www.sciencemag.org/cgi/content/full/science.aaa2494/DC1

Supplementary Materials for

Multiwavelength achromatic metasurfaces by dispersive phase

compensation

Francesco Aieta, Mikhail A. Kats, Patrice Genevet, Federico Capasso*

*Corresponding author. E-mail: capasso@seas.harvard.edu

Published 19 February 2015 on Science Express

DOI: 10.1126/science.aaa2494

This PDF file includes:

Materials and Methods

Supplementary Text

Figs. S1 to S10

Full Reference List

1

Supplementary Information:

Multiwavelength Achromatic Metasurfaces by Dispersive Phase Compensation

Francesco Aieta1, †, Mikhail A. Kats1, ‡, Patrice Genevet1, ꜜ and Federico Capasso1,*

1School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA

†Current affiliation: Hewlett-Packard Laboratories, Palo Alto, CA 94304, USA

‡Current affiliation: ECE Department, University of Wisconsin, Madison, WI 53706, USA

ꜜCurrent affiliation: SIMTech - Singapore Institute of Manufacturing Technology, 638075, Singapore

*capasso@seas.harvard.edu

This PDF file includes:

Materials and Methods

Supplementary Text

Figs S1 to S10

Materials and Methods

1. Fabrication of achromatic metasurfaces

The device was fabricated by depositing 400 nm amorphous silicon (a-Si) on a fused silica (SiO2) substrate at

300 ℃ by plasma-enhanced chemical vapor deposition (PECVD). The rectangular dielectric resonators (RDR)

were defined by electron-beam lithography using the positive resist ZEP-520A from ZEON diluted in Anisole

with a ratio of 1:1, exposed to a dose of 300 μC/cm2 (500 pA, 125kV) and developed for 50 sec at room

temperature in Oxylene. The silicon ridges were then obtained by dry etching using Bosch processing. At the

end of the process, the residual resist layer was removed with a one-hour bath in Remover 1165, rinsed in PG

2

Remover and exposed to 1 min of O2 plasma at 75 W. The sample used for the SEM image in Fig. 4A was

sputter-coated with 5 nm of platinum/palladium to eliminate charging in the SEM.

2. Optical properties of the amorphous silicon

We report the experimental data of the optical properties of the amorphous silicon layer in the wavelength range

of 400 nm – 850 nm obtained with an Imaging Ellipsometer “nanofilm_ep4” performed by Accurion. We used

the Cody-Lorentz dispersion model (23) to extrapolate the refractive index into the near infrared. The values

extracted were used for the numerical simulations.

Fig. S1. Ellipsometric characterization of the 400 nm thin a-Si film deposited with PECVD. The blue curve is obtained by

fitting the experimental data (in red) with the analytical model. The imaginary part of the refractive index is negligible at

the wavelengths of interest (1100 nm – 2000 nm).

3. Experimental setup

The measurement set-up consists of a supercontinuum laser (NKT “SuperK”) equipped with a set of acousto-

optic tunable filters (NKT “Select”) to tune the emission from 1100 nm to 2000 nm with a line-width of 15 nm.

3

The output of the laser is focused with a long focal distance lens (f = 20 cm, not shown in Fig. 3B) to guarantee

uniform illumination of the 240 μm x 240 μm metasurface. The intensity of the transmitted light as a function of

the angle 𝜃 is recorded by using a broadband InGaAs detector with active area 0.78 mm2 (Thorlabs DET10D)

mounted on a motorized rotation stage. Data are collected from -30° to 30° with 0.25° resolution. In Fig. 3C the

experimental data below λ =1300 nm are missing because of the low power of the source and low sensitivity of

the detector below that wavelength. For the measurement of the efficiency, the detector is replaced by the head

of a power meter (Ge photodiode sensor) with large active area. The efficiency values are normalized to the

power incident on the back of the device.

Supplementary Text

1. Rectangular dielectric resonator

Dielectric antennas are resonant elements that interact with electromagnetic waves via a displacement current

and can have both electric and magnetic resonances (24). Primarily used in the microwave frequency range,

dielectric antennas have recently been proposed in the optical regime as an alternative to metallic antennas

because of their low losses at shorter wavelengths (11, 25). Nanostructures made of a material with a large

refractive index exhibit resonances while remaining small compared to the wavelength of light in free-space,

similarly to what occurs in plasmonic antennas. In our demonstration the phase is independently controlled for

different wavelengths by taking advantage of the dense spectrum of resonances of subwavelength dielectric

resonators.

We studied the scattering properties of a rectangular dielectric resonator (RDR), a resonator with rectangular

cross-section in the x-z plane and infinite extent along y, as shown in the inset of Fig. S2. Despite the simple

geometry, an analytical closed-form for the electromagnetic fields does not exist for RDRs (24) therefore our

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designs are optimized using finite-difference time-domain (FDTD) simulations. However, in order to estimate

the spectral position of the resonant modes, an approximated solution based on the dielectric waveguide model

is derived (26). The model predicts the existence of TMmn and TEmn modes inside the resonator. TM modes are

excited by an electric field with polarization parallel to the side w of the RDR, while TE modes are activated by

an excitation polarized along the y-axis (Fig. S2). The subscripts m and n denote the number of field extrema in

the x- and z-directions. The derivation of the model and a detailed comparison with FDTD simulations are

reported in the next section. Fig. S2A shows scattering efficiencies calculated from FDTD simulations for an

isolated silicon RDR in vacuum with geometry w = t = 350 nm and excited with TM-polarized light (black line).

In analogy to the scattering of dielectric spheres rigorously described by Mie theory, the first two peaks

correspond to the magnetic and electric dipole resonances (25). This is confirmed by showing the scattering

spectra of the same RDR independently excited with an electric and a magnetic dipole placed at the center of

the resonator and oriented along the x and y axis, respectively. The grey arrows indicate the resonant

frequencies calculated with the analytical model for the first two modes (TM11 and TM21). The electric field

intensity distributions at the two resonances (Fig. S2, B and C) confirm the electric and magnetic dipole-like

scattering. At shorter wavelengths, many higher orders exist with multipole-like scattering.

Fig. S2. Scattering properties of an isolated silicon rectangular dielectric resonator with dimensions w = t = 350 nm

(infinite length along the y-axis) excited by a plane wave traveling at normal incidence along the z-axis. (A) Scattering

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efficiency Qscat (defined as ratio of the two-dimensional scattering cross-section, which has the dimension of a length, and

the geometric length w) for TM excitation (black), electric (blue) and magnetic (red) dipole excitation. The grey arrows

indicate the resonant frequencies calculated with the analytical model for the first two modes (TM11 and TM21). (B and C)

Electric field intensity distribution at the two resonant frequencies obtained with plane wave excitation. The white lines

give a schematic representation of the instantaneous electric field lines around the resonator.

2. Analytical model for rectangular dielectric resonators

A simple analytical expression based on dielectric waveguide model (DWM) is derived to estimate the resonant

frequencies of a rectangular dielectric resonator (RDR) (24). According to this model an isolated RDR is

assumed to be a truncated section of an infinite dielectric waveguide and the field pattern inside the resonator is

a standing wave along x inside the dielectric and decays exponentially outside (Fig. S3). If we truncate along z,

a standing wave pattern is setup along z as well. It can be assumed that the standing waves along x and z are

governed by the same equations. After writing the field components and imposing the boundary conditions we

can derive the transcendental equation from which the wave numbers kx and kz corresponding to the resonant

wavelengths can be calculated.

Fig. S3. Geometry and field distribution of a Rectangular Dielectric Resonator

6

TM modes are calculated by solving the Helmholtz equation (Fig. S3):

𝛻2𝐻𝑦 + 𝑘0𝜀𝑟𝐻𝑦 = 0

Assuming a harmonic field, from the Ampere law one has:

𝛻 × 𝐻 = 𝜀𝑟𝜕𝐸

𝜕𝑡→

{

𝐸𝑥 =

𝑗

𝜔𝜀𝑟

𝜕𝐻𝑦

𝜕𝑧

𝐸𝑧 = −𝑗

𝜔𝜀𝑟

𝜕𝐻𝑦

𝜕𝑥

Given the symmetry of the structure with respect to x=0, the expressions of the fields inside the resonator (-

w/2< x< -w/2 and -t/2< t< -t/2), and in the half-planes left (x>w/2), right (x< -w/2), up (z>t/2) and down (z< -t/2)

are:

-w/2< x< -w/2 and -t/2< t< -t/2:

{

𝐻𝑦 = 𝐴 cos(𝑘𝑥𝑥) cos(𝑘𝑧𝑧)

𝐸𝑥 = −𝑗

𝜔𝜀𝑟𝐴𝑘𝑧 cos(𝑘𝑥𝑥) sin(𝑘𝑧𝑧)

𝐸𝑧 =𝑗

𝜔𝜀𝑟𝐴𝑘𝑥 sin(𝑘𝑥𝑥) cos(𝑘𝑧𝑧)

z>t/2 ; z< -t/2:

{

𝐻𝑦 = 𝐵 cos(𝑘𝑥𝑥) 𝑒

−𝑘𝑧𝑜(𝑧−𝑡/2)

𝐸𝑥 = −𝑗

𝜔𝐵𝑘𝑧𝑜 cos(𝑘𝑥𝑥) 𝑒

−𝑘𝑧𝑜(𝑧−𝑡/2)

𝐸𝑧 =𝑗

𝜔𝐵𝑘𝑥 sin(𝑘𝑥𝑥) 𝑒

−𝑘𝑧𝑜(𝑧−𝑡/2)

x>w/2 ; x< -w/2:

{

𝐻𝑦 = 𝐶𝑒−𝑘𝑥𝑜(𝑥−𝑤/2) cos(𝑘𝑧𝑥)

𝐸𝑥 = −𝑗

𝜔𝐶𝑘𝑧𝑒

−𝑘𝑥𝑜(𝑥−𝑤/2) sin(𝑘𝑧𝑧)

𝐸𝑧 =𝑗

𝜔𝐶𝑘𝑥𝑜𝑒

−𝑘𝑥𝑜(𝑥−𝑤/2) cos(𝑘𝑧𝑥)

Where A, B and C are variables to be calculated. The boundary conditions at the edges of the RDR read:

{

𝐸𝑧,𝐼𝑁 = 𝐸𝑧,𝐿/𝑅 𝑥 = 𝑤/2

𝐸𝑥,𝐼𝑁 = 𝐸𝑥,𝑈/𝐷 𝑧 = 𝑡/2

𝜀𝑟𝐸𝑥,𝐼𝑁 = 𝐸𝑥,𝐿/𝑅 𝑥 =𝑤

2

𝜀𝑟𝐸𝑧,𝐼𝑁 = 𝐸𝑧,𝑈/𝐷 𝑧 = 𝑡/2

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Applying these conditions, finally one obtains:

{

𝐴 = 𝑘𝑥𝑜𝜀𝑟 , 𝐶 = 𝑘𝑥 sin(𝑘𝑥𝑤/2)

𝐵 =𝑘𝑥𝑜𝑘𝑧

𝑘𝑧𝑜sin(𝑘𝑧𝑡/2)

𝒌𝒙𝒘 = 𝒎𝛑− 𝟐 𝐭𝐚𝐧−𝟏(𝒌𝒙/(𝜺𝒓𝒌𝒙𝒐))

𝒌𝒛𝒘 = 𝒑𝛑 − 𝟐 𝐭𝐚𝐧−𝟏(𝒌𝒛/(𝜺𝒓𝒌𝒛𝒐))

(S1)

Using the expressions:

{

𝑘𝑥

2 + 𝑘𝑧2 = 𝜀𝑟 𝑘0

2

𝑘𝑥𝑜 = √(𝜀𝑟 − 1)𝑘02 − 𝑘𝑥

2

𝑘𝑧𝑜 = √(𝜀𝑟 − 1)𝑘02 − 𝑘𝑧

2

The last two equations of Eq. S1 can be solved to give the wavevectors along x and z corresponding to the

resonant modes.

For TE modes, the Helmholtz equation for the electric field is used, and following a similar procedure

the transcendental equations for the resonant wavevectors are obtained:

{𝒌𝒙𝒘 = 𝒎𝛑− 𝟐 𝐭𝐚𝐧−𝟏(𝒌𝒙/𝒌𝒙𝒐)

𝒌𝒛𝒘 = 𝒑𝛑 − 𝟐 𝐭𝐚𝐧−𝟏(𝒌𝒛/𝒌𝒛𝒐)

This model is useful to design a RDR because it helps us to predict the spectral positions of the resonant modes

for a given geometry. The predictions of the model were validated by comparing the results with FDTD

simulations. The scattering cross section of an isolated RDR for TM excitation as the one in Fig. S2 with w=400

nm and t=500 nm is simulated, allowing us to visualize the resonances in terms of the distribution of the electric

and magnetic fields inside the resonator. In the range of wavelengths from 1000 nm to 3400 nm the following

resonant modes TM11, TM12 and TM13 are observed (Fig S4A). In Fig S4B the resonant wavelengths in the

simulation are compared with those calculated the mode: the results are in good agreement, with an error of

±5% consistent with other works in literature (27).

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Fig. S4. (A) Scattering cross section for an infinitely long silicon rectangular dielectric resonator in vacuum of width

w=400 nm and height t=500 nm excited with TM polarization. (B) Comparison between theoretical model and FDTD

simulation of the resonant wavelengths for the first three modes (TM11, TM12, TM13).

We also performed a comprehensive comparison of the model with simulations, by calculating the first resonant

mode for TE and TM excitation for different geometries of the RDR. The results are reported in Fig. S5.

Fig. S5. (A and B) Comparison between theoretical model and FDTD simulations of the resonant wavelengths for TE

excitation for different width w and t=300 nm (A) and t=500 nm (B). (C and D) same comparison but for TM excitation.

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3. Coupled rectangular dielectric resonators

By placing two RDRs in close proximity such that their near fields overlap, a system of coupled resonators is

created which changes the spectral positions and widths of the resonances. We can thus utilize the gap size and

position as additional degrees of freedom to engineer the scattering amplitude and phase. Because of the lack of

an analytical solution for coupled RDRs, we rely on FDTD simulations to predict their optical response.

To demonstrate the mechanism of light control at different wavelengths, consider the phase response required at

one particular position of the beam deflector demonstrated in the main text. From Eq. 2 the target phase values

for the unit cell centered at the position x = 64 μm for λ1, λ2 and λ3 are calculated to be 𝜑𝑚1

=142°, 𝜑𝑚2 =25°, and

𝜑𝑚3 =141°. Figure S6 confirms that a unit cell with geometry s=1 μm, t=400 nm, w1=300 nm w2=100 nm and

g=175 nm gives a field with uniform transmitted intensity (|E|2) at the three wavelengths of interest (circles in

Fig. S6) and phases 𝜑𝑚1 , 𝜑𝑚

2 and 𝜑𝑚3 matching our design (crosses in Fig. S6). When different unit cells

composed of the two coupled resonators are placed close each other, we expect the mutual coupling between

neighboring resonators to partially modify the amplitude and phase response compared to the isolated cell.

However this does not significantly compromise the overall response of the structure as shown from the results

in the main text.

Fig. S6. Normalized intensity (solid line) and phase shift (dashed line) of the unit cell. The colored crosses represent the

required phase values calculated from Eq. 2 for λ1, λ2 and λ3. The circles correspond to the scattered intensities for the

same wavelengths.

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4. Design of the unit cells

To implement a given functionality of the achromatic metasurface, a particular wavelength-dependent phase

function (Eq. 1) is realized by designing the scattering properties of unit cells consisting of coupled dielectric

resonators. We fix the unit cell width s = 1 μm, the height of the silicon resonators t = 400 nm and the minimum

value for w and g at 100 nm to keep the aspect ratio of the structure compatible with the fabrication process and

we run a cycle of FDTD simulations for different geometries to obtain the desired phase response 𝜑𝑚(𝑥, 𝜆) and

quasi-uniform transmitted amplitude. We swept the parameters w1, w2, g in the range from 100 nm to 950 nm

with steps of 25 nm in all the possible combinations, enforcing the condition that the sum of w1, w2 and g does

not exceed the size of the unit cell s, and calculated the transmitted intensity and the phase shift imparted by the

metasurface. The phase shift was calculated as the phase of the field at a distance of 10 cm away on the vertical

to the interface minus the phase accumulated by the light via propagation through the glass slab and the air

above the unit cell. For each simulation, if the transmitted intensity is at least 35% of the total incidence power

and the difference between the calculated phase at each wavelength and the target value for a specific unit cell

is less than 60°, the set of parameter is saved for that specific unit-cell. The root-mean-square error (RMSE) of

the phase for the three wavelengths is also calculated and saved. Every time a new set of parameters passes the

check-test for transmitted intensity and phase shift for a specific unit cell, the geometry corresponding to the

minimum RMSE is retained. For the design of the beam deflector for three wavelengths demonstrated in this

paper, the average RMSE of the phase among all the unit cells for the three wavelengths at the end of the

optimization is about 30°. This causes an imperfect match with the design requirements that will somewhat

reduce the performance of the device (residual diffraction orders and background).

5. Geometrical aperiodicity of the metasurface

The achromatic beam deflector presented in this work does not feature any structural periodicity. While in

previous works, a metasurface functionally equivalent to a blazed-grating was designed by repeating a single

11

super cell (5-11), in our case all unit cells are different from each other because the three phase ramps necessary

to deflect different wavelengths do not have any periodicity. Figure S7A shows the Fourier spectrum of the

geometry used for the dispersion-free beam deflector (after converting the CAD file to a square wave function

describing the profile of the metasurface, we calculate the Fourier transform). While it shows a peak at 1 μm-1

corresponding to the subwavelength unit cell width (s), no notable spectral components are present at lower

frequencies that could give diffraction orders. Therefore we expect complete suppression of any residual -1

diffraction order at the angular position -θ0 =17° that may rise from the imperfect phase or amplitude profile of

the metasurface. This is confirmed by the measured far-field intensity distribution for the full range of angles -

30° to +30° (Fig. S7B). Although we see some peaks outside the desired beaming angle, no intensity peak is

measured at 17° (-θ0) confirming the absence of any residual effect due to periodicity.

Fig. S7. (A) Spatial Fourier transform of the geometry of the dielectric resonators used for the dispersion-free beam

deflector. (B) Far-field measurement of the beam deflector for λ1, λ2 and λ3. Since the structure does not present any

periodicity, there is no peak in correspondence of the -1 order (θ=17°). (Inset) Close-up around the angle θ0 and - θ0.

6. Angle of incidence dependence

The metasurfaces described in this paper are designed to work with light arriving at normal incidence. When the

incoming beam impinges at an angle, the symmetry of excitation of the unit cell is broken. As a consequence,

other modes will be excited in the two-coupled resonators system affecting the phase and amplitude response.

Under this condition the device does not perform in general as an achromatic metasurface.

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For angles of incidence in the range ±1°, achromatic deflection is preserved (Fig. S8, A and B). Figures

S8, C and D show the simulated far-field intensity distribution of the beam deflector for light incident from air

at -3° and 8° angles with respect to the normal. In these cases the angles of deflection are not constant for the

three wavelengths and do not match the angles that we would expect if the phase gradients were those described

in Eq. 2. While with the design used here it is hard to maintain the desired phase and amplitude response for

off-normal excitation, we expect that by reducing the thickness of the RDR or by choosing a different type of

resonators (i.e. plasmonic antenna), the effect of oblique illumination will have a smaller impact on the resonant

response thus enabling the achromatic operation for a wide range of angle of incidence.

Fig. S8. FDTD Simulation of the beam deflector performance for non-normal incidence. The incoming beam forms an

angle of -1° (A), +1° (B), -3° (C) and +8° (D) with respect to the normal. The orange arrows indicate the expected

deflection angles for an achromatic metasurface.

7. Efficiency

From the analysis of the FDTD simulations we can understand the origin of the limited efficiency and how we

can improve it. Optical losses are negligible, as expected given the low absorption coefficient of Si in the near

infrared. For the three wavelengths of interest the average transmitted power is about 40% of the incident power

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while the remaining 60% is reflected. The transmitted power that is not directed to the desired angle of

deflection goes into residual diffraction orders (note for example the intensity peak at 𝜃 = 0° for λ = 1100 nm

in Fig 3B. This is mainly due to the imperfect realization of the phase function and non-uniform resonators’

scattering amplitudes across the metasurface. We expect that a more advanced algorithm for the selection of the

resonators geometry (e.g. genetic algorithms, particle swarm optimization, etc.), optimization of the other

parameters (s and t) or choice of a different type of resonator, would yield a more accurate approximation of the

target phase function, which would bring the efficiency of the device up to 40%. The large reflected component

is a result of the strong directionality of the RDR scattering towards the half-plane with higher refractive index.

Therefore using a low index substrate (e.g. porous silica or even an aerogel) would increase the efficiency to

almost 50%. Recently, a stack of three metasurfaces has been proposed to provide complete phase control and

eliminate the reflected power leading to 100% transmission at a single wavelength (28, 29). An alternative

proposed approach is based on the control of the spectral position of electric and magnetic dipole resonances in

dielectric resonators to achieve impedance matching (21). It has indeed been shown that when these two

resonances have the exact same contribution to the scattering of a nanoparticle, the interference of the two

scattering channels with the excitation produces perfect transmission and zero reflection (25). The multi-polar

resonances observed in our RDRs can be separated in electric- and magnetic-type of resonances depending on

the distribution of the fields and the scattering properties (Fig. S2, B and C). By designing dielectric resonators

with multiple electric and magnetic resonances that overlap at the wavelengths of interests, multi-spectral

control of the wavefront with high transmitted power could be achieved.

8. From multi-wavelength to broadband

Figs. 3C and 4J show that the metasurfaces approaches the desired functionality (angle of deflection or focal

length) also for the wavelengths near the designed ones, suggesting that one can achieve full chromatic

correction for a continuous bandwidth by increasing the number of corrected wavelengths.

14

In order to prove it for the deflectors, five different devices are designed for an increasing number of

wavelengths by using the same type of unit cell based on dielectric resonators and an algorithm similar to the

one described above. The operation of the metasurfaces is modeled with FDTD simulations. Figure S9 shows

the intensity at the deflection angle as a function of wavelength and angle. Design A controls the deflection

angle only at one wavelength (1550 nm). Similarly to a conventional dispersive diffractive grating, the other

wavelengths are dispersively deflected at different angles. As the number of controlled wavelength increases

(up to five in design E), the peak of intensity is near the target angle (θ=-17°) almost throughout the entire

bandwidth (1300 nm to 1800 nm). The generalization of the design to a higher number of wavelengths, affects

the efficiency of the metasurface. While design A deflects at θ0 90% of the transmitted light for a single

wavelength, the average intensity at θ0 for λ1, λ2, λ3, λ4 and λ5 in design E is about 17% of the transmitted

power. The high number of constraints in design E leads to a less accurate phase function able to satisfy Eq. 2.

However by introducing a more complex design with a higher number of degrees of freedom (as discussed in

the previous paragraph) we expect to obtain a more efficient metasurface optimized for many wavelengths.

Fig. S9. Density plot showing the simulated diffracted intensity at the deflection angle as a function of wavelength and

angle for different designs. The beam deflectors are designed to deflect light at θ=-17° for an increasing number of

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wavelengths (A) λ1=1550 nm, (B) λ1=1450 nm, λ2=1650 nm, (C) λ1=1300 nm, λ2=1550 nm, λ3=1800 nm, (D) λ1=1300

nm, λ2=1467 nm, λ3=1633 nm, λ4=1800 nm, (E) λ1=1300 nm, λ2=1425 nm, λ3=1550 nm, λ4=1675 nm, λ5=1800 nm. The

intensities are normalized to the same reference value for all the panels. The white solid lines are calculated from Eq. 2 for

fixed phase gradients designed for θ0 and the controlled wavelengths.

9. Multiband filter

In the main text we described how our multiband beam deflector could find application as an optical filter with

multiple pass-bands. By illuminating the metasurface with broadband light, only the light at λ1, λ2 and λ3 will be

directed to the desired angle, creating a spatial filter. Fig. S10 shows the FDTD simulation of the intensity

monitored at the angle 𝜃 =-17°. While we notice a less uniform transmitted intensity at λ1, λ2 and λ3 compared

to the experimental data (Fig. 3D) (that we attribute to fabrication imperfections), the high suppression ratio

with respect to the other wavelengths is confirmed. We can also estimate the bandwidth of the filter by looking

at the full-width-half-maximum of the three peaks. As shown in the inset of Fig. S10 for the peak at λ2, the

FWHM is approximately 30 nm. By designing a more accurate phase function, the FWHM of the multi-pass

band filter can be reduced.

Fig. S10. FDTD simulation of the performance of the beam deflector as a multi-band filter. The inset shows a close up of

the peak corresponding to λ2 from which we can estimate the FWHM bandwidth of the filter.

Compared to conventional bandpass optical filters that often rely on thin film interference effects from multi-

layer stacks, a filter based on our achromatic metasurfaces is much thinner and can be created in a single step of

deposition, lithography, and etching.

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