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Cohen et al. Light: Science & Applications (2020) 9:200
Official journal of the CIOMP
2047-7538https://doi.org/10.1038/s41377-020-00411-7
www.nature.com/lsa
ART ICLE Open Ac ce s s
Generalized laws of refraction and reflection atinterfaces
between different photonic artificialgauge fieldsMoshe-Ishay Cohen
1,2, Christina Jörg 3,4, Yaakov Lumer1,2, Yonatan Plotnik1,2, Erik
H. Waller3, Julian Schulz3,Georg von Freymann 3,5 and Mordechai
Segev1,2
AbstractArtificial gauge fields the control over the dynamics of
uncharged particles by engineering the potential landscapesuch that
the particles behave as if effective external fields are acting on
them. Recent years have witnessed a growinginterest in artificial
gauge fields generated either by the geometry or by time-dependent
modulation, as they havebeen enablers of topological phenomena and
synthetic dimensions in many physical settings, e.g., photonics,
coldatoms, and acoustic waves. Here, we formulate and
experimentally demonstrate the generalized laws of refraction
andreflection at an interface between two regions with different
artificial gauge fields. We use the symmetries in thesystem to
obtain the generalized Snell law for such a gauge interface and
solve for reflection and transmission. Weidentify total internal
reflection (TIR) and complete transmission and demonstrate the
concept in experiments. Inaddition, we calculate the artificial
magnetic flux at the interface of two regions with different
artificial gauge fields andpresent a method to concatenate several
gauge interfaces. As an example, we propose a scheme to make a
gaugeimaging system—a device that can reconstruct (image) the shape
of an arbitrary wavepacket launched from a certainposition to a
predesigned location.
IntroductionSnell’s law and the Fresnel coefficients are the
corner-
stones of describing the evolution of electromagneticwaves at an
interface between two different media. Bycascading several such
systems, each with its own opticalproperties, it is possible to
design complex structures thatgive rise to various important
devices and systems, such aslenses, waveguides1, resonators,
photonic crystals2, andeven localization phenomena, when random
interfaces areinvolved3. The behavior of waves in the presence of
aninterface can exhibit fundamental features, e.g., totalinternal
reflection (TIR), back-refraction for negative-
positive refraction index interfaces4,5, and even confine-ment
of states to the interface itself, such as Tamm andShockley
states6,7, plasmon polaritons8,9, Dyakonovstates10,11 and
topological edge states12–14. Traditionally,the Fresnel equations
describe the reflection and trans-mission of electromagnetic waves
at an interface separ-ating two media with different optical
properties. Thesecan be two materials with different permittivities
or twodifferent periodic systems (photonic crystals) composed ofthe
same material, e.g., an interface between two dissim-ilar waveguide
arrays15. However, an interface can alsoseparate two optical
systems that differ only by the arti-ficial gauge fields created in
them. Generally, such a“gauge interface” marks a different
dispersion curve oneither side of the interface; hence, it must
affect thetransmission and reflection at the interface.Gauge fields
(GFs) are a basic concept in physics
describing forces applied on charged particles. Artificial
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Correspondence: Mordechai Segev ([email protected])1Physics
Department, Technion—Israel Institute of Technology, Haifa
32000,Israel2Solid State Institute, Technion—Israel Institute of
Technology, Haifa 32000,IsraelFull list of author information is
available at the end of the articleThese authors contributed
equally: Moshe-Ishay Cohen, Christina Jörg
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www.nature.com/lsahttp://orcid.org/0000-0003-1741-5926http://orcid.org/0000-0003-1741-5926http://orcid.org/0000-0003-1741-5926http://orcid.org/0000-0003-1741-5926http://orcid.org/0000-0003-1741-5926http://orcid.org/0000-0001-6187-0155http://orcid.org/0000-0001-6187-0155http://orcid.org/0000-0001-6187-0155http://orcid.org/0000-0001-6187-0155http://orcid.org/0000-0001-6187-0155http://orcid.org/0000-0003-2389-5532http://orcid.org/0000-0003-2389-5532http://orcid.org/0000-0003-2389-5532http://orcid.org/0000-0003-2389-5532http://orcid.org/0000-0003-2389-5532http://creativecommons.org/licenses/by/4.0/mailto:[email protected]
-
GFs are a technique for engineering the potential land-scape
such that neutral particles will mimic the dynamicsof charged
particles driven by external fields. With theadvent of the
particle-wave duality, artificial GFs havebeen demonstrated to act
on photons16–19, coldatoms20,21, acoustic waves, etc. These
artificial GFs aregenerated either by the geometry17 or by
time-dependentmodulation18 of system parameters. With the
growinginterest in topological systems22, which
necessitateGFs23,24, it was suggested that the interface between
tworegions of the same medium but with different GFs ineach region
can create an effective edge. In these systems,both sides of the
interface have the same basic dispersionproperties, altered only by
applying a different GF on eachside. Such a gauge edge was employed
to demonstrateanalogies to the Rashba effect25, optical
waveguiding26,27,
topological edge states28,29 and back-refraction30. In
thepresence of a different GF on either side of the interface,the
trajectories of waves crossing from one side to theother are
governed by the symmetries in the system,which are expected to
result in an effective Snell’s law,whereas the reflection and
transmission coefficients arisefrom the specific boundary.Here, we
theoretically and experimentally demonstrate
the effective Snell law governing the reflection andtransmission
of waves at an interface between regions ofthe same photonic
medium, differing only in the artificialgauge fields introduced on
either side. We show how thetransverse momenta of the reflected and
transmittedwaves change according to the interfacial change in
thegauge field, and demonstrate TIR and complete trans-mission.
Subsequently, we provide an approximate
axax
ayy
z
AGFinterface
a
c d
b
2�
10 μm
100 μm 20 μm
Z
z
x x
Fig. 1 Sketch of our artificial gauge interface. Two rectangular
arrays of waveguides (red: upper array, blue: lower array) are
stacked on top of eachother, creating an artificial GF interface in
the y-direction. The waveguide arrays are tilted by 2η with respect
to each other. Otherwise, the parametersfor the two arrays are
identical. a Front view. b Top view. The dashed box represents one
unit cell in the x–z-plane in the upper array. c SEM image ofthe
inverse fabricated waveguide sample from the side. The inset shows
a magnified region to visualize the hollow waveguides. d Microscope
imageof the infiltrated sample from the top
Cohen et al. Light: Science & Applications (2020) 9:200 Page
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-
calculation for the Fresnel coefficients for our exampleand
explain how to generalize the concepts. Finally, weshow how to
concatenate several “gauge interfaces” andpropose a design for a
gauge-based imaging system—adevice composed of several interfaces
between differentgauge fields, acting to reconstruct (image) an
arbitraryparaxial input wavepacket at a predetermined plane.
ResultsFor simplicity, consider first a simple system
constitut-
ing an artificial gauge interface: two 2D arrays of
eva-nescently coupled waveguides, where the waveguides ineach array
follow a different trajectory along the propa-gation axis z (Fig.
1). This model system serves to explainthe ideas involved, which
are later generalized. The GFs inour system are a direct result of
the trajectories of thewaveguides and do not require any temporal
modulationof the materials at hand. The upper array experiences
aconstant tilt in the x-direction from the propagation axisz, with
a paraxial angle η such that x(z)= x′+ ηz (para-xiality allows sin
η≃ η, where x′ is the original x-position),while the lower array is
tilted by −η (Fig. 1a). These twoarrays combined exhibit different
artificial GFs that can-not be gauged away by a coordinate
transformation. Thepropagation of light in this structure is
described by theparaxial wave equation
i∂zψ ~rð Þ ¼ � 12k0 ∇2?ψ ~rð Þ �
k0n0
Δn ~rð Þψ ~rð Þ ð1Þ
Here, ψ is the envelope of the electric field, k0 is theoptical
wavenumber inside the bulk material, n0 is theambient refractive
index, Δn ~rð Þ ¼ n ~rð Þ � n0 givesthe relative refractive index
profile, and ∇2? ¼ ∂2x þ ∂2y .Eq. (1) is mathematically equivalent
to the Schrödingerequation, where z plays the role of time, and Δn
~rð Þ playsthe role of the potential. This analogy between
theparaxial wave equation and the Schrödinger equation hasbeen used
many times in exploring a plethora offundamental phenomena, ranging
from Anderson locali-zation31 and Zener tunneling32 to
non-Hermitian poten-tials33 and Floquet topological
insulators23.The basic building block in our system is a two-
dimensional array of evanescently coupled straightwaveguides,
i.e., Δn ~rð Þ is a periodic function in both x andy, with periods
ax and ay, respectively, such that each unitcell consists of a
single waveguide.Consider first an array where the trajectories of
all the
waveguides are in the z-direction. Following coupledmode
theory25, the spectrum of light propagating in sucha 2D array of
waveguides is given by
β kx; ky� � ¼ β0 þ 2cx cos kxaxð Þ þ 2cy cos kyay� � ð2Þ
where β is the propagation constant of an eigenmode,defined by ψ
x; y; zð Þ ¼ ψ0 x; yð Þe�iβz , kx and ky are thespatial momenta of
the mode in the x- and y-directions, cxand cy are the coupling
strengths between adjacentwaveguides in the x- and y-directions
(taken to be realnegative numbers according to standard solid
statenotation) and β0 is the propagation constant of theguided mode
in a single isolated waveguide. Consider nowan array of waveguides
tilted at an angle η with respect tothe z-axis such that the
waveguides follow a trajectorydefined by x− ηz= constant. The
dynamics in an array oftilted waveguides are expressed by an
artificial GF, givenby the effective vector potential ~A zð Þ ¼
�k0ηx̂, with thefollowing spectrum25,27:
βη kx; ky� � ¼ β0 þ 2cx cos kx � k0ηð Þaxð Þ þ ηkx
� 12k0η
2 þ 2cy cos kyay� � ð3Þ
The shift of k0η in the cosine is the compensation dueto the
Galilean transformation of the waveguides. Thelinear ηkx shift term
appears because the spectrum in Eq.(3) is expressed in the
laboratory frame and not in theframe of reference in which the
waveguides are sta-tionary. The constant offset 12 k0η
2 results from theeffective elongation of the optical path
inside the tiltedwaveguides.Such a linear tilt of a waveguide array
is, in itself, a
trivial gauge field, as we can eliminate its effects bychanging
the frame of reference to the co-moving frame,i.e., a linear
coordinate transformation of the entire sys-tem can gauge it out.
This can also be understood byexamining the arising effective
magnetic field ~B ¼ ~∇ ´~A,which is zero for a constant vector
potential ~A. To have anontrivial gauge, we need the effective
gauge field to benonuniform (i.e., have either space or time
dependence).Such a nontrivial gauge is achieved by coupling two
2Darrays, each with a different tilt angle and therefore adifferent
gauge. Then, it becomes impossible to gaugeaway the effect of the
tilt when we combine two suchfields with different tilts. There is
no coordinate system inwhich both arrays would be simultaneously
untilted27.Here ~A ¼ ηk0bx at the upper section, and ~A ¼ �ηk0bx at
thelower section.With this in mind, consider a two-dimensional
array of
evanescently coupled waveguides divided into two regions—top and
bottom, as shown in Fig. 1a. The rows ofwaveguides in the top and
bottom regions are identical inevery parameter except for the tilt.
The overall GF is thengiven by
~A ~rð Þ ¼ 2Θ yð Þ � 1ð Þηk0x̂ ð4Þ
Cohen et al. Light: Science & Applications (2020) 9:200 Page
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-
where Θ(y) is the Heaviside function, which is 1 when y >0
and zero otherwise. This vector potential gives aneffective
magnetic field ~B ¼ �2δ yð Þηk0ẑ with a ΦB ¼�2ηk0ax magnetic flux
through a unit cell at the interface.The different gauge fields in
each subsystem result in a
different band structure (dispersion relation) for
eachsubsystem. The system has discrete symmetries in boththe x- and
z-directions, with periods of ax and
ax2η,
respectively (see Supplementary Information). Each ofthese
dictates a conservation law for the respectivemomentum (up to 2π
over the period), leaving only ky tobe modified as a wave crosses
between the two regions.Thus, launching an eigenmode (a Bloch wave)
with adefined wavevector (kx, ky,inc) on one side of the systemwill
result in refraction and reflection of the wave uponincidence at
the interface. According to the x- and z-translational symmetries
of the joint lattice, the wave-number in the second half plane will
have to satisfy
βη kx; ky;inc� � ¼ β�η kx; ky;tran� � ð5Þ
where ky,inc is the y-wavenumber of the incident beam, ky,tran
isthe y-wavenumber of the transmitted beam, and βη(kx, ky,inc)
is given by Eq. (3). Equation (5) acts as a generalized Snelllaw
for an interface between two regions of the samemedium but with
different artificial gauge fields on eachside in the specific
realization of titled photonic lattices.Note that Eq. (5) is
general and valid for any interface
that satisfies the symmetries in x and z (the plane normalto the
interface), even for a uniform dielectric medium.The main
difference between refraction from a dielectricinterface and
refraction from an AGF interface lies in thedispersion relation,
that is, the relation between the pro-pagation constant β and the
frequency. In uniform
dielectrics, a plane wave (which is an eigenmode of the
medium) obeys βdielectric
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinωc
� �2 �ðk2x þ k2y Þq
. On the
other hand, for an AGF medium, the dispersion can be
designed as almost any desired relation34,35. In the
specificcase of tilted waveguide arrays, the dispersion relation
isgiven by Eq. (3). The ability to design the dispersionallows for
interesting dynamics such as a negative groupvelocity for some kx
values, for example,2cxax sin kx � k0ηð Þaxð Þ
-
directions, as we suggest in the “Discussion” section(Fig. 7).
Notably, uniform dielectric interfaces requirematerials with
different optical properties on each side ofthe interface, whereas
an AGF interface can be engineeredeven when both sides have the
same optical properties (upto their gauge), achieving refraction
using the samematerials of the same composition and
configuration(periodicity).Figure 2 shows the band structures for
the upper (red)
and lower (blue) waveguide arrays. Depicted in threedimensions
(Fig. 2b), we note the sinusoidal shape of thedispersion along kx
as well as along ky. The projectiononto the kx-component (Fig. 2a),
however, helps us dis-play the ky-conversion between the two
arrays. In theprojected band structure, each band represents the
valuesof β (which plays the role of energy in the analogy to
theSchrödinger equation) for all values of ky associated withthat
band (see the Supplementary Information for a dis-cussion on band
replicas arising from the periodicity ofthe structure in x and z).
As an example, the solid lines inFig. 2a indicate the values of β
associated with somespecific ky. Note that the kx-range for total
reflection(dashed vertical lines in Fig. 2a) is not necessarily
sym-metric around kx= 0 (for a given ky). From this figure, itmay
seem that β is not periodic in kx, but a closer look at
the symmetries in the system reveals that the periodicity
ismaintained (see the discussion in the SupplementaryInformation).
Figure 2c displays a contour plot of equi-βas a function of kx and
ky. The group velocity of a wave-packet at each point is
perpendicular to the equi-β con-tour that goes through that
point.When a wavepacket crosses the artificial GF interface
between the lower array and the upper array, β is conservedsuch
that βη kx; ky;inc
� � ¼ β�η kx; ky;tran� �, as is the transversewavenumber,
kx,inc= kx,tran= kx. Graphically, this meansthat at this value of β
the red and blue bands in Fig. 2aoverlap. The quasi-energy β of the
red band may belong to adifferent ky than that of the blue band;
thus, ky,inc ≠ ky,tran(see solid colored lines in Fig. 2a).
Therefore, when the lightcrosses the artificial GF interface,
ky,inc must changeaccording to
cosðky;tran ayÞ � cos ky;inc ay� �
¼ cxcy
cos kx þ k0ηð Þaxð Þ � cos kx � k0ηð Þaxð Þ½ � � η kxcyð6Þ
We identify three different regimes, which depend on kx:
e7060
50
40
30
20Ref
ract
ion
an
dre
flec
tio
n
d f
0
0.1
0.2
0.3
0.4
0.5
0
0.2
0.4
0.6
0.8
1z = 600 μm
bz = 1900 μm
cz = 0 μm
y / μ
my
/ μm
y / μ
m
x / μm x / μm x / μm
70
60
50
40
30
20
k xa x
= 0
πk x
a x =
0.4
πk x
a x =
0.8
π
To
tal t
ran
smis
sio
n
a
6020–40
h
6020–40
70
60
50
40
30
20To
tal r
efle
ctio
n
6020 0 40–60 –20 0 40–60 –200 40–60 –20–40
g i
0
0.1
0.2
0.3
0.4
0.5
Inte
nsity
(no
rm.)
Inte
nsity
(no
rm.)
Inte
nsity
(no
rm.)
Fig. 3 Simulated dynamics of beams in the three regimes. Total
transmission (upper row), refraction and reflection (middle row),
and totalinternal reflection (bottom row). Each panel shows the
numerically calculated intensity distributions at three different
z-values along the propagationdirection: input facet (left column),
z= 600 μm (middle column) and z= 1900 μm (right column). The dashed
white line indicates the location of thegauge interface. The
initial y-wavenumber is always ky,incay= 0.5π. a–c An input beam
with kxax= 0 is transmitted completely across the interface.d–f An
input beam with kxax= 0.4π is partially reflected and partially
refracted at the interface. The refraction is highlighted by the
change in thetrajectory. g–i An input beam with kxax= 0.8π
undergoes total reflection, never crossing the interface. The
intensity is normalized separately in eachpanel, and the intensity
in the second and third rows is enhanced for better visibility. The
arrows are guides to the eye and indicate the
approximatetrajectories of the respective beams
Cohen et al. Light: Science & Applications (2020) 9:200 Page
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1. Total internal reflection (TIR): kx is such that theblue and
red bands do not intersect, hence nocoupling from the upper to the
lower array (and viceversa) is possible. Consequently, the
wavepacket iscompletely reflected (see Fig. 3g–i).
2. Perfect transmission for kx= 0: A wavepacketcrosses the
interface between the two arrayswithout changing its wave vector
componentswhile allowing all the light to be transmittedthrough the
interface (see Fig. 3a–c).
3. Refraction and reflection for all other values of kx:The red
and blue bands intersect, but for different ky,inc and ky,tran. As
the “energy” β and the transversewavenumber kx are conserved, ky
has to changeupon crossing the interface, resulting in both
arefracted wave and a reflected wave (Fig. 3d–f).
Examples of the dynamics of waves in these threeregimes are
given in Fig. 3, which shows the results ofdirect simulations of
Eq. (1) (using the commercialOptiBPM code), with parameters
corresponding to thoseused in the experiments. The figure shows the
intensity ofoptical beams at three different propagation planes
alongz. We probe the three different regimes by launchinginput
beams with a set ky,inc ay= 0.5π and selecting kxaxcorresponding to
total transmission (kxax = 0), refractionand reflection (kxax=
0.4π), and total internal reflection(kxax= 0.8π). Upon excitation
at z= 0 μm (Fig. 3a, d, g),the beams travel toward the interface
(indicated by thedashed white line), reaching it approximately
after z=600 μm (Fig. 3b, e, h). For the input beam with kxax= 0,the
beam is completely transmitted across the interface(Fig. 3c)
without any reflection. After passing through the
Pha
se
0
1
2
abs.
am
plitu
de
0
c ky,inc ay = 0.4�
TIRTIR
0
1
2
Pha
se
abs.
am
plitu
de
d kx ax = 0.4�
ky,inc ay = 0.4� kx ax = 0.4�
TIR
arg(t )
arg(r )
0
1
0.5
R TIRTIR
T
e f
0
1
0.5
–1 –0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1
–1 –0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1
TIRR
T
a 1
0.6|r |
|r |
|t |
|t |
0 ��
0 ��
0
0.2 (d),(f)
(c),(e)
ky,inc ay
ky,inc ay / �kx ax / �
ky,inc ay / �kx ax / �
ky,inc ay
k x a
x
k x a
x
–�
�
–�
0
�
–�
1
0.5
0
1.5
b
0
(d),(f)
(c),(e)–�
Fig. 4 Fresnel coefficients. Reflection amplitude (a) and
transmission amplitude (b) across the entire kx and ky range. The
line cuts in the middle rowshow the phase (blue lines) and
amplitude (red lines) of the reflection coefficient r (solid lines)
and the transmission coefficient t (dashed lines) alongthe cuts
indicated in (a) and (b): for ky,incay= 0.4π (c) and kxax= 0.4π
(d). Note that |t| can be greater than 1 for some (kx, ky,inc). e
and f show the
reflectance R= |r|2 (solid line) and transmittance T ¼ Re
vgy;tranvgy;incn o
tj j2 (dashed line) for the same wave vectors as in (c) and (d).
The total energy flux isconserved, as R+ T= 1 holds for all
regions
Cohen et al. Light: Science & Applications (2020) 9:200 Page
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interface, the beam strongly disperses (diffracts) in the
x-direction. For the input beam with kxax= 0.4π, part of thebeam is
reflected by the interface, returning to the lowerarray, while part
of it is refracted, as indicated by thechange in the slope of the
arrow (Fig. 3f). For the beamwith kxax= 0.8π, the beam undergoes
total reflection,never crossing the interface (Fig. 3i).Having used
symmetry and the dispersion relation to
find the general laws of refraction and reflection at agauge
interface, the next step is natural: finding thecoefficients for
reflection and transmission. However,similar to the Fresnel
coefficients at a dielectric interface,this calculation is system
specific, and the details dependon the interface between the two
regions. That is, unlikethe Snell-like law, the Fresnel
coefficients cannot begeneralized (conservation of power yields a
relationbetween the absolute values of the Fresnel coefficients,but
to obtain the coefficients, one must also employcontinuity at the
interface). With this in mind, we cal-culate the Fresnel-like
coefficients for our example of agauge interface constructed from
titled waveguides. Weuse an approximate model27 for the coupling
betweenthe two sections and derive an approximate formula forthe
coefficients. The details of the calculation are pre-sented in the
Supplementary Information. Figure 4 showsthe Fresnel-like
coefficients for our gauge interface forparameters corresponding to
those shown in Fig. 3. Weplot the amplitude and phase of the
transmitted and
reflected parts for ky,inc ay= 0.4π as a function of kx andfor
kxax= 0.4π as a function of ky,inc (Fig. 4c, d, respec-tively). As
explained above, the Fresnel coefficients arehighly dependent on
the specifics of the interface; hence, adifferent model for the
gauge field interface will yielddifferent coefficients. However,
the Snell-like law ofrefraction will not change, as it depends only
on thesymmetry and the dispersion relations on both sides ofthe
interface.To demonstrate the generalized Snell’s law in experi-
ments, we fabricate sets of tilted optical waveguides
cor-responding to the system described in Fig. 1. The samplesare
fabricated using direct laser writing to create hollowwaveguides,
which are subsequently infiltrated with ahigher index material
(Fig. 1c, d). For details on the fab-rication, see ref. 36. The
experimental measurement setupis sketched in Fig. 5. We reflect a
700 nm laser beam off aspatial light modulator (SLM) to excite a
Bloch mode witha given ky,inc while exciting the entire first
Brillouin zonein x (i.e., −π ≤ kxax ≤ π). To do this, the beam
reflected offthe SLM37 is shaped such that after a Fourier
transform(by a microscope objective), it consists of 5 lobes,
withtheir phase forming a linear ladder, commensurate withthe
chosen Bloch mode, while in x, the beam is simplyfocused into a
single row of waveguides. The beam passesthrough the sample, and
the output facet is imagedby a camera. Since we are interested in
investigating thepassage through the gauge interface, we scan the
value of
Stage
z
20 μm
y
x
Amplitude Phase
Waveguide structure
ky
1st order
NA
0.8
SLM 50x
20x
NA
0.4
Laser
Fouriertransform
Fig. 5 Experimental setup. A laser beam (wavelength 700nm) is
reflected off an SLM, which imprints a specific phase and amplitude
pattern ontothe beam. To shape the amplitude while using a
phase-only SLM, we overlay the phase pattern on the SLM with a
blazed grating that shifts parts ofthe reflected light into the
first diffraction order37. After Fourier transform by an objective,
the beam consists of five spots with a phase difference ofky,incay
between each of them and a Gaussian amplitude envelope (inset).
These spots are focused onto a row of waveguides below the
interface, asshown by the false color photograph of the beam on top
of the sample, with the interface marked by the dashed line. The
light then propagatesalong z, interacts with the interface, and
exits the sample after a propagation distance of z= 725 μm. The
intensity distribution at the output facet isimaged by a camera, as
is its spatial power spectrum (obtained by inserting an extra lens
at the focal distance to the camera)
Cohen et al. Light: Science & Applications (2020) 9:200 Page
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ky,inc by changing the relative phase between the lobeswhile
exciting the entire first Brillouin zone in x. Figure 5shows a
false color photograph of the input beam overlay aphotograph of the
sample, with the interface marked bythe dashed line. The light
propagates across the interface,and we measure the intensity of the
refracted and reflectedwaves at the output facet of the sample, as
well as theintensity at the Fourier plane (obtained at the focal
planeof a lens), which corresponds to the spatial
powerspectrum.Figure 6 shows the spatial power spectrum
(Fourier
space intensity) of the waves exiting the sample for aninput
wave with different ky,inc values but always launchedat the same
position in y, along with a comparison tobeam-propagation
simulations. The analytically calculatedvalues for ky,tran obtained
by Eq. (6) are marked by thegreen and blue dots on top of the
experimental andsimulated results. The beam travels toward the
artificialGF interface with a group velocity vgy, obtained from
thedispersion relation in Eq. (3) by taking the derivative
withrespect to kyay. In the first row (Fig. 6a, b), the beam
islaunched such that it moves away from the interface (kyay=−0.6π),
and never reaches the interface; hence, the outputbeam has the same
spatial spectrum as the input beam. AsFig. 6a, b show, the power
spectrum of the output beam islocated around the same kyay=−0.6π as
the input beam.In the second row (Fig. 6c, d), the beam is launched
withkyay= 0.1π. As these panels show, the output beam issplit: for
−0.1π < kxax < 0.6π, the beam is transmitted anddisplays
distinct refraction, according to the generalizedSnell’s law
expressed by Eq. (6), while in the regionsbeyond this range, the
beam experiences TIR. Note theprominent asymmetry between the
minimal and maximalkx boundaries between the regions of
transmission andTIR. In experiment (c), the measurement only
partiallyshows the results, as the group velocity in y for kyay=
0.1πis very low, and the beam has only partially passed
theinterface even upon reaching the output facet of thesample. In
the third row, Fig. 6e, f, the beam is launchedwith kyay= 0.5π. For
−0.5π < kxax < 0.5π, the beam istransmitted, while beyond
this range, the beam experi-ences TIR. Here, the asymmetry between
the minimal andmaximal kx boundaries between transmission and TIR
isless significant. The experiment (Fig. 6e) captures boththe
refraction and TIR regions. In the fourth row, Fig. 6g,h, the beam
is launched with kyay= π. For −0.7π < kxax <−0.1π, the beam
is transmitted. Beyond this domain, thebeam experiences TIR. Here,
both the incident andreflected beams have the same |ky|. This case
is similar toprism coupling at a grazing angle to couple a beam
into awaveguide where it will be bound by TIR.The comparison of the
experimental measurements and
numerical calculations with the analytical Eq. (6) (dotsin Fig.
6) shows good agreement for the refracted part.
–1
–0.5
Measurement
a b
c d
e f
g h
BPM
k y a
y / π
k y a
y / π
ky,inc ay = –0.6 π
ky,inc ay = 0.1 π
ky,inc ay = 0.5 π
ky,inc ay = π
k y a
y / π
k y a
y / π
kx ax / π kx ax / π
0.5
1
0.8
0.6
0.4
0.2
01
0
–1
–0.5
0.5
1
0
TIR
Refracted
Input
–1
–0.5
0.5
1
0
–1
–0.5
0.5
0.5 1–0.5–1 0 0.5 1–0.5–1 01
0
Fig. 6 Refraction and reflection by an artificial gauge
interface.Spatial power spectrum (intensity in Fourier space) for
an input wavewith several values of ky,inc always launched at the
same position iny. The left and right columns depict the
experimental and simulatedresults, respectively. The purple dashed
line shows the location of theinput beam in Fourier space and the
dots show the values analyticallycalculated from Eq. (6) (green for
refraction and blue for TIR). For ky,incay=−0.6π, the beam travels
away from the interface and does notrefract at all (a, b), while
for the other panels (c–h), we see both partialrefraction and
reflection. In the second row (c, d), the beam islaunched with
kyay= 0.1π. As these panels show, the output beam issplit: for
−0.1π < kxax < 0.6π, the beam is transmitted and
displaysdistinct refraction, according to the generalized Snell’s
law expressedby Eq. (6), while in the regions beyond this range,
the beamexperiences TIR. Note the prominent asymmetry between the
minimaland maximal kx boundaries between the regions of
transmission andTIR. In the third row (e, f), the beam is launched
with kyay= 0.5π. For−0.5π < kxax < 0.5π, the beam is
transmitted, while beyond this range,the beam experiences TIR. The
experiment (e) captures both therefraction and TIR regions. In the
4th row, (g, h), the beam is launchedwith kyay= π. For −0.7π <
kxax < 0.1π, the beam is transmitted. Beyondthis domain, the
beam experiences TIR. Here, both the incident andreflected beams
have the same |ky|. Altogether, the comparison of themeasurements
(left row), simulations (right row), and analyticalexpression from
Eq. (6) (green and blue dots) shows good agreementwith the expected
ky,tran-distribution. A movie showing the completeset of
measurements can be found in Supplementary Movie no. #1.
Cohen et al. Light: Science & Applications (2020) 9:200 Page
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-
As expected, the obtained k-distribution is broader thanthe
analytical curve due to finite size effects. Namely, theinput beam
with ky,inc has a finite width (see Fig. 5) of fivewaveguides. The
more waveguides are excited in realspace, the smaller the width of
the k-component inFourier space. However, we do not want the input
patternto excite waveguides in the other array across the
artificialGF interface; hence, we have to limit the size of the
inputbeam. In addition, the center position of the input
patternneeds to be close enough to the artificial GF interface
suchthat the beam can travel across the artificial GF interfacein
the given propagation distance. Therefore, we need tolimit the
number of excited waveguides. For excitation offive waveguides, the
width is Δky,inc ay ≈ 0.4π (see Fig. 6).As the same number of
illumination spots is chosen in theexperiments, the numerical
calculation reflects theexperimental conditions very well.
Altogether, the com-parison of the measurements, simulations, and
analyticalexpression shows good agreement with the expected
ky,tran-distribution. A movie showing the complete set
ofmeasurements can be found in Supplementary Movie #1.
DiscussionHaving demonstrated the Snell law for refraction
and
reflection at an interface between two different artificialGFs,
we move on to concatenating several gauge inter-faces and
constructing devices. As an example high-lighting the possibilities
that refraction by artificial GFsallows, we design a gauge-based
imaging system. By rea-lizing such a system with arrays of tilted
waveguides, wedesign a scheme that maps any (arbitrary)
wavepacketinput at the input facet to the output facet. In the
scheme
based on waveguide arrays, this corresponds to a systemwith
different rows of waveguides tilted at different angles(Fig. 7a)
mapping an input state from row y0 to row yimage.For every row of
waveguides with a tilt η(y) positioned aty, we find the propagation
constant βη yð Þ ¼ β0 þ2cx cos kx � k0η yð Þð Þaxð Þ þ η yð Þkx �
12 k0η2 yð Þ: To producean image, we need the phase accumulation by
all com-ponents to be identical. Thus, we require that when
awavepacket diffracts along y and propagate along z frominput row
y0 to output row yimage, the cumulative phaseaccumulation for each
of its kx constituents is the same.Therefore, the value of
R yimage0 βη yð Þ kxð Þdy should not
depend on kx. This can be written as
dd kxð Þ
Z yimage0
βη yð Þ kxð Þdy ¼ 0 ð7Þ
This requirement can be fulfilled in a 2D array ofstraight
waveguides, where each row (y) is tilted at a dif-
ferent angle such thatη yð Þ ¼ η0 sin 2πyyimage� �
. The require-
ment in Eq. (7) can be expressed by
Z 2π0
exp �ik0η0ax sin y0ð Þ� �
dy0 ¼ J0 k0η0ax� � ¼ 0 ð8Þ
where J0 is the zeroth-order Bessel function and y0 ¼
2πyyimageis now unitless. In other words, by designing the tilt
ofeach row properly, an arbitrary wavepacket f(x) at row y=0 is
reproduced at row yimage (apart from a global phase).Figure 7a
shows a sketch of the gauge-imaging waveguidestructure. Each row
has a different tilt angleη yð Þ ¼ η0 sin 2π y�y0ð Þyimage�y0
� �. Figure 7b compares the amplitude
4�
–4�–20 0
a
b c
x / μm x / μm20 –20 0 20
0.3
0.2
0.1
0
Am
plitu
de (
arb.
u.)
Phase at y0
Phase at yimage
Amplitude at y0
Amplitude at yimage
0
y0
z
x
y
yimage
Pha
se
Fig. 7 Gauge-based imaging system. a Sketch of a gauge-based
imaging system. The tilt angle η for each row as a function of y is
given by
η yð Þ ¼ η0 sin 2π y�y0ð Þyimage�y0� �
. We launch the beam at y0 and expect it to reconstruct at
yimage. In the calculations, we assume a periodicity in x of 32
waveguides. b, c The amplitude (pink) and phase (light blue) at
y= y0 and z= 0 (squares) compared to the amplitude (red) and phase
(dark blue) aty= yimage (circles). b The system parameters satisfy
Eq. (8), and reconstruction in both the amplitude and phase is
achieved. The output distance z ischosen such that the maximal
total power is obtained at the image plane (which occurs at z= 1000
μm). c The system parameters are the same as in(b) except for η0
such that now, it does not satisfy Eq. (8). Even at the z with the
best fit by cross-correlation (z= 926 μm), the original signal
differscompletely from the output signal
Cohen et al. Light: Science & Applications (2020) 9:200 Page
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and phase of the wavepacket launched at y= y0 and z= 0(pink and
light blue) to those of the imaged one at y=yimage and z= 1000 μm
(red and dark blue), revealing thatthe final and initial
wavepackets are essentially the same.One should note that the beam
diffracts in y as itpropagates along z; hence, the imaging is
one-dimensionalfor the field distribution in x only. Therefore, the
intensityreaching the row at yimage is limited by the 1D
discretediffraction in z such that the intensity at each row
(neglecting back reflections) is given approximately by
jJΔyayf zð Þð Þj2, where f(z) is a function of z that depends
on
the details of the coupling and JΔyayis the Bessel function
of
the order of row number Δyay . In our simulated example,
there are 29 rows of waveguides between y0 and yimage, so
the maximal intensity that can propagate to yimage islimited to
max J29j j2� 4:7% of the initial intensity. Inpractice, we obtain
approximately 2.9% due to backreflections (in y) and slightly
different effective couplingsalong y for each kx component. In the
simulation, weassume that the structure is periodic in x with a
period of32 waveguides. The output facet is chosen to
supportmaximal total power at the imaging row. Figure 7c usesthe
same system as Fig. 7b, changing only the size of η0such that Eq.
(8) is no longer fulfilled. We simulate thepropagation of the same
wavepacket in this (non-imaging)structure up to the distance that
gives the maximal cross-correlation between the input and the
output, whichoccurs at z= 926 μm, yet it is clear that the input
andoutput wavepackets do not overlap. Essentially, withproper
design of this simple gauge-based imaging system,we can transfer an
arbitrary wavepacket from an initialrow at the input facet of the
structure to a preselected rowat the output facet. This idea works
well as long as theBloch mode spectrum in x never projects onto
evanescentmodes (generated by TIR) throughout the propagation.This
example, albeit simple, demonstrates that it ispossible to
construct various optical devices and systemsby engineering
artificial gauge fields using just onedielectric material.To
summarize, we derived the laws of refraction and
reflection at an interface between two regions differingsolely
by their artificial gauge fields and demonstratedthe concepts in
experiments in 3D-micro-printed opticalwaveguide arrays.
Generalizing the concepts of refractionand reflection at a gauge
interface offers exciting possi-bilities for routing light and more
generally for con-structing photonic systems in a given medium
strictly bydesigning the local gauge. As an example, we proposedan
imaging system that maps any input state from oneplace at the input
facet to a predesignated other locationat the output facet by
cascading different artificial gaugefields.
AcknowledgementsG.v.F. acknowledges support by the Deutsche
Forschungsgemeinschaftthrough CRC/Transregio 185 OSCAR (project No.
277625399). M.S. gratefullyacknowledges support by an ERC Advanced
Grant, by the Israel ScienceFoundation, and by the German-Israel
DIP project.
Author details1Physics Department, Technion—Israel Institute of
Technology, Haifa 32000,Israel. 2Solid State Institute,
Technion—Israel Institute of Technology, Haifa32000, Israel.
3Physics Department and Research Center OPTIMAS, TUKaiserslautern,
67663 Kaiserslautern, Germany. 4Department of Physics,
ThePennsylvania State University, State College, PA 16802, USA.
5FraunhoferInstitute for Industrial Mathematics ITWM, 67663
Kaiserslautern, Germany
Author contributionsAll authors contributed significantly to
this work.
Data availabilityAll experimental data and any related
experimental background informationnot mentioned in the text are
available from the authors upon reasonablerequest.
Conflict of interestThe authors declare that they have no
conflict of interest.
Supplementary information is available for this paper at
https://doi.org/10.1038/s41377-020-00411-7.
Received: 28 May 2020 Revised: 14 September 2020 Accepted:
14September 2020
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Generalized laws of refraction and reflection at interfaces
between different photonic artificial gauge
fieldsIntroductionResultsDiscussionAcknowledgements