Ultrafast Fiber Bragg Grating Interrogation for Sensing in Detonation and Shock Wave Experiments

In our applications, we have used a variety of single-mode Type-I CFBG sensors with the shortest CFBG being 10 mm and the longest at 200 mm. All of the CFBGs are phase mask UV written on Ge-doped Corning SMF28e step-index-type fiber. The nominal fiber index is n = 1.4682. In addition, we used polyimide and acrylate-coated versions with a recoat process after the grating is written. For detonation fronts and strong shocks, the effects of coatings over the CFBG portion can be neglected due to the high pressures (and temperatures) present during destructive shock processing of the CFBG. Even when shock pressures are sufficiently low that an elastic strain response is present, the strains are still very high, exceeding 10${}^4$ $\mathsf{\mu }ϵ$, and it may be possible that the coating may be contributing. However, shock studies of FBG elastic response under conditions of high strain and high strain rate as in a shock wave are the subject of on going work [8,9,10]. Table 1 lists the various CFBGs used in this work. Typically, the C-band gratings are centered at λ = 1550 nm and are greater than 95% reflective across the wavelength range λ = 1533 nm–1568 nm. The C-band CFBGs are designed to fit within the flattest portion wavelength range of our ASE and mode-locked fiber laser sources. In addition to C-band CFBG, broader spectral C + L-band CFBGs are also used to increase spatial resolution or extend the CFBG length, for example to 200 mm-long sensors. The C + L-band CFBGs cover a wavelength range λ = 1533 nm–1595 nm and, therefore, also require broadband light sources that extend to the long wavelength end of the L-band, λ = 1605 nm. The vendors used for our custom-made CFBG sensors are Timbercon, Inc. (Tualatin, OR, USA), and O/E Land, Inc. (La Salle, QC, Canada).

Table 1 also lists the chirp rates used for the CFBGs. Depending on the chirp rate, we previously demonstrated that the positional sensitivity of the sensor can be adjusted [4]. The chirp of each grating used is measured before installation on an experimental test. The chirp is measured using a Luna Innovations frequency-swept optical backscatter reflectometer (Model No. OBR 3600). This ensures that every CFBG sensor chirp rate and length is calibrated before use. In Figure 4, we plot the reflection spectra and OBR measured chirp for several CFBGs. The Luna instrument measurement of the chirp on the same CFBG is repeatable to within 0.1%. However, due to manufacturing variability, LUNA OBR measurements of the chirp on CFBGs of the same fiber type can vary as much 1.5% from the design specification. We therefore characterize and measure the chirp of each CFBG using the Luna OBR and then use measured chirp for each individual CFBG to analyze its corresponding dataset. Often, we must trim the CFBG to length to match the experimental assembly, and we accomplish this using a femtosecond laser machining apparatus [1]. The OBR utilizes a frequency-swept laser interferometer to scan across the entire band of the CFBG and outputs 2× the group delay, because it measures the wavelength-dependent group delay in double pass. The slope of a linear fit line to the OBR group delay measurement is used to extract the chirp in units of ns/nm. To convert from units ns/nm to linear chirp in length per unit wavelength (i.e., mm/nm), we multiply the group delay slope by the factor (0.5 c/n), where c = 299.792 mm/ns is the speed of light and n = 1.4682 is the average fiber index. The 0.5 factor is to account for the fact that the Luna OBR measures the group delay in double pass.

After chirp calibration of the sensors, they are appropriately then trimmed in length to fit according to the experimental assembly requirements. Direct bonding of the CFBG sensor is done with a cyanoacrylate adhesive (M-Bond 200 from Vishay Precision Group, Inc.) that is compatible with most materials, including many of the plastic bonded explosives we have used in our tests. After bonding the CFBG sensors, we accurately measure the CFBG tip position to register a location on the device so that post-analysis hydrodynamic flow calculation models have a dataset for comparison. Typically, we measure the start tip position of each sensor to ±25 $\mathsf{\mu }$m.

An 8 inch-long × 8 inch-wide × 0.5 inch-thick slab of the plastic bonded high explosive (HE) PBX-9501 was uniformly initiated along one edge using an explosive line wave generator (LWG) [20]. The line wave generator is started at its apex point with a single SE-1 detonator. The goal of this test was to cross compare the performance of a series of detonation front position diagnostics along the top surface of the main charge. The nominal detonation velocity for PBX-9501 is 8.8 mm/$\mathsf{\mu }$s. The time-of-arrival electrical pin, electrical flat cable shorting switches, piezo pin and CFBG diagnostics were used to record the detonation front as it transits the length of the main HE charge. The time of arrival electrical shorting pins and piezo pins have been the standard for shock and detonation discreet position transducer measurements for many decades. They operate as coaxial electrical probes that incorporate a switch at the tip that produces an electrical signal when impacted by the detonation or shock front. The switch at the end of the probe may consist of a non-biased piezo crystal (piezo pin) or a voltage-biased unterminated air gap (shorting pin) that is crushed by the intense shock pressure to produce a high speed voltage transient indicative of the arrival of the shock front. The signal voltage is self-generated in the piezo pin under shock compression, and the bias voltage of the shorting pin was set to 200 V. The response time of these pins is 10 ns. The piezo and electrical shorting pins were procured from Dyansen, Inc. (Goleta, CA, USA), and the part numbers were CA-1037-C and CA-1135-SL, respectively. The electrical flat cable shorting switches are custom-made probes that operate in a similar fashion as the shorting pins, but instead have their transmission line and an array of metal disc pads embedded in a flat cable comprised of a multilayer cable strip of Kapton polyimide and copper. The overall thickness of the flat electrical shorting switches is only 0.003”, and the active metallic square shorting pads where the sensing occurs are only 200 $\mathsf{\mu }$m${}^2$ in area with a 110-$\mathsf{\mu }$m gap between the switch and grounding plane. The bias voltage for the shorting switches was 180 V, and the response time of each switch in the linear array is ∼4 ns.

In Figure 5a, we illustrate the geometry of the slab detonation experiment, and in Figure 5b, we show a photograph of the assembled experiment showing various locations of the diagnostics. A total of four 120 mm-long CFBGs (nominal chirp of 3.4 mm/nm) were fielded with two CFBGs straddling the center line on either side of a 16-point linear strip array of shorting switches. Two of the CFBGs were recorded using the spectrally-integrated approach, and two were recorded using the time-streak method. The CFBGs are laid into a 0.010” machined groove in the HE, and each CFBG is fixed in place using M-Bond 200 quick setting epoxy that is chemically compatible with PBX-9501. Careful measurements of the registration of the CFBG starting tip positions relative to the PBX-9501 and LWG edge were done using a microscope-equipped height gauge with a high resolution CCD imaging camera. Similar metrology of the other single-point time-of-arrival diagnostics was also done. Table 2 lists the various diagnostics fielded for inter-comparison. All pin and switch type diagnostics are point measurements, so an array of point-type measurements is used to measure detonation front position versus time. Only the CFBG yields a continuous measure of position versus time. Figure 5a shows the relative locations of the point-type measurements, and Table 2 shows the number of point measurements for each.

Figure 6a is a plot of the raw CFBG voltage versus time data recorded for the experiment for the two channels using the spectrally-integrated approach. The inset to Figure 6a is the CFBG spectra associated with each 120-mm CFBG channel. The nearly linear monotonic decrease in the detector voltage versus time is indicative of a detonation front that is traveling at a constant speed during the destructive process of the CFBG. Small deviations from linearity are a result of the non-flatness of the CFBG spectrum (see the inset to Figure 6a) that must be taken into account in the analysis. The plots in Figure 6b are the extracted CFBG length versus time during the transit of the detonation along the CFBG after appropriate data processing using the dynamic voltage trace, pre-shot measured spectrum and chirp value for each CFBG. We use the look-up table (LUT) data analysis method described previously [1] for the spectrally-integrated approach to map the measured voltage to the CFBG length. In the main portion of the detonation burn, CFBG1 and CFBG2 are in excellent agreement, but small deviations at the beginning and end of the runs indicate possible mechanical decoupling of the CFBG tip from the HE at the beginning the detonation and at the end where the CFBG sensor transitions to normal bare fiber inside of loose furcation tubing.

After showing the results for the two CFBGs that were fielded using the spectrally-integrated approach, we now show the results for the two 120 mm-long CFBGs that were fielded using the time-streak spectrally-resolved method. In Figure 7, we show four different 10-ns time slice windows for a single CFBG as recorded by the 25-GHz high speed digitizer (see Figure 2): Figure 7a, prior to arrival of the detonation front; Figure 7b, the detonation front reaches the beginning of the CFBG; Figure 7c, the detonation near the end of the CFBG; and Figure 7d, the detonation front past the end of the CFBG where it is fully destroyed. The time slice windows are 10 ns wide due to the 100-MHz interrogation rate determined by the mode-locked laser repetition rate. A fiber dispersion spool of 13 km of Corning SMF28e fiber was used to disperse the pulse reflected off the CFBG to a time-stretched spectrum fully contained in the 10-ns time slice window. For all of the plots in Figure 7, the top axis shows the wavelength band of the CFBG for 13 km of dispersion using the SMF28 fiber dispersion constant of D${}_f$ = 16.7 ps/nm/km. Post experiment data processing is used to convert every FBG interrogation time-slice window for the entire digitizer record into a 2D color contour plot of spectrum versus time. We use the recorded 100-MHz laser clock pulse as a reference inter-pulse timing fiducial to subdivide the entire CFBG waveform data record (which contains one thousand 10 ns-wide time-slice interrogation windows per 10 $\mathsf{\mu }$s recording period) into a series of time-resolved CFBG spectra, each separated by 10 ns. A sample contour plot of this 2D time-streak spectrum for the PBX-9501 slab detonation test (CFBG3) is shown in Figure 8a. The left y-axis of Figure 8a is the spectrum of the CFBG extracted for every time slice window (as was shown on each of the top x-axes of Figure 7). The right y-axis is the CFBG length that is calculated by multiplying the CFBG spectral bandwidth by the chirp rate value (3.43 mm/nm from Table 2). The intensity, or z-axis, is the recorded signal voltage level of the photoreceiver. The x-axis is time with a resolution of 10 ns determined by the 100-MHz interrogation rate of the laser. The y-axis, however, has been sampled at 100 GS/s, and at a 25-GHz signal recording bandwidth, the spectral resolution is estimated to be R${}_{\lambda }$ = 0.18 nm (using Equation (2)). An estimated spatial resolution of 0.62 mm is obtained after multiplying R${}_{\lambda }$ by the CFBG chirp. The PBX-9501 detonation front propagation along the length of the CFBG is clearly seen as a time-dependent reduction in the bandwidth of the CFBG beginning at t = 26.1 $\mathsf{\mu }$s and reaching the end of the grating t = 39.7 $\mathsf{\mu }$s. The detonation front position versus time is extracted by processing the 2D contour plot using a pulse width detection method at every interrogation time sample of the x-axis. Figure 8b contains plots of the length versus time results for both time-streak CFBGs that were fielded on the PBX-9501 slab test (CFBG3 and CFBG4). Comparison of the two time- streak CFBGs shows a slight excursion from overlap between t = 31 $\mathsf{\mu }$s and t = 35 $\mathsf{\mu }$s that we attribute to possible error in the edge extraction algorithm when processing the 2D time-streak spectra contour plots. Placing the entire dataset (CFBG and point-type) results on a single plot allows for inter-comparison of all diagnostics listed in Table 2. All of the data plotted in Figure 9 have been corrected for placement position on the PBX-9501 and their respective associated internal timing delays relative to SE-1 detonator initiation of the LWG. All data show excellent agreement of position versus time for detonation front position in the main HE charge, except for a peculiar timing anomaly in the shorting switch data that appears 1.05 $\mathsf{\mu }$s too late compared to the other diagnostic datasets. The shorting switch data have been globally shifted by −1.05 $\mathsf{\mu }$s to come into agreement with the rest of the datasets. The ordinate axis is the position on the main PBX-9501 charge where zero position is at the LWG/PBX-9501 interface. The entire length of the PBX-9501 slab is 8 inches (203.2 mm), and the first point sensors are located beyond 1” (25.4 mm) to allow for any overdrive of the PBX-9501 to dissipate before a detonation position is recorded. The CFBG sensors were limited to 120 mm in length, and therefore, Figure 9 shows that they only cover a portion of the entire detonation run position, i.e., from approximately 40 mm to just over 160 mm. Although pre-shot metrology of the measured uncertainty in the placement position of the electrical and CFBG diagnostics is ±25 $\mathsf{\mu }$m, the dynamic uncertainty of the position in the discreet electrical diagnostics is dominated by the time response when encountering a detonation front as it traverses the probe. For example, if the shorting or piezo pin response time is 10 ns, then for the detonation front traveling at 8.8 mm/$\mathsf{\mu }$s, the position uncertainty can be as large as 88 $\mathsf{\mu }$m. Similarly, for the shorting switches with a response time of a ∼4 ns, the uncertainty is 35 $\mathsf{\mu }$m. The CFBG position uncertainty is determined by the scatter level in the data of Figure 9 and can be estimated as ±200 $\mathsf{\mu }$m. The accepted value for a PBX-9501 detonation velocity in a rectangular slab is 8.80 mm/$\mathsf{\mu }$s [20]. Table 3 is a compilation of the extracted detonation velocity taking the entire dataset for each diagnostic. The listed CFBG average velocity and uncertainty in the table is calculated from four independent slope measurements for each CFBG. The CFBG data appear to be approximately 0.45% above the accepted value.

In a recent paper [4] we presented CFBG-based HE detonation front position measurements for a variety of high explosives consisting of a cylindrical stack of various HE-types in the form of a multi-HE rate stick with a final stage of shock wave tracking into an inert material section of PMMA (polymethyl methacrylate) plastic. The measurement of the strong shock that was launched by the final HE stage into the PMMA rapidly attenuated due to the unsupported drive from the HE after reaching the PMMA. Measurements using the spectrally-integrated CFBG interrogation approach were inadequate at tracking the full shock position versus time because for low shock pressures (i.e., p ≤ 5 GPa), the CFBG is not destructively processed and renders CFBG shock tracking measurement ineffective. Whereas measurements of detonation-based destructive-mode CFBG tracking of position are robust with the spectrally-integrated and time-streak spectrally-resolved interrogation approaches, accurate tracking is much more subtle and dependent on peak pressures when tracking inert shocks. A more detailed approach to measure CFBG response to weak shocks that perturb, but do not destroy, the CFBG are needed to correlate sensor behavior with shock dynamics. In these cases, we assert that the time-streak spectrally-resolved approach is essential, because this interrogation approach measures the full spectral response of a CFBG under a purely strain-induced effect. Here, we demonstrate an example situation where a decaying shock wave is generated with an impulse provided by the detonator that is then allowed to radially expand in a plastic case, while losing energy from the initial impulse. The 10 mm-long CFBG records the radial shock propagation, but then at a late time exhibits a strain-induced spectral blue shift once the pressure is sufficiently low enough and is no longer able to destructively shock process the CFBG. An illustration of the geometry used for this test is shown in Figure 10. A short (10 mm long) C-band CFBG is sandwiched between two plates of phenolic plastic. A 0.010” groove is machined into the bottom plastic case to accommodate the CFBG and bare single-mode fiber. The CFBG tip is placed at the outer radius of the LoFi (low fidelity) detonator (7.7 mm diameter), and the detonator is recessed into a hole of the upper plastic plate. The CFBG spectrum measured by the time-streak method and spectrometer is shown in Figure 3b. The time-streak spectrum is generated using 13 km of SMF28e fiber for chromatic dispersion, and the CFBG chirp is 0.35 mm/nm.

In Figure 11a, we plot the 2D time-streak color contour plot over a 5 $\mathsf{\mu }$s-long time window that displays the ensuing dynamics. On the left y-axis is the wavelength band of the CFBG detected and on the right y-axis is the corresponding length (or position) of the CFBG after applying the chirp value to the wavelength band. The destructive shock processing of the 10 mm-long CFBG is clearly visible, starting at t = 64.65 $\mathsf{\mu }$s, where the shock strikes the CFBG tip and proceeds in a curved nonlinear fashion to near complete consumption of the grating near t≈ 66 $\mathsf{\mu }$s. However, it is clearly visible that after the shock transits the entire length, a small portion (∼2–3 mm) of CFBG remains intact for times t > 66 $\mathsf{\mu }$s. The data support the explanation that as the shock pressure wave decays while moving away from the detonator/CFBG tip location, it weakens to the point where that it is no longer able to destructively process the CFBG. Instead, a portion of the CFBG survives and undergoes a blue shift as the weak shock pressure briefly compresses the final few mm portion of the grating. Although not shown, for very late times, there appears a series of complex relaxation oscillations over tens of microseconds that extend beyond the recording time window that shows a small portion of the CFBG in the 1537.5 nm–1539.2 nm range remaining reflective at the end of recording. The nonlinear shape of the shock position versus time is characteristic of a slowing wave as it attenuates. In Figure 11b, we show a line-symbol plot of the CFBG length versus time plot after processing the data from Figure 11a. The trace represents the shock location in the grating, and there is a breakdown in the recovery of the shock position near L = 3 mm when interference from the CFBG’s elastic wavelength shift response prevents clean extraction of the shock position. Each symbol is separated by 10 ns, the time resolution of the time-streaked interrogation system determined by the inverse time of laser repetition rate. The scatter of the data points in the ordinate axis during the pre-shock arrival time (i.e., t < 64.65 $\mathsf{\mu }$s) yields a measure of the position uncertainty after analysis. The average uncertainty in position is ±0.075 mm. The initial slope at shock entry (at t = 64.65 $\mathsf{\mu }$s) is nearly 12 mm/$\mathsf{\mu }$s. The first evidence of the spectral blue shift occurs at the position of 3.6 mm (at t = 65.6 $\mathsf{\mu }$s), and the wave speed has slowed to near the sound speed of silica fiber: c${}_o$ = 5.6 mm/$\mathsf{\mu }$s and in the expected range of plausible slope (velocity) measurement error of less than 2%. Therefore, we conclude that as the initial shock wave attenuates to acoustic conditions, the fiber survives and is no longer destructively processed, and instead, its elastic properties result in the dominant response at late times. Elastic strain sensing in silica single-mode uniform FBGs is known to have a strain-induced (ϵ) Bragg wavelength shift of −1.2 pm/$\mathsf{\mu }ϵ$ and a pressure-dependent sensitivity of −4 pm/MPa under compression [5]. The mean global wavelength blue shift in the data of $\mathrm{\Delta }\lambda$ = −16 nm at t = 67.5 $\mathsf{\mu }$s would indicate that the CFBG is sensing a peak strain and pressure of 0.013 ϵ and 4 GPa, respectively. The results here demonstrate that a high-speed strain/pressure measurement with CFBG sensors is possible if the pressures are sufficiently low enough to avoid destructive mode shock processing.

In our final example, we show CFBG results from an experimental test with a set of four ∼100 mm-long CFBGs mounted axially on the inside surface wall of a 6.1”-long (154.9 mm) by 4.75” (120.6 mm) outer diameter aluminum hollow cylinder (11.6-mm wall thickness). An axial propagating shock along the length of the wall is generated after loading from an HE PBX-9501 charge. Figure 12 illustrates the geometry used for this test. The HE drives a shock impulse into the wall of the hollow cylinder from one end. The four CFBGs were mounted such that their end tips were placed within 0.010” of the cylinder end closest to the HE (except for CFBG2, which was 0.31” away from the HE/Al interface). The four CFBGs are placed at azimuth angles separated by 90${}^{\circ }$ (i.e., CFBG1 at 30${}^{\circ }$, CFBG2 at 120${}^{\circ }$, CFBG3 at 210${}^{\circ }$ and CFBG4 at 300${}^{\circ }$). A photograph of the target assembly on the firing point is shown in Figure 12b. Anticipating a decaying shock structure because of lateral (radial) release and rarefactions inside the aluminum wall, we therefore fielded all four CFBGs using the time-streak spectrally-resolved approach. Figure 13 shows the 2D time-streak contour plot results for all four CFBGs (CFBG1–4). All of the CFBG results show a complicated response structure that would not have been otherwise possible if one had recorded the CFBG response using the spectrally-integrated approach. The CFBG data in Figure 13a, CFBG1, Figure 13c, CFBG3, and Figure 13d CFBG4, show that the detonation exits the HE, and the ensuing shock enters the aluminum cylinder at t = 21.5 $\mathsf{\mu }$s. Since CFBG2 (Figure 13b) is pulled back by 0.31” from the HE/Al interface, the shock arrives at this sensor slightly delayed at t = 22.8 $\mathsf{\mu }$s. In all of the data of Figure 13, although there is spectral-temporal signature of dynamic spectral shifting along much of the CFBG sensor-caused strain effects, the leading edge of the shock wave is clearly resolved if we properly analyze the 2D time-streak information. The shock-induced index modification of the CFBG allows for the detection of the edge of the shock position, but there is also large red shifting in large portions of the CFBG spectral band that can only be attributed to tensile strain in the sensor that we hypothesize is a result of the post shock release mechanics of the aluminum cylinder (i.e., initial stages of buckling) caused by lateral waves generated in the cylinder walls. Another possibility for the tensile strain observed behavior is the surface delamination process of the sensor as the M-Bond 200 adhesive used to cement the CFBGs to the cylinder yields to release the CFBG after the leading edge of shock has transited a localized shock processed region. To better understand the tensile strain dynamics, it would be useful to perform follow-on experiments with CFBG sensors located on both the inside and outside surfaces of the cylinder wall so that the wave exiting the inside and outside surfaces is simultaneously tracked. If we focus on the largest red shift that appears on channels CFBG1 (Figure 13a) and CFBG4 (Figure 13d) at t∼ 28 $\mathsf{\mu }$s, an estimate of the largest strain measured by the CFBG is extracted. The largest CFBG1 and CFBG4 measured spectral red shifts are $\mathrm{\Delta }\lambda$ = 27 nm and $\mathrm{\Delta }\lambda$ = 30 nm and occur at t = 26.4 $\mathsf{\mu }$s and t = 27.9 $\mathsf{\mu }$s, respectively. Using the known elastic strain-wavelength shift response coefficient of 1.2 pm/$\mathsf{\mu }ϵ$ yields a peak strain in the fiber of 0.023 ϵ and 0.025 ϵ for CFBG1 and CFBG4, respectively. After achieving peak strain, significant portions of the post shock regions in all of the CFBGs remain reflective (i.e., not destroyed). When the shock front reaches the end of each of the CFBGs at t∼ 37.5 $\mathsf{\mu }$s, the shock pressure places each grating in a compressed state with an average spectral blue shift of $\mathrm{\Delta }\lambda$∼−5 nm. Using the known pressure to wavelength shift conversion coefficient of −4 pm/MPa, an average peak pressure of 1.25 GPa is measured at the end of run in each grating. Late time signals are recorded to near 60 $\mathsf{\mu }$s when it appears that all of the signal is lost due to fiber breakage.

As was demonstrated for linear detonation front data of Figure 8 in Section 3.1, we also process the 2D contour plot data of Figure 13 to extract the shock wave position versus time. Despite having a complex spectral response, the time-streak spectrally-resolved approach is capable of identifying a precise measure of the shock position. This would not be true of the spectrally-integrated approach where undestroyed portions of the grating would lead to large uncertainty in the position measurements. In Figure 14a, we overlay the results of the CFBG shock position versus time line plots for the four CFBGs fielded. Excellent agreement in the dataset is observed. Late time CFBG shock position data after t = 37.5 $\mathsf{\mu }$s are difficult to resolve because of complications from the pressure-induced spectral shift that makes pulsed edge detection impossible. Nonetheless, the data in Figure 14a show that azimuthal symmetry in the shock launched along the sidewalls of the cylinder is very good. A linear fit to position versus time data computed for each CFBG yields: 6.43 ± 0.030 mm/$\mathsf{\mu }$s for CFBG1, 6.54 ± 0.022 mm/$\mathsf{\mu }$s for CFBG2, 6.43 ± 0.033 mm/$\mathsf{\mu }$s for CFBG3 and 6.42 ± 0.027 mm/$\mathsf{\mu }$s for CFBG4. It is not clear whether these differences are due to azimuthal asymmetry of the initial explosive drive and ensuing hydrodynamics or if this is due sensor error. Cross comparison with complementary diagnostics and multi-dimensional hydrodynamic simulations would be necessary to resolve the cause of these differences. Taking all four CFBG shock position line plots and averaging, we show in Figure 14b the average shock position versus time. A linear fit to the CFBG data in Figure 14b is also shown, and the computed slope yields an average shock velocity of 6.46 ± 0.024 mm/$\mathsf{\mu }$s. This value is slightly above the longitudinal speed of sound in aluminum 6061 (${\mathit{c}}_{\mathit{Al}}$ = 6.3 mm/$\mathsf{\mu }$s) and is consistent with a weak shock propagating at a speed slightly above ${\mathit{c}}_{\mathit{Al}}$.