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Abstract
To investigate the tolerance of laser linewidth, a stable and variable light source is essential. This study proposes a novel method for generating a variable linewidth and variable light source using a lithium niobite phase modulator, achieved through the application of white Gaussian noise and random walk noise with a limited wall. The proposed approach was thoroughly examined through equations, simulations, and experiments. Further, a practical application was demonstrated while investigating the effects of variable linewidth and shape on FMCW LiDAR performance. The results highlight the importance of optimizing the light source for LiDAR applications to ensure the maximum accuracy and resolution.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Laser linewidth is a crucial factor in various applications [1], including optical atomic clocks [2–5], optical coherent systems [6–9], and Frequency-Modulated Continuous-Wave (FMCW) Light Detection and Ranging (LiDAR) [10]. In such systems, the full width at half maximum (FWHM) of the laser determines its precision, range, and sensitivity [11–14]. Further, investigations on the tolerance of these systems to the laser linewidth is critical for reducing costs [15–17] and improving the performance [18,19]. The increasing demand for autonomous driving necessitates the use of inexpensive and high-quality sensors [18,19]. Consequently, to reduce costs, coherent FMCW LiDAR has been proposed; however, its tolerance can only be evaluated using laser diodes with different linewidths while maintaining a stable wavelength and optical power [20–22]. Consequently, ensuring consistent and controlled changes in the laser linewidth while maintaining the output power and wavelength stability is a significant challenge [23,24]. Two methods can be used to address this challenge. The first approach involves directly controlling the current and temperature of a laser diode [25]. However, the control of the laser current to produce a stable linewidth is difficult and requires a complex real-time feedback system [25]. Consequently, the cost of a variable-linewidth laser increases, which renders it not suitable for many systems. Furthermore, stable laser linewidths using this approach are limited to 53 MHz [25], which is not suitable for systems such as LiDAR, owing to the requirement of narrower linewidths ranging from kHz to MHz. An alternative approach involves the use of an external lithium niobite (LN) phase modulator (PM) and addition of white-frequency and white Gaussian noises [6,26–28]. However, this method is typically applicable only to a single noise source, which may not accurately simulate real-world noise conditions [6,27,28]. Consequently, the application of this method may not fully reflect the practical limitations of laser linewidth in various systems. Previous studies have applied random walk noise to generate a variable linewidth laser; however, the randomness of the walk steps presents a limitation. This is because an unlimited noise cannot be inserted into the input power of the LN PM [6]. The mirror method overcomes this input limitation but results in a new problem. The low-frequency part of the noise power is excessively low, thereby resulting in the center part of the laser linewidth in the spectrum being unmodulated [6,26]. To solve this issue, a new method that utilizes white Gaussian noise to compensate for low-frequency noise was proposed [26]. However, the previous method source of white Gaussian noise produced periodic noise, whereas the low-frequency phase noise in the laser was nonperiodic and contained various types of noise [26]. Thus, this was addressed by using a true random analog noise source mixed with white-frequency noise. Further, equations for different noises and laser linewidths in the spectrum are also required for precise laser linewidth control. Different quantities and types of noise affect the laser linewidth shape, which can affect the performance of the FMCW LiDAR [26,27,28]. In FMCW LiDAR configurations utilizing coherent receivers, signal aliasing is effectively mitigated, ensuring that distance does not hinder velocity detection [11]. As a result, the velocity signal is exclusively influenced by laser phase noise [26]. Investigating the effects of different laser linewidth shapes on LiDAR systems reveals the diverse noise types impacting FMCW LiDAR performance.
This study aims to thoroughly investigate the relationship between the laser linewidth and system performance in various applications, particularly in FMCW LiDAR. Thus, a novel approach using an external LN PM with added white Gaussian noise and random walk noise with a limited wall was proposed to generate a variable linewidth and a variable light source. This approach overcame the limitations of traditional methods such as the requirement of different laser types and introduction of errors because of the need to replace the light source between the experiments [29]. This study first derived the equations for white Gaussian noise and random walk noise with a limited wall on the laser linewidth. Subsequently, simulation and experimental results of different noise effects on the laser linewidth and a method for managing the laser linewidth were demonstrated. Finally, the effects of the variable linewidth and variable linewidth shape lasers on the coherent FMCW LiDAR were proven. By providing an in-depth investigation of laser linewidth tolerance, this research contributes to the wider application of FMCW LiDAR and other systems and paves the way for more effective and efficient design and optimization.
2. Theory
2.1 Equations
The laser linewidth spectrum is closely related to the phase noise of the laser, which is controlled by an external LN PM. The PM linearly increases the half-wave voltage within a specific frequency range. However, while the laser linewidth spectrum is examined in the frequency domain, the phase noise is typically discussed in the time domain. To bridge this gap and describe their relationship, the power spectral density (PSD) of phase: ${\textrm{S}_\phi }\textrm{}({\textrm{ra}{\textrm{d}^2}/\textrm{Hz}} )$ and frequency: ${\textrm{S}_\textrm{F}}\textrm{}({\textrm{H}{\textrm{z}^2}/\textrm{Hz}} )$ fluctuations were introduced [30]. ${\textrm{S}_\phi }$ and ${S_F}$ characterize the phase and frequency noise, respectively. The latter has been analyzed in the β-separation theory [31,32]. This theory suggests that a high-modulation index noise contributes to linewidth, whereas a low-modulation index noise contributes only to the wings [31,32]. However, there remains a discrepancy between the theory and the actual laser linewidth, and the relationship between the two types of noises and the laser linewidth must be determined quantitatively. To improve this theory, first, the noise sources must be confirmed, where the white frequency and white Gaussian noise contribute to the linewidth and wings of the laser linewidth spectrum, respectively. Random walk noise is a suitable model for white-frequency noise. The relationship between the PSD of the phase and frequency noises can be expressed as follows [30]:
Second, a laser light field should be proposed to connect to the laser linewidth. The laser light field is expressed as $\textrm{E}(\textrm{t} )= {\textrm{E}_0}\textrm{exp}[{\textrm{i}({2\mathrm{\pi }{\mathrm{\upsilon }_0}\textrm{t} + \phi (\textrm{t} )} )} ]\textrm{}$(complex representation), where ${\textrm{E}_0}$ is the amplitude of the optical field, ${\mathrm{\upsilon }_0}$ is the carrier frequency, and $\phi (\textrm{t} )$ is the phase fluctuation. Considering the Wiener-Khintchine theorem, the spectrum of the laser linewidth ${\textrm{S}_\textrm{E}}(\textrm{f} )$ according to the autocorrelation function after Fourier transform: $\mathrm{{\cal F}}({\textrm{E}(\textrm{t} )} )\times {\mathrm{{\cal F}}^\mathrm{\ast }}({\textrm{E}(\textrm{t} )} )$ can be written as [33]
Using Eqs. (4) and (5), Eq. (3) can be further expressed as
Furthermore, to facilitate mathematical analysis, the phase noise PSD: ${\textrm{S}_\phi }(\textrm{f} )$ can be examined in a dimensionless form. This can be achieved by combining the phase noise PSD with the fractional frequency fluctuation: ${\textrm{S}_\textrm{y}}(\textrm{f} )$, using the subsequent equation [30].
The PSD ${S_E}({v - {v_0}} )$ in carrier frequency domain can also be written as:
Random walk noise is a type of white-frequency noise. The white frequency noise in the PSD of the phase fluctuations ${\textrm{S}_\phi }(\textrm{f} )$ can be expressed as ${\textrm{v}_0}^2{\textrm{h}_0}{\textrm{f}^{ - 2}}$, where ${h_0}$ depends on the step length and the time interval of each step, and ${\textrm{f}^{ - 2}}$ implies that the noise processing obeys power laws with integer exponents. Therefore, when white frequency noise is added, the PSD ${S_E}({v - {v_0}} )$ in the carrier frequency domain can be written as:
The modest random walk is used as the noise, wherein each step has the possibility of 50% walking up and down, the step length is $\varDelta L$, and the time interval of each step is $\varDelta t$. Moreover, in order to transform the electrical signal into phase fluctuations, the electrical noise signal is defined as: ${\textrm{S}_{\textrm{VR}}}\textrm{}({{\textrm{V}^2} \cdot \textrm{H}{\textrm{z}^{ - 1}}} ).$ This signal is converted into phase noise via the PM, necessitating consideration of the half-voltage ${\textrm{V}_\mathrm{\pi }}$ of the PM. The relationship between the ${\textrm{S}_\phi }(\textrm{f} )$ and ${\textrm{S}_{\textrm{VR}}}\; $ can be articulated as follows [34]:
The PSD of the phase fluctuations, ${\textrm{S}_\phi }(\textrm{f} ),\textrm{}$ can be rewritten as ${\mathrm{\pi }^2}\varDelta {\textrm{L}^2}/({2{\mathrm{\pi }^2}\varDelta \textrm{t}{\textrm{V}_\mathrm{\pi }}^2{\textrm{f}^2}} )$. Using this equation for the phase fluctuations, the PSD in the carrier frequency domain can be expressed as a Lorentzian shape:
However, assuming that the cut-off frequency of the white Gaussian noise is ${f_c}$, the PSD of the white Gaussian noise as ${\textrm{S}_{\textrm{VW}}}\textrm{}({{\textrm{V}^2} \cdot \textrm{H}{\textrm{z}^{ - 1}}} )$, and combining it with Eqs. (2), (3), and (10), the PSD in the carrier frequency domain can be expressed as follows:
The relationship between the laser linewidth spectrum and the two types of noises is demonstrated by Eqs. (12) and (13). The use of white Gaussian noise in the laser linewidth is highly dependent on the cutoff frequency and output power. Moreover, the step length and time interval determine the Lorentzian linewidth spectrum when using random-walk noise. Further, the half-voltage of the LN PM also affects the linewidth spectrum. However, without limitations, a normal random walk has an infinite border. The mirror method was used to ensure a border for walking by fitting the input power limitation of the PM. The extremely large walk steps increase the reflections and result in the white frequency noise losing its low-frequency part. This results in the center part of the laser linewidth being unmodulated. To solve this problem, white Gaussian noise also plays the role of a low-frequency component compensation. In addition, different noise sources affect the laser linewidth shape, which is further discussed in Section 3.
2.2 Laser linewidth measurement setup
The experimental setup for measuring the laser linewidth is illustrated in Fig. 1. Two narrow-linewidth fiber laser diodes (NKT Photonics Koheras BASIK) with wavelengths of 1550.075 and 1550.090 nm were used. The optical power of the laser diodes was suitably attenuated using optical attenuators (Anritsu MN935A2) in one optical path, and the polarization was controlled using a polarization controller (PC). The light phase was modulated by an external LN PM (Sumitomo Osaka Cement T.PMH1.55 S). In addition, random walk and white Gaussian noises were applied to the PM, which was generated by an arbitrary waveform generator (Tektronix AWG 7102) and an analog noise source (Noisecom UFX 7107). Amplifiers (SHF 115 BP, NF BA4805, and FEMTO DHPVA-101) and electrical attenuators (Agilent 8494 B, Fairview microwave SA 4090) were used to regulate the noise output power. Further, a low-frequency variable bandpass filter (NF 3628), which included an electrical amplifier, was used to control the cutoff frequency of the white Gaussian noise. A power splitter (PS, Mini-Circuits, ZFRSC-183-S+) mixed the two types of noises, which constituted the linewidth control unit. Random walk noise with a limited wall (step length: 0.015 Vpp) and time interval (0.1 ns) was amplified with an amplifier from 1 to 23 dB, while white Gaussian noise was varied in terms of the cut-off frequency and power. Following the hybridization of the two optical paths by the optical coupler (OC), and based on coherent interference, a 0.015 nm wavelength was detected by the photodetector (PD, Sevensix Inc 12.5 Gb/s Optical Receiver). Furthermore, a real-time spectrum analyzer (RSA, Tektronix RSA 3308A) was used to transform the detected signal into a laser linewidth in the spectrum.
3. Results of laser linewidth
3.1 Simulation results
To highlight the significance of the numerical simulations in understanding the relationship between noise and laser linewidth, the following results were obtained under the assumption that the half-wave voltage of the PM is 1 V. Figure 2 (a) shows the laser linewidth results for different cut-off frequencies of the white Gaussian noise. The red line, with a cut-off frequency of 100 kHz, yielded a linewidth of 328 kHz, whereas the blue line, with a cut-off frequency of 50 kHz, resulted in a narrower linewidth of 162 kHz. Both noises had a PSD of -48 dBrad2/Hz and a resolution bandwidth (RBW) of 1 Hz. Thus, the higher the cutoff frequency of the white Gaussian noise, the wider the resulting laser linewidth. Figure 2 (b) shows the noise PSD of both noises, where the blue and red lines represent the 50 and 100 kHz cut-off frequency, respectively. The PSD resolution was 1 Hz. Figure 2 (c) shows the results of changing the noise PSD. The red and brown lines exhibited - 48 and -128 dBrad2/Hz PSD for white Gaussian noise, respectively. The corresponding noise PSD of these noises are shown in Fig. 2 (d). The results demonstrated that the carrier frequency of the laser linewidth was not modulated if the noise PSD was low, as indicated by the brown line. In contrast, the red line exhibited a linewidth of 328 kHz. Therefore, the noise PSD is crucial for determining the modulation ability of the carrier frequency of the laser linewidth. To achieve a laser linewidth with a central component, an adequate noise PSD is required. Furthermore, the cutoff frequency of the white Gaussian noise determines the width of the laser linewidth when there is sufficient noise PSD. Thus, a higher cutoff frequency led to a wider laser linewidth.
Fig. 2. Simulation results on (a) the laser spectrum of different cut-off frequency with same power, (b) the respective noise PSD, (c) the laser spectrum of different power with same cut-off frequency, (d) the respective noise PSD.
The variable linewidth laser also utilized random walk noise in the experiment. However, owing to the power limitation of the PM, random walk always has a limited wall. To investigate the effect of a random walk without a limited wall and perform comparisons with its limited wall counterpart, a simulation was performed involving millions of data points that required large memory. To accomplish this, high-performance computing using Amazon Web Services was used. The simulation results are shown in Fig. 3, where the step length and time interval were set to 0.01 V and 0.1 ns, respectively. As shown in Fig. 3(a), the laser linewidth span was 40 MHz and the RBW was 1 kHz. The brown line represents the random walk without the limited wall with a theoretical laser linewidth of 159.154 kHz; the simulation result was 155.927 kHz. There was a certain error between the theory and actual results owing to the use of random walk noise. This is because it is not a pure white frequency noise and rather was mixed with a bit of white noise. The noise produced by the PC was pseudorandom noise, which resulted in impure noise. In addition, owing to the randomness of the random walk, its border became unlimited with the increase in steps, resulting in high-voltage noise in the time domain that exceeded the power limitations of the LN PM. To address this issue, a mirror method was implemented to limit the border of the random walk. In this simulation, a 0.5 V limited wall was set, and the red line represents a random walk with a 0.5 V limited wall. The spectrum shows that the center part of the laser linewidth was not modulated, and a clear carrier frequency was present. The noise PSD in Fig. 3(b) reveals that the low-frequency part of the noise with a limited wall was a low PSD white noise, which is insufficient to modulate the carrier frequency of the laser, thereby resulting in the laser linewidth leaking into the center part. To compensate for this, low-frequency noise is necessary, and the best option was the noise PSD that resembled the random walk without a limited wall. However, the use of same noise PSD as the random walk without a limited wall results in a high voltage in the time domain that exceeds the maximum input power of the LN PM. To address this issue, a low-frequency compensation method was employed using white Gaussian noise as the compensation source. The experimental results of this compensation method are discussed in the next Section.
Fig. 3. Simulation results on (a) the laser spectrum of the random walk without limited wall and with a limited wall, and (b) the respective noise PSDs.
3.2 Experiments results
The effect of the modulation signal on the half-wave voltage of the PM is shown in Fig. 4(a), where an acousto-optic (AO) effect was induced in the range of 10–100 MHz. However, undesired interference between the AO and electro-optic (EO) effects was also observed, resulting in a strange change in the half-wave voltage of the PM in this range [35]. Furthermore, the impact of white Gaussian noise on the laser linewidth is shown in Fig. 4(b), where different powers of white Gaussian noise were applied to the laser system. The results revealed that when the power of the white Gaussian noise increased to 28 dBm, the carrier line of the laser linewidth was completely modulated. Whereas, decreasing the power of the white Gaussian noise caused the carrier line to remain unmodulated, with a larger margin observed at lower power levels. To modulate the center part of the laser linewidth spectrum, a suitable power of white noise connected to the half-wave voltage of the PM is required. Simultaneously, different cut-off frequencies of the white Gaussian noise significantly affected the laser linewidth spectrum, as shown in Fig. 4(c). At a constant power of 26 dBm, increasing the cut-off frequency of the white Gaussian noise from 100 to 300 kHz caused the laser linewidth to increase from 222 to 1443 kHz. In this experiment, noise power was transformed into phase fluctuations by linking it to the half-wave voltage of the PM. To convert the electrical signal noise into phase fluctuations in conjunction with the PM's half-wave, Eq. (10) can be employed to facilitate the conversion of the noise PSD.The power spectrum of white noise prior to amplification is depicted in Fig. 4(d). This precautionary measure safeguards the RSA from potential damage due to high-power, low-frequency noise. Consequently, the noise is measured before amplification and subsequently raised to a constant power of 26 dBm. To enhance the visibility of the noise cutoff frequency, the horizontal scale utilizes a logarithmic representation. Due to measurement limitations, only eight points are present below 80 kHz, highlighting minor fluctuations within the low-frequency domain and making them more discernible. Therefore, to obtain the central part of the laser linewidth spectrum, an appropriate white Gaussian noise power must be connected to the half-wave voltage of the white noise. The cutoff frequency of the white Gaussian noise is a decisive factor in determining the linewidth of the laser, which is consistent with the simulation results. Subsequently, the laser linewidth spectrum results of the random walk noise with a limited wall mixed with white Gaussian noise were analyzed. The laser linewidth spectrum is shown in Fig. 5(a), with white Gaussian noise having a cutoff frequency of 200 kHz. The Voigt fit model was used to fit the laser linewidth, resulting in a well-defined Voigt lineshape and linewidth of 898 kHz. The noise spectrum of the mixed noise before amplification is shown in Fig. 5(b) with a resolution of 10 kHz. The mixed noise exhibited a well-defined low-frequency component, which resulted in a well-defined Voigt lineshape when using the Voigt fit model.
Fig. 4. Results of (a) half-wave voltage of the phase modulator, (b) laser spectrum of different power of white Gaussian noises, (c) different cut-off frequency of white Gaussian noises, and (d) noise spectrum of different cut-off frequency.
Fig. 5. Laser spectrum of the (a) white Gaussian noise as low-frequency compensation, and (b) noise spectrum of the mixed noise.
3.3 Results of variable shapes laser linewidth
The PseudoVoigt model was used to measure the laser linewidth shapes [36]. The equation can be expressed as
The numerical simulation results are shown in Fig. 6. The simulation results showed that both the power and cutoff frequency of the white Gaussian noise affected the Lorentz components. These effects followed a hyperbolic trend, with the cutoff frequency mainly affecting the Lorentz components. This is because white Gaussian noise has a Gaussian distribution and the PM has a linear characteristic. Consequently, the linewidth after modulation became a Gaussian distribution, which was displayed as a Gaussian shape in the laser linewidth spectrum. In addition, a large cutoff frequency implied more white Gaussian noise components at low frequencies, which increased the Gaussian components and contributed to the laser linewidth. Further, a larger cutoff frequency resulted in fewer Lorentz components, while the power of the white Gaussian noise also affected the Lorentz components, although it was less than the cutoff frequency. This was because the quantity of the Gaussian distribution in the modulation could not be changed, and it mainly affected the linewidth shape. A higher power results in the power of the linewidth (A) in Eq. (14) being higher, and at the same linewidth, the peak of the Lorentz shape was higher than that of the Gaussian shape. Therefore, a higher power can cause the linewidth shape to incorporate more Lorentz components. The hyperbolic relationship originated from the effects of the cutoff frequency in the denominator of Eq. (14) and the effects of power in the numerator. Furthermore, the white Gaussian noise, subjected to low-pass filtering, led to a diminished power in the spectral domain. Specific instances of this reduction reveal unique attributes reminiscent of random walk noise. Within the experimental framework, these noise features contribute to a modest extent to the Lorentz components, albeit with a relatively minimal overall influence. However, the simulation results also showed that obtaining a pure Lorentz shape (100%) using the white Gaussian noise compensation method required high power and a low cut-off frequency of the white Gaussian noise, which is difficult to realize in actual experiments. The red star in the figure indicates the simulation results for a cut-off frequency of 210 kHz and power of 26 dBm, yielding a Lorentz component of 29% (0.29), which could be used in the experiment.
Fig. 6. Simulation results of different white Gaussian noise power and cut-off frequency on the Lorentz components.
The experimental results for the laser linewidth spectrum are shown in Fig. 7. When the cut-off frequency was 220 kHz and the power of the white noise was 26 dBm, the laser linewidth was 1.078 MHz and the Lorentz component was 1.2% (0.012). The center of the linewidth was consistent with the fit line, whereas the wings of the linewidth exhibited a significant difference in Fig. 7(a). This was because the amplifier (NF 3628) amplified only the low-frequency signal after filtering, resulting in a large gap between the wings of the linewidth and the fitted line. Further, the laser linewidth spectrum in Fig. 7(b) was obtained by using the white Gaussian noise with a cut-off frequency of 210 kHz and 26 dBm power mixed with random walk following 23 dB amplification. The linewidth was 1.017 MHz with Lorentz components of 29% (0.29). The center part of the linewidth matched well with the fit line, whereas the wings of the linewidth still exhibited certain errors. This was because of the lower power of the white Gaussian noise used in the experiment at a high frequency (wings of the linewidth) than the theoretical value, and the mixed noise at a high frequency (wings of the linewidth) was lower than the ideal one. Consequently, the wing linewidth could not be completely fitted with the PseudoVoigt model.
Fig. 7. Laser spectrum of (a) only white Gaussian noise and (b) white Gaussian noise mixed random walk noise.
4. Variable shapes linewidth laser and effects of the coherent FMCW LiDAR
4.1 Setup of variable shapes linewidth laser on the FMCW LiDAR
The experimental setup (Fig. 8) comprised an FMCW LiDAR unit integrated with a linewidth unit. The light was modulated using a Mach-Zehnder modulator (MZM, Sumitomo Osaka Cement T.MZH 1.5-10PD-ADC-101) at the null point, and the linear frequency modulated (LFM) signal, spanning 0.1-4.1 GHz, can be formulated as $\mathrm{\omega } = {\mathrm{\omega }_0} + 2\textrm{kt}$, where $\textrm{k}$ represents the rate of exponential change in frequency. In this configuration, ${\mathrm{\omega }_0}$ was set at 2 GHz, and the k was assigned a value of 3.183 × 1011. This signal was generated utilizing an AWG (Tektronix AWG7102B), which was integrated into the MZM. An erbium-doped fiber amplifier (EDFA, FITEL ErFA1215) and an optical bandpass filter (OBPF, Optoquest TFA-1550-S/F) were used to amplify the light signal and reduce amplified spontaneous emission (ASE). Further, to simulate the distance and velocity, the optical path was divided into two paths: the local optical (LO) and probed (PO) paths. An AO modulator (AOM; Gooch & Housego T-M110-0.2CJ-3-F2S) was used to simulate the Doppler frequency shifts (DFSs), which was shifted by 110 MHz to simulate a speed of 85.525 m/s. Next, a single-mode fiber (SMF) was used to simulate a distance of 431 m. In addition, EDFA (FITEL, ErFA11021B) and OBPF were used to maintain power and reduce ASE. Finally, the two paths were connected to a coherent receiver (Pure Photonics ECO-031837) and the balanced photodetectors were fed into a digital phosphor oscilloscope (DPO, Agilent DSO81204B) for measurement.
Fig. 8. Experimental scheme of phase-diversity coherent FMCW LiDAR using variable linewidth and variable linewidth shapes laser.
4.2 Effects of variable shape laser linewidth on coherent FMCW LiDAR
The linewidth of the laser was adjustable from 400 kHz to 1 MHz, and to achieve different linewidth shapes, two types of noises were introduced: only white Gaussian noise and the incorporation of both white Gaussian and random walk noises. Figure 9(a) illustrates the Lorentz components of these two noises and uses the PseudoVoigt model to fit the laser linewidth in the 10 MHz span and 20 kHz RBW spectrum. The Lorentz components were found to be below 1.4% (0.014) when using only white Gaussian noise. To achieve more Lorentz components, the cut-off frequency and power of the white Gaussian noise must be precisely controlled in conjunction with the random walk noise. However, owing to the limitations of filter precision and electrical attenuators, the maximum Lorentz components that could be produced in the 10 MHz span and 20 kHz RBW were approximately 25% (0.25). Consequently, it was possible to produce different Lorentz components of 25 and 1.4% while also varying the laser linewidth from 400 kHz to 1 MHz. These findings facilitated the examination of the different effects of the laser linewidth and different linewidth shapes on the velocity signal. The velocity signal, in this context, refers to the Doppler-shifted target signal obtained from the current of the coherent receiver after undergoing a Fast Fourier Transform (FFT). Figure 9(b) shows that the intensity of the velocity signal decreased from 0.034 to 0.018 a.u. with an increase in the laser linewidth for both 25% Lorentz component linewidths. Similarly, in the 1.4% Lorentz components linewidth, the velocity signal dropped from 0.05 to 0.02 a.u. with increase the laser linewidth. The intensity difference was attributed to the power fluctuation in the electrical signals, which were measured using DPO. Figure 9(c) demonstrates that the linewidth of the velocity signals in the 25% Lorentz components increased from 1.4 to 4.3 MHz, whereas in the 1.4% Lorentz components, it increased from 1.2 to 3.8 MHz. Both the velocity signals experienced an increase in linewidth with an increase in the laser linewidth, and the effects of the two types of Lorentz components on the velocity signal intensity and linewidth were not significantly different. However, the velocity position frequency varied between the two types of Lorentz-component linewidths. Here, the velocity position frequency refers to the central portion (220 MHz) of the velocity signal within the spectral domain. As shown in Fig. 9(d), the position frequency of the velocity signal fluctuated with an increase in the laser linewidth. In the 25% Lorentz components linewidth, the fluctuation range of the velocity position frequency was 1.26 MHz, while in the 1.4% Lorentz components linewidth it was 2.14 MHz. To determine the cause of this difference, the spectrum of the velocity signal must be examined. The velocity signal spectrum using only white Gaussian noise is shown in Fig. 10(a), whereas Fig. 10(b) displays the velocity signal spectrum using random walk and white Gaussian noise. As evident from the 10 kHz resolution spectrum, the white Gaussian noise produced a Gaussian-shaped velocity signal with a linewidth of 3.8 MHz. In contrast, the velocity signal shape was not entirely Gaussian when the noise contained both white and random walk noise. It contained both Lorentz and Gaussian components, resulting in a linewidth of 4.3 MHz. A pure Gaussian shape has a flat center in the spectrum, whereas a pure Lorentz shape has a sharp center. Therefore, as the noise increased, the velocity position fluctuated, and the flat center was more prone to shifting than the sharp center. This explains the difference in the fluctuation range. Thus, a linewidth with more Lorentz components exhibits less fluctuation in the velocity signal position frequency than a Gaussian shape.
Fig. 9. Relationship between laser linewidth and (a) laser linewidth shape, (b) intensity, (c) linewidth, and (d) position of velocity signal.
Fig. 10. Velocity signal spectrum of (a) white Gaussian noise and (b) white Gaussian noise mixed random walk noise.
5. Conclusion
This study presented the equations that describe the impact of different noises on the laser linewidth spectrum. The results showed that the power and cut-off frequency of white Gaussian noise affected the laser linewidth, with a higher power and frequency resulting in a larger linewidth. Similarly, a random walk with a limited wall affected the linewidth, with a larger step length and shorter time intervals leading to larger linewidths. However, a random walk with a limited wall resulted in a laser linewidth spectrum in the absence of center-part modulation, which can be compensated for using white Gaussian noise. The findings suggest that a combination of white Gaussian noise and random walk noise with a limited wall is required to achieve optimal laser linewidth. Furthermore, our investigation found that different types of noises affected the shape of the laser linewidth, with the 25% and 1.4% Lorentz components producing different shapes within the 0.4–1 MHz variable laser linewidth in the 10 MHz span and 20 kHz RBW. An increase in the laser linewidth resulted in a decrease in the intensity of the velocity signal and an increase in the linewidth of the velocity signal in both the 25% and 1.4% Lorentz components. The difference between the two components was owing to the position frequency of the velocity signal and the vibration range of the Gaussian shapes, which is larger than that of the Lorentz shape. These findings provide important insights into the impact of laser linewidth and noise on the performance of LiDAR systems and other applications and highlight the need for precise management and optimization of the laser linewidth.
It is imperative to emphasize that the presented methodology has not yet attained an unadulterated Lorentz linewidth profile, and the periphery of the laser linewidth spectrum exhibits discrepancies with the Pseudo-Voigt model. Future endeavors must focus on refining these aspects. Moreover, enhancements are necessitated to minimize expenses and augment user-friendliness, rendering the technique more viable for applications in the realm of optics.
Funding
National Institute of Information and Communications Technology (04901, 06001); Japan Society for the Promotion of Science (JP22H05196).
Acknowledgments
We would like to thank Editage for English language editing.
Disclosures
The authors declare no conflict 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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