DOI:.38/NNANO.25.86 Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission Amir Arbabi, Yu Horie, Mahmood Bagheri, and Andrei Faraon NATURE NANOTECHNOLOGY www.nature.com/naturenanotechnolog 25 Macmillan Publishers Limited. All rights reserved
DOI:.38/NNANO.25.86 S. ARBITRARY POLARIZATION AND PHASE TRANSFORMATION USING SYMMETRIC AND UNITARY JONES MATRICES Here we show that an arbitrar polarization and phase transformation can alwas be performed using a unitar and smmetric Jones matri. We prove this b determining the unitar and smmetric Jones matri T that maps a given input electric field E in to a desired output electric field E out. For polarization and phase transformations (i.e. no amplitude modification), the Jones matri should be unitar since the transmitted power is equal to the incident power and E out = E in. The general relation between the electric fields of input and output optical waves for normal incidence is epressed as E out = TE in. For a smmetric and unitar Jones matri we have where E in and E in T E in T E in + T E in = E out, (a) T T T E in = E out, (b) are the and components of the electric field of the input light, E out and E out are the and components of the electric field of the output light, T ij (i, j =, ) are the elements of the 2 2 Jones matri, and * represents comple conjugation. In deriving Eqs. a and b, we have used the smmetric properties T = T, and the unitar condition T T + T T =. B multipling Eq. a b T and Eq. b b T we obtain T 2 E in T 2 E in + T T E in T T E in = T E out, (2a) = T E out. (2b) B adding Eqs. 2a and 2b, using the unitar condition T 2 + T 2 =, and taking the comple conjugate of the resultant relation, we find T E out + T E out = E in. (3) Finall, b epressing Eqs. 3 and a in the matri form, we obtain Eout E in E out E in T = T Ein E out. (4) 2 NATURE NANOTECHNOLOGY www.nature.com/naturenanotechnolog 25 Macmillan Publishers Limited. All rights reserved
DOI:.38/NNANO.25.86 SUPPLEMENTARY INFORMATION Therefore, for an given E in and E out, we can find T and T from Eq. 4, and T and T from the smmetr and unitar conditions as T = T, T = ep(2i T )T. (5a) (5b) Thus, we can alwas find a unitar and smmetric Jones matri that transforms an input optical wave E in to an output optical wave E out. S2. REALIZATION OF ANY SYMMETRIC AND UNITARY JONES MATRICES USING A UNI- FORM BIREFRINGENT METASURFACE Here we show that an smmetric and unitar Jones matri can be realized using a uniform birefringent metasurface shown in Fig. 2a in the main tet, if φ, φ, and the angle between one of the principal ais of the metasurface and the ais (θ) could be chosen freel. An smmetric and unitar matri is decomposable in terms of its eigenvectors and eigenvalue matri ( ) as T = V eiφ V T = R(θ) R( θ), (6) e iφ where superscript T represents the matri transpose operation. V is a real unitar matri; therefore, it corresponds to an in-plane geometrical rotation R b an angle that we refer to as θ, and since V T = V, V T represents a rotation b θ. According to Eq. 6, the operation of a metasurface that realizes the Jones matri T can be considered as rotating the electric field of the input wave (E in ) b θ, phase shifting the and components of the rotated E in respectivel b φ and φ, and rotating back the rotated and phase shifted vector b angle θ. Equivalentl, T can be implemented using a metasurface that imposes phase shifts φ and φ to the components of E in along angles θ and 9 +θ, respectivel. Such a metasurface is realized b starting with a metasurface whose principal ais are along and directions and imparts φ and φ phase shifts to and -polarized waves, and rotating it anticlockwise b angle θ. Therefore, an smmetric and unitar Jones matri can be realized using a metasurface if its φ, φ, and in-plane rotation angle (θ) could be chosen freel. NATURE NANOTECHNOLOGY www.nature.com/naturenanotechnolog 3 25 Macmillan Publishers Limited. All rights reserved
DOI:.38/NNANO.25.86 S3. INDEPENDENT WAVEFRONT CONTROL FOR TWO ORTHOGONAL POLARIZATIONS In this section, we derive the necessar condition for the design of a device that imposes two independent phase profiles to two optical waves with orthogonal polarizations. The four elements of the Jones matri T are found uniquel using Eqs. 4 and 5, when the determinant of the matri on the left hand side of Eq. 4 is nonzero. Therefore, a devices that is designed to map E in to E out, converts an optical wave whose polarization is orthogonal to E in to an optical wave polarized orthogonal to E out. For eample, an optical element designed to generate radiall polarized light from polarized input light, will also generate azimuthall polarized light from polarized input light. have In the special case that the determinant of the matri on the left had side of Eq. 4 is zero we E out E in E out E in =, (7) and because T is unitar we have E in = E out ; therefore we find E out = ep(iφ)e in where φ is an arbitrar phase. This special case corresponds to a device that preserves the polarization ellipse of the input light, switches its handedness (helicit), and imposes a phase shift on it. In this case, the T matri is not uniquel determined from Eq. 4, and an additional condition, such as the phase profile for the orthogonal polarization, can be imposed on the operation of the device. Therefore, the device can be designed to realize two different phase profiles for two orthogonal input polarizations. SUPPLEMENTARY VIDEO LEGENDS Supplementar Video Polarization switchable phase hologram. Movie showing the evolution of the image generated b a polarization switchable hologram as the polarization direction (shown b an arrow on the bottom left) of the illumination light is changed. 4 NATURE NANOTECHNOLOGY www.nature.com/naturenanotechnolog 25 Macmillan Publishers Limited. All rights reserved
DOI:.38/NNANO.25.86 SUPPLEMENTARY INFORMATION SUPPLEMENTARY FIGURES µm µm db Scatterd light a-si post Glass substrate z k E i z z - Plane wave incident Supplementar Fig.. Large forward scattering b a single amorphous silicon post. Schematic illustration and finite element simulation results of light scattering b a single 75 nm tall circular amorphous silicon post with a diameter of 5 nm. The simulation results show the logarithmic scale energ densit of the light scattered b the single amorphous silicon post over the z and z planes. The energ densities are normalized to the energ densit of the 95 nm -polarized incident plane wave. NATURE NANOTECHNOLOGY www.nature.com/naturenanotechnolog 5 25 Macmillan Publishers Limited. All rights reserved
DOI:.38/NNANO.25.86 t 2 φ 4 4 (nm) 3 2 (nm) 3 2 2 3 4 2 3 4 D (nm) D (nm).75 t 2 φ.25.5 4 4 (nm) 3 2 (nm) 3 2 2 3 4 D (nm) 2 3 4 D (nm) Supplementar Fig. 2. Phase shifts and intensit transmission coefficients as a function of elliptical post diameters, used to derive data in Fig. 2b-e of the main tet. Intensit transmission coefficients ( t 2 and t 2 ) and the phase of transmission coefficients (φ and φ ) of and -polarized optical waves for the periodic arra of elliptical posts shown in Fig. 2a of the main tet as functions of the post diameters. 6 NATURE NANOTECHNOLOGY www.nature.com/naturenanotechnolog 25 Macmillan Publishers Limited. All rights reserved
DOI:.38/NNANO.25.86 SUPPLEMENTARY INFORMATION a Diffraction limited spot (NA=.6) 2 µm 2 µm Intensit (a.u.).5-2 - 2 (µm) Intensit (a.u.) b Measured focal spot Intensit (a.u.).5-2 - 2 (µm) Intensit (a.u.) Intensit (a.u.) c.8.6.4.2 Measured intensit Air pattern (NA=.58) -2-2 (µm) Supplementar Fig. 3. Diffraction limited focusing b device shown in Fig. 5c. a, Theoretical diffraction limited focal spot (Air disk) for a lens with numerical aperture (NA) of.6 at the operation wavelength of 95 nm. Inset shows the intensit along the dashed line.b, Measured focal spot for the device shown in Fig. 5c when the device is uniforml illuminated with right handed circularl polarized 95 nm light. Inset shows the intensit along the dashed line. c, Measured intensit along the dashed line shown in (b) and its least squares Air pattern fit which has an NA of.58. NATURE NANOTECHNOLOGY www.nature.com/naturenanotechnolog 7 25 Macmillan Publishers Limited. All rights reserved
DOI:.38/NNANO.25.86 t 2 t 2 φ φ.5 D = nm =2 nm.5 85 9 95.5.5 85 9 95 φ.8.6.4 D 45 D =8 nm =2 nm D =5 nm =3 nm.5.5 85 9 95 85 9 95.5.5.5 85 9 95 85 9 95 φ.2.2.4.6.8 φ.8.6.4 Diameter (nm) 35 25 5 65.2.5 D =85 nm =23 nm.5.5.2.4.6.8 φ 85 9 95 85 9 95 Supplementar Fig. 4. Transmission spectra of periodic arras of elliptical posts showing that the operation wavelength does not overlap with resonances. Wavelength dependence of the intensit transmission coefficients ( t 2 and t 2 ) and the phase of transmission coefficients (φ and φ ) of and -polarized optical waves for the periodic arras schematicall shown in Fig. 2a of the main tet. The spectra are shown for a few arras with different (D, ) combinations: ( nm, 2 nm), (8 nm, 2 nm), (5 nm, 3 nm), (85 nm, 23 nm). The corresponding phase shift values and post diameters for these arras are shown on the D and graphs on the right with black smbols. For brevit, onl the spectra for the arras with >D are shown. The transmission and phase spectra for the arras with D > (which are shown with red smbols on the D and graphs) can be obtained b swapping and in the spectra graphs. The desired operation wavelength (λ =95 nm) is shown with dashed red vertical lines in the spectra plots, and it does not overlap with an of the resonances of the periodic arras. 8 NATURE NANOTECHNOLOGY www.nature.com/naturenanotechnolog 25 Macmillan Publishers Limited. All rights reserved
DOI:.38/NNANO.25.86 SUPPLEMENTARY INFORMATION a Laser Polarization controller Fiber collimator lens Device Objective lens Tube lens Polarizer Camera b Laser Polarization controller Fiber collimator lens Device Pinhole Optical power meter Supplementar Fig. 5. Measurement setup. a, Schematic illustration of the measurement setup used for characterization of devices modifing polarization and phase of light. The linear polarizer was inserted into the setup onl during the polarization measurements. b, Schematic drawing of the eperimental setup used for efficienc characterization of the device shown in Fig. 4b. NATURE NANOTECHNOLOGY www.nature.com/naturenanotechnolog 9 25 Macmillan Publishers Limited. All rights reserved