## Abstract

The interaction of light with spinning objects can lead to a frequency shift as a result of the transferred angular momentum. Subwavelength plasmonic nanostructures on an optical surface provide an efficient platform for local light manipulation and thus can be utilized for such momentum exchanges. Here we demonstrate reflective-type plasmonic metasurface $q$-plates based on the geometric Pancharatnam-Berry phase that are capable of changing the spin and orbital angular momentum of light. When these metasurfaces rotate at a constant angular speed, we observe a rotational Doppler frequency shift of the reflected light, which depends on the total angular momentum transfer. The flexibility in the design of the metasurfaces even enables complex reflective phase masks that can transfer different orbital angular momenta, at the same time, to the beam. Our experiments show that such complex metasurface phase masks can be used to obtain spatially variant Doppler shift in the reflected beam profile.

© 2017 Optical Society of America

## 1. INTRODUCTION

The relative movement of an illuminated object or a light emitting source, with respect to a detector, leads to a change in the detected light frequency, which is commonly known as the optical Doppler effect [1]. In addition to the usual translational Doppler effect, used in applications like global positioning systems (GPS) [2,3] or ultrasound blood flow measurements [4], there is also a rotational Doppler effect of light for spinning objects. For the translational Doppler effect, the relative movement of an observer toward a light source causes an increased frequency of the detected wave. As an analogy, the rotational case can be understood in terms of a rotating observer with respect to a circular polarization state of light. The local electric field vector of circularly polarized light is rotating over time with the light’s frequency. If an observer rotates in the opposite direction, a higher effective rotation rate of the electric field vector and, therefore, a higher frequency of light will be observed.

For light matter interactions, the rotational Doppler effect can also be understood in the context of angular momentum transfer of light. In his seminal work, Poynting predicted that circularly polarized light carries an angular momentum, which can be transferred with a light-matter-interaction [5]. Allen experimentally observed such a momentum transfer, when he rotated a needle with circularly polarized microwaves [6]. In analogy to an elastic collision, the transferred angular momentum is accompanied by an energy transfer from or to the light that can be understood as a Doppler frequency shift.

Since optical frequencies exceed mechanical rotation frequencies by several orders of magnitude, the obtained rotational Doppler shift is generally very small and impedes practical applications, for example, in remote metrology for spinning objects. Hence, techniques that can increase the Doppler shift are highly appreciated and could potentially lead to a broad range of applications. It was shown that the exchange of orbital angular momentum of a light beam with a spinning $q$-plate leads to an additional momentum transfer and, therefore, to an increased Doppler shift [7]. Since in principle the orbital angular momentum carried by a beam can become very large, a transfer of such momentum can dramatically increase the frequency shift. Recently, Zhou *et al.* demonstrated that the rotational Doppler shift can be used to measure the orbital angular momentum distribution of light [8].

Here, we study the rotational Doppler effect in conjunction with plasmonic metasurfaces. In contrast to traditional optical elements, metasurfaces can enable local phase and amplitude control on a subwavelength scale [9–14]. Hence, they provide a flexible platform for altering beam propagation and the properties of light. For our study, we use the concept of the geometrical Pancharatnam-Berry phase [15] to encode phase information into the arrangement of plasmonic nanoantennas along a surface. These metasurfaces convert the circular polarization state of incident light into cross polarization [10], which corresponds to a change of the spin angular momentum (SAM) of light. Using this concept, ultrathin metasurface phase masks, which can transfer orbital angular momentum (OAM) to a light beam, were introduced [16–23]. Here, we followed the same concept and encoded an angular phase mask onto a reflective plasmonic metasurface, effectively turning it into a reflective-type $q$-plate [24]. Our experiments demonstrate that for a spinning metasurface, the added OAM for the reflected beam results in an increased Doppler shift that is proportional to the total amount of transferred angular momentum.

Although spinning $q$-plates based on form-birefringent glass slabs have shown the potential for manipulating the Doppler shift [7], plasmonic metasurfaces have the great advantage of easily realizing arbitrary phase profiles on a subwavelength scale, avoiding undesired diffraction effects. Furthermore, the transfer of arbitrary combinations of orbital angular momenta can be achieved, either in transmission or reflection, to study the influence of complex momentum coupling.

Using reflective metasurfaces, instead of non-uniform lenses, enables remote estimation of the angular speed of rotating opaque objects; the light source and the detector can be placed at the same location. Recently, an OAM spectral analyzer based on the rotational Doppler effect was demonstrated with a reflective-type spatial light modulator (SLM) [8]. However, SLMs cannot be used for the remote detection of angular speed of spinning objects with this approach.

## 2. DOPPLER SHIFT BY ANGULAR MOMENTUM TRANSFER

We will start with a brief discussion of the rotational Doppler shift for light interacting with spinning objects. For convenience, we will use circular polarization states throughout our description. Circularly polarized light with a radial symmetric intensity profile can be characterized by the circular polarization orientation $(\sigma =\pm 1)$ and the azimuthal phase number $l$. In the single-photon picture, the polarization corresponds to a spin, which carries a SAM of $\hslash \sigma $, while the angular phase number $l$ leads to an OAM of $\hslash l$. Both angular momenta can be added to a total angular momentum with the associated quantum number $j=\sigma +l$. Note that the SAM is limited to a single unit $\pm \hslash $ per photon, whereas the OAM can reach large multiples of that.

Starting from the energy and angular momentum conservation laws, one can demonstrate that the frequency shift only depends on the change of the total angular momentum number $\mathrm{\Delta}j$ induced by spin-orbit coupling and the rotation frequency. For simplicity, we consider a single-photon interaction on a spinning reflective object with a moment of inertia $I$ (Fig. 1). The angular momentum conservation is then given by

where $\mathrm{\Omega}$ is the angular rotation frequency of the object. Here, the indices 1 and 2 denote the quantities before and after the interaction, respectively. Note that the photon changes direction during reflection. Therefore, ${j}_{2}$ receives a minus sign in the balanced equation, which would not appear for the transmission case. Considering energy conservation, we obtainFor practical purposes, it is important to identify suitable systems that can provide a well-defined total momentum exchange with a photon. For example, a simple mirror is not sufficient, since an incoming beam would only reverse the direction after being reflected resulting in a convention based momentum sign switch. Due to optical isotropy along the surface, a momentum transfer is impossible. However, if the reflective surface can modify the SAM and/or the OAM, the photon should experience the corresponding rotational Doppler frequency shift.

For our demonstration, we use a reflective-type plasmonic metasurface, which consists of gold nanoantennas spaced by a thin ${\mathrm{MgF}}_{2}$ layer in close proximity on a gold mirror (Fig. 2). This material system behaves locally like a half-wave-plate in reflection, where the antenna orientation corresponds to the fast axis [26,27]. The antenna orientation enables a local phase control, which can be interpreted as a Pancharatnam-Berry phase (PBP). Hereby, an antenna rotation over an angle $\varphi $ leads to a local phase shift of $2\varphi $ in the reflected cross-polarized light. We use this concept to realize phase masks that act as $q$-plates for the reflected beam. By rotating the antennas $q$ times along one iteration of the position angle an angular phase gradient of $2q{\sigma}_{1}/2\pi $ is added to the interacting light beam profile, which increases the OAM number by $2q{\sigma}_{1}$. At the same time the PBP surface intrinsically switches the circular polarization state of light and therefore the SAM number. In sum, the reflected beam experiences a total angular momentum number change of

Note that the operation in reflection implies a propagation direction change of light, which corresponds to a coordinate system change. Therefore, one has to invert the sign after applying the momentum number changes to receive the momentum numbers presented in Fig. 2(b). By inserting the momentum number changes of Eq. (4) into Eq. (3), we obtain the Doppler frequency shift of $q$-plates of

## 3. RESULTS ON SPINNING METASURFACES

In our experiments, we use an indirect measurement technique by investigating the beating frequency of two interfering beams with opposite Doppler frequency shifts. For this, a linearly polarized beam is slightly focused on a spinning metasurface. The incoming light can be understood as a superposition of right and left circularly polarized beams, which have frequency shifts with opposite signs [see Eq. (5)]. After the reflection at the metasurface, a polarizing beam splitter projects both circular polarizations (with different frequency shifts) onto a linear polarization state perpendicular to the original incident linear polarization. The superposition of these two linearly polarized states results in a time-dependent intensity beating, determined by the difference frequency of the two beams Doppler shifted beams:

The obtained time-dependent signal from our detector is Fourier transformed to analyze the frequency spectrum. Figure 4 shows the obtained results for a rotation frequency of 13–14 Hz for the metasurface and different encoded topological charges. In the static case (no rotation), we obtain a “flower” like intensity distribution after the beam is reflected at the polarizing beam splitter (PBS) (Fig. 4 top row). The number of $4q$ leaves can be derived by looking at the antenna arrangement on the metasurface.

The PBS only transmits horizontally polarized light, whereas the reflected light is vertically polarized. The polarization conversion at the metasurface can only be achieved for antennas that are not vertically or horizontally orientated. Since the antennas are revolving $q$ times along one circle and each rotation includes two vertical and two horizontal orientations, we obtain $4q$ minima along the azimuthal angle $\varphi $ leading to a $4q$-fold symmetry. Likewise, this pattern can be analytically derived by E-field superposition. Both circular polarizations receive opposite azimuthal phase numbers of $\pm 2q$ after interacting with the metasurface $q$-plate. At the detection plane, this leads to a total intensity distribution of

In the dynamic case, the rotation of the metasurface leads to a rotation of the $4q$-fold beam profile. However, a full rotation of the metasurface does not correspond to a full rotation of the beam profile. This seemingly paradoxical fact arises from the geometric implementation of the phase mask. A full rotation of the $q$-plate leads to $q-1$ antenna turns at a local point. Therefore, the predicted beating frequency [Eq. (6)] has to be proportional to $q-1$. To confirm this prediction, the recorded time traces of the detector are Fourier transformed to obtain the frequency spectrum of the beating signal (Fig. 4 bottom row). The Fourier spectra show several peaks. These additional peaks at multiples of the rotation frequency $F=\mathrm{\Omega}/(2\text{\hspace{0.17em}\hspace{0.17em}}\pi )$ arise from experimental imperfections, as any beam fluctuation due to the rotation leads to a modulation at the rotation frequency. The resulting envelope function shows a “broadening” around the expected peak frequency, which is more dominant for higher topological charges $q$. While mechanical frequency fluctuation will only lead to a peak broadening, other sources of error, like a surface tilting relative to the rotation axis or an irregular motor movement, may cause additional peaks. Aside from those instances, the main peak is found at the position predicted by Eq. (6). For the special case $q=1$, where the antenna array is centrosymmetric, no frequency shift should be observed. For the measurement, we find no dominating frequency component, and the related Fourier spectrum only contains additional peaks due to the measurement errors. For $q=2$ and $q=3$, the main peaks at 4F and 8F are dominant and clearly visible in the Fourier spectra, while for higher $q$, the main peak is strongly broadened. Nevertheless, even for $q=20$, we find pronounced frequency components around $f=76F$ that are clearly visible in the spectrum.

To quantify the obtained results, the ratio of the main peak frequency and the motor frequency, depending on the used topological charge $q$, is plotted for the measurements (Fig. 5). Within the error tolerance of the measurement, the obtained frequency ratios confirm the linear dependence of the Doppler shift on the topological charge $q$, as predicted by Eq. (6).

In the next step, we study the influence of a combination of different topological charges from a more complex metasurface on the time-dependent signal. For that, we design and fabricate a reflective metasurface with a radial dependent phase profile. For this profile, the surface is now divided into a ring pattern with even widths. On these rings, alternating $q$-patterns for $q=2$ and $q=-2$ are encoded [Fig. 6(a)]. Since the resulting phase profile is no longer uniform in the radial direction, the observed static beam profile loses this property as well [Fig. 6(b)]. Now the spatial beam profile after the polarization beam splitter contains multiple 8-fold rings that evolve rather unintuitively with the sample’s rotation. Therefore, small changes in the illumination of the metasurface can result in different overall beam pattern. In our specific measurements, we utilized a beam waist radius of around 80 μm, while the ring width is approximately 22 μm. Hence, only the inner rings are contributing to the reflected beam.

Although the resulting beam shape has a complex structure, the rotational symmetry is predictable. A simple $q$-plate creates a $4q$-fold beam shape, which has ${C}_{|4q|}$ symmetry (see Fig. 4). This symmetry is closely connected to the phase mask’s symmetry, which is ${C}_{|2q|}$ for a $q$-plate. For mixed $q$-plates, the symmetry of the phase mask is determined by the greatest common divisor (GCD) of all appearing topological charges with ${C}_{2\xb7\mathrm{gcd}(\{{q}_{i}\})}$, which is also the minimal rotational symmetry of the beam pattern. However, in some cases the beam symmetry number is doubled, leading in our case to a ${C}_{8}$ symmetry.

Additionally, there is a third rotational symmetry regarding the metasurface structure itself (based on the arrangement of the nanoantennas). This symmetry determines after which rotation the identical beam pattern will reappear. For a $q$-plate metasurface, the intrinsic symmetry group is ${C}_{2(q-1)}$ [in Fig. 2(c); this can be verified for $q=2$]. For mixed $q$-plate metasurfaces, this relation translates to an intrinsic symmetry of ${C}_{2\xb7\mathrm{gcd}(\{{q}_{i}-1\})}$. Hence, for our specific mixed $q$-plate with $q=\{2,-2\}$ the beam pattern reappears after a rotation of 180°.

For the investigation of the dynamic behavior, the time-dependent signal on three different spatial locations [see Fig. 6(c)] was recorded and Fourier transformed. We observe predominant frequency peaks that appear at 4F, 8F, and 12F [Fig. 6(d)]. The obtained spectra can be interpreted by analyzing an ideal superposition of a $q=2$ and a $q=-2$ plate. The total E-field is the sum of the E-fields of individual $q$-plates, which oscillate at $2\xb7|q-1|\xb7\mathrm{\Omega}$. This approach leads to the following temporal intensity change:

## 4. CONCLUSION

In summary, we realized plasmonic metasurfaces, which modify the SAM and OAM of reflected light beams. Conceptually, this behavior is achieved by using a geometrical phase mask to encode an angular phase gradient resulting in an effective $q$-plate. The steady spinning of these metasurfaces shifts the frequency of reflected light for circular polarization states, which can be understood as a rotational Doppler effect. We experimentally determine the frequency shift by measuring a time-dependent interference signal. Our experiments confirm that the Doppler shift is proportional to the total angular momentum transferred to the beam, as well as to the rotational speed. Furthermore, we realize a complex $q$-plate consisting of space variant topological charges. Thereby, the effective superposed E-field components create additional beating frequencies. Due to strong light interactions, plasmonic metasurfaces with encoded phase profiles for transferring orbital angular momentum to a reflected beam might enable new ways to use remote metrology to determine angular speeds of spinning opaque objects. Reflective $q$-plate metasurfaces are well suited for tracking the rotational movement of objects with flat surfaces in a non-tactile way. Objects are often not transparent, and transmission techniques cannot be applied for detecting their rotational speed. Additionally, in reflection, the light source and detection system can be combined, enabling a compact solution for remote detection.

## 5. METHODS

The nanostructures are designed for the wavelength of the used fiber-coupled laser diode source at 780 nm. The efficiency of momentum transfer is calculated with rigorous coupled-wave analysis (RCWA) for a periodic pattern with 300 nm period. For reflected light, we obtain 70% cross polarization conversion, 9% co-polarization conversion and 21% loss. The input parameters assumed are: refraction index of $n(Au)=0.182+4.547\xb7i$ for gold and $n({\mathrm{MgF}}_{2})=1.375$ [30,31].

For the fabricated samples, the nanoantennas are placed on a circular grid with radial and azimuthal distances of 300 nm. Each nanoantenna is aligned under a certain angle to encode the required phase information to obtain the desired OAM. The fabrication process is based on e-beam lithography in conjunction with electron beam vapor deposition of the gold and a subsequent lift-off process.

## Funding

Deutsche Forschungsgemeinschaft (DFG) (ZE953/7-1).

## Acknowledgment

The authors would like to thank Cedrik Meier for providing his electron beam lithography system for the fabrication of the metasurfaces.

## REFERENCES

**1. **C. Doppler, *Ueber das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels: Versuch einer das Bradley’sche Aberrations-Theorem als integrirenden Theil in sich schliessenden allgemeineren Theorie* (Commission bei Borrosch & André, 1842).

**2. **N. J. Cafarelli Jr., “Doppler frequency position fixing method,” U.S. patent 2,968,034 (January 10, 1961).

**3. **S. C. Jasper, “Method of Doppler searching in a digital GPS receiver,” U.S. patent 4,701,934 (October 20, 1987).

**4. **D. L. Franklin, W. Schlegel, and R. F. Rushmer, “Blood flow measured by Doppler frequency shift of back-scattered ultrasound,” Science **134**, 564–565 (1961). [CrossRef]

**5. **J. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. London A **82**, 560–567 (1909). [CrossRef]

**6. **P. Allen, “A radiation torque experiment,” Am. J. Phys. **34**, 1185–1192 (1966). [CrossRef]

**7. **D. Hakobyan and E. Brasselet, “Optical torque reversal and spin-orbit rotational Doppler shift experiments,” Opt. Express **23**, 31230–31239 (2015). [CrossRef]

**8. **H. Zhou, D. Fu, J. Dong, P. Zhang, D. Chen, X. Cai, F. Li, and X. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on rotational Doppler effect,” Light Sci. Appl. **6**, e162512017. [CrossRef]

**9. **A. Pors, M. G. Nielsen, R. L. Eriksen, and S. I. Bozhevolnyi, “Broadband focusing flat mirrors based on plasmonic gradient metasurfaces,” Nano Lett. **13**, 829–834 (2013). [CrossRef]

**10. **L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett. **12**, 5750–5755 (2012). [CrossRef]

**11. **N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science **334**, 333 (2011). [CrossRef]

**12. **X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science **335**, 427 (2012). [CrossRef]

**13. **B. Walther, C. Helgert, C. Rockstuhl, F. Setzpfandt, F. Eilenberger, E.-B. Kley, F. Lederer, A. Tünnermann, and T. Pertsch, “Spatial and spectral light shaping with metamaterials,” Adv. Mater. **24**, 6300–6304 (2012). [CrossRef]

**14. **P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. **100**, 013101 (2012). [CrossRef]

**15. **Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. **27**, 1141–1143 (2002). [CrossRef]

**16. **J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. **13**, 4269–4274 (2013). [CrossRef]

**17. **F. Yue, D. Wen, J. Xin, B. D. Gerardot, J. Li, and X. Chen, “Vector vortex beam generation with a single plasmonic metasurface,” ACS Photon. **3**, 1558–1563 (2016). [CrossRef]

**18. **L. Huang, X. Song, B. Reineke, T. Li, X. Li, J. Liu, S. Zhang, Y. Wang, and T. Zentgraf, “Volumetric generation of optical vortices with metasurfaces,” ACS Photon. **4**, 338–346 (2017). [CrossRef]

**19. **G. Li, M. Kang, S. Chen, S. Zhang, E. Y.-B. Pun, K. W. Cheah, and J. Li, “Spin-enabled plasmonic metasurfaces for manipulating orbital angular momentum of light,” Nano Lett. **13**, 4148–4151 (2013). [CrossRef]

**20. **K. Y. Bliokh, F. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics **9**, 796–808 (2015). [CrossRef]

**21. **F. Cardano and L. Marrucci, “Spin-orbit photonics,” Nat. Photonics **9**, 776–778 (2015). [CrossRef]

**22. **E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. **3**, e167 (2014). [CrossRef]

**23. **Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric meta-reflectarray for broadband linear polarization conversion and optical vortex generation,” Nano Lett. **14**, 1394–1399 (2014). [CrossRef]

**24. **L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. **96**, 163905 (2006). [CrossRef]

**25. **G. Nienhuis, “Doppler effect induced by rotating lenses,” Opt. Commun. **132**, 8–14 (1996). [CrossRef]

**26. **A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Broadband plasmonic half-wave plates in reflection,” Opt. Lett. **38**, 513–515 (2013). [CrossRef]

**27. **G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. **10**, 308–312 (2015). [CrossRef]

**28. **J. Courtial, D. Robertson, K. Dholakia, L. Allen, and M. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. **81**, 4828–4830 (1998). [CrossRef]

**29. **B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate,” Opt. Commun. **31**, 1–3 (1979). [CrossRef]

**30. **A. D. Rakic, A. B. Djurišic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. **37**, 5271–5283 (1998). [CrossRef]

**31. **M. J. Dodge, “Refractive properties of magnesium fluoride,” Appl. Opt. **23**, 1980–1985 (1984). [CrossRef]