Letter pubs.acs.org/NanoLett
Dispersionless Phase Discontinuities for Controlling Light Propagation Lingling Huang,†,‡,⊥ Xianzhong Chen,†,⊥ Holger Mühlenbernd,§,⊥ Guixin Li,∥ Benfeng Bai,‡ Qiaofeng Tan,‡ Guofan Jin,‡ Thomas Zentgraf,*,§ and Shuang Zhang*,† †
School of Physics & Astronomy, University of Birmingham, Birmingham, B15 2TT, United Kingdom State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instruments, Tsinghua University, Beijing 100084, China § Department of Physics, University of Paderborn, Warburger Straße 100, D-33098 Paderborn, Germany ∥ Department of Physics, Hong Kong Baptist University, Hong Kong ‡
S Supporting Information *
ABSTRACT: Ultrathin metasurfaces consisting of a monolayer of subwavelength plasmonic resonators are capable of generating local abrupt phase changes and can be used for controlling the wavefront of electromagnetic waves. The phase change occurs for transmitted or reflected wave components whose polarization is orthogonal to that of a linearly polarized (LP) incident wave. As the phase shift relies on the resonant features of the plasmonic structures, it is in general wavelength-dependent. Here, we investigate the interaction of circularly polarized (CP) light at an interface composed of a dipole antenna array to create spatially varying abrupt phase discontinuities. The phase discontinuity is dispersionless, that is, it solely depends on the orientation of dipole antennas, but not their spectral response and the wavelength of incident light. By arranging the antennas in an array with a constant phase gradient along the interface, the phenomenon of broadband anomalous refraction is observed ranging from visible to near-infrared wavelengths. We further design and experimentally demonstrate an ultrathin phase gradient interface to generate a broadband optical vortex beam based on the above principle. KEYWORDS: Metamaterials, plasmonics, phase discontinuities, refraction, vortex beam
T
antennas with carefully designed geometries. The metasurface gives rise to anomalous refraction when illuminated by linearly polarized light and provides optical functionalities such as a phase modulator, beam steering, and a computer-generated hologram.9,10 A generalized version of Snell’s law encapsulating the gradient of the abrupt phase change at the metasurface was put forward to describe the refraction mechanism. Such metasurfaces were demonstrated in the mid-IR9 and nearIR10 regions by using V-shape gold antennas and extended to out-of-plane refraction with conical mounting incident condition.11 In practice, the parameters of the antennas need to be carefully engineered to cover the whole [0, 2π] phase range while maintaining uniform scattering amplitude. However, as the phase discontinuities rely on a delicate design of two plasmonic resonances along orthogonal directions,9−11 both the phase shift and the scattering amplitude are intrinsically wavelength-dependent. Away from the designed
he propagation of light is usually manipulated by phase control using optical elements with carefully designed geometries or a refractive index profile that introduces nonuniform phase accumulation.1 When light traverses across the boundary between two media, the refraction of light is determined by the conservation of the momentum along the interface. For homogeneous interfaces, the phase profile along the interface is continuous and linear; therefore, the light trajectory obeys Snell’s law. However, for patterned interfaces, such as gratings, the phase shift undergoes a periodic modulation, leading to scattering or diffraction.2 With the development of metamaterials that provide an unprecedented degree of freedom through tailoring the structure of the metaatoms, unusual phenomenon of light propagation at the interface may occur, such as negative refraction3,4 and cloaking.5−8 Recently, new types of structured plasmonic interfaces that introduce an abrupt phase discontinuity for shaping the phase front of electromagnetic waves have been demonstrated. They represent a promising alternative to control the bending of light at the interface. Such a metasurface consists of an array of metal © 2012 American Chemical Society
Received: August 14, 2012 Revised: October 4, 2012 Published: October 12, 2012 5750
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polarized (LCP) incident light. This phase change can be understood in the context of Pancharatnam−Berry (PB) phase that is acquired when the polarization state of light is changed.15,16 Specifically, the phase difference between the opposite circularly polarized light scattered by two dipole antennas of different orientations φ1 and φ2 can be viewed as half of the solid angle enclosed between two paths on the Poincare sphere, σ → L(φ1)→ σ−, and σ → L(φ2)→ σ−, where L(φi) represents the linear polarization states on the equator of the Poincare sphere.15 Note that the sign of the phase change is dependent on the polarization of the incident beam; that is, if the circular polarizations of the incident and transmitted beams are both reversed the sign of the phase change is flipped. On the other hand, the conversion efficiency to the opposite polarization is only determined by the geometry of the dipole antenna, but not its orientation. That is, the scattering amplitudes of light of the opposite helicity automatically satisfy the equal-amplitude condition, with the highest efficiency occurring at the peak of the resonance, and decreases when the wavelength of light is away from the resonance frequency. As the phase discontinuity does not depend on the dispersion property of the dipole antenna, the proposed structure can operate for a wide range of wavelengths with dispersion-less phase discontinuity. Hence, the proposed phase control scheme represents a robust and facile approach for controlling the phase and wavefront. By arranging the nanorod structures in an array with a lattice constant of S, and a constant gradient of the orientation angle φ along the interface, the phenomenon of broadband anomalous refraction is expected (Figure 1c,d). At relatively small incident and observation angles, an incident beam with a circularly polarization σ is primarily refracted into an ordinary refracted beam with the same polarization σ governed by the conventional Snell’s Law, and an anomalous refracted beam with the opposite circular polarization −σ, with the refraction angle determined by the gradient of the abrupt phase change (see the Supporting Information for details). Correspondingly, the generalized Snell’s law for anomalous refraction is expressed as,9
wavelength, the phase gradient and uniformity of the scattering amplitude of each antenna cannot be maintained, which may lead to the emergence of other diffraction orders. In this Letter, we demonstrate a plasmonic metasurface for generating an abrupt phase change and controlling the wavefront for circularly polarized light at the visible and nearinfrared wavelengths. The metasurface consists of an array of metallic nanorods with the same geometry but spatially varying orientations (Figure 1a). When a beam of circularly polarized
Figure 1. Images and schematic illustrations of a refract dipole array. (a) Schematic illustration of a representative dipole array considered as model in our calculations. A rectangular Cartesian coordinate system Oxyz is established so that the z-axis is parallel to the normal of the surface and the structure is periodically modulated in the x-direction and repeated in y-direction (with one unit cell highlighted in red color). θi and θt are the incident and transmission angles, respectively. (b) Excitation of dipole moment when illuminating one nanorod with the azimuth angle φ along the x-axis. (c−d) Schematic illustration of normal and anomalous refraction by dipole arrays when illuminated with σ and −σ polarized CP, respectively.
(CP) light is incident onto a dipole antenna, the scattered wave is partially converted into the opposite handedness of CP light with a phase change determined solely by the orientation of the dipole.12−14 At relatively small incident and observation angles, the scattered field can be approximated as (see Supporting Information), E Rad ≈
nt sin θt − ni sin θi =
(cos θ + 1)(cos ξ + 1) −σ +σi2φ⎤ Eu e ⎥ ⎦ 8
(2)
Interestingly, as the sign of the abrupt phase change is reversible by changing the handedness of the incident light, polarization-dependent anomalous refraction is expected, which exhibits switchable light bending properties. For a metasurface operating not in the deep subwavelength regime, that is, if the lattice constant S is not much less than the wavelength of light, anomalous diffraction may also occur, which is governed by a generalized diffraction equation that takes into account the contribution from the phase gradient:
cαek 2eikr ⎡ cos θ cos ξ + 1 σ Eu ⎢ 4 2 πr ⎣ 4 +
λ0 d Φ λ dφ =σ 0 2π dx π dx
(1)
where r is the distance from the antenna, k is the wave vector, αe is the dipole moment of the antenna, θ and ξ are, respectively, the incident and observation angle, and E±σ u = (cos ξ ex ± σi ey − sin ξ ez). Equation 1 shows that, within a certain range of incident angle around the surface normal, a circularly polarized beam is primarily scattered into waves of the same polarization as that of the incident beam without phase change, and waves of the opposite circular polarization with a phase change Φ = 2σφ twice the angle formed between the dipole and the x-axis, where σ = ± 1 correspond to the helicity of right- (RCP) and left-circularly
nt sin θt − ni sin θi = m
λ0 λ dφ +σ 0 S π dx
(3)
where m is the diffraction order. Equation 3 shows that the presence of a helicity switchable surface phase gradient offers extra flexibility in controlling light using diffractive optics. The proposed phase discontinuity and anomalous refraction are verified by a full wave numerical simulation. In the simulation, the orientation of a dipole antenna is linearly varied along the x-direction with a step size of π/8 to generate a constant phase gradient. The unit cell consists of eight gold rods, with a rod-to-rod spacing S of 400 nm in both x- and y5751
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directions, and a periodicity of 3.2 μm in the x-direction. The individual nanorod has a width of 50 nm, a length of 200 nm, and a thickness of 40 nm, which give rise to a plasmonic resonance around 970 nm along the rod and a resonance around 730 nm perpendicular to the rod. Full 3D finitedifference time-domain calculations by CST Microwave Studio were performed to model the reflection and refraction between air and glass substrate with a CP light incident onto the plasmonic interface located in the plane at z = 0 (Figure 2).
LCP, the direction of the refracted beam is switched to the other side of the surface normal relative to that of the RCP input, as a result of the reversed gradient of the phase discontinuity at the plasmonic interface (Figure 2d−f). By fixing the incident wavelength at λ0 = 810 nm, the electric field distributions at three different oblique incident angles of 5°, 15°, and 30° are calculated in Figure 2g−i, which further show robust performance of the metasurface in a broad incident angle range. Following the design of the plasmonic surface in the simulation, we fabricated the dipole antenna array on an ITO coated glass substrate with a standard electron-beam lithography and lift-off process (Figure 3a). The fabricated plasmonic surfaces with constant phase gradients are characterized by a continuously tunable laser source (Fianium-SC400-PP) for a number of wavelengths between 670 and 1100 nm. A quarter wave plate and a polarizer were inserted both before and after the sample to generate the incident CP beam and select the opposite handedness CP light for the transmission, respectively (detailed information is presented in the Supporting Information). The measurement results together with the theoretical predictions are summarized in Figure 3. We start with the investigation of the dependence of the observation angle on the incident angle for a fixed wavelength λ0 = 810 nm for RCP incidence and LCP detection (Figure 3b). There exist two refraction curves, corresponding to the ordinary refraction and anomalous refraction, respectively. The measured angles are in perfect agreement with the theoretical prediction. The anomalous refracted beam lies above the ordinary one, as the gradient of the abrupt phase change of the sample for this polarization combination is positive. As the ordinary refracted beam is largely filtered out in the measurement, the intensity of the detected normal refraction is much weaker than the anomalous one for the incident angle less than 60°. Thus, for this measurement scheme (RCP incidence/LCP detection), the anomalous refraction dominates over the normal refraction within a broad range of incident angle. As the wavelength of light in the substrate (glass) is not much greater than the lattice constant of the plasmonic antennas (400 nm), there exist both ordinary and anomalous diffractions at large incident angles. As shown by Figure 3b, the m = ±1 ordinary diffraction order starts to appear when the incident angle reaches θi = ±35°, consistent with a simple calculation based on the diffraction equation. On the other hand, the anomalous diffraction orders appear at different angles on the two sides of the surface normal, with θi = 18° on one side and θi = −56° on the other side. This asymmetry in anomalous diffraction orders agrees well with the generalized diffraction eq 3. Interestingly, when the incident and detection polarizations are interchanged, the anomalous refraction angle shifts below the regular refraction angle, as shown by Figure 3c. This shift is due to the change of the sign in the phase gradient when the polarizations are reversed. In addition, the incident angles where the anomalous diffractions start to appear are also shifted from 18° to 56° on one side, and from −56° to −18° on the other side. Note that, in the experiment, light is focused by an objective onto the sample. Due to the distance between the objective and the sample, the incident angle is limited between −50° and 60°. In addition, the substrate (glass) has an index of 1.5, giving rise to a total reflection angle around 41.8°. Any refraction angle greater than that angle cannot be transmitted through the back
Figure 2. Phase distributions for both incidence and refraction by full wave simulation using CST. (a−c) Phase distributions at three different wavelengths of λ0 = 670 nm, 810 nm, and 1020 nm, respectively, at normal incidence of RCP from air to glass substrate. We plot the incident phases and anomalous refraction phases of opposite handedness CP in the two half spaces, respectively. (d−f) Phase distributions by change the handedness of incident wave to LCP at normal incidence, corresponding to a−c. (g−i) phase distributions by fixing the incident wavelength, but changing the oblique incidence angles with θ = 5°, 15°, and 30°, respectively.
The simulations in Figure 2a−c were first carried out for normal incidence of RCP light at three different wavelengths of 670 nm, 810 nm, and 1020 nm, respectively. The constant phase fronts unambiguously demonstrate the presence of anomalous refraction. It should be noted that the field pattern at near field would vary for cutting planes at different y locations as the surface is inhomogeneous. However, the farfield pattern would not be affected since the period of the plasmonic structures is subwavelength along the y-direction. By changing the helicity of the incident circularly polarized light to 5752
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Figure 3. Analytical dipole radiation theoretical prediction and experimental verification of the refraction behavior of dipole arrays proposed in this paper. All of the results were obtained with LCP incidence and RCP output from air to glass substrate. (a) Scanning electron micrograph (SEM) of gold nanorods fabricated on an ITO coated glass substrate. (b,c) Investigation of reflected angle versus incident angle by fixing the incident wavelength λ0 = 810 nm when illuminated with σ = 1 and σ = −1 polarized CP, respectively. The marks show the experimental results of the refraction angle. The m corresponds to the symbol in eq 3, which indicates the diffraction order of these curves at large incident angles. (d,e) Investigation of reflected angle versus broadband wavelength range by fixing incident angle with θ = 0° and 30°, respectively.
at shorter wavelengths, ordinary and anomalous diffraction orders appear due to the fact that the lattice constant is comparable to the wavelength of light in glass. Based on the dispersionless phase gradient at the interface, we design and experimentally demonstrate an ultrathin metasurface to generate an optical vortex beam over a broad wavelength range. Such a plasmonic interface is capable of generating a helical phase front carrying orbital angular momentum (OAM) upon illumination by normally incident circularly polarized light with a Gaussian profile. Such a beam is
surface of the substrate and, consequently, cannot be detected. Thus, due to these restrictions, only the zeroth order normal and anomalous refraction are measured. To show the broadband operation of the metasurface, we measured refraction angles for both ordinary and anomalous refractions from 670 to 1080 nm for a beam with LCP incidence and RCP detection at two incident angles: 0° and 30° (Figure 3a−c). Again, the measured angles are in very good agreement with the theoretical predictions. At longer wavelengths, there only exists a single anomalous refraction, whereas 5753
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over a negligible propagation distance, in contrast to conventional methods. In the experiment, we measured the evolution of the vortex beam for different incident wavelengths and clearly observe the characteristic beam profile for all wavelengths (Figure 4b−e): an annular intensity distribution in the cross section, and a characteristic dark spot with zero intensity in the center. In summary, we experimentally realized a plasmonic metamaterial surface that exhibits switchable anomalous refraction for circularly polarized waves. As the abrupt phase change is independent of the resonance feature of the constituent plasmonic antennas, the metamaterial surface exhibits broadband operation with dispersionless phase discontinuities. Here the phase gradient is studied in the frame of generalized refraction, in contrast to previous works where it was treated as a diffraction order.22,23 More importantly, we show the applicability of the scheme to generalized refraction by investigating the effect at various incident angles and different wavelengths. We also experimentally showed the switchable anomalous refraction when the polarization of the incident wave is reversed. The approach reported herein demonstrates that the dipole arrays with welldesigned arbitrary 2D azimuthally oriented angle distribution can be generalized to various applications for ultrathin, robust and switchable optical devices in a broadband wavelength range without an extra burden on the fabrication. The metamaterial surface demonstrated here may serve as a platform for investigating novel optical phenomena arising from unconventional refraction, diffraction, and spin-state related effects.
known as the vortex beam, characterized by an exp(ilφ) azimuthal phase dependence, that is, the orbital angular momentum in the propagation direction has the discrete value of lℏ per photon, where l is the topological charge and can be integer or noninteger.17,18 These peculiar states of light are commonly generated by using a spiral phase plate or a computer-generated hologram and can be used as an optical tweezer, as well as for investigating quantum information and spin-related phenomena.19−21 As shown by Figure 4a, the plasmonic interface was created by arranging the dipoles with spatially varying orientation
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ASSOCIATED CONTENT
S Supporting Information *
Detailed derivation of phase discontinuity for circularly polarized light at the plasmonic interface, numerical method for calculating the field of anomalous refraction, and the information of the experimental setup. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected];
[email protected]. Author Contributions ⊥
These authors contributed equally to this work.
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work is partly supported by the Engineering and Physical Sciences Council of the United Kingdom. T.Z. and S.Z. acknowledge the financial support by the European Commission under the Marie Curie Career Integration Program. B.B. and L.H. acknowledge the support by the National Natural Science Foundation of China (Projects Nos. 11004119 and No. 61161130005). L.H. acknowledges the Chinese Scholarship Council (CSC, No. 2011621202) for financial support. The authors thank Professor R.E. Palmer for use of the Nanoscale Physics Research Laboratory Cleanroom and Dr. A.P.G. Robinson for useful conversations. The Oxford Instruments PlasmaPro NGP80 Inductively Coupled Plasma etching system used in this research was obtained through the Birmingham Science City project "Creating and Characterising Next Generation Advanced Materials" supported by Advantage
Figure 4. (a) Scanning electron microscopy image of the dipole array which was designed for generating an optical vortex beam. (b−e) Measured intensity distribution of the vortex beam patterns for different wavelengths from 670 to 1100 nm. Note that the shade of color is only a guide to the eye for visualizing the different wavelengths.
angles that rotate 180° when the location of the dipoles rotate by 360° around the singular point. Mathematically, the rotation angle φ is given by tan 2φ = (x/y) for y > 0 and tan 2(φ − (π/ 2)) = (x/y) for y < 0. The interface introduces a spiral-like phase shift with respect to the planar wavefront of the incident light, creating a vortex beam with topological charge l = 1. Because the azimuthal phase profile is abruptly introduced through the interaction with the dipole plane, the screw-like phase profile characteristic of the vortex beam can be created 5754
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West Midlands (AWM) and partially funded by the European Regional Development Fund (ERDF).
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REFERENCES
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