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Abstract
The flexible membranes used in MEMS tunable VCSELs are so small and light that thermally induced vibrations can impact laser performance. We measure the thermal vibration spectrum of such a membrane showing peaks at the spatial vibration mode resonant frequencies of the membrane/plate. These vibrations result in a theoretical floor to the linewidth of the VCSEL. Frequency domain LiDAR and optical coherence tomography systems can get around this thermal linewidth limit with adequate clock measurement and processing. Essentially an OCT/LiDAR sweep with a concomitantly measured clock is a feed-forward linewidth reduction scheme. This can be achieved because the membrane resonances are relatively low frequency. LiDAR ranging out to 9 meters has been demonstrated with a resolution of 13 μm, close to the transform limit for the 70 nm sampling range.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
As microelectromechanical systems (MEMS) technology used in tunable lasers is pushed to faster and faster speeds, additional technical considerations come to the fore. This is especially true of tunable VCSELs [1–3]. Two of the additional issues we cover here are: (1) higher order vibration modes [4,5], and (2) thermally-driven vibration noise [6,7]. Generally, higher-order vibrations are to be avoided and designed around. Thermally driven vibrations broaden the linewidth of the VCSEL and must be compensated for in long-range OCT and LiDAR applications. These methods are essentially feed-forward linewidth narrowing techniques [8].
Here we measure the vibration spectrum of a MEMS tunable VCSEL, recording many mechanical vibration resonances of the flexible silicon plate that the movable mirror is mounted on. Further, we directly image the spatial modes of these plates by OCT of a moving membrane. Finally, LiDAR ranging out to 9 meters has been demonstrated with a resolution of 13 $\mu$m at a 500 Hz sweep rate.
Another group has demonstrated a 1310 nm tunable VCSEL LiDAR swept at 100 kHz, obtaining 15 $\mu$m resolution with a 1.5-meter maximum range. [9] That work was made possible by a 25-GHz bandwidth photonic integrated circuit receiver and 16-GHz oscilloscope digitizer. In this work we were restricted to the 1 GHz bandwidths of a commercial balanced receiver and DAQ board. The value of our work is that we push to deeper depths and look into the limitations for this type of LiDAR from thermal jitter in the MEMS tunable VCSEL source.
2. Tunable VCSEL performance
Our tunable VCSEL has a design outlined earlier [1]. This design is routinely capable of a 90 nm sampled bandwidth, although the device here tunes only 70 nm. One goal here is to demonstrate OCT-like axial resolutions at LiDAR-like ranges. We show a resolution of 13 $\mu$m at a range of 9 meters, which is close to the theoretical number listed in Fig. 1. The system sweep was loosely structured to allow a 10-meter range with a 2 GS/s DAQ and 1 GHz bandwidth balanced receiver. These requirements suggested the sweep rate of 500 Hz.
Fig. 1. Sweep metrics for the VCSEL are plotted on the left. The scan plan table on the right fleshes out the numbers for the raw hardware clock and for the resampled data. Resolution limits are projected assuming a von Hann signal processing window.
The laser’s wavelength and power sweep trajectory are shown in Fig. 1(left) and the sweep parameters are tabulated in the “scan plan” in Fig. 1(right). Considering that frequency domain LiDAR is really just OCT at long range, the scan plan is laid out like an OCT application with sweep ranges, optical resolutions, and the maximum measurable distance or “Nyquist depth.” Apart from the low sweep speed and large number of sample points, the sweep parameters are not unlike a typical OCT system. Most of the numbers in the table refer to the slower main sweep, not the fast retrace. For example, the duty cycle refers to the 86% of the sweep that is devoted to data collection. The remaining 14% is dedicated to the turn-arounds and rapid snapback of the return sweep.
The clock interferometer is a fiber Mach-Zehnder with a 1370 mm added fiber path in one leg. That corresponds to a 2-meter air path and a free spectral range of 0.15 GHz. The clock transitions 126599 cycles in the 69.9 nm sweep. In software, the clock is multiplied 33.1$\times$ to 4194304 samples, which is $2^{22}$. Insisting on an even power of 2 samples dictates the Nyquist depth of 16.57 meters.
The section of the clock waveform shown in Fig. 2 illustrates the frequency jitter issue associated with MEMS VCSELs. It is a mostly sinusoidal waveform generated by the fiber Mach-Zehnder clock interferometer when probed by the sweeping the VCSEL at −11 GHz/$\mu$s. The accordioning of the waveform is due mostly to fundamental thermal vibrations of the MEMS membrane [3,6,7].
Fig. 2. Here is an expanded section of the clock waveform. It is a mostly sinusoidal waveform but shows evidence of tuning-rate jitter.
3. MEMS vibration spectrum
The tunable VCSEL used in this study is partly based on a silicon MEMS structure described previously [1,2]. The VCSEL is run as a swept source at a 500 Hz sweep rate. It sweeps from 1014 to 1090 nm with a 86% duty cycle. The clock has a 0.15 GHz free spectral range resulting in a clock frequency of about 74 MHz.
The signal processing to extract the thermal noise spectrum proceeds in the following four steps. Note that this clock phase extraction method [10] is commonly used in OCT for sweep linearization, but here is it also used here as a jitter measurement method.
- 1. A 1.3 ms length of the clock signal is digitized at 2 GS/s.
- 2. The clock phase is computed through a Hilbert transform method. [10]
- 3. The tuning rate is evaluated from the clock phase by Savitzky-Golay differentiation. [11]
- 4. The tuning rate power spectral density is found by Fourier transformation.
The fundamental mechanical resonance of the device in Fig. 3 occurs at 330 kHz, with second and third resonances at 1.58 and 3.68 MHz respectively. Many other mechanical resonances occur at even higher vibration frequencies. The $\delta$-function-like lines at 500 kHz, 1.06 MHz, 1.5 MHz and other frequencies are marked with black dot or triangle symbols. They are due to parasitic signals from switching regulators leaking into the high-voltage MEMS drive circuit. These parasitic sources prevent the VCSEL from obtaining the thermal linewidth limit, although only by a factor of 2.
Fig. 3. (a) Instantaneous and average VCSEL tuning rates in the time domain. (b) Tuning rate noise spectrum for the swept MEMS tunable VCSEL. Many vibration modes of the MEMS silicon plate are seen. The very sharp $\delta$-function-like peaks are parasitic electronic signals from the electronic drive circuitry marked by the black dot and triangle symbols. (c) Tuning noise spectrum. (d) RMS tuning noise found through integration of the tuning noise power spectrum.
With the MEMS mirror modeled as a mass on a spring, the following equation for the VCSEL thermal motion linewidth has been presented in several sources [3,6,7]:
Here $k$ is the spring constant, $FSR$ the free spectral range of the VCSEL, and $k_BT$ the thermal energy.We estimate a 1600 N/m spring constant for the main vibration mode of the MEMS structure. With $FSR = 24~THz$ and $\lambda = 1050~nm$, $\Delta \nu _{RMS}$ is estimated to be 73 MHz, in reasonable agreement with values presented in Figs. 3 and 4.
Fig. 4. Tuning rate fluctuations in the time domain. The slow ramp due to the laser sweep was subtracted off using a 12th order polynomial fit. The remainder we classify as “jitter”.
The data indicate optical frequency fluctuations of $\sim$0.2 GHz RMS, corresponding to a membrane movement of 0.03Å. About half of that is thermal noise, half technical noise. The thermal component is mostly due to the (1,0) vibration mode and most of the noise occurs at <1.2 MHz. Each vibration mode has thermal energy $k_BT$. Theoretically, for this device we expect a thermal motion of $\sqrt {k_BT/k}$ [1,3,6,7] or 0.016Å. The frequency-domain RMS noise estimate is confirmed in the time domain as seen in Fig. 4. The clock phase noise is estimated by subtracting out the average tuning phase trajectory, found from a 12th order polynomial fit, from the instantaneous phase.
Figure 3(a) shows the instantaneous and average tuning rates in $GHz/\mu s$. The tuning rate amplitude spectrum ($GHz/\mu s~/~\sqrt {Hz}$) is shown in Fig. 3(b) and the data is replotted as a frequency jitter spectrum ($GHz/\sqrt {Hz}$) in Fig. 3(c). The accumulated RMS frequency jitter in Fig. 3(d) is found by integrating up the jitter power spectrum and taking the square root. That plot shows that about half of the “jitter” is due to the thermal motion of the fundamental membrane mode and half due to the technical noise from the switching regulators powering the electronics board.
4. MEMS vibration spatial modes
There is a tendency to call the MEMS structures in tunable VCSELs “membranes.” Strictly speaking a membrane is a thin structure under tension such as a drumhead. These structures are actually “plates” of some thickness, for which there are closed-form solutions for circular symmetry [4,5].
Axsun’s tunable Fabry-Perot filters and tunable VCSELs [1] depend on electrostatically actuated movement of a silicon MEMS (Micro Electro Mechanical System) membrane to displace a dielectric mirror with submicron precision. These membranes resemble vibrating plates clamped at the edges [4,5] in that they can exhibit complex spatial vibration modes at specific resonant frequencies. We have observed these modes in sample structures using a standard 1310 nm, 100 kHz, swept source OCT scanner by imaging the membrane surface while driving it sinusoidally with high voltage. The MEMS driving frequency is typically higher than the laser’s sweep rate and nonsynchronous. The moving surface creates sidebands similar to FM radio. In the OCT image the sidebands look like phantom surfaces riding above and below the membrane. These phantom surfaces encode an image of the movement. Figure 5 shows idealized vibration modes from a circular plate with a fundamental resonance at 330 kHz. The nodes are blue and red, and the anti-nodes in white. Only the (n,0) modes are excited by the circularly symmetric electrodes.
Fig. 5. Low order spatial modes for a circular vibrating plate clamped at the edges. Only the (n,m) m=0 modes show up in the noise spectrum. The m>0 modes have an antinode at the center of the plate and do not contribute to piston motion of the mirror that tunes the laser.
This is not a stroboscopic snapshot of a vibrating membrane. Our laser sweeps at 100 kHz and the vibration resonances of interest are in the 100’s of kHz to several MHz range, much faster than the OCT sweep. However, the information we desire is encoded into the reflected signal. This is a phase-sensitive measurement, but it does not depend on intra-sweep phase stability. It depends on phase stability within the sweep provided by the natural, gapless, k-clocking process. The photodiode signal from a vibrating membrane is given by
Equation 3 consists of a main signal and two sidebands. In the OCT image, you see the reflection from the MEMS membrane at depth $d$ and two phantom images at $d\pm (cf)/(2R)$ on either side of the membrane. It is one of these phantom images that we display in an en face mode. Note that the sideband amplitudes are proportional to $z_0$, so the absolute value of the membrane amplitude is encoded in the image. This is how we make a picture of the vibration mode.
Figure 6 shows the MEMS structure and plate geometry along with pictures of (1,0)-like and (2,0)-like vibration modes. This is data from membrane design “A,” not the exact design used in our VCSEL, although it is similar.
Fig. 6. (a) Silicon MEMS structure, called a mirror on membrane (MOM). (b) Microscope photograph of a membrane of design "A". (c) En face experimental image of the (1,0)-like MOM vibration mode. (d) Experimental image of the (2,0)-like mode.
Along with the expected results of Fig. 6 we have measured the more complex vibration mode shown in Fig. 7. This is membrane design “B” that is also not used in the VCSEL. This asymmetric mode shape is due to the five short tethers used in this membrane design that destroy the reflection symmetry of the device.
Fig. 7. Measurement of a complex MOM vibration mode. (a) En face view of the phantom image at 2.343 MHz. (b) Photograph of membrane design "B" with five tethers. (c) Finite element analysis calculation of similar mode.
While finite element analysis is an important and often accurate tool for designing membranes, Fig. 7 shows that experiments can deviate from theory. The experimental image in Fig. 7(a) does not exactly correspond to Fig. 7(c), perhaps due to fabrication tolerances. An experimental tool can be helpful with analysis of very complicated membrane designs when membrane performance deviates from theoretical expectations. It is a simple measurement for those with access to an OCT scanner. In principle, such a method could be used to track down the other spatial modes associated with the resonant peaks in Fig. 3(b). They do not obviously correspond to any circular plate mode and it is not clear how accurately finite element analysis could model them.
The effective spring constants of each mode are an essential parameter in the thermal linewidth calculation. To the extent that the classic circular plate model approximates our situation, the spring constants can be calculated from that along with the resonant frequencies. First, the mean kinetic energy of the membrane, $K$, is calculated. Since this is equal to the mean potential energy $U$, the effective spring constant can be calculated.
Here $D$ is the density of silicon, and $T$ the MEMS membrane thickness, and $a$ the plate radius. The membrane deflection at $r=0$ is $x_0$ and the angular vibration frequency is $\omega$. The vibration mode [5] is
The values of $k$ are quantized for each $n$, and this $k$ should not be confused with the spring constants $k_n$ above. The $k$ values are the zeros of the following equation:
Many of these modes are plotted in Fig. 5. The following table lists the resonant frequencies and effective spring constants for this calculation:
Membrane resonances are very apparent in the tuning rate noise of Fig. 3(b), The stiffer effective spring constants at higher mode order seen in Table 1 account for the diminution of tuning noise (Fig. 3(c)) at higher mode orders. This is why most of the linewidth of the VCSEL is accounted for by just the fundamental vibration mode plus the technical noise.
Table 1. Effective spring constants.
5. LiDAR system
The optical layout of the LiDAR fiber-optic interferometer and scanner are shown in Fig. 8. The input from the VCSEL is split into the clock interferometer (0.15 GHz FSR) and the main interferometer consisting of a heavily attenuated reference leg and a signal leg directed out a telescope and galvo scanner. The first 200 mm of the LiDAR trace is occluded by fiber-optic Rayleigh scattering. This occluded range was minimized by cutting the blue output fiber short. The scattering is positioned to overlap in the main and conjugate images. The zero-delay point occurs 139 mm before the FC/APC facet of the output fiber (air length). The reference arm contains a polarization controller and the reference power is attenuated by two 10-dB-loss splices.
Fig. 8. Diagram of the main and clock fiber-optic interferometers for the LiDAR plus the free-space scanning optics.
The output "telescope" consists of a fiber-optic collimator (Thorlabs F260APC-1064) with a 3.37 mm $1/e^2$ beamwidth. The collimator is followed by a focusing lens of 1 meter or 2.5 meter focal length in the LiDAR experiments. The collection efficiency versus range is shown Fig. 9. An IR viewing card (Thorlabs VRC2) was used as a consistent Lambertian-like scattering target because the beam could be seen and centered on the card.
Fig. 9. LiDAR collection efficiency vs. range for a 1 meter and 2.5 meter focusing lens. The signal is plotted in dB above the noise floor. The target was an IR viewing card.
6. Signal processing
The LiDAR signals are constructed by software resampling utilizing clock and signal data digitized at 2 GS/s. The 0.15 GHz clock cycles are determined by the Hilbert method [10].
Here, H() is the Hilbert bandpass filter. The clock cycles are multiplied 33.1$\times$ into 0.45 MHz sampling instances for a Nyquist depth of 16.565 meters. An ad hoc filtering method that worked well was used. First, the clock is processed by a Hilbert-bandpass filter. For the bandpass, the frequency domain content is set to zero for f<50 MHz and f>110 MHz. The clock phase is unwrapped, multiplied 33.1$\times$ and divided by $2\pi$. This clock signal is further low pass filtered using a second order Savitsky-Golay with a time window of 0.50 $\mu s$. From there, the signal point density was multiplied 4$\times$ by band-limited interpolation. This results in more accurate resampling at ranges where the signal frequency approaches the 1 GHz Nyquist limit. Then the time signal is resampled at integer processed clock intervals.
The clock interferometer has 1370 mm of HI-1060 fiber in one leg, which has significant dispersion. The clock cycles were corrected by the following factor:
where the normalized optical frequency is which ranges from −1 to +1. The ends of the sampled tuning range are $\nu _1$ and $\nu _2$.The signal arm has its own dispersion from the 190 mm of HI-1060 fiber in the main interferometer. The dispersion is compensated by multiplying the resampled signal by
The factors of 8.8 in Eq. (9) and 0.94 in Eq. (11) were empirically derived but are close to that expected from the dispersion of fused silica. Presumably any difference has to do with the waveguide dispersion of the fiber.
The time delay between signal and clock need to be matched, which can be done in software. When all of these steps are taken, good signal reconstruction can be had as in Fig. 10(left). Note that the clock and signal traces need to be taken simultaneously because the VCSEL frequency jitter needs to be correlated in the signal and clock waveforms. Mismatched clock and signal traces result in poor point spreads. To illustrate this, the next clock sweep (2 ms later) was used to reconstruct the point spread in Fig. 10(center). There is very poor sweep correlation trace to trace. When the signal is used as its own clock in reconstruction of the point spread we call it “autoclocking.” A very clean point spread with low sidebands is obtained as in Fig. 10(right) because the signal and “clock” are perfectly correlated. Autoclocking is an analysis method useful for measuring coherence lengths from strong interference signals at a series of depths when a separate k-clock is not available. It is not useful for normal OCT practice.
Fig. 10. LiDAR point spread for a mirror reflection at 5 meter range in air. (left) Point spread with signal and clock digitized simultaneously. The small line at 7 meters is a detector reflection artifact. (center) Point spread reconstruction using a clock from the next sweep (2 ms later) after the signal. (right) “Autoclocked” point spread where the signal itself is used as the clock.
Eleven point spreads with ranges between 0.5 and 9 meters are shows in Fig. 11(left). In these measurements, there was only a collimator with no focusing lens at the output. Long paths were formed by bouncing the beam back and forth across a 2.5 meter optical table with gold-coated mirrors. Most of the decrease in peak readings with range are due to decreasing optical power returned to the receiver. The beam from the output collimator diverges significantly over the almost 20 meter round trip. The light red curve is a direct reading of returned optical power using a DC power meter. The black dotted line is the autoclocking result. By conventional OCT measures, the “coherence length” is in excess of 10 meters given only 3.7 dB difference between the autoclocked and conventionally clocked point spread reconstructions at 8.9 meters.
Fig. 11. (left) Eleven LiDAR point spreads vs. range. The light red line demarks DC power meter readings and the black dotted curve the autoclocking powers. (center) The time delay between signal and clock was adjusted to minimize point spread width. (right) Near transform-limited point spread widths were achieved out to 9 meters in range.
The signal waveform needs to be delayed in software because of the long time of flight in the LiDAR application. This ensures maximum correlation between the signal and clock waveforms and results in the maximum point spread amplitude and minimum sideband energy. The red dots of Fig. 11(center) show the empirically found best delays. The blue line plots the delay, $D$, vs. range, $R$.
The −3 ns offset accounts for the fiber delays in Fig. 8.
7. LiDAR ranging demonstration
High axial resolution at long range is demonstrated in Fig. 12. The height profile of the face of a Lincoln penny is less than 0.4 mm. A height map made with a convention OCT system is shown on the top line. That is compared with LiDAR images at 1.4 m range (center line) and 3.1 m range (bottom line). The axial resolutions mapped into spectral colors are similar for all the images. A VCSEL FMCW LiDAR system can have OCT-like axial resolution. However, the lateral resolution suffers at range because of beam divergence as can be seen from the speckle patterns in the rightmost column of the images.
Fig. 12. Penny height images from an OCT scanner and at two ranges with the LiDAR. This shows that LiDAR can have comparable axial resolution to OCT at long range, but that lateral resolution is sacrificed.
The data collection and computational load in this kind of ranging is significant. Each A-line starts off with the digitization of 3,239,936 signal and clock data points. That is two 1.6 ms time traces digitized at 2 GS/s. The signal data is resampled to 4,194,304 points. Each A-line required about 1 second processing time with unoptimized MATLAB code on a modest personal computer. The LiDAR images in Fig. 12 were 300$\times$300 pixels in size and took many hours to collect. That led to the vertical striping artifact in the bottom line of LiDAR images. On and off cycles of the HVAC system in the lab slowly changed the range on the order of 0.05 mm during the many hours of the measurement. The vertical axis was scanned faster than then horizontal. Rapidly collecting and cashing the data would remove these artifacts but that is a large task requiring specialized hardware. In principle, these images could be acquired in 180 seconds. These images would require 1.2 TB of storage and a streaming data rate 6.5 GB/s. This type of hardware is commercially available, but nonstandard.
Another aspect of the computation load is that the time delay to the target needs to be known for optimization of the signal processing as seen in Fig. 11(center). This may require two data processing runs. One to determine the range to target, and the second to optimize the image once the delay is known.
8. Summary
We expand the study of MEMS structures for tunable VCSELs to the behavior of higher order vibration modes. We image the modes and measure their thermal vibration noise spectrum. When one asks what is the “linewidth” of the VCSEL, it is a difficult question to answer; one needs to specify a timescale. Here we have data on long time scales where the VCSEL optical frequency jitters due to MEMS thermal vibration. Swept source OCT methods can track out these variations and are essentially a feed-forward linewidth reduction system [8]. Previously we have demonstrated LiDAR ranging out to 10 meters [1]. Here we present more detailed data out to 9 meters. The question remains under study as to the maximum practical range of an OCT/LiDAR system using a MEMS tunable VCSEL. Presumably this limit could be extended by engineering stiffer MEMS membranes or adopting other mechanical tuning mechanisms that would have lower thermal vibration amplitudes.
This work was an investigation on the physics of MEMS thermal vibrations. The LiDAR measurements were added as a demonstration of principle. Whether high-resolution imaging at long range is a compelling application remains to be seen. This would require adoption of high-performance digitization and storage hardware, highly parallelized computation, and optimized algorithms.
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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