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Symmetrical dual-sideband oppositely chirped differential FMCW LiDAR

Yanan Zhi, Yujiao Sun, Yu Zou, Bijun Xu, and Kehan Tian

Open Access Open Access

Abstract

A differential FMCW LiDAR for high-precision distance measurements of remote non-stationary targets is proposed and demonstrated experimentally. The required positive and negative symmetrically oppositely chirped laser beams are generated synchronously through a fixed-frequency laser by employing externally unified broadband optical phase modulation and symmetrical dual-sideband optical filtering. After coaxial transmission and reception, orthogonally polarized optical beat signals containing target distance and vector velocity data are de-chirped separately by optical in-phase and quadrature demodulations and then synchronously received by four-channel photoelectric balance detectors. After differential processing of the received beat signals and a fast Fourier transform, it is possible to implement real-time simultaneous range and vector velocity measurements. The inherent symmetrically oppositely chirped optical frequency make it possible to measure the target distance immune to the internal random phase noise introduced by the spectral linewidth of the frequency-swept laser and the external random phase noise introduced by atmospheric turbulence, speckle, and vibration. Meanwhile, the measurement of the target velocity is immune to the nonlinearity of the frequency-swept laser. These results encourage an approach to overcome the barriers of coherence length, nonlinearity, and external noise, and implement simultaneous real-time ranging and velocimetry of long-range, rapid-moving targets.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

As an active remote sensing technology, the frequency-modulated continuous-wave (FMCW) LiDAR (Light Detection and Ranging) technology, combines FMCW technology and coherent LiDAR technology, using linear frequency-modulated signal to carry out linear modulation of laser frequency, measure the target distance by the intermediate frequency (IF) of beat signal between echo beam and local oscillator (LO) beam, and measure the target velocity by Doppler effect. FMCW LiDAR benefits from high sensitivity, high resolution, high precision, excellent anti-interference capabilities, and high on-chip integration potential [1,2]. It has seen widespread application and played an essential role in high-precision photography, remote sensing, and autonomous driving [3]. The frequency-swept laser source, which can provide a broadband and precisely adjustable swept frequency, is the core component of the FMCW LiDAR system. The frequency-swept laser source can be classified into internal modulation [4], chirped pulse laser source [5], and external modulation [6]. During the FMCW Lidar working process, the linear frequency-modulated laser is split into two portions: one is used as the LO beam, and the other is utilized as the probe beam, which is irradiated to the target surface by a collimating and scanning system. After traveling via the optical circulator and entering the photodetector, the target's echo beam is mixed with the LO beam in the optical hybrid. Finally, the signal processing unit can derive the target distance and velocity from the IF of the beat signal at the same time, enabling 4-D [7] point cloud photography.

Figure 1 depicts the conventional triangular-waveform FMCW LiDAR operation concept, which can simultaneously conduct range and velocity measurements. For long-range, high-precision applications, a high-performance FMCW LiDAR should ideally satisfy narrow linewidth, high chirp linearity, and large time-bandwidth product (TBWP). Nonetheless, the universal FMCW LiDAR must surmount numerous obstacles.

 figure: Fig. 1.

Fig. 1. Schematic diagram for operation principle of the conventional triangular-waveform FMCW LiDAR. (a) Time-domain laser frequency waveform including probe beam, local oscillator and echo beam. ${f_0}$: optical frequency carrier frequency; $B$: chirp bandwidth; LO: local oscillator; ${f_D}$: Doppler shift; $T$: the period of triangular modulation; ${\tau _S}$: time delay between echo beam and probe beam; ${\tau _L}$: time delay between LO beam and probe beam. (b) Time-domain beat frequency waveform including up-chirp and down-chirp. ${f_{IF\_up}}$: IF of up-chirp; ${f_{IF\_down}}$: IF of down-chirp. (c) Time-domain FMCW waveform including up-chirp and down-chirp. (d) Frequency-domain beat frequency waveform including up-chirp and down-chirp.

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The chirp nonlinearity of a frequency-swept laser source is the initial obstacle. FMCW LiDAR's range and velocity measurements rely on the spectrum extraction of beat signals, which is extremely sensitive to chirp nonlinearity. The origins of chirp nonlinearity fall into two distinct categories: systematic nonlinearity and random noise. The systematic nonlinearity is the repetitive nonlinearity caused by the frequency-swept laser source's nonlinear dynamic response. The systematic nonlinearity can be characterized and compensated using iterative learning pre-distortion [8] or optical phase-locked loop (OPLL) [4,9]. The sources of random noise may include laser spontaneous emission, noise from the drive current, thermal fluctuations, etc. These sources of random noise collectively influence the frequency or phase stability of the frequency-swept laser, and can therefore be combined as internal random noise. Because the internal random noise is not repetitive, the pre-distortion method cannot be used to eliminate it.

The second challenge is the maximum ranging distance and precision [10] as imposed by the coherence length of laser sources. Due to the fact that the coherence length is directly proportional to the laser linewidth, the maximum distance can be increased by suppressing the laser's intrinsic noise to attain high spectral purity. Nevertheless, the approaches enhance the complexity and expense.

The third difficulty is external random phase noise introduced by atmospheric turbulence [11], speckle [12], movement [13], vibration [14] and nonlinear effects of optical elements [15]. External random phase noise can also degrade the long-range applications of FMCW LiDAR, resulting in severe detection precision and accuracy limitations.

The fourth difficulty is the deterioration of frequency variation rate as a result of lasers’ intrinsic relaxation oscillations and other inherent side effects when the sweep speed is high [16]. After changing its frequency, the laser requires a considerable amount of time to stabilize its optical output due to these effects.

The fifth difficulty is the simultaneous real-time ranging and velocimetry of fast accelerating targets [17], as well as the severe ambiguities caused by laser frequency chirping pulse length and high Doppler shifts [18,19].

The focus here is on internal random phase noise, whose presence degrades the overall performance of the system [20]. A high level of internal random phase noise can degrade laser coherence, resulting in a brief laser coherent time, a brief laser coherent length, and a broad spectral linewidth [21,22]. Consequently, the spectrum analysis will suffer from spectral energy diffusion, main lobe broadening, and side lobe lifting. Therefore, the internal random phase noise has become a significant factor degrading the detection sensitivity, resolution, precision and other critical performance indexes of FMCW LiDAR, particularly at a longer distance and with very weak target return photon levels [23]. It is emphasized how crucial it is to optimize the frequency-swept laser source.

The most direct approaches using narrow-linewidth laser source have been implemented. The approach based on directly frequency-modulated narrow-linewidth external cavity diode laser (ECDL) high repetition frequency tuning is proposed for FMCW LiDAR [24]. The narrow-linewidth semiconductor lasers based on an optical negative feedback scheme without external modulators is proposed [25]. A combination of a narrow-linewidth CW laser with external modulator can provide a straightforward approach to achieve frequency sweep with high linearity and low random phase noise [26,27]. A narrow-linewidth low-noise frequency-agile photonic integrated lasers is demonstrated [28]. Although the approaches based on narrow-linewidth laser are attractive, their operations are rather complex and the cost of the laser is rather high for the widespread application as a frequency-swept laser in FMCW LiDAR. Moreover, even for relatively narrow-linewidth laser sources, the detection range of the FMCW LIDAR is ultimately limited by the source's spectral purity if the coherence distance is treated as a hard constraint.

Therefore, FMCW LiDAR requires real-time phase noise compensation due to the fact that low-cost lasers can be utilized. A phase noise model for actively linearized FMCW semiconductor lasers is presented [29]. The conventional methods to eliminate phase noise from the heterodyne signal are based on feedforward [30] or feedback [31] mechanism. The method of post-processing phase noise compensation with auxiliary reference interferometer is proposed to extend the LiDAR detection range without requiring a laser with a narrow linewidth or a complex linewidth suppression configuration [32,33]. However, these schemes commonly have a relatively complicated structure, and require additional control circuits and high-performance hardware. Furthermore, due to the influence of the target motion on the phase-noise-compensation method through post-processing of sampled data, the phase noise of the echo beat signal does not match the phase noise obtained by the auxiliary interferometer, introducing additional residual phase noise to the echo beat signal. It has also been demonstrated that improving the spectral estimation algorithm substantially extends the range beyond the coherence distance [34,35]. However, these methods call for cumbersome data processing. In addition, the phase noise compensation effect will deteriorate when the object is moving.

A time-efficient hardware solution with rapid data processing is demanded. A differential detection scheme between the frequency-fixed carrier and frequency-modulated subcarrier which are generated from a single laser source is proposed [36]. A coherent dual-wavelength FMCW LiDAR utilizing dual-heterodyne mixing which permits efficient phase noise cancellation is proposed [37]. A dual interferometer method with two different optical delay lengths is proposed to increase distance measurement through multiple reference delay lines [38]. A monolithic integrated linear frequency-modulated dual-wavelength DFB laser chip is also designed and experimentally demonstrated [39]. A scheme of phase-noise-cancelled FMCW Lidar is proposed and experimentally demonstrated based on carrier-suppressed dual-sideband (CS-DSB) modulation and injection-locking technique [40].

In this paper, a differential FMCW LiDAR is proposed to surmount the limitations of coherence length, nonlinearity, and external noise. To generate the required FMCW laser signal, an externally modulated method employing an electro-optical phase modulator (EOPM) powered by a linearly swept RF signal and an optical band-pass filter (OBPF) is utilized. Positive and negative double chirped laser signals are generated synchronously through a fixed-frequency laser by symmetrical dual-sideband opposite chirps. The essence of the phase modulation and frequency filtering is the shifting and adoption of the swept-band. Because both positive and negative frequency-swept sidebands derive from the same laser source and phase modulation, the characteristics of the two chirped-frequency optical fields are always identical. Consequently, the common phase noise can be eliminated by differentially processing the received pulse signals during range and velocity measurements. Theoretical analysis and experimental verifications prove the superiorities of this differential FMCW LiDAR over universal FMCW LiDAR. These characteristics are ideal for simultaneous real-time ranging and velocimetry of long-range, rapid-moving targets.

2. Theoretical analysis

The symmetrical dual-sideband oppositely chirped differential FMCW LiDAR system has been designed and implemented, and the schematic diagram is depicted in Fig. 2

 figure: Fig. 2.

Fig. 2. System block diagram of the symmetrical dual-sideband oppositely chirped differential FMCW LiDAR. The solid redlines are optical signals. The solid blue lines are RF signals. The solid black lines are digital signals. RF, Radio Frequency; EOPM: electro-optical phase modulator; BPD: balanced photodetector; ADC: analog-to-digital converter; OBPF: optical band-pass filter; I-Channel: in-phase channel; Q-Channel: quadrature channel.

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2.1 Symmetrical dual-sideband opposite chirps

To generate the symmetric dual-sideband oppositely chirped optical field with a brief modulation period, a fixed-frequency continuous-wave laser source, RF driving module, EOPM, and OBPF are utilized. This method consists of the five phases listed below.

First, the unmodulated laser beam is produced by the fixed-frequency continuous-wave laser source, which can be expressed as follows:

$${E_0}(t )= {E_0}\exp [{j2\pi {f_0}t + j{\phi_{N\_SR}}(t )+ j{\phi_0}} ]$$
where ${f_0}$ is the carrier frequency of laser source. ${\phi _{N\_SR}}(t )$ is the phase noise introduced by the linewidth of the laser source; ${\phi _0}$ is the initial phase of the laser source; ${E_0}$ is the amplitude of the laser source; $t$ is the time;$j = \sqrt { - 1}$.

Second, after mixing, filtering and amplifying the linear frequency-modulated signal generated by the linear frequency generator and the sinusoidal fundamental frequency signal produced by the fundamental frequency generator, the RF driving signal can be expressed as follows:

$${V_{RF}}(t )= M{V_{RF\_FM}}{V_{RF\_B}}\cos \left[ {\pi \mathop f\limits^\cdot {t^2} + 2\pi {f_{RF\_B}}t + {\phi_{N\_RF}}(t )} \right]$$
where $M$ is the amplification factor of the RF circuit. ${V_{RF\_FM}}$ is the amplitude of the linear frequency signal. ${V_{RF\_B}}$ is the amplitude of the fundamental frequency signal. ${f_{RF\_B}}$ is the fundamental frequency. ${\phi _{N\_RF}}(t )$ is the phase disturbance introduced by RF equipment. $\mathop f\limits^\cdot{=} \frac{B}{T}$ is the chirp rate of linear frequency, B is the chirp bandwidth of linear frequency, and T is the modulation period.

Thirdly, the EOPM is driven by the RF driving signal, and its output optical field is:

$$\scalebox{0.89}{$\displaystyle{E_{PM}}(t )= {E_{0\_PM}}\exp \left\{ {j2\pi {f_0}t + j\beta \cos \left[ {2\pi \left( {{f_{RF\_B}} + \frac{1}{2}\mathop f\limits^\cdot t} \right)t + {\phi_{N\_RF}}(t )} \right] + j{\phi_{N\_SR}}(t )+ j{\phi_{N\_PM}}(t )+ j{\phi_0}} \right\}$}$$
where ${E_{0\_PM}}$ is the amplitude of the RF circuit. ${\phi _{N\_PM}}(t )$ is the phase noise introduced by EOPM. $\beta$ is the modulation index of EOPM, and $\beta = \pi \frac{{M{V_{RF\_FM}}{V_{RF\_B}}}}{{{V_\pi }}}$. ${V_\pi }$ is the half-wave voltage of EOPM.

The expansion of the corresponding Bessel function of the first kind can be expressed as:

$$\scalebox{0.89}{$\begin{aligned} {E_{PM}}(t )&= {J_0}(\beta ){E_{0\_PM}}\exp [{j2\pi {f_0}t + j{\phi_{N\_SR}}(t )+ j{\phi_{N\_PM}}(t )+ j{\phi_0}} ]\\ &+ \sum\limits_{n = 1}^\infty {{J_n}(\beta ){E_{0\_PM}}\exp \left[ {j2\pi {f_0}t + j{\phi_{N\_SR}}(t )+ j{\phi_{N\_PM}}(t )+ j{\phi_0} + j\frac{{n\pi }}{2}} \right]} \\ &\left\{ {\exp \left[ {j2\pi \left( {n{f_{RF\_B}} + \frac{1}{2}\mathop {nf}\limits^\cdot t} \right)t + jn{\phi_{N\_RF}}(t )} \right] + \exp \left[ { - j2\pi \left( {n{f_{RF\_B}} + \frac{1}{2}n\mathop f\limits^\cdot t} \right)t - jn{\phi_{N\_RF}}(t )} \right]} \right\} \end{aligned}$}$$
where ${J_n}(\beta )$ is the Bessel function of the first kind of the ${n^{\textrm{th}}}$ order.

The EOPM driven by a linearly swept RF signal plays the role of shifting and continuously sweeping the frequency of a beam emitting from the fixed-frequency continuous laser source. The first term of the formula above represents the optical frequency carrier of the laser source, while the remaining terms represent the sideband-modulated optical field, which includes positive and negative sidebands. The power of a single-frequency laser source is extended over modulated sidebands whose amplitudes are controlled by the corresponding order of the first kind of Bessel function with modulation index $\beta$ of EOPM, and carrier suppression is accomplished by varying the amplitudes of the RF driving signal. Figure 3 depicts the central optical frequency carrier and sideband-modulated optical fields output by the EOPM, respectively.

 figure: Fig. 3.

Fig. 3. Schematic diagram of the optical frequency domain generation of symmetrical dual-sideband opposite chirps. ${f_0}$: optical carrier frequency; $B$: chirp bandwidth; $k$: filtering order.

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Finally, an optical beam splitter divides the linear frequency-modulated laser into an upper-sideband FMCW channel and a lower-sideband FMCW channel. Figure 3 depicts the utilization of both positive and negative OBPF to generate the symmetric dual-sideband opposite chirps.

The upper-sideband FMCW channel adopts the $+ k$ OBPF to filter the residual carrier and other sidebands. As a result, only the $+ {k^{\textrm{th}}}$ order sideband optical field can pass through, which is expressed as follows:

$${E_{ + k}}(t )\textrm{ = }{E_{ + k}}\exp \left[ {j2\pi ({{f_0}\textrm{ + }k{f_{RF\_B}}} )t + j\pi k\mathop f\limits^\cdot {t^2} + j{\phi_{N\_SR}}(t )+ j{\phi_{N\_PM}}(t )+ jk{\phi_{N\_RF}}(t )+ j{\phi_0} + j\frac{{k\pi }}{2}} \right]$$
where ${E_{ + k}} = {J_{ + k}}(\beta ){E_{0\_PM}}$ is the amplitude of the upper-sideband FMCW channel. ${J_{ + k}}(\beta )$ is the first kind Bessel function of the $+ {k^{\textrm{th}}}$ order.

The lower-sideband FMCW channel adopts the $- k$ OBPF to filter the residual carrier and other sidebands. Consequently, only the $- {k^{\textrm{th}}}$ order sideband optical field can pass through, which is expressed as follows:

$${E_{ - k}}(t )\textrm{ = }{E_{ - k}}\exp \left[ {j2\pi ({{f_0} - k{f_{RF\_B}}} )t - j\pi k\mathop f\limits^\cdot {t^2} + j{\phi_{N\_SR}}(t )+ j{\phi_{N\_PM}}(t )- jk{\phi_{N\_RF}}(t )+ j{\phi_0} + j\frac{{k\pi }}{2}} \right]$$
where ${E_{ - k}} = {J_{ - k}}(\beta ){E_{0\_PM}}$ is the amplitude of the lower-sideband FMCW channel. ${J_{ - k}}(\beta )$ is the first kind Bessel function of the $- {k^{\textrm{th}}}$ order.

The time-domain optical frequency of generated positive and negative symmetrically oppositely chirped laser beams are depicted in Fig. 4(a).

 figure: Fig. 4.

Fig. 4. Schematic diagram for operation principle of the original FMCW LiDAR (a) Time-domain optical frequency of symmetrical dual-sideband oppositely chirped laser beams. (b) Time-domain beat frequency, which includes up-chirp and down-chirp. ${f_0}$: optical frequency carrier frequency; $B$: chirp bandwidth; $k$: filtering order; LO: local oscillator; ${f_D}$: Doppler shift; $T$: modulation period; ${\tau _S}$: time delay between echo beam and probe beam; ${f_{IF\_up}}$: IF of up-chirp; ${f_{IF\_down}}$: IF of down-chirp.

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2.2 Optical in-phase/quadrature demodulation

When the RF driving signal chirps in the positive direction, the upper- and lower-sideband FMCW channels can execute the frequency sweep in the positive and negative directions, respectively. Independent polarization controllers modify both laser beams to orthogonal polarizations, and then the two beams are multiplexed by a polarization beam combiner before being amplified by the laser amplifier. Beam splitters divide the optical outputs of upper- and lower-sideband FMCW channels into two portions, one of which serves as the LO beam for an optical hybrid and the other as the transmitted beam. Transmission (Tx) and reception (Rx) are coaxial, and a polarization-diversity optical circulator [41] is used to separate the transmitting beam from the receiving beam. In optical in-phase/quadrature (I/Q) demodulation, two copies of the LO are generated, one with a 90-degree delay relative to the other; these are the I-channel and Q-channel, respectively. The laser echo combines I-channel and Q-channel to produce two emission signals. Then the four output signals of the 2 × 4 90° optical hybrid are detected by two BPDs. Due to polarization diversity, it is possible to achieve crosstalk isolation between upper- and lower-sideband FMCW channels via polarization extinction. Alternatively, if the difference in fundamental frequency between the two channels is significantly greater than the IF of the pulse signal, crosstalk suppression is achieved by low-pass filtering.

In the upper-sideband FMCW channel, the four optical beat signals can be expressed as

$$\scalebox{0.8}{$\displaystyle\left\{ {\begin{array}{@{}c} {{i_{ + k}}{{(t )}_0} = \frac{1}{4}E_{S1}^2 + \frac{1}{4}E_{LO1}^2 + \frac{1}{2}{E_{S1}}{E_{LO1}}\cos \left[ {4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} + k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{ + k\_NS}}\textrm{ + }0} \right]}\\ {{i_{ + k}}{{(t )}_{90}} = \frac{1}{4}E_{S1}^2 + \frac{1}{4}E_{LO1}^2 + \frac{1}{2}{E_{S1}}{E_{LO1}}\cos \left[ {4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} + k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{ + k\_NS}}\textrm{ + }\frac{\pi }{2}} \right]}\\ {{i_{ + k}}{{(t )}_{180}} = \frac{1}{4}E_{S1}^2 + \frac{1}{4}E_{LO1}^2 + \frac{1}{2}{E_{S1}}{E_{LO1}}\cos \left[ {4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} + k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{ + k\_NS}}\textrm{ + }\pi } \right]}\\ {{i_{ + k}}{{(t )}_{270}} = \frac{1}{4}E_{S1}^2 + \frac{1}{4}E_{LO1}^2 + \frac{1}{2}{E_{S1}}{E_{LO1}}\cos \left[ {4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} + k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{ + k\_NS}}\textrm{ + }\frac{{3\pi }}{2}} \right]} \end{array}} \right.$}$$
where R is the distance of target. ${f_D}$ is the Doppler shift of target. c is the speed of light.${\phi _{N\_OA}}$ is the phase noise of the laser amplifier. ${\phi _{ + k\_NS}}$, the external random phase noise generated during the transmitting process in the upper-sideband FMCW channel. Two pairs of 180° phase shifted optical outputs are obtained: one pair including ${i_{ + k}}{(t )_0}$ and ${i_{ + k}}{(t )_{180}}$ is the in-phase channel (I-channel), and the other pair including ${i_{ + k}}{(t )_{90}}$ and ${i_{ + k}}{(t )_{270}}$ is the quadrature channel (Q-channel).

In the lower-sideband FMCW channel, the four optical beat signals can also be expressed as

$$\scalebox{0.8}{$\displaystyle\left\{ {\begin{array}{@{}c} {{i_{ - k}}{{(t )}_0} = \frac{1}{4}E_{S2}^2 + \frac{1}{4}E_{LO2}^2 + \frac{1}{2}{E_{S2}}{E_{LO2}}\cos \left[ { - 4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} - k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{ - k\_NS}}\textrm{ + }0} \right]}\\ {{i_{ - k}}{{(t )}_{90}} = \frac{1}{4}E_{S2}^2 + \frac{1}{4}E_{LO2}^2 + \frac{1}{2}{E_{S2}}{E_{LO2}}\cos \left[ { - 4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} - k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{ - k\_NS}}\textrm{ + }\frac{\pi }{2}} \right]}\\ {{i_{ - k}}{{(t )}_{180}} = \frac{1}{4}E_{S2}^2 + \frac{1}{4}E_{LO2}^2 + \frac{1}{2}{E_{S2}}{E_{LO2}}\cos \left[ { - 4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} - k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{ - k\_NS}}\textrm{ + }\pi } \right]}\\ {{i_{ - k}}{{(t )}_{270}} = \frac{1}{4}E_{S2}^2 + \frac{1}{4}E_{LO2}^2 + \frac{1}{2}{E_{S2}}{E_{LO2}}\cos \left[ { - 4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} - k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{ - k\_NS}}\textrm{ + }\frac{{3\pi }}{2}} \right]} \end{array}} \right.$}$$
where ${\phi _{ - k\_NS}}$ is the external random phase noise generated during the transmitting process in the lower-sideband FMCW channel. Two pairs of 180° phase shifted optical outputs are also obtained: one pair including ${i_{ - k}}{(t )_0}$ and ${i_{ - k}}{(t )_{180}}$ is the I-channel, and the other pair including ${i_{ - k}}{(t )_{90}}$ and ${i_{ - k}}{(t )_{270}}$ is the Q-channel.

As the transmitting process of upper- and lower-sideband FMCW channels is subjected to nearly identical target speckle, atmospheric turbulence, and vibration, the generated external random phase noise is regarded as identical. Consequently, the relevant parameters can be expressed as

$${\phi _{ + k\_NS}} = {\phi _{ - k\_NS}} = {\phi _{NS}}$$
By utilizing balanced photodetection, DC components can be eliminated. The I-channel and Q-channel photoelectric pulse signals can be expressed by
$$\left\{ {\begin{array}{c} {{i_{ + k\_I}}(t )= {\sigma_{in\_1}}{E_{S1}}{E_{LO1}}\cos \left[ {4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} + k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{NS}}} \right]}\\ {{i_{ + k\_Q}}(t )= {\sigma_{qu\_1}}{E_{S1}}{E_{LO1}}\sin \left[ {4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} + k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{NS}}} \right]} \end{array}} \right.$$
$$\left\{ {\begin{array}{c} {{i_{ - k\_I}}(t )= {\sigma_{in\_2}}{E_{S2}}{E_{LO2}}\cos \left[ { - 4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_\textrm{D}}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} - k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{NS}}} \right]}\\ {{i_{ - k\_Q}}(t )= {\sigma_{qu\_2}}{E_{S2}}{E_{LO2}}\sin \left[ { - 4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_\textrm{D}}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} - k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{NS}}} \right]} \end{array}} \right.$$
where ${\sigma _{in}}$ and ${\sigma _{qu}}$ are the photodetector responsivities of the balanced receivers of I-channel and Q-channel, respectively.

2.3 Original FMCW LiDAR

Following digital sampling by an ADC, all subsequent signal processing is performed in the digital domain.

First, the pulse signals can be processed with complex-value operations. Two channels’ complex-valued pulse signals can be expressed by

$$\scalebox{0.8}{$\displaystyle\left\{ {\begin{array}{@{}c@{}} \begin{array}{@{}c@{}} {I_{ + k}}(t )= {i_{ + k\_I}}(t )+ j{i_{ + k\_Q}}(t )= {\sigma_{in\_1}}{E_{S1}}{E_{LO1}}\cos \left[ {4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} + k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{NS}}} \right]\\ \textrm{ } + j{\sigma_{qu\_1}}{E_{S1}}{E_{LO1}}\sin \left[ {4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} + k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{NS}}} \right] \end{array}\\ \begin{array}{c} {I_{ - k}}(t )= {i_{ - k\_I}}(t )+ j{i_{ - k\_Q}}(t )= {\sigma_{in\_2}}{E_{S2}}{E_{LO2}}\cos \left[ { - 4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_\textrm{D}}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} - k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{NS}}} \right]\\ \textrm{ } + j{\sigma_{qu\_2}}{E_{S2}}{E_{LO2}}\sin \left[ { - 4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_\textrm{D}}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} - k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{NS}}} \right] \end{array} \end{array}} \right.$}$$
Second, The complex-valued pulse signals are then processed by fast Fourier transform (FFT), and the peak position of the spectrum is extracted using the gravity method to determine the IFs of up-chirp and down-chirp (depicted in Fig. 4(b)), which can be expressed as
$$\left\{ {\begin{array}{c} {{f_{IF\_up}} = 2k\mathop f\limits^\cdot \frac{R}{c} + {f_D}}\\ {{f_{IF\_down}} ={-} 2k\mathop f\limits^\cdot \frac{R}{c} + {f_D}} \end{array}} \right.$$

Finally, simultaneous range and vector velocity measurements can be obtained unambiguously from the aliasing-free beat signal, and be expressed as

$$\left\{ {\begin{array}{c} {R\textrm{ = }\frac{{({{f_{IF\_up}} - {f_{IF\_down}}} )c}}{{4k\mathop f\limits^\cdot }}}\\ {V\textrm{ = }\frac{{({{f_{IF\_up}} + {f_{IF\_down}}} )\lambda }}{4}} \end{array}} \right.$$
Nevertheless, the IFs of up-chirp and down-chirp from the original FMCW heterodyne beating can be degraded by the random phase noise including ${\phi _{N\_SR}}(t )$ introduced by the linewidth of the laser source, ${\phi _{N\_RF}}(t )$ introduced by RF devices, ${\phi _{N\_PM}}(t )$ introduced by EOPM, ${\phi _{N\_OA}}$ introduced by the laser amplifier, and the external random phase noise ${\phi _{NS}}$ introduced by the atmospheric turbulence, speckle and vibration. Precision and sensitivity can also deteriorate during range and velocity measurements.

2.4 Differential FMCW LiDAR

The differential FMCW LiDAR can be carried out by combining the received beat signals through basic mathematical operations. And the random phase disturbance can be eliminated effectively. The procedure consists of the following three stages.

First, the beat signals can be subjected to multiplication operations, with the results expressed as

$$\scalebox{0.7}{$\displaystyle\left\{ {\begin{array}{@{}c@{}} {{i_{ + k\_I}}(t ){i_{ - k\_I}}(t )\textrm{ = }{\sigma_{in\_1}}{\sigma_{in\_2}}{E_{S1}}{E_{LO1}}{E_{S2}}{E_{LO2}}\left\{ {\frac{1}{2}\textrm{cos}[{4\pi {f_D}t\textrm{ + 2}{\phi_{N\_SR}} + 2{\phi_{N\_PM}} - 2{\phi_{N\_OA}} - 2{\phi_{NS}}} ]\textrm{ + }\frac{1}{2}\textrm{cos}\left[ {8\pi k\mathop f\limits^\cdot \frac{R}{c}t\textrm{ + 2}k{\phi_{N\_RF}}} \right]} \right\}}\\ {{i_{ + k\_Q}}(t ){i_{ - k{\_}Q}}(t )\textrm{ = }{\sigma_{qu\_1}}{\sigma_{qu\_2}}{E_{S1}}{E_{LO1}}{E_{S2}}{E_{LO2}}\left\{ { - \frac{1}{2}\cos [{4\pi {f_D}t\textrm{ + 2}{\phi_{N\_SR}} + 2{\phi_{N\_PM}} - 2{\phi_{N\_OA}} - 2{\phi_{NS}}} ]\textrm{ + }\frac{1}{2}\cos \left[ {8\pi k\mathop f\limits^\cdot \frac{R}{c}t\textrm{ + 2}k{\phi_{N\_RF}}} \right]} \right\}}\\ {{i_{ + k\_I}}(t ){i_{ - k{\_}Q}}(t )\textrm{ = }{\sigma_{in\_1}}{\sigma_{qu\_2}}{E_{S1}}{E_{LO1}}{E_{S2}}{E_{LO2}}\left\{ {\frac{1}{2}\sin [{4\pi {f_D}t\textrm{ + 2}{\phi_{N\_SR}} + 2{\phi_{N\_PM}} - 2{\phi_{N\_OA}} - 2{\phi_{NS}}} ]- \frac{1}{2}\sin \left[ {8\pi k\mathop f\limits^\cdot \frac{R}{c}t + \textrm{2}k{\phi_{N\_RF}}} \right]} \right\}}\\ {{i_{ + k\_Q}}(t ){i_{ - k\_I}}(t )= {\sigma_{qu\_1}}{\sigma_{in\_2}}{E_{S1}}{E_{LO1}}{E_{S2}}{E_{LO2}}\left\{ {\frac{1}{2}\sin [{4\pi {f_D}t\textrm{ + 2}{\phi_{N\_SR}} + 2{\phi_{N\_PM}} - 2{\phi_{N\_OA}} - 2{\phi_{NS}}} ]\textrm{ + }\frac{1}{2}\sin \left[ {8\pi k\mathop f\limits^\cdot \frac{R}{c}t\textrm{ + 2}k{\phi_{N\_RF}}} \right]} \right\}} \end{array}} \right.$}$$
In an ideal scenario, the photodetector responsivities of balanced receivers can be optimized to be
$$\Lambda \textrm{ = }{\sigma _{in\_1}}{\sigma _{in\_2}}\textrm{ = }{\sigma _{qu\_1}}{\sigma _{qu\_2}}\textrm{ = }{\sigma _{in\_1}}{\sigma _{qu\_2}}\textrm{ = }{\sigma _{qu\_1}}{\sigma _{in\_2}}$$

Second, addition and subtraction operations can be conducted on the aforementioned results. The range measurement outputs can be expressed as

$$\left\{ {\begin{array}{c} {{I_{R\_\cos }}(t )= {i_{ + k\_I}}(t ){i_{ - k\_I}}(t )\textrm{ + }{i_{ + k\_\textrm{Q}}}(t ){i_{ - k{\_Q}}}(t )\textrm{ = }\Lambda {E_{S1}}{E_{LO1}}{E_{S2}}{E_{LO2}}\textrm{cos}\left[ {8\pi k\mathop f\limits^\cdot \frac{R}{c}t\textrm{ + 2}k{\phi_{N\_RF}}} \right]}\\ {{I_{R\_\sin }}(t )= {i_{ + k\_Q}}(t ){i_{ - k\_I}}(t )- {i_{ + k\_I}}(t ){i_{ - k{\_}Q}}(t )= \Lambda {E_{S1}}{E_{LO1}}{E_{S2}}{E_{LO2}}\textrm{sin}\left[ {8\pi k\mathop f\limits^\cdot \frac{R}{c}t\textrm{ + 2}k{\phi_{N\_RF}}} \right]} \end{array}} \right.$$
where ${I_R}(t )$ contains the desired distance and is FFT-processed. Using the gravity method, the optimum position of the spectrum is extracted to determine the range-related IF ${f_{IF\_R}}$.

The velocity measurement outputs can be expressed as

$$\scalebox{0.83}{$\displaystyle\left\{ {\begin{array}{@{}c@{}} {{I_{V\_\cos }}(t )= {i_{ + k\_I}}(t ){i_{ - k\_I}}(t )- {i_{ + k\_\textrm{Q}}}(t ){i_{ - k{\_Q}}}(t )\textrm{ = }\Lambda {E_{S1}}{E_{LO1}}{E_{S2}}{E_{LO2}}\cos [{4\pi {f_D}t\textrm{ + 2}{\phi_{N\_SR}} + 2{\phi_{N\_PM}} - 2{\phi_{N\_OA}} - 2{\phi_{NS}}} ]}\\ {{I_{V\_\sin }}(t )= {i_{ + k\_I}}(t ){i_{ - k{\_Q}}}(t )\textrm{ + }{i_{ + k\_\textrm{Q}}}(t ){i_{ - k\_I}}(t )\textrm{ = }\Lambda {E_{S1}}{E_{LO1}}{E_{S2}}{E_{LO2}}\sin [{4\pi {f_D}t\textrm{ + 2}{\phi_{N\_SR}} + 2{\phi_{N\_PM}} - 2{\phi_{N\_OA}} - 2{\phi_{NS}}} ]} \end{array}} \right.$}$$
$$\scalebox{0.92}{$\displaystyle\begin{array}{@{}l@{}} {I_V}(t )= {I_{V\_\cos }}(t )+ j{I_{V\_\sin }}(t )= \Lambda {E_{S1}}{E_{LO1}}{E_{S2}}{E_{LO2}}\{{\cos [{4\pi {f_D}t\textrm{ + 2}{\phi_{N\_SR}} + 2{\phi_{N\_PM}} - 2{\phi_{N\_OA}} - 2{\phi_{NS}}} ]} \\ \textrm{ } { + j\sin [{4\pi {f_D}t\textrm{ + 2}{\phi_{N\_SR}} + 2{\phi_{N\_PM}} - 2{\phi_{N\_OA}} - 2{\phi_{NS}}} ]} \}\end{array}$}$$
where ${I_{V\_\cos }}(t )$ and ${I_{V\_\sin }}(t )$ contain the desired velocity, complex-valued data can be processed using FFT. Using the gravity method, the peak position of the spectrum is extracted to determine the velocity-related IF ${f_{IF\_V}}$.

Finally, simultaneous range and vector velocity measurements can be obtained unambiguously from the aliasing-free beat signal, and be expressed as

$$\left\{ {\begin{array}{c} {R\textrm{ = }\frac{{{f_{IF\_R}}c}}{{4k\mathop f\limits^\cdot }}}\\ {V\textrm{ = }\frac{{{f_{IF\_V}}\lambda }}{4}} \end{array}} \right.$$
Please note that the ranging is immune to the random phase noise including ${\phi _{N\_SR}}(t )$ introduced by the linewidth of the laser source, ${\phi _{N\_PM}}(t )$ introduced by EOPM, ${\phi _{N\_OA}}$ introduced by the laser amplifier, and the external random phase noise ${\phi _{NS}}$ introduced by atmospheric turbulence, speckle and vibration. The only remaining phase noise ${\phi _{N\_RF}}(t )$ introduced by RF devices can be suppressed by optimizing RF devices. Meanwhile, the velocimetry is also immune to the nonlinearity of the frequency-swept laser introduced by RF devices.

3. Experimental results and discussions

3.1 Experimental setup

The experimental system includes a symmetrical dual-sideband oppositely chirped laser generator, optical amplifier, polarization controller, optical I/Q demodulation, data acquisition and processing. The 1.55 µm laser source has a nominal linewidth of approximately 100 kHz and an output power of 9.6 dB. The linear frequency-modulated RF signal from 1.8 GHz to 3 GHz depicted in Fig. 5 is generated by a digital arbitrary waveform generator (AWG) functioning as a linear frequency generator and following a pre-programmed sawtooth time-frequency spectrum in the RF domain. After combining, filtering, and amplifying the linear frequency-modulated signal generated by the AWG and the sinusoidal fundamental frequency signal generated by the fundamental frequency generator, the 9.4 GHz to 10.4 GHz sawtooth RF driving signal is produced. The pulse repetition frequency (PRF) can be adjusted between 1 and 400 kHz.

 figure: Fig. 5.

Fig. 5. (a) Spectrogram and (b) spectrum of AWG-generated sawtooth waveform. The PRF is 400 kHz. The corresponding chirp period is 2.5 µs. The duty cycle approaches 1.

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A high-speed fiber EOPM with bandwidth of 10 GHz is driven by the broadband RF signal, and the MATLAB simulation results are depicted in Fig. 6(a). The optical frequency carrier and symmetrical multi-sidebands -5th-order to +5th-order are obtained. The chirped bandwidth is 1.2 GHz, and the center spacing of adjacent sidebands is 10 GHz. The output of EOPM is split into two paths by a 50/50 fiber coupler. Two tunable fiber bandpass filters are used to adopt the ±1st OBPF to filter the residual carrier and other sidebands (Fig. 6(b)), respectively. Consequently, only the ±1st-order sidebands optical fields can pass through (Fig. 6(c)). The bandwidth of OBPF is 5 GHz and the central wavelength is around 1.55 µm. Two channels’ beams pass through the polarization controllers to achieve a pair of orthogonal polarization states. After combining two orthogonally polarized beams with a 50/50 polarization beam splitter (PBS), the optical power is amplified with an erbium-doped fiber amplifier (EDFA).

 figure: Fig. 6.

Fig. 6. Simulated optical spectrum of the output of high-speed EOPM driven by a broadband signal. (a) Schematic representation of optical frequency carrier and symmetrical multi-sidebands. (b) Schematic diagram of ±1st OBPF. (c) Schematic diagram of ±1st-order sidebands.

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In both +1st-order and -1st-order sideband FMCW optical channels, the outputs from the EDFA are split into two portions by a 1/99 fiber coupler: one small portion serves as the LO for a 2 × 4 90° fiber hybrid (Kylia, COH24), while the other serves as the transmitting beam. Transmission and reception are coaxial, and a self-developed polarization-diversity optical circulator is utilized to separate transmitting and receiving signals. The received FMCW signals are first aligned with the LO using polarization controller. In a 2 × 4 90° fiber hybrid, the laser echo mix with LO in I-channel and Q-channel, and four output signals are detected by two BPDs with responsivity of 0.9 A/W (Thorlabs PDB480C). The field-programmable gate array (FPGA) controls the critical processes, including waveform generation, high-speed AD acquisition, and digital signal processing.

3.2 Experimental results

The result of measuring the output of a high-speed EOPM with a commercial optical spectrum analyzer is depicted in Fig. 7(a). The optical frequency carrier and symmetrical multi-sidebands from -5th-order to +5th-order are obtained, which are consistent with the result shown in Fig. 5(a). After the two filtered beams with orthogonal polarizations are combined by PBS and amplified by EDFA, the ±1st-order sidebands optical spectrum is measured and illustrated in Fig. 7(b). The minimum side-mode suppression ratio (SMSR) of 30 dB is obtained.

 figure: Fig. 7.

Fig. 7. (a) Optical spectrum of the generated optical frequency carrier and symmetrical multi-sidebands. (b) Optical spectrum of the filtered ±1st-order sidebands. Based on an interferometric principle, the optical spectrum analyzer’ resolution is 140 MHz.

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The nonlinear frequency sweep is conducted by a self-homodyne unbalanced Mach-Zehnder interferometer with a short time delay [42]. The PRF is set to be 400 kHz, and the corresponding chirp period is 2.5 µs. The entire chirped optical bandwidth is 1.2 GHz. The length of the fiber delay line is 100 meters, and the delay time is approximately 0.5 µs. Therefore, only the data within 2.0 µs are displayed. Prior to the coherent reception, the optical frequency of the echo beam is shifted deliberately by amount of 50 MHz using an acousto-optic frequency shifter (AOFS) in order to simulate the Doppler shift.

The reconstructed up-chirped and down-chirped optical frequencies are depicted in Fig. 8(a) and Fig. 8(d), respectively. The residual frequency error of the LFM laser signal can be calculated by fitting the ideal time-varying instantaneous optical frequency line to the measured time-varying optical frequency and subtracted it using the least square method. Figure 8(b) and Fig. 8(e) illustrate the up-chirp and down-chirp residual frequency errors of the original FMCW without any differential processing.

 figure: Fig. 8.

Fig. 8. The nonlinear frequency sweep conducted by a self-homodyne unbalanced Mach-Zehnder interferometer with a short time delay. (a) The up-chirped optical frequencies reconstructed in time-frequency spectra. (b) The residual frequency defects of the original FMCW's up-chirp. (c) The residual frequency error measured by the differential FMCW range measurement. (d) The down-chirped optical frequencies in the reconstructed time-frequency spectra. (e) The residual frequency defects of down-chirp FMCW. (f) The residual frequency error measured by differential FMCW velocity measurement.

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Figure 8(c) also depicts the residual frequency errors measured by the range measurement of differential FMCW, which are the only remaining frequency errors introduced by RF devices. The nonlinearity of RF Devices is typically a repetitive nonlinearity with a maximal variation of 150 kHz due to the nonlinear dynamic response, which can be characterized and compensated by iterative learning pre-distortion of RF Devices. The nonlinearity can be calculated by dividing the chirp bandwidth by the maximal variation.

The residual frequency error measured by the velocity measurement of differential FMCW that is introduced by laser source, EOPM, and EDFA can be obtained, as shown in Fig. 8(f). Typically, random frequency errors are caused by laser spontaneous emission and thermal fluctuations, etc. It is therefore appropriate to use standard deviation (STD) to depict residual frequency error, and the obtained STD is 256.8 kHz.

The proposed differential FMCW LiDAR's functionality is validated using a 10 km fiber delay line. The PRF is adjusted to 8 kHz to accommodate the narrow bandwidth of BPDs. Figure 9(a) depicts the FFT comparison between original FMCW and differential FMCW. From the frequency-domain power spectra, it is evident that the original FMCW (blue curve) with a center frequency of approximately 495 MHz contains information about the laser source's intrinsic phase noise, EOPM, and EDFA. The observed significant improvement in noise performance as illustrated by differential FMCW (red curve) with a center frequency of approximately 990 MHz provides strong evidence for the efficacy of differential FMCW phase noise cancellation. As the coherence length has been significantly exceeded, the original FMCW spectral peak exhibits a Lorentzian shape. In accordance with the sweep range of 2.4 GHz FMCW, the range resolution of differential FMCW can achieve a transform-limit of 6.25 cm. The ranging precision is quantified by the standard deviation σ of the estimated distance across 4000 measurements. Figure 9(b) and (c) depict the histograms of range measurement and velocity measurement, respectively.

 figure: Fig. 9.

Fig. 9. (a) The FFT comparison of the original FMCW (blue) and the differential FMCW (red). (b) The range measurement histogram. (c) The velocity measurement histogram. The length of fiber delay line is 10 km.

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Figure 10(a) depicts the differential FMCW LiDAR photograph. As an essential performance indicator, the detection probability and ranging precision of differential FMCW LiDAR are measured, and the outdoor experimental scene is depicted in Fig. 10(b). The Lambertian object reflects 10% of light from a fixed distance of 100 meters. The relationship between the probability of detection and the SNR is depicted in Fig. 10(c). It can be seen that as SNR increases, the system's detection probability increases. When the SNR exceeds 8 dB, the probability of detection approaches one hundred percent. When the SNR is reduced to 3 dB, the probability of detection approaches 0. The probability of detection increases approximately linearly from 10% to 90% as SNR increases from 4 to 8 dB. Clearly, the SNR is one of the most significant factors influencing the detection probability. As shown in Fig. 10(d), the relationship between detection probability and range precision is also determined. With an increase in detection probability, the precision of range estimation will also improve. When the detection probability is less than 10%, the range accuracy degrades to 2.4 cm. When the detection probability is close to one hundred percent, the utmost range precision is 0.5 mm.

 figure: Fig. 10.

Fig. 10. (a) The differential FMCW LiDAR photograph. (b) The photograph of the experimental outdoor scene. (c) The measurements of the relationship between the probability of detection and the signal-to-noise ratio (SNR). (d) The relationship between detection probability and range precision as measured.

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3.3 Discussions

The symmetrical dual-sideband oppositely chirped differential FMCW LiDAR has the following advantages over the conventional triangular-waveform FMCW LiDAR:

First, the inherent opposite frequency chirp and opposite frequency offset to the optical frequency carrier of the two generated sidebands renders the range measurement immune to the internal phase noise introduced by the laser source, EOPM, and EDFA. In addition, the dual-frequency coherent detection renders the range measurement impervious to external sources, such as atmospheric turbulence, speckle, and vibration. By optimizing the RF device, the remaining phase noise (nonlinearity) introduced by RF devices can be suppressed. The noise is repetitive and can be compensated for by the pre-distorted RF drive voltage waveform, so its implementation is relatively simple. Nonlinearity in this work is approximately 1.25 × 10−4 and can be optimized to less than 5 × 10−5 in the future work. In this study, the coherence range of a laser with a 100 kHz linewidth in a standard single-mode fiber (SSMF) is 650 m, which corresponds to the maximal detection range (round trip is not considered). The 10 km fiber delay, which corresponds to approximately 15.38× intrinsic laser coherence length, still permits a range precision of 1.15 cm. Consequently, the utmost ranging precision, ranging sensitivity and ranging distance can be achieved for the long-range application.

Second, triangular FMCW LiDAR measures the distance and velocity of moving targets using frequencies with unequal difference. However, the unequal difference frequencies are actually located in distinct time slots, which makes determining the real-time velocity of rapidly accelerating targets more challenging [5,17]. The inherent opposite frequency chirps of differential FMCW LiDAR make it possible to implement simultaneous real-time measurements of range and vector velocity by frequency mixing without the need for complex post-digital signal processing. Additionally, the nonlinearity of the frequency-swept laser is not a concern for velocity measurement. However, the laser phase noise strongly impacts the received signal spectrum, which spreads from a narrow squared Cardinal Sine to a Lorentzian function for targets beyond the laser coherence length, making it uncertain the velocity estimations. Consequently, velocity measurement precision has deteriorated to 0.56 m/s in this work.

Thirdly, under the condition of guaranteeing synchronous range and velocity measurement, it is possible to double both the PRF and the effective chirped bandwidth. Consequently, both the point cloud density and the range resolution can be increased by a factor of two. In this study, only ±1st-order sideband optical fields are considered. The ±2nd-order, ± 3rd-order, and other higher symmetrical-order sidebands optical fields can also be utilized to achieve a greater chirped bandwidth and greater ranging resolution.

Finally, the universal FMCW LiDAR requires a high level of light source with a long coherence length for the long-distance measurement. The differential FMCW LiDAR is capable of overcoming obstacles. Admittedly, the required optical hybrid, BPD and ADC of the proposed system is twice than that of the conventional FMCW system. However, the hardware requirement is still acceptable to the FMCW Lidar system. The adopted method of low-cost lasers with wide linewidth, low-complexity unified EOPM modulation, and symmetric-order filtering can reduce implementation complexity despite being a time-efficient hardware solution.

4. Summary

In conclusion, a novel architecture for a differential FMCW LiDAR has been proposed, theoretically analyzed, and experimentally demonstrated. Positive and negative symmetrically oppositely chirped laser beams are simultaneously generated. A 1.55 µm all-fiber differential FMCW LiDAR prototype is demonstrated using a 9.4–10.6 GHz sawtooth FMCW RF signal with a minimum pulse period of 2.5 µs to drive the LiNbO3 waveguide EOPM. After symmetric sideband filtering and orthogonally polarized combining, a broadband oppositely chirped laser in the ±1st-order sidebands, is generated. Utilizing optical in-phase/quadrature demodulations, four-channel BPDs respectively receive orthogonally polarized optical beat signals comprising target distance and vector velocity information. The differential FMCW LiDAR can be implemented by combining the received beat signals through simple digital signal processing. Real-time spectral analysis yields the distance and velocity simultaneously, allowing for effective phase noise cancellation. Theoretical and experimental results demonstrate that the ranging is immune to the random phase noise introduced by the linewidth of the laser source, EOPM, laser amplifier, and external sources such as atmospheric turbulence, speckle and vibration. Consequently, the optimum ranging precision, range sensitivity and ranging distance can be achieved for the long-range application. Meanwhile, the velocimetry is immune to the frequency-swept laser's nonlinearity. In addition, the benefits of differential FMCW LiDAR over standard FMCW LiDAR are analyzed. This technique eliminates the need for lasers with a narrow linewidth, feedback circuits for laser linearization, and intensive post-processing, thereby reducing the complexity and cost of FMCW LiDAR systems for long-range applications. Further, we will develop differential FMCW LiDAR with high-speed and wide field-of-view scanning function for the airborne applications.

Funding

Natural Science Foundation of Zhejiang Province (LY20F050002); National Natural Science Foundation of China (61275110).

Disclosures

The authors declare no conflicts of interest.

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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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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Figures (10)
Fig. 1.
Fig. 1. Schematic diagram for operation principle of the conventional triangular-waveform FMCW LiDAR. (a) Time-domain laser frequency waveform including probe beam, local oscillator and echo beam. ${f_0}$: optical frequency carrier frequency; $B$: chirp bandwidth; LO: local oscillator; ${f_D}$: Doppler shift; $T$: the period of triangular modulation; ${\tau _S}$: time delay between echo beam and probe beam; ${\tau _L}$: time delay between LO beam and probe beam. (b) Time-domain beat frequency waveform including up-chirp and down-chirp. ${f_{IF\_up}}$: IF of up-chirp; ${f_{IF\_down}}$: IF of down-chirp. (c) Time-domain FMCW waveform including up-chirp and down-chirp. (d) Frequency-domain beat frequency waveform including up-chirp and down-chirp.

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Fig. 2.
Fig. 2. System block diagram of the symmetrical dual-sideband oppositely chirped differential FMCW LiDAR. The solid redlines are optical signals. The solid blue lines are RF signals. The solid black lines are digital signals. RF, Radio Frequency; EOPM: electro-optical phase modulator; BPD: balanced photodetector; ADC: analog-to-digital converter; OBPF: optical band-pass filter; I-Channel: in-phase channel; Q-Channel: quadrature channel.

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Fig. 3.
Fig. 3. Schematic diagram of the optical frequency domain generation of symmetrical dual-sideband opposite chirps. ${f_0}$: optical carrier frequency; $B$: chirp bandwidth; $k$: filtering order.

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Fig. 4.
Fig. 4. Schematic diagram for operation principle of the original FMCW LiDAR (a) Time-domain optical frequency of symmetrical dual-sideband oppositely chirped laser beams. (b) Time-domain beat frequency, which includes up-chirp and down-chirp. ${f_0}$: optical frequency carrier frequency; $B$: chirp bandwidth; $k$: filtering order; LO: local oscillator; ${f_D}$: Doppler shift; $T$: modulation period; ${\tau _S}$: time delay between echo beam and probe beam; ${f_{IF\_up}}$: IF of up-chirp; ${f_{IF\_down}}$: IF of down-chirp.

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Fig. 5.
Fig. 5. (a) Spectrogram and (b) spectrum of AWG-generated sawtooth waveform. The PRF is 400 kHz. The corresponding chirp period is 2.5 µs. The duty cycle approaches 1.

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Fig. 6.
Fig. 6. Simulated optical spectrum of the output of high-speed EOPM driven by a broadband signal. (a) Schematic representation of optical frequency carrier and symmetrical multi-sidebands. (b) Schematic diagram of ±1st OBPF. (c) Schematic diagram of ±1st-order sidebands.

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Fig. 7.
Fig. 7. (a) Optical spectrum of the generated optical frequency carrier and symmetrical multi-sidebands. (b) Optical spectrum of the filtered ±1st-order sidebands. Based on an interferometric principle, the optical spectrum analyzer’ resolution is 140 MHz.

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Fig. 8.
Fig. 8. The nonlinear frequency sweep conducted by a self-homodyne unbalanced Mach-Zehnder interferometer with a short time delay. (a) The up-chirped optical frequencies reconstructed in time-frequency spectra. (b) The residual frequency defects of the original FMCW's up-chirp. (c) The residual frequency error measured by the differential FMCW range measurement. (d) The down-chirped optical frequencies in the reconstructed time-frequency spectra. (e) The residual frequency defects of down-chirp FMCW. (f) The residual frequency error measured by differential FMCW velocity measurement.

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Fig. 9.
Fig. 9. (a) The FFT comparison of the original FMCW (blue) and the differential FMCW (red). (b) The range measurement histogram. (c) The velocity measurement histogram. The length of fiber delay line is 10 km.

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Fig. 10.
Fig. 10. (a) The differential FMCW LiDAR photograph. (b) The photograph of the experimental outdoor scene. (c) The measurements of the relationship between the probability of detection and the signal-to-noise ratio (SNR). (d) The relationship between detection probability and range precision as measured.

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Equations (20)

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$${E_0}(t )= {E_0}\exp [{j2\pi {f_0}t + j{\phi_{N\_SR}}(t )+ j{\phi_0}} ]$$
$${V_{RF}}(t )= M{V_{RF\_FM}}{V_{RF\_B}}\cos \left[ {\pi \mathop f\limits^\cdot {t^2} + 2\pi {f_{RF\_B}}t + {\phi_{N\_RF}}(t )} \right]$$
$$\scalebox{0.89}{$\displaystyle{E_{PM}}(t )= {E_{0\_PM}}\exp \left\{ {j2\pi {f_0}t + j\beta \cos \left[ {2\pi \left( {{f_{RF\_B}} + \frac{1}{2}\mathop f\limits^\cdot t} \right)t + {\phi_{N\_RF}}(t )} \right] + j{\phi_{N\_SR}}(t )+ j{\phi_{N\_PM}}(t )+ j{\phi_0}} \right\}$}$$
$$\scalebox{0.89}{$\begin{aligned} {E_{PM}}(t )&= {J_0}(\beta ){E_{0\_PM}}\exp [{j2\pi {f_0}t + j{\phi_{N\_SR}}(t )+ j{\phi_{N\_PM}}(t )+ j{\phi_0}} ]\\ &+ \sum\limits_{n = 1}^\infty {{J_n}(\beta ){E_{0\_PM}}\exp \left[ {j2\pi {f_0}t + j{\phi_{N\_SR}}(t )+ j{\phi_{N\_PM}}(t )+ j{\phi_0} + j\frac{{n\pi }}{2}} \right]} \\ &\left\{ {\exp \left[ {j2\pi \left( {n{f_{RF\_B}} + \frac{1}{2}\mathop {nf}\limits^\cdot t} \right)t + jn{\phi_{N\_RF}}(t )} \right] + \exp \left[ { - j2\pi \left( {n{f_{RF\_B}} + \frac{1}{2}n\mathop f\limits^\cdot t} \right)t - jn{\phi_{N\_RF}}(t )} \right]} \right\} \end{aligned}$}$$
$${E_{ + k}}(t )\textrm{ = }{E_{ + k}}\exp \left[ {j2\pi ({{f_0}\textrm{ + }k{f_{RF\_B}}} )t + j\pi k\mathop f\limits^\cdot {t^2} + j{\phi_{N\_SR}}(t )+ j{\phi_{N\_PM}}(t )+ jk{\phi_{N\_RF}}(t )+ j{\phi_0} + j\frac{{k\pi }}{2}} \right]$$
$${E_{ - k}}(t )\textrm{ = }{E_{ - k}}\exp \left[ {j2\pi ({{f_0} - k{f_{RF\_B}}} )t - j\pi k\mathop f\limits^\cdot {t^2} + j{\phi_{N\_SR}}(t )+ j{\phi_{N\_PM}}(t )- jk{\phi_{N\_RF}}(t )+ j{\phi_0} + j\frac{{k\pi }}{2}} \right]$$
$$\scalebox{0.8}{$\displaystyle\left\{ {\begin{array}{@{}c} {{i_{ + k}}{{(t )}_0} = \frac{1}{4}E_{S1}^2 + \frac{1}{4}E_{LO1}^2 + \frac{1}{2}{E_{S1}}{E_{LO1}}\cos \left[ {4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} + k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{ + k\_NS}}\textrm{ + }0} \right]}\\ {{i_{ + k}}{{(t )}_{90}} = \frac{1}{4}E_{S1}^2 + \frac{1}{4}E_{LO1}^2 + \frac{1}{2}{E_{S1}}{E_{LO1}}\cos \left[ {4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} + k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{ + k\_NS}}\textrm{ + }\frac{\pi }{2}} \right]}\\ {{i_{ + k}}{{(t )}_{180}} = \frac{1}{4}E_{S1}^2 + \frac{1}{4}E_{LO1}^2 + \frac{1}{2}{E_{S1}}{E_{LO1}}\cos \left[ {4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} + k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{ + k\_NS}}\textrm{ + }\pi } \right]}\\ {{i_{ + k}}{{(t )}_{270}} = \frac{1}{4}E_{S1}^2 + \frac{1}{4}E_{LO1}^2 + \frac{1}{2}{E_{S1}}{E_{LO1}}\cos \left[ {4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} + k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{ + k\_NS}}\textrm{ + }\frac{{3\pi }}{2}} \right]} \end{array}} \right.$}$$
$$\scalebox{0.8}{$\displaystyle\left\{ {\begin{array}{@{}c} {{i_{ - k}}{{(t )}_0} = \frac{1}{4}E_{S2}^2 + \frac{1}{4}E_{LO2}^2 + \frac{1}{2}{E_{S2}}{E_{LO2}}\cos \left[ { - 4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} - k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{ - k\_NS}}\textrm{ + }0} \right]}\\ {{i_{ - k}}{{(t )}_{90}} = \frac{1}{4}E_{S2}^2 + \frac{1}{4}E_{LO2}^2 + \frac{1}{2}{E_{S2}}{E_{LO2}}\cos \left[ { - 4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} - k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{ - k\_NS}}\textrm{ + }\frac{\pi }{2}} \right]}\\ {{i_{ - k}}{{(t )}_{180}} = \frac{1}{4}E_{S2}^2 + \frac{1}{4}E_{LO2}^2 + \frac{1}{2}{E_{S2}}{E_{LO2}}\cos \left[ { - 4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} - k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{ - k\_NS}}\textrm{ + }\pi } \right]}\\ {{i_{ - k}}{{(t )}_{270}} = \frac{1}{4}E_{S2}^2 + \frac{1}{4}E_{LO2}^2 + \frac{1}{2}{E_{S2}}{E_{LO2}}\cos \left[ { - 4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} - k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{ - k\_NS}}\textrm{ + }\frac{{3\pi }}{2}} \right]} \end{array}} \right.$}$$
$${\phi _{ + k\_NS}} = {\phi _{ - k\_NS}} = {\phi _{NS}}$$
$$\left\{ {\begin{array}{c} {{i_{ + k\_I}}(t )= {\sigma_{in\_1}}{E_{S1}}{E_{LO1}}\cos \left[ {4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} + k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{NS}}} \right]}\\ {{i_{ + k\_Q}}(t )= {\sigma_{qu\_1}}{E_{S1}}{E_{LO1}}\sin \left[ {4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} + k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{NS}}} \right]} \end{array}} \right.$$
$$\left\{ {\begin{array}{c} {{i_{ - k\_I}}(t )= {\sigma_{in\_2}}{E_{S2}}{E_{LO2}}\cos \left[ { - 4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_\textrm{D}}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} - k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{NS}}} \right]}\\ {{i_{ - k\_Q}}(t )= {\sigma_{qu\_2}}{E_{S2}}{E_{LO2}}\sin \left[ { - 4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_\textrm{D}}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} - k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{NS}}} \right]} \end{array}} \right.$$
$$\scalebox{0.8}{$\displaystyle\left\{ {\begin{array}{@{}c@{}} \begin{array}{@{}c@{}} {I_{ + k}}(t )= {i_{ + k\_I}}(t )+ j{i_{ + k\_Q}}(t )= {\sigma_{in\_1}}{E_{S1}}{E_{LO1}}\cos \left[ {4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} + k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{NS}}} \right]\\ \textrm{ } + j{\sigma_{qu\_1}}{E_{S1}}{E_{LO1}}\sin \left[ {4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_D}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} + k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{NS}}} \right] \end{array}\\ \begin{array}{c} {I_{ - k}}(t )= {i_{ - k\_I}}(t )+ j{i_{ - k\_Q}}(t )= {\sigma_{in\_2}}{E_{S2}}{E_{LO2}}\cos \left[ { - 4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_\textrm{D}}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} - k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{NS}}} \right]\\ \textrm{ } + j{\sigma_{qu\_2}}{E_{S2}}{E_{LO2}}\sin \left[ { - 4\pi k\mathop f\limits^\cdot \frac{R}{c}t + 2\pi {f_\textrm{D}}t\textrm{ + }{\phi_{N\_SR}} + {\phi_{N\_PM}} - k{\phi_{N\_RF}} - {\phi_{N\_OA}} - {\phi_{NS}}} \right] \end{array} \end{array}} \right.$}$$
$$\left\{ {\begin{array}{c} {{f_{IF\_up}} = 2k\mathop f\limits^\cdot \frac{R}{c} + {f_D}}\\ {{f_{IF\_down}} ={-} 2k\mathop f\limits^\cdot \frac{R}{c} + {f_D}} \end{array}} \right.$$
$$\left\{ {\begin{array}{c} {R\textrm{ = }\frac{{({{f_{IF\_up}} - {f_{IF\_down}}} )c}}{{4k\mathop f\limits^\cdot }}}\\ {V\textrm{ = }\frac{{({{f_{IF\_up}} + {f_{IF\_down}}} )\lambda }}{4}} \end{array}} \right.$$
$$\scalebox{0.7}{$\displaystyle\left\{ {\begin{array}{@{}c@{}} {{i_{ + k\_I}}(t ){i_{ - k\_I}}(t )\textrm{ = }{\sigma_{in\_1}}{\sigma_{in\_2}}{E_{S1}}{E_{LO1}}{E_{S2}}{E_{LO2}}\left\{ {\frac{1}{2}\textrm{cos}[{4\pi {f_D}t\textrm{ + 2}{\phi_{N\_SR}} + 2{\phi_{N\_PM}} - 2{\phi_{N\_OA}} - 2{\phi_{NS}}} ]\textrm{ + }\frac{1}{2}\textrm{cos}\left[ {8\pi k\mathop f\limits^\cdot \frac{R}{c}t\textrm{ + 2}k{\phi_{N\_RF}}} \right]} \right\}}\\ {{i_{ + k\_Q}}(t ){i_{ - k{\_}Q}}(t )\textrm{ = }{\sigma_{qu\_1}}{\sigma_{qu\_2}}{E_{S1}}{E_{LO1}}{E_{S2}}{E_{LO2}}\left\{ { - \frac{1}{2}\cos [{4\pi {f_D}t\textrm{ + 2}{\phi_{N\_SR}} + 2{\phi_{N\_PM}} - 2{\phi_{N\_OA}} - 2{\phi_{NS}}} ]\textrm{ + }\frac{1}{2}\cos \left[ {8\pi k\mathop f\limits^\cdot \frac{R}{c}t\textrm{ + 2}k{\phi_{N\_RF}}} \right]} \right\}}\\ {{i_{ + k\_I}}(t ){i_{ - k{\_}Q}}(t )\textrm{ = }{\sigma_{in\_1}}{\sigma_{qu\_2}}{E_{S1}}{E_{LO1}}{E_{S2}}{E_{LO2}}\left\{ {\frac{1}{2}\sin [{4\pi {f_D}t\textrm{ + 2}{\phi_{N\_SR}} + 2{\phi_{N\_PM}} - 2{\phi_{N\_OA}} - 2{\phi_{NS}}} ]- \frac{1}{2}\sin \left[ {8\pi k\mathop f\limits^\cdot \frac{R}{c}t + \textrm{2}k{\phi_{N\_RF}}} \right]} \right\}}\\ {{i_{ + k\_Q}}(t ){i_{ - k\_I}}(t )= {\sigma_{qu\_1}}{\sigma_{in\_2}}{E_{S1}}{E_{LO1}}{E_{S2}}{E_{LO2}}\left\{ {\frac{1}{2}\sin [{4\pi {f_D}t\textrm{ + 2}{\phi_{N\_SR}} + 2{\phi_{N\_PM}} - 2{\phi_{N\_OA}} - 2{\phi_{NS}}} ]\textrm{ + }\frac{1}{2}\sin \left[ {8\pi k\mathop f\limits^\cdot \frac{R}{c}t\textrm{ + 2}k{\phi_{N\_RF}}} \right]} \right\}} \end{array}} \right.$}$$
$$\Lambda \textrm{ = }{\sigma _{in\_1}}{\sigma _{in\_2}}\textrm{ = }{\sigma _{qu\_1}}{\sigma _{qu\_2}}\textrm{ = }{\sigma _{in\_1}}{\sigma _{qu\_2}}\textrm{ = }{\sigma _{qu\_1}}{\sigma _{in\_2}}$$
$$\left\{ {\begin{array}{c} {{I_{R\_\cos }}(t )= {i_{ + k\_I}}(t ){i_{ - k\_I}}(t )\textrm{ + }{i_{ + k\_\textrm{Q}}}(t ){i_{ - k{\_Q}}}(t )\textrm{ = }\Lambda {E_{S1}}{E_{LO1}}{E_{S2}}{E_{LO2}}\textrm{cos}\left[ {8\pi k\mathop f\limits^\cdot \frac{R}{c}t\textrm{ + 2}k{\phi_{N\_RF}}} \right]}\\ {{I_{R\_\sin }}(t )= {i_{ + k\_Q}}(t ){i_{ - k\_I}}(t )- {i_{ + k\_I}}(t ){i_{ - k{\_}Q}}(t )= \Lambda {E_{S1}}{E_{LO1}}{E_{S2}}{E_{LO2}}\textrm{sin}\left[ {8\pi k\mathop f\limits^\cdot \frac{R}{c}t\textrm{ + 2}k{\phi_{N\_RF}}} \right]} \end{array}} \right.$$
$$\scalebox{0.83}{$\displaystyle\left\{ {\begin{array}{@{}c@{}} {{I_{V\_\cos }}(t )= {i_{ + k\_I}}(t ){i_{ - k\_I}}(t )- {i_{ + k\_\textrm{Q}}}(t ){i_{ - k{\_Q}}}(t )\textrm{ = }\Lambda {E_{S1}}{E_{LO1}}{E_{S2}}{E_{LO2}}\cos [{4\pi {f_D}t\textrm{ + 2}{\phi_{N\_SR}} + 2{\phi_{N\_PM}} - 2{\phi_{N\_OA}} - 2{\phi_{NS}}} ]}\\ {{I_{V\_\sin }}(t )= {i_{ + k\_I}}(t ){i_{ - k{\_Q}}}(t )\textrm{ + }{i_{ + k\_\textrm{Q}}}(t ){i_{ - k\_I}}(t )\textrm{ = }\Lambda {E_{S1}}{E_{LO1}}{E_{S2}}{E_{LO2}}\sin [{4\pi {f_D}t\textrm{ + 2}{\phi_{N\_SR}} + 2{\phi_{N\_PM}} - 2{\phi_{N\_OA}} - 2{\phi_{NS}}} ]} \end{array}} \right.$}$$
$$\scalebox{0.92}{$\displaystyle\begin{array}{@{}l@{}} {I_V}(t )= {I_{V\_\cos }}(t )+ j{I_{V\_\sin }}(t )= \Lambda {E_{S1}}{E_{LO1}}{E_{S2}}{E_{LO2}}\{{\cos [{4\pi {f_D}t\textrm{ + 2}{\phi_{N\_SR}} + 2{\phi_{N\_PM}} - 2{\phi_{N\_OA}} - 2{\phi_{NS}}} ]} \\ \textrm{ } { + j\sin [{4\pi {f_D}t\textrm{ + 2}{\phi_{N\_SR}} + 2{\phi_{N\_PM}} - 2{\phi_{N\_OA}} - 2{\phi_{NS}}} ]} \}\end{array}$}$$
$$\left\{ {\begin{array}{c} {R\textrm{ = }\frac{{{f_{IF\_R}}c}}{{4k\mathop f\limits^\cdot }}}\\ {V\textrm{ = }\frac{{{f_{IF\_V}}\lambda }}{4}} \end{array}} \right.$$
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