Nonreciprocal resonant transmission/reflection based on a one-dimensional photonic crystal adjacent to the magneto-optical metal film

We report the design of nonreciprocal resonant transmission/reflection using a one-dimensional photonic crystal (1DPC) adjacent to the magneto-optical (MO) metal film. The nonreciprocal surface modes are found at the interface between the PC and MO metal within the forbidden band of native PC structure. Breaking time-reversal symmetry using external magnetic field gives rise of such unique nonreciprocal properties. Quantitatively understanding of nonreciprocal resonant optical transmission/reflection behavior is performed using an effective admittance-matching theory. With excitation of the nonreciprocal surface modes, light can transmit and be reflected in one-way. Such design offers promising potential in realizing the optical diode.

The topic of one-way wave propagation has attracted much interest in both optics and acoustics. Several ways have been proposed, such as nonreciprocal linear, nonlinear, phase shift and diffractive process 1-10 . Symmetry-breaking plays a crucial role in realizing these one-way phenomena, no matter which mechanism is used. To break Time-reversal (T) symmetry, it is a general way to introduce MO medium in the presence of an external magnetic field. For example, various types of Giga-Hertz one-way electromagnetic surface modes in MO dielectric PCs have been both theoretically and experimentally studied 3,4,[11][12][13][14][15] . Surface plasmon polaritons (SPPs) of MO metal can also exhibit nonreciprocal behaviors 16 . The one-way waveguide modes at the interface of the two-dimensional dielectric PC and MO metal has been proposed due to the different cutoff frequency between forward and backward SPPs 17 . After applying external magnetic field on the MO metal to break T symmetry, nonreciprocal SPP waves could be excited to realize the one-way propagation.
On the other hand, dielectric-metal and metal metamaterials have shown their abilities in manipulating evanescent waves to achieve super resolution 18,19 . Evanescent waves, characterized by the imaginary propagating constant without carrying energy, play a crucial role in these models, which could be enhanced in negative refractive index medium or coupled to propagating mode using artificial structures. Surface modes, with imaginary propagating constant perpendicular to the boundary, could perform similar coupling effects as that of evanescent waves, such as extraordinary optical resonant transmission [20][21][22][23] . However, it is unclear whether the nonreciprocal surface modes can be enhanced or coupled into free space, and this coupling mechanism still needs to be studied.
In this Letter, focusing on exciting, enhancing and coupling the nonreciprocal surface modes, we presented a type of nonreciprocal resonant transmission/reflection model based on 1DPC-metal structure. An effective admittance-matching theory based on such nonreciprocal model was proposed to determine the operating frequency quantitatively. With external magnetic field, the forward and backward admittances of MO metal become different to excite nonreciprocal SPP modes, which can be enhanced and coupled into free space, realizing nonreciprocal light propagation in the band-gap of native PC. This model is useful to realize the optical diode in experiment for its advantage in the low loss, small area of an external magnetic field, wide operating frequency window, and compatible technique in industry.
Herein, we constructed a 1DPC-metal structure with 8 periods symmetric TiO 2 /SiO 2 /TiO 2 PC attached a thin MO metal film as shown in Fig. 1(a). The lattice constant of PC is 0.
where p  is the bulk plasmon frequency, represents the cyclotron frequency, and e is the electron charge and m is the electron mass. Metal loss is indicated by decay time . To calculate the optical admittance (the ratio of the total magnetic to electric H/E field) of this nonreciprocal metal, we assume the loss to be infinitesimal. For TE mode (the electric field along z direction), the optical admittance of MO metal can be written as where ) /( On the other hand, the optical admittance of such three layers symmetric PC as in Fig. 1(a) has been well-discussed [24][25][26][27] . The total characteristic matrix can be represented mathematically by a single equivalent characteristic matrix 28 . The optical admittance of PC for TE mode is where subscript p and q represents the TiO 2 and SiO 2 layer respectively. , is the optical phase of a single layer, is the wave vector along x direction, d is the thickness, and n is the index of refraction.
The condition of existence of the surface mode at the interface of metal-1DPC is Since the nonreciprocal character of optical admittance of MO metal in Eq. (2), surface modes at the interface of metal-1DPC with a pair of counter incident waves would be nonreciprocal. These surface modes would locate in the band-gap of native PC, where both optical admittance of PC and metal are pure imaginary. So we just need to analyze the imaginary part of optical admittance.
Due to the thin thickness of metal, the wave attenuation distance in the metal is short. This 1DPC-metal model has a good merit of the low loss. Considering a typical loss of metal, The nonreciprocal effects are attributed to the T symmetry broken under the external magnetic field. On the other hand, spatial inversion symmetry in real space is relative to x direction, and in reciprocal space the parity is relative to y k direction. That is to say that a pair of incident waves symmetry relative to y direction is the parity-time symmetric case, while a pair of counter incident waves is one-way transmission and a pair of symmetric incident waves relative to x direction is one-way reflection 29 .
In summary, we have designed a 1DPC-MO metal model to realize the nonreciprocal resonant transmission/reflection, which stems from the excitation of nonreciprocal surface modes at the interface between the PC and metal, in the band-gap of pure PC with the broken time-reversal T symmetry applying an external magnetic field. An effective nonreciprocal admittance-matching theory is proposed. The advantage of this 1DPC-metal model with of the low loss (propagate through the thin metal film), small area with an external magnetic field (just need to cover the thin metal film), wide operating frequency window (dependent on the incident angle), and compatible technique in industry, may be very useful to design some optical nonreciprocal