Abstract

The physics of a photonic structure is commonly described in terms of its apparent geometric dimensionality. On the other hand, with the concept of synthetic dimension, it is in fact possible to explore physics in a space with a dimensionality that is higher as compared to the apparent geometrical dimensionality of the structures. In this review, we discuss the basic concepts of synthetic dimension in photonics, and highlight the various approaches toward demonstrating such synthetic dimensions for fundamental physics and potential applications.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (15)

L. Yuan, M. Xiao, Q. Lin, and S. Fan, “Synthetic space with arbitrary dimensions in a few rings undergoing dynamic modulation,” Phys. Rev. B 97, 104105 (2018).
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A. Schwartz and B. Fischer, “Laser mode hyper-combs,” Opt. Express 21, 6196–6204 (2013).
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A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
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Figures (9)

Fig. 1.
Fig. 1. (a) Physical states labeled by consecutive integers. (b) Introducing coupling between nearest-neighbor physical states creates a one-dimensional system. (c) Introducing long-range coupling between different states generates a two-dimensional system [9,13].
Fig. 2.
Fig. 2. (a) A dynamically modulated ring resonator can be described by a tight-binding model of a photon along a one-dimensional lattice in the synthetic frequency dimension [11,51]. (b) A phase-matched modulation along a dielectric waveguide can achieve a one-dimensional lattice formed from waveguide modes at different frequencies [52,53].
Fig. 3.
Fig. 3. A cavity that is degenerate for optical beams with different angular momenta (solid line). Within this cavity, an auxiliary cavity (dashed line) is incorporated, where two spatial light modulators couple a beam at angular momentum l to beams at angular momenta l±1, respectively. Such a cavity can be described by a synthetic lattice along the direction of the angular momenta [10]. Detailed illustration of a sophisticated design about the optical cavity can be found in Refs. [66,68].
Fig. 4.
Fig. 4. (a) Two fiber loops connected by a 50/50 coupler. PM denotes a phase modulator. (b) An equivalent lattice network that describes a one-dimensional synthetic lattice (n) evolves along the time (m) [7,8].
Fig. 5.
Fig. 5. Evolution of the light in time for each resonant mode n, along the frequency axis of light, exhibits the spectral Bloch oscillation [51].
Fig. 6.
Fig. 6. (a) One-dimensional array of ring resonators undergoing dynamic modulations [11,12]. The modulators in the rings have different modulation phases ϕ. (b) Projected band structure for the system in (a) with 21 rings, assuming that the lattice is infinite along the synthetic frequency dimension [11]. kf is the wave vector reciprocal to the synthetic frequency dimension.
Fig. 7.
Fig. 7. (a) Two-dimensional honeycomb array of ring resonators undergoing dynamic modulations; (b) band structures show Weyl points in three-dimensional synthetic space. Left (right), the band structure in kx(ky)kf plane at ky=0 (kx=4π/33d) [54].
Fig. 8.
Fig. 8. (a) One-dimensional photonic crystal in Ref. [117] with each unit cell including four layers where the thickness of each layer depends on parameters p and q; (b) band structure of the crystal with p=q=0; (c) band structure in the (p, q) space with k=π/2(da+db); (d) reflection phase for light incident upon such a photonic crystal from air, in the (p, q) space [117].
Fig. 9.
Fig. 9. (a) Band structure of the Aubry–André model described by Eq. (30) composed of 99 sites with g=1, V=0.5, and b=(5+1)/2; (b) intensity distributions in the experiment show the edge state in the gap. Here ϕ=0.5π. (c) Waveguide array where spacings between waveguides are slowly modified along the z direction, as described by Eq. (31); (d) intensity distributions versus the propagation distance z. Light is injected into the rightmost waveguide into a structure similar to (c). ϕ is changed from 0.35π to 1.75π adiabatically [119]. (b) and (d) are adapted from Ref. [119].

Equations (31)

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ε(r,t)=εs(r)+δε(r,t),
××E+μ02t2εsE=μ02Pt2,
P=δεE
××Em=μ0εsωm2Em,
E=mamEmeiωmt,
ωm=ω0+mΩR.
δε(r,t)=δ(r)cos(Ωt+ϕ),
P=δ(r)Em2am(eiϕeiωm+1t+eiϕeiωm1t).
idamdt=geiϕam1+geiϕam+1.
H=gm(eiϕcm+1cm+eiϕcmcm+1),
Ek=Ek(x,y)eikz.
ωω0=vg(kk0),
δε=δ(x,y)cos(ΩtKz+ϕ),
Ω=Kvg.
E=mam(z)Em(x,y)ei(ωmtkmz),
P=δ(x,y)Em(x,y)2am[eiϕei(ωm+1tkm+1z)+eiϕei(ωm1tkm1z)].
idamdz=geiϕam1+geiϕam+1.
H=gl(eiϕclcl1+eiϕclcl+1),
unm+1=12(un+1m+ivn+1m),
vnm+1=12(iun1m+vn1m)eiϕ(n).
H=gi,j(eiϕij(t)cicj+eiϕij(t)cjci),
ijA·dr=ϕij(t),
B=1SplaquetteA·dr
E=At
H=m,ng(einϕcm,ncm+1,n+einϕcm+1,ncm,n)+κ(cm,ncm,n+1+cm,n+1cm,n),
H(R)|Ψ(R)=E(R)|Ψ(R).
γ=CiΨ(R)|R|Ψ(R)·dR.
A(R)=iΨ(R)|R|Ψ(R).
B(R)=R×A(R).
H=mg(amam+1+amam1)+mamamVcos(2πbm+ϕ),
H=gm[1+Vcos(2πbm+ϕ)]amam+1+[1Vcos(2πb(m1)+ϕ)]amam1,