We set the output surface of the collimator as the zero point in ranging. Then the stand-off distance Rmea can be expressed as:
$${R_{mea}}=\frac{{{L_{ex}} - {L_f}}}{n}$$
3
where Lf stands for the OPL in fiber and n for the refractive index. A series of experiments are performed to verify the validity for ranging noncooperative targets.
A. Effectiveness of Compensation
The performance of the compensation method based on DC-LFI is evaluated first. To simulate the unwanted OPL drift in propagation, we use a vibrating aluminum sheet as the ranging target, the effective reflectivity of which is 10− 7, calibrated in experiments with a Ф42.5mm collecting aperture. The sheet is attached to another piece of PZT actuator (CoreMorrow, Inc. XMT 150), and placed 9.6 meters away from the collimator of the system. A power amplifier (Aigtek, ATA-4051) drives the sinusoidal vibration of PZT as well as the target, and the amplitude is 2µm. Figure 3(a) lists the ranging results with or without compensation under various frequencies from 1Hz to 1kHz, and the details can be seen in the right column. High-frequency components of the OPL vibration make the bandwidth of the signal broadened obviously, which leads to measurement error. Contrarily, with compensation, the signal peaks can be recognized easily, but also with identical linewidth and peak positions. The proposed method is effective for the unwanted OPL drift from Hz to kHz-scale, which covers the frequency band of environment noise in most cases.
Another experiment is performed to test the validity in remote target ranging. The target is 162.2m away from the system. Figure 3(b) shows the results of ranging in comparison, and Fig. 3(c) is the corresponding OPL drift recorded by DC-LFI during measuring, caused by the environmental factors. The compensated signal peak, with higher SNR and narrower bandwidth, confirms the effectiveness of our method for OPL drift compensation.
B. Precision and Linearity
To evaluate the precision of the system, repeated measurements are conducted with OPL drift compensation. The target is the aluminum sheet mentioned above, and it is fixed on a displacement stage (PI, Inc. M511), 152.76m away from the collimator. The measurements are repeated 10 times, and the standard deviation is calculated. Then, the stage, as well as the target, moves 500µm in each step and 5mm in total. In each position, similar tests are performed. The total data are displayed in Fig. 4(a)., where the experimental data indicates the step motion clearly. Meanwhile, the standard deviation of 10 measurements in 10 positions is also analyzed respectively. They all possess a standard deviation, σ, no more than 0.067mm, corresponding to 1.3×10− 6 relative precision, calculated by 3σ/Rmea.
The linearity test of the system is also carried out, which demonstrates the nonlinear error in measurements. The precision of the stage is 50nm, which can be taken as the standard. The stage drives the aluminum sheet forward by 10 cm, covering the whole travel range of the stage. Meanwhile, the ranging results from the FSFI are recorded. The acquired data, linearly fitting, and residual error are marked in Fig. 4(b). The maximum residual error is 83µm, corresponding to 8.3×10− 4 linearity within 10 cm at 152.76m.
C. Resolution
In most previous research, the resolution of the ranging system refers to the ability to distinguish between several simultaneously present targets [3], and it is quantified as the full width at half maximum (FWHM) of the signal peak. The resolution, ΔR = c/2B in theory, is determined by the frequency swept bandwidth B. Similar tests are performed in our system. The aluminum sheet works as the non-cooperative target, and stands 152m away. The signal peak is plotted in Fig. 5(a). Numerically, the FWHM is 0.94 mm, slightly worse than the theory, 0.75mm with 200GHz tuning. This deviation originates mainly from the fiber dispersion of the delay line in the auxiliary interferometer.
Besides, we evaluate the actual resolution of the system, ranging two targets at different distances at the same time. As Fig. 5(b) shows, the probe beam illuminates the aluminum sheet and its iron support. The thickness of the sheet provides a distance difference. The ranging result is illustrated in Fig. 5 (c), where the two-peak signal is recognizable, and the resolution is realized in a real sense. In Fig. 5(d), the gap value fluctuates within 87µm in 10 times repetitions, and the average is 1.076 mm. The system resolves two non-cooperative targets, even after hundreds of meters of propagation, and the experimental resolution is better than 1.1mm.
D. SNR
By Eq. (1), the beat signal can be amplified in the laser cavity. On the other hand, in the band of the significant amplification, the laser intensity (LI) noise also increases remarkably, which is manifested as the RO peak. Additionally, the PD noise is another important source, contributing to the total noise. The normalized power spectra in dB of the signal and noise versus the beat frequency are shown in Fig. 6 (a). The analysis of the SNR is conducted in two cases. (1) In the absence of PD noise, the SNR of FSFI is independent of the beat frequency and only shot-noise limited [31], which can be expressed as:
$$SN{R_{FSFI}}=\frac{{P{R_{fb}}}}{{2h\nu \Delta F}}$$
4
where P and ΔF represent the laser output power and demodulation bandwidth. ν is the frequency of the laser, and it can be approximated as the central frequency of sweeping. h is Planck's constant. (2) With the PD noise, presumed to be white noise, the SNR can also reach shot-noise limitation within a frequency range close to the RO frequency, where the intensity of RO is several orders stronger than the PD noise. In this frequency band, the PD noise will not affect the SNR, even if it is much stronger than the shot noise. The SNR of the beat signal with and without the PD noise is shown in Fig. 6 (b). Due to the enhancement, the system gets rid of the limitation of PD noise in the LN noise-dominating band. Beyond the band, the SNR is also enhanced, merely smaller in value. Totally, the system exhibits ultra-high detection sensitivity and satisfying SNR for weak echo signals, even with low probe-beam power.
In our system, the noise equivalent power (NEP) of PD is 69.5pW/Hz1/2, which is tested in experiments. The beam power received by PD is 22µW, which corresponds to 2.3pW/Hz1/2 shot noise [43]. Compared with the PD noise, shot noise hardly contributes to the total noise. In the frequency band close to the RO peak, the LI noise is dominant, up to several nW/Hz1/2. According to the theory above, when the beat frequency fb is located in this band, it gets remarkable amplification and provides a satisfying SNR. We evaluate the minimum detected feedback power in this regime. A cube mirror is used as the target to calibrate the power attenuation. An adjustable attenuator is installed before the collimator, which can change the optical attenuation of the probe beam. The demodulation bandwidth ΔF is 17Hz. The SNR-attenuation curve is shown in Fig. 6 (c). The slope of the fitting is 1.027, which verifies the linear relationship in Eq. (4). Meanwhile, the fitting results predict the ideal detection limit (i.e. SNR = 1) is -127.33dB. The output power of the laser is 230µW, corresponding to the minimum echo power of 0.0425fW. The theoretical detection limit is -137.22dB corresponding to 0.0043fW, calculated by Eq. (4). The results from experiments are comparable with the theory, considering the experimental loss, like mode mismatch between the feedback beam and local oscillator.
It is verified that the proposed ranging system exhibits high sensitivity to echo signals with sub-milliwatt output power. This performance demonstrates the system has the potential in ranging non-cooperative targets farther. Figure 6(d) provides the results of ranging with a 460m stand-off distance. An iron block is selected as the target, with 8.3×10− 7 effective reflectivity under a Ф42.5mm collecting aperture. The signal peak is obvious compared with the noise baseline, and the SNR is over 20dB.
E. Comparison with Conventional FSI-based Ranging system
To have a clear comparison with a conventional FSI-based ranging system (i.e. without optical feedback), we perform another experiment. An FSFI-based and a conventional FSI-based system are set up, and they target the same object under identical conditions, where they employ the same laser source, collimator, auxiliary interferometer, detector, and measure with equal probe beam power. More details of the setup are demonstrated in Supplementary 1 Note 4. The red and black solid lines stand for the PD noise and LI noise respectively, and the LI noise is stronger than the PD noise. The power spectra of the ranging signals are shown in Fig. 7(a)-(b). The SNR of the conventional ranging system is 8.8dB, while the FSFI-based is 39.4dB. The over 103 SNR enhancement verifies the high echo-signal sensitivity of the proposed system when the beat frequency fb is within the LI-noise dominating frequency band.
Additionally, another experiment is performed when fb is out of the LI-noise dominating band. We replace another PD with higher NEP, where the PD noise is dominant. The power spectra of two systems in Fig. 7(c)-(d). In this case, the beat signal fb in the conventional system is submerged in the noise, while the SNR is 25.2dB in the FSFI-based system. The results show the FSFI also exhibits better SNR than conventional LFI in the detector-noise dominating band.