A Brewster incidence method for shocked dynamic metrology of transparent materials and its error evaluation

The shock etalon method with normal incidence is an effective method to extract the shocked dynamic parameters of transparent materials. In order to eliminate the sample surface reflection, additional efforts on the sample preparation are usually introduced, which may limit the application of the method. Here, we proposed a Brewster incidence method to carry out the shock compression experiment on transparent materials. By utilizing the p -polarized light as the probe pulse at Brewster incidence, the sample surface reflection can be directly eliminated, which consequently simplified the experiment preparations. The errors of the proposed method have been evaluated using a set of virtual experiments. The results show that the shocked dynamic parameters can be accurately and robustly retrieved even when the nominal refractive index of the sample deviates 0.01 from its true value, or the incident angle bias 0.33 ○ from the Brewster angle. Finally, a set of shock compression experiments on a polycarbonate film sample are carried out, and shocked dynamic parameters, such as shock velocity, particle velocity, and shocked refractive index, are successfully measured, for demonstration


I. INTRODUCTION
2][3][4] With the development of ultrashort and high-power laser technology, the laser-driven shock loadings make it available to carry out the dynamic compression experiments using the tabletop setups, which have a relatively low cost and need less preparation time compared to the traditional methods, such as the light-gas gun, high-explosivedriven planar shock, and the nuclear explosives. 1 Distinguishing from the shock loading experiments on metallic samples, which usually only need to measure the free surface velocity, 5,6 optical property changes in the test sample need to be taken into consideration for the experiments on transparent materials.This is because the reflections from different interfaces, such as the sample surface, the shock front, and the interface between the sample and the ablators, are simultaneously collected by the detector.Consequently, either a more complex optical system or an appropriate optical model of the sample is usually introduced in the shocked dynamics metrology of transparent materials. 7,8In 2007, the research group from Los Alamos National Laboratory developed an ultrafast dynamic ellipsometry (UDE) and simultaneously achieved the shocked response of transparent materials, such as the initial and shocked optical constants, shock and particle velocities, and the initial thickness of the film. 7In their configuration, two probe arms with different incident angles are employed and each probe beam has both the ppolarized component and the s-polarized component, and thus, four spectral interferograms can be achieved in a single shot.In 2010, Armstrong et al. proposed the shock etalon method for the metrology of shocked transparent materials, 8 which has a relatively simple optical path structure compared to UDE.Since the shock front and the metallic ablator surface can be regarded as a scanning optical etalon when a steady wave propagates in the transparent films, the optical beating between the etalon will result in a high frequency oscillation in the phase.According to the first principle, a correlation between the finite differences in these phase oscillations and the shocked dynamic parameters, such as shock velocity, particle velocity, and shocked refractive indices, can be used to directly extract the shocked dynamics.][11][12][13][14][15][16] Although the shock etalon method is an effective method to extract the shocked dynamic parameters of transparent materials, the reflection from the upper surface of the sample may significantly degrade the measurement accuracy and should be eliminated.Therefore, extra efforts on the sample preparation are usually accompanied.For example, in the metrology of the pre-compressed liquid argon, 8 a slight tilt between the sample surface and the ablator/sample interface has been intentionally introduced.For the solid transparent samples, such as energetic materials or polymers, an extra thin uncured polymer layer is added 16 and acts as an index matching fluid to remove the unwanted probe reflections.However, the index matching process complicates the sample preparation, and it may be a challenge to find a proper index matching material for some solid transparent materials as well.
In this work, we proposed a Brewster incidence method to measure the shocked dynamic parameters of transparent materials.By utilizing the characteristics of zero reflection of p-polarized chirped pulse probing with the angle of incidence as the Brewster angle, the sample surface reflection can be eliminated directly.Therefore, the proposed configuration can reduce the complexity of the sample preparations that are required in the traditional shock etalon method presented in Ref. 8. We provide the theoretical formulas to extract the dynamic parameters that are derived on top of the shock etalon method.We have evaluated the measurement error of the proposed method via a set of virtual experiments, with the emphasis on the error of sample refractive index and Brewster angle setting, which may be critical error sources accompanied with the new configuration.The results show that, even when the given nominal sample refractive index deviates 0.01 from its true value, or the incident angle has 0.33 ○ bias from the Brewster angle, which can be easily controlled in the practical measurement system, the measurement errors of the shock velocity, the particle velocity, and the shocked refractive index are less than 1%, 3%, and 4%, respectively, demonstrating the accuracy and robustness of the proposed method.Finally, practical shock compression experiments on a polycarbonate film sample have been carried out for demonstration, and the shocked dynamic parameters are obtained successfully.

II. EXPERIMENT SCHEME AND METHOD
A. Experiment scheme When a p-polarized beam projects onto the surface of a transparent film with the Brewster angle, all the optical energy will transmit into the materials without surface reflection.By utilizing such a phenomenon, we propose a measurement scheme based on the ppolarized light and Brewster incidence.The proposed configuration can be schematically depicted in Fig. 1.
In our experiment, a Ti:sapphire laser (Newport Corporation, SOL-ACE35F1K-HP) is employed as the light source, which can output Gaussian limit pulses with ∼5.5 mJ single pulse energy, ∼35 fs pulse width, ∼42 nm band width, and 800 nm center wavelength.Then, a home-made grating-based pulse stretcher converts the fs pulses to linear chirped pulses.The linear chirped characteristics of the stretched pulse that can be expressed as λ = 0.1124 * t + 786.2 have been obtained using the stationary phase point method, making the stretched ps chirped pulse interfere with the unstretched fs pulse in the spectral domain.More details about the pulse stretcher and the chirped coefficient measurement method can be found in our other work. 17Before entering the pump-probe optical path, the chirped pulse is shaped first by clipping the front part spectrum with a knife edge in the pulse stretcher to produce a sharp rise and a flat top in the intensity profile.Then, the shaped chirped pulse is split into two beams by an 8:2 non-polarizer beam splitter.The beam with 80% pulse energy acts as the pump beam and is focused to a spot size of ∼80 μm diameter at the rear surface of the sample to create a steady shock wave.The beam with 20% pulse energy transits a polarizer first to generate a p-polarized light, which is further split into the probe beam and the reference beam with a 5:5 beam splitter.The probe beam is adjusted to project onto the sample with the Brewster incident angle.An achromatic lens focuses the probe beam to a spot size of ∼325 μm diameter, which completely covers the shocked area.With another achromatic lens, the shocked area is imaged onto the slit of an imaging spectrometer (Horiba, iHR550) with about ∼17 magnification.To balance the two arms of the interferometer, identical lenses are placed in the reference beam, as shown in Fig. 1.Combining the probe beam and the reference beam with a slight angle, space-shifted spectral interferograms (SSSI) will be recorded on a CCD detector (Horiba, Syncer-1024 × 256).The spectral range of the CCD detector is about 31.7 nm, corresponding to the ∼282 ps measurement range in a single shot according to the previous measured chirped coefficient.
It is worth noting that although the proposed setup looks similar to the configuration of ultrafast dynamic ellipsometry proposed in Ref. 7, it is not a simple simplification from the latter.The proposed p-polarized light and Brewster incident probe angle measurement scheme is a modified work on top of the shock etalon method presented in Ref. 8, whose principles and methods are totally different to those in Ref. 7.

B. Method to extract the shocked dynamic parameters
When a steady shock wave propagates in a transparent film, the thickness of the shock wave front is relatively smaller compared with the film thickness and can be regarded as a steep interface that separates the shocked region and the unshocked region. 6,18,19Therefore, a double-layer thin film model of the transparent film can be introduced, as shown in Fig. 2. On top of the shock etalon method, 8 we further derived the theoretical equations for the proposed measurement configuration.Since the sample surface reflectivity of the ppolarized beam at the Brewster incident angle is zero, the probe light only beats between the shock front and the sample/ablator interface.As the first several beams reflected by the sample dominate the vast majority energy of the probe beam, we can approximate the reflected probe beam with the first three reflected beams E 1 , E 2 , and E 3 , as shown in Fig. 2.
For a steady shock wave, the shock velocity, the particle velocity, and the shocked refractive index can be regarded as constants. 7,8hen, the thickness of unshocked layer h 1 and shocked layer h 2 can FIG.2. The double-layer thin film model when a steady shock wave propagates in a transparent film.E 0 , p-polarized probe beam; θ 0 , Brewster incident angle; us, shock velocity; up, particle velocity; n, refractive index; h, film thickness.

be expressed as
where h 0 is the initial thickness of the transparent film, t is the time after the shock begins, and us and up are the shock velocity and particle velocity, respectively.Since the shock rise time does not affect the extraction of the shocked parameters, it has been ignored to simplify the analysis, and the reason will be further discussed in Sec.IV, combined with the experiment results.Then, the phase changes of the probe beam after a round trip in the shocked layer and the unshocked layer, which are denoted as δ 1 and δ 2 , respectively, can be written as where λ is the probe wavelength, n 1 and n 2 are the refractive indices of the unshocked layer and the shocked layer, respectively, and θ 1 and θ 2 are the incident angles of the probe beam when transmitting in the unshocked layer and the shocked layer, respectively.Here, we use r 1 to denote the reflectivity of the shock front when a beam transmits from the unshocked layer into the shocked layer and use r 2 to denote the reflectivity of the metal ablator.According to the Fresnel law of reflection and refraction, we can express the first three reflected beams E 1 , E 2 , and E 3 as where E 0 is the electric field of the probe beam.For a transparent film, there is only a small difference between the shocked refractive index n 2 and the initial refractive index n 1 , so the absolute value of the shock front reflectivity r 1 is relatively small and we can assume that 1 − r 2 1 ≈ 1.Based on this analysis, the total reflectance E total can be approximated as According to the analysis in work of Armstrong et al., 8 the coefficient term in Eq. ( 4) has the sinusoidal oscillation character and can be approximately rewritten as where τ is the oscillation period and γ is the oscillation amplitude, which can be further expressed as On the other hand, δ 1 + δ 2 in Eq. ( 4) can be rewritten in the following form according to Eq. ( 2): where a 0 and a 1 can be expressed as Then, combining Eqs. ( 4), (5), and ( 8), the total reflectivity of the probe beam can be approximated as In a practical SSSI experiment, the fixed value −a 0 in Eq. ( 10) can be eliminated by subtracting the reference measurement without shock.So, the measured dynamic phase shift induced by the shock wave is Since the bandwidth of the probe chirped pulse is usually small compared to h 1 and h 2 , λ can be regarded as an invariable value in Eqs. ( 6) and (9b).It can be found that φ consists of a linear variation and a sinusoidal variation, while a 1 is the coefficient of the linear variation and γ is the amplitude of the sinusoidal variation.The dynamic phase shift φ can be retrieved from the measured space-shifted spectral interferograms.If we use Δφ to denote the finite difference of φ for a time interval Δt, we can observe that Δφ has a sinusoidal profile with an oscillation offset.Then, Δφ can be expressed as For a metallic ablator, its reflectivity can be approximated to be r 2 ≈ 1.So, we have the approximation (1 + r 2 2 )/r 2 ≈ 2 in Eq. (7).Then, the oscillation amplitude γ ′ and the oscillation offset σ of Δφ can be obtained from Eq. (11), From Eq. ( 12), we can get the approximated value of the shock front reflectivity r 1 as According to Fresnel's law of p-polarized light, we can get another expression of the shock front reflectivity r 1 as For the Brewster incident angle, we have the relationship θ 0 + θ 1 = 90 ○ .Then, we can rewrite Eq. ( 15) as Using Fresnel's law of refraction sin(θ 2 ) = (n 0 /n 2 ) × sin(θ 0 ) and the trigonometric function 1 − sin 2 (2θ 2 ) = cos 2 (2θ 2 ) = [1 − 2 sin 2 (θ 2 )] 2 , and combining Eqs. ( 14) and ( 16), we can get the shocked refractive index n 2 , According to Eqs. ( 6), (13), and ( 17), the shock velocity us and the particle velocity up can be extracted with the following equations: Thus, in a steady shock compression experiment on transparent film samples, if the dynamic phase shift induced by the shock wave can be achieved, the shocked dynamic parameters can be directly extracted using the proposed method.

III. ERROR EVALUATION OF THE PROPOSED METHOD
During the derivation of the theoretical equations in Sec.II B, the wavelength of the chirped pulse with a limited bandwidth is regarded as an invariable value, and several approximations have been adopted in Eqs. ( 4), ( 5), (7), and (12) as well.Thus, inherent theoretical calculation errors will be inevitably introduced into the extracted dynamic parameters.Besides, the measurement noise, the nonideal adjustment of the measurement system, or the deviations between the given nominal value and their actual values of the measuring configuration parameters may also bring measurement errors.Here, we use a set of virtual experiments to evaluate the measurement accuracy and errors for the proposed method.

A. Errors due to calculation
Supposing we are carrying out a shock compression experiments on a polycarbonate film sample.Assume that the shock velocity, the particle velocity, and the shocked refractive index of the sample are us = 5.3 km/s, up = 2.0 km/s, and n 2 = 1.7, respectively.All the simulation parameters in the virtual experiment are relisted in Table I and are selected to be consistent with the parameters used in the practical experiments in Sec.IV.
According to the double-layer sample model shown in Fig. 2, the dynamic phase shift caused by the shock wave can be calculated theoretically with Fresnel's law of refraction and reflection, which will be regarded as measured data in the virtual experiment, as shown in Fig. 3(a).Figure 3(b) shows the finite difference of the phase shift for a time interval of 5 ps.With the proposed method, the three shocked dynamic parameters are obtained, and their relative errors are listed in Table II, which are all less than 2%.The measurement results in Table II are extracted from an ideal experiment without noise, and thus, it can be regarded as the inherent calculation errors of the proposed method.For a probe pulse with a limited bandwidth, the oscillation characteristics of the phase finite difference will change against with the wavelength, which will further lead to a deviation in the sine fitting.When fixing the wavelength of the probe beam at 800 nm and repeating the above virtual experiment to generate the dynamic phase shift, all the measurement errors of the three dynamic parameters are less than 0.05%, indicating that the limited bandwidth of the chirped pulse is the dominant source of calculation errors, while the approximations adopted in the theoretical equations contribute little.

B. Uncertainty induced by the noise
The light source noise, the roughness of the sample surface, and the camera noise can raise measurement noise in the spectral interferogram.Assume their total effects can be approximated using a Gaussian distribution.To evaluate the influence of the noise, we added Gaussian noise on the dynamic phase shift obtained in the above virtual experiment, as shown in Fig. 3.The noise rms is set as about 0.03 rad, the same noise level as the measured phase shift in our practical experiment.Repeating the virtual experiment with noise several times, we found that the increments of the uncertainties of the three extracted dynamic parameters are all less than 1% compared to the experiment without noise, showing the robustness of the proposed method.Usually, a smaller time interval corresponds to a more obvious periodical oscillation profile of the phase finite difference, which can lead to a better sinusoidal fitting and a better measurement accuracy theoretically.However, the phase finite difference is more sensitive to noise when the time interval is small, which can reduce the accuracy in turn.The virtual experiments show that a time interval between 3 ps and 10 ps can be an optimal range for the final measurement accuracy.Therefore, a time interval of 5 ps is selected in the virtual experiments.

C. Errors due to deviation of configuration parameters
According to the expressions of Eqs. ( 17)-( 19), it can be found that only two measuring configuration parameters can affect the measurement result, such as the initial refractive index n 1 of the sample and the incident Brewster angle θ 0 of the probe beam.To evaluate the measurement errors introduced by the deviations of n 1 and θ 0 between their given nominal values and actual values, virtual experiments with different deviations of n 1 and θ 0 have been carried out.
Assume the given nominal value of n 1 is 1.50, and its corresponding Brewster incident angle θ 0 can be calculated as 56.31 ○ .Since the measurement error of the refractive index can be easily ensured less than 0.01 when a Muller matrix ellipsometry is used, 20 the setting varying interval [1.49, 1.51] of n 1 is rational.Meanwhile, the optical path adjustment error needs to be taken into consideration, and we assume the practical angle of incidence θ 0 may vary within [55.98 ○ , 56.64 ○ ], which is because an angle deviation of 0.33 ○ can be easily guaranteed by the mechanical design in our practical measurement system.Then, we use the practical values to simulate the measured dynamic phase shift and use the given nominal values to extract the dynamic parameters.The comparison of the simulation results and corresponding errors is shown in Fig. 4. It can be observed that even in the worst case, i.e., existing the maximum initial refractive index deviation and the maximum incident  angle adjustment error, the measurement errors of n 2 , us, and up are less than 1%, 3%, and 4%, respectively.The results indicate the accuracy and robustness of the proposed method.It is worthy to point out that differences of the measurement errors with and without the noise injected are all less than 1% for all the shock dynamic parameters, showing the robustness of the proposed method as well.
In the virtual experiments, the shock velocity is assumed to be constant when transmitting in the transparent film.However, in actual shock compression experiments, it is hard to achieve such an ideal shock profile.Therefore, how to generate an ideal steady shock wave is critical to keep the measuring accuracy in the acceptable range.

IV. EXPERIMENT AND RESULTS
To verify the validity of the proposed method, shock compression experiments on a polycarbonate film sample have been carried out.A 2 μm aluminum thin film ablator is prepared first, which is vapor deposited on a 0.17 mm glass substrate.Then, we dissolved polycarbonate powder in the 8% tetrachloroethane solution and spin cast the mixed solution on the above 2 μm aluminum film.Measured with a Mueller matrix ellipsometer (ME-L, Wuhan Eoptics Technology Co.), the thickness of the polycarbonate film is about h 0 = 3.974 μm, and its refractive index near the wavelength of 800 nm is about n 1 = 1.500 ± 0.007, corresponding to a Brewster angle θ 0 of 56.31 ○ ± 0.12 ○ .
By changing the laser output power, a series of shock experiments are carried out with the pump pulse energy ranging from ∼1.0 mJ to ∼1.8 mJ.The static (without pump) and dynamic (with pump) space-shifted spectral interferograms recorded on the imaging spectrometer are shown in Fig. 5.With the fast Fourier transformation (FFT) method, we extracted the phase maps from the static and dynamic interferograms, respectively.More details about  the data processing method of the phase extraction can be found in the other works. 21,22Subtracting the static phase map from the dynamic one, we have obtained the dynamic phase shift caused by the shock wave, as shown in Fig. 6(a), whose spatial domain has been converted to the actual size of the sample and its spectrum is also mapped into the time domain.Figure 6(b) shows the dynamic phase shift curve corresponding to the shocked center in Fig. 6(a).Figure 6(c) shows the finite difference of the phase map for a time interval of 5 ps, and Fig. 6(d) shows the phase finite difference of the shocked center.The experiments on the 2 μm aluminum films with the same measurement configuration show that the rise time of the shock wave is about 50 ps-75 ps.According to the profile of the phase difference in Fig. 6(d), it can be found that the period oscillation profile started at about 75 ps after the shock wave begins, also showing that the moment of the steady shock wave appears.Thus, only the middle data that correspond to a steady shock wave are adopted to perform the sinusoidal fitting, as shown in Fig. 6(d).Therefore, the shock rise time does not affect the measurement results, as previously mentioned in Sec.II B. With the periodic oscillation parameters extracted from the curve fitting, such as the period τ, the amplitude γ ′ , and the offset σ, we directly calculated the shocked refractive index, the shock velocity, and the particle velocity using Eqs.( 14)-( 19) in Sec.II B.
The measured data us vs up obtained with different pump energies are shown in Fig. 7, which also shows the Hugoniot data measured by the Los Alamos National Laboratory (LASL) 23 for comparison.Good linearity of the data measured by the proposed method can be found and demonstrates its effectiveness.Although the linearity is good, the linear fitting results are different to the results reported by the LASL.Such a difference can be attributed to the difference of the sample.The results achieved by the LASL utilized the traditional high explosive or the light gas gun to generate the shock wave and adopted the buck material as the sample.Since the laser-driven shock wave loading experiments can be easily affected by the laser pulse profile and the sample quality, our data may have larger uncertainties than the LASL data.However, considering the low cost and short experiment period, the proposed method still has great potential in the shocked dynamic parameter metrology of transparent materials.
Using ρ 1 and ρ 2 to denote the initial density and the shocked density of the sample, respectively, the ratio of ρ 2 to ρ 1 can be expressed as The shocked refractive index n 2 of the polycarbonate sample against the density ratio ρ 2 /ρ 1 is plotted in Fig. 8.It can be found that the measured data appear to have good linear correlation.In Bolme's work, 7    However, the measurement results in our experiments do not follow the Gladstone-Dale relation, which may rise from the different sample preparations.

V. DISCUSSION
In this work, a Brewster incidence method has been proposed to measure the shocked dynamic responses of transparent materials.With the proposed configuration, our solution can be adapted to the simplest solid transparent samples and no extra effort is needed on sample preparation.Error analysis with virtual experiments shows the robustness of the proposed method against noise and the measuring configuration parameters, and practical shock compression experiments on a polycarbonate thin film verified the validity of the proposed method.
In practice, a satisfactory steady shock wave is difficult to achieve, and usually, a successful test may need several shots to guarantee.An important benefit of the proposed method is that it can provide a quick evaluation of the validity of the shock test by observing whether the phase finite difference has a good periodical oscillation profile.Additionally, the initial sample thickness and the refractive index of the ablator have little influence on the measurement results, which reduces the measurement error sources.However, it is worth noting as well that the proposed method has its inherent limitation.For example, a steady shock wave and an obvious change in sample refractive index due to shock are necessary.Besides, due to the different refractive indices among various samples, the Brewster incident probe angle of the measurement optical path should be readjusted for each measured sample, which increases the experimental workloads to a certain extent.

FIG. 3 .
FIG. 3. Dynamic phase shift (a) and its finite difference (b) achieved in the virtual experiment.The data with and without noise have been vertically offset for clarity.

FIG. 4 .
FIG. 4. The measurement errors of the three shocked dynamic parameters against the deviation of Brewster incident angle θ 0 when the given nominal values of n 1 and θ 0 are 1.5 ○ and 56.31 ○ , respectively.(a)-(c) are the simulation results when the actual values of n 1 are 1.49, 1.50, and 1.51, respectively.Dotted lines denote the simulation results without noise, while solid lines denote the results when adding a noise rms of 0.03 rad on the dynamic phase shift.

FIG. 5 .
FIG. 5. (a) The static interferogram without pump and (b) the dynamic interferogram with pump obtained in the shock compression experiment on a polycarbonate film sample.

FIG. 6 .
FIG. 6.(a) The 2D dynamic phase shift map caused by the shock wave, (b) the dynamic phase shift against time corresponding to the shocked center in (a), (c) the 2D phase finite difference map for a time interval of 5 ps, and (d) the sine fitting on the phase finite difference.

FIG. 7 .
FIG. 7.The comparison between the shocked Hugoniot data measured with the proposed method and the LASL data.

FIG. 8 .
FIG. 8.The shocked refractive index against the density ratio of the polycarbonate film before and after the shock loadings.

TABLE I .
The measuring configuration parameters in the virtual shock compression experiments.The symbols denote the same meanings as those in Sec.II B.

TABLE II .
The extracted dynamic parameters and the relative errors in the virtual experiment.