## VI. CONCLUSION

Fiber Bragg Gratings (FBGs) are gaining acceptance as velocity or pressure gauges in the field of detonation and shock physics on account on their high sensitivity, passive nature, small size, flexibility and electromagnetic immunity. High-bandwidth signal acquisition is required (digitizing oscilloscopes) and the SNR mainly depends on the acquisition electronics. FBGs provide direct temperature and strain sensing, the latter parameter being converted into stress with the help of a mechanical stress-to-strain model. The wavelength-encoded capability enables multiplexing (WDM) and provides reliable measurements even in harsh conditions. Additional benefits of FBGs in the domain of shock physics are the low shock-induced self-heating and the high Hugoniot Elastic Limit (HEL) of silica (8.8 GPa). For intermediate range of impact stress (i.e. less than the HEL value but albeit higher than conventional mechanical stress levels (several 100 MPa)), the practical consequences are a high strain sensitivity and the absence of temperature compensation of the Bragg signal.

While regular FBGs are usually considered as point sensors up to several-mm long, chirped FBGs (CFBGs) are investigated to provide wavelength-to-position discrimination with the purpose of monitoring pressure and velocity profiles over distance ranges of typically 100 mm. The practical use of CFBGs for monitoring detonation velocity of High Explosives (HEs) has been investigated by several research teams$^{6-10}$ and is now well established. The crucial advantage of CFBGs in comparison with conventional FBGs is the distributed measurement capability that simplifies both sensor deployment, data retrieval and increases the number of data points which in turn improves the metrological resolution. CFBGs have found a useful application in the determination of Shock-to-Detonation Transition (SDT).

In this paper, the use of CFBGs as *in situ* pressure/velocity gauges is investigated theoretically and experimentally under inert planar shock loading conditions with a gas gun launcher and Aluminum flyer and target. Earlier shock monitoring experiments were investigated with several mm-long FBGs$^{11-15}$ and CFBGs have been investigated only recently for this purpose.$^{17-18}$ The principle is the same as for velocity measurement except that the CFBG is not broken immediately by the shockwave, but few microseconds later by the release wave. Hence, as the shockwave travels through the grating, the length of the pristine part (out of the shock) decreases at the shockfront speed while the grating part within the shock is under compression. Therefore, the Full Width at Half-Maximum (FWHM) of the spectrum of the pristine part shrinks progressively as the shockwave comes forward. The amount of light reflected back thus provides a measurement of the shockwave position as a function of time, from which the shockwave velocity may be estimated by linear regression, in a similar manner as previous experiments with HEs. Within metallic materials as targets (e.g. aluminum), the shockfront is associated with an elastic precursor that travels faster than the plastic shockwave. The Bragg spectrum of the compressed part is shifted to lower wavelengths (i.e. blue-shifted), the Bragg shift being stress-dependent. The amplitude of Bragg reflection spectrum provides
a complementary information related to the length of the grating part under sustained stress. At first glance, the reflectivity is proportional to the square of the grating length for weakly-reflecting FBGs. In the intermediate range of shock stress (i.e. less than the HEL of silica), CFBGs thus provide a simultaneous measurement of both shockwave velocity and state-of-stress within the target material. However, the estimation of the state-of-stress is less straightforward than the velocity as the wavelength-to-stress relation linking the Bragg wavelength shift to the shock stress requires both stress-to-strain and wavelength-to-strain relationships. A complete model requires to (i) know the state of stress within the target material, (ii) calculate the stress coupling coefficient due to imperfect impedance matching between the target material and the silica fiber, (iii) convert the state-of-stress into a state-of-strain within the silica fiber taking into account Hooke's laws and finally, (iv) translate the strain data into observable Bragg wavelength shifts (taking into account photo-elasticity and density change). Moreover, many constitutive parameters of silica (Young modulus, Poisson coefficient, refractive index) and of usual metals (yield strength Y) are stress-dependent which brings additional complexity.

In this work, a theoretical analysis of the Bragg wavelength-to-stress sensitivity is provided in the intermediate stress range [0 – 4 GPa], below the Hugoniot Elastic Limit (HEL) of silica. Starting from the linear Bragg constitutive equations, the state-of-stress within the target material (Aluminum 6061-T6) is firstly determined for both elastic (shear stress $τ < Y/2$) and hydrodynamic domain (shear stress $τ > Y/2$), taking into account the SCG model. Then, the state-of-stress within the silica fiber is estimated taking into account a constant coupling coefficient ($K = 0.78$), determined by analytical estimation (for impact stress less than 0.6 GPa) and by hydrodynamic code FEM modelings (OURANOS). The state-of-strain in the Silica fiber is then calculated considering that silica behaves elastically in this pressure range of interest ([0 – 4 GPa]), from which both the evolution of Bragg shift and sensitivities are determined with respect to impact stress for both domains. In the elastic domain of Aluminum, the wavelength-to-stress sensitivity of the CFBG is quite high (between -9 nm/GPa and -12 nm/GPa according to impact stress) on account of the orientation of the stress ellipsoid, towards the direction of shock, along the fiber axis. In the hydrodynamic domain however, the Bragg sensitivity is much lower (~ -7 to -8 nm/GPa according to impact stress) and tends to decrease with increasing impact stress until reaching a null value at ~ 3.6 GPa. The Bragg sensitivity of the CFBG embedded within Aluminum targets is therefore strongly nonlinear with impact stress.

In order to compare with the theoretical approach, an experimental calibration of CFBG under planar shock loading configuration has been performed. 6061-T6 Aluminum flyers have been launched at several velocities by a gas gun over transfer plates (targets) of similar material. CFBGs (zero-stress wavelength = 1605 nm, FWHM = 4 nm) were located within the target and bonded along the axis with glue.
a complementary information related to the length of the grating part under sustained stress. At first glance, the reflectivity is proportional to the square of the grating length for weakly-reflecting FBGs. In the intermediate range of shock stress (i.e. less than the HEL of silica), CFBGs thus provide a simultaneous measurement of both shockwave velocity and state-of-stress within the target material. However, the estimation of the state-of-stress is less straightforward than the velocity as the wavelength-to-stress relation linking the Bragg wavelength shift to the shock stress requires both stress-to-strain and wavelength-to-strain relationships. A complete model requires to (i) know the state of stress within the target material, (ii) calculate the stress coupling coefficient due to imperfect impedance matching between the target material and the silica fiber, (iii) convert the state-of-stress into a state-of-strain within the silica fiber taking into account Hooke's laws and finally, (iv) translate the strain data into observable Bragg wavelength shifts (taking into account photo-elasticity and density change). Moreover, many constitutive parameters of silica (Young modulus, Poisson coefficient, refractive index) and of usual metals (yield strength Y) are stress-dependent which brings additional complexity.

In this work, a theoretical analysis of the Bragg wavelength-to-stress sensitivity is provided in the intermediate stress range [0 – 4 GPa], below the Hugoniot Elastic Limit (HEL) of silica. Starting from the linear Bragg constitutive equations, the state-of-stress within the target material (Aluminum 6061-T6) is firstly determined for both elastic (shear stress $\tau < Y/2$) and hydrodynamic domain (shear stress $\tau > Y/2$), taking into account the SCG model. Then, the state-of-stress within the silica fiber is estimated taking into account a constant coupling coefficient ($K = 0.78$), determined by analytical estimation (for impact stress less than 0.6 GPa) and by hydrodynamic code FEM modelings (OURANOS). The state-of-strain in the Silica fiber is then calculated considering that silica behaves elastically in this pressure range of interest ([0 – 4 GPa]), from which both the evolution of Bragg shift and sensitivities are determined with respect to impact stress for both domains. In the elastic domain of Aluminum, the wavelength-to-stress sensitivity of the CFBG is quite high (between -9 nm/GPa and -12 nm/GPa according to impact stress) on account of the orientation of the stress ellipsoid, towards the direction of shock, along the fiber axis. In the hydrodynamic domain however, the Bragg sensitivity is much lower (~ -7 to -8 nm/GPa according to impact stress) and tends to decrease with increasing impact stress until reaching a null value at ~ 3.6 GPa. The Bragg sensitivity of the CFBG embedded within Aluminum targets is therefore strongly nonlinear with impact stress.

In order to compare with the theoretical approach, an experimental calibration of CFBG under planar shock loading configuration has been performed. 6061-T6 Aluminum flyers have been launched at several velocities by a gas gun over transfer plates (targets) of similar material. CFBGs (zero-stress wavelength = 1605 nm, FWHM = 4 nm) were located within the target and bonded along the axis with glue.
Two spectrum analyzers were used for the experiments, the first one used a free-space Czerny-Turner (CT) spectrometer (channel spacing ~ 6.2 nm) and the second one used an Arrayed-Waveguide Grating (AWG) spectrometer designed in integrated optics (channel spacing ~ 1.7 nm). The optical setup also involves a broadband light source (ASE), transimpedance amplifiers and digitizing oscilloscopes. The AWG-based spectrometer provided the most accurate results on account of smaller channel spacing and also unnoticeable interchannel crosstalk.

With the aim to calibrate the wavelength-to-stress response of CFBGs, both the elastic precursor and the sustained part of the plastic shock were analyzed with respective periods of time of 0.6 µs and 5 µs post-impact. The shock velocity that is measured by progressive extinction of the Bragg signal reflected back by the pristine CFBG is that of the elastic precursor and agrees well with expected value. For each portion of sustained plastic shockwave, the Bragg spectra were reconstructed from signal data and Gaussian-fitted with satisfactory goodness-of-fit. The experimental Bragg wavelength shifts were then plotted with respect to impact stress along with theoretical models for both elastic and hydrodynamic domains of 6061-T6 Al.

Experimental data agree well with theoretical predictions in the intermediate range of stress [0 – 4 GPa] investigated here, therefore validating the model for planar shock loading of aluminum flyer/target. The deviation between theoretical calculation and experimental data recorded with the AWG is less than ± 2%. This work opens large perspectives for the future of shock physics either for inert materials or for HEs.

First of all, our investigations were restricted to the intermediate stress range [0 – 4 GPa] for which data were available about pressure dependence of constitutive parameters, especially the evolution of the Young modulus of Silica under hydrostatic loading. Extended data on broader range are therefore needed in order to extend the stress range of interest.

Secondly, silica may withstand compressive stress up to 16 GPa and therefore, CFBG are likely to be used up to this stress value despite the occurrence of a phase transition (permanent densification) undergone by silica above its HEL value. To our knowledge, no such experiment has been reported yet. An important question arises whether linear photo-elasticity is still a valid assumption up to this stress value or if nonlinear photoelasticity must be considered instead.

Thirdly, this study highlights the nonlinear behavior of the Bragg shift *vs* applied stress. Since the CFBG remains in its elastic domain, the nonlinearity arises from constitutive parameters, mainly refractive index, Poisson coefficient and Young modulus of Silica. Another nonlinear contribution of lesser importance is strain hardening of 6061-T6 Aluminum (i.e. increase of the yield strength with applied stress).

Fourthly, an inverse calculation of the stress distribution from Bragg spectra is an ill-posed problem as phase information is missing from spectrometric acquisitions. Nevertheless, stress distributions and velocities may be modeled *a priori* in simplified configurations (e.g. hydrostatic or planar shock loading) and stress data may be retrieved with the help of the experimental wavelength-to-stress relations provided in this work.
Two spectrum analyzers were used for the experiments, the first one used a free-space Czerny-Turner (CT) spectrometer (channel spacing ~ 6.2 nm) and the second one used an Arrayed-Waveguide Grating (AWG) spectrometer designed in integrated optics (channel spacing ~ 1.7 nm). The optical setup also involves a broadband light source (ASE), transimpedance amplifiers and digitizing oscilloscopes. The AWG-based spectrometer provided the most accurate results on account of smaller channel spacing and also unnoticeable interchannel crosstalk.

With the aim to calibrate the wavelength-to-stress response of CFBGs, both the elastic precursor and the sustained part of the plastic shock were analyzed with respective periods of time of 0.6 µs and 5 µs post-impact. The shock velocity that is measured by progressive extinction of the Bragg signal reflected back by the pristine CFBG is that of the elastic precursor and agrees well with expected value. For each portion of sustained plastic shockwave, the Bragg spectra were reconstructed from signal data and Gaussian-fitted with satisfactory goodness-of-fit. The experimental Bragg wavelength shifts were then plotted with respect to impact stress along with theoretical models for both elastic and hydrodynamic domains of 6061-T6 Al.

Experimental data agree well with theoretical predictions in the intermediate range of stress [0 – 4 GPa] investigated here, therefore validating the model for planar shock loading of aluminum flyer/target. The deviation between theoretical calculation and experimental data recorded with the AWG is less than ± 2%. This work opens large perspectives for the future of shock physics either for inert materials or for HEs.

First of all, our investigations were restricted to the intermediate stress range [0 – 4 GPa] for which data were available about pressure dependence of constitutive parameters, especially the evolution of the Young modulus of Silica under hydrostatic loading. Extended data on broader range are therefore needed in order to extend the stress range of interest.

Secondly, silica may withstand compressive stress up to 16 GPa and therefore, CFBG are likely to be used up to this stress value despite the occurrence of a phase transition (permanent densification) undergone by silica above its HEL value. To our knowledge, no such experiment has been reported yet. An important question arises whether linear photo-elasticity is still a valid assumption up to this stress value or if nonlinear photoelasticity must be considered instead.

Thirdly, this study highlights the nonlinear behavior of the Bragg shift *vs* applied stress. Since the CFBG remains in its elastic domain, the nonlinearity arises from constitutive parameters, mainly refractive index, Poisson coefficient and Young modulus of Silica. Another nonlinear contribution of lesser importance is strain hardening of 6061-T6 Aluminum (i.e. increase of the yield strength with applied stress).

Fourthly, an inverse calculation of the stress distribution from Bragg spectra is an ill-posed problem as phase information is missing from spectrometric acquisitions. Nevertheless, stress distributions and velocities may be modeled *a priori* in simplified configurations (e.g. hydrostatic or planar shock loading) and stress data may be retrieved with the help of the experimental wavelength-to-stress relations provided in this work.
Finally, the results of this paper may be further extended to other configurations such as hydrostatic shock loading and detonation. The extension to pure planar shock loading requires the use of relationships from Barker and Hollenbach<sup>56</sup> (Eq. 38) and Setchell<sup>29</sup> (Eq. 34) for the Young modulus and RI change respectively in lieu of the relationships provided by Kondo et al.<sup>57</sup> (Eq. 40)) and Zha et al.<sup>53</sup> (Eq. 35) that are valid for hydrostatic loading. We are also looking forward to using this model with HEs showing elasto-plastic behavior and high-HEL materials such as sapphire or fused silica. A significant increase in Bragg sensitivity (for a CFBG aligned along the direction of shock) is expected due to a more favorable stress ellipsoid by comparison to the hydrodynamic configuration. The choice of fused Silica as target material should also improve the stress coupling to the silica fiber on account of ideal impedance matching. In all case, fiber bonding is necessary for both practical handling and elimination of air bubbles but it requires accurate FEM modelings to be performed in order to estimate the pressure coupling coefficient that reveals itself as an important parameter of the model.

## ACKNOWLEDGMENTS

This work has been performed within the ANDROMEDE Project, funded by the DGA (Direction Générale de l'Armement), in the framework of the ASTRID Program of the ANR (Agence Nationale de la Recherche).

The authors would like to thank C. Collet and L. Blarasin (ArianeGroup, Vert-le-Petit, France) for their contribution to the ANDROMEDE Project.

## REFERENCES

<footnote n="1"><sup>1</sup>D.R. Goosman, J. Wade, R. Garza, G. Avara, T. Crabtree, A. Rivera, D. Hare, D. Tolar and B. Bratton, 26<sup>th</sup> Int. Congress On High Speed Photography and Photonics, Alexandria (USA), September 19-24, 2004, Proc. SPIE **5580**, 517 (2004).</footnote>

<footnote n="2"><sup>2</sup>Z. Rosenberg, D. Yaziv and Y. Partom, J. App. Phys. **51**, 3702 (1980).</footnote>

<footnote n="3"><sup>3</sup>H. Vantine, J. Chan, L. Erickson, J. Janzen, R. Weingart and R. Lee, Rev. Sci. Instr. **51**, 116 (1980).</footnote>

<footnote n="4"><sup>4</sup>L.M. Barker, R.E. Hollenbach, J. Appl. Phys. **43**, 4669 (1972).</footnote>

<footnote n="5"><sup>5</sup>R.L. Gustavssen, S.A. Sheffield and R.R. Alcon, J. Appl. Phys. **99**, 114907 (2006).</footnote>

<footnote n="6"><sup>6</sup>J. Benterou, C.V. Bennett, G. Cole, D.E. Hare, C. May and E. Udd, SPIE Defense, Security and Sensing (DSS), Orlando FL, USA, Proc. SPIE **7316** (2009).</footnote>

<footnote n="7"><sup>7</sup>S. Magne, A. Lefrançois, J. Luc, G. Laffont and P. Ferdinand, 5<sup>th</sup> European Workshop on Optical Fiber Sensors (EWOFS), Krakow (Poland), Proc. SPIE **8794**, 87942K (2013).</footnote>

<footnote n="8"><sup>8</sup>G. Rodriguez, R.L. Sandberg, Q. McCulloch, S.I. Jackson, S.W. Vincent and E. Udd, Rev. Sci. Instrum. **84**, 015003 (2013).</footnote>

<footnote n="9"><sup>9</sup>S. Gilbertson, S.I. Jackson, S.W. Vincent and G. Rodriguez, Appl. Opt. **54** (13), 3849 (2015).</footnote>

<footnote n="10"><sup>10</sup>P. Wei, H. Lang, T. Liu and D. Xia, Sensors **17**, 2552 (2017).</footnote>

<footnote n="11"><sup>11</sup>P.G. Van't Hof, L.K. Cheng, J.H.G. Scholtes and W.C. Prinse, 27<sup>th</sup> Int. Congress on High-speed photography and Photonics, Proc. SPIE **6279**, 62791Y (2007).</footnote>

<footnote n="12"><sup>12</sup>X. Deng, G. Chen, Q. Peng, Z. Li, J. Meng and J. Liu, Rev. Sci. Instrum. **82**, 103109 (2011).</footnote>

<footnote n="13"><sup>13</sup>G. Rodriguez, R.L. Sandberg, B.M. Lalone, B.R. Marshall, M. Grover, G. Stevens and E. Udd, Proc. SPIE **9098**, Fiber Optic Sensors and Applications XI, 90980C (2014).</footnote>

<footnote n="14"><sup>14</sup>R.A. Sandberg, G. Rodriguez, L.L. Gibson, D.M. Dattelbaum, G.D. Stevens, M. Grover, B.M. Lalone and E. Udd, 18<sup>th</sup> APS-SCCM and 24<sup>th</sup> AIR APT, J. Phys. Conf. Series **500**, 142031 (2014).</footnote>

<footnote n="15"><sup>15</sup>E. Shafir, S. Zilberman, A. Ravid, B. Glam, G. Appelbaum, A. Fedotov Gefen, Y. Saadi, N. Shafir, G. Berkovic, 23<sup>rd</sup> International Conference on Optical Fiber Sensors (OFS 23), Santander (Spain), Proc. SPIE **9157**, 915713 (2014).</footnote>

<footnote n="16"><sup>16</sup>G. Rodriguez, M. Jaime, F. Balakirev, C.H. Mielke, A. Azad, B.M. La Lone, B. Henson and L. Smilowitz, Opt. Exp. **23** (11), 14219 (2015).</footnote>
Finally, the results of this paper may be further extended to other configurations such as hydrostatic shock loading and detonation. The extension to pure planar shock loading requires the use of relationships from Barker and Hollenbach⁵⁶ (Eq. 38) and Setchell²⁹ (Eq. 34) for the Young modulus and RI change respectively in lieu of the relationships provided by Kondo et al.⁵⁷ (Eq. 40)) and Zha et al.⁵³ (Eq. 35) that are valid for hydrostatic loading. We are also looking forward to using this model with HEs showing elasto-plastic behavior and high-HEL materials such as sapphire or fused silica. A significant increase in Bragg sensitivity (for a CFBG aligned along the direction of shock) is expected due to a more favorable stress ellipsoid by comparison to the hydrodynamic configuration. The choice of fused Silica as target material should also improve the stress coupling to the silica fiber on account of ideal impedance matching. In all case, fiber bonding is necessary for both practical handling and elimination of air bubbles but it requires accurate FEM modelings to be performed in order to estimate the pressure coupling coefficient that reveals itself as an important parameter of the model.

## ACKNOWLEDGMENTS

This work has been performed within the ANDROMEDE Project, funded by the DGA (Direction Générale de l'Armement), in the framework of the ASTRID Program of the ANR (Agence Nationale de la Recherche).

The authors would like to thank C. Collet and L. Blarasin (ArianeGroup, Vert-le-Petit, France) for their contribution to the ANDROMEDE Project.

## REFERENCES

<footnote n="1">¹D.R. Goosman, J. Wade, R. Garza, G. Avara, T. Crabtree, A. Rivera, D. Hare, D. Tolar and B. Bratton, 26th Int. Congress On High Speed Photography and Photonics, Alexandria (USA), September 19-24, 2004, Proc. SPIE **5580**, 517 (2004).</footnote>

<footnote n="2">²Z. Rosenberg, D. Yaziv and Y. Partom, J. App. Phys. **51**, 3702 (1980).</footnote>

<footnote n="3">³H. Vantine, J. Chan, L. Erickson, J. Janzen, R. Weingart and R. Lee, Rev. Sci. Instr. **51**, 116 (1980).</footnote>

<footnote n="4">⁴L.M. Barker, R.E. Hollenbach, J. Appl. Phys. **43**, 4669 (1972).</footnote>

<footnote n="5">⁵R.L. Gustavssen, S.A. Sheffield and R.R. Alcon, J. Appl. Phys. **99**, 114907 (2006).</footnote>

<footnote n="6">⁶J. Benterou, C.V. Bennett, G. Cole, D.E. Hare, C. May and E. Udd, SPIE Defense, Security and Sensing (DSS), Orlando FL, USA, Proc. SPIE **7316** (2009).</footnote>

<footnote n="7">⁷S. Magne, A. Lefrançois, J. Luc, G. Laffont and P. Ferdinand, 5th European Workshop on Optical Fiber Sensors (EWOFS), Krakow (Poland), Proc. SPIE **8794**, 87942K (2013).</footnote>

<footnote n="8">⁸G. Rodriguez, R.L. Sandberg, Q. McCulloch, S.I. Jackson, S.W. Vincent and E. Udd, Rev. Sci. Instrum. **84**, 015003 (2013).</footnote>

<footnote n="9">⁹S. Gilbertson, S.I. Jackson, S.W. Vincent and G. Rodriguez, Appl. Opt. **54** (13), 3849 (2015).</footnote>

<footnote n="10">¹⁰P. Wei, H. Lang, T. Liu and D. Xia, Sensors **17**, 2552 (2017).</footnote>

<footnote n="11">¹¹P.G. Van't Hof, L.K. Cheng, J.H.G. Scholtes and W.C. Prinse, 27th Int. Congress on High-speed photography and Photonics, Proc. SPIE **6279**, 62791Y (2007).</footnote>

<footnote n="12">¹²X. Deng, G. Chen, Q. Peng, Z. Li, J. Meng and J. Liu, Rev. Sci. Instrum. **82**, 103109 (2011).</footnote>

<footnote n="13">¹³G. Rodriguez, R.L. Sandberg, B.M. Lalone, B.R. Marshall, M. Grover, G. Stevens and E. Udd, Proc. SPIE **9098**, Fiber Optic Sensors and Applications XI, 90980C (2014).</footnote>

<footnote n="14">¹⁴R.A. Sandberg, G. Rodriguez, L.L. Gibson, D.M. Dattelbaum, G.D. Stevens, M. Grover, B.M. Lalone and E. Udd, 18th APS-SCCM and 24th AIR APT, J. Phys. Conf. Series **500**, 142031 (2014).</footnote>

<footnote n="15">¹⁵E. Shafir, S. Zilberman, A. Ravid, B. Glam, G. Appelbaum, A. Fedotov Gefen, Y. Saadi, N. Shafir, G. Berkovic, 23rd International Conference on Optical Fiber Sensors (OFS 23), Santander (Spain), Proc. SPIE **9157**, 915713 (2014).</footnote>

<footnote n="16">¹⁶G. Rodriguez, M. Jaime, F. Balakirev, C.H. Mielke, A. Azad, B.M. La Lone, B. Henson and L. Smilowitz, Opt. Exp. **23** (11), 14219 (2015).</footnote>
¹⁷Y. Barbarin, A. Lefrançois, S. Magne, K. Woirin, F. Sinatti, A. Osmont and J. Luc, Proc. SPIE **9960** (Interferometry XVIII), San Diego (CA), 99600U (2016).

¹⁸G. Rodriguez and S.M. Gilbertson, Sensors **17**, 248 (2017).

¹⁹T. Erdogan, J. Lightwave Technol. **15** (8), 1277 (1997).

²⁰A.D. Kersey, M.A. Davis, H.J. Patrick, M. Leblanc, K.P. Koo, S.G. Askins, M.A. Putnam and E.J. Friebele, J. Lightwave Technol. **15** (8), 1442 (1997).

²¹H. Sugiura, K. Kondo and A. Sawaoka, J. Appl. Phys. **52**, 3375 (1981).

²²F. Malaise, J-M. Chevalier, I. Bertron and F. Malka, J. Phys. IV **134**, 929 (2006).

²³J. Wackerle, J. Appl. Phys., **33** (3), 922 (1962).

²⁴Y. Barbarin, A. Lefrançois, G. Zaniolo, V. Chuzeville, L. Jacquet, S. Magne, J. Luc and A. Osmont, Proc. SPIE **9480**, 9480-25 (2015).

²⁵A. Ravid, E. Shafir, S. Zilberman, G. Gerkovic, B. Glam, G. Appelbaum and A. Fedotov Gefen, 18th APS-SCCM and 24th AIR APT, J. Phys. Conf. Series **500**, 142029 (2014).

²⁶L. Bouyeron, A. Lefrançois, J. Luc, H-C. Seat, *13<sup>ème</sup> Colloque International Francophone sur les Méthodes et Techniques Optiques pour l'Industrie* (CMOI), Orléans, France (2013) (in french).

²⁷LASL Shock Hugoniot data, *Los Alamos Series on dynamic material properties*, S. P. Marsh editor, University of California Press, (1980).

²⁸M.G. Xu, L. Reekie, Y.T. Chow and J.P. Dakin, Electron. Lett. **29** (4), 398 (1993).

²⁹R.E. Setchell, J. Appl. Phys. **50**, 8186 (1979).

³⁰D.J. Steinberg, J. Phys. IV, **C3**, 837 (1991).

³¹D.J. Steinberg, Report No. UCRL-MA-106439, Lawrence Livermore National Laboratory, Livermore, CA (1991).

³²Z. Rosenberg and Y. Partom, J. Appl. Phys. **58** (8), 3072 (1985).

³³B.L. Glushak and S.A. Novikov, Fizika Goreniya i Vzryva, **28** (1), 84 (1991) [Reproduced in Combust. Expl. Shock Waves **28** (1), 79 (1992)].

³⁴Y.V. Bat'kov, B.L. Glushak and S.A. Novikov, Fizika Goreniya i Vzryva, **25** (5), 126 (1988) [Reproduced in Combust. Expl. Shock Waves **25** (5), 635 (1989)].

³⁵J. Peng, F. Jing, D. Li and L. Wang, J. Appl. Phys. **98**, 013508 (2005).

³⁶D.J. Steinberg, S.G. Cochran and M.W. Guinan, J. Appl. Phys. **51** (3), 1498, (1980).

³⁷W.D. Reinhart, J.R. Asay, C.S. Alexander, L.C. Chabildas and B.J. Jensen, J. Dynamic Behavior Mater. **1**, 275 (2015).

³⁸J. Schroeder, J. non Cryst. Solids **40**, 549 (1980).

³⁹G.B. Hocker, Appl. Opt. **18** (9), 1445 (1979).

⁴⁰K. Vedam and R. Srinivasan, Acta Crysta. **22**, 630 (1967).

⁴¹W.W. Morey, G. Meltz and J.M. Weiss, 8th Optical Fiber Sensor (OFS) Conference, Monterey (CA), Postdeadline Paper PD-4.4 (1992).

⁴²F. Julich, L. Aulbach, A. Wilfert, P. Kratzer, R. Kuttler and J. Roths, Meas. Sci. Tech. **24**, 94007 (2013).

⁴³N.K. Sinha, Phys. Chem. Glass. **19** (4), 69 (1978).

⁴⁴A.J. Barlow and D.N. Payne, IEEE J. Quant. Electron. **19** (5), 834 (1983).

⁴⁵K. Matusita, C. Ihara, T. Komatsu and R. Yokota, J. Am. Ceram. Soc. **67** (10), 700 (1984).

⁴⁶Y. Namihara, J. Lightwave Technol. **3** (5), 1078 (1985).

⁴⁷A. Bertholds and R. Dändliker, J. Lightwave Technol. **6** (1), 17 (1988).

⁴⁸P.D. Dragic, J. Ballato, S. Morris and T. Hawkins, J. Opt. Soc. Am., **30** (2), 244 (2013).

⁴⁹M.S. Pambianchi, M. Dejneka, T. Gross, A. Ellison, S. Gomez, J. Price, Y. Feng, P. Tandom, D. Bookbinder and M.J. Li, "Corning Inc : Designing a new future with glass and optics", Ch. 1 in Material Research for Manufacturing, L.D. Madsen and E.B. Svedberg Ed., Springer Series in Materials Science **224** (2016).

⁵⁰N.F. Borelli and R.A Miller, Appl. Opt. **7** (5), 745 (1968).

⁵¹D.P. Dandekar, J. Appl. Phys. **84** (12), 6614 (1998).

⁵²J. Arndt and D. Stöffler, Phys. Chem. Glass., **10** (3), 117 (1969).

⁵³C.S. Zha, R.J. Hemley, H.K. Mao, T.S. Duffy and C. Meade, Phys. Rev., **50** (11), 13105 (1994).

⁵⁴H.M. Cohen and R. Roy, J. Am. Ceram. Soc., **44** (10), 523 (1961).

⁵⁵R. Roy and H.M. Cohen, Nature **190**, 798 (1961).

⁵⁶L.M. Barker and R.E. Hollenbach, J. Appl. Phys. **41**, 4208 (1970).

⁵⁷K. Kondo, S. Iio and A. Sawaoka, J. Appl. Phys. **52** (4), 2826 (1981).

⁵⁸K. Vedam, E.D.D. Schmidt and R. Roy, J. Am. Ceram. Soc. **49** (10), 531 (1966).

⁵⁹C. Meade and R. Jeanloz, Phys. Rev. B, **35** (1), 236 (1987).

⁶⁰X.F. Liu, R.J. Wang and W.H. Wang, Scripta Mat. **62**, 254 (2010).

⁶¹T. Rouxel, H. Ji, T. Hammoud, and A. Moréac, Phys. Rev. Lett. **100**, 225501 (2008).

⁶²W.M. Isbell and D.R. Christman, NASA Report 2419, MSL-69-60 (1970).

⁶³H. Kogelnik, Bell Sys. Tech. J. **55** (1), 109 (1976).
¹⁷Y. Barbarin, A. Lefrançois, S. Magne, K. Woirin, F. Sinatti, A. Osmont and J. Luc, Proc. SPIE **9960** (Interferometry XVIII), San Diego (CA), 99600U (2016).

¹⁸G. Rodriguez and S.M. Gilbertson, Sensors **17**, 248 (2017).

¹⁹T. Erdogan, J. Lightwave Technol. **15** (8), 1277 (1997).

²⁰A.D. Kersey, M.A. Davis, H.J. Patrick, M. Leblanc, K.P. Koo, S.G. Askins, M.A. Putnam and E.J. Friebele, J. Lightwave Technol. **15** (8), 1442 (1997).

²¹H. Sugiura, K. Kondo and A. Sawaoka, J. Appl. Phys. **52**, 3375 (1981).

²²F. Malaise, J-M. Chevalier, I. Bertron and F. Malka, J. Phys. IV **134**, 929 (2006).

²³J. Wackerle, J. Appl. Phys., **33** (3), 922 (1962).

²⁴Y. Barbarin, A. Lefrançois, G. Zaniolo, V. Chuzeville, L. Jacquet, S. Magne, J. Luc and A, Osmont, Proc. SPIE **9480**, 9480-25 (2015).

²⁵A. Ravid, E. Shafir, S. Zilberman, G. Gerkovic, B. Glam, G. Appelbaum and A. Fedotov Gefen, 18th APS-SCCM and 24th AIR APT, J. Phys. Conf. Series **500**, 142029 (2014).

²⁶L. Bouyeron, A. Lefrançois, J. Luc, H-C. Seat, *13<sup>ème</sup> Colloque International Francophone sur les Méthodes et Techniques Optiques pour l'Industrie* (CMOI), Orléans, France (2013) (in french).

²⁷LASL Shock Hugoniot data, *Los Alamos Series on dynamic material properties*, S. P. Marsh editor, University of California Press, (1980).

²⁸M.G. Xu, L. Reekie, Y.T. Chow and J.P. Dakin, Electron. Lett. **29** (4), 398 (1993).

²⁹R.E. Setchell, J. Appl. Phys. **50**, 8186 (1979).

³⁰D.J. Steinberg, J. Phys. IV, **C3**, 837 (1991).

³¹D.J. Steinberg, Report No. UCRL-MA-106439, Lawrence Livermore National Laboratory, Livermore, CA (1991).

³²Z. Rosenberg and Y. Partom, J. Appl. Phys. **58** (8), 3072 (1985).

³³B.L. Glushak and S.A. Novikov, Fizika Goreniya i Vzryva, **28** (1), 84 (1991) [Reproduced in Combust. Expl. Shock Waves **28** (1), 79 (1992)].

³⁴Y.V. Bat'kov, B.L. Glushak and S.A. Novikov, Fizika Goreniya i Vzryva, **25** (5), 126 (1988) [Reproduced in Combust. Expl. Shock Waves **25** (5), 635 (1989)].

³⁵J. Peng, F. Jing, D. Li and L. Wang, J. Appl. Phys. **98**, 013508 (2005).

³⁶D.J. Steinberg, S.G. Cochran and M.W. Guinan, J. Appl. Phys. **51** (3), 1498, (1980).

³⁷W.D. Reinhart, J.R. Asay, C.S. Alexander, L.C. Chabildas and B.J. Jensen, J. Dynamic Behavior Mater. **1**, 275 (2015).

³⁸J. Schroeder, J. non Cryst. Solids **40**, 549 (1980).

³⁹G.B. Hocker, Appl. Opt. **18** (9), 1445 (1979).

⁴⁰K. Vedam and R. Srinivasan, Acta Crysta. **22**, 630 (1967).

⁴¹W.W. Morey, G. Meltz and J.M. Weiss, 8th Optical Fiber Sensor (OFS) Conference, Monterey (CA), Postdeadline Paper PD-4.4 (1992).

⁴²F. Julich, L. Aulbach, A. Wilfert, P. Kratzer, R. Kuttler and J. Roths, Meas. Sci. Tech. **24**, 94007 (2013).

⁴³N.K. Sinha, Phys. Chem. Glass. **19** (4), 69 (1978).

⁴⁴A.J. Barlow and D.N. Payne, IEEE J. Quant. Electron. **19** (5), 834 (1983).

⁴⁵K. Matusita, C. Ihara, T. Komatsu and R. Yokota, J. Am. Ceram. Soc. **67** (10), 700 (1984).

⁴⁶Y. Namihara, J. Lightwave Technol. **3** (5), 1078 (1985).

⁴⁷A. Bertholds and R. Dändliker, J. Lightwave Technol. **6** (1), 17 (1988).

⁴⁸P.D. Dragic, J. Ballato, S. Morris and T. Hawkins, J. Opt. Soc. Am., **30** (2), 244 (2013).

⁴⁹M.S. Pambianchi, M. Dejneka, T. Gross, A. Ellison, S. Gomez, J. Price, Y. Feng, P. Tandom, D. Bookbinder and M.J. Li, "Corning Inc : Designing a new future with glass and optics", Ch. 1 in Material Research for Manufacturing, L.D. Madsen and E.B. Svedberg Ed., Springer Series in Materials Science **224** (2016).

⁵⁰N.F. Borelli and R.A Miller, Appl. Opt. **7** (5), 745 (1968).

⁵¹D.P. Dandekar, J. Appl. Phys. **84** (12), 6614 (1998).

⁵²J. Arndt and D. Stöffler, Phys. Chem. Glass., **10** (3), 117 (1969).

⁵³C.S. Zha, R.J. Hemley, H.K. Mao, T.S. Duffy and C. Meade, Phys. Rev., **50** (11), 13105 (1994).

⁵⁴H.M. Cohen and R. Roy, J. Am. Ceram. Soc., **44** (10), 523 (1961).

⁵⁵R. Roy and H.M. Cohen, Nature **190**, 798 (1961).

⁵⁶L.M. Barker and R.E. Hollenbach, J. Appl. Phys. **41**, 4208 (1970).

⁵⁷K. Kondo, S. Iio and A. Sawaoka, J. Appl. Phys. **52** (4), 2826 (1981).

⁵⁸K. Vedam, E.D.D. Schmidt and R. Roy, J. Am. Ceram. Soc. **49** (10), 531 (1966).

⁵⁹C. Meade and R. Jeanloz, Phys. Rev. B, **35** (1), 236 (1987).

⁶⁰X.F. Liu, R.J. Wang and W.H. Wang, Scripta Mat. **62**, 254 (2010).

⁶¹T. Rouxel, H. Ji, T. Hammoud, and A. Moréac, Phys. Rev. Lett. **100**, 225501 (2008).

⁶²W.M. Isbell and D.R. Christman, NASA Report 2419, MSL-69-60 (1970).

⁶³H. Kogelnik, Bell Sys. Tech. J. **55** (1), 109 (1976).