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Complete Control of Polarization and Phase of Light with High Efficiency and Sub-wavelength
Spatial Resolution
Amir Arbabi,1Yu Horie,1Mahmood Bagheri,2and Andrei Faraon1, ∗
1T. J. Watson Laboratory of Applied Physics, California Institute of Technology, 1200 E California Blvd., Pasadena, CA 91125, USA
2Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
Meta-surfaces are planar structures that locally change polarization, phase, and amplitude of light, thus en-
abling flat, lithographically patterned free-space optical components with functionalities controlled by design.
Several types of meta-surfaces have been reported, but low efficiency and the inability to provide simultane-
ous phase and polarization control have limited their applications. Here we demonstrate a platform based on
high-contrast dielectric elliptical nano-posts providing complete and efficient control of polarization and phase
with sub-wavelength spatial resolution. The unprecedented freedom in manipulating light not only enables re-
alization of conventional free-space transmissive optical elements such as phase-plates, wave-plates and beam-
splitters, but also elements with novel functionalities such as general polarization switchable phase holograms
and arbitrary vector beam generators which will change the design paradigms for free-space optical systems.
Polarization, phase and amplitude completely characterize
monochromatic light. In free space optical systems, polariza-
tion is modified using wave retarders, polarizers, and polariza-
tion beam splitters, phase is shaped using lenses, curved mir-
rors or spatial phase modulators, and amplitude is controlled
via neutral density absorptive or reflective filters. Control-
ling these properties using ultra-thin flat elements, thus real-
izing the functionalities provided by conventional free space
optical components, will enable vertical integration of op-
tical systems, and will provide massively parallel and low
cost production of optical elements using micro-fabrication
techniques. Several thin flat optical components for shaping
phase [1–9, 11, 12] and polarization [12–17] have been real-
ized but poor efficiencies and the inability to realize various
optical elements with a single technology, limited their adop-
tion for practical applications. Devices based on plasmonic
metasurfaces were demonstrated but have limited efficiencies
because of fundamental limits [18] and metal absorption loss
[17, 19, 20]. Components based on 1D high contrast gratings
have higher efficiencies but do not provide high spatial resolu-
tion for realizing precise phase or polarization profiles in the
direction along the grating lines. The majority of flat elements
have been realized using a platform that provides only phase
control [4, 6, 7, 11, 21] (in most cases only for a fixed in-
put polarization), or only a limited polarization modification
capability [12, 14, 15, 17]. Here, for the first time, we pro-
pose and experimentally demonstrate a unified platform for
achieving simultaneous full control over the polarization and
phase of light with high efficiency. The platform allows for
realization of all possible highly transmissive optical compo-
nents based on passive monochromatic phase and polarization
control. As we show, our approach not only allows for effi-
cient ultra-thin devices with similar functionalities as already
existing bulk optical elements, but also enables novel optical
elements that do not have a bulk component counterpart, and
their functionalities can only be achieved by using optical sys-
tems composed of multiple cascaded optical components.
Figures 1A and B show schematic illustrations of a planar
device based on this platform. It is composed of a single layer
array of amorphous silicon (aSi) elliptical posts with different
sizes and orientations, resting on a fused silica substrate. The
posts are placed at the sites of a periodic hexagonal lattice
with sub-wavelength lattice constant. Each of the elliptical
posts exhibits a polarization dependent scattering response.
Normally incident light undergoes a spatially varying phase
shift and polarization modification as it passes through these
elliptical posts.
A uniform array of elliptical posts with one ellipse axis
aligned to one of the hexagonal lattice vectors is shown in
Fig. 1C. Due to symmetry, normally incident light linearly
polarized along one of the axes of the elliptical posts does
not change polarization and only acquires phase as it passes
though the array. By proper selection of the major and minor
diameters of elliptical posts, any combination of phase shifts
for the two linear polarizations aligned to the ellipse axes can
be simultaneously obtained while keeping the transmission of
both polarizations high. Figures 1D and E show the two di-
ameters of the elliptical posts for achieving any combination
of phase shifts φ1and φ2for light polarized along the two
ellipse axes. Figure 1F shows that the average transmission
(Supplementary Information S.1) for all the values of φ1and
φ2remains high.
Next, we show that by properly selecting and rotating these
elliptical posts, incident light with any polarization and phase
can be transformed with high transmission efficiency to any
arbitrary polarization and phase. This transformation is the
most general that can be physically achieved for monochro-
matic light. We first show that such transformation can be
achieved using a symmetric and unitary Jones matrix, and
then we present a method to realize the unitary and symmetric
Jones matrix by elliptical posts. The general relation between
the electric field of the input and output light for normal in-
cidence is expressed using a Jones matrix Tas Eo=TEi,
where Eiand Eoare the electric fields of the input and output
optical waves, respectively. Considering highly transmissive
devices, any arbitrary transformation between the input (Ei)
and output Eofield can be achieved using a Tmatrix, whose
coefficients can be found using the following equation (see
arXiv:1411.1494v1 [physics.optics] 6 Nov 2014
2
Transmission
I1 (2S)
I2 (2S)
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
I1 (2S)
I2 (2S)
00.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
65
150
250
350
450
D2
D1
I1 (2S)
I2 (2S)
00.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
65
150
250
350
450
C
FD
AB
T
x
y
D2
D1
x
y
E
E1eiI2
E2
E1
D1
D2
E2eiI1
FIG. 1. Illustration and simulation results of polarization and phase control. (A) Schematic illustration of a device which locally modifies
the polarization and phase of light. (B) Top view of the device depicted in (A) showing the periodic hexagonal lattice structure and elliptical
posts with different diameters and orientations. (C) A uniform array of elliptical posts with one of the ellipse axes aligned with one of the
lattice vectors. This uniform array only imposes phase shifts to the components of incident light along the two ellipse axes. (D) and (E)
Color coded simulated values of the diameters of the elliptical posts shown in (C) as a function of the desired phase shifts for the two linear
polarizations. (F) Color coded values of the average transmission (see Supplementary Information S.1) of the light through the device shown
in (C) with the corresponding diameters presented in (D) and (E).
Supplementary Information S.4):
Eo
x
∗Eo
y
∗
Ei
xEi
yTxx
Tyx =Ei
x
∗
Eo
x(1)
where Ei
xand Ei
yare the xand ycomponents of the elec-
tric field of the input light, Eo
xand Eo
yare the xand ycom-
ponents of the electric field of the output light, Tij (i, j =
x, y) are the elements of the 2×2 Jones matrix, and * rep-
resents complex conjugation. We also set Txy =Tyx and
Tyy =−exp(2i∠Tyx )Txx
∗so the Tmatrix becomes sym-
metric and unitary. Therefore, any input optical wave Eican
be transformed to any output optical wave Eoby a unitary and
symmetric Jones matrix. Now, we show that any symmetric
and unitary Jones matrix can be realized using elliptical posts.
Any symmetric and unitary matrix is decomposable in terms
of its eigenvectors and eigenvalue matrix (∆) as
T=Veiφ10
0 eiφ2VT=R(θ)∆R(−θ),(2)
where superscript Trepresents the matrix transpose operation.
Vis a real unitary matrix; therefore, it corresponds to an in-
plane geometrical rotation Rby an angle that we refer to as
θ, and since VT=V−1,VTrepresents a rotation by −θ.
According to Eq. 2, the operation of a device that realizes the
Jones matrix Tcan be considered as rotating Eiby −θ, phase
shifting the xand ycomponents of the rotated Eirespectively
by φ1and φ2, and rotating back the rotated and phase shifted
vector by angle θ. Equivalently, Tcan be implemented using
a device that imposes phase shifts φ1and φ2to the compo-
nents of Eialong angles θand 90◦+θ. As we mentioned
before, a uniform array of elliptical aSi posts with one of the
ellipse axis aligned with one of the lattice vectors (as shown
in Fig. 1C) can be designed to impose any combinations of
the phase shifts φ1and φ2to normally incident light waves
polarized along the ellipse axes. As a result, we can realize a
Jones matrix decomposed according to Eq. 2 using a uniform
array of elliptical posts similar to the one shown in Fig. 1C
which is rotated such that one of its lattice vectors makes an
angle θwith the xaxis.
The high refractive index contrast between the aSi posts and
their surroundings leads to weak coupling among the posts so
the light scattered by each post is primarily affected by its own
geometrical parameters, and has negligible dependence on the
position and orientations of the neighboring posts. As Fig. 2
shows, rotating each individual aSi post about the post axis
while keeping the lattice fixed is well approximated by rotat-
ing the entire array by the same angle. This is a result of the
concentration of the optical energy inside the posts (as can be
seen in Fig. 2B) and is an indication that the array operates
in the high contrast transmitarray regime [2]. Therefore, any
Jones matrix decomposed according to Eq. 2 can be locally
3
A
CD
B
0 20406080
0.0
0.5
1.0
Transmission
T (degree)
0 20406080
-0.2
0.0
0.2
Phase (2S)
T (degree)
0
50
100
500 nm
x
y
TyxEx
TxxEx
Ex
T
|Txx|2
|Tyx|2
Txx
Tyx
Array rotation
Post rotation
Array rotation
Post rotation
FIG. 2. Weak coupling among the elliptical posts. (A) A uniform array of elliptical posts with the major axis of the ellipse making an angle
θwith the xaxis converts part of an x-polarized light to a y-polarized one. (B) Top view (on the left) and side view (on the right) of the color
coded magnetic energy density when an x-polarized light with magnetic energy density of 1 is normally incident on the posts from the top. The
dashed black lines depict the boundary of the posts. (C) and (D) Simulated transmissions and phase of transmission coefficients as a function
of θ. The solid curves represent the results when the major axis of the elliptical posts are along one of the lattice vectors and the entire device
is rotated by θwhile the dashed lines correspond to keeping the lattice fixed and rotating the elliptical posts by θabout the post axis.
realized by using an elliptical post whose diameters are se-
lected using Figs. 1D and E to impose phase shifts of φ1and
φ2for light waves polarized along the ellipse’s two axes, and
is rotated counterclockwise by angle θalong the post axis (as
shown in the inset of Fig. 1B). As a result, a locally varying
Jones matrix Tthat generates a desired electric field distri-
bution Eofrom a given input electric field distribution of Ei
can be sampled with sub-wavelength resolution at the lattice
sites, and realized using dissimilar elliptical posts of the same
heights located at those sites.
The freedom provided by the proposed platform to simul-
taneously control the polarization and phase of light allows
for implementation of a wide variety of optical components.
Polarization insensitive optical elements can be designed us-
ing circular posts; a particular case of this platform has been
recently demonstrated [2, 11]. To demonstrate the versatility
and high performance of this technology to simultaneously
control polarization and phase, we fabricated and character-
ized two categories of flat optical elements enabled by the
proposed platform. Details of the device fabrication and mea-
surement are presented in (see Supplementary Information S.2
and S.3).
The first category of devices generate two different wave-
fronts for two orthogonal input polarizations. This function-
ality can be achieved if the device does not change the polar-
ization ellipse of the two orthogonal polarizations it has been
designed for, and only changes their handedness (see Supple-
mentary Information S.4). A special case is when both of the
input polarizations are linear. Simulation and experimental
measurement results, as well as the optical and scanning elec-
tron microscope images of three types of devices in this cat-
egory are shown in Fig. 3. A polarization beam splitter that
deflects the xand ypolarized portions of light by 5◦and -5◦is
presented in Fig. 3A. We measured 72% and 77% efficiencies
for the xand ypolarized input light, respectively. A polar-
ization beam splitter which separates and focuses the xand y
polarized light at two different positions is presented in Fig.
3B. An example of the general form of the devices of this cat-
egory is a polarization switchable phase hologram. Figure 3C
shows an example of this device which generates two distinct
patterns for xand ypolarized lights. We measured efficien-
cies of 84% and 91% for this device for the xand ypolarized
incident light.
The second category of the devices that we consider gener-
ate light with a desired arbitrary phase and polarization distri-
bution from an incident light with a given polarization. An ex-
ample of such devices that converts an x-polarized light into
a radially polarized light is shown in Fig. 4A. An incident
x-polarized incident Gaussian beam is converted into a radi-
ally polarized Bessel-Gauss beam, and y-polarized incident
Gaussian beam is transformed into an azimuthally polarized
Bessel-Gauss output beam. We measured a transmission effi-
4
B
C
A
Simulation Measurement
Simulation Measurement
Simulation Measurement
0
0.5
1
0
0.5
1
200 Pm
1 mm
1.5 mm
2 Pm
2 Pm
0
0.5
1
500 nm
100 Pm
100 Pm
100 Pm
200 Pm
100 Pm
100 Pm
FIG. 3. Independent control over two polarizations. (A) A polarization beam splitter which separates the xand ypolarized portions of light
and deflects them by different angles. (B) A device that separates xand ypolarized portions of light and focuses them to two different points.
(C) A polarization switchable phase hologram which generates two arbitrary patterns for xand ypolarized light. Schematic illustration of the
devices are depicted on the left, simulation and measurement results are presented in the middle, and optical (bottom) and scanning electron
(top) microscope images of the devices are shown on the right.
ciency of 96% and 97% for the xand ypolarizations, respec-
tively. Measured intensity profiles for different polarization
projections are also shown in Fig. 4. When the polarization
of the incident Gaussian beam is linear but not aligned with
the xor yaxis, a generalized cylindrical vector beam is gen-
erated by this device. It has been recently shown that cylindri-
cal vector beams show unique features such as focus shaping
when focused with a high numerical aperture lens [22]. Fur-
thermore, the same device in Fig. 4A generates light with
different orbital angular momentum depending on the helic-
5
A
C
B
Simulation Measurement
0
0.5
1
500 Pm
Simulation Measurement
0
0.5
1
Simulation Measurement
0
0.5
1
200 Pm
200 Pm
100 Pm
100 Pm
100 Pm
500 nm
2 Pm
50 Pm
2 Pm
2 Pm
50 Pm
2 Pm
2 Pm
2 Pm
FIG. 4. Vector beam generation. (A) A device that generates radially and azimuthally polarized cylindrical vector beams from xand y
polarized Gaussian beams, respectively. (B) A device that generates and focuses radially and azimuthally polarized light respectively from x
and ypolarized lights. (C) Focal spot control with polarization. The device focuses the right handed circularly polarized lights to a diffraction
limited spot, and focuses left circularly polarized light to a doughnut shaped spot with m= 2 units of orbital angular momentum. Schematic
illustration of the devices are depicted on the left, simulation and measurement results are presented in the middle, and optical (bottom) and
scanning electron (top) microscope images of the device are shown on the right. The black double sided arrows shown in the right panel
of the measurement results represents the direction of the transmission axis of a linear polarizer inserted into the measurement setup (see
Supplementary Information S.3).
6
ity of the input beam; right and left handed circularly polar-
ized beams will acquire m= 1 and m=−1units of orbital
angular momentum as they pass through this device, respec-
tively. Both the generation and focusing of cylindrical vector
beams can be performed using a single device based on the
proposed platform. Such a device that generates and focuses
radially and azimuthally polarized light is shown in Fig. 4B.
Incident optical waves linearly polarized along xand ydirec-
tions are respectively converted into radially and azimuthally
polarized waves and are focused. Similar to the device shown
in Fig. 4A, due to the polarization conversion, right and left
handed polarized beams acquire plus or minus one units of
orbital angular momentum as they pass through the device.
As a result, by adding a sinusoidal dependence in the form of
exp(iφ)to the phase profile of the device, the total orbital an-
gular momentum of the right and left handed circularly polar-
ized light after passing through the device will become m= 0
and m= 2, respectively. A device with such a phase and po-
larization profile is shown in Fig. 4C. As can be seen from the
simulation and measurement results, a right handed circularly
polarized incident beam is focused to a diffraction limited spot
while a left handed circularly polarized beam is focused into
a doughnut shaped intensity pattern. Therefore, the focal spot
shape can be modified by changing the polarization of the in-
cident beam. This is particularly interesting since the polariza-
tion state of the incident beam can be switched rapidly using
a phase modulator.
The functionalities provided by the optical devices we
demonstrated here can currently be achieved by using a com-
bination of multiple bulk optical components. The unprece-
dented complete and simultaneous control over the polariza-
tion and phase profiles of light offered by the proposed plat-
form and the design technique, enables realization of novel
optical components that may lead to advances in a wide range
of fields in optics and photonics including optical imaging and
microscopy, optical communications, biophotonics, quantum
optics, and astrophysics. As most other diffractive optical el-
ements, these devices have an optical bandwidth of several
percent of the design wavelength [23]. Thus, they could di-
rectly replace conventional optics in applications employing
narrow-band light sources like communications or two-photon
microscopy. The optical components realized based on this
technology are sub-wavelength thick, low weight and can be
mass produced at low cost using a single lithography step and
standard silicon micro-fabrication techniques. Their high ef-
ficiency and planar geometry make these devices suitable for
cascading multiple elements and realizing free space optical
systems on a chip. This platform provides all possible passive
control for polarization and phase of monochromatic light.
Further improvements are expected from using materials with
optical nonlinearities and gain that might extend the spectral
bandwidth of operation and provide tunability.
ACKNOWLEDGEMENT
This work was supported by Caltech/JPL president and di-
rector fund (PDF) and DARPA. Yu Horie was supported as
part of the DOE ”Light-Material Interactions in Energy Con-
version’ Energy Frontier Research Center under grand de-
sc0001293 and JASSO fellowship. The device nanofabrica-
tion was performed in the Kavli Nanoscience Institute at Cal-
tech. The authors thank to Dr. David Fattal and Dr. Charles
Santori for useful discussion.
∗Corresponding authors: A.F: faraon@caltech.edu, A.A:
amir@caltech.edu
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8
SUPPLEMENTARY INFORMATION FOR “COMPLETE CONTROL OF POLARIZATION AND PHASE OF LIGHT WITH
HIGH EFFICIENCY AND SUB-WAVELENGTH SPATIAL RESOLUTION”
S.1 Simulations and Design
To obtain the simulation results presented in Figs. 1D-E, we computed the transmission coefficients txand tyof xand y
polarized light for a uniform array similar to the one shown schematically in Fig. 1C by using the rigorous coupled wave
analysis (RCWA) technique using a freely available software package [1]. The simulations were performed at λ= 915 nm.
The aSi posts (refractive index of 3.56 at 915 nm) are 715 nm tall, and rest on a fused silica substrate. We computed these
transmission coefficients for all mutual values of the ellipse diameters D1and D2in the range 0.1ato 0.7a, where a= 650
nm is the lattice constant. Next, using the computed transmission coefficients, for all combinations of the phases φxand φywe
found the diameters D1and D2that maximizes the average transmission coefficient defined as
T= 1 −q|tx−eiφx|2+|ty−eiφy|2.(3)
The simulation results presented in Figs. 2B-D were also computed using the RCWA technique. The simulation parameters used
to obtain Figs. 2B-D are the same as the ones used in Fig. 1, and the diameters of the elliptical posts are 300 nm and 150 nm.
To design the devices presented in Fig. 3 which impose two distinct phase profiles for the xand ypolarized lights, the optimum
phase profiles for xand ypolarized optical waves that generate the desired patterns were first determined by back propagating
the desired pattern to the plane of the device and finding the phase difference between the back propagated wave and the incident
wave. This method is discussed in details in [2]. After finding the desired phase profiles for both of the polarizations, the
profiles were sampled at the lattice sites and elliptical posts with major and minor diameters that impart the required phases and
polarization rotations on the transmitted beam were placed on those sites.
Optical elements shown in Fig. 4 that simultaneously modify polarization and phase of light were designed to generate a
desired spatially varying optical wave from a given input optical wave profile. We sampled the input and output optical waves at
the lattice sites to obtain Eiand Eoat each of the sites. Next, by using Eq. 1, the symmetry condition, and the unitary condition,
we computed the Jones matrix Tat each of the lattice sites. We then decomposed the Tmatrix at each lattice site according
to Eq. 2 to determine the phase shifts along the major and minor axes of the ellipse (i.e. φ1and φ2) and the rotation angle θ.
Finally, from Figs. 1D and E, we found the diameters of the elliptical posts imposing the target φ1and φ2phase shifts, rotated
them counterclockwise by their θ, and placed them at their corresponding lattice sites.
We computed the simulation results presented in Figs. 3 and 4 by assuming that the devices perform the polarization and
phase conversions ideally, and with λ/15 spatial resolution. For these simulations, the input light was assumed to be uniformly
polarized Gaussian beams. The output light was computed at each point on a rectangular grid assuming ideal polarization and
phase conversion by the device, and then propagated to the planes of interest using the plane waves expansion technique [3].
S.2 Sample Fabrication
We fabricated the devices shown in Figs. 3 and 4 on a fused silica substrate. We deposited 715 nm aSi using plasma enhanced
chemical vapor deposition (PECVD) with a 5% mixture of silane in argon at 200◦C. Next, we spun 300 nm positive electron
beam resist (ZEP-520A) and patterned it using electron beam lithography. A 70 nm thick aluminum oxide layer was then
deposited on the developed resist and patterned by lifting off the resist. The patterned aluminum oxide was subsequently used
as a hard mask for dry etching of aSi in a 3:1 mixture of SF6and C4F8. Finally, the aluminum oxide mask was removed using
a 1:1 mixture of ammonium hydroxide and hydrogen peroxide heated to 80◦C.
Fiber collimator Objective lens
Tube lens
Camera
Polarizer
Laser
Polarization
controller
Device
FIG. S1. Measurement setup. Schematic illustration of the measurement setup used for characterization of devices modifying polarization
and phase of light. The linear polarizer was inserted into the setup only during the polarization measurements.
9
S.3 Measurement Procedure
We characterized the devices using a setup that is schematically shown in Fig. 5. Light from a 915 nm fiber coupled semicon-
ductor laser was passed through a fiber polarization controller and collimated to generate a Gaussian beam. The objective lens,
the tube lens (Thorlabs LB1945-B), and the camera (CoolSNAP K4 from Photometrics) comprise a custom built microscope.
We used three different objective lenses to achieve different magnifications. Measurement results shown in Fig. 3A were ob-
tained using a 20X objective lens (Olympus UMPlanFl, NA=0.4), results shown in Figs. 3B and C, and Fig. 4A were recorded
using a 50X objective lens (Olympus LCPlan N, NA=0.65), and the ones presented in Figs. 4B and C were obtained using a
100X objective lens (Olympus UMPlanFl, NA=0.95). The overall microscope magnification for each objective lens was found
by imaging a calibration sample with known feature sizes. The polarizer (Thorlabs LPNIR050-MP) was inserted into the setup
to confirm the polarization state of the incident light (after removing the device) and the output light. Efficiency values were
obtained by integrating the light intensity on the camera and normalizing it to the integrated intensity recorded when the device
was removed.
S.4 Determination of the Jones Matrix
Here we show that elements of a unitary and symmetric Jones matrix that maps a given input electric field Eito a desired
output electric field Eosatisfy Eq. 1. Since Eo=TEiwe have
TxxEi
x+TyxEi
y=Eo
x,(4a)
TyxEi
x−Tyx
Tyx
∗Txx
∗Ei
y=Eo
y,(4b)
where we have used the symmetric properties Txy =Tyx, and the unitary condition Txx Txy
∗+Tyx
∗Tyy = 0. By multiplying
Eq. 4a by Txx
∗and Eq. 4b by Tyx
∗we obtain
|Txx|2Ei
x+TyxTxx
∗Ei
y=Txx
∗Eo
x,(5a)
|Tyx|2Ei
x−TyxTxx
∗Ei
y=Tyx
∗Eo
y.(5b)
By adding Eqs. 5a and 5b, using the unitary condition |Txx|2+|Ty x|2= 1, and taking the complex conjugate of the resultant
relation, we find
TxxEo
x
∗+TyxEo
y
∗=Ei
x
∗.(6)
Finally, Eqs. 6 and 4a in the matrix form, we obtain Eq. 1
Eo
x
∗Eo
y
∗
Ei
xEi
yTxx
Tyx =Ei
x
∗
Eo
x.(7)
The four elements of the Jones matrix Tare found uniquely using Eq. 7, the symmetry condition Txy =Tyx , and the unitary
condition Tyy =−exp(2i∠Tyx )Txx
∗, when the determinant of the matrix on its left hand side is nonzero. Since a unitary matrix
preserves the angle between two vectors, an optical wave whose polarization is orthogonal to Eiwill be converted to an optical
wave polarized orthogonal to Eo. As a result, an optical element designed to generate radially polarized light from xpolarized
input light, will also generate azimuthally polarized light from ypolarized input light.
In the special case that the determinant of the matrix on the left had side of Eq. 7 is zero we have
Eo
x
∗Ei
y−Eo
y
∗Ei
x= 0,(8)
and because Tis unitary we have |Ei|=|Eo|; therefore we find Eo= exp(iφ)Ei∗where φis an arbitrary phase. This special
case corresponds to a device that preserves the polarization ellipse of the input light, switches its helicity, and imposes a phase
shift on it. In this case the Tmatrix is not uniquely determined from Eq. 7, and an additional condition can be imposed on the
operation of the device. For example, the device can be designed to realize two different phase profiles for two orthogonal input
polarizations.
∗Corresponding authors: A.F: faraon@caltech.edu, A.A: amir@caltech.edu
