The study of light-field manipulation has had immeasurable effects on the development of physics theory and practical applications. As an important aspect of light-field manipulation, the realization of spatial beam shifts attracts a great deal of attention. The horizontal shift of a spatial light beam along the plane of incidence, explained by the reflection and interference of light, is known as the Goos–Hänchen (GH) shift^[1]. Conversely, the vertical shift of a light beam perpendicular to the plane of incidence can be attributed to the spin Hall effect of light [SHEL, referred to as the Imbert–Fedorov (IF) shift]^[2–4]. These works contribute to advancements in optics, electrodynamics, quantum mechanics, and related fields, demonstrating potential applications in optical sensing^[5–8], reciprocal lenses^[9], quantum and classical information processing^[10], optical encoder design, and optical imaging detection^[11].

To date, considerable work has been done to explore the underlying physics and expand the application potential of beam shifts, including angular GH and IF shifts, as well as beam shifts under various incident conditions, such as introducing orbital angular momentum, changing incident angles, and coherence. Beam shifts have also been studied in structures such as photonic crystals (PhCs)^[12–15], plasmonic structures^[16–19], waveguides^[20,21], and anisotropic materials^[7,11,22]. However, recent beam-shifting devices are static, which means that once the device is fabricated and the incident conditions are set, the degree of beam shift remains fixed. If we can control the light beam in both space and time, i.e., shift the beam dynamically, it can bring more flexibility and broaden the range of possible applications^[23].

Here, we present a method to dynamically tune the degree and direction of beam shifts by changing a system’s refractive index by simulation. As shown in Fig. 1(a), we define the $XY$ plane with the incident light position as the origin and define four quadrants (I–IV) in the PhC slab. First, we construct four different bound states in the continuum (BICs) in the PhC to introduce cross-polarized phase and Pancharatnam–Berry (PB) phase, achieving beam shifts in four quadrants. The maximum beam shift ($|r|$) is about 12.32 µm when the beam radius is only 8 µm. Then we investigate the dynamics of the beam shifting, focusing on the cases where the system’s refractive index ($n$) changes gradually over a small range [Fig. 1(a)] and jumps over a large range [Fig. 1(b)]. We study the cross-polarized phase gradient that arises as the wave vector and radiation loss of the resonant mode change. When the refractive index changes by 0.06, the direction change ($\theta$) of the beam shift can exceed 50°. If the refractive index changes by only the order of ${10}^{-3}$, which is close to the value of electro-optic materials under the Pockels effect, the change in beam shift ($|\mathrm{\Delta }r|$) is more than 500 nm. We also study the interaction of light with the far-field polarization of different BICs under the refractive index jumps of the phase-change materials. In this case, large changes in the PB phase gradients will cause the beam to shift between different quadrants, leading to a 61.30° change in direction and a 15 µm change in displacement.

There are material-independent spatial transverse shifts related to cross-polarization that can increase the beam shift^[24]. The geometric phase helps us construct a connection between the spatial space and the momentum space of the light field, making it possible to generate spatial beam shifts by designing the states of polarization (SOPs) of light and geometric phase (also called the PB phase)^{[9,14,15,25,26]}. The operators for the coordinate momentum can be written in the momentum representation as $$\widehat{r}=i\frac{\partial }{\partial k},\widehat{p}=k,$$which reveals that since the momentum space and the spatial space are a pair of reciprocal spaces, spatial beam shifts can be realized by a phase gradient constructed in the momentum space. According to coupled mode theory (Fig. S1 in the Supplementary Material), the process of light reflection around the BICs can be described from momentum space. Considering an in-plane electromagnetic wave with a certain ${\mathit{k}}_{||}$, the reflected wave can be written as$$|E_{\mathrm{out}}⟩=S|E_{\mathrm{in}}⟩=\left[\begin{array}{cc}{r}_{ll}({\mathit{k}}_{||})& r_{lr}({\mathit{k}}_{||})e^{2i\theta ({\mathit{k}}_{||})}\\ r_{rl}({\mathit{k}}_{||})e^{-2i\theta ({\mathit{k}}_{||})}& r_{rr}(k_{||})\end{array}\right]|E_{\text{in}}⟩,$$(1)where $S$ is the general scattering matrix of the reflection; the subscripts $l$ and $r$ represent the left-handed circularly polarized (LCP) light and right-handed circularly polarized (RCP) light, respectively; $|E_{\mathrm{in}}⟩$ and $|E_{\mathrm{out}}⟩$ refer to the Jones vectors of the incident and reflected lights; $r_{ll}$, $r_{lr}$, $r_{rl}$, and $r_{rr}$ are the reflection coefficients; and $\theta (k_{||})$ is defined as the azimuthal angle of the SOPs of corresponding resonant radiative mode. When an LCP light is incident on the slab, the cross-polarized light (RCP light) will be affected by the reflection phase originating from the upper right element of the matrix. The additional phase part $e^{-2i\theta (k_{||})}$ arises from the PB phase and the part $r_{rl}(k_{||})$ originates from the cross-polarized phase from LCP light to RCP light. So, the reflection phase can be written as$${\phi }_{rl}({\mathit{k}}_{||})=\mathrm{angle}[r_{rl}({\mathit{k}}_{||})]-2\theta ({\mathit{k}}_{||}).$$(2)

Due to the reciprocal relationship between the spatial space and momentum space, large momentum-space phase gradients will result in spatial-space lateral shifts. For a beam with a finite size, the spatial beam shift can be written as$$⟨R_{\mathrm{RCP}}⟩=-⟨\frac{\partial {\phi }_{rl}({\mathit{k}}_{||})}{\partial k}⟩,$$(3)$$R_{\mathrm{RCP}}=\widehat{X}+\widehat{Y}=\frac{2{\alpha }_{v_g}\gamma }{4{\gamma }^2+{(|{\mathit{k}}_{||}|-k_0)}^2}\widehat{x}+\frac{2\partial \theta ({\mathit{k}}_{||})}{\partial k_y}\widehat{y},$$(4)where $⟨R_{\mathrm{RCP}}⟩$ is the coordinates of the field centroid of the cross-polarized converted RCP light with an incident LCP light with an in-plane ${\mathit{k}}_{||}$ along the $\mathrm{\Gamma }$-X direction; $\widehat{x}$ and $\widehat{y}$ represent the unit vectors that are horizontal (X) and vertical (Y) to the incident plane, respectively. $k_0$ is the wave vector of the resonant radiative mode. $\gamma$ refers to the radiation loss, and ${\alpha }_{v_g}$ is the group velocity factor of the radiative mode, where ${\alpha }_{v_g}=1(-1)$ for the positive (negative) group velocity.

As shown in Fig. 1(a), we define the $XY$ plane with the incident light position as the origin and define four quadrants (I–IV) in the PhC slab. We input LCP light (blue beam) at the center of the PhC and detect the reflected RCP light (red beam). A position vector $r$ ($x$, $y$) is defined to describe the relative beam offset between the input and output circular polarized beams. The $X$-direction shift is achieved by introducing cross-polarized light, which can be induced by inputting and collecting light with different chirality, which is related to the group velocity factor and radiation loss of the resonant mode. The $Y$-direction beam shift is achieved by introducing a geometric phase via the topological vortices generated by the BIC, which is especially sensitive to its polarization states. By carefully designing the structure parameters such as the hole diameter ($d$), lattice constant ($a$), film thickness ($h$, $t$), and refractive index ($n$) [Fig. 1(c)], the group velocity factor of the photonic bands and polarization states near the BIC can be precisely tuned, thereby achieving beam shifts in the four quadrants.

We design two PhC slabs with $C_{4v}$ symmetry, as shown in the inset of Fig. 2. Such structures will introduce symmetry-protected BIC modes centered at the $\mathrm{\Gamma }$ point, also called $\mathrm{\Gamma }$-BIC modes. BICs are the topological singularities in momentum space, and the SOPs will change greatly around them, resulting in a significant PB phase that can be used to introduce beam shifts. In the first PhC slab, the lattice constant $a$ is 1100 nm, the air hole diameter $d$ is 760 nm, and the film thickness is 200 nm. The refractive index is set to 2.4. In this structure, we find two TE-like Γ-BIC modes with different ${\alpha }_{v_g}$ [Fig. 2(a)]. For mode 1, the group velocity is negative, and the topological charge of the central BIC is $+1$ [Fig. 2(b)]; hence, there is a negative $X$-direction offset ($x_1$) and a positive $Y$-direction offset ($y_1$) [Fig. 2(c)] according to Eq. (4), while for mode 2, the group velocity is positive, and the topological charge of the central BIC is $-1$. Hence, there is a positive $X$-direction shift ($x_2$) and a negative $Y$-direction shift ($y_2$). We further simulate the beam shift in a $100\times 100$-unit device and find that when the wavelength of LCP light is 1620 nm, the light interacting with mode 1 shifts towards quadrant II [Fig. 2(d)]. By calculating the coordinates at the center of the incident and exiting light spots, we find the displacements in the $X$ and $Y$ directions are $x_1=-10.58\mathrm{\mu m}$ and $y_1=6.32\mathrm{\mu m}$, respectively. When the input light wavelength is changed to 1408 nm, the light interacting with mode 2 will shift to quadrant IV with a displacement of $x_2=8.53\mathrm{\mu m}$ and $y_2=-4.23\mathrm{\mu m}$, respectively.

The refractive index of the second PhC slab is 3.3, with a thickness of 300 nm [Fig. 2(e)]. The lattice constant $a$ and hole diameter $d$ are 1400 and 880 nm, respectively. We can also find another two TE-like Γ-BIC modes with different ${\alpha }_{v_g}$. After a similar analytical process [Figs. 2(f)–2(g)], we simulate the displacement in an $86\times 86$-unit device. As shown in Fig. 2(h), the incident light interacting with mode 3 and mode 4 will shift to quadrant III and quadrant I, respectively. The displacements in mode 3 are $x_3=-6.17\mathrm{\mu m}$ and $y_3=-4.81\mathrm{\mu m}$, and those in mode 4 are $x_4=6.35\mathrm{\mu m}$ and $y_4=5.94\mathrm{\mu m}$. In simulations, the structures are both shone on with an LCP light consisting of two orthogonal Gaussian beams with a phase difference of 90° at the center of the slabs, whose incident angle is 2.5° for better exciting the BIC modes and whose divergence angle is 1.3°. To sum up, the position vectors of the output beam in four modes are $r_1$ ($-10.58\mathrm{\mu m}$, 6.32 µm), $r_2$ (8.53 µm, $-4.23\mathrm{\mu m}$), $r_3$ ($-6.17\mathrm{\mu m}$, $-4.81\mathrm{\mu m}$), and $r_4$ (6.35 µm, 5.94 µm), respectively. The corresponding absolute values of the displacements are $|r_1|=12.32\mathrm{\mu m}$, $|r_2|=9.52\mathrm{\mu m}$, $|r_3|=7.82\mathrm{\mu m}$, and $|r_4|=8.70\mathrm{\mu m}$, and this is achieved when the incident light radius is only 8 µm.

The above four cases illustrate the beam shift when the structural parameters are fixed. Based on the characteristics of these modes, we further analyze the response of the beam shift to the change in the system refractive index. We will focus on two common situations in the field of electro-optics. First, we will study the dynamic beam shift under a small-range gradient of the refractive index, which is common in electro-optic effects and in refractive index sensing. Second, we will analyze the optical response under refractive index jumps, which is common in the field of light control based on phase-change materials.

It is important to study the signal response of optical devices with small changes in the refractive index. For example, in the electro-optic effect, people use an external excitation to change the effective refractive index of the system, thereby changing the phase, amplitude, and other parameters of the incident light^[27–30]. In general, the changes in the refractive index ($\mathrm{\Delta }n$) are only about ${10}^{-3}$ (e.g., ${\mathrm{LiNbO}}_3$) and ${10}^{-4}$ (e.g., GaAs) due to the weak light–matter interactions in the electro-optic effect (such as the Pockels effect). Another example is in the field of sensing, where a slight change in the refractive index of an optical system causes a change in the optical response, allowing for precise detection of the target substance. Therefore, it would be interesting and necessary to study the dynamic beam shift when the refractive index undergoes small changes.

We use mode 4 above to analyze this effect, with the wavelength of the incident LCP light being 1725 nm. It is worth noting that here we are concerned with the amount of change in the beam offset during the refractive index change, i.e., $\mathrm{\Delta }r$ ($\mathrm{\Delta }x$, $\mathrm{\Delta }y$) [as indicated by the red marks in Fig. 1(a)]. First, we control the refractive index ($n$) to vary gradually by the order of ${10}^{-2}$ from 3.28 to 3.34. Figure 3(a) shows the displacements in the $X$ and $Y$ directions as a function of refractive index ($n$). When $n$ is changed from 3.3 to 3.31, the beam shift in the $X$ direction changes by $\mathrm{\Delta }x=2.55\mathrm{\mu m}$, while the displacement change in the $Y$ direction ($\mathrm{\Delta }y$) is almost negligible. This is because the displacement in the $Y$ direction is related to the PB phase, which is directly related to the far-field polarization distribution of the BIC [Fig. 2(f)]. When the refractive index changes within a small range, the far-field polarization remains almost unchanged, so its effect on the $Y$-direction displacement is very small. However, the $X$-direction displacement is related to the cross-polarized phase gradient, i.e., $2{\alpha }_{v_g}\gamma /4{\gamma }^2+{(|{\mathit{k}}_{||}|-k_0)}^2$ in Eq. (4). When the refractive index changes, the radiation loss ($\gamma$) and the wave vector ($k_0$) of the resonant modes change, leading to a change in the cross-polarized phase gradient distribution [Fig. 3(b)]. Therefore, when fixing the incident conditions, a slight change in the refractive index results in a significant change in the displacement in the X direction.

We define the beam shift direction as the angle $\theta$ between the direction of the position vector $r$ ($x$, $y$) of the output beam and the $X$ axis [Fig. 1(a)], which can be expressed as $$\theta =\arctan \left(\frac{y}{x}\right).$$(5)

With a slight change in the refractive index, we calculate the beam shift angle $\theta$, as indicated by the red triangles in Fig. 3(c). We can see that when the refractive index rises from 3.28 to 3.34, the shift direction $\theta$ tunes from 84.99° to 31.45°, with a change $\mathrm{\Delta }\theta$ of about 53.54°. It can be observed that even a slight change in refractive index results in a significant shift angle. We also calculate the total displacement ($|r|$) in response to small changes in the refractive index, as indicated by the blue squares in Fig. 3(c). For a refractive index change $\mathrm{\Delta }n$ of 0.01, the displacement change $|\mathrm{\Delta }r|$ is about 2.46 µm. The effect of refractive index adjustment can be analogous to the effect of the incident wavelength change (Fig. S3 in the Supplementary Material). Furthermore, we simulate the intensity distribution of the RCP light at the refractive indices of 3.28 and 3.33 with the incidence of LCP light at the wavelength of 1725 nm, as shown in Figs. 3(d)–3(e). We further investigate the amount of change in displacement as the refractive index varies on the order of ${10}^{-3}$, which is the order of magnitude of the change in refractive index when lithium niobate undergoes the Pockels effect. As shown in Fig. 3(f), the change in refractive indices from 3.304 to 3.316 shows a near-linear increase in displacement in the X direction. For every 0.002 increase in refractive index, the change in displacement ($|\mathrm{\Delta }r|$) is still more than 500 nm.

Based on the above discussion, we can find that beam shifts can be significantly changed, even with a slight change in refractive index. Added to the influence of structural parameters and incident conditions, we can carefully design these parameters to achieve the desired displacement angle with the request of sensing or light-field control.

In addition to the small-range changes in refractive index, the response of optical devices to large-scale jumps in refractive index is also a focus of attention. Among them, dynamic light control based on phase-change materials is an important research direction. Phase-change materials can switch between amorphous and crystalline states in response to thermal, optical, or electrical stimuli^[31]. This transition is accompanied by a significant change in refractive index. Various optical devices have been realized using this property, such as beam-steering devices^[32], optical switches^[33], reconfigurable metasurfaces^[34], and programmable silicon photonics^[35]. Therefore, whether dynamic beam shifts can be achieved using phase-change materials is an important issue. Here, we choose ${\mathrm{Sb}}_2{\mathrm{Se}}_3$ for its low loss, non-volatility, and considerable nonlinear variation in the near-IR range (Fig. S2 in the Supplementary Material). We establish a practical multilayer PhC and substitute the ${\mathrm{Sb}}_2{\mathrm{Se}}_3$ material into the calculation.

Mode 1 and mode 2 of Fig. 2 are used to analyze the effect. We modulate the photonic band structure by designing structural parameters to ensure that the incident light can interact with different BICs. In this process, we only change the crystalline state of the ${\mathrm{Sb}}_2{\mathrm{Se}}_3$ and not the incident conditions. We determine the structure to be a 100 nm-thick ${\mathrm{Si}}_3{\mathrm{N}}_4$ ($\mathrm{refractive}\mathrm{index}=2$) PhC slab covered with a 30 nm-thick ${\mathrm{Sb}}_2{\mathrm{Se}}_3$ layer. The lattice constant of the slab is 1000 nm, and the diameter of the air hole is 540 nm. We simulate a $100\times 100$-unit device with LCP light incident at a wavelength of 1438 nm.

As schematically illustrated in Fig. 1(b), the ordered crystalline phase exhibits higher real and imaginary parts of the refractive index compared to the disordered amorphous phase; therefore, the excited modes undergo a switch before and after the phase transition. When the ${\mathrm{Sb}}_2{\mathrm{Se}}_3$ layer is in its amorphous state, the incident light will interact with modes in the photonic band near 1438 nm [the intersection of the red dashed line with the energy band in Fig. 4(a)]. At this time, the topological charge of $\mathrm{\Gamma }$-BIC is $+1$, and the far-field polarization state of these modes [Fig. 4(b)] will produce an obvious phase gradient, leading to a positive $Y$-direction displacement of 3.62 µm. In addition, this TE-like mode exhibits a negative group velocity dispersion ${\alpha }_{v_g}$, suggesting it will lead to a negative $X$-direction displacement of $-16.55\mathrm{\mu m}$. The beam shift of the device in the amorphous state is shown in Fig. 4(c), and the output beam is in the second quadrant. The results will change greatly when ${\mathrm{Sb}}_2{\mathrm{Se}}_3$ transforms into a crystalline phase, where the band structure moves toward higher wavelengths. At this time, the same incident light will interact with modes in another photonic band. Since the topological charge of the second BIC is $-1$, its far-field polarization state will change significantly, and thus its phase gradients will also change greatly. The displacement in the $Y$ direction is negative of $-6.52\mathrm{\mu m}$, and the displacement in the $X$ direction is $-7.49\mathrm{\mu m}$. The intensity of output RCP light will be distributed mainly in quadrant III [Fig. 4(f)].

In summary, the phase transition of ${\mathrm{Sb}}_2{\mathrm{Se}}_3$ will change the effective refractive index of the system, which will bring about an obvious change in the photonic band. This change will result in a significant alteration of the PB phase gradient, which enables the beam to dynamically shift between different quadrants. The result is a dynamic change of 61.30° in beam direction ($\mathrm{\Delta }\theta$) and 15 µm in beam displacement ($|\mathrm{\Delta }r|$). In addition, by carefully designing the structure and utilizing phase-change materials, we can enable dynamic movement of the light beam between any two quadrants.

In conclusion, we achieve dynamic beam shifts on a PhC slab for the first time, to the best of our knowledge. Simulation results show that a small change in refractive index can cause a significant change in the direction of beam displacement. For a change in refractive index of only 0.06, the angle of direction shift exceeds 50°. In addition, the significant changes in refractive index brought about by phase-change materials allow for dynamic control of beam shifts over different quadrants. Furthermore, all displacement amounts ($>6\mathrm{\mu m}$) of our devices are comparable to the radius of the incident beam ($\sim 8\mathrm{\mu m}$), a characteristic that outperforms traditional IF and GH shifts. By optimizing the structural parameters and incident light configurations, this approach enables large-angle beam scanning across all quadrants. There is no need to change the incident conditions in our device; therefore, it is amenable to chip integration. This innovative technique holds promise for various applications, including optical sensing, quantum and classical information processing, optical encoder design, and optical imaging detection.

Numerical simulation: The SOPs, the band structure, and the radiation loss are calculated using finite element analysis. Periodic boundary conditions are applied in the X and Y directions of the PhC slab, while scatter boundary conditions are applied in the $Z$ direction. The simulation of the actual devices uses finite-difference time-domain analysis. In such simulation, perfect match layers are applied in all directions. The simulated real devices are all shone upon with an LCP incidence consisting of two orthogonal Gaussian beams with a phase difference of 90° and a waist radius of 8000 nm. The incident angle is 2.5° for better exciting the BIC modes, and the divergence angle is 1.3°. To probe the reflective beam, we apply a field monitor on the same side with the incidence at the height of one wavelength. The simulation data are extracted and processed using MATLAB or ORIGIN.