Surface electromagnetic waves (SEW) propagate along the interface between two media with different properties and can be strongly confined with huge field enhancement at the surface. One of the best-known examples of SEW is surface plasmon polaritons (SPPs), which are guided along the interfaces between dielectrics and metals. The SPP has been attracted great attention in the last decade due to the wide applications in surface enhanced Raman scattering (SERS) [1][2], enhanced nonlinear optical effects [3][4], highly sensitive biological and chemical sensors [5][6]. Alternatively, Bloch surface waves (BSW) can be formed at the surface of a one-dimensional photonic crystal (1DPC). The BSW exploited the band gap of the 1DPC to obtain the guiding of a surface wave thus giving rise to a surface sensitive evanescent field [7][8]. Due to the use of dielectric materials with low loss, the BSW have much longer propagation distance compared to that of SPPs. As a promising alternative to the plasmonic device, BSW has become one of researching hotspots in photonics with a wide range of applications ranging from fluorescence enhancement, label-free sensing, to nano waveguide and more [9]-[13].

Although the 1DPC can sustain the propagation of BSW with TE and TM polarization, many attentions are currently concentrated on the TE BSW. One of the reasons is the difference of band gap between TE and TM polarization in the 1DPC. In the one-dimensional binary PC consisted by alternating the dielectric layers with high and low refraction index, the band gap with TM polarization will be disappeared at the Brewster angle due to the decoupling between the incident and reflected waves. Then, the TM BSW cannot be formed in this region for the one-dimensional binary PC, which prevents the applications of TM BSW as that of TE BSW. However, TM polarized surface waves have richer intrinsic properties, such as spin momentum locking effect [14][15], which can be used to mimic the quantum spin Hall effect and generate new applications.

The one-dimensional ternary periodic structure can be formed by sandwiching the third material between the two materials with different properties. Then, there are three material layers constituting a period of lattice in the one-dimensional ternary PC. Due to the increased interface reflection as the waves propagation through a period of lattice, the one-dimensional ternary PC structures demonstrated some different electromagnetic behaviors compared to that of the one-dimensional binary PC. The ternary periodic structures were used to enlarge the bandgap frequency range of the omnidirectional total reflection [16]-[18] and extend the zero-effective-phase photonic bandgap [19]. However, there aren’t any papers discussing the properties of BSW sustained by a truncated one-dimensional ternary PC structure, although the BSW are widely studied in the binary periodic structure. In this work, the BSW with TM polarization sustained by a truncated one-dimensional ternary PC structure are theoretically studied. Different from the BSW in the binary PC, the dispersion of TM BSW is not broken off at the Brewster angle, and can be engineered in wide frequency range as that of the TE BSW in the ternary PC by adjusting the material compositions and geometrical size of the lattice. Therefore, the TM BSW in the ternary PC can be used to mimic the similar electromagnetic phenomena related to the SPPs due to the same polarization state.

The considered ternary dielectric structure is shown in Figure 1. The truncated one-dimensional periodic ternary structure consists of thin films of three dielectric materials, with refractive indices of n₁, n₂, and n₃ and thicknesses of d₁, d₂, and d₃, respectively. The unit period Λ = d 1 + d 2 + d 3 . At the top of the structure, there is a dielectric layer with a refractive index of n₂ and a thickness of d_a to support the propagation of BSW. General features of the photonic band structure of the semi-infinite periodic structure can be understood through the properties of the infinite periodic structure. In the infinite periodic structure, the dielectric function ε ( r ) varies periodically with z and is invariant in the y and x directions. Then, for the TM polarization (the magnetic field is perpendicular to the x-z plane), the magnetic field is given in the Bloch-wave form.

Figure 1. Schematic diagram of truncated one-dimensional periodic ternary dielectric structure. The unit cell of the structure consists of three different dielectric layers (refraction index n j , film thickness d j ). An additional layer with refraction index n₂ and thickness d_a is introduced on top of the structure to support the propagation of BSW.

where K is the Bloch wave number, β is the wave vector parallel to the interface, H y , K ( z ) is periodic function with a period Λ . The magnetic field in each layer of the nth cell can be written as the combination of two plane waves propagating along the opposite directions [2].

where γ j = k 0 2 n j 2 − β 2 is the wave vector along the z direction.

Applying the continuity boundary condition at z = n Λ + d 1 , z = n Λ + d 1 + d 2 and z = ( n + 1 ) Λ , the Bloch wave satisfies the following eigenvalue equation.

where the matrix element A, B, C and Dare obtained as:

where g i j = 1 / g j i ( i , j = 1 , 2 , 3 ) . For TM polarization g i j = ( γ i n j 2 ) / ( γ j n i 2 ) ，For TE polarization g i j = γ i / γ j . According to the eigenvalue Equation (4), the dispersion relation for the propagation of the electromagnetic wave in the infinite periodic structure can be obtained as

The Bloch wave number K can be real or imaginary values, corresponding to the propagating or evanescent Bloch wave, respectively. Then, the solution of Equation (9) defines the photonic band structure for the infinite periodic structure, ω ( K , β ) . Regimes where | cos ( K Λ ) | > 1 , K has an imaginary part, and the Bloch wave is evanescent. It corresponds to the stop bands for the periodic medium. When | cos ( K Λ ) | < 1 correspond to the real K and the Bloch wave is propagating.

In simulation, the dielectrics are lossless, isotropic, and dispersion-free over the analyzed frequency range. The unit period Λ is fixed at 230 nm for the binary and ternary periodic structure for the convenience of comparing photon energy bands.

Figure 2(a) shows the projected photonic band structure for a binary periodic structure. The cyan regions denote the phase space where the K is imaginary, that is, regions of the evanescent states, whereas carmine areas represent the regions of propagating states. It is noted that the stop band shrink to zero as β = k 0 n 1 sin θ B with θ B = arctan ( n 3 / n 1 ) as the Brewster angle, since at this angle the incident wave is not reflected at the interface of film. Then, the TM BSW cannot be supported at this angle.

In Figure 2(b) we show the projected photonic band structure for a ternary periodic structure. In the unit cell, there are three dielectric layers with two interfaces. As the electromagnetic wave is incident at the Brewster angle of adjacent dielectric layer, the wave will be reflected at the other interface. Then, the incident

Figure 2. (a) Projected photonic band structure of a binary periodic structure with the light line and Brewster line. The film parameters are n₁ = 1.46, n₃ = 3.22, d₁ = 160 nm and d₂ = 70 nm. (b) Projected photonic band structure of a ternary periodic structure together with the light line and Brewster line. The film parameters are n₁ = 1.46, n₂ = 2.3, n₃ = 3.22, and the thickness d₁ = 120 nm, d₂ = 60 nm, d₃ = 50 nm. The cyan regions denote the photonic stop band of the structure in which light cannot propagate through the multilayer. The unit cell is shown in the inset.

and reflected waves will be coupled, and formed the bandgap. The extent of bandgap is well related to the contrast of the refraction indices in the unit cell. For example, as the wave is incident at the Brewster angle, the exact expression for the band edges is obtained as:

where α = n 2 2 / n 2 2 ( n 1 2 + n 3 2 ) − n 1 2 n 3 2 . As the thickness d 2 → 0 (i.e. binary multilayer structure), the upper and lower band edges will be intersected, and the corresponding frequency in the intersection point in the first harmonic is f = c n 1 2 + n 3 2 / [ 2 n 1 2 ( Λ − d 3 ) + 2 n 3 2 d 3 ] . For the finite thickness d 2 , the upper and lower band edge will be separated due to the Bragg reflection. The separated frequency is determined by solving the Equation (10). A dimensionless parameter η is used to quantify the extent of separation between upper and lower band edges, and is defined as η = 2 ( ω H − ω L ) / ( ω H + ω L ) , where ω H and ω L are determined by solving the Equation (10).

Figure 3 shows the ratio η as a function of the n 2 / n 1 and n 3 / n 2 . The contours in this figure represent various bandgap ranges for different refractive index parameters. It is noted that the photonic bandgap width increases with the increase of refractive index contrast. The refractive index contrast and bandgap width of the ternary materials used in this work are represented by dashed lines and symbol circles in the figure, respectively. It is noted that the frequency separation η can reach 15.8%. Then, a bandgap at the Brewster angle can be observed in the ternary periodic structure as shown in Figure 2(b).

Figure 3. The range to midrange ratio η as a function of the n 2 / n 1 and n 3 / n 2 . The results were obtained at the Brewster angle in the first photonic bandgap. Here, the thicknesses of the layers were set as one sixth wavelength.

In Figure 1, the semi-infinite periodic structure is created by replacing the part of the infinite periodic structure lying in the region − ∞ < z < 0 by dielectric layer with refractive indices n₂ and film thickness d_a. The BSW are solutions of Maxwell’s equations in which the waves are localized at surface of the semi-infinite periodic structure, and the field amplitude is evanescent in both periodic structure and environmental dielectric with refractive index ne. According to the continuity boundary conditions, the dispersion equation of TM-BSW in truncated ternary multilayers can be obtained as

where Z B is the equivalent impedance of dielectric layer on the top of structure, Z G is the equivalent impedance of the truncated ternary multilayers. In Equation (13), the coefficientsA and B are given in Equations (5) and (6).

Figure 4(a) shows the dispersion relation of TM-polarized BSW by solving the Equation (11). The dispersion curve of TM BSW is sited in the bandgap region and below the light line. It contributes to the evanescent behavior of field in the multilayer and the environmental medium. Moreover, similar to the TE BSW, the dispersion of TM BSW can be engineered in wide frequency range without the influence of the Brewster effect. The TM BSW can be sustained by the truncated ternary multilayers above 385 THz as shown in Figure 4(a). The electric field amplitude distribution of TM BSW at wavelength of 760 nm is shown in Figure 4(b). As expected, the electric field is strongly peaked at the interface between the multilayer and air. The attenuation of wave in the multilayer depends on the refractive index contrast in the unit cell. Whereas the exponential decay in the

Figure 4. (a) Projected band structure and BSW dispersion relation (solid blue) for a semi-infinite ternary multilayer with d_a = 120 nm for TM polarization. (b) Electric field amplitude distribution in the ternary multilayer for the TM BSW at wavelength of 400 THz.

environmental medium is only determined by the refraction index ne as the parallel wave vector β is fixed. Thus, the TM-BSW exhibits evanescent behaviors in both the environmental medium and the dielectric multilayers, which has the intrinsic spin-momentum locking effect as that of surface plasmon polaritons (SPPs) [20].

In summary, we have theoretically studied the dispersion behaviors of BSW in a truncated one-dimensional periodic ternary multilayer. Different from the binary structure, the Brewster effect is inhibited in the ternary structure due to the additional interface reflection of waves. The photonic bandgap can also be formed around the Brewster angle. The bandgap width can be engineered by changing the refractive index contrast in the ternary multilayer. The BSW with TM polarization can be sustained in wide frequency range as that of TE-BSW in the truncated ternary multilayer. Our work has potential applications in various areas, such as polarization transformation devices, biosensing, and imaging, among others.