Experimental detection of transverse particle movement with structured light

Abstract

One procedure widely used to detect the velocity of a moving object is by using the Doppler effect. This is the perceived change in frequency of a wave caused by the relative motion between the emitter and the detector, or between the detector and a reflecting target. The relative movement, in turn, generates a time-varying phase which translates into the detected frequency shift. The classical longitudinal Doppler effect is sensitive only to the velocity of the target along the line-of-sight between the emitter and the detector (longitudinal velocity), since any transverse velocity generates no frequency shift. This makes the transverse velocity undetectable in the classical scheme. Although there exists a relativistic transverse Doppler effect, it gives values that are too small for the typical velocities involved in most laser remote sensing applications. Here we experimentally demonstrate a novel way to detect transverse velocities. The key concept is the use of structured light beams. These beams are unique in the sense that their phases can be engineered such that each point in its transverse plane has an associated phase value. When a particle moves across the beam, the reflected light will carry information about the particle's movement through the variation of the phase of the light that reaches the detector, producing a frequency shift associated with the movement of the particle in the transverse plane.

Introduction

The classical Doppler effect, i.e., the longitudinal Doppler effect, is a key ingredient in myriad of laser remote sensing systems, which are widely used to monitor the location and velocity of moving targets¹. In a monostatic standard laser remote system, where the transmitter and the receiver are at the same location, the electromagnetic signal that is incident on the moving target is usually characterized as a Gaussian beam which does not show any appreciable phase dependence in the transverse spatial coordinates. The beam reflected by the target will have a time-varying phase given by Φ(r, t) = 2kz(t), where k = 2πf/c is the wavenumber, f is the frequency of the incident light beam, c is the velocity of light in vacuum and z(t) is the time-dependent relative displacement along z between the emitter and the target. If the target is moving with velocity v, the reflected signal will show an optical frequency shift Δf = 2|v|cosθ/λ, where θ is the angle between the velocity of the target and the direction of propagation of the light beam. Therefore, the classical Doppler effect does not provide information about the components of the velocity perpendicular to the direction of propagation of the light beam (θ = 90°).

To detect the full vector velocity, including transverse components, one can perform many Doppler measurement along the line of sight for a large set of directions². Moving Doppler instruments can map the velocity field over large areas by alternating the pointing direction or by scanning the beam during the measurement. Various velocity retrieval techniques have been developed to estimate 2D and 3D vector fields from Doppler longitudinal data. Algorithms range from computationally intensive variational data assimilation techniques to simpler and faster methods based on volume velocity processing. In general, however, all these algorithms suffer poor spatial and temporal resolution and tend to lose local information about the velocity field due to the averaging involved. Moreover, these schemes requires fast mechanical realignment of the direction of propagation of the laser beam, which render its implementation more complicated.

Relativity theory shows that the Doppler effect is indeed sensitive to transverse velocities as well - the relativistic transverse Doppler effect³. Unfortunately, it yields relative frequency shifts of the order of ~v²/c², which renders the sought-after frequency shifts staggeringly small in all applications of interest in current laser radar systems. Moreover, it can not distinguish between different directions of movement in the transverse plane, giving all of them the same frequency shift for a given transverse velocity.

Here, we demonstrate experimentally a method⁴ for directly measuring the transverse velocity component of a moving target. The method we present here can be added to the toolbox of currently used detection methods based on the Doppler effect. The velocity and position measure of a moving target can be fully measured in a more straightforward manner, with the help of a fixed single laser beam. The key idea is the use of a structured light beam where each point in its transverse plane is associated with a particular value of the phase of the field. The light reflected back from a target located at a specific position will carry information about the phase at that specific location. When the particle moves across the beam, it reflects the position-dependent varying phase imprinted on the laser beam, producing a signal with a time-dependent phase at the receiver. This can also be thought of as a Doppler shift.

Along these lines, a scheme based on a rotating off-axis aperture was used to elucidate experimentally the orbital angular momentum discrete spectrum of an arbitrary light signal⁵. In this experiment, a small detector which moves across the beam captures the local electric field at the detector location, revealing information about the phase of the beam at that location. The movement of the detector translates such phase gradient into a Doppler frequency shift. In contrast to this, the so-called rotational Doppler effect^6,7 is related to a time-varying global phase change. It is generated when a light beam with orbital angular momentum traverses a rotating element which introduces a time varying global phase. The rotating element can be a Dove prism⁷, or an ensemble of atoms put previously into rotation by another light beam with orbital angular momentum⁸.

Currently available technology offers different possibilities to efficiently generate arbitrary spatially-shaped light beams⁹. Appropriately designed spiral phase plates¹⁰, computer-generated holograms^11,12 and q-plates¹³ play an outstanding role and can be used to produce the required phase distribution, especially for generating light beams embedded with orbital angular momentum. A suitable combination of astigmatic optical elements can also be used to generate light with different types of phase profiles¹⁴. Spatial light modulators have been increasingly useful in the last few years. They allow on-demand modulation of the phase of the light: one can generate and modify complex spatial phase and amplitude light patterns in a prompt and efficient manner. In our experiments we make use of a spatial light modulator.

Results

A modified Mach-Zehnder interferometer, shown in Fig. 1 (see also Methods section) is used to demonstrate the feasibility and usefulness of the method proposed. The continuous wave (CW) Helium-Neon laser light source (wavelength λ = 633 nm, power P ~ 15 mW), is spatially filtered and collimated to produce a beam with a Gaussian intensity profile. The first beam splitter (BS) separates the beam into a reference beam and a signal beam. The reference beam will be needed to extract the sought-after phase information from the signal beam. The signal beam acquires the required phase profile after reflection from the Spatial Light Modulator (SLM), so that the electric field amplitude of the signal beam writes

where r_⊥ = (x, y) are transverse coordinates, I(r_⊥) is intensity of the beam and Φ(r_⊥) is the spatially-varying phase that has been imprinted onto the beam by the SLM. This spatially-varying phase is the key tool that allows retrieval of the transverse component of the velocity of a particle. Even though arbitrary phase profiles can be generated, prior information about the specific characteristics of the movement can make the use of certain phase profiles more convenient. For instance, a rectilinear movement calls for the use of linear phase gradients, while rotational movements might be better analyzed by azimuthally-varying phase gradients.

Figure 1

The experimental set-up.

A beam from a He-Ne laser (λ = 632.8 nm) is collimated by a telescope T1 and divided by a non-polarizing beam splitter (BS) into a reference beam (green line) and a probe beam. The probe beam impinges on a Spatial light Modulator (SLM) wherein a chosen phase profile is imprinted with a computer generated hologram. The beam (blue line) now has a structured phase profile after the SLM. The structured light is then made to shine onto a Digital Micromirror Device (DMD). The DMD is controlled to mimic a moving particle by setting which mirrors in specific positions are in the ‘ON’ state at a particular time. In our experiments, we turn a lump of 7 × 14 adjacent mirrors to the ‘ON’ state. This translates to a reflecting microcircle with a radius of 35 μm. We vary the speed of the movement of the particle by changing the time interval to switch to the next set of mirrors. Light reflected by the particle (orange line) is made to interfere with the reference beam at the photodetector (PD) that is attached to a digital oscilloscope (DO). The SLM, the DO and the DMD are all connected to a computer for control and for faster acquisition and analysis of data. The power ratio between the reference and the probe beam is adjusted for maximum interference fringe visibility at the PD.

The structured light beam illuminates a scatterer that reflects back the signal beam with a phase that depends on the specific location of the scatterer. In order to mimic different types of movements, we use a Digital Micromirror Device (DMD). A 35 μm radius disk-like particle is made of an array of 7 × 14 micromirrors. Each micromirror can be switched on and off with the use of a software provided by the manufacturer. By controlling which specific mirrors are in the ON or OFF states and the timing between these states, we can simulate different types of physical trajectories and velocities of particles. This is equivalent to having a reflecting particle that is moving in the transverse plane. At each position where the particle would be located, light is reflected back to the detector, while no light is reflected elsewhere. This system is very convenient to demonstrate the feasibility of the scheme put forward here. It allows simulating different types of movements with great control of the experimentally relevant parameters such as the specific trajectory and velocity.

We retrieve the time-varying phase of the reflected signal beam by observing the time-varying intensity modulation of the interference between the reference and the signal beams. A typical record is shown in Figs. 2(a) and (b), where a signal beam of the form E(ρ, φ, z) = I^1/2(ρ) exp(ikz + imφ − i2π ft) illuminates a particle which follows a uniform circular movement. Here ρ is the radial coordinate in cylindrical coordinates, φ is the azimuthal angle and m is the winding number of the beam¹⁵, the number of 2π jumps of the phase as one goes around φ.

Figure 2

Signals.

Raw signals detected by the photodetector as acquired by the oscilloscope when the particle moving in a circular motion of ω = 16.36 s⁻¹, is being illuminated by a beam with a helical phase Φ = mφ with topological charge (a) m = 1 and (b) m = 4. Their respective power spectra (c) and (d) were obtained with an FFT algorithm. The peaks in (c) and (d) correspond to the Doppler frequency shifts of mω/(2π) = 2.60 Hz and 10.41 Hz, respectively. See text for further details.

Ideally, the structured optical beam is designed so that the movement of the scatterer under investigation takes place in a region where I₀(r_⊥) is approximately constant, so that only the spatially-varying induced phase differences produce time-varying intensity modulations at the receiver side. Finally, one obtains frequency spectra similar to Figs. 2(c) and (d) after detection, filtering and postprocessing of the signal detected to remove noise and unwanted signals. The spectra give the Doppler frequency shift associated with the velocity of the scatterer.

For a given phase profile, the Doppler frequency shift due to the transverse velocity of the scatterer writes⁴

where v is the transverse velocity of the scatterer and ∇_⊥Φ is the transverse gradient of the phase of the signal beam. Notice that Eq. (2) allows retrieval not only of the magnitude of the transverse velocity |v|, but its direction in the transverse plane as well, since one can always produce a phase profile with a gradient along a selected direction. Fig. 3 shows results for the case of a particle moving with constant rectilinear velocity v shined by a signal beam with a uniform phase gradient profile (Φ(x) = γx). This uniform gradient phase profile, generated by the SLM, tilts the incident Gaussian beam into different angles¹⁶. The Doppler shift expected from Eq. (2) is Δf = γv/(2π), which shows a linear dependence on both velocity and phase gradient of the light beam. Fig. 3(a) shows the dependence of the frequency shift on the velocity of the particle for a constant phase gradient γ = 17.92 mm⁻¹ and Fig. 3(b) shows the dependence of the frequency shift for different phase gradients, for a particle that moves with velocity v = 4.68 mm/s.

Figure 3

Detected frequency shifts when the target moves in a rectilinear path.

(a) The target is set to move at different rectilinear velocities when illuminated by a beam with a linear phase gradient of γ = 17.92 mm⁻¹. (b) The target moves under the illumination of a beam with different linear phase gradients γ at a constant linear velocity of v = 4.68 mm/s.

Fig. 4 shows the case of a particle moving in a circular path with a constant angular velocity ω. In this case, the most convenient phase profile to retrieve the value of the angular velocity is the one corresponding to a Laguerre-Gauss beam with winding number m. The phase profile given by Φ(φ) = mφ has an expected Doppler shift of Δf = mω/(2π) from Eq. (2). The Doppler frequency shift shows a linear dependence on the angular velocity and the winding number m. Fig. 4(a) shows the linear dependence of the frequency shift on the angular velocity of the target for m = 3 and Fig. 4(b) shows the dependence on m for a target that moves with angular velocity ω = 16.36 s⁻¹. In general, one can detect arbitrary transverse velocities by engineering phase profiles. Figs. 3 and 4 are examples of how a choice of the appropriate phase profile for a particular type of movement can give simple relationships between the velocity and the Doppler frequency shift.

Figure 4

Detected frequency shifts when the target moves in a circular path.

(a) The target is set to move at different circular velocities when illuminated by a beam with a helical phase of Φ = 3φ, corresponding to m = 3. (b) The target moves at a constant circular velocity of ω = 16.36 s⁻¹. The particle is illuminated with a phase gradient Φ = mφ, where m is the winding number.

Discussion

Laser Doppler instruments are widely accepted by the remote sensing community for the study of flow and particle's velocity due to its high accuracy, non-intrusiveness, directional sensitivity and high spatial and temporal resolution. However, the classical Doppler principle only allows direct estimation of velocities along the line of sight. This research has arisen out of the need to expand the capabilities of current systems to accommodate a larger variety of applications which may benefit from information about the transverse component of the velocity not measured by classical Doppler methods.

Along these lines, the scheme demonstrated here might be added to current laser radar systems to expand their functionalities, allowing the detection of both longitudinal (v_||) and transverse components (v_⊥) of the velocity without the need to introduce fast changes of the pointing direction of laser beams or use several light beams. In this scenario, a properly engineered structured light beam illuminates a moving target and the Doppler shift of the back reflected light is analyzed. Doppler signals will be generated in two different frequency bands. On one hand, the longitudinal component of the velocity will generate frequency shifts ~v_||/λ. On the other hand, the transverse component will produce frequency shifts in a frequency band determined by the phase gradient of the light. For instance, a particle moving in a circular path with radius R and constant angular velocity will have a frequency shift of Δf = mv_⊥/(2πR) due to the transverse velocity v_⊥. Notice that the frequency band can be tuned by changing the phase profile, which cannot be done with the classical longitudinal Doppler effect.

Notice that the magnitude of the frequency shift can be increased by changing the phase profile. In the cases shown here, the frequency can still be increased by using a higher γ for the translational movement and larger m for the rotational movement. This flexibility enables better velocity estimation as one can have a frequency that is more appropriate for different detection regimes.

The use of a spatially dependent phase gradient, which could be achieved with current technology for shaping light beams, opens the possibility to use the Doppler effect discussed here to detect not only transverse velocities, but transverse positions as well. This is because a unique phase gradient can be associated with a particular location in the transverse plane. Thereby a certain frequency shift can only come from the presence of the particle in the location with that specific phase gradient.

Recently while this paper was under review, Lavery et al.¹⁷ demonstrated detection of the angular frequency of a spinning object by using light with orbital angular momentum. In general, the Doppler frequency shift generated by a moving surface can be written as

where λ is the wavelength of the incident light with unit vector , is the unit vector of the scattered light and v is the velocity of the moving surface¹⁶. For a paraxial incident beam whose vector potential¹⁸ A is of the form , the Poynting vector S writes

where η is the vacuum impedance. Since for a paraxial beam, the longitudinal component of S is much larger than the transverse component, one can write

From Eqs. (3) and (5), the Doppler shift observed for is

Note that this is equivalent to Eq. (4) of Belmonte and Torres⁴, which is Eq. (2) in this paper, if we substitute k = 2π/λ in Eq. (6). Furthermore, we can define α ≡ 1/k∇_⊥Φ. Using λf = c, Eq. (6) reduces to Eq. (2) in Lavery et al.¹⁷ for a one-dimensional case. Different from the usual longitudinal Doppler effect, the frequency shift given by Eq. (4) in Belmonte and Torres⁴, or Eq. (2) in Lavery et al.¹⁷, is independent of the frequency of the illuminating light beam and only depends on the velocity and trajectory of the object moving under structured illumination. If we illuminate the moving object with a Laguerre-Gauss beam with winding number m, the phase gradient is . For a spinning object with velocity , substitution into Eq. (6) of the values of the phase gradient and velocity yields Eq. (6) in Belmonte and Torres⁴ and Eq. (3) in Lavery et al.¹⁷.

The approach proposed here can also be considered for other types of measurements, as in the case of the motility of single-cells or simple multicellular organisms. For a typical value of motility of a biological specimen of tens of micrometers per second¹⁹, a local spatial modulation of ~1 μm⁻¹ would yield a Doppler frequency shift of some tens of hertz. For instance, a typical value of the motility of sperm cells is 20 μm/s. Moreover, this scheme can also be used to measure fluid flows in live tissue, where the possibility of inducing tiny phase gradients that would not affect the in vivo system under study can be of great interest. The diagnosis of certain important eye diseases can be assessed by observing abnormal retinal blood flow²⁰. Typical blood flow velocities in the retina are in the range of some tens of mm/s. Using phase gradients of some ~0.1 μm⁻¹ would result in Doppler shifts of several kilohertz.

Methods

Experimental procedure

Our experimental setup is a modified Mach-Zehnder interferometer with a 15 mW Helium-Neon laser (Melles-Griot, λ = 632.8 nm) light source. The beam is collimated and expanded by a telescope T1 made with a lens combination of focal lengths f = 25 mm and f = 100 mm for the front and back lens, respectively. It is spatially cleaned with a 30 μm pinhole placed at the middle focus of the telescope. The beam is then divided by a beam splitter into two, a reference beam (green line in Fig. 1) and a signal beam. The signal beam impinges onto a spatial light modulator (SLM, Hamamatsu LCOS-SLM) where it acquires the desired phase profile via a 2π-modulo phase wrapped computer generated hologram (CGH) displayed on the SLM. The CGH is calculated from the interference of a beam with our desired phase and a tilted plane wave to generate a hologram with a carrier period of 84.85 μm. In our experiments, we produce beams with helical and linear phases. The helical phase is made with a phase that changes as 2πm as one goes around the azimuth, where m is an integer, the winding number of the beam. The linear phase is done by putting a small constant tilt in the beam. The carrier period makes the separation and filtering of the desired structured beam easier. This is done by appropriately placing another telescope T2 with lens combination of focal lengths f = 50 mm for the front lens, f = 30 mm for the back lens and a 100 μm pinhole. Note that the beam size is reduced to fit into the active area of the Digital micromirror device (DMD, Texas Instruments).

The structured signal beam (blue line in Fig. 1) is then sent to the controllable DMD where a tiny circle composed of an array of micromirrors simulates a 35 μm radius particle. By manipulating the position and the time in which the mirrors are in the ON state, the micromirror ensemble can mimic a particle that is moving with different paths and velocities (see details at the Simulation of particle and its movement section). Light reflected (orange line in Fig. 1) from the DMD contains information about the velocity and position of the particle. The signal beam is then made to interfere with the reference beam at the photodetector (PD, PDAA36A-EC, Thorlabs). The PD is connected to an oscilloscope (TDS2012C, Tektronix) attached to a computer for faster data acquisition and analysis.

Simulation of particle and its movement

We control a Digital micromirror device (DMD) from a DLP Lightcrafter (Texas Instruments) to mimic a particle and its movement. We remove the RGB LED light engine to expose the DMD display. Our DMD is composed of an array of 608 × 684 micromirrors arranged in a diamond geometry. A single micromirror has a diagonal side length of 10.8 μm and can be switched to an ON and OFF state. A set of 1-bit depth images are uploaded in the DMD software; a 0 corresponds to the ON state and a 1 to the OFF state. The time in which the micromirrors are in the ON or OFF state can also be controlled by the software. A micromirror in the ON state will have a tilt of +12° while the OFF state has −12°. Thereby, only micromirrors which are on the ‘on’ state will reflect light in the correct direction with a properly aligned DMD. All other micromirrors reflect light in another direction. These stray lights are blocked. We make sure that the ‘ON’ state reflects light parallel to the optical axis of the incident beam. We simulate a 35 μm radius particle with an array of 7 × 14 micromirrors. This ensemble of micromirrors are manipulated such that a constant array size are turned ‘on’ at specific positions in a particular interval of time while all other micromirrors are switched ‘OFF’. With this, the array seems to be moving similar to a moving particle. We vary the speed of the movement by changing the time interval to switch to another set of micromirror array.

Data analysis

Understanding how the strength of our signal is distributed in the frequency domain, relative to the strengths of other unwanted ambient signals, is central to the design of any sensor system intended to estimate the signal Doppler shift. While many methods for spectrum estimation are discussed in the statistical literature, we use here only the overlapped segmented averaging of modified periodograms. In our case, a periodogram is the discrete Fourier transform (DFT) of one segment of the signal time series that has been modified by the application of a time-domain window function. It has been averaged to reduce the variance of the spectral estimates. While its practical implementation involves a number of nontrivial details -such as equal binning of frequencies, Hanning windowing and filtering of unwanted residual amplitude modulations- our data processing and analysis is rather straightforward and computes a spectrum or spectral density starting from a digitized time series, typically measured in Volts at the input of the A/D-converter.