Coherent detection of the rotational Doppler effect measurement based on triple Fourier transform

Hongyu Yan; Hongyu Yan; Yaohui Fan; Yaohui Fan; Ziyan Huang; Ziyan Huang; Ruoyu Tang; Ruoyu Tang; Shuyu Ma; Shuyu Ma; Yu Lei; Yu Lei; You Ding; You Ding; Xiangyang Zhu; Xiangyang Zhu; Tong Liu; Tong Liu; Zhengliang Liu; Zhengliang Liu; Yuan Ren; Yuan Ren

doi:10.1364/OE.520674

1. Introduction

The linear Doppler effect (LDE) is widely used as a universal physical phenomenon in our lives, such as traffic monitoring [1–3], weather forecasting [4–6], space exploration [7–9], etc. It can detect motion in the direction of the wave vector, but not perpendicular to it.

Recently, with the development of structured light [10], the vortex beam (VB) has gradually attracted wide attention [11–16]. It has a spiral wavefront and a central phase singularity and carries orbital angular momentum (OAM), and its special properties come from the spiral phase factor $exp (il\theta )$, where l refers to the topological charge (TC) that represents the number of times that the cross-sectional phase changes from 0 to $2\pi$ for one rotation around the beam cross-section, $\theta $ represents the angular coordinates on the beam cross-section. Due to this light structure, there is a skew angle between the Poynting vector and the wave vector [17–19]^. When the VB irradiates a rotating object, the rotational velocity vector of the object has a component in the direction of the Poynting vector, resulting in “relative motion” between the VB source and the rotating object [20] and thus a frequency shift of light, which can be called the rotational Doppler effect (RDE)[21–26]. Lavery et al. (2013) used two VBs of equal TC and opposite sign to form a superposed-state VB and realized the rotational speed measurement of the object based on RDE[27]. Fu et al. (2017) utilized the self-reconstructing characteristics of Bessel-Gauss beams to achieve RDE measurements under the condition that obstacles exist in the probe beam path [28]. Zhang et al. (2018) realized RDE measurement in 120 m free space at photon counting level [29], which was an important step from the theory to practical applications. Zhai et al. (2020) used the superposed-state vortex beam with various TCs from $l ={\pm} 25$ to $l ={\pm} 45$ as the probe beam to achieve RDE measurements in tens of meters detection distances [30]. Qiu et al. (2022) utilized the principle of OAM carried by each scattered photon as a quantized quantity to obtain the rotational speed of the object by extracting the difference value between two adjacent RDE frequency shift signals [31]. Tang et al. (2022) proposed a dual Fourier analysis method to transform the broadened RDE frequency spectrum to a spectrum with a single peak, which reduces the requirements of RDE measurements on beam quality and incidence conditions [32].

Currently, there are two established methods for RDE measurements: the fringe method using superposed-state VB as the probe beam and the heterodyne method using single-state VB to interfere with local oscillator light. Anderson, et al. (2020) demonstrated that only the heterodyne method is sensitive to the rotating object’s phase. The fringe method’s signal-to-noise ratio (SNR) is inversely proportional to the number of rotating particles, producing the strongest SNR in the presence of a single particle. The heterodyne method’s SNR is proportional to the number of particles in the beam [33]. but they used a hologram rotated on the SLM to simulate a rotating object rather than an actual object with physical scattering properties, and their experiment required the alignment between the VB axis and the rotating axis of the object. In practical applications, rotating objects often consist of rough surfaces with a large number of scattering points on the object’ surface, which is very similar to the large number of randomly distributed rotating particles mentioned in the Anderson, et al. (2020) work [33]. The signal light scattered back to the detector from each point is very weak and can be coupled with each other, so the heterodyne method, which is sensitive to the phase of rotating objects, i.e., the coherent detection method, is more advantageous in practical RDE remote sensing. Meanwhile, in practical measurements, some problems can make the RDE frequency spectrum broaden, which further reduces the SNR, e.g. the purity reduction of the VB mode caused by atmospheric turbulence [34,35], partial obscuration of VB during propagation[36,37], and misalignment of the optical axis of VB and the rotational axis of the rotating object [38–40], etc. Therefore, it is an urgent issue to effectively extract the rotational speed of an object from the weak and broadened scattered light signal.

To address the above problems, we proposed a coherent detection method of the RDE measurement based on triple Fourier transform. The weak scattered light signal is amplified by using the balanced homodyne detection method, then the amplified signal remains broadened in the frequency domain with a low SNR, just like the original signal. By Fourier transforming the amplified signal, we calculated the single-sided amplitude spectrum and input it for the next Fourier transform, and this process is repeated three times to extract the broadened RDE frequency shift signal by means of the coherent amplification. First, we analyzed the theory of coherent detection and explained the signal amplification mechanism. Then we theoretically analyzed the processing of the triple Fourier transform for scattered light signal and revealed its mathematical and physical significance. Finally, we conducted an experimental demonstration to verify the correctness and practicality of the coherent detection of the RDE measurement based on the triple Fourier transform. This method can significantly improve the SNR of RDE measurement and facilitate the accurate extraction of rotational speed. Furthermore, it may help enlarge the RDE detection range and promote its practical application.

2. Theoretical analysis

2.1 Coherent detection of weak RDE signal

When the VB is used as the probe beam for direct detection, the wave function of the scattered light signal can be given by,

(1)$${E_S}(t) = {E_S}\cos (2\pi {f_S}t + l\theta + {\varphi _S})$$

where $E_s$ is the amplitude of the scattered light signal, $f_s$ is the frequency of the scattered light signal, and $\varphi _s$ is the initial phase of the scattered light signal.

As the light frequency is much larger than the detector bandwidth, the detector can only respond to the average power of the light, rather than the frequency of the light signal. The average optical power of the detector responding to the scattered light signal can be obtained via,

(2)$${P_S} = \overline {{{|{{E_S}(t)} |}^2}} = \frac{1}{2}{E_S}^2$$

where $\overline {{{|{{E_S}(t)} |}^2}}$ is the average value of ${|{{E_S}(t)} |^2}$. The output photocurrent of the detector ${I_S}$ is,

(3)$${I_S} = \alpha {P_S} = \frac{1}{2}\alpha {E_S}^2$$

where $\alpha $ denotes the responsivity of the detector. If the load resistance of the detector is ${R_L}$, the power of the electrical signal output from the detector can be expressed by,

(4)$${P_D} = I_S^2{R_L} = {(\alpha {P_S})^2}{R_L} = {\alpha ^2}P_S^2{R_L}$$

During the balanced homodyne detection, the probe beam is the VB and the local oscillator light is the fundamental mode Gaussian beam whose wave function can be described by,

(5)$${E_L}(t) = {E_L}\cos (2\pi {f_L}t + {\varphi _L})$$

The scattered light is coherent with the local oscillation light at the coupler, which causes the photocurrent given by,

(6)$${I_D}(t) = \alpha \overline {{{|{{E_S}(t) + {E_L}(t)} |}^2}} = {I_S} + {I_L} + 2\alpha \sqrt {{P_S}{P_L}} \cos [{2\pi ({f_S} - {f_L})t + l\theta + ({\varphi_S} - {\varphi_L})} ]$$

where ${I_S}$ represents the DC component caused by the scattered light signal and ${I_L}$ represents the DC component caused by the local oscillation light. The third part in the above equation result is the intermediate frequency (IF) signal ${I_{IF}}$ that the detector can respond to, and the electric power output of IF signal can be depicted by,

(7)$${P_{IF}} = \overline {I_{IF}^2} {R_L} = 2{\alpha ^2}{P_S}{P_L}{R_L}$$

Compared to the direct detection, the conversion gain G of the coherent detection can be expressed as follows,

(8)$$G = \frac{{{P_{IF}}}}{{{P_D}}} = \frac{{2{P_L}}}{{{P_S}}}$$

We can see that under a certain power of the local oscillation light, the weaker the power of the scattered light signal, the higher the conversion gain. In practical detection, the scattered light signal power is often very weak, so the conversion gain is much greater than 1. Thus, the coherent method is more applicable to long-range detection. This is because direct detection is the direct measurement of the power of the scattered light signal, whereas coherent detection utilizes the local oscillator light to convert the scattered light signal to an IF signal by making use of the coherence between the scattered light and the local oscillator light.

2.2 Triple Fourier transform method for RDE signal processing

After coherent amplification, the weak scattered light signal carrying the RDE frequency shift is still broadened in the frequency domain with a low SNR, making it difficult to extract the rotational speed information by searching for a single peak signal in the frequency domain. Since the broadening of the RDE frequency spectrum is essentially the broadening of the OAM spectrum, and the OAM carried by each scattered photon is a quantized quantity, the difference in TC between different peaks of the OAM spectrum is minimized to 1, which makes the frequency difference between neighboring peaks of the broadened RDE frequency spectrum with $\varOmega /2\pi $, and is exactly the rotational speed frequency of the object [31,40]. Such a characteristic makes the broadened RDE signal in the frequency domain essentially a set of periodic signals with a period of the rotational speed frequency $\varOmega /2\pi $, and the rotational speed can be obtained by extracting the signal period. Utilizing such a characteristic, we proposed a rotational speed extraction method based on the triple Fourier transform, which no longer obtains the rotational speed by searching for a single peak signal in the frequency domain, but rather obtains that by extracting the period of the broadened RDE signal. This method allows us to successfully convert the unfavorable conditions during the conventional RDE measurements into favorable ones.

We derived the entire flow of the triple Fourier transform and assumed that the signal is a broadened RDE signal with OAM mode ranging from m to n. It can be expressed in the following equation,

(9)$${x_1}(t)=\sum\limits_{k = m}^n {{A_k}\sin (k\varOmega t)}$$

where $A_k$ denotes the relative intensity coefficient in the broadened RDE signal.

Meanwhile, we used an ideal noise-free broadened RDE signal as an example with $m = 29$, $n = 35$, $A = [{A_m},{A_{m + 1}},\ldots \ldots ,{A_{n - 1}},{A_n}] = [0.3,0.5,0.7,1,0.7,0.5,0.3]$ to simulate when a single-state VB with TC $l = 32$ is used as the probe beam to irradiate the object with rotational speed $\Omega = 60\pi rad/s$, the produced broadened RDE signal centered at $\ell \Omega /2\pi = 960Hz$, which is shown in Fig. 1(a).

Fig. 1. The simulation of the ideal and noise-free broadened RDE signal: (a) the signal, (b) the first Fourier spectrum, (c) the second Fourier spectrum, and (d) the third Fourier spectrum.

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The Fourier transform of the above equation can be given by,

(10)$${F_1}(if) = {\cal F}[{{x_1}(t)} ]= \frac{1}{{2i}}\left\{ {\sum\limits_{k = m}^n {{A_k}\left[ {\delta (f - \frac{{k\varOmega }}{{2\pi }}) - \delta (f + \frac{{k\varOmega }}{{2\pi }})} \right]} } \right\}$$

where ${\cal F}$ represents the Fourier transform, $\delta$ is the impulse function and i denotes the imaginary number.

During the process of Fourier transform, the input signal is a real signal, and the output signal is a complex signal. In order to facilitate observation, researchers often take the modulus of the output complex signal to extract the signal amplitude. Since the output signal of the Fourier transform is symmetric about the origin, half of the signal is redundant, we can get all the information about the signal by getting half of it. Half of the signal's amplitude spectrum is called the single-sided amplitude spectrum. The result of taking single-sided amplitude spectrum for the above equation is calculated by,

(11)$${X_1}(f) = |{{F_1}(if)} |=\sum\limits_{k = m}^n {\frac{{{A_k}}}{2}\delta (f - \frac{{k\varOmega }}{{2\pi }})} $$

Meanwhile we took the single-sided amplitude spectrum of the simulated signal, the result is shown in Fig. 1(b). Notably, in the whole process of triple Fourier transform, we used linear coordinate data for calculation. In order to facilitate the demonstration of experimental data, we used logarithmic coordinates in the plot.

Since we aimed for each Fourier transform to extract the frequency information of the input signal, we calculated the single-sided amplitude spectrum of the signal after the Fourier transform and input it to the next Fourier transform, which makes each Fourier transform of the input signal is a real signal. The Fourier transform of the above equation can be given by,

(12)$${F_2}(it) = {\cal F}[{{X_1}(f)} ]=\sum\limits_{k = m}^n {\frac{{{A_k}}}{2}} \exp ( - ik\varOmega t)$$

We can expand it using Euler's formula as,

(13)$${F_2}(it) = \sum\limits_{k = m}^n {\frac{{{A_k}}}{2}} [{\cos ( - k\varOmega t) + i\sin ( - k\varOmega t)} ]$$

The single-sided amplitude spectrum is,

(14)$${X_2}(t) = |{{F_2}(it)} | =\sqrt {\left\{ {{{\left[ {\sum\limits_{k = m}^n {\frac{{{A_k}}}{2}\sin (k\varOmega t)} } \right]}^2} + {{\left[ {\sum\limits_{k = m}^n {\frac{{{A_k}}}{2}\cos (k\varOmega t)} } \right]}^2}} \right\}} t \ge 0$$

Expanding the above equation using the product-to-sum formula, and combining like terms yields,

(15)$${X_2}(t) = \sum\limits_{j = 0}^{m - n} {{B_j}\cos (j\varOmega t)} t \ge 0$$

where $B_j$ denotes the coefficient of each combined term after using the product-to-sum formula.

The second Fourier spectrum of the simulated signal is shown in Fig. 1(c).

Likewise, we took the single-sided amplitude spectrum of the second Fourier transform as the input signal of the third Fourier transform, and the Fourier transform of the above equation can be given by,

(16)$${F_3}(if) = {\cal F}[{{X_2}(t)} ]= \sum\limits_{j = 1}^{m - n} {\frac{{{B_j}}}{2}\left[ {\delta (f - \frac{{j\varOmega }}{{2\pi }}) + \delta (f + \frac{{j\varOmega }}{{2\pi }})} \right]}$$

The result of the single-sided amplitude spectrum can be described by,

(17)$${X_3}(f) = |{{F_3}(if)} |=\sum\limits_{j = 1}^{m - n} {\frac{{{B_j}}}{2}\delta (f - \frac{{j\varOmega }}{{2\pi }})} $$

The third Fourier spectrum of the simulated signal is shown in Fig. 1(d).

Moreover, by adding a Gaussian white noise with mean value 0 and variance 100 to the simulated ideal broadened RDE signal, we analyzed the performance of the triple Fourier transform method under strong background noise, as shown in Fig. 2. It is evident that the background noise completely submerges the signal, making it impossible to find signal characteristics. However, the third Fourier spectrum can still extract the rotational speed information. Notably, at this time the first Fourier spectrum has peak signals, and the RDE frequency shift can be obtained by extracting the horizontal coordinates of the peak signals, which is due to the fact that in order to simplify the complexity of the mathematical derivation, we did not set the broadened RDE signal with numerous OAM modes. We merely want to explain the theory of the triple Fourier transform. Meanwhile, the triple Fourier transform method obtains the rotational speed of the object by extracting the period of the broadened RDE signal, and the fewer OAM modes the signal contains, the weaker the period of the signal is. The currently used signal is more unfavorable for the triple Fourier transform method to extract the rotational speed, but we demonstrated the usability of the method in harsh conditions through simulation experiments. In actual measurements, the broadened RDE signal would consist of numerous OAM modes, and the signal would be more periodic, which is more favorable to extract the rotational speed by using the triple Fourier transform method.

Fig. 2. The simulation of the broadened RDE signal under strong background noise: (a) the broadened RDE signal under strong background noise, (b) the first Fourier spectrum, (c) the second Fourier spectrum, and (d) the third Fourier spectrum.

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We analyzed the changes of each parameter during the triple Fourier transform process, as shown in Table 1. It is evident that the frequency resolution of the previous Fourier transform becomes the sampling interval for the next Fourier transform. As the single-sided amplitude spectrum of the previous Fourier transform is the input signal for the next Fourier transform, each transform conducted halves the number of sampling points. Besides, the frequency resolution of the third Fourier spectrum is affected by the number of sampling points and sampling frequency of the original signal.

Table 1. Parameters of triple Fourier transform

View Table

In the triple Fourier transform process, the first Fourier spectrum extracts the frequency of the original signal, the peak horizontal coordinate corresponds to the RDE frequency shift of the object, and the second Fourier spectrum extracts the periodic information of the first Fourier spectrum, which is essentially an improved cepstrum analysis method, and the peak horizontal coordinate of the second spectrum corresponds to the period of the object's rotational speed. At this time, the second spectrum is the result of the calculation of the dual Fourier transform method proposed in Qiu, et al. (2020)[36]. Notably, combining the physical meaning of the horizontal coordinates of the second Fourier transform with the formula for the frequency resolution of it given in Table 1, we can see that the process of the second Fourier transform constitutes a low-pass filter, and all signals with the frequency larger than ${f_S}/2$ would be filtered out. Unlike the fringe method, which uses the superposition of two conjugate VBs as the probe beam, can eliminate the common mode noise, the remaining high-frequency common-mode noise in the coherent detection method can have an impact on the rotational speed extraction. Whereas the low-pass filter property of the second Fourier transform can filter out the high-frequency common-mode noise that cannot be eliminated due to the use of coherent detection methods, which is more favorable for extracting the RDE signals with frequency shifts at the kHz level. Meanwhile, the period of the first Fourier spectrum may be $n\varOmega /2\pi $($n$ is a positive integer) in addition to the rotational speed frequency $\varOmega /2\pi $, which shows a series of equally spaced wave packets on the second Fourier spectrum, and the interval between the peaks of these wave packets is minimized as the reciprocal of the rotational speed frequency, which provides a prerequisite for the computation of the third Fourier spectrum. Similarly, the third Fourier spectrum extracts the period information of the second Fourier spectrum, namely, the reciprocal of the reciprocal of the first Fourier spectrum period, and the peak horizontal coordinate of the third Fourier spectrum corresponds to the rotational speed frequency of the object. According to the physical quantities expressed by the horizontal coordinate in the triple Fourier transform processing combined with the Nyquist–Shannon sampling theorem, it can be known,

(18)$$\left\{ \begin{array}{l} \frac{{{f_S}}}{2} = \frac{{l\Omega }}{{2\pi }} \to \Omega = \frac{{\pi {f_S}}}{l}\\ \frac{N}{{2{f_S}}} = \frac{{2\pi }}{\Omega } \to \Omega = \frac{{4\pi {f_S}}}{N}\\ \frac{{{f_S}}}{4} = \frac{\Omega }{{2\pi }} \to \Omega = \frac{{\pi {f_S}}}{2} \end{array} \right.$$

The first Fourier spectrum and the third Fourier spectrum together determine the maximum value of the measurable rotational speed, and the second Fourier spectrum determines the minimum value, from which it follows,

(19)$$\left\{ \begin{array}{ccccc} & \frac{{4\pi {f_S}}}{N} \le \Omega \le \frac{{\pi {f_S}}}{l} & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} l \ge 2\\ & \frac{{4\pi {f_S}}}{N} \le \Omega \le \frac{{\pi {f_S}}}{2}{\kern 1pt} & {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} l < 2 \end{array} \right.$$

3. Experimental setup and results

To verify the effectiveness of the coherent detection of RDE measurement based on triple Fourier transform, a measurement experiment was designed. The experimental setup can be divided into two branches. The yellow one represents the fiber optical path, composed of single-mode fibers, which transmits the local oscillator light for coherent amplification, and receives the zero-order OAM component. The two lights of different components are input into a $2 \times 2$ fiber coupler through the fiber for coherent mixing. The red one is the free-space optical path, which transmits the vortex beam used for detection, as shown in Fig. 3.

Fig. 3. Experimental setup. QWP: quarter-wave plate. VHP: vortex half-wave plate. LP: linear polarizer. BS: beam splitter.

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First, the laser produces a linearly polarized Gaussian beam with a narrow linewidth at a wavelength of $1550nm$, which is split into two beams, one output through a fiber to the $2 \times 2$ fiber coupler, and the other output through the fiber collimator to free space. By adjusting the fast axis orientation of the quarter-wave plate (QWP), the polarization state of the beam incident on the vortex half-wave plate (VHP) can be controlled. Thus, we can choose to produce either a single-state VB or a superposed-state VB. Subsequently, the linear polarizer (LP) is used to transform the beam into the linearly polarized state. Finally, after passing through the beam splitter (BS), the beam is directed onto the rotating object. The light signal scattered back from the rotating object passes through the BS again and the fiber coupler, and then it is input into the $2 \times 2$ fiber coupler by utilizing the property of single-mode fibers to only receive zero-order OAM component. Within this coupler, the scattered light signal, after coherent mixing with the local oscillator light, is divided into two equal parts (50:50) and input into the balanced detector. After subtracting the two signals to eliminate common-mode noise, the output signal is used for triple Fourier transform to extract the rotational speed. To simulate the surface roughness of the actual object, we attached the rubbed metal reflective material to the surface of the rotating object. Meanwhile, we deliberately misaligned the optical axis of the VB with the rotational axis of the object to simulate the relative pose relationship between the probe beam and the target during actual measurements. The position of the rotating object can be precisely adjusted using a positioning stage, and its rotation speed can be controlled by a controller.

In the first step of the experimental demonstration, we turned off the local oscillator light to simulate the situation without coherent amplification. The fast axis of the QWP is adjusted to $0^\circ $ with the linearly polarized beam vector emitted by the laser. This allows the beam to enter the VHP in the linearly polarized state, producing a superposed-state VB. At this time, the polarization direction of the LP can be arbitrarily placed, as long as it can unify the polarization states of the two beams forming the superposed-state VB. Next, we set the TC of the VB to $l ={\pm} 32$, and the rotational speed of the object to $60\pi rad/s$, the sampling frequency of the signal to 100000 Hz, and the sampling points to 100000. The results are shown in Fig. 4, where (a)-(d) represent the scattered light signal received, the first Fourier spectrum, the second Fourier spectrum, and the third Fourier spectrum, respectively, when using the superposed-state VB as the probe beam. Notably, there is no significant peak in the first Fourier spectrum, due to the fact that the optical axis of the VB misaligns with the rotational axis of the object, including tilt and eccentricity and other relative poses, which broadens the RDE spectrum with a single peak originally. The broadening of the RDE frequency spectrum significantly reduces the SNR of RDE signal, submerging the RDE signal in noise. Although we collected the signal, the information could not be effectively extracted.

Fig. 4. Experimental results with and without coherent amplification. (a)-(d) represent the scattered light signal, the first Fourier spectrum, the second Fourier spectrum, and the third Fourier spectrum, respectively, without coherent amplification. (e)-(h) represent the scattered light signal after coherent amplification, the first Fourier spectrum, the second Fourier spectrum, and the third Fourier spectrum, respectively, with coherent amplification.

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Subsequently, we turned on the local oscillator light to simulate the situation with coherent amplification. The fast axis of the QWP is adjusted to $45^\circ $ with the linearly polarized beam vector emitted by the laser. This allows the beam to enter the VHP in the circularly polarized state, producing a single-state VB. At this time, the polarization direction of the LP is needed to be consist with the polarization direction of the local oscillator light to ensure the coherence efficiency between the local oscillator light and the probe beam. We then set the TC of the VB to $l = 32$ and the rotational speed of the object to $60\pi rad/s$, the sampling frequency of the signal to 100000 Hz, and the sampling points to 100000. The results are shown in Fig. 4, where (e)-(h) represent the scattered light signal received after coherent amplification with the local oscillator light, the first Fourier spectrum, the second Fourier spectrum, and the third Fourier spectrum, respectively, when using the single-state VB as the probe beam. As seen in Fig. 4(e), the intensity of the scattered light signal after coherent amplification is significantly enhanced, making it more favorable for RDE signal extraction. From Fig. 4(f), it can be seen that the weak scattered light signal carrying the RDE frequency shift after coherent amplification is still broadened in the frequency domain with low SNR. Except for the intrinsic noise of the system at 86 Hz and its second harmonic at 172 Hz, there are no other peak signals, making it difficult to extract the rotational speed information. At this point, the minimum interval among each RDE signal peak is the rotational speed frequency $\varOmega /2\pi $ of the object. In Fig. 4(g), the second Fourier spectrum appears as a series of equally spaced wave packets, with the smallest interval between the peaks corresponding to the reciprocal of the rotational speed frequency $\varOmega /2\pi $. Figure 4(h) reveals the third Fourier spectrum has multiple peaks, corresponding to the horizontal coordinates of $n\varOmega /2\pi $($n$ is a positive integer). At this point, the horizontal coordinate of the first peak is the rotational speed frequency $\varOmega /2\pi $. It is evident that after triple Fourier transform, the SNR is significantly improved, which enlarges the detection range and promotes its practical application.

Furthermore, to investigate the effect of rotational speed and TC on the RDE measurement, we used the coherent detection method by changing the rotational speed of the object from $4\pi rad/s$ to $100\pi rad/s$, and obtained the third Fourier spectrum of Gaussian beam and two kinds of VBs at different rotational speeds when they are used as the probe beam, respectively. Figure 5(a), (b) and (c) respectively represent the third Fourier spectrum of the Gaussian beam, $l = 16$ and $l = 32$ at different rotational speeds. It can be clearly seen that the first peak of the third Fourier spectrum under different rotational speeds is also the highest peak value corresponding to the rotational speed frequency $\varOmega /2\pi $ of the object, and the faster the rotational speed, the stronger the third Fourier spectrum signal intensity. Even when the Gaussian beam is used as the probe beam, it also conforms to this rule, but compared with the use of the VB as the probe beam, the peak intensity is much lower due to the following factors. When the Gaussian beam is used as the probe beam, the third Fourier transform only detects the scintillation signal caused by the rotation of the object itself, so its intensity is low. When the VB is used as the probe beam, the scintillation signal and the broadened RDE signal can be detected, and the intensity of the broadened RDE signal is much stronger than the scintillation signal, which makes peak value of the third Fourier transform much higher. Figure 5(d) shows the absolute errors of the RDE measurement of the Gaussian beam and the two kinds of VBs at different rotational speeds, and it can be seen that the absolute error of the rotational speed frequency is less than 1rps. This means that the accuracy of the triple Fourier transform method is not affected by the number of TC of the probe VB and the rotational speed of the object to be measured, but rather depends on the number of peak intervals of the broadened RDE signal in the first Fourier transform, the more the peak intervals are, the more periodic the signal is, and the more helpful it is to extract the rotational speed of the object.

Fig. 5. The third Fourier spectrum and absolute errors of different types of beams at different rotational speeds. (a) The third Fourier spectrum of the Gaussian beam. (b) The third Fourier spectrum of $l = 16$. (c) The third Fourier spectrum of $l = 32$. (d) Absolute errors of the three types of beams at different rotational speeds.

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4. Conclusion

This paper proposed a coherent detection method of the RDE measurement based on triple Fourier transform. This method amplifies the weak light signal scattered back from the rough surface of a rotating object by utilizing the coherence between the scattered light and the local oscillator light, then the amplified signal remains broadened in the frequency domain with a low SNR, just like the original signal. After the amplified signal is Fourier transformed, we used the single-sided amplitude spectrum as the input signal for the next Fourier transform, and this process is repeated three times to extract the broadened RDE frequency shift signal after coherent amplification. Through this method, we successfully detected the weak RDE signal that traditional methods could not detect using superposed-state VB as the probe beam.

Considering the impact of rotational speed and TC on the triple Fourier transform, the results show that the accuracy of the triple Fourier transform method is not affected by the number of TC of the probe VB and the rotational speed of the object to be measured, and the accuracy mainly depends on the number of peak intervals of the broadened RDE signal in the first Fourier transform.

In sum, we reported a novel RDE measure method, which enhances the detection range of RDE measurement. The effectiveness and feasibility of this method has been verified through experiments. This proposal breaks the limitation imposed by the broadening of RDE frequency spectrum on RDE measurement, and reduces the requirements for the power and alignment of the probe VB, while the broadening better matches the characteristic of RDE signal during the actual measurement. The method has significant potential for application in RDE remote sensing.

Funding

National Natural Science Foundation of China (61805283, 62173342).

Disclosures

The authors declare no conflicts of interest regarding this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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	Sampling interval	Sampling frequency	Sampling points	Sampling period	Frequency resolution
First Fourier spectrum	$Δ t$	$f_{S}$	$N$	$\frac{N}{f_{S}}$	$\frac{f_{S}}{N}$
Second Fourier spectrum	$\frac{f_{S}}{N}$	$\frac{N}{f_{S}}$	$\frac{N}{2}$	$\frac{f_{S}}{2}$	$\frac{2}{f_{S}}$
Third Fourier spectrum	$\frac{2}{f_{S}}$	$\frac{f_{S}}{2}$	$\frac{N}{4}$	$\frac{N}{2 f_{S}}$	$\frac{2 f_{S}}{N}$

Coherent detection of the rotational Doppler effect measurement based on triple Fourier transform

Abstract

1. Introduction

2. Theoretical analysis

2.1 Coherent detection of weak RDE signal

2.2 Triple Fourier transform method for RDE signal processing

3. Experimental setup and results

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Tables (1)

Equations (19)

Optics Express