Introduction

In recent years, topological photonics have attracted extensive attention owing to its unique characteristics to control the propagation and manipulation of light waves robustly and immune to disorder and defects1,2,3,4,5,6. Up to now, a diverse range of topological phenomena have been proposed in topological photonics, including topologically protected interface states7,8,9, corner states10,11,12,13, and one-way propagated edge states14,15,16. These discoveries have paved the way for numerous device applications, such as pseudo-spin-based light splitters17,18,19, high-quality factor topological interface state cavity lasers9, sensitivity topological optical sensors20, and high-speed topological electro-optic modulators21. In topological photonics, topological phases play a crucial role in predicting topological states and designing various topological structures. However, the generation of non-trivial topological phases has mainly relied on precise structural designs in the most existing literature. This limitation restricts their adaptability due to the fixed structures. To overcome it, researchers have proposed some tunable structures22,23, such as magneto-optical photonic crystals (PCs)24, programmable structures25, and single-chip control structures26. Otherwise, researchers have demonstrated that incident angles can provide a new degree of freedom for regulating topological phases27,28,29,30, offering greater flexibility than adjusting structural parameters.

Light, a transverse electromagnetic wave, exhibits distinct transverse-electric (TE) and transverse-magnetic (TM) polarized modes. In classical31 and quantum32,33,34 optical experiments, splitting TE and TM polarized modes is of great significance for efficiently separating, analyzing, controlling light signals, and optimizing the utilization of light sources. The significance of TE-TM splitting has led to the development of various conventional structures to achieve this effect, such as birefringent crystal materials35,36,37, grating waveguide couplers38,39,40, and multimode interference couplers41,42,43. However, the large size of these conventional structures, often ranging from tens to hundreds of microns, impedes their integration into chip-scale systems. To overcome this, inverse design methods have been employed to create ultra-small TE-TM mode splitters44,45,46. Despite their compact size, these designs tend to be complex and sensitive to disturbances, which can reduce their efficiency and performance. In response, some researchers have explored the use of topological valley photonic crystal slabs to design TE-TM mode splitters that are more robust to certain perturbations47. Nevertheless, research on TE-TM splitting using topological interface states, especially within the visible light spectrum, remains scarce.

In this study, we explore the incident angle-induced topological phase transitions in one-dimensional (1D) PCs system. Our study systematically investigates the band structure evolution for both TE and TM polarized modes. Results reveal that they can simultaneously undergo topological phase transitions at identical critical angles and frequencies. Additionally, the TM polarized mode solely undergoes a unique topological phase transition process with the Brewster angle as the critical transition angle. Fascinatingly, when these two categories of transition angles are precisely adjusted to be equal, the corresponding bandgap of the TM polarized mode remains unchanged in its topological properties, even when it undergoes the “open-close-reopen” process. Through precise incident angle adjustments, we theoretically and experimentally demonstrate the achievement of the TE-TM splitting effect in transmission spectra in the visible light range. Our research introduces an innovative mechanism for achieving topological interface states with the TM-TE splitting effect, enabling the selection and control of the polarization of light. This research enhances the understanding of topological phase transition in photonics and provides new possibilities for developing optical devices, such as polarization beam splitters, filters, and optical quantum gates.

Results and discussion

Band structure and topological characterization

We begin our study with a 1D PC composed of two distinct dielectric materials, labeled as A and B, as depicted in Fig. 1a, b, respectively. By employing the transfer matrix method (TMM), the band structures in the case of oblique incidence for TE and TM polarized modes obey the following relationships48,49:

Fig. 1: The evolution of bandgaps for transverse-electric (TE) and transverse-magnetic (TM) polarized modes.
figure 1

Schematic illustration of photonic crystal (PC) composed of dielectric layers A and B for TE (a) and TM (b) polarized modes. The red dashed line marks the unit cell we consider. The evolution of bandgaps as functions of the refracted angle θa for TE (c) and TM (d) modes. The thickness of layers A and B are \({d}_{a}=286.36{{{{{\rm{nm}}}}}}\) and db = 63.64 nm, and the refractive indices of material A and B are na = 1 and nb = 3, respectively. Here, c denotes the wave speed in the vacuum, \(\Lambda ={d}_{a}+{d}_{b}\) denotes the lattice constant of the PC. The yellow areas denote the bandgaps while the blue number marks the number of bandgaps. The green up triangles and purple down triangles represent the critical topological phase transition points of bandgaps. And black labels (\(\xi > 0,\xi < 0\)) denote the \({{{{{\mathrm{sgn}}}}}}[{\xi }^{(m)}]\) for bandgaps. The dashed blue lines in (d) correspond to θa = 60° and 75°, whose band structures and eigenfield distributions at band edges are revealed in Fig. 2.

Full size image

For TE polarization:

$$\cos (k\Lambda )=\,\cos ({\delta }_{a}+{\delta }_{b})-\frac{1}{2}\left(\frac{{n}_{a}\,\cos {\theta }_{a}}{{n}_{b}\,\cos {\theta }_{b}}+\frac{{n}_{b}\,\cos {\theta }_{b}}{{n}_{a}\,\cos {\theta }_{a}}-2\right)\sin {\delta }_{a}\,\sin {\delta }_{b}$$
(1)

For TM polarization:

$$\cos (k\Lambda )=\,\cos ({\delta }_{a}+{\delta }_{b})-\frac{1}{2}\left(\frac{{n}_{a}\,\cos {\theta }_{b}}{{n}_{b}\,\cos {\theta }_{a}}+\frac{{n}_{b}\,\cos {\theta }_{a}}{{n}_{a}\,\cos {\theta }_{b}}-2\right)\sin {\delta }_{a}\,\sin {\delta }_{b}$$
(2)

Here, \({\delta }_{i}=\omega {n}_{i}{d}_{i}\,\cos {\theta }_{i}/c\), where \({n}_{i}\), \({d}_{i}\), \({\theta }_{i}\) denote the refractive index, thickness, and refracted angle for materials A (\(i=a\)) and B (\(i=b\)), respectively. \(k\) represents the Bloch wave number, \(\Lambda ={d}_{a}+{d}_{b}\) denotes the lattice constant of PC. The refracted angles \({\theta }_{a}\) and \({\theta }_{b}\) obey Snell’s law, \({n}_{a}\,\sin {\theta }_{a}={n}_{b}\,\sin {\theta }_{b}\). For simplicity, hereafter we assume that the light is emitted from material A and then \({\theta }_{a}\) is the incident angle.

To validate the appearance of the topological interface states in a 1D PC system, the well-defined Zak phase is utilized to determine the topological properties of bulk bands50,51,52. The Zak phase for the nth isolated band can be expressed as53:

$${\theta }_{n}^{{{{{\rm{Zak}}}}}}={\int }_{-\pi /\Lambda }^{\pi /\Lambda }\left[i{\int }_{unitcell}dx\cdot \varepsilon (x)\cdot {u}_{n,k}^{\ast }(x){\partial }_{k}{u}_{n,k}(x)\right]dk$$
(3)

Here, \(\varepsilon (x)\) represents the dielectric function, and \({u}_{n,k}\) is the periodic-in-cell component of the normalized eigenfunction associated with the nth band at wave number k. The topological property of the mth bandgap depends on the summation of Zak phases for all bulk bands lying below the bandgap, which can be defined as:

$${{{{{\mathrm{sgn}}}}}}[{\xi }^{(m)}]={(-1)}^{m}\exp \left({i}{\sum }_{n=0}^{m-1}{\theta }_{n}^{{{{{{\rm{Zak}}}}}}}\right)$$
(4)

According to the bulk-interface correspondence principle, topological interface states will emerge within the two PC’s common bandgap but with different \({{{{{\mathrm{sgn}}}}}}[{\xi }^{(m)}]\). Thus, by calculating \({{{{{\mathrm{sgn}}}}}}[{\xi }^{(m)}]\), we can explore the dependency of topological properties on the incident angle, and further control the generation and elimination of topological interface states by tuning the incident angle.

Topological properties of band structure for TE and TM polarized modes

The topological phase transition, i.e., open-close-reopen process in the band structure, is one of the hallmarks of the appearance of the two distinct topological bandgaps. Then, to search for the topological phase transition point, i.e., the critical state that two bulk bands cross at the center or boundary of the Brillouin zone, is a convenient and effective method to find topologically non-trivial states (in 2D or 3D systems, this critical state is the widely studied Dirac points or Weyl points). To investigate the topological phase transition point, we begin by analytically examining the dispersion relation for TE and TM polarized modes with respect to the incident angle.

When \({\delta }_{a}+{\delta }_{b}=m\pi\), \(m\in {N}^{+}\), the Eqs. (1) and (2) can be reduced as:

For TE polarization:

$$\cos (k\Lambda )={(-1)}^{m}+{(-1)}^{m}\frac{{\sin }^{2}{\delta }_{a}}{2}\left(\frac{{n}_{a}\,\cos {\theta }_{a}}{{n}_{b}\,\cos {\theta }_{b}}+\frac{{n}_{b}\,\cos {\theta }_{b}}{{n}_{a}\,\cos {\theta }_{a}}-2\right)$$
(5)

For TM polarization:

$$\cos (k\Lambda )={(-1)}^{m}+{(-1)}^{m}\frac{{\sin }^{2}{\delta }_{a}}{2}\left(\frac{{n}_{a}\,\cos {\theta }_{b}}{{n}_{b}\,\cos {\theta }_{a}}+\frac{{n}_{b}\,\cos {\theta }_{a}}{{n}_{a}\,\cos {\theta }_{b}}-2\right)$$
(6)

It is apparently that \(|\cos (k\Lambda )|\ge 1\). If the second terms in Eqs. (5) and (6) are not equal to zero, i.e., \(|\cos (k\Lambda )| > 1\), k doesn’t have a real root and this state is in the bandgap. However, when the second terms in Eqs. (5) and (6) equal to zero, i.e., \(|\cos (k\Lambda )|=1\), that denotes the phase transition points where two bulk bands cross at the Brillouin zone boundaries or center49,53, which is what we focus on. Now if \(\sin {\delta }_{a}=0\), and combining \({\delta }_{a}+{\delta }_{b}=m\pi\), then \({\delta }_{a}={m}_{1}\pi\), \({\delta }_{b}={m}_{2}\pi\) where \(m={m}_{1}+{m}_{2}\) and \({m}_{1},{m}_{2}\in {N}^{+}\). In this event, in the mth bandgap, both TE and TM polarized modes have identical critical transition frequencies and critical transition angles:

$$\left\{\begin{array}{c}{\omega }_{1}^{({m}_{1},{m}_{2})}=\frac{\pi c}{{d}_{a}{d}_{b}}\sqrt{\frac{{{m}_{2}}^{2}{d}_{a}^{2}-{{m}_{1}}^{2}{d}_{b}^{2}}{{n}_{b}^{2}-{n}_{a}^{2}}}\hfill \\ {\theta }_{a1}^{({m}_{1},{m}_{2})}={\cos }^{-1}\left(\frac{{m}_{1}{d}_{b}}{{n}_{a}}\sqrt{\frac{{n}_{b}^{2}-{n}_{a}^{2}}{{m}_{2}^{2}{d}_{a}^{2}-{m}_{1}^{2}{d}_{b}^{2}}}\right)\end{array}\right.$$
(7)

Obviously, this kind of critical transition state has a complex dependence on many parameters (\({n}_{a},{n}_{b},{d}_{a},{d}_{b},{m}_{1},{m}_{2}\)) and may not always exist in every bandgap.

Besides, only for TM-polarized mode, when \({n}_{a}\,\cos {\theta }_{b}={n}_{b}\,\cos {\theta }_{a}\), there is another kind of critical transition state. Combining Snell’s law \({n}_{a}\,\sin {\theta }_{a}={n}_{b}\,\sin {\theta }_{b}\), we can solve the critical transition frequencies and critical transition angle in the mth bandgap:

$$\left \{\begin{array}{c}{\omega }_{2}^{m}=\frac{m \pi c\sqrt{{n}_{b}^{2}+{n}_{a}^{2}}}{{d}_{a}{n}_{a}^{2}+{d}_{b}{n}_{b}^{2}}\\ {\theta }_{a2}={\tan }^{-1}(\frac{{n}_{b}}{{n}_{a}})\end{array} \right.$$
(8)

It’s interesting that these critical states will exist in all bandgaps and the critical transition angle is the Brewster angle, which only relies on the refractive indices na and nb. It should be noted that this kind of critical state never appears for TE polarized modes, that because the equation \({n}_{a}\,\cos {\theta }_{a}={n}_{b}\,\cos {\theta }_{b}\) have no solutions under the constraints of Snell’s law \({n}_{a}\,\sin {\theta }_{a}={n}_{b}\,\sin {\theta }_{b}\).

To verify the analytic derivation, we design a special 1D PC and numerically calculate its band structure to study the topological transition process caused by the incident angle. Figure 1c, d illustrate the evolution of bandgaps as the functions of the refracted angle θa for TE and TM modes, respectively. It is evident that for both TE and TM polarized modes, the bandgaps close at the same critical angles: \({\theta }_{a1}^{(1,1)}=49.86^{\circ}\) for bandgap 2, \({\theta }_{a1}^{(1,2)}=71.57^{\circ}\) for bandgap 3, \({\theta }_{a1}^{(2,2)}=49.86^{\circ}\) and \({\theta }_{a1}^{(1,3)}=77.87^{\circ}\) for bandgap 4, marked as the green up triangles in Fig. 1c, d. The bandgaps on the two sides of each critical state are topologically different due to the flip of \({{{{{\mathrm{sgn}}}}}}[{\xi }^{(m)}]\), indicating the generation of a topological transition process.

Next, solely for TM-polarized mode, there is a unique topological transition process with the critical state at the Brewster angle \({\theta }_{a2}=71.57^{\circ}\), marked by purple down triangles in Fig. 1d. Since the Brewster angle only depends on the refractive indices of materials A and B, this kind of topological transition process happens in all bandgaps. In other words, when the incident angle is equal to the Brewster angle, all bandgaps will be closed simultaneously. By skillfully adjusting da and db, the first kind of critical angle can be tuned to align with the Brewster angle (\({\theta }_{a1}={\theta }_{a2}\)), as illustrated in the third bandgap in Fig. 1d. The overlap of these two distinct topological transition processes leads to a seemingly counterintuitive phenomenon: the bandgaps on either side of critical state share identical topological characteristics. To validate it, we calculate the Zak phases of the lowest five isolated bands, represented by the purple numbers in Fig. 2 and further calculate \({{{{{\mathrm{sgn}}}}}}[{\xi }^{(m)}]\) for the lowest four bandgaps. Furthermore, the parity analysis method, which involves observing the parities of the eigenmodes at the band edges (center or boundary of the Brillouin zone) both above and below the \(m{{{{{\rm{th}}}}}}\) bandgap, serves as an alternative tool to determine the \({{{{{\mathrm{sgn}}}}}}[{\xi }^{(m)}]\)53,54,55,56.

Fig. 2: The band evolution and field distribution of transverse-magnetic (TM) polarized mode.
figure 2

a–c The simulated band structures and field distributions of band edge states for the TM mode at θa = 60°, 71.57°, and 75°, respectively. Here, the yellow and blue areas in (a) and (c) denote the bandgaps while the blue numbers (1–4) mark the number of bandgaps. Dark green numbers (1–4) denote the numbers of the bulk bands. And purple labels (0, π) denote the Zak phase of each individual band. The parities of the field distribution of eigenmodes at band edges are marked with the blue (odd parity) and pink (even parity) dots, while the cross points marked with dark green dots.

Full size image

Firstly, at the Brewster angle of 71.57°, all bandgaps close at either the center or boundary of the Brillouin zone, as shown in Fig. 2b. When \({\theta }_{a}=60^{\circ} \, < \, 71.57^{\circ}\), the crossed bulk bands are opened to be bandgaps, as shown in Fig. 2a. Numerical evaluated Zak phases for the lowest four bulk bands are “\(\pi -0-\pi -0\)”. Consequently, the \({{{{{\mathrm{sgn}}}}}}[{\xi }^{(m)}]\) for bandgaps 1, 2, 3, and 4 are positive, negative, negative, and positive (\({\xi }^{(1)} > 0\), \({\xi }^{(2)} < 0\), \({\xi }^{(3)} < 0\), and \({\xi }^{(4)} > 0\)), respectively. In contrast, when \({\theta }_{a}=75^{\circ} \, > \, 71.57^{\circ}\), the Zak phases of the lowest four bands transition to “\(0-0-0-\pi\)”, as depicted in Fig. 2c. Therefore, the corresponding \({{{{{\mathrm{sgn}}}}}}[{\xi }^{(m)}]\) for bandgaps 1, 2, 3, and 4 switch to negative, positive, negative, and negative (\({\xi }^{(1)} < 0\), \({\xi }^{(2)} > 0\), \({\xi }^{(3)} < 0\), and \({\xi }^{(4)} < 0\)). And the \({{{{{\mathrm{sgn}}}}}}[{\xi }^{(m)}]\) reverses in bandgaps 1, 2, and 4, while it remains unchanged in bandgap 3 (marked with blue areas in Fig. 2). This implies that the topological property of bandgap 3 remains unchanged as the incident angle varies from \({\theta }_{a} < 71.57^{\circ}\) to \({\theta }_{a} > 71.57^{\circ}\). This conclusion can be confirmed by examining the parities of the eigenfield distributions at the band edges of all bandgaps52. Bandgap 3 consistently exhibits an odd parity field distribution at its lower band edge and an even parity field distribution at its upper edge, regardless of θa being less than or greater than 71.57°. Conversely, the parities of eigenfield at band edges of bandgap 1, 2, and 4 are reversed for \({\theta }_{a} < 71.57^{\circ}\) and \({\theta }_{a} > 71.57^{\circ}\) cases. Apparently, the topological phase transition process occurs on the bandgap 1, 2, and 4, contrasting with the unchanged status of bandgap 3, aligning with the calculations of Zak phases and \({{{{{\mathrm{sgn}}}}}}[{\xi }^{(m)}]\). This seemingly abnormal phenomenon can be understood as that although bandgap 3 undergoes only one open-close-reopen process, \({{{{{\mathrm{sgn}}}}}}[{\xi }^{(m)}]\) has been flipped twice, and then the topological property remains unchanged. Certainly, it cannot happen for TE polarized mode.

Observation of incident angle-induced topological interface states and TE-TM splitting effect

Regarding the above theoretical demonstration, it is convenient to design two distinct PCs with common bandgaps but different topological properties. According to the bulk-interface correspondence51, when these two PCs are integrated to form a composite structure, topological interface states will exist at the interface between the two PCs. To observe the topological interface states in the visible light range and investigate their dependence on incident angle, we design PC1 and PC2 with specific parameters: material A is SiO2 with refractive index na = 1.5 and material B is TiO2 with refractive index nb = 2.3, layer thicknesses are da1 = 260 nm, db1 = 115 nm for PC1, and da2 = 120 nm, db2 = 190 nm for PC2. Four periods PC1 and PC2 are interfaced to be the composite PCs, as shown in Fig. 3a. Figure 3b, c illustrate the bandgap evolution of TE and TM polarized modes for PC1 and PC2, respectively. In Fig. 3c, d, we focus on the second bandgap of PC1 and PC2, as shown in purple areas. Evidently, PC1’s TE-polarized second bandgap undergoes a topological transition process at the critical angle 55.02°, while PC2’s bandgap remains consistently open. Differently, for TM polarized mode, the second bandgap of PC1 undergoes two topological transition processes with critical angles 55.02° and 56.89° (Brewster angle), whereas that in PC2 only experiences the latter one. When \({\theta }_{a} < 55.02^{\circ}\), \({{{{{\mathrm{sgn}}}}}}[{\xi }^{(2)}]\) of PC1 and PC2 are different for both TE and TM polarized modes. Consequently, this discrepancy ensures the existence of topological interface states within a wide incident angle range of [0°, 55.02°].

Fig. 3: Simulation of the incident angle-induced topological interface states and transverse-electric-transverse-magnetic (TE-TM) splitting effect.
figure 3

a Schematic of the 1D composite photonic crystals (PCs) formed by interfacing PC1 and PC2 composed of dielectric layers A and B. Here, light obliquely incident into the composite PCs from the air with an incident angle θ0. b, c reveal the evolutions of bandgaps of PC1 and PC2 for TE and TM polarized modes, respectively. Here, the yellow and purple areas in (b) and (c) denote the bandgaps. d is the transmission spectra of the composite PCs as functions of the refracted angle θa for TE and TM polarization modes, respectively. e is the numerically calculated spatial distribution of the electric field of the topological interface states. f exhibits the splitting of the topological interface states for TE and TM polarized modes as the increase of θa.

Full size image

Then, we calculate the transmission spectra of the composite PCs for both TE and TM polarized modes with the increase of incident angle, as is shown in Fig. 3d. Considering in experiments, light is emitted from the air (n0 = 1) to the composite PCs, the refracted angle θa is within the range [0°,41.8°] corresponding to incident angle in air \({\theta }_{0}\in [0^{\circ} ,90^{\circ}\,\, ]\). Apparently, for both TE and TM polarized modes, the transmission peaks appear in the common bandgap (see Supplementary Note 1 for more details) in the whole incident angle range, implying the existence of topological interface states. Furthermore, Fig. 3e displays the absolute electric field distributions at the transmission peak with θa = 0° and θa = 41°. They consistently exhibit the typical field distribution pattern of interface states: the field has maximum intensity on the interface and rapidly decays away from the interface. Moreover, the localization of the topological interface states can be improved by the increase of the number of periods (see Supplementary Note 2 for more details). Additionally, for the normal incident light, TE and TM waves have identical dispersion relations (Eqs. (1) and (2)), and then their topological interface states have the same frequencies. Whereas, as the incident angle increases, the topological interface states for TE and TM polarized modes will gradually split, as indicated by the red (TM) and blue (TE) lines in Fig. 3f. By precisely adjusting the geometric parameters (\({d}_{a1},{d}_{b1},{\Lambda }_{1},{d}_{a2},{d}_{b2},{\Lambda }_{2}\)) and the refractive indices (na and nb) of PC1 and PC2, we can effectively manipulate the effect of transmission peaks splitting for TE and TM modes’ topological interface states, as shown in Supplementary Note 3. These results provide a good means to fine-tune and optimize the TE-TM splitting of the topological interface states.

Next, we experimentally demonstrate the TE-TM splitting effect of topological interface states. Figure 4a presents the experimentally measured transmission spectra of TE and TM polarized modes as functions of the refracted angle θa in the range of [0°, 41°] (corresponding to the incident angle \({\theta }_{0}\in [0^{\circ} ,80^{\circ} \; ]\) in air). Apparently, the topological interface states definitely exist for both TE and TM polarized modes, exhibiting transmission peaks in the common bandgap, which agrees well with the numerical simulation results shown in Fig. 3d. To visually illustrate the TE-TM splitting effect, we present Fig. 4b–e, which reveal the experimental measurement of transmissivity at the refracted angles θa of 0°, 10°, 28°, and 36° (corresponding to incident angle in air θ0 of 0°, 15°, 45°, and 62°), respectively. For the normal incidence, both TE (blue lines) and TM (red lines) waves exhibit transmission peaks at the same frequency about 477.5 THz. As the increase of incident angle, the topological interface states, exhibiting as transmission peaks, of TE wave and TM wave, will gradually split. For example, at the large incident angle (\({\theta }_{a}=36^{\circ}\) corresponding to \({\theta }_{0}=62^{\circ} \,\)), the transmission peak of the TE wave is at the frequency 546.6 THz, while that of the TM wave is at the frequency 539.5 THz. The negligible deviations between the simulated and experimental values may stem from the errors in monitoring the layer thicknesses during the deposition process.

Fig. 4: Experimental observation of topological interface state with the transverse-electric-transverse-magnetic (TE-TM) splitting effect.
figure 4

a is the experimentally measured transmission spectra of the composite photonic crystals (PCs) as functions of the refracted angle θa for TE and TM polarization modes, respectively. b–e are the transmission spectra of TE (blue lines) and TM (red lines) waves at θa = 0°, 10°, 28°, and 36°, respectively.

Full size image

Conclusions

This study investigates the effects of oblique incident angles on the topological phase transitions and the TE-TM splitting in a 1D composite PC system. We show that there are two types of critical angles for the topological phase transition processes. One type shared by TE and TM polarized modes depends on the structural parameters of the 1D PC, while the other type (the critical angle is the Brewster angle) is unique to the TM polarized modes and only depends on the refractive indices of the PC’s components. When these two critical angles are tuned to be identical, the topological properties of the TM polarized modes remain unchanged, even when the open-close-reopen process happens. Based on theoretical study, we design the composite PCs composed of interfaced two PCs having a common bandgap but with distinct topological characteristics. By manipulating the incident angle, we achieve topological interface states and split the TE and TM polarized modes in a large incident angle region within the visible light spectrum. Considering that the sample is in the air during the experiment, we discuss and rule out the potential influence of Tamm states57,58,59 at the interface of air and PCs (see Supplementary Note 4 for more details). Our experimental results agree well with the theoretical predictions.

This study primarily manipulates the topological properties of the band structure and the polarization of topological interface states in a 1D PC system by adjusting the incident angle. This unique dependence on the incident angle not only simplifies the structural design but also enhances the stability and tunability of the device, making it potentially applicable in highly integrated optical systems. Moreover, our method can be employed to obtain pure polarized topological interface state cavity lasers9, further broadening its application scope. Certainly, our approach is generalized, meaning it is not only applicable to AB superlattices but also to other more complex PCs. This approach can also be extended to realize Dirac nodal line semimetals29, Weyl points30, and other associated topological phenomena in 1D PCs. Furthermore, manipulating the incident angle in 2D or 3D PC systems enables modulation of polarization distribution in momentum space, offering avenues for optical phenomena, including suppressing radiation, beam shifting, and wave-front reshaping60. These advances open up new possibilities for designing compact and multifunctional optical devices based on topological photonics.

Methods

Numerical calculations

The calculation details in this study include band structure calculation, Zak phase evaluations, eigenmode distributions, and transmission spectra calculation. All numerical simulations were conducted employing the theoretical TMM and the commercial software MATLAB.

Experiments

Figure 5a is the photograph of the experimental setup. A halogen lamp (XD-301) emitting 380–1100 nm light is used as a light source, and the light is input through a coupled fiber. Next, the incident light is spatially filtered by the combination of a lens and an iris diaphragm. Then a linear polarizer is used to control the polarization direction of light. Thus, the linearly polarized light impinges on the sample, which is clamped on a plate holder, and the incident angle is adjusted by rotating the plate holder. Finally, the transmitted light is collected by an output fiber, and then the transmission spectrum is taken by a fiber-coupled spectrometer (Ocean Optics USB2000+).

Fig. 5: The optical setup and sample for transmission spectrum measurement.
figure 5

a The photograph of the experimental setup. The scale, represented by white lines, indicates the distance between two holes on the optical platform. b Cross-sectional scanning electron microscope (SEM) image of the sample consisting of 4 periods PC1 and PC2 on each side of the interface.

Full size image

Sample preparation

We employ the electron-beam evaporation deposition technique to fabricate the composite PC sample. Our sample is composed of 4 periods interfaced PC1 and PC2, both of which are composed of alternating SiO2 and TiO2 layers. The thicknesses of each SiO2 and TiO2 layer are identical to those in Fig. 3a. We alternately deposited the SiO2 and TiO2 layers on a glass substrate (40 mm × 40 mm × 1 mm), and the thickness of each layer can be accurately controlled by the depositing time. Our fabricated sample has a high degree of film thickness uniformity with a precision of ±5%, which is visually presented in the sample’s cross-sectional scanning electron microscope image, as is shown in Fig. 5b.