therefore, 8 wavelengths are enough for a 150 nm
bandwidth here, and increasing this number may not result in a
considerable improvement in the performance.
Reflection amplitude and phase of the meta-atoms were
found using rigorous coupled wave analysis technique [
1
]. For
each meta-atom size, a uniform array on a subwavelength lattice
was simulated using a normally incident plane wave. The sub-
wavelength lattice ensures the existence of only one propagating
mode which justifies the use of only one amplitude and phase
for describing 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 indices were set as follows in the simula-
tions: 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 Fig.
S15
.
The FDTD simulations of the gratings (Figs. 4(e-h)) were per-
formed using a normally incident plane-wave illumination with
a Gaussian amplitude in time (and thus a Gaussian spectrum) in
MEEP [
2
]. 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. 4(e-h) 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 de-
posited 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 sam-
ple 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 Fig.
S8
(a). Light emit-
ted from a tunable laser source (Photonetics TUNICS-Plus) was
collimated using a fiber collimation 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 or-
der to decrease the measurement error (similar to Fig.
S8
(b)).
The light reflected from the device was redirected using the
same beamsplitter, and imaged using a custom built microscope.
The microscope consists of a 50X objective (Olympus LMPlanFL
N, NA=0.5), a tube lens with a 20 cm focal distance (Thorlabs
AC254-200-C-ML), and an InGaAs camera (Sensors Unlimited
320HX-1.7RT). The grating deflection angle was found by calcu-
lating 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 approxi-
mately 40
μ
m iris in the object plane) and a photodetector (Thor-
labs 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 wave-
length. 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 grating efficien-
cies, the setup shown in Supporting Information Fig.
S8
(b) 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.
S2. CHROMATIC DISPERSION OF DIFFRACTIVE DE-
VICES.
Chromatic dispersion of a regular diffractive grating or lens is
set by its function. The grating momentum for a given order of
a grating with a certain period is constant and does not change
with changing the wavelength. If we denote the size of the
grating reciprocal lattice vector of interest by
k
G
, we get:
sin
(
θ
) =
k
G
2
π
/
λ
θ
=
sin
1
(
k
G
2
π
/
λ
)
,
(1)
where
θ
is the deflection angle at a wavelength
λ
for normally
incident beam. The chromatic angular dispersion of the grating
( d
θ
/d
λ
) is then given by:
d
θ
d
λ
=
k
G
/2
π
1
(
k
G
λ
/2
π
)
2
=
tan
(
θ
)
λ
.
(2)
and in terms of frequency:
d
θ
d
ω
=
tan
(
θ
)
ω
.
(3)
2
Therefore, the dispersion of a regular grating only depends on
its deflection angle and the wavelength. Similarly, focal distance
of one of the focal points of diffractive and metasurface lenses
changes as d
f
/d
λ
=
f
/
λ
(thus d
f
/d
ω
=
f
/
ω
([3–5]).
S3. CHROMATIC DISPERSION OF MULTIWAVELENGTH
DIFFRACTIVE DEVICES.
As it is mentioned in the main text, multiwavelength diffractive
devices ([4–6]) do not change the dispersion of a given order in
a grating or lens. They are essentially multi-order gratings or
lenses, where each order has the regular (negative) diffractive
chromatic dispersion. These devices are designed such that at
certain distinct wavelengths of interest, one of the orders has
the desired deflection angle or focal distance. If the blazing
of each order at the corresponding wavelength is perfect, all
of the power can be directed towards that order at that wave-
length. However, at wavelengths in between the designed wave-
lengths, where the grating or lens is not corrected, the multiple
orders have comparable powers, and show the regular diffrac-
tive dispersion. This is schematically shown in Fig.
S1
(a). Figure
S1
(b) compares the chromatic dispersion of a multi-wavelength
diffractive lens to a typical refractive apochromatic lens.
S4. GENERALIZATION OF CHROMATIC DISPERSION
CONTROL TO NONZERO DISPERSIONS.
Here we present the general form of equations for the dispersion
engineered metasurface diffractive devices. We assume that the
function of the device is set by a parameter
ξ
(
ω
)
, where we
have explicitly shown its frequency dependence. For instance,
ξ
might denote the deflection angle of a grating or the focal
distance of a lens. The phase profile of a device with a desired
ξ
(
ω
)
is given by
φ
(
x
,
y
,
ξ
(
ω
)
;
ω
) =
ω
T
(
x
,
y
,
ξ
(
ω
))
,
(4)
which is the generalized form of the Eq. (1). We are interested in
controlling the parameter
ξ
(
ω
)
and its dispersion (i.e. derivative)
at a given frequency
ω
0
.
ξ
(
ω
)
can be approximated as
ξ
(
ω
)
ξ
0
+
∂ξ
/
∂ω
|
ω
=
ω
0
(
ω
ω
0
)
over a narrow bandwidth around
ω
0
.
Using this approximation, we can rewrite
4 as
φ
(
x
,
y
;
ω
) =
ω
T
(
x
,
y
,
ξ
0
+
∂ξ
/
∂ω
|
ω
=
ω
0
(
ω
ω
0
))
.
(5)
At
ω
0
, this reduces to
φ
(
x
,
y
;
ω
)
|
ω
=
ω
0
=
ω
0
T
(
x
,
y
,
ξ
0
)
,
(6)
and the phase dispersion at
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
.
(7)
Based on Eqs. (6) and (7) the values of
ξ
0
and
∂ξ
/
∂ω
|
ω
=
ω
0
can be set independently, if the phase
φ
(
x
,
y
,
ω
0
)
and its deriva-
tive
∂φ
/
∂ω
can be controlled simultaneously and independently.
Therefore, the device function at
ω
0
(determined by the value
of
ξ
0
) and its dispersion (determined by
∂ξ
/
∂ω
|
ω
=
ω
0
) will be
decoupled. The zero dispersion case is a special case of Eq. (7)
with
∂ξ
/
∂ω
|
ω
=
ω
0
=
0. In the following we apply these results
to the special cases of blazed gratings and spherical-aberration-
free lenses (also correct for spherical-aberration-free focusing
mirrors).
For a 1-dimensional conventional blazed grating we have
ξ
=
θ
(the deflection angle), and
T
=
x
sin
(
θ
)
. Therefore the
phase profile with a general dispersion is given by:
φ
(
x
;
ω
) =
ω
x
sin
[
θ
0
+
D
(
ω
ω
0
)]
,
(8)
where
D
=
∂θ
/
∂ω
|
ω
=
ω
0
=
ν
D
0
, and
D
0
=
tan
(
θ
0
)
/
ω
0
is the
angular dispersion of a regular grating with deflection angle
θ
0
at the frequency
ω
0
. We have chosen to express the generalized
dispersion
D
as a multiple of the regular dispersion
D
0
with a
real number
ν
to benchmark the change in dispersion. For in-
stance,
ν
=
1 corresponds to a regular grating,
ν
=
0 represents
a dispersionless grating,
ν
=
1 denotes a grating with positive
dispersion, and
ν
=
3 results in a grating three times more dis-
persive than a regular grating (i.e. hyper-dispersive). Various
values of
ν
can be achieved using the method of simultaneous
control of phase and dispersion of the meta-atoms, and thus
we can break this fundamental relation between the deflection
angle and angular dispersion. The phase derivative necessary
to achieve a certain value of
ν
is given by:
∂φ
(
x
;
ω
)
∂ω
|
ω
=
ω
0
=
x
/
c
sin
(
θ
0
)(
1
ν
)
,
(9)
or in terms of wavelength:
∂φ
(
x
;
λ
)
∂λ
|
λ
=
λ
0
=
2
π
λ
0
2
x
sin
(
θ
0
)(
1
ν
)
.
(10)
For a spherical-aberration-free lens we have
ξ
=
f
and
T
(
x
,
y
,
f
) =
x
2
+
y
2
+
f
2
/
c
. Again we can approximate
f
with its linear approximation
f
(
ω
) =
f
0
+
D
(
ω
ω
0
)
, with
D
=
f
/
∂ω
|
ω
=
ω
0
denoting the focal distance dispersion at
ω
=
ω
0
. The regular dispersion for such a lens is given by
D
0
=
f
0
/
ω
0
. Similar to the gratings, we can write the more gen-
eral form for the focal distance dispersion as
D
=
ν
D
0
, where
ν
is some real number. In this case, the required phase dispersion
is given by:
∂φ
(
x
,
y
;
ω
)
∂ω
|
ω
=
ω
0
=
1
c
[
x
2
+
y
2
+
f
0
2
+
ν
f
0
2
x
2
+
y
2
+
f
0
2
]
,
(11)
which can also be expressed in terms of wavelength:
∂φ
(
x
,
y
;
λ
)
∂λ
|
λ
=
λ
0
=
2
π
λ
0
2
[
x
2
+
y
2
+
f
0
2
+
ν
f
0
2
x
2
+
y
2
+
f
0
2
]
.
(12)
S5. MAXIMUM META-ATOM DISPERSION REQUIRED
FOR CONTROLLING CHROMATIC DISPERSION OF
GRATINGS AND LENSES.
Since the maximum achievable dispersion is limited by the meta-
atom design, it is important to find a relation between the maxi-
mum dispersion required for implementation of a certain meta-
surface device, and the device parameters (e.g. size, focal dis-
tance, deflection angle, etc.). Here we find these maxima for the
cases of gratings and lenses with given desired dispersions.
For the grating case, it results from Eq. (10
) that the maximum
required dispersion is given by
max
(
∂φ
(
x
;
λ
)
∂λ
|
λ
=
λ
0
) =
k
0
X
sin
(
θ
0
)
λ
0
(
1
ν
)
,
(13)
3
where
X
is the length of the grating, and
k
0
=
2
π
/
λ
0
is the
wavenumber. It is important to note that based on the value of
ν
, the sign of the meta-atom dispersion changes. However, in
order to ensure a positive group velocity for the meta-atoms, the
dispersions should be negative. Thus, if 1
ν
>
0, a term should
be added to make the dispersion values negative. We can always
add a term of type
φ
0
=
kL
0
to the phase without changing the
function of the device. This term can be used to shift the required
region in the phase-dispersion plane. Therefore, it is actually the
difference between the minimum and maximum of Eqs.
10
and
12
that sets the maximum required dispersion. Using a similar
procedure, we find the maximum necessary dispersion for a
spherical-aberration-free lens as
φ
max
=
k
0
f
λ
0
Θ
+
ν
Θ
1
ν
ν
<
1
Θ
+
ν
Θ
2
ν
1
<
ν
<
Θ
(
1
ν
)
2
Θ
<
ν
<
Θ
(
Θ
+
ν
Θ
1
ν
)
Θ
<
ν
,
(14)
where
f
is the focal distance of the lens, and
Θ
= (
f
2
+
R
2
)
/
f
2
=
1
/
(
1
NA
2
)
( R: lens radius, NA: numerical aper-
ture).
log
[
φ
max
/
(
k
0
f
/
λ
0
)]
is plotted in Fig.
S2
(a) as a function
of NA and
ν
. In the simpler case of dispersionless lenses (i.e.
ν
=
0), Eq. (14) can be further simplified to
φ
max
=
k
0
R
λ
1
1
NA
2
NA
≈−
k
0
R
NA
2
λ
(15)
where
R
is the lens radius and the approximation is valid for
small values of NA. The maximum required dispersion for the
dispersionless lens is normalized to
k
0
R
/
λ
0
and is plotted in
Supporting Information Fig.
S2
(b) as a function of NA.
S6. FERMAT’S PRINCIPLE AND THE PHASE DISPER-
SION RELATION.
Phase only diffractive devices can be characterized by a local
grating momentum (or equivalently phase gradient) resulting in
a local deflection angle at each point on their surface. Here we
consider the case of a 1D element with a given local phase gradi-
ent (i.e.
φ
x
=
∂φ
/
x
) and use Fermat’s principle to connect the
frequency derivative of the local deflection angle (i.e. chromatic
dispersion) to the frequency derivative of
φ
x
(i.e.
∂φ
x
/
∂ω
). For
simplicity, we assume that the illumination is close to normal,
and that the element phase does not depend on the illumina-
tion angle (which is in general correct in local metasurfaces and
diffractive devices). Considering Fig.
S16
(a), we can write the
phase acquired by a ray going from point A to point B, and
passing the interface at x as:
Φ
(
x
,
ω
) =
ω
c
[
n
1
x
2
+
y
A
2
+
n
2
(
d
x
)
2
+
y
B
2
] +
φ
(
x
,
ω
)
(16)
To minimize this phase we need:
Φ
(
x
,
ω
)
x
=
ω
c
[
n
1
x
x
2
+
y
A
2
+
n
2
(
d
x
)
(
d
x
)
2
+
y
B
2
] +
φ
x
=
0.
(17)
For this minimum to occur at point O (i.e.
x
=
0):
φ
x
(
ω
) =
ω
c
n
2
d
r
=
n
2
ω
c
sin
(
θ
(
ω
))
(18)
which is a simple case of the diffraction equation, and where
r
=
d
2
+
y
B
2
is the OB length. At
ω
+
d
ω
, we get the following
phase for the path from A to B’ [Fig.
S16
(b)]:
Φ
(
x
,
ω
+
d
ω
) =
ω
+
d
ω
c
[
n
1
x
2
+
y
A
2
+
n
2
(
d
x
+
d
x
)
2
+ (
y
B
+
d
y
)
2
] +
φ
(
x
,
ω
+
d
ω
)
(19)
where we have chosen B’ such that OB and OB’ have equal
lengths. Minimizing the path passing through O:
φ
x
(
ω
+
d
ω
) =
ω
+
d
ω
c
n
2
(
d
+
dx
)
r
=
n
2
(
ω
+
d
ω
)
c
sin
(
θ
(
ω
+
d
ω
))
(20)
subtracting
18
from
20, and setting
φ
x
(
ω
+
d
ω
)
φ
x
(
ω
) =
∂φ
x
∂ω
d
ω
, we get:
∂φ
x
∂ω
=
n
2
c
sin
(
θ
(
ω
)) +
d
θ
d
ω
n
2
ω
c
cos
(
θ
(
ω
))
.
(21)
One can easily recognize the similarity between
21
and
7.
S7. RELATION BETWEEN DISPERSION AND QUALITY
FACTOR OF HIGHLY REFLECTIVE OR TRANSMISSIVE
META-ATOMS.
Here we show that the phase dispersion of a meta-atom is lin-
early proportional to the stored optical energy in the meta-atoms,
or equivalently, to the quality factor of the resonances supported
by the mata-atoms. To relate the phase dispersion of transmis-
sive or reflective meta-atoms to the stored optical energy, we
follow an approach similar to the one taken in chapter 8 of [
7
]
for finding the dispersion of a single port microwave circuit. We
start from the frequency domain Maxwell’s equations:
∇×
E
=
i
ωμ
H
,
∇×
H
=
i
ωe
E
,
(22)
and take the derivative of the Eq. 22 with respect to frequency:
∇×
E
∂ω
=
i
μ
H
+
i
ωμ
H
∂ω
,
(23)
∇×
H
∂ω
=
i
e
E
i
ωe
E
∂ω
.
(24)
Multiplying Eq.
23
by
H
and the conjugate of Eq.
24
by
E
/
∂ω
,
and subtracting the two, we obtain
∇·
(
E
∂ω
×
H
) =
i
μ
|
H
|
2
+
i
ωμ
H
∂ω
·
H
i
ωe
E
∂ω
·
E
.
(25)
Similarly, multiplying Eq.
24
by
E
and the conjugate of Eq.
23
by
H
/
∂ω
, and subtracting the two we find:
∇·
(
H
∂ω
×
E
) =
i
e
|
E
|
2
i
ωe
E
∂ω
·
E
+
i
ωμ
H
∂ω
·
H
.
(26)
Subtracting Eq. 26 from Eq.
25
we get:
∇·
(
E
∂ω
×
H
H
∂ω
×
E
) =
i
μ
|
H
|
2
+
i
e
|
E
|
2
.
(27)
Integrating both sides of Eq.
27, and using the divergence theo-
rem to convert the left side to a surface integral leads to:
V
(
E
∂ω
×
H
H
∂ω
×
E
) =
i
V
(
μ
|
H
|
2
+
e
|
E
|
2
)
dv
=
2
iU
,
(28)
4
where
U
is the total electromagnetic energy inside the volume
V
,
and
V
denotes the surrounding surface of the volume. Now we
consider a metasurface composed of a subwavelength periodic
array of meta-atoms as shown in Fig.
S17
. We also consider
two virtual planar boundaries
Γ
1
and
Γ
2
on both sides on the
metasurface (shown with dashed lines in Fig.
S17
). The two vir-
tual boundaries are considered far enough from the metasurface
that the metasurface evanescent fields die off before reaching
them. Because the metasurface is periodic with a subwavelength
period and preserves polarization, we can write the transmitted
and reflected fields at the virtual boundaries in terms of only one
transmission
t
and reflection
r
coefficients. The fields at these
two boundaries are given by:
E
1
=
E
+
rE
H
1
=
ˆ
z
×
(
E
η
1
r
E
η
1
)
E
2
=
tE
H
2
=
t
ˆ
z
×
E
η
2
(29)
where
E
is the input field,
E
1
and
E
2
are the total electric fields
at
Γ
1
and
Γ
2
, respectively, and
η
1
and
η
2
are wave impedances
in the materials on the top and bottom of the metasurface.
Inserting fields from Eq.
29
to Eq.
28, and using the unifor-
mity of the fields to perform the integration over one unit of
area, we get:
r
∂ω
r
|
E
|
2
η
1
+
t
∂ω
t
|
E
|
2
η
2
=
i
̃
U
(30)
where
̃
U
is the optical energy per unit area that is stored in
the metasurface layer. For a loss-less metasurface that is totally
reflective (i.e.
t
=
0 and
r
=
e
i
φ
), we obtain:
∂φ
∂ω
=
̃
U
P
in
,
(31)
where we have used
P
in
=
|
E
|
2
/
η
1
to denote the per unit area
input power. Finally, the dispersion can be expressed as:
∂φ
∂λ
=
∂φ
∂ω
∂ω
∂λ
=
ω
λ
̃
U
P
in
.
(32)
We used Eq.
32
throughout the work to calculate the disper-
sion from solution of the electric and magnetic fields at a single
wavelength, which reduced simulation time by a factor of two.
In addition, in steady state the input and output powers are
equal
P
out
=
P
in
, and therefore we have:
∂φ
∂λ
=
1
λ
ω
̃
U
P
out
=
Q
λ
(33)
where we have assumed that almost all of the stored energy is
in one single resonant mode, and
Q
is the quality factor of that
mode. Therefore, in order to achieve large dispersion values,
resonant modes with high quality factors are necessary.
5
(a)
(b)
Apochromatic
Multiwavelength
f [a.u.]
Wavelength [a.u.]
0.8
1
1.2
0.8
1
1.2
f
Short wavelength
f
Middle wavelength
Long wavelength
Multi-wavelength
Regular negative
Apochromatic
f
Fig. S1.
Comparison of regular, multi-wavelength, and apochromatic lenses. (a) Schematic comparison of a regular, a multi-
wavelength, and an apochromatic metasurface lens. The multi-wavelength lens is corrected at a short and a long wavelength to
have a single focal point at a distance
f
, but it has two focal points at wavelengths in between them, none of which is at
f
. The apoc-
hromatic lens is corrected at the same short and long wavelengths, and in wavelengths between them it will have a single focus
very close to
f
. (b) Focal distances for three focal points of a multiwavelength lens corrected at three wavelengths, showing the
regular dispersion (i.e.
f
1
/
λ
) of each focus with wavelength. For comparison, focal distance for the single focus of a typical
apochromatic lens is plotted.
Exact
NA/2
0
0
1
1
0.5
0.5
NA
2R
(b)
NA
ν
=D/D
0
(a)
log[
φ
max
/
(-k
0
f/
λ
)]
[
φ
max
/
(-k
0
R/
λ
)]
log[
φ
max
/
(-k
0
f/
λ
)]
0.5
0
1
-2
1
4
-2
-1
0
1
Fig. S2.
Maximum required dispersion of meta-atoms for lenses. (a) Maximum meta-atom dispersion necessary to control the
dispersion of a spherical-aberration-free lens. The maximum dispersion is normalized to
k
0
f
/
λ
0
and is plotted on a logarithmic
scale. (b) Normalized (to
k
0
R
/
λ
0
) maximum dispersion required for a dispersionless lens.
R
is the radius,
f
is the focal distance,
and NA is the numerical aperture of the lens.
6