Generating optical vortex beams by momentum-space polarization vortices centered at bound states in the continuum
Bo Wang, Wenzhe Liu, Maoxiong Zhao, Jiajun Wang, Yiwen Zhang, Ang Chen, Fang Guan, Xiaohan Liu, Lei Shi, Jian Zi
GGenerating optical vortex beams by momentum-space polarization vortices centeredat bound states in the continuum
Bo Wang , ∗ Wenzhe Liu , † Maoxiong Zhao , Jiajun Wang , YiwenZhang , Ang Chen , Fang Guan , Xiaohan Liu , Lei Shi , ‡ and Jian Zi § State Key Laboratory of Surface Physics, Key Laboratory of Micro- and Nano-Photonic Structures(Ministry of Education) and Department of Physics, Fudan University, Shanghai 200433, China and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
An optical vortex (OV) is a beam with spiral wave front and screw phase dislocation. Thiskind of beams is attracting rising interest in various fields. Here we theoretically proposed andexperimentally realized a novel but easy approach to generate optical vortices. We leverage theinherent topological vortex structures of polarization around bound states in the continuum (BIC)in the momentum space of two dimensional periodic structures, e.g. photonic crystal slabs, toinduce Pancharatnam-Berry phases to the beams. This new class of OV generators operates inthe momentum space, meaning that there is no real-space center of structure. Thus, not only thefabrication but also the practical alignment would be greatly simplified. Any even order of OV, whichis actually a quasi-non-diffractive high-order quasi-Bessel beam, at any desired working wavelengthcould be achieved in principle. The proposed approach expands the application of bound states inthe continuum and topological photonics.
An optical vortex (OV) is a light beam with spiralphase front and a zero-intensity point at the beam center.OVs are proved to be carrying orbital angular momen-tum (OAM) [1–6], which is a new degree of freedom oflight and has greatly broadened the fields such as opticalmicroscopy [7], optical micromanipulation [8–14], opticalcommunications [15–17] and quantum information pro-cessing [18–20]. As a result, generating OVs becomesa very hot topic in different wavelength ranges like vis-ible light, microwaves and radio waves [21–24]. In lowfrequency range, spiral phase plates [25] and phased an-tenna arrays [26] are the most commonly used approachesto generate OVs. However, both those two kinds of struc-tures can hardly be made into compact and integrabledevices when the working wavelength becomes smallerapproaching the terahertz or visible range, due to thelimitations on device thickness and feed network arrange-ments. Recently, with rapid development of metasurfaces[27–30], planar OV generators in micrometer or nanome-ter scale are shown possible, still requiring intricate de-signing of the individual units, complexity in fabricationto introduce helical phases and tough alignment in prac-tice.In this paper, we propose that, instead of artificiallyarranging resonators with winding configurations in thereal space, we can induce Pancharatnam-Berry (PB)phases [31] to beams by taking the advantage of wind-ing topologies of resonances which naturally exist in themomentum space near bound states in the continuum(BIC) [32–37]. We realize such kind of OV generators ∗ These authors contributed equally to this work. † [email protected] ‡ [email protected] § [email protected] with photonic crystal (PhC) slabs [38] which are verysimple in structure. Operating in the momentum space,the proposed OV generators have no center of structureto be aligned at the incident beam center. The gener-ated OVs are proved to have ring-like profiles in the mo-mentum space, thus they are quasi-Bessel beams whichhave quasi-diffraction-free behavior [39] (their momen-tum space profiles are rings with finite peak widths ratherthan δ -function rings as perfect Bessel beams). By chang-ing only the symmetry and the scale of the unit cells, dif-ferent orders of OVs at visible and near-infrared workingwavelengths are experimentally achieved.As plotted in Fig. 1(a), a 4-fold rotational-symmetricphotonic crystal slab can be viewed as one example ofour proposed OV generator. Unlike various metasurface-based OV generators, it seems counterintuitive that,there are neither space-variant resonators nor windingtopologies in the system to induce the extra spiral phasefactors. Actually, there ARE underlying winding topolo-gies in these radiative periodic systems. They exist inthe momentum space. Recently, vortex structures of res-onances are theoretically studied [32, 37, 40, 41] and ex-perimentally observed [34–36] in the momentum spaceof 2d periodic structures such as PhC slabs, two dimen-sional plasmonic crystals and gratings. It is of great im-portance that those vortex topologies are tightly relatedto the fascinating optical phenomenon, BIC [32–37, 40–57], and they are believed to result from topological prop-erty of the system. For example, let us consider a PhCslab with rotational symmetry higher than 2-fold [4-foldin Fig. 1(a)]. In such a system, any singlet at the Γpoint must be a BIC. In the vicinity of the BIC, the reso-nant guided modes are of high quality factors. Moreover,the states of polarization (SOPs) of far-field radiationfrom these guided resonances, which are almost linear,are momentum-space-variant and forced by symmetry to a r X i v : . [ phy s i c s . op ti c s ] S e p k x k y w BICS S S qW = 4 q (b)(c) ° q xy z(a) RCPLCP FIG. 1.
Concept of the proposed optical vortex gen-eration method. (a) A schematic view of proposed opticalvortex (OV) generating approach. (b) A schematic view of aparabolic band of a photonic crystal (PhC) slab, which has abound state in the continuum (BIC) in the center. The vortexstructures formed by nearly-linear guided resonances at differ-ent frequencies close to the BIC frequency are projected ontothe momentum space plane. (c) The working principle of theOV generator. Different linear states of polarization (SOPs)correspond to different azimuthal positions on the equator ofthe Poincar´e sphere, which will determine the trajectory ofthe SOPs and result in different geometric phase factors. form vortex topologies, shown as Fig. 1(b). In otherwords, the whole PhC slab will behave as different po-larized resonators with different incident wave vector k near a at-Γ BIC. These “momentum-space resonators”certainly could play the role that the real-space windinggeometry played in the previous works.Choosing a working frequency close to the at-Γ BICand regarding the guided resonances of which the SOPsare almost linear as wave plates [oriented in the direction θ ( k (cid:107) ) to the x axis], the transmitted field of a certain k (cid:107) incidence can be formulated as [58] | E out (cid:105) = 12 [ t x ( k (cid:107) ) + t y ( k (cid:107) )] | E in (cid:105) +12 [ t x ( k (cid:107) ) − t y ( k (cid:107) )] e − iθ ( k (cid:107) ) (cid:104) E in | R (cid:105) | L (cid:105) +12 [ t x ( k (cid:107) ) − t y ( k (cid:107) )] e iθ ( k (cid:107) ) (cid:104) E in | L (cid:105) | R (cid:105) on a helical basis. Here t x , t y are the transmittance co-efficients of the k (cid:107) guided resonance with the polariza-tion parallel and perpendicular to the efficient fast axis. | E in (cid:105) , | E out (cid:105) are the Jones vectors of the incident andtransmitted light, while | L (cid:105) , | R (cid:105) denote the left- andright-handed circularly polarized (LCP & RCP) unit vec-tors (0 , T & (1 , T . From the formula it is clear that, if we choose the incident light to be circularly polarized,the transmitted light would be composed of a trivial partwith the same polarization of the incidence, along withanother non-trivial cross-polarized part converted by theresonant process. This part of light would gain a geomet-ric phase factor, i.e. Pancharatnam-Berry (PB) phase[31], depending on the orientation of the SOP of theguided resonance, which can also be understood morephenomenologically by introducing the Poincar´e spherepicture [See Fig. 1(c)]. The PB phase equals to half thesolid angle enclosed by the trajectory of SOPs on thePoincar´e sphere. With the starting point and the endpinned at the opposed poles, the trajectory (also the PBphase) would vary with the intermediate point on theequator which correspond to the orientation of the reso-nance in the k -space. As a result, when we normally shinea right-handed circularly-polarized (RCP) and slightlydivergent beam at the corresponding working wavelengthonto the PhC slab, the different k components of thebeam would interact with different k (cid:107) resonances, thenthe transmitted left-handed circularly-polarized (LCP)beam would gain the desired spiral phase front of whichthe topological charge is l = − × q ( q here is the polarization charge of the BIC). The de-tailed proof could be found in the Supplemental Mate-rial (SM) [59]. Furthermore, if we choose the guidedresonances to be on a parabolic (or conical) band, theamplitude distribution will be a circular ring (the same l = 532 nm500550600650 F r equen cy ( T H z ) a r t(a) (b) k x a/2 p k y a / p FIG. 2.
Simulated band structure and polarization dis-tribution of the designed C v structure. (a) Simulatedband structure of the designed sample near the Γ point. Theband we focus on (TE-like 2) is colored with opaque darkturquoise, while the other TE-like bands are translucent. Theband surfaces are sliced in Γ − X and Γ − M direction. Theworking wavelength (532 nm) is pointed out with a translu-cent dark plane. Inset: the designed structure. (b) Theiso-frequency contours (dotted loops) with the SOPs (orangedouble-sided arrows) marked on them. The contours corre-spond to 532 (the working wavelength, marked red), 535, 538nm from inside to outside. The background is the flattenedband surface of TE-like 2, of which the different colors corre-spond to different frequencies. maxmin p-p
RCPinci.LCPinci . G p F r equen cy ( T H z ) BIC
532 nm l /2 CP1 l /2CP2Pol. OL1OL2M1L1L2 Laser CCD FIG. 3.
Experimental setup and the measured results of the fabricated 4-fold symmetric sample. (a) Theexperimental setup. The lens L1 in the reference light path could be moved to modify the reference wave’s direction and shapeof wave front. (b) The dispersion of the sample measured as angle-resolved transmittance spectra. The working wavelengthis marked with a green dashed line. The vanishing point of the transmittance signal on TE-like 2 corresponds to the centralBIC. The white regions on the two sides are regions limited by the numerical aperture which cannot be measured. Inset: thescanning electron microscopy image of the sample (scale bar: 400 nm). (c) The measured beam profile, interference patternand phase distribution of the transmitted cross-polarized beams. The angular range of the plots are about 13 ◦ . Note that L1is moved a bit making the reference beam spherical to show the vortex-shaped interference pattern more obviously. shape as iso-frequency contours) in the k -space, indi-cating that the generated OV is actually a high-orderquasi-Bessel beam. The quasi-non-diffractive nature ofthe quasi-Bessel beam is very meaningful. By applyinga lens, the transmitted beam can be Fourier transformedinto a quasi-Laguerre-Gaussian one.Note that, we here only present the theory in trans-mitting mode. However, the proposed approach worksfine in reflecting mode with the theory almost the same.Our OV generating approach basing on Bloch modes onlyrequires symmetry and periodicity. Adding a substrateeven a mirror or changing the basing material to evenmetal won’t matter as long as the Bloch modes exist.The wide choice of materials and the simple structurewould dramatically simplify the designing and fabrica-tion for practical uses. Accounting that our proposedOV generators work in the momentum space, no real-space alignment according to the optical axis is neededin application. In addition, periodic structures like PhCslabs can mostly be scaled up or down arbitrarily to workin different wavelength ranges like microwaves.To experimentally realize the proposed approach, wefirstly designed a PhC slab working at 532 nm, whichis in the visible range. We obtained its band structureand SOP distribution in the momentum space by simula-tions in order to assure that our proposition would work.Fig. 2 shows the structure along with the simulated re-sults. The structure shown as the inset of Fig. 2(a) isa periodically etched freestanding silicon nitride (Si N , refractive index n ≈ .
02) slab, of which the thickness t is 120 nm. The lattice is square and C symmetric, andthe periodicity a is 380 nm. The etched holes are circu-lar with their radius r equal to 140 nm. The calculatedband structure is illustrated in Fig. 2(a), and SOPs atthree different wavelengths on band TE-like 2 are shownin Fig. 2(b). One can clearly see that this band we fo-cus on is a paraboloid, making the iso-frequency contoursnear the Γ point circularly shaped as we wish. More im-portantly, the SOPs on the iso-frequency contours showpredicted winding behaviors corresponding to the centralBIC, which allow us to induce geometric phases. Thetotal winding angle of the SOPs is 2 π along a counter-clockwise loop around the polarization singularity, i.e.the topological charge of the BIC is q = +1. The SOPsare all close to linear polarization, which verifies our ap-proximation considering the guided resonances as waveplates. It needs to be emphasized again that, althoughour structure has extra symmetries such as mirror sym-metries and is made of a dielectric material, the only nec-essary condition to design such kind of OV generators isa i -fold ( i >
2) rotational symmetry, which sustains theexistence of parabolic bands and BICs on them.We then fabricated the designed prototype of our pro-posed OV generator. A freestanding Si N layer win-dow on a silicon substrate is periodically etched applyingreactive-ion etching technique. The parameters are thesame as designed, with the total number of unit cells be-ing 260 × k -variant guided resonances close to the BIC, generatinga cross-polarized counterpart with a phase vortex. Then,the transmitted beam is filtered by the orthogonally po-larized CP2 and fourier-transformed into far field by an-other objective lens (OL2). We finally obtain the far-field profiles of the cross-polarized beams from the CCD,plotted as the first column of Fig. 3(c). One can seethe beam profiles in far field are donut-shaped, confirm-ing that they are quasi-Laguerre-Gaussian beams afterFourier transformation, and are originally quasi-Besselbeams.In order to verify whether the filtered beams are OVs,we switched our system to the interferometer by intro-ducing a reference beam. The reference beam is linearpolarized and made a little divergent using a set of convexlenses (L1 & L2) to show the interference pattern moreclearly, as plotted in the second column of Fig. 3(c). Wefind that there are two spiral arms in each interferencepattern, proving the beams to be OVs with their topolog-ical charges l = ∓ ∓ × q . Exchanging the polarizerand analyzer, the spiral arms will change their directions.This corresponds to the fact that, the geometric phasevortex will change its sign when the start and end pointsof the trajectory on the Poincar´e switch their locations.Also, we apply the method mentioned in Ref. [60] basedon Fourier component filtering to measure the phase dis-tribution. The measured distributions are illustrated asthe third column of Fig. 3. The transmitted beam withRCP incidence apparently have a topological charge of-2, while the charge of the LCP one is +2, as predicted.The separation of the phase singularity is due to the de-fects in the sample and the imperfectness of the CPs. It is worth repeating that, because the working resonancesare in the momentum space, the beam is not required tobe focused on the center of the sample. As a proof, wehave tried to move the sample in the experiment, findingthe far-field beam profile unchanged. The result could befound in the SM [59]. We also simulated the propagat-ing behavior of the generated beam, confirming that thebeam could maintain diffraction free after a 7.5-micron(about 14 times the working wavelength) propagation, ofwhich the results could also be found in the SM [59].Considering the simple working principle we proposed,OVs with higher topological charges can also be achievedby applying structures with higher symmetry. We heredesigned another sample with C v symmetry, and theworking wavelength is modified to be in the near-infraredrange (923 nm). The thickness of the slab is modified to100 nm, while the periodicity is 850 nm and the radiusof holes is 265 nm. Similarly, we get the band structureand the polarization distribution of the C v sample bysimulation, shown in Fig. 4(a). The applied band hereis TE-like 2, of which the winding number of the central p F r equen cy ( T H z ) F r equen cy ( T H z ) G K
923 nm
BICMka/2 pG KM (b)(a) Profile Phasemaxmin p-p k x a/2 p k y a / p
923 nm
FIG. 4.
The measured results of the fabricated 6-foldsymmetric sample generating a higher-order opticalvortex. (a) The simulated band structure of the C v PhCslab and the polarization distribution on its band TE-like 2.In the band structure diagram, the band TE-like 2 is markedby purple and the working wavelength is marked by the reddashed line. In the polarization plot, the background is theflattened band surface of the studied band, while the dashedlines are the iso-frequency contours of 923 nm (marked or-ange) and 947 nm. (b) The measured band structure of thePhC slab in the form of transmittance spectra and the pro-file & phase distribution of the generated beam with RCPincidence.
BIC is -2. Thus, the produced OVs shall be with topo-logical charge l = ∓ × q = ±
4, which we are also ableto measure experimentally. The experimental results areplotted as Fig. 4(b). With RCP incidence, the gener-ated OV has the charge of +4. For systems with highersymmetry, we can even change the topological order ofthe generated OV by switching the working wavelength.Experimental results switching charge of OV from ± ∓ [1] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, andJ. P. Woerdman, Phys. Rev. A , 8185 (1992).[2] K. Y. Bliokh, Phys. Rev. Lett. , 043901 (2006).[3] S. Franke-Arnold, L. Allen, and M. Padgett, Laser &Photon. Rev. , 299 (2008).[4] M. R. Dennis, K. O’Holleran, and M. J. Padgett, in Progress in Optics , Vol. 53 (Elsevier, 2009) pp. 293–363.[5] K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, andA. Aiello, Phys. Rev. A , 063825 (2010).[6] A. M. Yao and M. J. Padgett, Adv. Opt. Photon. , 161(2011).[7] S. Frhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, Opt. Lett. , 1953 (2005).[8] A. T. ONeil and M. J. Padgett, Opt. Commun. , 139(2000).[9] A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett,Phys. Rev. Lett. , 053601 (2002).[10] D. G. Grier, Nature , 810 (2003).[11] J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. , 169 (2002).[12] J. E. Curtis and D. G. Grier, Phys. Rev. Lett. , 133901(2003).[13] J. Ng, Z. Lin, and C. T. Chan, Phys. Rev. Lett. ,103601 (2010).[14] M. Padgett and R. Bowman, Nat. Photon. , 343 (2011).[15] G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov,V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express , 5448 (2004).[16] G. Walker, A. S. Arnold, and S. Franke-Arnold, Phys.Rev. Lett. , 243601 (2012).[17] A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed,G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, et al. , Adv. Opt.Photonics , 66 (2015).[18] A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature , 313 (2001).[19] A. Vaziri, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. , 240401 (2002).[20] G. Molina-Terriza, J. P. Torres, and L. Torner, Nat.Phys. , 305 (2007).[21] E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Up-ham, and R. W. Boyd, Light Sci Appl , e167 (2014).[22] K. Zhang, Y. Yuan, D. Zhang, X. Ding, B. Ratni, S. N.Burokur, M. Lu, K. Tang, and Q. Wu, Opt. Express ,1351 (2018).[23] F. Tamburini, E. Mari, B. Thid, C. Barbieri, and F. Ro-manato, Appl. Phys. Lett. , 204102 (2011).[24] P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M.Litchinitser, and L. Feng, Science , 464 (2016).[25] M. Beijersbergen, R. Coerwinkel, M. Kristensen, andJ. Woerdman, Opt. Commun. , 321 (1994).[26] Q. Bai, A. Tennant, and B. Allen, Electron. Lett. ,1414 (2014).[27] L. Huang, X. Chen, H. Mhlenbernd, G. Li, B. Bai,Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, Nano Lett. , 5750 (2012).[28] A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, Sci-ence , 1232009 (2013).[29] N. Yu and F. Capasso, Nat. Mater. , 139 (2014).[30] L. Liu, X. Zhang, M. Kenney, X. Su, N. Xu, C. Ouyang,Y. Shi, J. Han, W. Zhang, and S. Zhang, Adv. Mater. , 5031 (2014).[31] M. Berry, J. Mod. Opt. , 1401 (1987).[32] B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljai,Phys. Rev. Lett. , 257401 (2014).[33] C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos,and M. Soljai, Nat Rev Mater , 16048 (2016).[34] Y. Zhang, A. Chen, W. Liu, C. W. Hsu, B. Wang,F. Guan, X. Liu, L. Shi, L. Lu, and J. Zi, Phys. Rev.Lett. , 186103 (2018).[35] H. M. Doeleman, F. Monticone, W. den Hollander, A. Al,and A. F. Koenderink, Nat. Photon. , 397 (2018).[36] A. Chen, W. Liu, Y. Zhang, B. Wang, X. Liu, L. Shi,L. Lu, and J. Zi, Phys. Rev. B , 180101 (2019).[37] W. Chen, Y. Chen, and W. Liu, Phys. Rev. Lett. ,153907 (2019).[38] S. Fan and J. D. Joannopoulos, Phys. Rev. B , 235112(2002).[39] P. Vaity and L. Rusch, Opt. Lett. , 597 (2015).[40] Z. Sadrieva, K. Frizyuk, M. Petrov, Y. Kivshar, andA. Bogdanov, Phys. Rev. B , 115303 (2019).[41] W. Chen, Y. Chen, and W. Liu, Phys. Rev. Lett. (2019), 10.1103/physrevlett.122.153907.[42] E. N. Bulgakov and A. F. Sadreev, Phys. Rev. B ,075105 (2008).[43] C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson,J. D. Joannopoulos, and M. Soljai, Nature , 188(2013).[44] E. N. Bulgakov and A. F. Sadreev, Phys. Rev. A ,053801 (2014).[45] Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, Phys.Rev. Lett. , 037401 (2014). [46] J. Gomis-Bresco, D. Artigas, and L. Torner, Nat. Pho-tonics , 232 (2017).[47] Y. Guo, M. Xiao, and S. Fan, Phys. Rev. Lett. ,167401 (2017).[48] E. N. Bulgakov and D. N. Maksimov, Phys. Rev. A ,063833 (2017).[49] Y. Song, N. Jiang, L. Liu, X. Hu, and J. Zi, Phys. Rev.Applied , 064026 (2018).[50] S. Dai, L. Liu, D. Han, and J. Zi, Phys. Rev. B ,081405 (2018).[51] Y. He, G. Guo, T. Feng, Y. Xu, and A. E. Mirosh-nichenko, Phys. Rev. B , 161112 (2018).[52] J. Jin, X. Yin, L. Ni, M. Soljaˇci´c, B. Zhen, and C. Peng,arXiv preprint arXiv:1812.00892 (2018).[53] K. Koshelev, S. Lepeshov, M. Liu, A. Bogdanov, andY. Kivshar, Phys. Rev. Lett. , 193903 (2018).[54] K. Koshelev, A. Bogdanov, and Y. Kivshar, Science Bul-letin , 836 (2019).[55] Y. Guo, M. Xiao, Y. Zhou, and S. Fan, Adv. OpticalMater. , 1801453 (2019).[56] A. Cerjan, C. W. Hsu, and M. C. Rechtsman, Phys. Rev.Lett. (2019), 10.1103/physrevlett.123.023902.[57] K. Koshelev, G. Favraud, A. Bogdanov, Y. Kivshar, andA. Fratalocchi, Nanophotonics , 725 (2019).[58] Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, Opt.Lett. , 1141 (2002).[59] See Supplementary Material.[60] N. Carlon Zambon, P. St-Jean, M. Milievi, A. Lematre,A. Harouri, L. Le Gratiet, O. Bleu, D. D. Solnyshkov,G. Malpuech, I. Sagnes, S. Ravets, A. Amo, and J. Bloch,Nat. Photon. , 283 (2019).[61] A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman,and B. Kant, Nature , 196 (2017).[62] B. Bahari, F. Vallini, T. Lepetit, R. Tellez-Limon,J. Park, A. Kodigala, Y. Fainman, and B. Kante, arXivpreprint arXiv:1707.00181 (2017).[63] S. T. Ha, Y. H. Fu, N. K. Emani, Z. Pan, R. M. Bakker,R. Paniagua-Domnguez, and A. I. Kuznetsov, Nat. Nan-otechnol. , 1042 (2018). ACKNOWLEDGEMENTS
We thank Prof. Che Ting Chan, Dr. Hai-wei Yin for helpful discussions. The work was sup-ported by 973 Program and China National Key BasicResearch Program (2015CB659400, 2016YFA0301100,2016YFA0302000 and 2018YFA0306201) and NationalScience Foundation of China (11774063, 11727811 and91750102). The research of L. S. was further supportedby Science and Technology Commission of Shanghai Mu-nicipality (17ZR1442300, 17142200100).
AUTHOR CONTRIBUTIONS
W. L., L. S. and J. Z. conceived the basic idea for thiswork. W. L. gave the theoretical explanation. B. W.and W. L. designed the structures, carried out the finiteelement method and the finite-difference time-domainmethod simulations, and analyzed the simulated and measured data. B. W. and J. W. performed the sam-ple fabrications. B. W., M. Z. and J. W. performed theoptical measurements. M. Z. and Y. Z. constructed themeasurement system. L. S. and J. Z. supervised the re-search and the development of the manuscript. W. L.wrote the draft of the manuscript and all authors tookpart in the discussion, revision and approved the finalcopy of the manuscript.
DATA AVAILABILITY
The data that support the findings of this study areavailable from the authors on reasonable request, see au-thor contributions for specific data sets.
METHODS
Theoretical analysis.
Please see the SupplementalMaterials for the derivations and discussions.
Simulations.
The eigen-mode simulations and thepolarization analysis were done using finite elementmethod and finite-difference time-domain method. Pe-riodic (Bloch) boundary conditions were applied in x, y direction, while perfect matching layers were applied in z direction. Sample fabrication.
The samples were fabricatedbasing on commercial silicon nitride windows, of whichthe silicon nitride layers are 100 ∼
120 nanometers thick.The silicon nitride layers resided on center-windowed sili-con substrates, whose thicknesses are about 200 microns.To fabricate a designed structure, the raw sample wasfirstly spin-coated with a layer of positive-tone electron-beam resist (PMMA950K A4, MicroChem). An addi-tional layer of conductive polymer (AR-PC 5090.02) wasalso attached to avoid charging effects during electron-beam lithography (EBL). Then, a hole array mask pat-tern was defined onto the PMMA layer using EBL (ZEISSSigma 300), followed by developing in a 1:3 mixtureof methyl isobutyl ketone (MIBK) and isopropyl alco-hol (IPA). The periodically EBL etched PMMA layerwould act as a mask in the subsequent reactive-ion etch-ing (RIE) process. Anisotropic etching of the periodicstructure was achieved using a mixture of CHF and O .After ensuring that the freestanding part of the siliconnitride layer had been etched through, the PMMA maskwas eventually removed by RIE using O . The over-all sizes of the fabricated samples are approximately 100microns ×
100 microns.
Experimental technique.
The Fourier-optics-basedmomentum-space spectroscopy system has three operat-ing modes: a spectrometer mode, an imaging mode, and an interferometer mode. The illustration of the systemcould be seen in Fig. 3, while the schematic views of theother two modes could be found in Ref. [34, 36].To switch the system to the spectrometer mode, thereference beam should be blocked, and the light sourceshould be replaced with a broadband one. An spectrom-eter with a slit should be plugged in front of the chargecoupled device (CCD) to resolve the frequencies, and thecircular polarizers should be removed. Operating in thismode, the dispersion of the system could be obtained inthe form of angle-resolved transmittance spectra.Meanwhile, the imaging mode would enable us to ob-tain the beam profile. To use this mode, the lightsource should be a monochromatic laser, the spectrom-eter should be removed, and the polarizers should beplugged back in. Focused by an objective lens (OL1) andpassing through the incident circular polarizer (CP1), theincident beam would be convergent and circularly polar-ized. With the analyzer (CP2), the cross-polarized partof the transmitted beam is filtered out and its profilewould be captured by the CCD.Thirdly, the interferometer mode of the system couldgive us the interference fringes and the phase distribu-tions. The system could be switched to this mode byintroducing the reference beam. When the lenses L1 andL2 were confocal, the reference beam should be planewave like. Shifting the lens L1 away from confocal con-dition on-axis would make the reference beam sphericaland the interference patterns would thus be more vividvortex-like ones. Otherwise, if the lens L1 is shifted off-axis, the reference beam would gain an extra transversewave vector, which would separate the phase distributioninformation from the zeroth-order Fourier component ofthe interference fringes to the ± −−