of 5
1
Supplementary
Figure
2 |
Measurement
results
of
HCTA
beam
deflectors.
(
a
)
Schematic
illustration
of
an
ideal
beam
deflector
.
(
b
)
The
HCTA
pattern
that
implements
a
uniform
phase
ramp
along
the
horizontal
direction.
The
polarization
direction
for
the
TE
and
TM
polarizations
are
also
shown.
(
c
)
Measured
deflection
ef
ficienc
y
of
a
set
of
beam
deflectors
for
the
transverse
electric
and
magnetic
polarized
incident
beam
as
a
function
of
deflection
angle.
Supplementary
Figure
1 |
Optimum
transmissive
mask
design
for
shaping
an
incident
light
to
a
desired
tangential
form
.
(
a
)
The
light
from
the
sources
and
scatterers
in
the
half
space
(1)
passes
through
the
transmission
mask
(
,
)
,
and
its
tangential
electric
field
on
the
target
plane
is
represented
by
tan
.
(
b
)
The
equivalent
magnetic
surface
current
density
s
emits
the
same
tangential
electric
field
tan
on
the
target
plane.
(
c
)
The
magnetic
field
d
(
2
)
is
emitted
by
an
electric
surface
current
density
s
which
is
located
on the
target
plane.
2
Supplementary
Note
1:
Optimum
t
ransmissive
m
ask
d
esign
We
consider
the
general
case
of
using
a
transmissive
mask
to
modify
the
optical
wavefront
emitted
by
given
sources
to
a
desired
form.
The
light
is
generated
by
sources
located
in
the
half
space
(1) as
shown
schematically
in
Supplementary
Fig.
1a.
The
tangential
component
of
the
electric
field
of
the
incident
light
just
before
passing
through
the
transmissive
mask
is
represented
by
i
(
1
)
.
The
incident
field
might
be,
for
example,
a
diverging
beam
from
a
semiconductor
laser
or a
collimated
Gaussian
beam.
The
desired
output
wavefront
can
be
chosen
arbitrarily
(examples
include
a
beam
that
is
matched
to
a
mode
of
an
optical
fiber
,
a
Bessel
beam,
or
a
tightly
focused
beam).
Since
the
propagation
is
governed
by
the
Maxwell’s
equations,
the
desired
output
beam
is
fully
described
by
the
tangential
components
of
its
electric
field
on
a
target
plane
parallel
to
the
trans
missive
mask.
We
represent
this
desired
tangential
component
by
d
(as
shown
in
Supplementary
Fig.
1a).
The
output
beam
formed
by
the
transmissive
mask
is
in
general
different
from
the
desired
beam,
and
its
tangential
electric
field
on
the
target
plane
tan
is
an
approximation
for
the
desired
tangential
electric
fiel
d.
Our
main
objective
is
to
determine
the
transmissive
mask
(
,
)
such
that
tan
is
the
best
possible
approximation
for
d
.
A
useful
measure
for
quantifying
the
accuracy
of
the
approximation
is
the
norm
of
the
projection
integral
defined
as
|
<
tan
,
d
>
|
=
|
tan
d
d
|
,
(1)
where
represents
the
complex
conjugate
operation;
and
the
surface
integral
is
evaluated
over
the
target
plane.
For
the
best
approximation,
this
norm
should
be
maximized.
To
relate
the
projection
integral
to
the
transmission
mask
we
use
the
equivalence
principle
and
the
reciprocity
theorem.
According
to
the
definition
of
a
transmissive
mask,
the
tangential
electric
field
of
the
light
just
after
it
passes
through
the
transmissive
mask
is
given
by
i
(
1
)
.
Using
the
equ
iv
alence
principle,
a
magnetic
sur
f
ace
current
density
s
=
2
i
(
1
)
×
̂
located
at
the
output
plane
of
the
transmissive
mask
and
emitting
in
vacuum,
will
generate
the
same
beam
in
the
region
(1) as
the
original
sources
and
the
transmissive
mask
(see
Supplementary
Fig.
1b).
Next,
we
consider
an
electric
surface
current
density
with
s
=
on
the
target
plane
which
is
emitting
in
3
vacuum.
We
denote the
magnetic
field
emitted
by
s
at
the
output
plane
of
the
transmissive
mask
by
d
(
2
)
. Using
the
reciprocity
theorem
[1],
we
can
write
tan
s
d
=
d
(
2
)
s
d
.
(2)
From
(1)
and
(2)
we
obtain
|
<
tan
,
d
>
|
=
2
|
d
(
2
)
(
i
(
1
)
×
̂
)
d
|
,
(3)
Using (3), we see that the best approximation to a desired output beam is achieved when
|
|
=
1
and
=
(
d
(
2
)
(
i
(
1
)
×
̂
)
)
(4)
In other words, the best transmissive mask is a phase mask, and we can determine its phase
profile
as
follows.
First,
we
find
the
tangential
component
of
the
incident
light
at
the
location
of
the
transmissive
mask (
i
(
1
)
)
;
this
can
be
done
either
analytically
or
numerically
depending
on
the
type
of
the
excitation.
Next,
we
consider
an
electric
surface
current
density
with
the
same
spatial
distribution
as the
complex
conjugate of the
desired tangential electric field at the
target
plane which is emitting in the free
space
, and
w
e
find
the
magnetic
field
emitted
by
this
current
at
the
location
of
the
transmission
mask
(
d
(
2
)
).
Since
s
is
planar
and
is
radiating
in
free
space,
its
fields
can
be
obtained
using
a
simple method
such
as
the
plane
wave
expansion
technique
[2].
Finally,
we
obtain
the
phase
profile
of
the transmission
mask
using
(4).
To
achieve
a
lens
with
tight
focus,
the
desired
field
should
be
set
to
a
uniform
electric
field
confined
inside
a
circle
with
deep
subwavelength
radius
located
at
the
plane
of
focus.
Supplementary
Note
2:
Under
-
sampling
of
the
phase
profile
Considering
a
micro
-
lens
in
the
geometrical
optics
picture
offers
an
intuitive
understanding
of
how
the
under
-
sampling
of
the
phase
profile
affects
the
lens
performance.
According
to
the
geometrical
optics,
a
ray
that
is
propagating
parallel
to
the
lens’s
optical
axis
is
deflected
by
the
lens
toward
its
focal
point.
The
rays
that
are
propagating
farther
away
from
the
optical
axis
are
4
deflected
by
larger
angles,
and
the
lens’s
NA
represents
the
sine
of
the
largest
deflection
angle.
Therefore,
a
lens
can
be
considered
as
a
deflect
or
whose
local
deflection
angle
gradually
increases
from
zero
at
the
center
of
the
lens
to
its
maximum
(which
is
given
by
sin
1
(
NA
)
)
at
the
perimeter
of
the
lens.
We
show
that
,
due
to
the
under
sampling
of
the
phase
profile
by
the
HCTA
lattice,
the
deflection
ef
ficienc
y
of
the
HCTA
deflectors
decreases
by
increasing
their
deflection
angle.
The
lower
deflection
ef
ficienc
y
at
larger
deflection
angles
leads
to
the
lower
focusing
ef
fic
iency
of
the
micro
-
lenses
with
higher
NA.
A
uniform
deflector
functions
similar
to
a
blazed
grating
and
deflects
a
monochromatic
nor
mally
incident
light
by
a
fix
ed
angle
(as
shown
in
Supplementary
Fig.
2a).
The
uniform
de
flector
has
a
linearly
varying
phase
profile
whose
slope
is
proportional
to
the
sine
of
its
deflection
angle.
A
schematic
illustration
of
an
HCTA
uniform
deflector,
with
phase
profile
varying
linearly
along
the
horizontal
direction,
is
depicted
in
Supplementary
Fig.
2b.
Nine
400
μ
m
diameter
uni
form
deflectors
with
different
deflection
angles
were
fabricated
using
the
same
family
of
periodic
HCTAs
used
for
the
high
NA
micro
-
lenses
and
the
same
fabrication
process.
The
deflectors
were
illuminated
with
a
linearly
polarized
collimated
laser
beam
with
beam
ra
dius
of
approximately
100
μ
m.
The
deflected
power
was
measured
using
a
photodetector
located
10 cm
away
from
the
deflector along
the
expected
deflection
direction.
The deflection
ef
ficienc
y
was
obtained
by
dividing
the
measured
deflected
power
by
the
incident
power.
The
measured
deflection
efficiencies
for
two
linear
orthogonal
polarizations
are
depicted
in
Supplementary
Fig
.
2c.
The
directions
of
polarization
for
the
TE
and
TM
polarized
lights
are
shown
in
Supple
mentary
Fig.
2b.
The
TE
polarization
corresponds
to
the
transverse
electric
polarized
deflected
light,
while
the
deflected
light
is
transverse
magnetic
polarized
for
the
TM
polarization.
As
Supplementary
Fig.
2c
shows,
the
deflection
ef
ficienc
y
of
an
HCTA
deflector
decreases
as
its
deflection
angle
increases.
The
deflection
ef
ficienc
y
drop
is
faster
for
the
TM
polarized
incident
light
compared
to
the
TE
one.
We
attribute
the
ef
ficienc
y
reduction
to
the
under
sampling
of
the
phase
profile
of
the
deflectors
with
large
deflection
angles.
The
desired
phase
profile
of
a
deflector
with
deflection
angle
of
is
sampled
by
=
/
(
푠푖푛
(
)
)
unit
cells
over
2
phase
variation,
where
is
the
lattice
constant
of
the
HCTA.
For
the
HCTA
used
in
this
study,
the
lattice
constant
is
roughly
equal
to
a
half
of
a
wavelength;
therefore,
for
a
40°
d
eflector,
the
full
phase
range
of
2
is
sampled
by
approximately
three
unit
cells.
Similar
diffraction
ef
ficienc
y
reduction
due
to
5
phase
sampling
and
quantization
error
is
encountered
in
the
design
and
implementation
of
Fresnel
lenses
with
a
limited
number
of
levels
[3].
One
approach
to
increase
the
ef
ficienc
y
of
a
high
NA
micro
-
lens
is
to
use
an
HCTA
with
a
smaller
lattice
constant.
This
leads
to
a
finer
sampling
of
the
desired
micro
-
lens
profile.
For
example,
as
Fig.
1b
in
the
main
manuscript
show
s
,
we
could
use
the
lattice
constant
of
650
nm
instead
of
800
nm
and
still
achieve
the
full
2
transmission
phase
range
by
changing
the
post
diameters.
Supplementary
References
[1]
Harrington, R. F. Time Harmonic Electromagnetic Fields (Wiley
-
IEEE Press, 2001
).
[2]
Born, M. & Wolf, E. Principles of Optics (Cambridge University Pr
ess, Cambridge, 1999),
7th edn.
[3]
Wyrowski,
F.
Efficiency of quantized diffractive phase
elements.
Optics
Communications
92
,
119
126,
(1992).