A Variable Carrier Generation for Heterodyne LDV with an Optical Phase-locked Loop

A heterodyne laser Doppler vibrometer (LDV) requires a frequency shifter to generate a carrier for the vibration information in the detector signal. The carrier frequency should be carefully selected in order to obtain the intended measurement range with a demodulation bandwidth that avoids existing noise sources. The traditional frequency shift in heterodyne LDV is realized with an acoustooptical modulator, which can only generate a fixed carrier frequency. In this paper, a variable carrier generation method based on an optical phase locked loop (OPLL) is demonstrated. Our setup implements a feedback loop to control the phase of the second laser to synchronize a reference laser. In lock-in status the carrier for the vibration information associated with a local oscillator signal. In this paper, we obtain the laser diode parameters relevant to performance and on the design of the photodetector and loop filter. Finally, the performance of the lock-in OPLL, variable carrier as well as velocity measurement is reported and the reliability of the proposed method is evaluated.


Introduction
Laser Doppler vibrometry (LDVy) shows excellent potential in the measurement of velocity and vibration and it was firstly reported for fluid velocity measurements by Yeh and Cummins in 1964 [1].Proportional to the velocity of the device under test, the Doppler frequency shift is the basic principle of LDVy as light is scattered by a moving device and coherent with a reference beam.An LDV employs the optical coherent detection method and is widely used in industry, vehicles, robots, aerospace, biomedical field, etc. [2][3][4][5][6][7][8][9].Compared with the contact sensors, the LDV attracts widespread attention because it enables non-contact measurements.Depending on the frequency difference between the reference beam and measurement beam, the LDV is usually divided into two types, homodyne and heterodyne [10,11].Compared to the homodyne LDV, the heterodyne LDV is preferred because of its sensitivity to photodetector nonlinearities and reflectivity modulations, as well as its better noise performance.However, a frequency shifter is required to obtain the carrier.The common method to obtain the carrier in heterodyne LDVy is the deployment of an acoustooptical modulator (AOM, also called Bragg cell) which uses the acoustooptical effect to diffract and shift the frequency of light using sound waves [12,13].However, the carrier obtained from the Bragg cell is stationary based on the efficiency of acoustooptical effect and limited by its design and material.The minimum detectable capability of LDVy is restricted by the noise floor, and in most cases, it is limited by the photon shot noise of the detected measurement light [4].In the process of the measurement, the noise floor suffers from the ambient environment and electric-magnetic radiation in the air, which may not be so perfect.Consequently, a variable carrier for LDVy can avoid an unexpected high noise level in a certain frequency range.In addition, the Bragg cell is also a complex component that limits the cost of a LDV.
In order to overcome these disadvantages, a variable carrier generation for heterodyne LDVy is proposed by means of a compact OPLL.The OPLL is a sort of feedback-controlled system which is capable to synchronize the phase of two lasers [14] which can have a different frequency (heterodyne OPLL).It contains a designed PLL, in this case, the detector signal together with the two-laser system corresponds to a voltage-controlled oscillator.Namely, with the deployment of a heterodyne OPLL, the heterodyne carrier is achieved by offset-locking two individual laser diodes in which the offset carrier is controlled by a local oscillator [15].An OPLL-LDV system with a GHz carrier was reported in [15] by Robert Kowarsch et al. which is built by two DBR lasers combined with a commercial control system.Time delays in the feedback loop increase the phase delay and reduce the possible bandwidth and should be avoided.Sin Hyuk Yim et al. realized a dual loop to achieve OPLL with 8 MHz bandwidth yet suffering a delay length of 3 meters.Besides, the first-order loop filter cannot provide sufficient phase compensation to overcome the phase lag from the electronic components or loop delay [16].Julien Le Gouët et al. use an EOM to increase the OPLL bandwidth, but the employment of the EOM and related components increases the cost of the system as well [17].Yuxiang Feng et al. achieve a dual-frequency laser Doppler velocimeter via OPLL [18].Most of the work focuses on the fixed carrier, and their bandwidth is limited by the loop delay or other restrictions of the optic and electronic components.In this paper, we present an OPLL-LDV with a selectable carrier frequency in the range from 5 MHz to 50 MHz.Besides, a photodetector and a high-order controller are designed, so that a high bandwidth and an application-optimized circuit can be achieved.For building an OPLL fit to LDV, we analysed the effect of the laser on the performance of the OPLL and the measurement of critical parameters of laser diodes.Based on the property of the selected laser a corresponding photodetector, and controller were designed to complete the feedback loop.In addition, the optimization of loop parameters is implemented to improve the noise performance.Then the result of the built OPLL and the lock-in status was shown.Finally, the vibration of the carrier was shown within the bandwidth of the photodetector.And the measurement result of the velocity by means of the OPLL-LDV was demonstrated and compared to the reference value, proving the reliability of the system.

OPLL and LDV principle
The heterodyne LDV introduces a frequency shift on one of the laser beams forming a carrier at the vibration frequency for the photocurrent signal.Usually, the Bragg cell is employed to shift the beam so that a frequency difference (between reference beams and measurement beams) can be achieved.Consequently, the photo-current obtained on the photodiode in the heterodyne LDV is where the k is the responsivity of the photodiode, Pr and Pm the power of reference beam and measurement beam respectively, ωc the carrier frequency obtained by the difference of the ωr and ωm the circular frequency of reference beam and measurement beam, and φ(t) the phase modulation from the specimen.Since the carrier arises from the Bragg-cell frequency shift whose efficiency is sensitive to a special frequency (range), the carrier is restricted.To overcome this shortcoming, we deploy the OPLL-LDV configuration shown in Figure1 (a).Unlike the common heterodyne LDV, a second laser is introduced in OPLL-LDV.The carrier is generated via a feedback system that offset locks one laser frequency to another.In lock-in status, the slave laser synchronizes with the master laser, and the frequency difference equals the frequency of the local oscillator, which is detected by photodetector 1 (PD 1).Meanwhile, the slave laser beam is phase-modulated by the specimen and coherent with the master laser generating the carrier signal with vibration information, which is detected by photodetector 2 (PD 2).The small-signal model is a standard technique for the analysis of OPLL systems.The small-signal model for our problem is sketched in figure1 (b), since in lock-in status the OPLL is able to be regarded as linear system (sin(φ) ≈ φ).Specifically, the phase of master laser and slave laser is first detected on the photodetector PD, and then amplified by transimpedance amplifier (TIA) with a gain of kpd yielding the beat signal.The beat signal is further mixed with the LO signal yielding the error signal.The error signal is sent to loop filter generating the control signal.Finally, the control signal is feedback to the slave laser to complete the loop.
with kpd the gain of the photodetector (including the gain of the TIA), kmixer the gain or loss of the mixer depending on the type of the mixer, F(s) the transfer function of loop filter, ksl the gain of the slave laser, namely the CTC, and the τopll the delay over the loop.The main loop performance, such as the hold-in range, is determined by the transfer function.The stability criterion for OPLL can be evaluated by the Bode diagram of the open loop transfer function ().Specifically, the phase margin should be positive at the frequency the amplitude response of the open transfer function crosses () = 0 dB, which is described by with φpm the phase margin, ( 0 )| ( 0 )=0 the phase at the frequency open transfer function G(s) crosses 0 dB, s0 the 0 dB crossing frequency.The graphical representation of the equation ( 3) is shown in Figure 2. Once the phase delay exceeds -180 ° at the frequency of the zero-crossing point in amplitude frequency response, the negative feedback system becomes a positive feedback one.The error signal can be no longer reduced and the system oscillates.

Setup realization
The design of the OPLL-LDV system has several requirements which are discussed in this chapter.

Laser requirement
For the traditional heterodyne LDV, the reference beam is coherent with the measurement beam shifted by the Bragg cell.Since the optical path difference in the intended applications are rather short compared to the coherence length of a laser diode with even a relatively large linewidth, the realization of a reliable system has average requirements on the linewidth [19].Most important for the OPLL is that the linewidth  is lower than the modulation response bandwidth.The remaining phase noise is a minor requirement for robustness because the delayed self-heterodyne linewidth measurement requires a / = 300 m long fiber to achieve decoherence even at a 1 MHz linewidth [20].The speed of light is  ≈ 3 × 10 8 /.For the OPLL-LDV systems, the laser has a strong influence on the robustness of the carrier generation.Since the OPLL cannot suppress the laser phase noise completely, the laser selection directly affects the noise performance of the whole system.Laser requirements are divided into 7 items below based on the OPLL-performance requirements.1. Linewidth 1.1 For a certain loop-propagation delay, the laser linewidth restricts the phase-error variance of the loop.(Here an intuitive and simple understanding of the phase-error variance could be obtained from the analysis of the phase difference between slave laser and master laser in locking stage.)Based on the relation between linewidth and loop delay [21], to achieve a locking state with phase error variance of 0.03 rad 2 , a summed linewidth of 3 MHz is achievable for 1 ns loop delay [22].1.2 In order to catch up with the master laser, all frequency components of the slave laser should be transmitted.So, the summed linewidth should be less than the total bandwidth of the loop.

Frequency modulation response
The frequency modulation (FM) response of the laser directly affects the loop transfer function.In calculation, we always assume the FM response is flat so that the gain of the slave laser could be regarded as a constant.However, in practice, the FM response of the laser (especially DFB laser) is not uniform and exhibits different characteristics depending on the range of the modulation frequency.For a typical single-section DFB laser, due to the competition between the thermal and the carrier-induced effects, the FM response of a single section SCL exhibits a characteristic π phase reversal in the frequency range 100 kHz~10 MHz [23].In order to catch up with the phase of the master laser, the error signal with a combined linewidth should be allowed to pass through the feedback loop.Hence, the FM response of the slave laser must be substantially larger than the combined linewidth of slave laser and master laser.

Output power
From the view of stability, a higher output power can achieve a higher loop gain.This benefits a more stable OPLL system.In addition, given the relation between laser power and linewidth from the Schawlow-Townes equation, a high power corresponds to a narrower linewidth for a readymade laser.However, the maximum tolerance value of the selected photodiode should also be considered for power selection.

Intensity noise and intensity modulation
Intensity noise and intensity modulation should be low, as any fluctuation in output power will be converted to phase noise by the loop.Meanwhile, a balanced photodetector can be considered to mitigate this effect.5. Temperature sensitivity Temperature sensitivity should be low to reduce the influence of the ambient environment.Temperature deviation is one of the most common factors that cause the OPLL to lose its lock.Besides, the current tuning coefficient also suffers from the temperature [24].

Current tuning coefficient
The CTC should be large as it would be regarded as the gain of slave laser and then converted into the loop gain.7. Laser package Given the limitation from propagation delay and linewidth as discussed in the first requirement, the package type TO CAN is recommended for large linewidth laser since it has no extra delay from the pigtail.A butterfly package with a pigtail is also acceptable for a linewidth in the order of kilohertz.One needs to mention that some requirements, such as 3 and 5, are not absolute.Because there are some convenient methods to compensate, for example, a large loop gain and precise temperature control are easy to achieve.So, it gives us a broader margin for laser selection.Besides, the choice of laser power also suffers from the requirement of an LDV system as well as an eye-safe requirement.In this project, two lasers from Eblana and JDSU are selected with high power and narrow linewidth, as well as a high modulation bandwidth.

CTC requirements
After the laser selection, the next step is the exploration of laser properties to support OPLL-LDV design.The current tunning coefficient should be tested since it acts as a key role in loop gain and loop stability.The traditional method to measure the CTC is recording laser wavelength and corresponding injection current, and then calculating or fitting the slope of wavelength-current curve.The wavelength is measured over the current range from 25 mA to 60 mA to obtain the CTC. Figure 3 shows the fitting curve in blue and the measurement data with spot.The CTC is obtained by calculating the slope of fitting curve of the slave laser.The CTC results in 494.3 MHz/mA corresponding to the Ksl in equation 2. The CTC measurement method is restricted by the resolution bandwidth of the optical spectrum analyzer.We implemented the CTC measurement with a more convenient and precise method in the process of building OPLL (The accuracy of this method is restricted by the resolution bandwidth of the spectrum analyzer, normally smaller than that of the optical spectrum analyzer), shown in Figure 4(a).The first step for the OPLL design is the observation of the beat signal between the master laser and the slave laser.The free-running beat signal moves with the variation of the injection current accordingly, and then the CTC could be obtained.Additionally, That is to say, the beat signal does a reciprocating motion when the slave laser is frequency modulated by a sinusoidal signal.With the increase in the variation frequency of the injection current, the movement of the beat signal will form a continuous range on the spectrum shown in Figure 4 (b), which can be used to estimate the frequency response of the slave laser.

FM response
DFB lasers are wildly used in different optic fields because of its high wavelength stability, narrow linewidth, and small size.However, for the construction of OPLL, one of the special characteristics should be clear and overcome, which is the nonuniform FM response.The nonuniform characteristic origin from the counteraction between the thermal shift and the carrier shift and induces a π phase reversal in the frequency range from hundreds of kilohertz to several megahertz [25].The common method to obtain the FM response of a DFB laser is based on the relationship between intensity modulation (IM) and frequency modulation of the laser, which is driven by a sinusoidal signal [26].
where the k0 is the CTC in the DC case, b is related to the ratio thermal-induced effect and carrierinduced effect, fc is structure-dependent and is typically located at 10 kHz-10 MHz and j is the imaginary unit .A corresponding diagram is illustrated in Figure 5. Based on the setup obtaining the CTC shown above, a convenient method is proposed to measure the FM response of the slave laser.The FM response is a function of the CTC and the modulation signal.As demonstrated above, the motion range of the beat signal refers to the CTC.The modulation frequency can be obtained from the signal generator.So once the amplitude of the modulation signal remains the same and the modulation frequency is manually tuned, a change of the CTC can be observed.The measured result is shown in Figure 6.The x-axis is the modulation frequency which is tuned from 1 kHz to 5 MHz.The y-axis and z-axis are the frequency of the spectrum and power spectrum density of the signal respectively.With the increase of the modulation frequency, the tuning range of the beat signal changes accordingly.That is to say the FM characteristic changes with modulation frequency.In order to observe the tendency of the change more obviously, the top view is shown in Figure 6 (a).Likewise, the bright part refers to a high amplitude of the beat signal in Figure 6   It is obvious to see the CTC reduces until 2.5 MHz with the increase of the modulation frequency and then increase with the modulation frequency.So it forms a "waist" at the frequency range from 1.5 MHz to 2.5 MHz which corresponds to the phase reversal point.The intensity modulation is suppressed in this measurement, due to the balanced photodetector as well as the tuning range of the beat signal is influenced only by the CTC and input modulation signal.The tuning range was extracted and drawn with its corresponding modulation frequency thus forming the FM response as shown in Figure 7.It can be seen that the FM response of the slave laser drops down around 1 MHz-3 MHz and then tends to be uniform.Yet the phase response cannot be obtained by this method so far.A proper estimation of 30-130°phase delay could be obtained by equation 3 as shown in Figure 5. Based on the dropped part of FM response, a properly designed loop filter can compensate for it.

Photodetector
The main design specification of the phase detector is the bandwidth since it determines the maximum offset frequency that the OPLL can achieve.However, the OPLL can be locked outside the PD bandwidth by adjusting proper loop parameters and the PD bandwidth is still regarded as the main restriction for the maximum achievable offset frequency.For an OPLL achieving an offset frequency beyond GHz, a very high bandwidth photodetector is needed [16].A balanced PD was designed to suppress the additional noise, which is shown in Figure 8(a).The bandwidth is set to around 40 MHz, in order to adjust the offset frequency within that range.A capacitor was put at the output of the PD to meet the requirement of the mixer in order to enable the mixer to operate in its linear operational range.

Loop filter
The loop filter acts as the controller for the OPLL and converts the error signal into the control signal.The control signal serves as feedback to the slave laser.The main challenge for designing a loop filter for OPLL realized with DFB lasers is the compensation of nonuniform FM response.Regarding the existence of the phase delay from the nonuniform FM response, a phase-lead loop filter would be taken into account rather than a lead-lag filter.In addition, in order to achieve a large OPLL bandwidth, a high-order passive loop filter was designed.The passive and active loop filter is both acceptable for achieving a stable status.In this paper, a second-order passive loop filter was designed since the amplification is compensated in TIA and CTC.It was designed by two first-order lead loop filters in series shown in Figure 9.The frequency response of the designed loop filter is shown in Figure 10.By adjusting the parameters on the loop filter, the bandwidth and compensation property could be changed accordingly.The transfer function is measured by a vector network analyser (VNA).The maximum phase compensation can achieve more than 100 degrees at 2-4 MHz.Although the amplitude response is a type of high pass filter, the most additional noise at a high-frequency range would be suppressed by the photodetector and laser driver which shows a low-pass characteristic.In addition, considering the very high CTC value of the DFB laser, the loop gain is high enough to achieve stability.At next, a first measurement was conducted on a specimen of a fixed mirror.The measurement light reflected by the fixed mirror interferes with the reference beam (from the master laser) yielding a beat signal which is detected by a balanced photodetector for the LDV part.The detected signal on the LDV photodetector and on the OPLL photodetector is shown in Figure 12.Due to the power loss of the reflection, the amplitude of the signal on LDV photodetector is smaller than that of the OPLL photodetector, yet the locking status and the loop bandwidth remains the same.This validates the method, as the beat signal from the LDV photodetector compares with the signal from the OPLL photodetector.

Motion measurement
In order to evaluate the validation of the LDV-OPLL system for speed detection, a vibration measurement is conducted.A vibration system driven by a motor is employed since the precise speed can be obtained as a reference, which is shown in Figure 14.A retro elective tape was attached to the measured surface in order to improve the returned light power.A pulse signal drove the motor.The rotation frequency frot equalling to the vibration frequency fvib could be calculated by the frequency of the pulse signal fpulse,   =   =   (360 /) −1 , β is the step angle with a value of 1.8 degree.The amplitude of the straight-line harmonic motion A is determined by the distance from the fixing screw to the center of the circle.The reference and the measurement velocity can be obtained by where vr and vm are the measured and the reference velocity respectively, fvib vibration frequency equalling the rotation frequency, A is the amplitude of the rotation motion referring to the distance from the screw to the circle center, ΔfD the Doppler shifted frequency, λ the wavelength of the laser, the vr-max and vm-max the maximum measured and reference velocity respectively, ΔfD-max the maximum Doppler shift (maximum frequency deviation).The velocity at maximum amplitude vr-max was measured as the maximum frequency shift ΔfD-max could be identified more precisely and a simple harmonic's maximum velocity is a better reference (it is fixed).A 338 Hz pulse signal was sent into the motor (which is set to 200 pulses per revolution.)corresponding to a 338/200 rotation frequency, namely the vibration frequency fvib.The amplitude of the rotation motion  is measured and equals  = 1.02 cm.Therefore, the maximum velocity vr-max of 0.1083 m/s could be calculated as a reference.The result of the Doppler shifted spectrum is shown in Figure 15.The measured velocity was obtained by the maximum Doppler shifted frequency ΔfD-max of 0.13 MHz, which equals to a maximum measured velocity vm-max of 0.1008 m/s.

Conclusions
In this paper, we presented an OPLL to realize an adjustable and controllable carrier frequency in a heterodyne LDV.The feedback control system with OPLL can generate flexible frequency shifts and may replace the state-of-the-art acoustic-optical modulator, Bragg cell, in LDV.Compared with the Bragg cell, the generated carrier via OPLL can vary continuously within the photodetector bandwidth, which corresponds to approximately 50 MHz in our setup.Compared with the state of the art, the OPLL solution can adapt the carrier frequency to the specific requirements of the measurement task and other hardware restrictions like the photo-detector bandwidth.An integration into one single PCB could benefit the reduction of loop delay and, thus, could lead to a better lock in the future.However, our implementation of the OPLL introduces more phase noise close to the carrier as a Bragg cell.The further reduction of this noise remains a task for future research.The loop bandwidth can still be improved by adjusting the structure and parameters of the loop filter which would be conducted in the future.

Figure 1 .
Figure 1.(a) Schematic diagram of OPLL-LDV system.BS is the beam splitter, M the mirror, PBS the polarizing beam splitter, PD1 and PD2 photodetector1 and photodetector 2, L lens, FC fiber coupler, TIA transimpedance amplifier, LO the local oscillator, LPF the low pass filter, LF loop filter, LDC the laser driver controller, OSA the optical spectrum analyzer, SA spectrum analyzer.(b) Small-signal model of OPLL.Where the φm and φs are the phase of master laser and slave laser respectively, φbeat the phase of beat signal, φlo the phase of local oscillator, φe the phase of error signal, the kpd the gain of the photodetector (including the gain of the TIA), kmixer the gain or loss of the mixer depending on the type of the mixer, ksl the gain of the slave laser, namely the current tuning coefficient (CTC), and the τopll the delay over the loop.Based on the small-signal model, the transfer function of OPLL can be explored.The open-loop transfer function (), closed-loop transfer function () (system transfer function) and error transfer function () are described by

Figure 2 .
Figure 2. Bode diagram of open transfer function.The phase margin φpm is indicated in red line.

Figure 3 .
Figure 3. Measured (dot) and fitting result (line) of the CTC determination.

Figure 5 .
Figure 5.The nonuniform FM response of DFB laser, a simulation result obtained with Eq. 4. The parameter b is set to 1.6 and fc is set to 1.7 respectively.A drop-down occurs at around 2 MHz, and the 30-130°phase delay can be observed accordingly. (b).

Figure 6 .
(a)CTC changes with modulation frequency from 400 kHz to 5 MHz.The colour bar indicates the colour distribution of the power spectrum in dBm (b) Top view of Figure 6 (a).A "waist" occurs at around 2 MHz.The motion range moves as the temperature of the laser is changed with the ambient temperature.(RBW:100 kHz, VBW:3 kHz)

Figure 7 .
Figure 7. FM response of the slave laser.A drop occurs in the FM response corresponding to the waist in Figure 6(b).It refers to a maximum phase delay point.

Figure 8 .
Figure 8.(a) Schematic diagram of a balanced photodetector (the driven circuit and a part of noise suppression circuit is ignored for brief).(b) The picture of the self-built photodetector.

Figure 9 .
Figure 9. (a) Schematic diagram of second order loop.(b) Designed loop filter with different parameters.

Figure 13 .
Figure 13.Variable carrier from 10 to 35 MHz (RBW: 100 kHz, VBW: 3 kHz).The lock-in status can maintain within the bandwidth of the photodetector.When the carrier is moved near the bandwidth of photodetector, an amplitude reduction occurs owing to the drop of the transfer function of the photodetector.

Figure 14 .
Figure 14.Specimen under test.The vibration information can be calculated as a reference.

Figure 15 .
Figure 15.Doppler shifted spectrum (RBW: 30 kHz, VBW: 3 kHz).The dark blue curve is the static spectrum of the locked beat signal on LDV'PD.The orange curve is the Doppler shifted curve which carrier is shifted from the static one due to the Doppler Effect.Since the vibration frequency is low, the carrier is changed according to the movement of the specimen. 1

Table 1 .
Parameter of different loop filters