## Doppler Effect with a Twist

The Doppler shift is a familiar and well-understood effect in acoustics. Radar guns use the same effect to determine the speed of moving vehicles. Applied to a rotating object side-on, however, a linear Doppler effect would register no movement. Using twisted light, whereby photons are imprinted with a given amount of optical angular momentum, **Lavery et al.** (p. 537; see the Perspective by

**Marrucci**) detected rotation with an analogous angular Doppler shift, which may be useful for remote sensing and observational astronomy.

## Abstract

The linear Doppler shift is widely used to infer the velocity of approaching objects, but this shift does not detect rotation. By analyzing the orbital angular momentum of the light scattered from a spinning object, we observed a frequency shift proportional to product of the rotation frequency of the object and the orbital angular momentum of the light. This rotational frequency shift was still present when the angular momentum vector was parallel to the observation direction. The multiplicative enhancement of the frequency shift may have applications for the remote detection of rotating bodies in both terrestrial and astronomical settings.

The spin angular momentum of light is manifested as circular polarization and corresponds to the spin angular momentum of the photon, *ħ* (Planck’s constant divided by 2π). More than 20 years ago, it was recognized that light beams with a helical phase structure described by exp(*i*ℓϕ), where ℓ is an integer and ϕ is the azimuthal coordinate, also carry an orbital angular momentum corresponding to ℓ*ħ* per photon (*1*). Since then, this orbital angular momentum (OAM) has been studied in various contexts such as optical micromanipulation and quantum optics (*2*).

Consideration has been given to the use of OAM in imaging and remote sensing, where the detection of the angular momentum may reveal the structure or potentially the motion of the object (*3*–*7*). When light is scattered from a spinning object, we find that the rotation rate of the object can be measured by analyzing frequency shifts in the OAM of the light. This method of remote sensing has applications in both terrestrial and astronomical arenas.

The Doppler shift is a well-known phenomenon in which the relative velocity *v* between a wave-emitting source and an observer gives a frequency shift Δ*f* of that wave. Such an effect is readily seen for audio waves, where the pitch of the sound changes with the speed of the source. For a light beam, the resulting frequency shift is Δ*f* = *f*_{0}*v/c*, where *f*_{0} is the unshifted frequency and *c* is the speed of light. Less well-known than this linear effect is the rotational, or angular, Doppler effect (*8*–*11*). This frequency shift has also been considered for the scattering of light from atomic (*12*) or macroscopic (*13*) objects rotating around the axis of a helically phased laser beam. For a beam with helical phase fronts, a rotation of angular frequency Ω between the source and observer shifts the frequency by (1)where σ = ±1 for right- and left-handed circularly polarized light and 0 for linearly polarized light, hence (ℓ + σ)*ħ* is the total angular momentum per photon (*14*). All of this previous rotational work has been based on pure OAM states, explicitly rotated using specialist optical elements.

The standard linear Doppler shift applies when the relative motion between source and observer is along the direction of observation. For motion transverse to the direction of observation, a reduced Doppler shift can still be observed in the light scattered at an angle α from the surface normal. For small values of α, this reduced Doppler shift is given by (2)This frequency shift occurs because the form of the scattered light is not determined solely by the laws of reflection. The roughness of the surface means that some of the light normally incident on the surface is scattered at angle α. The observation of this frequency shift in light scattered from a moving object is the basis of speckle (i.e., laser Doppler) velocimetry, and it is used for the remote sensing of the transverse velocity of moving surfaces (*15*) or fluids (*16*) (Fig. 1A).

It is also possible to understand speckle velocimetry in the time domain. In illumination mode, the resulting interference between the two beams at ±α creates straight-line fringes with period Λ = λ/2α (small α, where sin α ≈ α). When a rough surface translates across this fringe pattern, the inhomogeneity of the surface results in a slight modulation in the intensity of the scattered light. The frequency of this modulation is *f*_{mod} = *v*/Λ, which is exactly the same rate as anticipated from the differential Doppler shifts of the two beams given by Eq. 2. The depth of the modulation depends on the period of the fringes relative to the period of the surface roughness, where the depth is maximized when the two periods match. These two complementary explanations of speckle velocimetry have an angular equivalent.

In a helically phased beam, the Poynting vector, and hence the optical momentum, has an azimuthal component at every position within the beam. The angle between the Poynting vector and the beam axis is α = ℓλ/2π*r*, where *r* is the radius from the beam axis (*17*) (Fig. 1B). In the frequency domain, for a helically phased beam illuminating a spinning object, we can see from Eq. 2 that the Doppler frequency shift of the on-axis (ℓ = 0) scattered light is given as (3)Note also in Eq. 3 that there is no dependence on polarization because, unlike the OAM, the polarization is largely unchanged by the process of scattering.

When the illumination comprises two helically phased beams of opposite values of ℓ, their scattering into a common detection mode gives opposite frequency shifts and an intensity modulation of frequency (4)If the common detection mode is not ℓ = 0, then the frequency shifts of the two beams are different, but their frequency difference remains the same. Hence, even if the detection is multimodal, all of the detected modes experience the same modulation frequency.

Within the time domain, the interpretation is that the superposition of two helically phased beams with opposite values of ℓ creates a beam cross section with a modulated intensity of 2ℓ radial petals (*18*). Therefore, the light scattered from the rough surface of a spinning disc will undergo an intensity modulation also given by Eq. 4.

Linear optical systems tend to be reciprocal in that the source and detector can be interchanged. Consequently, we would expect the frequency shift produced by the spinning surface to be observed either in the case of illumination by a beam containing OAM and the on-axis detection of the scattered light, or for on-axis illumination and detection of an OAM component in the scattered light.

For illumination of the spinning object with light of specific OAM modes, a diode laser at 670 nm is coupled to a single-mode fiber, the output of which is collimated and used to illuminate a phase-only spatial light modulator (SLM). The SLM is programmed with a kinoform to produce a superposition of two helically phased beams with opposite signs of ℓ. The phase contrast is adjusted over the SLM cross section such that the radial intensity structure of the diffracted beam is a single annulus corresponding to a *p* = 0 Laguerre-Gaussian mode (*19*). The plane of the SLM is reimaged using an afocal telescope to illuminate the spinning object. Relay mirrors allow the axis of the illuminating beam to be precisely aligned to the rotation axis of the object. The diameter of the beam superposition on the object is about 18 mm, which gives a petal period of about 2 mm for typical values of ℓ = ±18. This petal beam illuminates a metallic surface attached to a plastic rotor, which is driven at speeds ranging from 200 to 500 radians/s. The rough nature of the surface means that irrespective of the modal composition of the illumination light, the light scattered from the surface comprises a very wide range of modes. The modal bandwidth of this scattered light is a function of both the size of the illuminating beam and the range of angles over which the light is scattered or, more important, detected. The bandwidth is usefully approximated in terms of the Fresnel number of the optical system. A lens and a large-area photodiode are used to collect light scattered from the spinning surface. The output of the detector is digitized and Fourier-transformed to give the frequency components of the detected intensity modulation (see supplementary materials).

For the light scattered from the rotating surface when illuminated with a Laguerre-Gaussian superposition of ℓ = ±18 (Fig. 2B), the resulting power spectra were obtained from a data collection period of 1 s. A clearly distinguishable peak was observed at a frequency matching that predicted by Eq. 4. To further test the relationship predicted in Eq. 4, we varied the rotation speed and value of |ℓ| and compared the result to the predicted results (Fig. 2C). The subsidiary peaks at higher frequencies arise from a cross-coupling to different mode indices corresponding to Δℓ = 37, Δℓ = 38, etc. As adjacent peaks correspond to Δℓ = ±1, they are separated in frequency from each other by the rotation speed Ω. Most likely is that this cross-coupling arises from a slight misalignment in the experiment between the rotation and detection axes (*20*, *21*).

The relative frequency shift between the +ve and –ve components of OAM gives rise to relative energy shifts. However, the rotation of the disk does not change the OAM spectrum of the scattered light. This can be understood as a consequence of the fact that the OAM per photon is a discrete or quantized quantity, given by the topological charge of the vortex for the mode. This discrete quantity cannot be changed continuously, of course, and so cannot depend on the rotation frequency of the disk, which can take a continuum of possible values. Equivalently, a mode with an ℓ-fold rotational symmetry has the same symmetry when viewed in a rotating frame and hence the same OAM per photon.

The underlying mechanism introducing this frequency shift can be understood with respect to either the laboratory or the rotating frame. In the laboratory frame, the change in local ray direction, α, between the incident and detected light means that there is an azimuthal reaction force acting on the scattering surface. Doing work against this force is the energy input required to shift the frequency of the light, not dissimilar in origin to the mechanism associated with the rotation of a mode converter (*22*). The +ve and –ve OAM components undergo shifts in frequency (up and down, respectively), which interfere to give the modulation in intensity recorded at the detector. In the rotating frame, the incident beams are themselves seen as rotating and hence are subject to a rotational Doppler shift, with the +ve and –ve OAM components again experiencing frequency shifts up and down, respectively. Scattering centers on the surface radiate both of the frequencies back to the detector, where they again interfere to give the observed modulation in intensity.

The equivalent interpretation in terms of a Doppler shift or patterned projection applies both to OAM illumination and OAM detection. Consequently, it is possible to interchange the laser and detector. In this alternative configuration, the spinning object is illuminated directly with the expanded laser beam. Some of the scattered light is incident on the SLM, which is programmed with an identical kinoform as previously, to couple a superposition of ±ℓ into the single-mode fiber. The SLM and the fiber are now acting as a mode filter to select only the desired superposition from the many modes within the scattered light. The power of light in these desired modes is a small fraction of that illuminating the object, so the light transmitted through the fiber is only of low intensity. We use a photomultiplier to measure the light transmitted through the fiber; the output is Fourier-transformed as in the previous configuration. To enhance the signal, the metal surface was lightly embossed with a pattern of 18-fold rotational symmetry, resulting in a dominant overlap with the modal superposition of ℓ = ±18 (Δℓ = 36).

Figure 3 shows the frequency spectrum of the intensity modulation in the detected light as obtained over a data collection period of 5 min. Again the frequency of this peak can be predicted by Eq. 4 to reveal the rotational speed of the object.

We have shown that an OAM-based analysis of the scattered light makes it possible to infer the rotation speed of a distant object, even though the rotation axis is parallel to the observation direction. The high mode number of the OAM state, and the resulting rotational symmetry, means that the recorded frequency is higher than the rotation frequency itself by a factor of 2ℓ. A similar advantage in using OAM has been noted previously for fixed-angle measurement in both classical (*23*) and quantum (*24*) regimes. Of course, the maximum value of OAM mode that can be used is set by the modal bandwidth of the scattered light. However, increasing the bandwidth also reduces the fraction of the scattered light that falls within the scattered mode. Consequently, the degree of OAM enhancement of the rotation detection is a complicated function of the experimental conditions.

Although the Doppler shift, Doppler velocimetry, and their application to the remote measurement of transverse velocity are well known, our study recognizes that these phenomena have an angular equivalent. An analysis in terms of the OAM gives a clear and intuitive understanding of the angular case. This understanding indicates possible applications in multiple regimes. Two application areas of particular promise are the potential for the remote sensing of turbulence in backscattered light and the possible application to astronomy for the remote detection of rotating bodies.

## Supplementary Materials

www.sciencemag.org/cgi/content/full/341/6145/537/DC1

Materials and Methods

Figs. S1 and S2

Tables S1 and S2