Response Time Characteristics of the Fast-2D Optical Array Probe Detector Board (2025)

Ideally the comparator output state is digital one when the amplifier voltage is greater than 50% of its average voltage, and digital zero when the amplifier falls below 50% of its average voltage. Practically, the comparator changes state when the voltage on the noninverting terminal crosses the voltage on the inverting terminal. The presence of positive feedback from the comparator output generates hysteresis in the comparator circuit, so the output state does not oscillate when the input signal is near the threshold voltage. This also results in a bias term in the threshold voltage (from the perspective of the amplifier driving the comparator input) where the comparator changes state. The presence of this bias means that the comparator state transition is not strictly a function of shading but also dependent on the amount of signal from the amplifier. The threshold voltage for the comparator input (the test point in Fig. 2) is given by

where the comparator changes its output state when the test point voltage crosses

and is referred to here as the threshold voltage,

is the voltage on the inverting terminal of the comparator [one-half the average test point voltage for a standard diode state comparator circuit and one-third the average test point voltage for a depth-of-field (DOF) comparator circuit],

is the initial output voltage of the comparator (digital one or zero, which is typically on the order of 3.3 or 0 V, respectively),

is the input resistor (7.5 kΩ on this board), and

is the positive feedback resistor (

on this board). Equation (1) is used to calculate the step response voltage waveform of the total detector system. Figure 3 shows the amount of shading required to cause a comparator transition as a function of the mean test point voltage (as measured on the analog test point right before the input resistor in Fig. 2). The circuit is designed to transition when the test point voltage crosses the 50% shading threshold. It asymptotically approaches this value as the optical power on the diode increases, but it always requires greater than 50% shading for any physically realizable mean test point voltage.

Table 1.

Measured and derived variables and their sources.

a. Minimum pulse width

In the first measurement, a diode laser illuminates the photodiode array. The laser is normally on except for very short periods where the laser is turned off to simulate shading from a particle passing through the beam. We progressively reduce the duration where the laser is off until the detection circuit does not respond fast enough to cause a change in state in the digital output. Contributions of zero-mean noise will blur this exact boundary, so we estimate the fraction of input pulses that result in a change in comparator output state and report here the 50% point (ideally where the limit would be if noise were absent). The histogram in Fig. 5 shows the distribution of minimum response times for all 64 channels on a Fast-2D board. The sizable spread would seem to suggest that there is significant variation in the response characteristics of the different detection channels. Furthermore, it seems to suggest the board does not meet its 50-ns time constant specification. We will show in the next section that an unexpected slow decay term is having a detrimental effect on the detector performance.

b. Total voltage waveform

In our second test, we measure an effective voltage-versus-time waveform for a case when the laser is initially on and turned off very rapidly (with a bandwidth of 100 MHz, the laser has a fall time on the order of 2 ns). This step-down input is functionally described as , where is a unit step function. In this measurement, we change the voltage on the inverting terminal of the comparator by changing the duty cycle of the laser (the fraction of time during the waveform period where the laser is on divided by the waveform period) and recording the time at which the comparator changes state. Thus, where voltage waveforms are typically sampled at known time intervals, in this case, we are recording the time at known voltage intervals.

Based on the duty cycle of the laser, we determine the effective threshold voltage at which the comparator changes state using Eq. (1) and

where D is the laser duty cycle,

is the analog test point voltage when the laser is on, and

is the amplifier voltage when the laser is off. This calculation assumes that the pulse repetition rate of the laser (10 kHz) is significantly above the cutoff frequency of the low-pass (LP) filter (250 Hz) in the reference leg of the comparator circuit (see Fig. 2).

The waveform resulting from a step-down input appears to have two different components associated with it (see data plotted in Fig. 6). One is clearly the result of the electronic response time, which is an exponential with a time constant on the order of 50 ns. The other dynamic term has a very slow response characteristic and does not conform well with an exponential decay function. The total detected waveform is the sum of the two decay terms.

The physical effect responsible for the slow decay remains unclear. Capacitive coupling was considered, but we can change the amount of slow decay in the waveform by adjusting the laser angle of incidence (near-normal incidence significantly reduces the amplitude of the slow decay).

The slow term could be the result of scattering between a reflective mask over the photodiode array and the matte white surface on which the diode array is mounted. However, the relative separation between the two surfaces is on the order of millimeters, so a decay function on the order of 1 μs would have to experience approximately 300 000 reflections and require a surface reflectivity larger than 0.999 997. Also, the decay function of most scattering is expected to conform to an exponential form.

Finally, it may be possible the laser excites electron-hole pairs in an inactive region of the detector and the carriers have very slow diffusion rates. If these regions are near the edges of the photodiode, it might explain the dependence on the angle of incidence. Whatever the physical cause of the slow decay function, it remains a significant factor in the characterization and analysis of the detector system of the Fast-2D.

We did not observe this slow decay term on the DET10A that was monitoring the laser optical output, so we are confident that it is an attribute of the 2D board and not an artifact of the test equipment.

Giving consideration to the different components of the threshold-time versus threshold-voltage waveforms, we developed the following fit function to describe the detector response to a step-down input:

See Also

In The Setup Shown Above, A Student Uses Motion Detector 1 To Measure The Speed V, Of A Cart With MassThe Pen On Seismograph Swings Freely.

This equation can be broken into three parts. The first term is exponential and is the electronic response function with exponential amplitude (A) and electronic response time (τ). The second term is the slow decay term with slow decay amplitude (C) and slow decay rise time (

) and a slow decay time offset (

). Finally, the third term, B, is a dc bias. A unit step function

is used to indicate the dynamic terms of this function are only valid after

. The electronic response function is in the form expected for first-order linear circuits and was explicitly used in Baumgardner and Korolev (1997), Jensen and Granek (2002), and Strapp et al. (2001), and implicitly in Lawson et al. (2006). The form of the second term is based on observations of the voltage waveforms, where it was noted that the data exhibited a long decay tail with a

behavior. The second term in Eq. (3) is not based on first principles or any physical effect known to the authors; however, it produces the best fits, decoupling electronic time constant retrievals from variability in the relative amplitudes of the two waveforms.

The energy of the total signal is split between the fast response electronic exponential and the slow decay function. The term slow fraction will be used to describe the fraction of signal power coming from the slow term in Eq. (3) and is represented in this work by the variable

. The slow fraction can be determined by evaluating the ratio of the slow term to the total signal (less the bias term B) at the time of transition in the step response function in Eq. (3). Evaluating this ratio at

produces the relationship between

and C as follows:

and

The relationship in Eqs. (4) and (5) is useful for simulation, where often

is the parameter of interest, but C is needed for model implementation.

An example of the fit waveform is shown in Fig. 6. We fit Eq. (3) to all 64 detector channels of a Fast-2D board. Table 2 contains a summary of the fit parameters from Eq. (3). All amplitude terms in Table 2 are normalized (except ) to the initial voltage prior to the step function (), so they are reported in unitless in the table. Note the electronic time constant τ is a very narrow distribution, but the mean and standard deviation are largely affected by a small number of outliers. Those outlier measurements of the electronic time constant correspond to a few cases of large fit error or instances of a large slow fraction, causing the fit to be insensitive to the electronic response.

A scatterplot of the electronic time constant versus slow fraction is shown as blue dots in Fig. 7. It is important to note that the electronic time constant is independent of the slow fraction. A poor fit function would likely couple these two terms. For comparison, Fig. 7 also shows the corresponding minimum detectable pulse widths described in the previous section. These pulse widths tend to be significantly larger than the electronic time constants and show a clear correlation with the slow fraction. This is expected, because as the slow fraction increases it will dominate the decay waveform and the effective response time of the system increases.

Our observations of the Fast-2D detector system (optical input to digital output) did not significantly differ from the response characteristics we would have obtained from analyzing the amplifier output at the analog test point. However, as the analog front end of the detection system is pushed to faster response times, the response characteristics of the digitizer will likely become more significant. The analysis shown here is important, because the response characteristics of a digitizer can be difficult to measure directly. A complete analysis should include all of the analog components, including the digitizer.

We have found that the electronic time constant of our Fast-2D boards is near its design specification of 50 ns. However, the effective response time of the detector is slower due to an additional response characteristic. Were the slow decay term not a factor, we would expect the minimum detectable pulse width to be less than the electronic time constant. Instead, we typically observed minimum pulse widths in the range of 70–150 ns.

There is substantial scatter in the slow fraction seen in Fig. 7. During the acquisition of the time constant data, we did not maintain a constant angle of incidence. Cursory observations seemed to suggest that the angle of incidence was a significant factor in the scattering fraction. To further investigate this, we took data on the slow fraction as a function of the angle of incidence for a few channels using the setup shown in Fig. 8. The high-bandwidth laser is collimated by lens L0 (40-mm focal length) and passes through an afocal beam expander composed of lenses L1 (30-mm focal length) and L2 (200-mm focal length). An iris is placed at the focal plane of the lens L1, and a beam splitter is placed between the iris and lens L2. The light exiting L2 is approximately 1-cm diameter. The beam position and angle relative to the Fast-2D board is adjusted using the periscope mirrors M3 and M4, and the test point is monitored with an oscilloscope. The laser is controlled using a function generator set in transistor–transistor logic (TTL) pulse mode. The initial reflection off the beam splitter is detected using the DET10A and is used as the reference channel in the oscilloscope. Upon impinging on the Fast-2D, most of the light is reflected by the reflective mask over the detector. The reflected light passes back through the optical system. The beam splitter picks off a portion of the reflected light after it passes through lens L2 and directs it onto a ruled slide that is located precisely at the focus of L2 (also the conjugate plane of the iris). A complementary metal–oxide–semiconductor (CMOS) camera (not shown) is used to image the focused spot on the slide and read off the displacement of the beam. The incident angle of the beam is given by

where

is the angle of incidence (rad),

is the displacement of the beam on the ruled slide, and

is the focal length of lens L2. From the oscilloscope, we estimate the slow fraction by recording

,

, and the approximate point where the slow decay curve dominates the waveform. A plot of these observations is shown in Fig. 9.

Setup for measuring the scattering fraction as a function of the angle of incidence. The bright red arrow indicates light directed onto the detector. The darker red arrow is light reflected by the mask over the detector.

Citation: Journal of Atmospheric and Oceanic Technology 33, 12; 10.1175/JTECH-D-16-0062.1

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Plot of the estimated slow fraction as a function of the angle of incidence. (left) The raw measurements are taken over ±0.3° and are substantially scattered for the different channels. (right) An optimization routine is used to adjust the offset angles of each channel to obtain a quadratic function resulting in the scatterplot shown here (offsets used are given in Table 3). The variation in offset angles between channels may be the result of the small misalignment between the mask and detector on the Fast-2D.

Citation: Journal of Atmospheric and Oceanic Technology 33, 12; 10.1175/JTECH-D-16-0062.1

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In previous investigations we had found that adjacent even and odd array photodiodes seemed to have different angular sensitivity, suggesting that the mask over the detector array was slightly offset. In addition to the difference in angular sensitivity, it also seems that the scattering fraction has offsets that may be a similar result of a mask offset. Channels 17, 18, and 24 all seem to show a minimum in scattering fraction but at different angles of incidence. Meanwhile, channel 45 appears to be significantly offset from the minimum. All of the data were taken at the same absolute angles of incidence (shown in the left plot of Fig. 9). We fit the data to a quadratic function and allow the optimizer to adjust the angular offsets of each channel. This provides a more coherent picture of the angle of incidence versus the scattering fraction shown in the right plot of Fig. 9. The angle offsets of the tested channels are given in Table 3.

The Fast-2D scattering fraction seems to be quite sensitive to the angle of incidence. To obtain a scattering fraction of less than 0.1, it would seem that there is about a 1° tolerance in the angle of incidence. This assumes that the detected sensitivity is symmetric, which we could not confirm. The test setup cannot measure a wide range of incidence angles (limited to approximately ±0.3°), so we are subject to the alignment of the detector mask on this particular board. Thus, these results are not fully conclusive with regard to detector tolerancing, but it does suggest that the boards are quite sensitive in at least one incidence direction.

We should not expect all detectors, even on the same Fast-2D card, to have the same slow fraction. It would appear that slight deviations in mask alignment to the detector determine where the incident beam lies on the angle of incidence versus the slow fraction curve. The angle of incidence that obtains a small slow fraction in one channel is very likely suboptimal for another channel.

a. Simulation description

The 2D probe simulation used here is largely composed of a diffraction model and includes a number of prior determined characteristics. It models the incident wave front as an elliptical Gaussian beam with 0.2-mrad vertical divergence and 0.6-mrad horizontal divergence based on laboratory measurements of the beam. Prior published OAP 2D simulations have been limited to plane-parallel waves. It also includes the optical receiver point spread function (PSF), which has been characterized as a 10-μm-diameter (null to null) diffraction-limited spot (or Airy disk). The PSF represents the smallest resolvable feature and is the spatial analog to the impulse response of a dynamic system. The PSF results in blurring of an image such that features smaller than the PSF typically cannot be resolved. Prior published OAP simulations do not include the PSF, which may be important in assessing the sample volume of small particles consisting of only a few pixels.

Particles are simulated passing through the probe sample volume as opaque disks as was done in Korolev et al. (1991). In this study, the electric field at the object plane is determined using a Fourier optics method detailed in Goodman (2005), where propagation of a field is calculated by multiplying the field’s Fourier transform by the transfer function of freespace such that the exiting wave’s Fourier transform is given by

where

is the Fourier transform of the input electric field;

is the Fourier transform of the electric field after propagating a distance z;

and

are the spatial frequency axes corresponding to spatial dimensions x and y, respectively; and

is the transfer function of freespace, which is

where λ is the wavelength of the illuminating laser, which is 660 nm in this work.

This method of propagating waves is practically similar to the Fresnel convolutions performed in Korolev et al. (1991). The Fourier transform method tends to be faster when Fresnel convolution kernels are near the same size as the total grid area, but it has the drawback of aliasing (where diffracted patterns that exit the edge of the modeled area wrap around).

The initial simulation runs at a much smaller spatial grid spacing than what is actually measured by the probe. Optical diffraction calculations are performed on a grid with a spacing of 1.5625 μm. The grid spacing details are included in the simulation table.

We describe the electric field after the particle (modeled as an opaque circular disk here) is illuminated by the laser using

where

is the total field after the laser is occulted by the particle. The particle has a position

along the optic axis;

along the detector array direction; and

orthogonal to the array, which is a function of t, reflecting that the particle is moving along this axis. The function

is the transmission mask describing the particle centered about

and

;

is a Dirac delta function centered at

and

; and

is the electric field of the incident laser beam at

, which is described as an elliptical Gaussian beam in this simulation. For the purposes of this simulation, we assume the laser is centered on the detector array.

The elliptical Gaussian beam is described as the product of two 1D Gaussian beams (see Yariv and Yeh 2007, section 2.11),

where k is the wavenumber and is given by

For brevity, we present only the definition of

, where

follows the same form. The electric field definition is

with the following definitions:

beam radius:

beam waist:

where

is the beam divergence along the x-coordinate axis;

Rayleigh range:

beam radius of curvature:

In the case of the Gaussian beam description above, the term

refers to the location of the x-dimension beam waist. Notably this might be different from

when the beam is astigmatic. Note that in general, the y axis will have its own separate definition for the beam radius, waist, divergence, Rayleigh range, and radius of curvature.

The definition for an elliptical Gaussian beam provided above assumes the major and minor axes are aligned to x and y coordinates, respectively. Rotating the axes can be accomplished by substituting a rotated coordinate system into the above-mentioned definitions.

Once the electric field after the particle is obtained from Eq. (11), it is propagated, using Eq. (9), to the object plane of the probe (assumed to be in the center of the two probe arms) at

to obtain

. The intensity on the detector is given by

where

is the optical system PSF that accounts for the optical system resolution. The PSF was measured directly by placing a charge-coupled device (CCD) in the object plane of the instrument, illuminating the detector with a 660-nm laser and measuring the Airy pattern from a small feature on the detector array. The first null in the Airy pattern is at a radial position of 5 μm. The PSF for this simulation is given by

where

is a Bessel function of the first kind, order 1,

, and

is the radius of the first null in the PSF. For a diffraction limited system, the radius of the Airy disk is generally approximated using

where λ is the wavelength of the light and

is the numerical aperture of the collection optics.

Once the intensity on the detector is obtained from Eq. (19), we account for the circular apertures in the mask in front of each detector and perform spatial sampling (decimation) to obtain the intensity signal on the detectors as a function of time using

where

is the time domain signal on each detector at positions separated by

, and

is the description of the mask aperture function (a

function 20 μm in diameter). The spatial variable y is replaced with n for the channel number. The Dirac delta function

represents the fact that, as a linear array, the detectors only sample the intensity at

. The function

is a periodic train of impulses positioned wherever the argument is an integer such that

This function is used to represent the discrete spatial sampling of the 64 detectors [for more information, see chapter 2 in Goodman (2005)].

The photodiode spacing on the instrument is measured directly by placing a CCD in the object plane of the instrument and illuminating the detector array with a 660-nm laser. We have found the most effective way to measure the spacing is by Fourier transforming the CCD line along the reimaged detector array and measuring the location of the peak in spatial frequency.

Finally, the electrical signal includes the response time of the probe detector and electronics, and the 25-μm horizontal resolution through the following equation:

where

is the impulse response of the detection chain as determined in this work, and the comb function represents the decimating of the signal at

, where

is the airspeed of the aircraft. The term

is scanned in our analysis to account for possible variations in the particles’ arrival time relative to the clock on the probe, where for a particular particle

and is uniformly distributed. For our analysis, we break this up into eight evenly spaced samples to numerically approximate averaging across all possible values of

.

Examples of a particle through the stages of analysis are shown in Fig. 10, where , and Fig. 11, where . The top-left panel shows for a 137.5-μm particle with , , and when it is illuminated by the elliptical Gaussian beam. The top-right panel shows the resulting intensity pattern on each detector recorded as the particle moves across the array in time. The bottom-left panel shows the resulting voltage waveform after sampling at the diode array resolution and imparting the detector response characterized in this work. Finally, the bottom-right panel shows the 1-bit digitized signal after 50% shading requirements are imposed. The difference between is subtle between the two cases (the case has slightly more horizontal blurring), but the resulting 1-bit images have noticeable differences. Perhaps more significant, the case will trigger the DOF flag but the will not. The result is that the case is registered as in the sample volume and the case is not. The threshold for where the particle is counted is at about . Figure 12 shows the response waveforms on pixel 32 (near the center) for no time response effects, , , and , with the diode state and DOF shading thresholds shown as dashed lines. As increases, the modulation depth in the signal is reduced, so the particle’s signal is less likely to cross the DOF threshold.

In this analysis we use the DOF flag (requires 67% shading) to determine whether a particle is in the probe sample volume. If at least one pixel triggers the DOF flag, we count the particle as being in the sample volume. We then count the number of particles in the sample volume and divide by the total number of particles simulated to provide the probability of detection. Note that we do not require the particle be sized correctly, as this will depend on the specific OAP processing code. To determine the sample area of a particular particle size, we multiply the probability of detection by the total simulated area of the particles. The parameters used for the 25-μm Fast 2D-C simulation are described in Table 4.

b. Simulation results

The sample area as a function of particle size is shown in Fig. 13 for different slow fractions. These slow fractions correspond to the range of values observed during the initial detector characterization. The black line in the figure is the case where there are no response time effects in the probe. For comparison, the waveforms of those impulse responses are shown in Fig. 14.

As the amount of energy in the slow decay term increases, it can have a significant effect on particle sample volume. Figure 15 shows the fractional difference in sample volume compared to the case where response time is not included in the simulation. In this case the slow fractions used are more realistic for operational 2D probes with values between 0 and 0.1. Figure 15 shows the significant challenge in estimating the sample area (and therefore particle concentrations), particularly at small sizes.

We expand the simulation of our 25-μm 2D-C probe to include airspeeds from 80 to 250 m s⁻¹ with slow fractions between 0 and 0.1. We calculate the mean sample area as a function of particle diameter and airspeed, and the corresponding fractional spread in sample area () due to variation in slow fraction. The results are plotted in Fig. 16, where the surface height corresponds to mean sample area and the color scheme corresponds to fractional error. Not surprisingly, the uncertainty due to slow fraction is larger for smaller particles; however, the airspeed dependence is relatively small. It is notable that the uncertainty due to slow fraction tends to exceed changes in sample volume due to airspeed. Overall, the uncertainty in sample area caused by uncertainty in slow fraction is on the same order or less than the slow fraction uncertainty or about 10%.

The fractional uncertainty due to slow fraction will add in quadrature with other fractional uncertainty terms (e.g., sizing error, Poisson statistics, and any sample volume uncertainty arising from other unknowns in the probe operation). This means the error shown in Fig. 16 represents a base fractional uncertainty in particle concentration. While the results reported here raise some basic concerns about sizing particles in the smallest size bins, it also offers some assurance that the error in particle concentration caused by response time uncertainty is less than 10% for particles larger than the third bin (diameter of 87.5 μm and larger). Furthermore, the uncertainty due to slow fraction is not fundamental. If the slow fraction could be accurately determined for each individual channel on a specific probe, and we were confident the slow fractions would not change during flight, then the effects could be accounted for in the sample volume estimate. At this time, this does not seem practical because the analysis should include the probe’s illuminating laser, which does not currently have sufficient bandwidth for this analysis.