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Resonator-free realization of effective magnetic field for photons

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2015 New J. Phys. 17 075008

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New J. Phys. 17 (2015) 075008 doi:10.1088/1367-2630/17/7/075008

PAPER

Resonator-free realization of effective magnetic field for photons

Qian Lin1,3 and Shanhui Fan2

1 Department of Applied Physics, StanfordUniversity, Stanford, CA 94305,USA2 Department of Electrical Engineering, StanfordUniversity, Stanford, CA 94305,USA3 Author towhomany correspondence should be addressed.

E-mail: linqian@stanford.edu

Keywords: effectivemagnetic field for photons, waveguide network, integer quantumHall effect

AbstractWepropose to create an effectivemagnetic field for photons in a two-dimensional waveguide networkwith strong scattering at waveguide junctions. The effectivemagneticfield is realized by imposing adirection-dependent phase along eachwaveguide link. Such a direction-dependent phase can beproduced by dynamicmodulation or by themagneto-optical effect. Compared to previous proposalsfor creating an effectivemagneticfield for photons, this scheme is resonator-free, thus potentiallyreduces the experimental complexity.We also show that such awaveguide network can be used toexplore photonic analogue of integer quantumHall effect formassless particles.

1. Introduction

The concepts of an effective gauge potential and effectivemagnetic field for photons has been extensivelyexplored in the past few years [1–3]. Such concepts have strong connections to the emerging interests intopological photonics [4–8], and have led to the developments of new capabilities formanipulating light, such ason-chip optical isolators [9], topologically protected one-way edge states [10–16], as well as gauge-field inducednegative refraction [17] andwaveguiding [18].

In typical proposals for achieving effectivemagnetic field for photons [2, 3, 15, 16], one consider an array ofresonators as described by a tight-bindingHamiltonian, with the coupling constants between the resonatorsexhibiting direction-dependent phases. In these proposals, the effectivemagnetic field in fact arises solely fromthe direction-dependent phases, and the resonators themselves play no essential role with respect to the effectivemagnetic field, other than perhaps simplifying theoretical treatments by allowing the use of a tight-bindingmodel. On the other hand, in experimental systems, the use of resonator array places a very strong constraint onthe practical implementation and applications. For example, the resonators themselves typically impose astringent constraint on the operating bandwidth.

In this paperwe propose a resonator-free scheme for generating an effectivemagnetic field for photons.Weshow that thewaveguide network, as proposed in [19], can be used as a platform for implementing themechanism for creating an effective gaugefield. Using the transfermatrix formalism that describes thewaveguide network system,we demonstrate robust one-way edge states andHofstadter butterfly spectrum asevidence of an effectivemagnetic field for photons in these waveguide networks. This proposal eliminates thepotential experimental complexity introduced by the resonators.

Furthermore, the network systemprovides additional flexibility in exploring newphysics associatedwithphotonic gauge field. For example, by designing the scatteringmatrix at thewaveguide junction, the networkmay exhibit aDirac-like dispersion in the absence of an effectivemagnetic field [19]. As a result, the behavior ofthe photonic statesmimics the quantumHall effect of a linear dispersion system (such as a single layergraphene), and differs from that of a quadratic dispersion systemone usually get from a tight bindingmodel on asquare lattice.

This paper is organized as follows: section 2 describes the physicalmodel of thewaveguide network system,and possible implementations of thewaveguide junction and the direction-dependent phase onwaveguide links.Section 3 summarizes the band structure of a reciprocal waveguide network system. Section 4 introduces a

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uniformmagnetic field into thewaveguide network by adding direction-dependent phases onwaveguide links.We present the dispersion and transport behavior of the resulting topologically protected edge states. Section 5calculates theHofstadter butterfly for thewaveguide network system, and explores its connection to theDirac-like underlying dispersion.We summarize in section 6.

2. Physicalmodel forwaveguide networks

Throughout this paper wewill consider a class of waveguide network system as shown schematically infigure 1(a). The system consists of a square lattice of four-port waveguide junctions. These junctions areconnected into a network via a set of waveguide links. Such a systemhas been analyzed previously byFeigenbaumandAtwater in [19], who envision awaveguide network systembased onmetal-insulator-metalwaveguide structures (figure 1(b)) [20]. Similar waveguide networks can also be constructed for radio frequency(RF) photons using transmission lines ormicrostrip lines, or in the optical frequency range using dielectricwaveguides [21]. Unlike Feigenbaum andAtwater, here we seek to create an effectivemagnetic field for photons,by introducing a non-reciprocal direction-dependent phase along eachwaveguide link.

In this work, we consider a four-port junction described by the following scatteringmatrix S:

a x y

b x y

c x y

d x y

S

a x y

b x y

c x y

d x y

( , )( , )( , )( , )

·

( , )( , )( , )( , )

, (1)

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

′′′′

=

Figure 1. (a) Conceptual drawing of thewaveguide network. The pink dots represent waveguide junctions acting as four-port powersplitters. The black lines represent waveguide links. The blue boxes represent components that generate a direction-dependent phasefor photons propagating through thewaveguide link. (b) Apossible implementation of awaveguide networkwithout the non-reciprocal phase using a plasmonicmetal–insulator–metal wave network, based on the original proposal in [19]. The yellow regionsrepresentmetal. Thewhite regions represent a dielectric such as air. (c) Thewaveguide junction and the labeling of various input andoutput wave amplitudes at the waveguide junction. (d) The labelling of input and output amplitudes at a waveguide link and thedefinition of the reciprocal ( rϕ) and non-reciprocal ( nrϕ ) phase shift. (e) The dynamicmodulation scheme for the generation of adirection dependent phase. Twomixers (circles with crosses) are driven by local oscillators with phases difference nrϕ . The differencebetween the local oscillator phase defines the direction-dependent phase. (f) Themagneto-optical scheme for the generation ofdirection-dependent phase. Amagneto-opticalmaterial is incorporated into a normalwaveguide and externalmagnetic field isapplied to split the forward and backward propagatingmodes.

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New J. Phys. 17 (2015) 075008 QLin and S Fan

S

t r t t tt t r t tt t t r tt t t t r

ii

ii

. (2)

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=

− +− +

− +− +

Here, (x,y) labels the position of the junction. The incoming and outgoing amplitudes along fourwaveguidesconnected to the junction at (x,y) are denoted as a x y( , ), b x y( , ), c x y( , ), d x y( , ) and a x y( , )′ , b x y( , )′ ,c x y( , )′ , d x y( , )′ , respectively, as shown infigure 1(c). Thewaveguide junction is describedmathematically by ascatteringmatrix S in equation (2).Here t ri− + is the reflection coefficient, and t is the transmission coefficientinto a single outgoingwaveguide. r and t are both assumed to be real. Herewe assume a fully-symmetric Smatrix. The formof the S-matrix is further constraint by energy conservation and time-reversal symmetry.Energy conservation requires that r t4 12 2+ = .

In thewaveguide network considered here, when the power splitting into all four output ports arecomparable in amplitude, any closed loop of connectedwaveguides demonstrate strong interference andresonance effect. The coupling of these non-local interferences and resonances enriches the network dynamicsand tunability. Such power splitting behavior in thewaveguide junction can be easily achievedwhen the crosssection of thewaveguide and the dimension of the junction is smaller than thewavelength, as in the cases ofplasmonicmetal–insulator–metal waveguides [19, 20], RF transmission lines andmicrostrip lines. In thesesystems the junction acts like a point-scatterer with equal scattering amplitude into all output ports.Furthermore, such a scatteringmatrix can be achievedwith design of dielectric waveguide structures, for whichboth thewaveguide and the junction are at the single wavelength scale [21].

Sincemost previous works on the effective gaugefield for photons have utilized arrays of resonators[2, 3, 15], here we briefly comment on the connection between awaveguide network and a resonator array. Thewaveguide junction as shown infigure 1(a) can be realizedwith a resonator coupled equally to the fourwaveguide ports. In this case, the transmission and reflection can be calculated using coupledmode theory [22],which relates the incoming and outgoingwave amplitude si

+ and si− in the four ports to the amplitude of the

resonantmode b

b

tb b s

s s b i

d

di

4 2,

2, 1, 2, 3, 4. (3)

i

i

i i

0

1

4

∑ωτ τ

τ

= − +

= − + =

=

+

− +

Here 0ω is the resonance frequency. 1 τ is the amplitude decay rate of the resonator to one of thewaveguides.From equation (3) we can solve for si

− in termof si+ at a given frequencyω

s s s i12

i( ) 4

2

i( ) 4, 1, 2, 3, 4 (4)i i

j j i

j0 1:

4

0

⎛⎝⎜⎜

⎞⎠⎟⎟ ∑τ

ω ω ττ

ω ω τ= − −

− ++

− +=− +

= ≠

+

and then obtain the scatteringmatrix of the junction. For the resonant case 0ω ω= , equation (4) simplifies to

s s s i1

2

1

2, 1, 2, 3, 4.i i

j j i

j

1:

4

∑= − + =− +

= ≠

+

The scatteringmatrix therefore has exactly the same form as in equation (2)with t=0.5 and r=0. Thus, thebehavior of thewaveguide network can be realized alternatively by considering a resonator array operating nearresonance. In previous works on the effective gaugefield for photons using a resonator array, the system is alwaysoperating close to resonance. However these resonators have very limited bandwidth. Thewaveguide junctionstudied here reproduces the behavior of a resonator operating on-resonance, but has amuch larger bandwidthsince the scatteringmatrix can be largely frequency-independent.

For the resonator case, at frequencies away from 0ω , if we define t 2

i( ) 40= ∣ ∣τ

ω ω τ− + and absorb the phase of

the coefficient2

i( ) 40

τω ω τ− + into the definition of s j

+, equation (4) becomes

s t s ts i1 i2

, 1, 2, 3, 4.i i

j j i

j0

1:

4

⎜ ⎟⎛⎝

⎞⎠ ∑ω ω

τ= − +

−+ =− +

= ≠

+

This is equivalent to the scatteringmatrix in equation (2) oncewe define

r t2

.0ω ωτ

≡ −−

Thus thewaveguide junctionwith t 0.5< represents a resonator operating off-resonance. The scatteringmatrixin equation (2) used to describe thewaveguide junction can alsomap to resonator arrays operating on and offresonance.

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New J. Phys. 17 (2015) 075008 QLin and S Fan

Nextwe look at thewaveguide link.Herewe assume that thewaveguide link has no scattering between thetwo propagation directions, such that

( )

( )

a x y b x y

b x y a x y

( 1, ) e ( , ),

( , ) e ( 1, ), (5)

i ( )

i ( )

r

r

nr

nr

+ = ′

′ = ′ +

ϕ ω ϕ

ϕ ω ϕ

+

−

where k a( ) ( )rϕ ω ω= is the reciprocal phase, as was also used in [19]. Here a is the length of eachwaveguidelink. In this work, for simplicity we use a linear dispersion k vω= where v is the phase velocity in thewaveguide. Amore general dispersion relationwill not change the essence of our result. Unlike [19], however,here we also include a non-reciprocal phase shift nrϕ . The two directions of propagation have opposite non-reciprocal phases. The non-reciprocal phases are introduced to create an effectivemagnetic field for photons inthewaveguide network.

Experimentally, the non-reciprocal phase can be introduced either with the use of dynamicmodulation orwith the use ofmagneto-optical effects.With the dynamicmodulation scheme, one achieves a non-reciprocalphase by cascading twomixers. The difference in the phases of the local oscillators provides the non-reciprocalphase (figure 1(d)). This schemewas theoretically proposed in [23] and experimentally demonstrated in[9, 24, 25]. Alternatively, the non-reciprocal phase can be achieved usingmagneto-optical waveguides(figure 1(e)), as discussed theoretically in [26] in the context of photonic gaugefield, and demonstratedexperimentally in [27].We refer the readers to the cited references for details on the experimentalimplementation. To conclude this sectionwe note that the basic ingredients of ourwaveguide network, i.e. thejunction and thewaveguide link that provides a non-reciprocal phase, have all been implementedexperimentally before. Onemay therefore anticipate the possibility of integrating these components togetherinto a larger network.

3. Summary ofwaveguide network properties

In this sectionwe provide a summary of the photonic band structure of thewaveguide network shown infigure 1(a), in the absence of an effectivemagnetic field. In the rest of this paper wewill refer to such a bandstructure as the underlying band structure. A detailed discussion of the properties of such awaveguide networkcan be found in [19].Here we focus only on those aspects that wewill need for subsequent discussions.

Figure 2 shows the photonic band structure of a square lattice of waveguide networks.We assume thewaveguide links are reciprocal by setting 0nrϕ = in equation (5).We consider two different cases correspondingto t=0.5 and t=0.4 in equation (2). In both cases, the network supports four photonic bands, since each unit cellhas four degrees of freedom corresponding to a x y( , ), b x y( , ), c x y( , ) and d x y( , ) in equation (2). Among thefour bands, two of them are flat, which correspond to localized resonant states in the network [30]. In thefollowingwewill focus on the two non-flat bands, whichwewill refer to as the upper and lower bands,respectively.

For t=0.5 (figure 2(a)), the upper and lower bands touch at theΓ point, and thewhole band structure isgapless. Furthermore, the dispersion is linear near theΓ point. For t 0.5< (figure 2(b)), the degeneracy at theΓpoint between the lower and upper bands is lifted, and the spectral width of both the upper and lower bands iscompressed as compared to the casewhere t=0.5. The band structure is gapped, and the dispersion near theΓpoint becomes quadratic. Tuning the scattering amplitude t of thewaveguide junction thus allow one to exploresystemswith photons that are eithermassless ormassive near theΓ point.

4. Effectivemagneticfield and edge state transport inwaveguide networks

In this section, we calculate the band structure of an semi-infinite stripe of waveguide networkwith a uniformeffectivemagnetic field.We further demonstrate the edge state transport in afinite structure. In this section, weassume t=0.5 in equation (2).

A uniform effectivemagnetic field for photons is introduced by imposing a specific arrangement of non-reciprocal propagation phases onwaveguide links as shown infigure 3(a). Such an arrangement corresponds tothe vector potential in the Landau gauge for a uniformmagnetic field. All waveguide links along the x-directiondo not have non-reciprocal phase, i.e. 0nrϕ = . All waveguide links along the y-direction at the same xhave thesame non-reciprocal phase, and such a phase increases linearly as a function of x, i.e. xnrϕ ϕ= . This produces an

effectivemagnetic flux ofϕ per unit cell, and an effectivemagnetic field a zB ˆeff2ϕ= .

The key effect of an effectivemagnetic field is the creation of topologically robust one-way edge states.Therefore, we consider the stripe geometry shown infigure 3(a). The stripe has a finite width in the x-direction,and is infinite in the y-direction. Thewaveguide links at the edges of the structure in the x-direction are

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New J. Phys. 17 (2015) 075008 QLin and S Fan

truncatedwith a perfectly reflecting boundary condition.With our choice of spatial distribution of the non-reciprocal phases, the structure remains periodic in the y-direction. Therefore, its eigenstates can becharacterized by a band diagram relating the frequenciesω to thewavevector ky along the y-direction. Such aband diagram is shown infigure 3(b), assuming that the stripe has awidth of a20 , and 2 3ϕ π= , such that themagnetic flux through each unit cell is 1 3 of theflux quanta 2π . Theflat bands from figure 2 remain unchanged,since they correspond to localized resonances in thewaveguide network, and are unperturbed by the effectivemagnetic field. Since themagnetic flux through each unit cell is 1 3flux quanta, the corresponding infinitestructure will have a periodicity of a3 along the x-direction. As a result the lower and upper non-flat bands infigure 2 each splits into three groups ofmagnetic subbands. Band gaps occur between thesemagnetic subbands.

For each band gap between a pair ofmagnetic subbands, there are two one-way edge states whose dispersionspan the entire gap as can be seen infigure 3(b). These two states have opposite group velocities. The spatialprofile of these edge states can be visualized by calculating the distribution of powerflux in the y-direction,which is c x c x( ) ( )2 2∣ ∣ − ∣ ′ ∣ using the labeling convention infigure 1(b). This powerflux distribution is plotted infigure 3(b).We see that the powerflux is strongly localized on the edge, and that the two edge states in the sameband gap are located on opposite edges of the stripe and propagate in opposite directions. Thesemodes are thephotonic analogue to the one-way edge states of electrons in the integer quantumHall effect.

Having established the existence of the one-way edge states, we now consider their transport properties. Forthis purpose, we consider a finite waveguide network structure with a dimension of a10 by a10 , as shown infigure 4(a). The network structure is subject to a uniform effectivemagnetic field corresponding to 2 3ϕ π= .We terminate all waveguide ports at the edge of the structure with a perfectly reflecting boundary condition,except for the two ports at the upper-left and the lower-right corners, which act as the input/output ports.Weinject thewave into the input port and extract the transmittedwave from the output port. For bettervisualization of the transport properties, we introduce a small loss at eachwaveguide link. The transmission persingle pass through a link is assumed to be 99%,which corresponds to a single-pass loss of 0.087 dB per link. Thetransmission spectrum is plotted infigure 4(b). Transmission is significant at frequencies that lie within theband gaps spanned by edge states, and is suppressed at frequencies that lie either in themagnetic subbands or inthe topologically trivial gaps.

Figures 4(c)–(f) shows the powerflux distribution at four different frequencies. These frequencies areindicated by arrows infigure 4(b). The dashed lines indicate thewaveguides, and the arrows indicate the inputand output ports. The powerflux along eachwaveguide link is represented by a colormap superimposed on top

Figure 2.Band structure of an infinite waveguide network in the absence of the direction-dependent phase, in the first Brillouin zone(left) and along the symmetry directions of a square lattice (right). (a) t=0.5 and r=0.Note the triply degenerate Dirac-like point attheΓ point [28, 29] consisting of two crossing linear bands and aflat band. (b) t=0.4 and r=0.6.Note the existence of gap at theΓpoint.

5

New J. Phys. 17 (2015) 075008 QLin and S Fan

of a schematic of the physical structure. Using the labeling convention in figure 1(c), the powerflux in thehorizontal waveguide links is defined as a x y a x y( , ) ( , )2 2∣ ∣ − ∣ ′ ∣ , and the power flux in the vertical waveguidelinks is defined as c x y c x y( , ) ( , )2 2∣ ∣ − ∣ ′ ∣ . The powerflux is represented in redwhen it goes downor right, andbluewhen it goes up or left.

When the excitation frequency lies within a gap spanned by edge states (figures 4(c) and (e)), the power fluxis well-confined to the edge of the block and efficiently transmitted to the output port. In contrast, when thefrequency lies within themagnetic subbands (figure 4(f)), the flux diffuses into the bulk of the block and isattenuated significantly.When the frequency lies within a trivial gap that does not have edge states (figure 4(d)),inputflux barely penetrates into the bulk of the block and ismostly reflected.

In the left panel offigure 4(c), when light is injected from the top left corner and extracted from the bottomright corner, the power flux follows the left-bottom edge of the block, and it goes down and right. In the rightpanel offigure 4(c)where the input and output ports are switched from that of the left panel, the power fluxfollows the right-top edge of the stripe, and it goes up and left. This simulation result verifies that the two edgestates lyingwithin the same topologically non-trivial gap are located on opposite edges and travel in oppositedirections, in consistencywith the analysis of the semi-infinite stripe infigure 3.

Figure 4(c) is calculated for a frequencywithin the bandgap between the two lower subbands of the uppernon-flat band of the underlying band structure. In contrast, figure 4(e) is calculated for a frequencywithin thebandgap between the two upper subbands of the lower non-flat band of the underlying band structure. The edge

Figure 3. (a) A semi-infinite waveguide networkwith a uniform effectivemagneticfield. The phase on each vertical link denotes thenon-reciprocal phase a photon accumulates as it propagates upward along thewaveguide link. Downward propagation accumulatesan opposite phase. The ends of the horizontal waveguides are truncatedwith a perfectly reflectingmirror. (b) Band structure and edgestate power flux in the y-direction of a semi-infinite waveguide networkwith awidth of a20 , andwithmagneticflux 2 3ϕ π= perunit cell for t=0.5.Using the labeling convention infigure 1(c), the powerflux in the y-direction is c x c x( ) ( )2 2∣ ∣ − ∣ ′ ∣ .

6

New J. Phys. 17 (2015) 075008 QLin and S Fan

states in these two bandgaps have different chiralities. As a result the powerfluxflows anti-clockwise infigure 4(c) and clockwise infigure 4(e).

The presence of small amount of loss does not change the nature of the topological edge states. This isverified by the simulation shown infigure 4, inwhich a small loss on eachwaveguide link is included. In general,to demonstrate the effect predicted here, the propagation length needs to be on the order of a few structuralperiods. For plasmonicmetal–insulator–metal waveguide, the propagation length of guidedmode in near IR canbe a few tens ofmicrons [20]. Thus the periodicity needs to be on a single wavelength scale. Loss can bemuchlower in dielectric waveguide for optical frequencies ormicrostrip line for RF frequencies, which should allowstructures with larger periodicities thatmight be easier to construct.

To summarize this section, by computing both the bandstructure and the transport properties we show thattopologically protected edge states can be created using awaveguide network subject to a uniform effectivemangetic field. The results show that non-trivial topology in photonic states can be achieved in a systemwithoutresonators.

5.Dirac-like dispersion

In section 3we show that by tuning the scattering parameter t of thewaveguide junction, the underlyingbandstructure near theΓ point can be tuned fromquadratic to linear and from gapped to gapless. Thewaveguidenetwork can therefore be used to study the effect of a gaugefield in bothmassive andmassless systems. In thissection, we contrast the behavior of thewaveguide network under a uniform effectivemagneticfield between themassive andmassless cases, by computing theHofstadter butterfly pattern [31] in both cases.

Figure 4.Transport through a a10 by a10 block of waveguide network. (a) Illustration of the structure. All waveguide terminals areperfectly reflecting, except the input and output ports, as indicated by the green arrows. (b) Transmission as a function of frequency.The yellow regions indicate the frequency ranges of themagnetic subbands. Thewhite regions correspond to the band gap regions.(c)–(f) Power flux distribution for forward (left) and backward (right) excitation, with input frequencymarked by arrows in (b).Arrows in (c)–(f) indicate the input and output ports. Dash lines indicate waveguide links. Power flux is shown in redwhen goingdown or right, and bluewhen going up or left.

7

New J. Phys. 17 (2015) 075008 QLin and S Fan

Infigure 5we show theHofstadter butterfly pattern for thewaveguide network, under an uniform effectivemagnetic field. The horizontal axis is themagnetic fluxΦnormalized to theflux quanta 20Φ π= . For each

p q0Φ Φ = where p q is a irreducible fraction, the system is periodic with amagnetic unit cell consisting of qunit cells of thewaveguide network lattice in the absence of the effectivemagnetic field. Such a periodicityfacilitates numerical calculations of the frequencies of the eigenstates.We plot the eigen frequencies infigure 5,with each black dot in figure 5 representing an eigen state of thewaveguide network under a specific effectivemagnetic field. Clusters of black dots formmagnetic subbands, separated bywhite patches representingbandgaps. In the latticeHofstadtermodel, the lowestmagnetic subband at small effectivemagnetic field (e.g.small 0Φ Φ ) should behave similarly to the lowest Landau level calculated from a continuousmodel. At small

0Φ Φ , the lowestmagnetic subbands therefore should have near-zero bandwidth, as we indeed observe infigure 5.

Figures 5(a) and (b) are calculatedwith different waveguide junction scattering amplitude t. Figure 5(a)corresponds to the case of t= 0.5, for which the underlying dispersion isDirac-like (linear) at theΓ point asshown infigure 3(a). Consequently the frequency of the lowestmagnetic subband at a small effectivemagneticfield is proportional to the square root of the effectivemagnetic field, e.g. 0ω Φ Φ∝ as shownwith the redcurve infigure 5(a), in consistencywith [32–34]which explored integer quantumHall effect in graphene. Incontrast, figure 5(b) corresponds to the case of t=0.4, of which the underlying dispersion is gapped and isquadratic near theΓ point as shown infigure 3(b). This system then behaves as amassive system, and the lowestmagnetic subband has linear dependence on the effectivemagnetic field or 0Φ Φ , as shown by the red curve infigure 5(b). This behavior reproduces the standard behavior in integer quantumHall effect for a free electronundermagnetic field. Thewaveguide network system thus enables one to explore the effect of an effectivemagnetic field, for bothmassless andmassive photons.

6. Summary

In summary, we propose a two-dimensional waveguide networkwith non-reciprocal phases as a resonator-freeplatform for realizing an effectivemagnetic field for photons, and demonstrate photonic analogue to the integerquantumHall effect through band structure calculations and edge state transport simulations. In addition, weshow that such awaveguide network can be used to study the effect of an effectivemagnetic field, on eithermassless ormassive photons, by adjusting the power splitting ratio at thewaveguide junction.

Thewaveguide networkmay prove to be a versatile platform for experimentally demonstrating the effectivemagnetic field for photons. It is free of some of the experimental constraints imposed by resonators, and itpossesses rich tunability in its underlying dispersion.

This work is supported by anAFOSRMURI program,GrantNo. FA9550-12-1-0488.

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New J. Phys. 17 (2015) 075008 QLin and S Fan

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New J. Phys. 17 (2015) 075008 QLin and S Fan