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 (msegev@technion.ac.il)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:msegev@technion.ac.il

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 2 of 11

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 3 of 11

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

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bz = 1900 μm

cz = 0 μm

y / μ

my

/ μm

y / μ

m

x / μm x / μm x / μm

70

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k xa x

= 0

πk x

a x =

0.4

πk x

a x =

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To

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ran

smis

sio

n

a

6020–40

h

6020–40

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6020 0 40–60 –20 0 40–60 –200 40–60 –20–40

g i

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0.5

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nsity

(no

rm.)

Inte

nsity

(no

rm.)

Inte

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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 5 of 11

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 6 of 11

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 7 of 11

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

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a b

c d

e f

g h

BPM

k y a

y / π

k y a

y / π

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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 8 of 11

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 9 of 11

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