1
Supporting Information for
Giant Helical Dichroism of Single Chiral Nanostructures
with Photonic Orbital Angular Momentum
Jincheng Ni,†,‡ Shunli Liu,† Guangwei Hu,‡ Yanlei Hu,† Zhaoxin Lao,† Jiawen Li,†,* Qing Zhang,‡ Dong Wu,†,* Shaohua Dong,‡ Jiaru Chu,† and Cheng-Wei Qiu‡,*
†CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Precision Machinery and Precision Instrumentation, University of Science and
Technology of China, Hefei, Anhui 230027, China.
‡Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117583, Singapore.
*Corresponding author. Email: [email protected] (J.L.); [email protected] (D.W.); [email protected] (C.-W.Q.)
This PDF file includes:
Section S1 to S5 Figs. S1 to S10 SI References
2
Section S1. Circular dichroism in single nanostructures
Here, we discuss the chiroptical properties in single structures by circularly polarized light. Chiral nanostructures normally exhibit a strong resonance with a comparable or smaller size to the wavelength of incident circularly polarized light. Therefore, the optical activity in the form of CD is extremely weak in our experiment with large structures (diameter: 4 μm) but using laser at visible wavelength. We simulated the CD signals of the planar-chiral nanostructures, as shown in Figure S2. For simplicity, the refractive index of structure is fixed to 1.51 and incident Gaussian beam waist is 10 μm. The simulated CD signal is less than 1.5% at the wavelength range from 600 to 1,000 nm.
It is worth noting that the beam waist cannot be further decreased in CD measurements at the visible wavelength. When the circularly polarized light is tightly focused under a high numerical-aperture objective, the spin angular momentum is converted to orbital angular momentum,1 implying that the chiroptical signal is no longer associated with CD. To enhancing the CD signals, one may consider to shrink the dimension of nanostructure. However, a single chiral nanostructure cannot provide a dimension-matching area for high-efficiency light-matter interaction, leading to the requirements of a large array of nanostructures (Figure S7). Therefore, one can achieve a pronounced spin-dependent chiroptical response with large amounts of chiral nanostructures. On the contrary, chiroptical response by utilizing optical vortices can be directly excited in a single nanostructure due to the dimensional matching between
3 vortex beams and nanostructures.
Section S2. Theoretical analysis of the mirror-symmetric helical dichroism on left-
and right-handed planar nanostructures
Planar-chiral nanostructures such as the Archimedean spiral exhibit a reversed sense of rotation if illuminated in the opposite direction, thereby producing an asymmetric transmission effect. We consider an incoming vortex beam propagating in -z direction
( , ) = l l
l ( ) (S1)
where k0 is the wavevector, ω is the frequency, and ±l describes the amplitude distributions of vortex beams with topological charge ±l . Distinguishing from circularly polarization with only two states, the light can be projected on infinite-dimensional OAM eigenstates characterized by Hilbert factor l , which demonstrates as digital spiral spectrum.2 After interacting with planar-chiral metamaterials, the transmitted field is then given by
( , ) = l l
l ( ) (S2)
The forward transmission matrix relates the generally complex amplitudes of the incident field to the complex amplitudes of the transmitted field as
= (S3)
where denotes the complex transmission coefficient of the l helical component under l helical wavefronts illumination.3 For the sake of simplicity, we
4
omit the topological charge l after the signs. Applying the Lorentz reciprocity theorem delivers the transmission matrix for propagation in z or the mirror images of planar nanostructures,
= = (S4)
Therefore, the asymmetric transmission can be defined as the difference between the transmitted intensities for opposite handedness of vortex beams as
= 2 ×( ) + ( ) − ( ) − ( )
( ) + ( ) + ( ) + ( ) = − (S5) The forward and backward transmission models can also be achieved by changing the handedness of planar-chiral nanostructures. Note that the discussion is also valid for other topological charges in the discrete OAM spectra, although we use the original topological charge l here (see Figure S6). The similar relationship can also be derived for reflection with = 1 − since there is no loss in the material. To avoid the influence of substrate, we fabricated two Archimedean nanospirals with opposite handedness on the substrate.
Section S3. Digital spiral spectra of chiral optical fields
The linearly incident light beam carrying OAM can be described as a complete, orthogonal, and infinite-dimensional basis for the solutions of paraxial wave equation in a high-dimensional Hilbert space.2 The optical field can be decomposed by the complete basis l . Assuming a discrete Fourier relationship between the azimuthal
5
angular distribution ( , ) and the angular momentum ratios l( ), we can express the function as,
l( ) = 1 √2 ( , ) l (S6) ( , ) = 1 √2 l( ) l l (S7) where ( , ) describes an arbitrary field distribution, and l( ) is the corresponding OAM spectrum.
The simulated discrete OAM spectra of electromagnetic fields on left-, right-handed and achiral nanostructures are shown in Figure S6. The OAM spectra are stronger on a multiple of 4 by incident OAM beams with topological charge l=±4 on the C4 symmetric nanostructures, which controls the rotational symmetry of electric field distributions. For the right-handed nanostructure, there is an obvious OAM-mode transformation to left side ( > , > ) for topological charge l=±4. On the contrary, an OAM-mode transformation to right side is observed for the left-handed nanostructure due to the opposite chirality. This directional transformation of OAM spectra is induced by chiral nanospirals, which can also yield non-zero OAM beams by incident zero-order OAM beam (Gaussian beam) as an OAM beam generator.4-6 For the achiral nanostructure, the OAM spectra are mirror-symmetric for two optical vortices with opposite topological charges l=±4. As a result, the optical chiral response on nanospirals by OAM beams can also be considered as the mirror-symmetric breaking on discrete OAM spectra after interacting with chiral nanostructures.
6
Section S4. Simulation methods
We performed numerical simulations of electromagnetic fields in the nanostructures using a commercial finite difference time-domain software (Lumerical FDTD Solutions, Inc). The geometric parameters of structures were extracted from the SEM images of experimental nanostructures. The simulated area is set as 40 μm × 40 μm× 2.6 μm to ensure the valid electromagnetic field distributions by OAM beams even with maximum topological charge l=±20. According to the spectral range in the experiments, for each configuration we calculated the reflectance on nanowires from the topological charge l from -20 to 20. The substrate has not been included in the simulations. The reflectance of glass substrate is around 10% based on the experimental measurements.7
In Figure S10, we also show additional numerical results for Au nanowires exhibiting not only HD but also helical dichroism absorption due to the lossy materials in comparison to the lossless polymer used in our experiment. The permittivity of gold nanostructures was from the build-in material database within the software, i.e., Au (gold)-Johnson and Christy. Considering the high absorption of Au, the thickness of nanostructure is decreased to 200 nm, which still yields a significant vortex-dependent chiroptical signal.
7
Section S5. Angular-momentum flux of the electromagnetic field
Here we inspect the orbital angular momentum flux of the paraxial electromagnetic field, started by the z-component angular-momentum flux density for the vortex beam through a transversal plane oriented in the z-direction:
= 1
2 [ (
∗+ ∗ ) − ( ∗+ ∗ )] (S8)
where E and H are the electric and magnetic field, and and are the permittivity and permeability of the medium, respectively. For simplicity, we can separate the spin and orbital angular momentum fluxes proposed in previous literatures.8-9 The total angular momentum flux density can be separated as = + . Specifically, the two angular momentum flux are given by:
= 1 2 [ ∗+ ∗] (S9) = 1 4 [ ∗ − ∗ + ∗ − ∗ ] (S10)
where is the angular frequency of the electromagnetic field.10 Utilizing FDTD simulations, we excite the nanostructures with x-polarized vortex beams propagating along the -z axis. Analogous to the spin angular momentum flux,11-12 the simulated OAM flux is bounded by the conservation law of optical chirality in free space. After interacting with nanostructures, the OAM flux demonstrates the expected behavior with dissipation of chirality |F|<1. For achiral systems, the OAM flux density is related by mirror symmetry, as shown in Fig. S8. However, the symmetry of OAM flux density is broken in chiral structures, resulting in the OAM-related optical activity.
8
Figure S1. Design of planar nanostructures. (a) Schematic diagram of right-handed,
cross-shaped, and left-handed nanostructures. The Archimedean nanospirals are modeled by the function of r=2/π×rmax×mod(θ, π/2), θ∈[0,2π]. The linewidth and maximum radius are w=420 nm and rmax=2 μm, respectively. (b) Top-view SEM images of metamaterials with varying thickness from 200 to 3,000 nm at a step of 200 nm. (c) Oblique-view SEM images. To avoid any influence on chiroptical signals from other nanostructures, the intervals between adjacent nanostructures are 10 μm by engineering the detailed geometric parameters of arrays.
9
Figure S2. Circular dichroism of single nanostructures. (a) Schematic of chiroptical
interaction in a single nanostructure by circularly polarize light. (b) Simulated CD of single planar-chiral nanostructures. The spin-dependent chiroptical response is weak for single chiral nanostructures with the operating wavelength from 600 to 1,000 nm. The structural parameters of nanostructures are identical with realistic structures. For clear comparison, the circular dichroism is defined as CD = 2 × ( − )/( + ), where the subscripts indicate the right- and left-handed polarized states of light.
10
Figure S3. C4 symmetry of optical fields on the right-handed nanostructure
illuminated by cylindrical vector vortex beams. (a) Schematic of the vortex beam
with radial polarization. (b and c) Electric field distributions of the right-handed nanostructure under radial polarized vortex beams with topological charges l=±4. (d to
f) Corresponding to (a to c) with azimuthal polarized vortex beams. The optical fields
and Poynting vectors demonstrate C4-symmetry distributions illuminated by cylindrical vector vortex beams with radial or azimuthal polarization.
11
Figure S4. Reflectance and helical dichroism on single right-handed nanostructures with different rotational symmetry. (a to c) Simulated reflectance
on right-handed nanostructures with three- (a), four- (b), and five-fold (c) rotational symmetry, respectively. (d) Calculated HD spectrum of the different right-handed nanostructures, which demonstrates a robust chiroptical response throughout the investigated regime. All the nanostructures have the same structural parameters of Archimedean spiral but different number of spiral lobes.
12
Figure S5. The simulated Poynting vectors map of OAM beams. (a and b) The
rotational Poynting vector distributions on vortex beams with topological charge l=±4, which demonstrates their helical wavefronts. Insets shows the azimuthal phase distributions. White arrows indicate Poynting vectors on the transversal plane. The radial components of Poynting vectors demonstrate the divergency of light beam away from the beam waist. (c and d) Electric field distributions of cross-shaped nanowires under vortex beams with topological charge l=±4. The simulated areas of optical fields (intensity and phase distributions) are 6 μm×6 μm.
13
Figure S6. The simulated discrete OAM spectra for optical vortices interacting with the planar nanostructures. (a to c) The digital spiral spectra of right-handed (a),
left-handed (b) and achiral cross-shaped (c) nanostructures illuminated by vortex beams with topological charge l=±4.
14
Figure S7. Circular dichroism on wavelength-scale nanostructures. (a) Schematic
of a single wavelength-scale right-handed nanostructure. (b) A large array of nanostructures can effectively interact with circularly polarized light to yield an obvious CD signal. (c) A single nanostructure can only modulate a small area of incident light beams, leading to a weak CD signal.
15
Figure S8. Non-chiral interaction between OAM beams and the cross-shaped nanostructure. (a) Schematic of the cross-shaped nanowires illuminated by OAM
beams with topological charge l=±4. (b) Simulated time-averaged OAM flux by OAM beams with opposite topological charge l=±4 at varying propagating distance. The x-y plane (z=0 μm) locates at the top surface of nanowire. The color scale indicates the magnitude of the OAM flux, with red (blue) indicating a positive (negative) value. Scale bars, 1 μm.
16
Figure S9. Numerical simulations of HD spectra on chiral nanostructures with varying thickness. (a) Schematic of right-handed nanospirals with increased thickness
from 0.8 to 2.4 μm at a step of 0.4 μm. (b) Simulated HD spectra of right-handed planar nanostructures with different thickness. The little difference between experiment and simulation in the small thickness is attributed to the inhomogeneous geometrical linewidth of nanostructures fabricated with elliptical focus as deduced from SEM images.
17
Figure S10. Numerical simulations of vortex-assisted chiroptical response on lossy materials. (a) HD spectrum on Au nanowires with different chirality. (b) Helical
dichroism absorption of the corresponding Au nanowires. The geometric parameters of metasurfaces are the linewidth of w=420 nm, thickness of h=200 nm, and rmax=2 μm.
REFERENCES
(1) Zhao, Y. Q.; Edgar, J. S.; Jeffries, G. D. M.; McGloin, D.; Chiu, D. T. Spin-to-Orbital Angular Momentum Conversion in a Strongly Focused Optical Beam.
Phys. Rev. Lett. 2007, 99, 073901.
(2) Chen, L.; Lei, J.; Romero, J. Quantum Digital Spiral Imaging. Light Sci. Appl.
2014, 3, e153.
(3) Menzel, C.; Helgert, C.; Rockstuhl, C.; Kley, E.-B.; Tünnermann, A.; Pertsch, T.; Lederer, F. Asymmetric Transmission of Linearly Polarized Light at Optical Metamaterials. Phys. Rev. Lett. 2010, 104, 253902.
(4) Spektor, G.; Kilbane, D.; Mahro, A.; Frank, B.; Ristok, S.; Gal, L.; Kahl, P.; Podbiel, D.; Mathias, S.; Giessen, H. Revealing the Subfemtosecond Dynamics of Orbital Angular Momentum in Nanoplasmonic Vortices. Science 2017, 355, 1187-1191.
(5) Yang, Y.; Thirunavukkarasu, G.; Babiker, M.; Yuan, J. Orbital-Angular-Momentum Mode Selection by Rotationally Symmetric Superposition of Chiral States with Application to Electron Vortex Beams. Phys. Rev. Lett. 2017, 119, 094802.
(6) Sun, W.; Liu, Y.; Qu, G.; Fan, Y.; Dai, W.; Wang, Y.; Song, Q.; Han, J.; Xiao, S. Lead Halide Perovskite Vortex Microlasers. Nat. Commun. 2020, 11, 1-7.
(7) Cai, Z.; Liu, Y.; Zhang, C.; Xu, J.; Ji, S.; Ni, J.; Li, J.; Hu, Y.; Wu, D.; Chu, J. Continuous Cubic Phase Microplates for Generating High-Quality Airy Beams with Strong Deflection. Opt. Lett. 2017, 42, 2483-2486.
18
Opt. 2002, 4, 7-16.
(9) Bliokh, K. Y.; Dressel, J.; Nori, F. Conservation of the Spin and Orbital Angular Momenta in Electromagnetism. New J. Phys. 2014, 16, 093037.
(10) Fang, Y.; Han, M.; Ge, P.; Guo, Z.; Yu, X.; Deng, Y.; Wu, C.; Gong, Q.; Liu, Y. Photoelectronic Mapping of the Spin–Orbit Interaction of Intense Light Fields.
Nat. Photon. 2020, 1-6.
(11) Gilroy, C.; Hashiyada, S.; Endo, K.; Karimullah, A. S.; Barron, L. D.; Okamoto, H.; Togawa, Y.; Kadodwala, M. Roles of Superchirality and Interference in Chiral Plasmonic Biodetection. J. Phys. Chem. C 2019, 123, 15195-15203.
(12) Poulikakos, L. V.; Gutsche, P.; McPeak, K. M.; Burger, S.; Niegemann, J.; Hafner, C.; Norris, D. J. Optical Chirality Flux as a Useful Far-Field Probe of Chiral Near Fields. ACS Photon. 2016, 3, 1619-1625.
