Mode 1 – PRF scan

In this mode, the interferometer has the reference arm fixed and the pulse repetition frequency of the laser diode is swept over the tuning range. The distance measurement is performed using the frequencies associated to the two successive correlation peaks. This configuration has no moving parts and its performance depends mainly on the uncertainty of the measurement of the exact frequency at which the correlation peaks occur.

The governing equation can be simply expressed in the following manner:

Note that the quantity 2L corresponds to the interferometer optical path difference.

This last expression shows that the absolute distance can be obtained directly by measuring the frequency sweep range between two consecutive correlation events.

The accuracy of the process is directly related to the ability of measuring the beat frequency accurately, ultimately dependent on the stability of the reference oscillator and on the error finding the correlation peak.

Mode 2 – OPD scan

The OPD scan mode implies moving the reference arm of the interferometer from position l₁ to position l₂ while the PRF is fixed. The absolute distance measurement is obtained indirectly by evaluating the distance travelled by the reference arm stage between two successive correlation peaks, which are obtained for two different (fixed and stabilized) pulse repetition frequencies. Although presenting the “inconvenience” of exhibiting moving parts, this method is more relaxed in terms of the uncertainty requirements on the frequency measurement. The distance estimate is obtained by the following set of expressions:

In this last expression, Δl = l₁-l₂, Δf=f₁-f₂ and n is the number of impulses travelling in the measurement arm of the interferometer.

There are three critical characteristics of the MLLD that enable the technique presented in this work, and these are common for the two operating modes: PRF tuning range, PRF stability and pulse width.

The first one, associated to the capability of changing the PRF of the laser would affect mainly the closest and the farthest distance for which the system is able to perform measurements. As explained before, this technique relies on the capability of changing the pulse repetition frequency (or reciprocally, the pulse period) in order to perform the pulse overlap at the detector on a Michelson Interferometer arrangement. Once the optimum time overlap at the detector is achieved (represented by a maximum signal in the cross-correlator detector), we should be able to measure the frequency at which it has occurred with a specific uncertainty, which is directly dependant on the PRF stability.

It is then crucial to be able to tune the laser pulse repetition frequency over a certain range (dictated by the minimum and maximum measurement ranges) and maintain a reduced RF linewidth – directly related to the PRF stability level, in order to be able to measure the instantaneous PRF with a low uncertainty.

The pulse width produces a similar effect with respect to the jitter associated to frequency instability. Longer pulse widths would produce wider cross correlation events and thus increasing the difficulty to find the frequency of optimal overlap in the presence of detection noise.

Stabilization of PRF

One of the most direct techniques for PRF stabilization is running the laser in hybrid mode, where the laser is driven with constant current at the gain section level, but the saturable absorber is locked to an external RF source, by super-imposing a AC signal over the DC bias⁶.

A very simple schematic of hybrid mode-locking is shown in figure 2.a)

Fig. 2.

a) General connection diagram for hybrid mode b) Alan variance results for passive and hybrid mode

Figure 2.b) compares the variation of the PRF in both passive and hybrid mode. A good measure of the level of stabilization that can be achieved is by means of the root Alan variance, allowing to discriminate the stabilization achieved for different integration periods.

Although the increase in stability is not dramatic for shorter integration times (below millisecond level), “slow” variations in frequency show a significant damping when working in hybrid mode-locking. This can be explained by the fact that the RF frequency source we use is a commercial grade device, with fair performance in the short period stabilization.

Correlation detection

The correlation detector is a critical component for this metrology concept as it is responsible for determining the degree of time overlap of the pulses travelling in both arms of the interferometer. In terms of the experimental setup, we implemented two different cross-correlator schemes:

The first setup is a conventional intensity correlator, using a SHG crystal as the nonlinear element. However, the very low pulse energy of the QD laser (<100 pJ), make the pulse correlation a very difficult task, especially if we remember that the second harmonic generation in thin crystals exhibits an efficiency on the order of 10^-5.

The second setup is a technique for measuring the degree of polarization (DOP) of the resulting electrical field during the pulse overlap process. In this case, the Michelson interferometer produces two orthogonally polarized copies of the original pulse by means of a polarized beam splitter, one travelling over the reference arm, the other over the measurement arm. When overlapping at the polarimeter detector, they produce an electrical field with a degree of polarization that is proportional to the pulse width and to the degree of pulse overlap⁷.

Experimental setup

The next figure (figure 3) shows the setup built for the testing of the instrument concept, using in this case the use of DOP correlation detection. A spool of polarization maintaining fiber with different patch lengths was used to simulate different object distances, from 3 meters up to 100 meters. An additional translation stage located in the measurement arm was introduced to produce small displacements on top of the distance offset resulting from the use of the optical fiber. The images in Figure 3 show the implementation of the metrology concept tested in our labs.

Fig. 3.

Setup of the distance measurement system using the ‘degree of polarization’ technique as a correlation detector

It is important to note that all the components associated to the metrology system, including the laser, interferometer, optics and proximity electronics exhibit a small footprint, with a maximum volume of 3 Liter, weighting less than 3 kg. The power consumption is estimated to be less than 5 W, if we exclude the measurement equipment.

Mode 1

One of the first conclusions to draw from the expression that allows the evaluation of the absolute distance measure (eq.1) is that the tuning range required is independent of the nominal PRF of the laser used for this application. The required tuning capability only depends on the range of distances to be measured. In general terms, an error analysis regarding the accuracy of the distance measurement shows that it depends only on the accuracy of the PRF measurement,

where εL is the error in the distance measurement, c the speed of light, ε_f the tuning range and ε_f the error in the PRF measurement.

Figure 5 shows the distance measurement accuracy as function of the distance to the object, for several PRF measurement accuracies:

Fig. 4.

Metrology concept tested and detail of the laser employed in this experience

Fig. 5.

Measurement error and tuning range for mode1

This figure shows that accuracies below 100 micrometer can be achieved up to a 150m measurement range with a PRF measurement accuracy of 1 Hz. This level of measurement uncertainty can be maintained for longer distances if we are able to determine the beat frequencies down to the sub Hertz level. The measurement accuracy of 1Hz requires a stability level of 10^-9 (at millisecond level period) for a laser with a PRF of 1 GHz.

The vertical axis on the right indicates the amount of tuning needed to be able to perform measurements at different distances.

It is also important to evaluate the amount of frequency tuning that is necessary to produce a complete correlation event.

For a fixed object distance, the total variation of the pulse position (in time due to the jitter) at the correlator detector level results from the sum of all period increments associated to the N pulses travelling in the measurement arm of the interferometer. Thus, frequency tuning requirements for the cross-correlation process decrease with increasing distance, since N increases for longer distances. This fact implies also that mode-locked pulses with shorter pulse width require less PRF tuning producing a complete correlation.

The relevance of the tuning requirements is expressed mainly in the laser stability requirements and in the capability of finding the frequency corresponding to the correlation peak with the highest accuracy. In practical terms, and as expected, narrower pulse widths are more favorable, as the error associated to peak finding is minimized when decreasing the correlation width.

Mode 2

In terms of the measurement accuracy, we can evaluate the average error on the measurement from the first expression of Equation 2.

Again, Δl = l₁-l₂ and Δf=f₁-f₂, ε_f is the error associated to the measurement of the PRF and ε₁ is the error associated to the measurement of the location of the correlation peak.

The next graph (Figure 6) shows the error in the distance measurement for a baseline frequency of 5GHZ, for two magnitudes of frequency stability (10^-8 and 10^-7). The error on the location of the correlation peaks is assumed to be constant over distance and, in the case presented in the graphs, equal to 5 micrometer.

Fig. 6.

Uncertainty on L measurement for mode 2, for baseline frequency of 5 GHz. ε_l is assumed to be of 5μm.

It is important to note that the error due to the accuracy of the PRF measurement is smaller than the error due to the location of the correlation peaks, for approximately half of the working range. The contribution of the PRF measurement accuracy is in fact very low, even for lower PRF stabilities for ranges up to 150-200 meter. In fact, regarding the overall error budget, the entire weight in terms of distance measurement accuracy is referring to determining the location of the correlation peak, relaxing the requirements regarding the PRF stability.

Even though the value assumed for the correlation peak location error is relatively low, the final accuracy is always topped by this value, with the error associated to the PRF uncertainty only being significant for longer ranges. A simple “back of the envelope” calculation evidences this assumption:

Considering a frequency stability of 10^-8 we solve the second expression of Equation 2 in order to estimate the influence of the PRF on ε_l.

Using error propagation to evaluate the error due to the frequency stability, we will get a value close to 1 nm for n=1.

This corresponds to the error in position location as if the correlation detector worked as an auto-correlator.

If we consider it working as a cross-correlator, the overlap is produced after n>>1 pulses and the error in ε_l due to the PRF instability can be evaluated in the following manner (assuming that the timing jitter of the individual pulses is uncorrelated):

Where ε_{l, n} is the error after n pulses and ε_{l, 1} is the error associated to jitter after the first pulse

For a large value of n, associated to long distances, the error due to this parameter is still at the micrometer level.

The major source of uncertainty is then associated to the error on the location of the correlation peak, which has two main components: the repeatability of the translation stage and the error associated to the numerical fitting employed to locate the correlation peak. Even if we assume an error of say, 50 micrometers, this will always be much larger than the error due to the PRF measurement uncertainty and we can consider this as the maximum error for the L measurement in Mode 2.

The choice of the baseline PRF will affect directly the minimum requirements in terms of the travel range of the reference arm of the interferometer, since the ε_l defines the minimum distance between the two correlation peaks (one for baseline PRF, the other for the shifted PRF). As an example, a baseline PRF of 10 GHz would require a travel range of a few millimeters while 1 GHz would require tens of centimeters. Clearly, lower PRF present the disadvantage of requiring long OPD travels, surely not compatible with practical metrology systems for space.

Update rate

The update rate is strictly limited by two intrinsic factors, associated to the characteristics of the laser.

The first one is the laser stability. We have seen earlier that the requirements in terms of stability are slightly different for Mode 1 and Mode 2, but generally we can statethat the measurement accuracy of the laser PRF should be in the range of 10^-9 in Mode 1 and 10^-8 for Mode 2, for measurement ranges up to 150 meter. The averaging time necessary to achieve this accuracy depends on the intrinsic frequency stability of the laser.

Considering an accuracy of this order of magnitude, at a millisecond time interval (kHz rate), this would represent the minimum acquisition period for each point forming the cross-correlation waveform.

The total number of points necessary to define correctly the correlation waveform depends itself on the pulse width. Furthermore, the uncertainty in finding the position of the waveform peak is itself dependent on the width of the pulse. This would suggest the use of shorter pulse widths.

Pulse Repetition Frequency

This parameter is not critical in terms of the metrology system performance, since the basic principle of measurement is directly related to the frequency difference between two correlation events and not to the nominal laser pulse repetition frequency itself. However, since a fixed amount of PRF tunability is required, the fractional tuning (in relation to the nominal frequency) is higher for lower repetition rate lasers making it more difficult to achieve. In Mode 2, a higher frequency is also more beneficial; as it minimizes the travel range (Δl) required obtaining the two correlation peaks. A generic tradeoff points out to PRFs in the range of 5 to 10 GHz

Tunability

This parameter is directly related to the nominal PRF, but has also a strong implication on the shortest range the system can measure (in Mode 1). The shorter the range, the wider should be the tuning range.

A 10 to 15MHz PRF tuning range should be sufficient to allow measurements down to a 10 meter distance range.

PRF stability

The PRF stability needs to be discussed under two aspects: the stability related to the accuracy obtained when measuring the laser PRF at the correlation peaks, and the maximum period allowed to make this measurement and integrate frequency readings, directly related to the update rate required for the metrology system.

The accuracy analysis for both Mode 1 and Mode 2 results in two different stability requirements. While Mode 1 requires 1 Hz accuracy in the frequency measurement to obtain a measurement precision of better than 100 microns for distances up to 150 meters (with the advantage of having no moving parts), Mode 2 allows to relax this requirement down to a 10^-8 or less fractional frequency measurement accuracy for the same distance accuracy requirement.

Mode 2 requirements for PRF stability can be considered less restrictive, specifically because the frequency measurement can be averaged over longer periods, depending only on the speed of displacement of the travel stage in the reference arm of the interferometer.

Pulse Duration

The pulse duration has direct implications on the range of frequencies (Mode 1) or on the travel displacement (Mode 2) necessary to produce a correlation event. The accuracy required to define the position of the correlation peak in the Hz range (Mode 1) or with micrometer resolution (Mode 2), in the presence of noise or for low level signals, tends to push for shorter pulse durations.

The basic argument is associated to the fact that shorter pulses produce narrower correlation waveforms (both in Mode 1 and in Mode 2), and for the same sampling the error associated to the location of the peak is lower.

For this particular MLLD characteristic the laser pulses should have the narrowest width possible. Available data on existing MLLDs show that 500 fs width pulses are already attainable.

Output power

The output power is to be considered critical. The main driver here is related to the efficiency of the correlator and of the SHG process (non-linear crystal or two photon absorption), which exhibit efficiencies at the 10^-5 level.

The laser used in our experiments had a 600mW average power for a 5 GHz PRF, which leads to an energy per pulse at the 100 pJ level, which is considered a low value to have a good SNR at the correlator detector.

Wavelength

In general there is no specific requirement in terms of wavelength. The correlator and the 2nd order detection process is wavelength dependent and some restrictions may arise from this subsystem.

Coherence

Any requirement regarding this property depends on the type of cross-correlation detector used. If an intensity correlation method is employed there are no specific restrictions regarding coherence since it’s a second order process and phase independent. However, the Degree of Polarization method as a first order process is radiation phase dependent.

Polarization

The issue of polarization is direct related to the type of cross-correlation detector used. For both of the cross-correlation methods used a linearly polarized laser is needed.