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We study the nonreciprocal Tamm plasmon polaritons (TPPs) inside the light cone, which can be directly excited at the interface between the one-dimensional photonic crystal (1DPC) and magneto-optical (MO) metal film. Applying an external magnetic field on the MO metal, the broken time-reversal symmetry gives rise to such nonreciprocal electrons oscillation mediated plasmon mode. Separately exciting the forward and backward TPPs, light can be transmitted and reflected in one-way. An analytic dispersion relation based on admittance-matching approach is obtained. This design offers promising potential in realizing the optical diode.
© 2013 Optical Society of America
1. Introduction
The topic of one-way wave propagation has attracted much interest in both optics and acoustics. Several different approaches have been explored, including nonreciprocal linear, nonlinear, phase shift, and diffraction process [1–10], in which symmetry-breaking plays the crucial role in realizing these one-way phenomena. Particularly, MO medium in the presence of an external magnetic field offers a convenient way to break the time-reversal (T) symmetry. For example, a broad range of Gigahertz one-way electromagnetic surface modes in MO dielectric PCs have been theoretically and experimentally studied [3, 4, 11–15]. Similarly, surface plasmon polaritons (SPPs) associated with collective electrons oscillation at the surface of MO metal can also exhibit nonreciprocal behaviors under external magnetic field [16]. However, it is hard to realize the one-way propagation of SPP without using prisms or diffractive gratings. Except the T symmetry-breaking, the existence of propagating mode is another key factor in realizing one-way propagation. One compromise is to fold the SPP dispersion curve into the light cone using a two-dimensional periodic structure which can demonstrate the one-way waveguide modes [17]. Several limitations of this model might be accounted, such as the propagation loss of SPP, large external magnetic field region and difficulty to fabricate.
On the other hand, dielectric-metal and metal metamaterials have shown their abilities in manipulating SPPs, such as super resolution and extraordinary optical resonant transmission [18–23]. Recently, a kind of one-way TPPs in MO photonic crystal-metal oxides structure was proposed, in which the nonreciprocal TPPs are located between light curves for conducting metal oxides and free space [24]. Therefore, the large incidence angle (over 70°) is needed to realize one-way character and the external magnetic field should cover the whole photonic crystal. However, TPP as a kind of SPPs at the interface between the metal and a material with stop band (1DPC for example), could have a large range of wave vectors inside the light cone, which can be excited by direct illumination, especially for the incidence of small angles [21, 25–27]. Unlike the folding mechanism [17], 1DPC-metal structure can confine the field at the interface due to the band-gap of 1DPC and negative permittivity of metal, which can directly support the propagating plasmon modes inside the light cone. Such electrons oscillation mediated plasmon modes can further be nonreciprocal when T symmetry is broken using MO metal. One-way transmission of a pair of counter incident light could be realized by separately exciting the nonreciprocal TPPs. The forward and backward electromagnetic field amplitudes are remarkably high at the interface of the 1DPC-metal at different frequencies, then attenuated and transmitted through the boundary, finally radiate to the free space in one-way.
In this Letter, focusing on directly exciting the nonreciprocal TPPs, we presented a type of nonreciprocal resonant transmission/reflection model based on 1DPC-MO metal structure. To study the excitation of the nonreciprocal TPPs, an effective admittance-matching approach was proposed, which can quantitatively determine the operating frequency. With external magnetic field, the forward and backward admittances of MO metal are pure imaginary and different. The band-gap of 1DPC could provide desirable pure imaginary admittance to support the nonreciprocal TPPs as propagating modes, which can be directly excited to realize the one-way transmission. This design might offer promising potential in realizing optical components, such as optical diode [28], nonreciprocal filters [29] and absorbers [30].
2. Model
Herein, we constructed a 1DPC-MO metal structure with 8 periods symmetric TiO2/SiO2/TiO2 layers PC attached to a thin MO metal film as shown in Fig. 1. The lattice constant of PC is 0.6λp with thickness of each layer 0.2λp, and the thickness of MO metal film is 0.4λp,where λp is the bulk plasmon wavelength in free space. The refractive indices of SiO2 and TiO2 are 1.46 and 2.25 respectively. By applying external magnetic field B in z direction (covering the thin metal film), the dispersive dielectric function of MO metal yields [31]
In Eq. (1), is unit matrix, ωp is the bulk plasmon frequency, ωb = eB/m represents the cyclotron frequency, and e is the electron charge and m is the electron mass. Metal loss is indicated by decay time τ. To calculate the optical admittance (the ratio of the magnetic to electric Hz/Ey field) of this nonreciprocal metal, we assume the loss to be infinitesimal. For TM mode (the magnetic field along z direction), the optical admittance of MO metal can be written asIn Eq. (2), , , . c is the speed of light in vacuum, η0 is the admittance of vacuum, and ky represents the wave vector along the y direction. The optical admittance of metal is reciprocal η+ = η- (εf = 0) without the external magnetic field. By applying the external magnetic field, T symmetry is broken. Then for a pair of counter incident waves, where + ky and -ky have opposite signs in Eq. (2), the admittance of these two cases are different, which would excite the nonreciprocal TPPs and realize the nonreciprocal transmission/reflection. It should be noticed that for normal incident wave (ky = 0), the nonreciprocal phenomena would not occur even with applying external magnetic field. The difference of admittance between two counter incident waves could be enlarged by increasing the angle of incidence or the intensity of the applied external magnetic field.
Fig. 1 Schematic of 1DPC-MO metal model including 8 periodical TiO2/SiO2/TiO2 layers PC and MO metal film.
On the other hand, the optical admittance of three-layer symmetric PC in Fig. 1 has been well discussed [25, 26, 32, 33]. The total characteristic matrix can be represented mathematically by a single equivalent characteristic matrix [34]. The optical admittance of PC for TM mode is
where subscript p and q represent the TiO2 and SiO2 layer, respectively. is the optical admittance of a single layer, is the optical phase of a single layer, and. is the wave vector along x direction, d is the thickness, and n is the index of refraction.The condition of existence of the TPPs at the interface of 1DPC-metal yields [25]
Due to the nonreciprocal character of optical admittance of MO metal in Eq. (2), TPPs at the interface of 1DPC-MO metal with a pair of counter incident waves would be nonreciprocal. These TPPs 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. In Fig. 2, we show the frequency dependent imaginary part of optical admittance of the MO metal and PC with a pair of ± 30° incident waves. The nonzero region of the imaginary optical admittance of PC indicates the first band-gap from 0.3854ωp to 0.4755ωp. The imaginary optical admittances of MO metal correspond to forward ( + y direction) and backward (-y direction), where the cyclotron frequency (induced by external magnetic field) is assumed to be ωb = 0.1ωp. The different frequencies of intersection 0.4227ωp and 0.4269ωp correspond to the forward and backward TPPs, respectively.Fig. 2 The imaginary part of optical admittances of the PC (black line) and MO metal (red and blue lines) with a pair of ± 30° incident waves (external magnetic field ωb = 0.1ωp). The dot-dashed line is the optical admittance of metal without external magnetic field.
3. Results and discussion
With applying external magnetic field, optical admittances for forward and backward waves would split. The real part of permittivity of metal is negative in optical window, and the effective index of refraction is imaginary. At the interface of MO metal-air, the nonreciprocal SPP along the interface could be excited as reported before [17]. To make such SPP radiate into free space, another material with stop band is needed. The region of the band-gap of 1DPC could be chosen as an excellent candidate, owing to its suitable operating frequency window and compatible technique in industry [35]. The field amplitude is remarkably high at the interface of 1DPC-metal due to the excitation of TPPs, which would realize nonreciprocal resonant transmission. Furthermore, some other models might also achieve the similar process, such as metamaterial-metal and structured MO metal models.
We calculate the TPPs at the interface of 1DPC-MO metal in the first band-gap according to Eq. (4) as shown in Fig. 3, where ωb = 0.1ωp. According to the boundary condition of Maxwell equations, the incidence angle exciting the TPPs can be defined as , where k and ω correspond to the wave vectors and frequency of TPPs as in Fig. 3. Fixing an incidence angle, the forward and backward TPPs would be excited at different frequencies to realize the one-way property. With normal incidence, the forward and backward TPPs locate at the same frequency. Increasing the incidence angle could make the nonreciprocal frequency blue shift. Due to the large range of one-way wave vectors, our design can realize one-way transmission/reflection in large range of incidence angles (from 0° to 85° in theory). The TPPs inside the light cone can be directly excited by incident light from air. To excite the TPPs outside the light cone, some high index of refraction medium or structured surface would be needed to obtain the large ky.
Fig. 3 The nonreciprocal TPPs at the interface of 1DPC-MO metal of the first band-gap (ωb = 0.1ωp). The shadow regions are the projected band of 1DPC along ky direction.
To verify the admittance-matching method, we use various methods to get the same results shown in Fig. 4. The solid lines are calculated analytically with Eq. (4). The open circles are calculated by solving the eigen values of supercell structure. Such supercell structure is constructed by air-1DPC-MO metal-air model, where the thickness of each air layer is 3λp. The pluses are obtained by extracting the peak values from transmission spectra. The little differences come from the finite layers of 1DPC and thickness of metal film [21]. All the numerical results are calculated by using a finite element method software package (COMSOL MULTIPHYSICS 3.3).
Fig. 4 Zoom-in TPPs are calculated by various methods: the solid lines by analytic Eq. (4), the open circles by supercell method and the pluses by transmission spectra.
Then, we plot the nonreciprocal transmission spectra in Fig. 5(a) with ωb = 0.1ωp. Three cases of incidence at ± 15° (dashed lines), ± 30° (solid lines) and ± 45° (open circles) are calculated. The maximum transmission almost reaches 100%. The relative nonreciprocal frequency (Δω/ω) is approximately 15%. The one-way bandwidth has a range from 0.05ωb to 0.1ωb dependent on the incidence angles. The maximum value of contrast ratio ((T+-T-)/(T+ + T-)) is around 80%, where T ± is the transmission). Both the peak values and shapes of transmission can be manipulated by changing the thickness of the interface layer TiO2 between 1DPC and MO metal [21, 25]. Moreover, the contrast ratio could be enlarged by increasing the external magnetic field. As shown in Figs. 5(b), for ± 30° incidence, if the applied external magnetic field is ωb = 0.3ωp, the relative nonreciprocal frequency would be over 30% [solid line in Fig. 5(b)], and the contrast ratio is over 95%. Since we use the metal thin film, the wave attenuation distance in the metal is short. This 1DPC-MO metal model has a good merit of the low loss. Considering a typical loss of metal, 1/τ = 0.01ωp, the transmission can still reach 50% and the contrast ratio can reach 90% [open circles in Fig. 5(b)].
Fig. 5 (a) The nonreciprocal transmission spectra of forward (blue lines) and backward (red lines) waves with incidence angles ± 15° (dashed lines), ± 30° (solid lines), and ± 45° (open circles), the external magnetic field ωb = 0.1ωp. (b) The transmission spectra of ± 30° incident waves, where dashed (ωb = 0.1ωp) and solid (ωb = 0.3ωp) lines represent lossless condition, open circles represent ωb = 0.1ωp and 1/τ = 0.01ωp.
The out-of-plane magnetic field distributions with incidence waves are shown in Fig. 6 [open circles in Fig. 5(b)]. Only the forward TPP could be excited at frequency 0.4175ωp. Therefore, only the incident waves from lower part (left-down and right-down) with the wave vector along + y direction ( + ky component), could be transmitted as shown in Fig. 6(a). The electromagnetic field amplitude is enhanced toward to the interface of metal-PC, and then transmitted through the boundary and get attenuated, at last transmitted into the air. A pair of counter incident electromagnetic waves would have different propagating phenomena: one is almost completely transmitted and the other is totally reflected [28]. The one-way property would be reversed at frequency 0.4293ωp as shown in Fig. 6(b). Only the incident waves from upper part (left-up and right-up) with the -ky component could excite the backward TPP and be transmitted. Furthermore, these two cases of one-way propagation would be reversed by flipping the applied external magnetic field.
Fig. 6 Out-of-plane field magnetic distributions with ± 30° incident waves [open circles in Fig. 5(b)]. (a) Exciting the forward TPP at frequency 0.4175ωp. (b) Exciting the backward TPP at frequency 0.4293ωp. Red and blue color represents the positive and negative phase, respectively.
The one-way bandwidth of our model (from 0.05ωb to 0.1ωb dependent on the incidence angle) is proportional to the strength of the external magnetic field. Such bandwidth is about 2 GHz by applying a static field of 1 Tesla, which is sizable in the optical window [17]. On the other hand, such nonreciprocal property can be realized using metamaterials with an effective plasmon frequency in the GHz range [36]. Thus, more modest static magnetic field can be applied.
The nonreciprocal effects are attributed to the broken T symmetry 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 ky direction. That is to say a pair of incident wave 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 [28].
4. Conclusion
In summary, we have designed a 1DPC-MO metal model to realize the nonreciprocal resonant transmission/reflection, which stems from the direct excitation of nonreciprocal TPPs at the interface between the PC and metal, in the band-gap of native PC with the broken T symmetry by applying an external magnetic field. An effective nonreciprocal admittance-matching approach is proposed. The advantages of this 1DPC-MO metal model with the low loss (just need to 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 incidence angle), and compatible technique in industry, may be very useful to design optical nonreciprocal devices such as optical diodes.
Acknowledgments
The work was jointly supported by the National Basic Research Program of China (Grant No. 2012CB921503 and No. 2013CB632702) and the National Nature Science Foundation of China (Grant No. 1134006). We also acknowledge the support from Academic Program Development of Jiangsu Higher Education (PAPD) and China Postdoctoral Science Foundation (Grant No. 2012M511249 and No. 2013T60521).
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