The Rotational Doppler Effect of Twisted Photons in Scattered Fields

Along with broad applications of the linear Doppler effect, the rotational Doppler effect (RDE) of a structured light source carrying orbital angular momentum (OAM) has attracted significant attention for applications ranging from optical sensors to Doppler cooling. However, the high‐purity structured source's low energy efficiency and unknown optimal OAM parameters have significantly degraded the RDE's performance in previous work. Here, instead of utilizing an optical vortex source, the Doppler features are analyzed in scattered twisted photons carrying OAM with a common light source, and it is demonstrated that the RDE induced by a typical rotator can be extracted using a spiral phase spatial filter (SPSF) at the receiver. This model reveals that a rotating rough surface scatters abundant twisted photons carrying varied OAM values, and the OAM spectrum distribution is modulated by its angular coherence of spatial signature. Furthermore, common surfaces with different autocorrelation structures on received signals using the SPSF method are analyzed theoretically and experimentally. Such a scheme facilitates rotator detection with a generalized nonvortex simple source and dozen–fold improved efficiency with robustness against noncoaxial problems. These demonstrations open a path for studying and applying scattered twisted photons during light detection.


Introduction
Electromagnetic and acoustic waves typically travel in a straight line and carry a linear momentum.The well-known linear DOI: 10.1002/lpor.3][4][5][6] More recently, scientists discovered that structured light could travel in a twisted trajectory and possess orbital angular momentum (OAM). [7]When twisted photons meet a rotator, a novel and distinguishing rotational Doppler effect (RDE) arises in scattered light. [8]The rotational Doppler frequency shift is associated with the rotational angular velocity and the OAM index of the probe beam, which is predicted by Nienhuis and then verified by Allen and Padgett using a rotating Dove prism. [8,9]12] Considerable attention has been paid to exploiting this scheme in astrophysics, [13] nonlinear optics, [14] Doppler cooling, [15] nonreciprocity, [16] vectorial Doppler metrology, [17] Quantum measurement, [18] and so on. [19,20][40][41][42] An OAM probe beam with wavelength  and the topological charge (TC) equal to l possesses an angle  = l/(2r) between the Poynting vector and the beam axis, resulting in the RDE for rotator detection (see Figure S1a in the Supporting Information).However, the OAM sources increase the complexity and costs of the practical detection systems, and the determinations of the optimal probe beams' TCs for various rotators lack theoretical guidance.Therefore, the new RDE-based detection scheme using a generalized nonvortex simple light source is highly anticipated.[45][46][47][48][49] As long as the RDE can be directly measured from scattered twisted photons, we can replace the complex structured light source with a simple light source for broad applications.The scientific question involved is that the scattered RDE induced by a typical rotator with an ordinary light source remains elusive.The challenges concern the scattering field properties of rotating rough surfaces in terms of the OAM spectrum's dispersion and the received RDE's signals, and an effective method for extracting the RDE of twisted photons in scattered light fields.
Here, we modify a physical model describing RDE, even hybrid-Doppler effects, when a light beam illuminates a moving rough surface.The conclusion illustrates that, despite using a common light source, the scattered twisted photons in rotational random fields still carry RDE, and the RDE's spectral characteristics are induced by the angular coherence function of typical roughness.Then, inspired by computational imaging, [50] which reconstructs the high-dimensional information by the polarization or phase coding process of an imaging system, [50,55] we develop a computational rotator detection scheme by decoding the OAM in the scattered field.First, we put a customized couplemode spiral phase mask in the speckle field to extract a specific twisted mode for a rotational Doppler frequency shift.Then, the RDE is identified in proof-of-concept experiments by speckle and coherent detection methods in forward and backward scattering processes.Furthermore, the scattered RDE's spectroscopic characteristics of different rotating rough surfaces (Gaussian type, exponential type, anisotropy type) characterized by statistical parameters (such as height standard deviation and coherence length) are quantified at different distances.We also demonstrate that the spiral phase spatial filter (SPSF) method can achieve large energy efficiency improvements with robustness against the noncoaxial problems in remote detection.By considering OAM encoding in scattered fields, our method exploits a new dimension for computational optical detection with the capability of multidimensional velocity measurement.

The RDE of Twisted Photons in Dynamic Scattered Fields
The general surface has a roughness for light waves, which can be written as h(r, ), and the spinning object's phase modulation function is exp (iФ(r, , t)) = exp (i4h(r, , t)/).The normally incident Gaussian beam with initial frequency f expressed by u = u 0 (r, )exp(−i2ft) at the z = 0 plane is modulated as U s = u exp (iФ(r, , t)) in cylindrical coordinates.[49] To study the scattered spectroscopic characteristics of RDE, the OAM spectrum of a random light field U s (r, , z, t) can be written in azimuthal Fourier expansion form. [19,43,44]s (r, , which comprises discrete modes characterized by the angular index l and angular velocity Ω.The azimuthal Fourier coefficient a l (r, z) is the complex amplitude of the l-th harmonic at radial position r and distance z.Due to the invariant a l (r, z), a certain mode's proportion in a scattered light field is constant, independent of Ω.A significant change induced by Ω is that each mode with index l has a specific time-dependent phase factor term as shown in Figure 1a, yielding a frequency shift concerning the initial incident frequency as Δf Ω = lΩ/(2).The strength P(l) of a surface's OAM channel can be described as Thus, the twisted mode in the scattered field is mapped to the Doppler frequency shift.Here, the rotational Doppler shift of the fundamental mode is zero, and twisted photons carry nonzero rotational Doppler shifts.
The common surfaces with different structure distributions will result in the various OAM spectrum's dispersion and scattered spectroscopy.The key is to clear the scattered OAM distribution P(l) induced by general surfaces.Here, inspired by OAM dispersion in atmospheric turbulence, [56] we establish a model in Note S1 (Supporting Information) to describe OAM scattering by a rough surface and finally find that the rotational correlation function C Φ (r, Δ) of the surface directly leads to the average strength <P(l)> as follows R(r, z, t 0 ) is the radial power distribution of the beam at distance z when t = t 0 , and Δ is the angular deviation.Thus, for a given radial beam profile R(r, z, t 0 ), the average strength <P(l)> is determined by the circular harmonic transform (Fourier series expansion) of C Φ (r, Δ), which is related to the surface's spatial coherence properties.The result presented in Equation ( 2) is valid for any rough surface.Usually, the rotational field correlation is a function of the second-order spatial statistics of the complex amplitude fluctuations, [57] which can be written as where D Φ (r) is the phase structure function of the rough surface and C(r) is the normalized autocorrelation function of the surface with its height standard deviation w, correlation length , and wave number k.In general, according to the normalized autocorrelation function, a rough surface can be divided into different types, for example, the Gaussian type C(r) = exp(−|r| 2 / 2 ), exponential type C(r) = exp(−|r|/), and Lorentz type C(r) =  4 /( 2 +|r| 2 ) 2 ). Figure 1b-d shows the three rough surfaces with different correlation types with the same values of w =  and  = 10.Then, substituting three normalized autocorrelation functions sequentially into the rotational field correlation in Equations (3) and (4), the corresponding average of P(l) induced by varied rough surfaces can be acquired by the angular Fourier transform of C Φ (r, Δ) with Equation (2).For simplicity, supposing R(r, 0, t 0 ) is Laguerre-Gaussian (LG) base mode LG 00 at z = 0.Moreover, a broader physical model as an extension is discussed, to describe hybrid Doppler effects containing rotational, linear, and even micro-Doppler effects.When the scattering surface has a transverse velocity that can be split into a uniform angular velocity Ω and a constant radial velocity v r in Figure S1b (Supporting Information), each mode (n, l) has a transverse Doppler frequency shift Δf ⊥ (l, n) = lΩ/(2)+nv r /T.There are two terms, the rotational Doppler frequency shift term and the radial Doppler frequency shift, [58] associated with the mode indices l and n of the scattered light field, respectively (see Note S2 in the Supporting Information in detail).

The SPSF Method for Measuring the RDE
The scattered twisted photons with nonzero indices carrying the transverse Doppler frequency shifts should be captured to measure the transverse velocity.In detail, we develop the SPSF method to detect the RDE.We can employ a spiral phase plate (with index l 0 ) in the scattered or transmission path in Figure 2a.After the superposition of the spiral phase on each mode, the base mode with index 0 in scattered light is transferred to the vortex mode l 0 with a spiral phase distribution.Meanwhile, the vortex mode along the z-axis with index −l 0 carrying rotational Doppler frequency shift −l 0 Ω/(2) is transferred to the base mode with a planar phase distribution.After utilizing a Kepler telescope (L1 and L2), the base mode with the RDE focuses on the center of the lens's postfocus Fourier spectrum due to its zero spatial frequency.Other modes with nonzero l are near the center due to their nonzero spatial frequencies.Finally, the base mode is filtered in the spatial frequency domain through an optical hole or a single-mode fiber.The rotational Doppler shift Δf = −l 0 Ω/(2) can be measured from the hybrid frequency shifts via coherent detection, and the echo signals' strengths induced when using diverse SPSFs are proportional to P(l).Thus, the angular velocity of the rotating body is inversely derived.Furthermore, more than a particular mode, multiple rotational Doppler frequency shifts can be filtered simultaneously when the scattered photons with indices of l s (s = 1, 2, 3…) pass through the multiple-mode SPSF.
The scattered fields associated with the distance z are obtained after Kirchhoff diffraction (see Experimental Section and Note S3 in the Supporting Information).The time-dependent intensities and phases of speckle fields at the focal plane, with and without the SPSF with an index of l 0, are shown in Figure 2b,c.The center point A represents the base mode in the initial speckle field without a transverse frequency shift.After the spiral phase filter with an index of l 0 , the base mode represented by point B in the center is transferred by the twisted mode with an index of −l 0 carrying the rotational frequency shift −l 0 Ω/(2).In this case, the intensity of point B corresponding to P(l 0 ) is much less than that of point A corresponding to P(0).To study each harmonic component in the natural scattered light, Figure 2d,e shows the OAM spectra of typical Gaussian rough surfaces, characterized by standard deviation w and correlation length  of its height fluctuation (see Experimental Section).It can be seen that the OAM distributions are modulated by the statistical parameters w and , the longer the correlation length , the more concentrated the OAM scattering.
To quantitatively and statistically describe the relation of the scattered OAM spectrum with w and , Figure 2f,g shows the average strength <P(l)> scaled against the reciprocal of correlation length 1/ (w = ) and the standard deviation w ( = 50).For low values of 1/ and w, the OAM scatter is small, but it increases rapidly as 1/ and w increase until each OAM channel reaches the same stable value.Figure 2h,i shows that the behaviors of <P(l)> for the three kinds of rotational correlation surfaces are similar.The larger the angular correlations are in a certain range, the more low-order components there are.When the angular correlation decreases, all orders are consistent.Thus, it is theoretically proven that the OAM scatter modulated by the rotational correlation function is ubiquitous in the scattered field.

Experimental Detection of Angular Velocity in Scattered Light and Transmitted Light with the SPSF Method
When passing through a couple-mode helical phase (with indices ±l) plate, scattered photons with indices of ±l carrying frequency shifts of ±lΩ/(2) are converted into fundamental modes or more extensive modes with indices of (±2l, 0).The base modes with a redshift and a blueshift are filtered with an iris diaphragm together with a 4-f system.Then, the beating signal between the two base modes can be detected.Since the axial velocity may have the same linear Doppler effect on twisted photons with indices of ±l, the beating signals in this case only show the RDE without the influence of the axially linear Doppler effect.
In the proof-of-concept experiment, as shown in Figure 3a, the couple-mode (±l) helical phase on a spatial light modulator (SLM) is used to filter the RDEs in scattering fields and detect a rotating rough surface (see the Experimental Section).When we load a grey plate on the SLM in Figure 3b, dynamic speckle images induced in the focal plane by a Gaussian beam illuminating the rough surface are shown in Figure 3e.The rotational angular velocities of the speckle images in different locations are uniform and consistent with the rotation speed of the rotator.Specifically, when the mirror surface rotator is fixed in Figure 3c, the petal flare is immobile in Figure 3f.Then, a couple-mode vortex phase plate with indices of ±10 in Figure 3d is added to the SLM.The time-dependent intensity of the speckle image reflected by the rough surface with high reflectivity and random phase structures is shown in Figure 3g.We observe that the light intensity changed faster as the location approached the centre.The videos of Figure 3e,f are shown in Video S1 in the Supporting Information.The time-dependent intensities in the speckle's centre and the frequency spectra are shown in Figure 3h,i.The central peak frequency on Point B in the spectra is equal to 0.334 Hz for l = ±30, consistent with the theoretical value of Point A in Figure 3h, under a certain rotational velocity of 0.035 rad s −1 .The blue line describes the echo signal corresponding to a velocity equal to 0 rad s −1 for comparison.This indicates that the RDE in speckle fields can be detected by the SPSF method.
We record dynamic speckle images (see Video S2 in the Supporting Information) and the frequency shifts using SPSF indices of ±20, ±30, and ±40 in Figure 4a, b with the same velocity of 0.035 rad s −1 .As shown in Figure 4e, the signals' velocities change over time and increase with the SPSF index, while the intensity decreases as the SPSF index increasing, due to the larger the index, the more serious the divergence.Despite the spectrum broadening effect, according to the peak values in the frequency spectra, the theoretical rotational speeds should be equal to 0.195, 0.320, and 0.420 Hz.In comparison, the experimental values are 0.222, 0.334, and 0.445 Hz, respectively.This implies that detecting rotators using different SPSFs is feasible.Additionally, we employ different OAM values (l = ±10, ±20, ±30, ±40) and various rotational velocities.Figure 4h shows that the filtered Doppler frequency shift is proportional to the rotational velocity, and the slope of the Δf -v r curve is proportional to the SPSF index.This reveals that the scattered twisted photons carrying RDEs under different rotational velocities can be captured.Furthermore, we consider a more complicated vortex phase filter structure containing three angular harmonic components (l 0 , l 1 , and l 2 ) to acquire multiple RDEs.The moves corresponding to the varying speckle images are shown in Figure 4c  Then, we modify the experimental diagram (see Figure S2 and Experimental Section in the Supporting Information ) to verify whether the RDE of transmitted light could be obtained by the SPSF method.Unlike a scattered reflecting surface, we use a scattered transmission surface (polished glass with random phase structures and high transmittance).Then, the time-dependent intensity speckle image obtained after a couple-mode (±30) vortex phase plate in the transmitted light is shown in Figure 4d and Video S4 in the Supporting Information.The frequency shift value of 0.325 Hz in Figure 4g is the same as that of the scattered light in Figure 3i under the same velocity of 0.035 rad s −1 .In addition, the rotational Doppler frequency shift of the transmission light in Figure 4i is also proportional to the rotational velocity and the SPSF index.Consequently, we experimentally confirm that the SPSF method works in both forward and backward scattering with a maximum measurement error of ≈0.05 Hz.

SPSF Detections of Various Rough Surfaces Characterized by Different Statistical Parameters are Quantified at Different Distances
To analyze the received RDE's signals using the SPSF method when the common surface contains different structure distributions characterized by different statistical parameters, we perform extensive statistical experiments using the various types of simulated rough surfaces with different  and w (For details, see the Experimental Section).Different realizations by using different random complex matrixes are employed when computing the rough surface screen.Figure 5a shows the echo rotational Doppler signal intensities with the SPSF index l = 5 in three Gaussion-type rough surfaces.One can see that the received relative power is around 0.15 when by  = 23.25 mm, w = 1.73 mm, around 0.036 when  is reduced to 18.6, or around 0.036 when w is increased to 2.6 mm.To further investigate the received RDE's signals employing different SPSF indexes in different structure distribution sufficiently, Figure 5b-d   received energy distribution is.Induced by height standard deviation w, the mean variance of the echo signals described by the blue line is changing.
To clarify the performance of the SPSF method in remote detection, we conduct numerical simulations of speckle fields induced by different types of rotating bodies with multiple propagation distances using the schematic diagram in Figure 6a.6c indicate that the relative average echo energy decreases with distance and that the number of scattered photons with a larger OAM value seems fewer than that of scattered photons with a small OAM value due to more considerable divergence.To be more convincing, Figure 6d shows the rotational Doppler signals (l = ±10) induced by different rotators at different distances, for which the SPSF method works successfully.Thus, in addition to the near-field speckles, the SPSF method works in less scattered photons in the far field.

Discussion and Conclusion
The robustness against noncoaxial problems in SPSF detection should be considered.Even though relevant research is proliferating, the required coaxial condition as a strict and troubling limitation, even denied the performance of the previous RDE in practical application scenarios.The displacement leads to the initial OAM mode's dispersion, which in turn deactivates the measurement results.Significantly, the SPSF method is robust against off-axial situations.This is because the probe mode carrying a zero OAM value with a widely expanded beam radius w(z) compared to the displacement  can minimize the effect of  in the scattered OAM dispersions as shown in Figure S5a-f in the Supporting Information.Another point, according to Equations S23-S24, the scattered twisted photons with the rotational axis can be captured when displacement  occurs.Rather than the probe light beam's axis, we adjust the SPSF process neatly to make the rotation axis and the spiral phase plate coaxial in Figure S5g in the Supporting Information.After that, we can acquire the ideal signals reducing the impact of noncoaxial effects, such as the full peak value at 400 Hz when  = 3 mm with a beam radius of 15 mm in Figure S5h (Note S5 in the Supporting Information provides the details).
Regarding efficiency, although optical vortices can be generated by various methods, [39][40][41][42][60][61][62][63] the enormous energy in engineering is still so hard that the detection range needs to be im-proved. The hig-purity vector vortex state laser has a low slope efficiency (1-2%), which is dozens of times lower than that of traditional base-mode emission (50-75%) in Table 1 (details in Note S6 in the Supporting Information).In addition, the base-modes have small divergence angles and can reach further distances under the same energy.Thus, the SPSF method for rotator detection uses a simple light source with high energy efficiency and is very suitable for energy-limited scenarios.
The scattered photons emitted from moving media have many extraordinary properties. [59]Mapping a general surface's rotational correlation function C Φ (r, Δ) to the OAM spectrum modulates the RDE's spectral characteristics.These results have rich physics, offering practical means for obtaining clear statistics or the dynamical properties of the random light field.This scheme also inspires the multidimensional velocity measurement, compensating for a traditional Lidar-only linear velocity measurement (Note S7, Supporting Information).The following issues are interesting to be considered: the influences of the coherence and polarization properties of the probe light and rotator on the measurement results.
In summary, we have derived a generalized physical diffraction model to analyze the order-mapping hybrid Doppler effects modulated by the angular coherence of a rotating surface and demonstrated that the SPSF method can filter the RDE of scattered photons for rotator detection (only by adding a vortex phase filter at the receiver, which turns speckle noise in coherent

OAM source
In resonant cavity Q plate [60] 1-2% normal normal Metasurface [42] 1-2% normal high (90%) Intracavity spherical aberration [61] 10-20% normal normal Out of resonant cavity Vortex plate [62] ∖ low normal √ 2l + 1 0 SPSF method ∖ Gaussian base source [63] 50-75% high high  0 detection into helpful information in principle).Compared with previous works, this scheme has the advantages of excellent energy efficiency, robustness against noncoaxial problems, and remote rotator detection with a simple light source.The above conclusions can benefit all wave fields (acoustic, radio, and light).We anticipate that the proposed scheme will greatly promote computational optical detection through OAM encoding for extensive rotators detection.

Experimental Section
The Generation Method for Rough Surfaces: A random rough surface model following a Gaussian distribution is the most common approach for creating a modified rough surface, which accords with the height distributions of many engineered surfaces.A 2D random surface can be represented by a square matrix h(x, y) where W is a statistical filter equal to √ F[(x, y)] and (x, y) = w 2 exp(−(x 2 + y 2 )∕ 2 )is the auto-correlation function of the surface with a height fluctuation correlation length of  and a standard deviation of w.F and F −1 are the Fourier transform and inverse Fourier transform, respectively, and g(k x , k y ) is the Fourier transform of a sequence of independent random numbers.When the autocorrelation function of the surface is the exponential type, satisfying (x, y) = w 2 exp(− √ x 2 + y 2 ∕).According to Equations ( 9) and ( 5), we can obtain the OAM spectra of rough surfaces in Figure 2d,e.Thus, different types of rough surfaces characterized by  and w can be acquired by modulating the (x, y).Various realizations of rough surfaces with the same parameters  and w can be acquired by using different random complex matrixes g(k x , k y ).
The OAM spectrum of rough surfaces: The OAM spectrum of a phase distribution described by exp(i4h(r, )/) can be obtained by expanding the spiral wave of the light field.Similar to Fourier expansion, the phase distribution is extended by the spiral harmonic term exp(il) where the expansion coefficient a l (r) can be obtained by The strength of an angular harmonic exp(il) is The relative intensity of the spiral harmonic in the phase distribution can be written as Detecting RDE in Scattered Light: The entire characterization apparatus is shown in Figure 3a.Gaussian beams with wavelengths of 1.6 μm were generated by a laser diode and collimated to a diameter of 3 mm.The half wave plate (HWP) and the polarized beam splitter (PBS1) behind the collimator were used to change the power and polarization of each beam.A 45°quarter wave plate (QWP) in front of PBS2 ensures that PBS2 can reflect the scattered light.Then, the probe beam was collimated by a Kepler telescope lens (L1 (f = 100 mm) with a diameter of 25.4 mm and L2 (f = 300 mm) with a diameter of 50.8 mm) before launch and reception.Then, the scattered light was modulated by the SLM (CAS Microstar, FSLM-2K55-P04).The generated speckles were selected as the first diffraction order by an iris diaphragm with a 4-f system.HWP2 matches the horizontal polarization requirement of the phase-only modulation of the SLM.The output signals were recorded by a photodiode (THORLABS, PDA20CS) and analyzed by an oscilloscope (TektronixTBS, 2204B).The angular velocity of the rotator was controlled by a motion controller, which offered a variable voltage for the stepper motor.The scattering rotator (THORLABS, DG10-600-M01) had high reflectivity and isotropic polarization, so the rotational Doppler frequency shift was only associated with the OAM rather than the SAM in this condition.To observe the intensity profiles of the probe beams, an infrared CCD camera (InGaAs, WiDy Swir 320U) in conjunction with a lens L5 was located behind BS2.
Experimental Configuration of the SPSF for the RDE in Transmitted Light: In Figure S2a (Supporting Information), Gaussian beams with wavelengths of 1.6 μm were generated by a laser diode and collimated to a diameter of 3 mm.The half wave plate (HWP) and the polarized beam splitter (PBS1) behind the collimator were used to change the power and polarization of the beams.Then, the SLM (Holoeye, PLUTO-TELCO-013-C) modulates the light transmission process.The generated signal light was selected as the first diffraction order by an iris diaphragm together with a 4-f system.The output signals were recorded by a photodiode (THOR-LABS, PDA20CS) and analyzed by an oscilloscope (TektronixTBS, 2204B).The angular velocity of the rotator was controlled by a motion controller.The rotator was a ground glass scatter sheet with isotropic polarization, so the rotational Doppler frequency shift was only associated with the OAM rather than the SAM in this condition.An infrared CCD camera (InGaAs, WiDy Swir 320U) was used to observe the intensity profiles of received signals.Figure S2b,c in the Supporting Information shows the high transmittance of the rotator in Figure S2a (Supporting Information) and the high reflectivity of the rotator in Figure 3a.
SPSF Detection Simulations from Various Roughness Levels at Different Distances: In the simulation, the waist and the wavelength of the fundamental Gaussian beam were chosen as w 0 = 5 cm and  = 1550 nm, respectively.Then, the beam passed through the simulated rough surface based on the Gaussian-and exponential-type models (Methods), resulting in a distorted light field after a certain distance.The simulated process was performed by employing the MATLAB platform, in which the screen width was 20 cm, and the number of discrete points N was equal to 540.
The diffraction distance can be set to thousands of meters under reasonable parameters.At the receiver, the radius of the receiving screen was set to 10 cm.When the incident light wave was a Gaussian base beam, the evolution of the scattered light intensity distribution diverged as the scattering distance increases, and the detector received fewer scattered photons, as shown in Figure 6a.Then, the echo signal intensity after the helical phase filter was the core issue in detection.When w = 23.25 mm and  = 2.6 mm, Figure S3 showed the signals in time and frequency domains for a filtered frequency shift equal to 200 Hz using a double-helix phase plate with indices of ±10 at different distances z; then, according to RDE, the angular velocity was derived as 628 rad s −1 .Naturally, due to divergence in distance, the echo signal became weaker.Different helical phase plates with indices of ±5, ±10, and ±15 were successfully employed to filter the corresponding twist photons at 100 m in Figure 6b.The corresponding frequency shift values were 100, 200, and 300 Hz.Furthermore, Figure 6c displayed the change curves of the relative average echo energy after performing the SPSF method with l values equal to 5, 10, and 15 as the distances increased.The curves in Figure 6c,d were multiple (50 times) statistical averaging values.

Figure 1 .
Figure 1.The average OAM spectra depend on the rotational correction functions of rough surfaces.a) The schematic representation of RDE of twisted photons in scattered fields.b-d) The Gaussian surface (b), exponential surface (c), and Lorentz surface (d) when w =  and  = 10.e-g) The corresponding average OAM spectra of three types of rotational correction functions, Gaussian (e), exponential (f), and Lorentz (g), with different values of w and  (w =  and  = 10, w =  and  = 50, w = 0.5 and  = 10).

Figure 1e shows the
Figure 1e shows the average OAM distributions scattered by a Gaussian correlation-type rough surface with different values of w and  at z = 0.It indicates that the longer the correlation length is , the more concentrated the OAM scattering.Due to the occurrence of standard deviation w, angular harmonics are widely present in various Gaussian rough surfaces, offering sufficient scattered twisted photons for rotator detection.Figure 1f,g displays the OAM spectra of exponential and Lorentz correlationtype rough surfaces.It shows that when the values of w and  are given, different correlation types influence the type line of spectral distribution.Thus the common rough surface scattered abundant twisted photons carrying varied OAM, and the scattered OAM distribution is modulated by the angular coherence of the spatial signature.Then, the OAM spectra at distance z > 0 in detail are shown in Section 3.2.Moreover, a broader physical model as an extension is discussed, to describe hybrid Doppler effects containing rotational, linear, and even micro-Doppler effects.When the scattering surface has a transverse velocity that can be split into a uniform angular velocity Ω and a constant radial velocity v r in FigureS1b(Supporting Information), each mode (n, l) has a transverse Doppler frequency shift Δf ⊥ (l, n) = lΩ/(2)+nv r /T.There are two terms, the rotational Doppler frequency shift term and the radial Doppler frequency shift,[58] associated with the mode indices l and n of the scattered light field, respectively (see Note S2 in the Supporting Information in detail).

Figure 2 .
Figure 2. The SPSF method for measuring the RDE.a) Schematic of the SPSF system with a Kepler telescope (L1 and L2) and optical holes (H).b,c) Speckle intensity (b) and phase distribution (c) at the focal plane before and after the spiral phase plate (with an index of l 0 = 10).d,e) OAM spectra of rough surfaces with different roughness levels parameterized by w and .The horizontal axis is the OAM value, and the vertical axis is the normalization proportion coefficient.f,g) The curves of the scattering coefficients versus the reciprocal of the correlation length 1/ (w = ) and height standard deviation w ( = 50) for different OAMs (l = 0, 1, 2).h,i) The curves of the scattering coefficients versus 1/ and w for different OAMs (l = 0, 1) scattered by different correlation surfaces.
and Video S3 in the Supporting Information.The frequency shifts of the beating signals in Figure 4f corresponding to |l 0 -l 1 |, |l 0 -l 2 |, and |l 2 -l 1 | are equal to 0.11, 0.11, and 0.22 Hz for ≈l 0 = 0, l 1 = 20, and l 2 = −20 (see Note S4 in the Supporting Information for details).

Figure 3 .
Figure 3. Experimental configuration of the SPSF method for the RDE in scattered light.a) Schematic of the proof-of-concept system for testing the SPSF method.LD, laser diode; PBS1&PBS2, polarized beam splitter; HWP1&HWP2, half wave plate; QWP, quarter wave plate; L1&L3&L4&L5&L6 lenses with focal lengths of 100 mm.The L2 lens has a focal length of 300 mm.BS1&BS2, beam splitter; SLM, liquid crystal spatial light modulator; CCD, an infrared CCD camera; R, reflector.The rotator with an optical rough surface has a high scattering power and a diameter of 25.4 mm.b) The grey plate.c) Mirror surface of a rotator with silver skin.d) Vortex phase plate with an OAM value of ±10.e-g), Dynamic speckle images recorded by CCD, including the initial speckle image e) obtained without the SPSF method, the speckle image f) of the mirror surface obtained with the SPSF method, and the speckle image g) of the rough surface obtained with the SPSF method.h,i) Theoretical value (h) and experimental value (i) of the echo signal in the time domain diagram (horizontal axis: time, vertical axis: the received relative intensity I(t)), and frequency domain (horizontal axis: frequency f(Hz), vertical axis: the power of Fourier transform).The red line describes the echo signal when the rotator velocity equals 0.035 rad s −1 , and the blue line corresponds to a speed equal to 0 rad s −1 .Point A denotes a theoretical frequency value of 0.324 Hz, and Point B represents an experimental frequency value of 0.324 Hz.
shows the received average powers of the SPSF indexes l in the range −10 to 10.The spectral energy distribution varies with the coherence length .The larger the coherence length is, the closer near the base mode the Laser Photonics Rev. 2023,17, 2201022

Figure 4 .
Figure 4. Experimental data processing.a) Spiral filter phase plates (l = ±20, ±30, ±40).b) The reflected speckle images (l = ±20, ±30, ±40).c) The triple helical filter phase plate (l = 0 and ±20) and the related speckle images.d) Dynamic transmitted speckle images obtained with an SPSF index of ±10.e) The time-domain and frequency-domain plots of the central light intensity employing different filter phase plates (l = ±20, ±30, ±40).f) Experimental values of the time domain diagram and frequency domain diagram when l = 0 and ±20 with a velocity of 0.017 rad s −1 .g) The time and frequency domains of the received signal in the transmitted light obtained with the SPSF method (l = 30).h,i) Measurements of the reflected h) and transmitted i) rotational Doppler frequency shifts versus the rotational velocity Ω. Δf depends on Ω.Other values of l result in different slopes for the Δf -Ω curves.Squares: experimental results; solid lines: theoretical results.The scale bar in (a), (b) also applies to (c), (d).
Figure 6b illustrates that rotational Doppler signal processing with the SPSF index (l = ±5, ±10, ±15) can be achieved at 200 m, and the details of the signals at different distances are shown in Figure S5 in the Supporting Information.Furthermore, the statistical averaging simulation results when  = 23.25 mm and w = 2.6 mm in Figure

Figure 5 .
Figure 5. SPSF detections of various rough surfaces.a) The received RDE's signals intensities using the SPSF method (l = 5) in different rough surface realizations all characterized by  = 23.25 mm, w = 1.73 mm;  = 23.25 mm, w = 2.6 mm;  = 18.6 mm, w = 2.6 mm (horizontal axis: the number of realizations, vertical axis: the received relative power).Dashed lines represent the mean values.b-d) Statistical means of the echo signals employing different SPSF indexes (in the range l = −10 to l = 10),  = 23.25 mm, w = 1.73 mm (b);  = 23.25 mm, w = 2.6 mm (c);  = 18.6 mm, w = 2.6 mm (d).Bars represent average data.Every bar's value is given by the average of 50 random rough surfaces with the same w and , whereas the blue lines are the received signals' variances.

Figure 6 .
Figure 6.SPSF detection in distances.a) Rough rotating surface scattering in the space, in which the scattered intensity diffuses with distance (z = 200, 400, 600, 800 m).The details of the simulation process (such as the parameters of the beam size and the surface) are provided in the Supporting Information.b) Time and frequency domain signal with l equal to ±5, ±10, and ±15 at 200 m; the corresponding frequency shift values are 100, 200, and 300 Hz, respectively.Here, the rough surface has w = 2.6 mm and  = 23.25 mm.c) The relative echo energy average levels of different ls at different distances.d) The relative echo average energy levels induced by rough surfaces with other mean square deviations w and coherence lengths  at different distances.Every point on the curve in (c) and (d) corresponds to the average value of 50 various rough surface realizations.

Table 1 .
The energy efficiency of rotator detection based on the OAM source and the SPSF method.