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Symmetry‐Protected Photonic Chiral Spin Textures by Spin–Orbit Coupling

Shi, Peng, Du, Luping, Li, Mingjie, Yuan, Xiaocong·2021·Laser & Photonics Reviews
DOI 10.1002/lpor.202000554
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Abstract

Chiral spin textures are widely researched in condensed matter systems and have shown potential for use in spintronics and storage applications. Photonic counterparts of these textures are observed in various optical systems with broken inversion symmetry. Unfortunately, the resemblances are only phenomenological. This work proposes a theoretical framework based on the variational theorem to show that the photonic chiral spin textures in an optical interface originate from the system's symmetry and relativity. Analysis of the rotational symmetry in optical systems indicates that the conservation of the total angular momentum is preserved from the local variations of spin vectors. Specifically, although the integral spin momentum does not carry net energy, the local spin momentum distribution, which determines the local subluminal energy transport and minimization variation of the square of the total angular momentum, results in the chiral twisting of the local spin vectors. The findings of this study deepen the understanding of the symmetries, conservative laws, and energy transport in optical systems, reveal the similarities in the formation mechanisms and the geometries of photonic and condensed‐matter chiral spin textures, and have potential applications in chiral and spin photonics.

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Figure 1 Geometry of photonic skyrmions. Projections of the skyrmion-like spin textures in cylindrical coordinates with quantum numbers (a) l =+1 and (b) l >+1 to a specific spherical coordinate; (c) corresponding projection of the normalized spin texture (top panel) to a specific spherical coordinate. These projections indicate that the skyrmion numbers of the spin textures for photonic skyrmions with positive integer quantum numbers are -1 uniformly. The directional vector of the iSAM is in the outer direction along the connecting line between the centre (0, 0) and red rings and its relative intensity is proportional to the length of connecting line. Projection of the skyrmion-like spin texture in cylindrical coordinates with quantum numbers (d) l =-1 and (e) l <-1 to a specific spherical coordinate; (f) corresponding projection of the normalized spin texture (top panel) to a specific spherical coordinate. These projections indicate the skyrmion numbers of the spin textures for photonic skyrmions with negative integer quantum numbers are +1. The directional vector of iSAM is in the inner direction along the connecting line between the centre (0, 0) and blue rings. It should be noted that the geometries of photonic skyrmions with quantum numbers l >+1 or l <-1 are approximately identical. Therefore, only (b) and (e) are exhibited here. The skyrmion-like spin texture is cylindrically symmetrical and thus the 2D contours in the xz -plane are shown here. The photonic skyrmions are constructed by the p -polarized light at air-gold interface.
Figure 1 Geometry of photonic skyrmions. Projections of the skyrmion-like spin textures in cylindrical coordinates with quantum numbers (a) l =+1 and (b) l >+1 to a specific spherical coordinate; (c) corresponding projection of the normalized spin texture (top panel) to a specific spherical coordinate. These projections indicate that the skyrmion numbers of the spin textures for photonic skyrmions with positive integer quantum numbers are -1 uniformly. The directional vector of the iSAM is in the outer direction along the connecting line between the centre (0, 0) and red rings and its relative intensity is proportional to the length of connecting line. Projection of the skyrmion-like spin texture in cylindrical coordinates with quantum numbers (d) l =-1 and (e) l <-1 to a specific spherical coordinate; (f) corresponding projection of the normalized spin texture (top panel) to a specific spherical coordinate. These projections indicate the skyrmion numbers of the spin textures for photonic skyrmions with negative integer quantum numbers are +1. The directional vector of iSAM is in the inner direction along the connecting line between the centre (0, 0) and blue rings. It should be noted that the geometries of photonic skyrmions with quantum numbers l >+1 or l <-1 are approximately identical. Therefore, only (b) and (e) are exhibited here. The skyrmion-like spin texture is cylindrically symmetrical and thus the 2D contours in the xz -plane are shown here. The photonic skyrmions are constructed by the p -polarized light at air-gold interface.CC-BY-NC-ND · From the PDF
Figure 2 Dynamical properties of photonic skyrmions. (a) Orbital/canonical MD, (b) spin MD, (c) 1D contours of kinetic MD (red line), orbital/canonical MD (blue line), spin MD (magenta line), (d) orbital/canonical group velocity and (e) group velocity of photonic skyrmion with vortex topological charge l =-2; (f) orbital/canonical MD, (g) spin MD, (h) 1D contours of kinetic MD (red line), orbital/canonical MD (blue line), spin MD (magenta line), (i) orbital/canonical group velocity and (j) group velocity of photonic skyrmion with vortex topological charge l =+3. The dotted circles indicate the region where l / k 0 r =1. In the region where l / k 0 r >1, the spin MD must be inverted with respect to the orbital/canonical MD, which leads to the subluminal group velocity of the photonic skyrmion. The opearting wavelength is 633nm and the MDs are normalized with respect to the maximal absolute value of the corresponding kinetic MD. Here, the photonic skyrmions are constructed by the p -polarized light at air-gold interface.
Figure 2 Dynamical properties of photonic skyrmions. (a) Orbital/canonical MD, (b) spin MD, (c) 1D contours of kinetic MD (red line), orbital/canonical MD (blue line), spin MD (magenta line), (d) orbital/canonical group velocity and (e) group velocity of photonic skyrmion with vortex topological charge l =-2; (f) orbital/canonical MD, (g) spin MD, (h) 1D contours of kinetic MD (red line), orbital/canonical MD (blue line), spin MD (magenta line), (i) orbital/canonical group velocity and (j) group velocity of photonic skyrmion with vortex topological charge l =+3. The dotted circles indicate the region where l / k 0 r =1. In the region where l / k 0 r >1, the spin MD must be inverted with respect to the orbital/canonical MD, which leads to the subluminal group velocity of the photonic skyrmion. The opearting wavelength is 633nm and the MDs are normalized with respect to the maximal absolute value of the corresponding kinetic MD. Here, the photonic skyrmions are constructed by the p -polarized light at air-gold interface.CC-BY-NC-ND · From the PDF
Figure 3 Comparison of theoretical and variational results for the skyrmion-like spin textures. 1D contours of Sz and Sx for the photonic skyrmions with quantum numbers (a) l =+1; (b) l =-1; (c) l =+2; and (d) l =-2. The Sz and Sx characteristics of the photonic skyrmions formed by Bessel-type surface waves are labelled by blue and red lines, respectively, while the corresponding Sz and Sx characteristics solved using the variational Equation 3 are shown as black and magenta lines, respectively (top panels). The middle panels show the spin textures of the photonic skyrmions formed by Bessel-type surface waves with various quantum numbers. The bottom panels show the spin textures given by solving the variational Equation 3 . The 2nd-order differential equation is solved using the Runge-Kutta method or Euler method and the boundary conditions are: Θ(0.001)=0, Θ′(0.001)=1×10 -3 for l =+1 and Θ(0.001)=π, Θ′(0.001)=-1× 10 -3 for l =-1; Θ(0.001)=0, Θ′(0.001)=5.5×10 -4 for l =+2 and Θ(0.001)=π, Θ′(0.001)=-5.5× 10 -4 for l =-2. Here, the photonic skyrmions are constructed by the p -polarized light at air-gold interface. The operating wavelength is 632.8nm. The horizontal axis of vectorgraph is consistent with the corresponding 1D contours of spin vectors.
Figure 3 Comparison of theoretical and variational results for the skyrmion-like spin textures. 1D contours of Sz and Sx for the photonic skyrmions with quantum numbers (a) l =+1; (b) l =-1; (c) l =+2; and (d) l =-2. The Sz and Sx characteristics of the photonic skyrmions formed by Bessel-type surface waves are labelled by blue and red lines, respectively, while the corresponding Sz and Sx characteristics solved using the variational Equation 3 are shown as black and magenta lines, respectively (top panels). The middle panels show the spin textures of the photonic skyrmions formed by Bessel-type surface waves with various quantum numbers. The bottom panels show the spin textures given by solving the variational Equation 3 . The 2nd-order differential equation is solved using the Runge-Kutta method or Euler method and the boundary conditions are: Θ(0.001)=0, Θ′(0.001)=1×10 -3 for l =+1 and Θ(0.001)=π, Θ′(0.001)=-1× 10 -3 for l =-1; Θ(0.001)=0, Θ′(0.001)=5.5×10 -4 for l =+2 and Θ(0.001)=π, Θ′(0.001)=-5.5× 10 -4 for l =-2. Here, the photonic skyrmions are constructed by the p -polarized light at air-gold interface. The operating wavelength is 632.8nm. The horizontal axis of vectorgraph is consistent with the corresponding 1D contours of spin vectors.CC-BY-NC-ND · From the PDF
Figure 4 Comparison of domain wall-like spin texture between theoretical and variational results. (a) the theoretical and (b) variational results as the in-plane wave number component ky = kr sin(15º ). The corresponding vector vectorgraphs for (c) theoretical (red) and (d) variational (blue) results in (a) and (b), respectively. (e) the theoretical and (f) variational results as the in-plane wave number component ky = kr sin(165º ). The corresponding vector vectorgraphs for (g) theoretical (red) and (g) variational (blue) results in (e) and (f), respectively. The theoretical and variational results match very well. Here, the photonic domain wall-like textures are constructed by the p -polarized light at air-gold interface. The wavelength is 632.8nm. The horizontal axis of vectorgraph is consistent with the corresponding 1D contour of spin vectors.
Figure 4 Comparison of domain wall-like spin texture between theoretical and variational results. (a) the theoretical and (b) variational results as the in-plane wave number component ky = kr sin(15º ). The corresponding vector vectorgraphs for (c) theoretical (red) and (d) variational (blue) results in (a) and (b), respectively. (e) the theoretical and (f) variational results as the in-plane wave number component ky = kr sin(165º ). The corresponding vector vectorgraphs for (g) theoretical (red) and (g) variational (blue) results in (e) and (f), respectively. The theoretical and variational results match very well. Here, the photonic domain wall-like textures are constructed by the p -polarized light at air-gold interface. The wavelength is 632.8nm. The horizontal axis of vectorgraph is consistent with the corresponding 1D contour of spin vectors.CC-BY-NC-ND · From the PDF

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Metadata

Authors
Shi, P., Du, L., Li, M., & Yuan, X.
Venue
Laser & Photonics Reviews
Publisher
Wiley
Published
2021-07-29
Volume
15
Issue
9
ISSN
  • 1863-8880
  • 1863-8899
DOI
10.1002/lpor.202000554
Cited by
39
References
6
Language
en
Licence
creativecommons.org/licenses/by-nc-nd/4.0/ (vor)
Funders
  • National Natural Science Foundation of China (U1701661, 61935013, 62075139, 61427819, 61622504, 61705135)
  • Natural Science Foundation of Guangdong Province (2016A030312010)
  • Shenzhen Peacock Plan (KQTD20170330110444030)

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