Nonreciprocal diffraction of light based on double-transition-assisted photonic Aharonov-Bohm effect

We propose a nonreciprocal diffraction system based on the photonic Aharonov-Bohm effect. The implementation utilizes the simultaneous up and down photonic transition of Bloch modes in a dielectric grating created by time-harmonic dielectric constant modulation. This double transition process generates opposite effective magnetic fluxes for photons in symmetric and antisymmetric modes, which gives rise to nonreciprocal spatial interference between them. With the broken time-reversal symmetry, this system is possible to exhibit unidirectional highly efficient diffraction, which enables grating-based nonmagnetic isolation and circulation of free space light, and integrates the functions of gratings and isolators.

Figure 1

Schematic of the diffraction of a TE-polarized plane wave incidence through a 1D periodic structure. The blue transparent plane denotes the incidence plane and the green double-arrows denote the electric field.

Figure 2

Illustration of the Mach-Zehnder interferometer-like diffraction process.

Figure 3

(a) Electric field profile of the first two Bloch modes in 1D periodic structure. (b) Diffraction efficiency under Littrow angle illumination obtained with Eq. (6) (solid line) and RCWA (dashed line). The parameters are ${\lambda }_0=a, f=0.5, \mathrm{\Lambda }=0.2a, n_{\mathrm{I}}=3.5, n_{\mathrm{II}}=n_0=3.3, \theta =49.{25}^{\circ }$, where $a$ is a length unit. The effective indices of the Bloch modes are $n_{\mathrm{eff},1}=2.3993$ and $n_{\mathrm{eff},2}=2.2116$.

Figure 4

Dispersion relations of the symmetric (solid curve) and antisymmetric (dashed curve) Bloch modes. The structure parameters are the same as Fig. 3, and the modulation frequency is $\mathrm{\Omega }=0.0375{\omega }_0$.

Figure 5

Schematic of the refractive index modulation. The modulation profile is the antisymmetrical step function $\left[h\right(x)=\mathrm{\Delta }\epsilon$ for $-3\mathrm{\Lambda }/8<x<-\mathrm{\Lambda }/8, h\left(x\right)=-\mathrm{\Delta }\epsilon$ for $\mathrm{\Lambda }/8<x<3\mathrm{\Lambda }/8$, and $h\left(x\right)=0$ for other regions], and the modulation strength is $\mathrm{\Delta }\epsilon =0.55=0.045{\epsilon }_{\mathrm{I}}$.

Figure 6

Double photonic transition process (the structure and modulation parameters are the same as Fig. 5). (a) Photon number flux at frequencies ${\omega }_1={\omega }_0$ and ${\omega }_3={\omega }_0+\mathrm{\Omega }$. (b) Photon number flux at ${\omega }_1$ and ${\omega }_2={\omega }_0-\mathrm{\Omega }$. The circles are the results obtained by FEM, and the curves are the analytical results obtained by coupled mode theory. The brown circles are the total photon number flux $N_{{\omega }_1}+N_{{\omega }_2}+N_{{\omega }_3}$.

Figure 7

Normalized photon number flux spectrum at the maximum conversion point $\kappa L=\pi /2$ ($L=9.25a$).

Figure 8

Schematic of nonreciprocal diffraction. (a) Structure illustration of the modulated grating. (b1) Incidence from the left side. (b2) Incidence from the right side. The red and blue arrows in (b1) and (b2) represent red and blue shifts of frequency, respectively.

Figure 9

Characterization of nonreciprocal diffraction ($L=9.25a,l=1.407a$ such that ${\mathrm{\Theta }}_1-{\mathrm{\Theta }}_2=7.5\pi$). (a1) and (a2) show the diffraction efficiencies for 0th and $-1\mathrm{st}$ orders, respectively. The circles are calculated by numerical FEM, and the solid lines are obtained through analytical model. (b) Maximum nonreciprocity with $\mathrm{\Delta }\mathrm{\Phi }=\pi /4$. The upper and lower figures represent the left-to-right and right-to-left diffractions, respectively. The results are calculated by FEM, and the length is compressed by a factor of 1/3.

Figure 10

Photon number flux spectrum for the diffraction process shown in Fig. 9 (left to right diffraction). (a) and (b) represent the input and output signals, respectively.

Figure 11

The frequency response of the system. (a) $a_1\to b_3$ photonic transition efficiency versus frequency. (b) Contrast ratio versus frequency. The results are calculated with different grating periods, i.e., $\mathrm{\Lambda }={\lambda }_0$ (red solid curves), $\mathrm{\Lambda }=2{\lambda }_0$ (blue dashed curves), $\mathrm{\Lambda }=3{\lambda }_0$ (green dotted dashed curves).

Figure 12

The extinction ratio spectra with different grating periods (same as Fig. 11). In decibel unit, the extinction ratio is 10lg $({\eta }_0^{L\to R}/{\eta }_0^{R\to L})$. The dashed black line indicates the 20 dB extinction ratio.

Figure 13

Characterization of walk-off effect. (a) Effective index of the symmetric and antisymmetric Bloch modes as a function of Floquet wave vector $k_x$. (b1) and (b2) are electric field profiles $|E_y\left(x\right)|$ for the left to right and right to left diffraction processes, respectively. The red solid curve represents the incidence field, and the black dashed line and the dotted-dashed line are the output fields.

Figure 14

Three possible applications of the system.