Symmetry-breaking enabled topological phase transitions in spin-orbit optics

Jiahao Cheng; Zan Zhang; Wang Mei; Yong Cao; Xiaohui Ling; Ying Chen; Ying Chen

doi:10.1364/OE.494534

1. Introduction

Light can carry spin angular momentum (SAM) and orbital angular momentum (OAM). The latter can be further divided into intrinsic OAM (IOAM) and extrinsic OAM (EOAM) [1–5]. The SAM, IOAM and EOAM are related to the circular polarization, optical vortices within the beam, and beam trajectories with respect to a given reference point or axis, respectively. The interaction and coupling between the SAM and the two types of OAMs are known as the two main spin-orbit interactions (SOI) in optics [3–6], i.e., spin-dependent vortex generation and photonic spin-Hall effect (PSHE). The SOIs are inherent in many basic optical processes, such as reflection, refraction, propagation, focusing, and imaging [4–16], and have attracted wide attention in recent years, playing an important role in the fields of nanophotonics [17–19], topological photonics [20–22], and plasmonics [23–25].

A transition between these two types of SOIs can take place in some optical systems, which is called the topological phase transitions (TPT) [26,27]. The TPT provides a new perspective for unifying the two types of SOIs and could be leveraged for the generation and manipulation of structured optical fields. The TPT phenomenon in spin-orbit photonics has been demonstrated in some systems, such as light beam reflection and refraction at optical interfaces [26–29] and light beam propagation in uniaxial crystals [30,31]. When a light beam illuminates an optical interface normally or propagates along the optical axis in a uniaxial crystal, an optical vortex is generated. As the incident angle is gradually increased, the vortex is evolved to an off-axis one, and then to non-vortex states with a transverse shift (i.e., the PSHE), manifested by an incident-angle-driven TPT.

In focusing systems, the coupling between the SAM and the two types of OAMs, such as the spin-dependent vortex generation and the PSHE, may also occur. Particularly, when a circular-polarization beam is focused, a fraction of the transverse focused beam is spin-flip and acquires a vortex phase with a topological charge of ±2, whereas the longitudinal mode (z-polarization component of the light field) gets a vortex phase with a topological charge of ±1 [10,14,15,32–36]. This indicates a SAM-to-IOAM conversion in the focusing system. When an asymmetrical circular-polarization beam is tightly focused, the beam centroid is shifted against the focal point. This effect is regarded as a manifestation of the PSHE, indicating a SAM-to-EOAM conversion in the focusing process [37–44]. Since there are two different types of SOIs in the focusing system, one is natural to wonder whether they can also be unified from the viewpoint of the TPT. Moreover, the intrinsic connection between the SOIs in the focusing system and at the optical interface is still obscure.

In this work, based on a full-wave theory, the focusing of off-axis and partially masked circular-polarization Gaussian beams is systematically examined, respectively. With the increases of the off-axis distance or masked area, the focused field exhibits a transition from the spin-dependent vortex generation to the PSHE, i.e., two different types of SOIs can be unified by the TPT. The underlying mechanism is attributed to the symmetry-breaking of the focusing system with respect to the central axis. Such type of symmetry-breaking enabled TPT is further explored from the perspective of vortex mode decomposition and is compared with the beam scattering at optical interfaces or propagation in uniaxial crystals. From our work, it is suggested that the breaking of the cylindrical symmetry can lead to the TPT between two different types of SOIs, providing a unique opportunity for a unified understanding of the TPT phenomenon in spin-orbit photonics systems.

2. Full-wave theory of beam focusing

A full-wave theory is first employed to describe the SOI when the beam is focused. The E-field distribution of the light beam on the reference plane under circular polarization basis can be written as follows (superscripts a={i,f} represent the incidence or focusing, respectively):

(1)$${\mathbf{E}^a}(\mathbf{r}_ \bot ^a) = \int {{d^2}\mathbf{k}_ \bot ^a\,\exp (i{\mathbf{k}^a} \cdot {\mathbf{r}^a})\sum\limits_{\xi ={+} , - ,z} {U_\xi ^a({\mathbf{k}^a}){{\hat{\mathbf{V}}}_\xi }({\mathbf{k}^a})} } ,$$

where $U_\xi ^a({\mathbf{k}^a})$ denotes the transverse and the longitudinal patterns of the ath beam, ${\hat{\mathbf{V}}_ \pm }({\mathbf{k}^a}) = (\hat{\mathbf{x}} \pm i\hat{\mathbf{y}})/\sqrt 2 $ refers to the unit vector of circular polarizations in the transverse direction of the beam, and ${\hat{\mathbf{V}}_z}({\mathbf{k}^a}) = \hat{\mathbf{z}}$. A 3 × 3 matrix $\hat{M}$ can be used to relate the focused field with the initial field [12,14]:

(2)$$\hat{M} = \frac{1}{2}\left[ {\begin{array}{ccc} {\cos {\theta^i}\cos {\theta^f} + 1}&{\exp ( - 2i\varphi )(\cos {\theta^i}\cos {\theta^f} - 1)}&{ - \sqrt 2 \exp ( - i\varphi )\cos {\theta^f}\sin {\theta^i}}\\ {\exp (2i\varphi )(\cos {\theta^i}\cos {\theta^f} - 1)}&{\cos {\theta^i}\cos {\theta^f}}&{ - \sqrt 2 \exp (i\varphi )\cos {\theta^f}\sin {\theta^i}}\\ { - \sqrt 2 \exp (i\varphi )\cos {\theta^i}\sin {\theta^f}}&{ - \sqrt 2 \exp ( - i\varphi )\cos {\theta^i}\sin {\theta^f}}&{2\sin {\theta^i}\sin {\theta^f}} \end{array}} \right]$$

where ${\theta ^f}$ represents the angle between the wave vector of any focused plane wave and the z-axis, and $\varphi $ signifies the azimuth angle of the plane wave component in the spherical coordinate system.

It is assumed that the incident beam is a left-handed circular-polarization beam, i.e., $U_\textrm{ + }^i({\mathbf{k}^i}) = \exp [ - {(k_ \bot ^iw)^2}/4]$, $U_ - ^i({\mathbf{k}^i}) \equiv 0$, and $U_z^i({\mathbf{k}^i}) \equiv 0$, where w denotes the half-width of the Gaussian beam waist. Under the normal incidence and paraxial approximation (${\theta ^i} \approx 0$), the following expression can be derived:

(3)$$\left[ {\begin{array}{c} {U_ +^f}\\ {U_ -^f}\\ {U_z^f} \end{array}} \right] = \hat{M}\left[ {\begin{array}{c} {U_ +^i}\\ {U_ -^i}\\ {U_z^i} \end{array}} \right] = \frac{1}{2}\left[ {\begin{array}{c} {(\cos {\theta^f} + 1)}\\ {\exp (2i\varphi )(\cos {\theta^f} - 1)}\\ { - \sqrt 2 \exp (i\varphi )\sin {\theta^f}} \end{array}} \right]U_\textrm{ + }^i.$$

As can be observed, a part of the transverse field retains its circular polarization handedness (called the normal mode), another part undergoes spin-flip transformation and carries a vortex phase with a topological charge of 2 (called the abnormal mode), while the longitudinal mode carries a vortex phase with a topological charge of 1. For the incidence of the right-handed circular-polarization beam, the abnormal and longitudinal modes exhibit vortex phases with opposite topological charges of -2 and -1, respectively.

Following Richards and Wolf's method [45], the E-field distribution of the focused beam is obtained by Debye integral:

(4)$$E_\xi ^f(\rho ,\phi ,z) ={-} \frac{{ikL\,\exp ( - ikL)}}{{2\pi }}\int\limits_{\theta _1^f}^{\theta _2^f} {\int\limits_{{\varphi _1}}^{{\varphi _2}} {U_\xi ^f} } \exp (ikz\cos {\theta ^f})\exp [ik\rho \sin {\theta ^f}\cos (\varphi \textrm{ - }\phi )]\sin {\theta ^f}d\varphi d{\theta ^f},$$

where L signifies the focal distance, k denotes the wave number, ρ and φ represent the radial coordinate and azimuthal angle in the focal area, respectively. When the integral range of ${\theta ^f}$ and $\varphi$ is 0 to $\theta _{\max }^f$ (i.e., the maximum exit angle) and 0 to 2π, respectively, the E-field distribution of the incident beam is cylindrically symmetric about the z-axis. If other values for the integral range of ${\theta ^f}$ and $\varphi$ are used (e.g., $\theta _1^f$ to $\theta _2^f$, ${\varphi _1}$ to ${\varphi _2}$), the E-field distribution of the incident light beam is no longer cylindrically symmetric about the z-axis.

Equation (4) provides the Debye integral in cylindrical coordinates. By substituting ${\theta ^f} = {\tan ^{ - 1}}[{({u^2} + {v^2})^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}}}/L]$, $\varphi = {\tan ^{ - 1}}(v/u)$, $\rho = {({x^2} + {y^2})^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}}}$, and $\phi = {\tan ^{ - 1}}(y/x)$ into Eq. (4), the integral in Cartesian coordinates can be obtained, where (u, v) are the Cartesian coordinates of the focusing system’s output pupil, and (x, y) are Cartesian coordinates in the focal region.

3. Symmetry-breaking induced TPT in focusing

The focusing of three types of circular-polarization beams, i.e., the off-axis, sector-masked, and horizontally masked beams, is considered here. The intensity patterns and beam shifts after focusing are calculated by selecting the following parameters: w = 40λ, L = 100λ, and k = 2π/λ, where λ is the working wavelength.

3.1 Off-axis beam focusing

Figure 1(a) schematically illustrates the focusing effect of an off-axis beam by a thin lens. The incident beam is a left-handed circular-polarization Gaussian beam whose centroid is located at a distance d from the z-axis [1st column in Fig. 1(b)]. The focusing field has three components: a normal mode $E_ + ^f$ with its spin handedness the same as the incident one, a spin-flip abnormal mode $E_ - ^f$, and a longitudinal mode $E_z^f$ (z-polarization component), as depicted in the 2nd to 4th column in Fig. 1(b). When the off-axis distance is d = 0, the abnormal mode exhibits a vortex phase with a topological charge of 2 and the longitudinal mode has a vortex phase with a topological charge being 1 [10]. As d is increased, the intensity patterns of the abnormal and longitudinal modes significantly deform, which gradually evolves from a perfect vortex state to crescent-shaped vortex states, and finally to non-vortex states with spin-Hall shifts. This is manifested as a TPT process from one type of SOI to another. Although the incident beam is off-axis, no beam shift and intensity deformation in the normal mode take place, as the vortex and beam shift only occur in the abnormal and longitudinal modes orthogonal to the incident one.

Fig. 1. The TPT of light in off-axis beam focusing. (a) Schematic of the problem studied. (b) The light intensity patterns of the normal, abnormal, and longitudinal modes after focusing by a thin lens when d = 0, 0.25w, 0.75w, and 1.25w. (c) The dependence of the beam centroid shift Δx of the abnormal and longitudinal modes on the off-axis distance d.

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The shifts of the beam centroid positions can be visually observed from the third and fourth columns of Fig. 1(b), which are manifested as the PSHE. The direction of the beam shift is always orthogonal to the off-axis direction of the incident light. That is, the off-axis direction of the incident light is along the y-axis, then the directions of the beam shifts is along the x-axis, and vice versa. This is due to the wavefront rotation of the vortex beam upon propagation, which is usually attributed to the Gouy phase [37,46]. To further understand the TPT induced by the off-axis distance, the following equation is used to calculate the shift Δx of beam centroid [26]:

(5)$$\Delta x = \frac{{\int\!\!\!\int {x|E_\xi ^f{|^2}dxdy} }}{{\int\!\!\!\int {|E_\xi ^f{|^2}dxdy} }}.$$

The shift Δx can be controlled by changing the off-axis distance d. The dependencies of Δx of the abnormal and longitudinal modes on the off-axis distance d after focusing are given in Fig. 1(c). With the increases of d, the values of Δx of the abnormal and longitudinal modes increase from 0, then decrease after reaching a maximum value. This TPT is very similar to that of a beam scattering by a sharp interface, where the beam shift has a peak in the transition region [26]. This is a remarkable feature of this TPT, which originates from the competition between intrinsic OAM and extrinsic OAM.

The beam shifts of the abnormal and longitudinal modes exhibit the same evolutionary behavior. However, the shift magnitude of the abnormal mode is obviously larger than that of the longitudinal mode since the two modes possessed vortex phases with different topological charges, which have different beam sizes. It is also worth noting that the beam shift directions of the right-handed and the left-handed circular-polarization incident light are opposite, as can be ascertained from Fig. 2 [37,46–49].

Fig. 2. The TPT of light under the incidence of left- and right-handed circular-polarization light. The first and second rows: intensity patterns of the incident, normal, abnormal, and longitudinal modes when the incident beam is left and right-handed circular-polarization, respectively.

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3.2 Sector-masked beam focusing

Another kind of asymmetric beam is taken into account, i.e., the sector-masked beam. Figure 3(a) schematically shows the focusing effect of a sector-masked beam by a thin lens. The incident beam is a left-handed circular-polarization Gaussian beam, a sector of which is masked by a sharp obstacle at an angle α. The obstacle is symmetric along the y-axis, and its center locates at the z-axis.

Fig. 3. The TPT of light in sector-masked beam focusing. (a) Schematic of the problem studied. (b) The light intensity patterns of the normal, abnormal, and longitudinal modes after focusing by a thin lens when α=0, π/2, π, and 3π/2. (c) The dependence of the beam centroid shift Δx of the abnormal and longitudinal modes on the masked angle α.

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When the incident beam is not masked (i.e., α=0), the focused field is actually exactly the same as that of the on-axis focusing [1st row of Fig. 3(b)]. When the mask angle is increased, the intensity patterns of the abnormal and longitudinal modes are significantly deformed [50]. Furthermore, their centroids are displaced away from the coordinate origin, manifesting as a topological transition from the vortex generation to the PSHE. Although the light intensity patterns of the normal mode are also deformed due to the partial masking of the incident beam, no beam shift is detected in the normal mode.

During the process of the TPT, the beam centroid is shifted to the x-axis direction, manifested as the PSHE. To further understand the influence of the masked angle α of the incident beam on the TPT, Eq. (5) is also used to calculate the beam shifts Δx. The dependences of the abnormal and longitudinal modes Δx on the masked angle are displayed in Fig. 3(c). Thus, the shift Δx can be controlled by adjusting the masked angle α. By increasing the value of α, the values of Δx of the abnormal and longitudinal modes increase accordingly. It's worth noting that since the field distribution in the focal region is entirely determined by the far-field, the effects of diffraction are not considered.

3.3 Horizontally masked beam focusing

The focusing effect of a left-handed circular-polarization Gaussian beam that is horizontally masked by a sharp obstacle is also further considered [Fig. 4(a)]. The edge of this sharp obstacle is parallel to the x-axis, and the masked height h represents the distance from the edge of the sharp obstacle to -w on the y-axis.

Fig. 4. The TPT of light in horizontally masked beam focusing. (a) Schematic of the problem studied. (b) The light intensity patterns of the normal, abnormal, and longitudinal modes after focusing by a thin lens when h = 0, 0.5w, w, and 1.5w. (c) The dependence of the beam centroid shift Δx of the abnormal and longitudinal modes on the masked height h.

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The first row of Fig. 4(b) indicates the light intensity patterns when the beam is not masked (i.e., h = 0), which is the same as that when the incident beam is on-axis or not masked by the sector-masked sharp obstacle. The masked height h is gradually increased, and the first to fourth rows of Fig. 4(b) show the light intensity patterns after focusing when h = 0, 0.5w, w, and 1.5w are used, respectively. The light intensity patterns of the horizontally masked beam are similar to those of the sector-masked beam. The TPT phenomenon exists in the abnormal and longitudinal modes when the horizontally masked beam is focused. The spin-Hall shift Δx increases with h, as can be ascertained from Fig. 4(c). On top of that, the dependency of Δx on the masked height h is very similar to that of Δx on the masked angle α in Fig. 3(c).

3.4 Comparison and discussion

From the above analysis, the two types of SOIs in the focusing system, namely the spin-dependent vortex generation and the PSHE, are unified from the viewpoint of the TPT. When the focusing system has cylindrical symmetry, both the abnormal and longitudinal modes acquire vortex phase with topological charges of ±2 and ±1, respectively. However, when the cylindrical symmetry of the system is broken, such as when the incident beam is off-axis or partially obscured, the abnormal and longitudinal modes in the focused beam exhibit a TPT from the spin-dependent vortex generation to the PSHE. This TPT phenomenon is very akin to that produced by the light beam scattering at optical interfaces [26–29] or propagating in uniaxial crystals [30,31]. When a beam is scattered by a sharp interface, the scattered light exhibits a transition from the spin-dependent vortex generation at normal incidence to the PSHE at oblique incidence. A beam propagating in a uniaxial crystal has a similar TPT as the angle between the beam propagation direction and the optical axis gradually increases. Thus, a common origin of the TPT phenomenon is found, which is associated with the breaking of the system’s cylindrical symmetry. In other words, the breaking of cylindrical symmetry will drive a spin-orbit optical systems manifesting from one type of SOI to another. From this point of view, we can describe the TPT in spin-orbit photonics in a unified form from the perspective of symmetry breaking.

4. Understanding the TPT based on vortex mode decomposition

Vortex mode decomposition provides an alternative perspective for understanding the two types of SOIs and is considered an effective method for describing the TPT phenomenon. The vortex mode contents of the abnormal mode are further examined here. The longitudinal mode has a similar behavior and will not be discussed. The abnormal mode can be decomposed into a series of Laguerre–Gaussian (LG) modes. More specifically, $C_p^m$ denotes the normalized weight coefficient of the mth-order azimuthal LG mode with radial index p [27,51,52]:

(6)$$C_p^m = \frac{{\int {\int {E_{LG_p^m}^ \ast E_ - ^fdxdy} } }}{{{{(\int {\int {\textrm{|}{E_{LG_p^m}}{\textrm{|}^2}} } dxdy)}^{1/2}}{{(\int {\int {\textrm{|}E_ - ^f{\textrm{|}^2}} } dxdy)}^{1/2}}}},$$

where ${E_{LG_p^m}}$ stands for the E-field distribution of the LG beam.

Figures 5(a)–5(c) show the normalized vortex mode spectra of the focused optical field under the illumination of three types of left-handed circular-polarization beams, i.e., the off-axis, sector-masked, and horizontally masked beams, respectively. When the incident beam is cylindrically symmetric, the abnormal mode has only a single vortex mode with a topological charge of m = 2 [1st column in Fig. 5]. Multiple vortex modes appear as the symmetry of the incident beam is broken (i.e., 2nd column to 4th column in Fig. 5), competing and superposing with m = 2 vortex mode. Then, the cylindrical symmetry breaking causes the transformation of the vortex mode from a single mode to multimode, which significantly deforms the intensity patterns. Unlike the focusing system in this work, the TPT process of the previously reported two systems [26,27,30] can be described by the competition and superposition of three vortex modes with topological charges of only m = 2, 1, and 0 (or -2, -1, and 0) [27].

Fig. 5. Normalized vortex mode spectra of the focusing field of the (a) off-axis beam, (b) sector-masked beam, and (c) horizontally masked beam.

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It can also be found that the envelopes of the vortex mode spectra in Fig. 5(a) possess a Gaussian distribution when the cylindrically symmetric incident beams are focused [53]. The envelopes of the spectra in Fig. 5(b) and 5(c) can be described by using sinc functions, which is similar to the single slit diffraction of light [54]. In the single slit diffraction, a restriction of position due to a single slit causes a sinc envelope of the light intensity in the transverse linear momentum or the far field position. Here, the restrictions of the angular position and the horizontal position by the mask cause the sinc envelopes of the vortex mode spectra in the focused optical field. Unlike the vortex spectrum of the TPT produced when the beam is scattered at the optical interface or propagates in the uniaxial crystal, which has only three components [27], the vortex spectra here possess more modes and the shape of the spectra is symmetric about m = 2. The vortex mode spectra of the longitudinal modes exhibit similar behaviors to those of the abnormal modes. It is also worth noting that the vortex mode spectra of the right-handed and the left-handed circular-polarization beams are symmetric about m = 0.

5. Conclusions

The focusing of three asymmetric beams with the full-wave theory has been thoroughly investigated in this work, and a unified description of the two different types of SOIs in the focusing system from the perspective of TPT was presented. The intrinsic mechanism of TPT is attributed to the cylindrical symmetry-breaking of the system. The TPT in the focusing system was compared with that in the previous two systems, i.e., beam scattering at optical interfaces and propagation in uniaxial crystals. It was found that the incidence-angle-driven TPT in the previous two systems can also be considered to be induced by the symmetry-breaking of the systems. Furthermore, the TPT phenomena in the focusing system was explained, in terms of vortex mode decomposition, indicating that it can be accurately described by the competition and superposition of multiple vortex modes. Our work paves the way for understanding the two types of SOIs in other systems, providing an opportunity for the development of a unified theoretical framework of the TPT phenomenon in various spin-orbit photonics systems. More importantly, since SOIs play an important role in the fields of nanophotonics [17–19], topological photonics [20–22], and plasmonics [23–25], our findings can be applied to optical vortex metrology and diagnostics [50], structured light communication [55], and particle manipulation [56].

Funding

National Natural Science Foundation of China (12174091); Hunan Provincial Innovation Foundation for Postgraduate (CX20221288).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Symmetry-breaking enabled topological phase transitions in spin-orbit optics

Abstract

1. Introduction

2. Full-wave theory of beam focusing

3. Symmetry-breaking induced TPT in focusing

3.1 Off-axis beam focusing

3.2 Sector-masked beam focusing

3.3 Horizontally masked beam focusing

3.4 Comparison and discussion

4. Understanding the TPT based on vortex mode decomposition

5. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (6)

Optics Express