Observing half and integer polarization vortices at band degeneracies
Ang Chen Wenzhe Liu Yiwen Zhang Bo Wang Fang Guan Xiaohan Liu Lei Shi Ling Lu Jian Zi
physics.optics








These authors contributed equally.

These authors contributed equally.

These authors contributed equally.




lshi@fudan.edu.cn

linglu@iphy.ac.cn

jzi@fudan.edu.cn
^1 Department of Physics, Key Laboratory of Micro- and Nano-Photonic Structures (Ministry of
Education), and State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China
^2 Institute of Physics, Chinese Academy of Sciences/Beijing National Laboratory for Condensed Matter Physics, Beijing 100190, China
^3 Collaborative Innovation Center of Advanced Microstructures, Fudan University, Shanghai 200433, China







Far-field polarization vortices were recently found on singlet bands in the momentum-space of two-dimensional photonic lattices, also known as the dark states and bound states in continuum. Here, we theoretically proposed and experimentally verified the existence of the polarization vortices at the degenerate points of photonic dispersions, whose vortex cores can be radiative bright states.
Half-charged vortices were generated from the Dirac points of π Berry phase and integer-charged vortices were generated from a quadratic degeneracy. Using a home-made polarization-resolved momentum-space imaging spectroscopy, we observed the complete evolution of the splitting from one quadratic point to a pair of Dirac cones by tracking the winding of the polarization vectors and the full spectrum of iso-frequency contours.


Observing half and integer polarization vortices at band degeneracies
    Jian Zi^1,3
    
=====================================================================



Quantized vortices, the most fundamental topological excitations <cit.>, have recently been observed in the Brillouin zone (BZ) of two-dimensional periodic photonic <cit.> and plasmonic <cit.> slabs. Such vortices can exist at multiple momentum points across the entire BZ and their topological charges characterize the winding numbers of far-field polarization vectors in momentum space <cit.>. The configuration of vortices, a fascinating topological structure, gives a new perspective view on understanding properties of photonic states, such as the topological nature of bound states in the continuum <cit.>. So far, the vortices centered at non-degenerate states are widely discussed, while those centered at degenerate states have not been well investigated yet. Due to the extensive existence of degenerate states in photonic bands <cit.>, researching on those vortices is of fundamental significance. It could complement the general picture of polarization vortices and dig out new physical phenomena.

Degenerate states are classified into two categories according to the dispersions in vicinity: Quadratic degeneracy points and linear degeneracy points (Dirac points), as shown in Fig. <ref>. Note that Dirac points induced by accidental degeneracy <cit.> are not included here .Those states are topologically different according to their Berry phase which is a phase change acquired over a cycle evolution of Bloch states. Berry phase of quadratic degeneracy points are trivial (0 or 2π), while that of Dirac points are nontrivial (π) <cit.>. Meanwhile, as the far-field polarization vector represents the zero-order Fourier component of the Bloch state, far-field polarization vortices mentioned above should carry the fingerprint of Berry phase. Therefore, the winding of polarization vectors will be an integer multiple of π, implying that the polarization vector has to be parallel or anti-parallel to itself after a closed loop centered at the degenerate state. For the quadratic case, it is even, forming a integer vortex that corresponds to trivial Berry phase. On the other hand, the Dirac cones are half vortices corresponding to nontrivial Berry phase.





In this paper, we theoretically and experimentally demonstrated such vortices, integers at quadratic degeneracy points and half ones at Dirac points in the plasmonic structures. The plasmonic structures studied here  are periodically corrugated thin PMMA layer covered flat silver substrates <cit.>.
Due to periodicity of the air hole arrays, the surface plasmon polaritons (SPPs) could be excited, forming well-defined band structures.
Because of the specific symmetries (C_4v or C_6v), a pair of bands can be degenerate quadratically at high-symmetry point, such as the center (Γ point) of the BZ. This quadratic degeneracy can be further regarded as a pair of linear degeneracies continuable to Dirac points, which can be obtained by symmetry breaking <cit.>.
Besides, Dirac points can also be found at C_3v-symmetry point, such as K point in the honeycomb lattice <cit.>.
Here we took the former approach to observe the evolution from a quadratic degeneracy point splitting into a pair of linear degeneracy points. The corresponding integer vortices splitting into pairs of half vortices is also shown. The sum of those half vortices after symmetry breaking equals to that of integer vortices, approving the law of charge conservation.





The band structures along Γ–M direction and the corresponding polarization vectors of quadratic and linear degeneracies are calculated by using finite element method (COMSOL Multi-physics), as shown in Fig. <ref>. The structures studied are illustrated in Fig. <ref>(a) and (b). The silver substrates are 200 nm thick, ensuring that there is no light transmitted. The thickness of PMMA is 80 nm, and the period (a) of square arrays of rectangular air holes is 600 nm. The length-width ratio l/w (l, w are the length and width of a rectangle hole respectively and the direction of l is along the diagonal lines of the square lattice) of air holes equals 1 and 1.48 in Fig. <ref> (a) and (b), respectively. In Fig. <ref>(a), when l/w = 1, the upper and lower bands are quadratically degenerate at Γ point, which is protected by C_4v symmetry. The corresponding polarization vectors plotted on a small region of BZ near Γ point for both two bands are shown aside. It should be noted that since one cannot distinguish the electric-field direction with its opposite direction due to the time-harmonic oscillations, we represent the polarization vector fields using line segments without arrows, which still allows us to precisely exhibit the vortices. Clearly, the topological charges for those bands at Γ point are both -1. When the length of the rectangle hole is increased to l/w = 1.48 (all other parameters fixed), the symmetry reduces to C_2v. Consequently, the quadratic degeneracy protected by C_4v splits into a pair of Dirac cones along Γ–M direction, see Fig. <ref>(b). From the polarization distributions shown aside, it is found that the topological charge of -1 disappears at Γ point and a pair of singularities appear along Γ–M. Those singularities corresponds to Dirac points. The polarization vector is anti-parallel to itself after a closed loop centered at those singularities, indicating π winding of the polarization vectors, namely half vortices. The same results are also observed in the case of reducing the C_6v symmetry of the hexagonal lattice, shown in Supplemental Material (SM) <cit.>. In general, along with splitting the quadratic degeneracy into pairs of Dirac degeneracy by symmetry breaking, the topological evolution of an integer vortex splitting into a pair of half vortices, consistent with charge conservation, is clearly exhibited, showing their different topological nature.




To experimentally verify above discussions, we performed angle-resolved reflectivity measurements via a momentum-space imaging spectroscopy <cit.> on a series of
samples with different l/w fabricated using electron-beam lithography. As shown in Fig. <ref>, we plotted the band structures defined by the peaks of the extinction spectra (one minus reflectivity) as a function of wavelength along Γ–M direction.
When l/w = 1, the quadratic degeneracy at Γ point is protected by the C_4v symmetry of the structure and thus there is only one peak of the extinction spectra in the wavelength range of interest; see Fig. <ref>(a).

When l/w ≠ 1, the C_4v symmetry of the structures is broken to C_2v. At Γ point, there are two peaks instead of one peak in the C_4v case. Meanwhile, another two peaks emerge along Γ–M, corresponding to the pair of linear degeneracy points, as shown in Fig. <ref>(b) and (c).
When l/w = 1.33, the Dirac points whose wavelength is 706 nm appear at wave vector k_∥ = ± 0.023×2π/a along Γ–M.
As l/w becomes larger (l/w = 1.48), the Dirac points will move outwards along Γ–M and appear at k_∥ = ± 0.027×2π/a, with the wavelength 702 nm. The experimental results along Γ–M direction agree well with theoretical ones (Fig. <ref>).
The plotted experimental data along other directions are provided in SM <cit.>.

To directly observe the vortices at band degeneracies, we measured the polarization-resolved iso-frequency contours and summed them with a range of wavelengths <cit.>.
In Fig. <ref>, we plotted the summed iso-frequency contours of the lower bands of the C_4v-symmetry structure (l/w = 1) and C_2v-symmetry structure (l/w = 1.48), as shown in Fig. <ref>(a) and (b), respectively. The center one is measured under unpolarized illumination, while the ones outside are polarization resolved.
Signal will diminish in the polarization resolved plots as dark areas, due to perpendicular far-field polarization of such states to the polarizer. Here, distribution of polarization vectors forming vortices are plotted in the central figure, and vortex centers are shown. In the case of C_4v-symmetry structure, as the angle change of polarization vectors is -2 π along a closed loop counterclockwise around the vortex center <cit.>, this indicates a vortex of -1. Meanwhile, we obtain -1/2 vortices in the case of C_2v symmetry.

These vortices can be viewed as rotating dark patterns along with the polarizer in summed iso-frequency contours, which gives a more vivid picture of winding polarization vectors. The shape, spinning speed and spinning direction of such patterns directly reveal the sign and magnitude of the charges carried by the vortices.
For the C_4v-symmetry structure, we can see that there is one dark strip spinning around the Γ point. Note that the state at the degeneracy point is always excited under any polarization, thus making the dark strip appearing disconnected.
The strip rotates π clockwise when the polarizer rotates π counterclockwise, indicating the charge of the degeneracy point is -1.
As for the C_2v-symmetry structure, there are two spinning dark radials rather than one strip in the C_4v case. And the state at Γ point can be either excited or not under different polarizations, which is distinguished from the degenerate state of C_4v symmetry.
The radials in C_2v system rotate 2π clockwise when the polarizer rotate π counterclockwise, showing the charges here are both -1/2.
Evidently when symmetry of the structure is changed from C_4v to C_2v, the -1 charge at the quadratic degeneracy point splits into two -1/2 charges at the linear degeneracy points, approving charge conservation. Two animations showing those spinning patterns are present in SM <cit.>. For an intuitive understanding, we give a schematic view of the spinning dark patterns for the half vortices; see SM <cit.>.
Last but not least, we emphasize that vortices near both quadratic and linear degeneracies don't necessarily lead to bound states in the continuum, even though the polarizations are ill-defined at those degeneracies.

In summary, we have experimentally observed topological integer and half vortices at band degeneracies of the plasmonic structures in momentum space. Integer vortices could appear near the quadratic degeneracy while half vortices near the linear degeneracy. Via controlling the symmetry of the structure, the evolution from one quadratic degeneracy point to a pair of linear degeneracy points is clearly exhibited, consistent with charge conservation. These unique vortices at band degeneracies enrich the various topological phenomena in photonics <cit.> and can accessibly enable half vector-vortex beams <cit.> or even lasers <cit.>.

We thank Yihua Wang, Shaoyu Yin for helpful discussions.
The work was supported by 973 Program (2013CB632701 and 2015CB659400), China National Key Basic Research Program (2016YFA0301100 and 2016YFA0302000) and NSFC (11404064). The research of L. S. was further supported by Science and Technology Commission of Shanghai Municipality (17ZR1442300), Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, and the
Recruitment Program of Global Youth Experts (1000 plans).
L.L. was supported by the National key R&D Program of China under Grant No. 2017YFA0303800, 2016YFA0302400 and supported by NSFC under Project No. 11721404.

Note.–A related paper <cit.> on topological half charges in the photonic crystal slab came to our attention when the preparation of this work.












34
fxundefined [1]
 ifx#1

fnum [1]
 #1firstoftwo
 secondoftwo
 
fx [1]
 #1firstoftwo
 secondoftwo
 





noop [0]secondoftwo

ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]

rl [1]href #1












@bib@innerbibempty

[Mermin(1979)]mermin1979topological
  author author N. D. Mermin, @noop  journal journal
  Reviews of Modern Physics volume 51, pages 591 (year 1979)NoStop
[Kosterlitz and Thouless(1972)]kosterlitz1972long
  author author J. Kosterlitz and author D. Thouless, @noop  journal journal
  Journal of Physics C: Solid State Physics volume
  5, pages L124 (year 1972)NoStop
[Hsu et al.(2013)Hsu,
  Zhen, Lee, Chua,
  Johnson, Joannopoulos, and Soljacic]hsu2013observation
  author author C. W. Hsu, author B. Zhen, author J. Lee, author
  S.-L. Chua, author
  S. G. Johnson, author
  J. D. Joannopoulos,  and author M. Soljacic, @noop 
  journal journal Nature volume 499, pages 188 (year
  2013)NoStop
[Zhang et al.(2017)Zhang,
  Chen, Liu, Hsu, Guan, Liu, Shi, Lu, and  Zi]zhang2017observation
  author author Y. Zhang, author A. Chen,
  author W. Liu, author
  C. W. Hsu, author F. Guan, author X. Liu, author L. Shi, author L. Lu,  and author
  J. Zi, @noop    (year
  2017), note arXiv: 1709.03648NoStop
[Zhen et al.(2014)Zhen,
  Hsu, Lu, Stone, and  Soljačić]zhen2014topological
  author author B. Zhen, author C. W. Hsu,
  author L. Lu, author
  A. D. Stone,  and author
  M. Soljačić, @noop 
  journal journal Physical Review Letters  volume 113, pages 257401 (year 2014)NoStop
[Hsu et al.(2016)Hsu,
  Zhen, Stone, Joannopoulos,  and Soljačić]hsu2016bound
  author author C. W. Hsu, author B. Zhen, author A. D. Stone, author
  J. D. Joannopoulos,  and author M. Soljačić, @noop
   journal journal Nature Reviews Materials  volume 1, pages 16048 (year
  2016)NoStop
[Sakoda(2005)]sakoda2005optical
  author author K. Sakoda, @noop  title Optical Properties of
  Photonic Crystals (publisher Springer Heidelberg, address New York, year 2005)NoStop
[Huang et al.(2011)Huang,
  Lai, Hang, Zheng, and  Chan]huang2011dirac
  author author X. Huang, author Y. Lai, author Z. H. Hang, author
  H. Zheng,  and author
  C. T. Chan, @noop  journal journal Nature Maretials volume 10, pages 582 (year 2011)NoStop
[Sakoda(2012)]sakoda2012proof
  author author K. Sakoda, @noop  journal journal Optics
  Express volume 20, pages 25181
  (year 2012)NoStop
[Zhen et al.(2011)Zhen,
  Hsu, Igarashi, Lu,
  Kaminer, Pick, Chua,
  Joannopoulos, and Soljačić]zhen2015spawning
  author author B. Zhen, author C. W. Hsu,
  author Y. Igarashi, author L. Lu, author
  I. Kaminer, author A. Pick, author S.-L.  Chua, author J. D. Joannopoulos,  and author M. Soljačić, @noop  journal journal Nature Maretials volume 10, pages 582 (year 2011)NoStop
[Zhang et al.(2005)Zhang,
  Tan, Stormer, and Kim]zhang2005experimental
  author author Y. Zhang, author Y.-W. Tan,
  author H. L. Stormer,  and  author P. Kim, @noop 
  journal journal Nature volume 438, pages 201 (year
  2005)NoStop
[Geim and Novoselov(2007)]geim2007the
  author author A. K. Geim and author K. S. Novoselov, @noop  journal journal
  Nature Materials volume 6, pages 183
  (year 2007)NoStop
[Haldane and Raghu(2008)]haldane2008possible
  author author F. D. M.  Haldane and author S. Raghu, @noop  journal journal Physical Review Letters volume 100,  pages 013904 (year 2008)NoStop
[Han et al.(2006)Han,
  Wu, Li, Xu, Liu, and Zi]han2006transmission
  author author D. Han, author F. Wu, author X. Li, author
  C. Xu, author X. Liu,  and author J. Zi, @noop  journal journal Applied Review Letters volume 89,  pages 091104 (year 2006)NoStop
[Chong et al.(2008)Chong,
  Wen, and Soljačić]chong2008effective
  author author Y. D. Chong, author X.-G. Wen,   and author M. Soljačić, @noop  journal journal Physical Review B volume 77, pages 235125 (year 2008)NoStop
[Yu et al.(2016)Yu,
  Sun, Ni, Wang, Yan, He, Liu, Feng,
  Lu, and Chen]yu2016surface
  author author S.-Y. Yu, author X.-C. Sun,
  author X. Ni, author
  Q. Wang, author X.-J. Yan, author C. He, author X.-P.  Liu, author L. Feng, author M.-H. Lu,  and  author Y.-F. Chen, @noop
   journal journal Nature Materials volume 15, pages 1243 (year
  2016)NoStop
[Dong et al.(2017)Dong,
  Chen, Zhu, Wang, and  Zhang]dong2016valley
  author author J.-W. Dong, author X.-D. Chen,
  author H. Zhu, author
  Y. Wang,  and author
  X. Zhang, @noop  journal journal Nature Materials volume 16, pages 298 (year 2017)NoStop
[]sm See Supplemental Material.
[Lu et al.(2014)Lu,
  Joannopoulos, and Soljačić]lu2014topological
  author author L. Lu, author J. D. Joannopoulos,  and author M. Soljačić, @noop  journal journal Nature Photonics volume 8, pages 821 (year 2014)NoStop
[Xiao et al.(2014)Xiao,
  Zhang, and Chan]xiao2014surface
  author author M. Xiao, author Z. Q. Zhang,   and author C. T. Chan,  @noop  journal journal Physical Review
  X volume 4, pages 021017 (year 2014)NoStop
[Gao et al.(2015)Gao,
  Lawrence, Yang, Liu,
  Fang, Béri, Li, and  Zhang]gao2015topological
  author author W. Gao, author M. Lawrence,
  author B. Yang, author
  F. Liu, author F. Fang, author B. Béri, author J. Li,  and  author S. Zhang, @noop 
  journal journal Physical Review Letters  volume 114, pages 037402 (year 2015)NoStop
[Leykam and Chong(2016)]leykam2016edge
  author author D. Leykam and author Y. D. Chong, @noop  journal journal
  Physical Review Letters volume 117, pages 143901 (year 2016)NoStop
[Leykam et al.(2016)Leykam,
  Rechtsman, and Chong]leykam2016anomalous
  author author D. Leykam, author M. C. Rechtsman,  and author Y. D. Chong, @noop  journal journal
  Physical Review Letters volume 117, pages 013902 (year 2016)NoStop
[He et al.(2016)He,
  Sun, Liu, Lu, Chen, Feng, and Chen]he2016photonic
  author author C. He, author X.-C. Sun,
  author X.-P. Liu, author M.-H. Lu, author
  Y. Chen, author L. Feng,  and author Y.-F.  Chen, @noop  journal journal Proceedings of the National Academy of Sciences volume 113, pages 4924 (year
  2016)NoStop
[Wang et al.(2017)Wang,
  Xiao, Liu, Zhu, and  Chan]wang2017optical
  author author Q. Wang, author M. Xiao, author H. Liu, author
  S. Zhu,  and author
  C. T. Chan, @noop  journal journal Physical Review X volume 7, pages 031032 (year
  2017)NoStop
[Guo et al.(2017a)Guo, Yang, Xia, Gao,
  Liu, Chen, Xiang, and  Zhang]guo2017three
  author author Q. Guo, author B. Yang, author L. Xia, author
  W. Gao, author H. Liu, author J. Chen, author Y. Xiang,  and  author S. Zhang, @noop 
  journal journal Physical Review Letters  volume 109, pages 213901 (year 2017a)NoStop
[Guo et al.(2017b)Guo, Xiao, and Fan]guo2017topologically
  author author Y. Guo, author M. Xiao,  and  author S. Fan, @noop 
  journal journal Physical Review Letters  volume 119, pages 167401 (year 2017b)NoStop
[Yao and Padgett(2011)]yao2011orbital
  author author A. M. Yao and author M. J. Padgett, @noop  journal journal
  Advances in Optics and Photonics volume 3,  pages 161 (year 2011)NoStop
[Bauer et al.(2015)Bauer,
  Banzer, Karimi, Orlov,
  Rubano, Marrucci, Santamato,
  Boyd, and Leuchs]bauer2015observation
  author author T. Bauer, author P. Banzer,
  author E. Karimi, author S. Orlov, author
  A. Rubano, author L. Marrucci, author E. Santamato, author R. W.  Boyd,  and author G. Leuchs, @noop  journal journal Science volume 347, pages
  964 (year 2015)NoStop
[Noda et al.(2001)Noda,
  Yokoyama, Imada, Chutinan,  and Mochizuki]noda2001polarization
  author author S. Noda, author M. Yokoyama,
  author M. Imada, author A. Chutinan,  and author M. Mochizuki, @noop 
  journal journal Science volume 293, pages 1123 (year
  2001)NoStop
[Iwahashi et al.(2011)Iwahashi, Kurosaka, Sakai, Kitamura, Takayama, and Noda]iwahashi2011higher
  author author S. Iwahashi, author Y. Kurosaka,
  author K. Sakai, author K. Kitamura, author
  N. Takayama,  and author
  S. Noda, @noop  journal journal Optics express volume
  19, pages 11963 (year 2011)NoStop
[Kodigala et al.(2017)Kodigala, Lepetit, Gu, Bahari, Fainman, and Kanté]kodigala2017lasing
  author author A. Kodigala, author T. Lepetit,
  author Q. Gu, author
  B. Bahari, author Y. Fainman,  and author B. Kanté, @noop  journal
  journal Nature volume 541, pages 196 (year 2017)NoStop
[Bahari et al.(2017)Bahari,
  Ndao, Vallini, Amili,
  Y., and Kanté]bahari2017nonreciprocal
  author author B. Bahari, author A. Ndao,
  author F. Vallini, author A. E. Amili, author
  F. Y.,  and author
  B. Kanté, @noop  journal journal Science volume
  358, pages 636 (year 2017)NoStop
[Zhou et al.(2017)Zhou,
  Peng, Yoon, Hsu,
  Nelson, Fu, Joannopoulos,
  Soljačić, and Zhen]zhou2017observation
  author author H. Zhou, author C. Peng, author Y. Yoon, author
  C. W. Hsu, author K. A. Nelson, author L. Fu, author J. D.  Joannopoulos, author M. Soljačić,  and author
  B. Zhen, @noop    (year 2017), note arXiv: 1709.03044NoStop




