1. Introduction

Laser applications such as cutting, drilling, welding, annealing, dicing, shock peening, surface patterning, and many others are an important part of modern manufacturing [1, 2]. Even though the mentioned techniques are different in many ways, there is one thing in common: all of them are driven by the absorbed energy of optical radiation that results in a number of different physical phenomena. The perturbation of an electronic subsystem may be followed by photochemical reactions or excited state relaxation. In turn, the absorbed energy is thermalized leading to thermo-optical and thermo-elastic effects (thermal lensing and high-pressure generation) and phase transitions (melting, boiling, evaporation, ablation, resolidification) [1]. The outcome of light-matter interaction is governed by a number of factors that fall into three main categories, namely:

Such variety of parameters opens up new possibilities for tailoring the mentioned laser processes, but at the same time numerous degrees of freedom complicate the theoretical models describing light-matter interactions. Even though the general principles of theory describing light-matter interaction are known [3–5], a vast majority of available models are merely phenomenological descriptions with limited precision and often involve unknown parameters. More robust theoretical models start with the estimation of absorbed energy and simulate the ensuing consequences using specific dissipation pathways. Recently it was reported that critical deposited energy remains almost constant in a wide range of experimental conditions and is therefore an appropriate criterion for predicting the damage threshold in transparent materials [6]. Estimation of deposited energy is also critical when optimizing efficiency of laser ablation process in micromachining. Vorobyev et al. [7] showed that the residual deposited energy is significant even during the femtosecond ablation of metals, usually considered to be a “cold process”. Cumulative heating was also observed at higher repetition rates of femtosecond pulses [8, 9]. It was shown, that the magnitude of induced refractive index significantly rises when the time interval between successive laser pulses is below the time for thermal diffusion [10]. In summary, the estimation of deposited energy is very important for practical applications. The methods for such estimation fall into four different categories, depending on the exploited physical phenomena and the number of required laser pulses as indicated in Table 1.

Table 1. Reported approaches used to estimate the laser energy deposited into material.

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The deposited energy is eventually turned into heat, therefore thermal effects play an important role in the estimation of deposited energy. These effects have been studied for many decades. From the physical viewpoint, deposited thermal energy is a long-lived type of excitation, able to stay around the photoexcited zone for an appreciable amount of time. The lifetime of such lattice excitation typically lies in μs-ms range and is ultimately determined by the characteristic time of thermal diffusion. In the case of ultrafast laser excitation, thermal equilibration time is much longer than the typical duration of interaction between the material and the laser pulse [4]. The rise of temperature, initial heat distribution and its dissipation dynamics all contain important information about the absorbed energy and relevant material parameters. In the simplest case, temperature distribution can be monitored using conventional thermographic sensors monitoring the infrared radiation from the heated zone [11, 15]. However, this method cannot offer sufficient spatial nor temporal resolution (the latter is determined by the camera frame rate). Furthermore, as commercially available femtosecond laser systems operate with pulse energies ranging from few nJ to mJ, the amount of energy absorbed from a single pulse is rather low and the processes under consideration happen on very small spatio-temporal scales. This puts the characterization of heat affected zone below the sensitivity level of many conventional thermometry methods. Averaged temperature fields become directly measurable only when operating at high average powers (either continuous wave, or high repetition rate laser sources). Such time-integrated approach provides averaged estimates, however, they are not always appropriate, because material properties above damage threshold are being altered with every arriving laser pulse (the so-called fatigue effect) [16]. In order to trace the dynamics of heat distribution induced by single laser pulse, more sensitive time-resolved methods are necessary. Seeking a deeper insight into the physics of relevant processes, recent studies have addressed the characterization of absorption efficiency, energy deposition and control [6, 13, 14] as well as the phenomena of thermal and acoustic energy relaxation [17, 18]. The development of better detection methods with improved spatial and temporal resolution has a strong impact on these studies and has enjoyed a steady progress in recent years [19]. The purpose of this study is to demonstrate how time-resolved digital holography (TRDH) can trace the thermal dynamics in the bulk of transparent material after single pulse femtosecond excitation and quantitatively evaluate the absorbed energy remaining in the sample. Pump-probe imaging of TRDH is an interferometric method employing a short probe pulse propagated through the heat-affected zone of photoexcited material where the refractive index is altered. The main advantage of this method is the unique combination of its high sensitivity and spatio-temporal resolution with the ability to retrieve absorption and refraction changes in the material [20]. In our previous works [21, 22], as well as in other sources found in literature [23, 24] it was shown that TRDH is a reliable tool for both qualitative and quantitative measurements of material parameters. In this paper, we propose a new application of TRDH: to measure the fraction of the absorbed laser energy (eventually turned into localized heat) as a function of incident energy. Using this technique, the dynamics of heat diffusion in bulk transparent materials can be directly visualized. The estimations of deposited energy fraction were compared with laser-induced damage threshold measurements performed on the surface of the same sample. This way damage criteria could be expressed as a function of absorbed energy.

 

Fig. 1 The layout of the experimental setup. PBS - polarizing beam splitter, M - mirror, BS - neutral beam splitters, OBJ - microscope objective, SMP - sample, TL - telescope, L - lens, CCD1 - holography registration, CCD2 - object view.

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2. Materials and methods

2.1. Experimental set-up

The experimental setup is depicted in Fig. 1. The core of the setup is a Mach-Zehnder interferometer used for holographic study of heat dynamics in a wide bandgap dielectric materials. Two electronically synchronized lasers, namely - a femtosecond Yb:KGW laser (“Pharos”, Light Conversion, Lithuania) producing 450 fs duration pulses (FWHM) at the wavelength of 1030 nm and a nanosecond Nd:YVO₄ laser (“NL640”, Ekspla, Lithuania) producing 4 ns pulses (FWHM) at the wavelength of 1064 nm were used as pump and probe sources, respectively. Both lasers were set to produce pulses on demand (single-shot triggered mode). The polarizations of both laser pulses were vertical. High intensity pump pulses were attenuated to appropriate energy level and focused into the sample at normal incidence using a microscope objective with 0.4 NA (20x magnification), 400 μm below the surface. The transmittance of the focusing objective was carefully checked as a function of incident pulse energy across the entire range of energies used in the experiments and no evidence for the self-action of the pump beam occurring inside the objective was found (data not shown). The second harmonic (532 nm) of nanosecond laser was generated in an external LiIO₃ crystal and used as a probing pulse that was directed to the interferometer. Using the first non-polarizing beam splitter BS, the probe beam was split into object and reference arms. The probe light in the object arm was focused by a lens (f=75 mm), intersected with the volume of the sample excited by the pump beam and then collected by a microscope objective with 0.4 NA (20x magnification) affording a 400 μm field of view. The focal spot of the focusing lens was adjusted to be located before the sample, ensuring that the entire field of view was illuminated more or less uniformly by divergent probe light. The out-of-focus object image was interfered with the reference beam on the CCD1 camera using the second BS and the fringe pattern was recorded. The digitized interference pattern was used as a digital hologram for numerical reconstruction. CCD2 was used as a secondary in-focus camera to control the image plane of the object inside the sample. The magnification of the system and the spatial dimensions of the excited zone were estimated by replacing the sample with 1951 USAF resolutions target. The lines with 2.19 μm spacing could be resolved clearly. Thus, 2.19 μm or better is considered to be the spatial resolution of this pump-probe setup, while a single pixel of reconstructed image corresponds to 0.1 μm of the recorded object.

The probe delay was varied by electronically adjusting the trigger time of probe laser. The timing signals were produced by timing electronics module of “Pharos” laser (pump) that allows generating TTL pulses with programmable delay with respect to its optical output. These pulses were used to fire the Q-switch on the Nd:YVO₄ probe laser, providing the delay range between pump and probe from -200 ns to +200 μs.

The heat deposited into the lattice by the pump pulse, modifies the local optical properties of the material through thermo-optical or thermo-elastic effects. They induce a phase change in the probe beam passing through the excited sample. Time-dependent phase and amplitude of the probe are detected by the means of the short-probe digital holography. The probe laser is electronically triggered at selected time delay after the pump pulse. Two holographic images are acquired - first without and then with pump pulse. Subsequent numerical reconstruction based on convolution approach [25] gives the quantitative information of the amplitude and phase of the object field. This way, the temporal evolution of heat (thermal lensing) in various transparent materials can be investigated.

A fused silica slab, polished on all sides (5x20x50 mm) was chosen as a sample. Two types of data sets were obtained. First of all, time-dependent dissipation of the thermal lens was recorded at three different pump pulse energies. Later, the response of material to wider range of energies was investigated at selected probe delays. The main parameters of pump and probe laser pulses are summarized in Table 2.

Table 2. The experimental conditions of absorbed heat evaluation

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The pump beam profile was close to Gaussian, and its diameter was estimated at $e^{-2}$ level. The estimate was obtained directly by imaging the focal field with an 20 x 0.4 NA microscope objective and a camera, while the value in parenthesis was calculated using the formula for diffraction limit. All the tests were performed in a single laser shot regime in order to avoid possible accumulation or incubation effects, as well as the impact of previous damage. Each experiment was performed at a fresh sample point (translating the sample between the measurements).

During the investigation of the thermal response, the heat evolution in fused quartz was recorded in the time range from 1 ns to 110 μs for 3 discrete pump pulse energies above the damage and ablation thresholds. The pulse energies before the microscope objective were 10, 13, and 16 μJ per pulse, which, accounting for the reflection losses both in the microscope objective (transmittance T=0.61 at all energies) and at the sample’s front surface (T=0.96), translated to the energies of 5.86, 7.61 and 9.37 μJ.

Optical damage thresholds for fused quartz were measured at the same laser conditions (450 fs, 1030 nm). The measurements were done using a lens with a focal distance of +400 mm. Linearly polarized light was focused into a 138.7 ± 0.85 μm spot on the surface of the sample. The diameter of the beam was measured at $1/e^2$ level of intensity using a CCD camera. According to the ISO 21254 standard, a series of single shot irradiations were conducted to determine the 1-on-1 thresholds. The fluence was varied from 0.07 to 4.48 J/cm² with a step size of 0.07 J/cm². 5 shots were taken at each energy level to confirm that the damage mechanism was deterministic. The sample was then removed and inspected using a microscope with high resolution and broadband illumination in order to determine the fluence levels corresponding to optical damage. Two thresholds were found: one for color center formation and one for ablation. The obtained threshold values are 3.54 J/cm² for discoloration and 4.09 J/cm² for ablation. These values are close to the data (3.68, 4.47 J/cm²) found in the literature for similar conditions (300 fs, 800 nm) [26]. The small differences could be due to different manufacturers of the glass, sample polishing, laser wavelength or pulse duration.

3. Results and discussion

3.1. Experimental data

An example of a typical reconstructed phase shift map of a dynamically changing thermal lens (heat-affected zone) at different delays after femtosecond pulse excitation is provided in Fig. 2. In all the pictures of this type, the direction of light propagation is upwards. The rise of the induced signal is limited by the duration of the probe pulse (4 ns). It is followed by a 300 ns time frame, during which the signal remains approximately constant. After this, the observed phase shift pattern starts to expand. In approximately 4 μs, the sharp needle-like structure transforms to a smoother, more Gaussian shape. Due to heat conduction this is followed by further expansion of the thermal lens until the distribution expands beyond the area of interest and the signal amplitude dwindles below the noise floor of the experiment.

 

Fig. 2 A typical phase shift map in radians for fused quartz at different time delays, pump energy - 5.86 μJ.

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This experiment was followed by the investigation of the thermal response to different energies in the range of 0.59–11.71 μJ within the sample. The experiments were performed at four discrete time delays: 0.01, 2.5, 5, 10 μs. The material response to different pump pulse energies is shown in Fig. 3. The main difference observed when changing incident power values is apparent in the size of the heat-affected zone, the magnitude of the peak and the shape of the affected area. The excitation at low energies creates a needle like structure. As the incident power increases, this structure grows in the direction opposite to beam propagation and the trailing part of the structure expands. This may be explained by the generation of free electrons. Initially, the electrons only form at the focal point, but as energy increases the intensity prior to the focal point becomes sufficient for ionization. The growth of the thermal trace is asymmetric with respect to the focal point due to the screening of electron plasma - most of the energy is absorbed or reflected. After passing the focal point, the intensity decreases below the ionization threshold, therefore no similar processes are observed. Catastrophic optical damage, that grows in size with fluence, can be noticed at the central part of the excitation. It is interesting to note, that the most evident damage at higher fluences forms prior to the focal point, while the damage at the focal point is comparatively small and appears at all investigated energies.

The obtained phase information was used to evaluate the time-dependence and power-dependence of the thermal lens and assess the absorbed energy density. In order to evaluate the absorbed energy, one needs to find the correspondence between the optical phase shift and the spatial heat distribution.

 

Fig. 3 A typical phase shift map in radians for fused quartz at 5 μs delay, for different pump energies in μJ.

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3.2. Analysis

Space- and time-dependent phase change data were analyzed globally using a commercial fitting software CarpetView (Light Conversion). This technique is routinely applied for the analysis of time-resolved spectra [27, 28], fluorescence lifetime imaging data [29] and 2D electronic spectra [30].

Since the sign of phase change has the same sign (negative) at all investigated delay times, we chose a parallel decay model for data parametrization. Such model fits the time profiles at each spatial position data as a sum of exponential decays, the lifetimes τ_i of which are shared by all image pixels and can be described by Eq. (1):

In addition to the globally determined lifetimes, the fitting routine produces amplitude maps for each lifetime (decay-associated images, DAI), which indicate the nature of signal change corresponding to each lifetime. In our case, negative features in DAI correspond to the decay of the (negative) observed signal, whereas the positive features show the corresponding growth. Four exponential components were enough to adequately describe the experimental results. The corresponding DAIs are plotted and the fitted lifetimes are given in Fig. 4. Each frame represents the changes occurring with the timescale indicated at the top of the slice, therefore the four DAIs provide a concise description of entire dataset consisting of more than 50 time-gated phase maps.

 

Fig. 4 Decay-associated images resulting from the global analysis of the experiment with 5.86 μJ pump energy. The lifetimes corresponding to each image are given at the top of each frame. Negative values, in radians, represent the decay, whereas positive values show the corresponding growth of the induced thermal lens effect.

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The obtained decay times also give some insight into what processes may be taking place in the observed time frames. As the experiments are done with pulse energies above the damage threshold, the first decay component, (τ₁ = 0.25 μ s) could be attributed to the phase transitions between solid and liquid states - the release of energy due to solidification and changes in thermal capacity, at the site of the focal spot. The distance over which the heat is transported during the first component issmall (<2 μm), and significant changes are observed only in the front part of the affected volume (the point of the needle), whereas the trailing part of the affected volume remains largely unchanged. In later frames, the peak phase decreases as the diameter of the affected zone increases in all 3 coordinates. The second exponential decay (τ₂ = 3.5 μs) coincides with energy transfer from the originally exited zone to the close surrounding (<6 μm). The third component with (τ₃ = 23 μs) shows the typical time necessary for heat affected zone to expand to the edges of the area of interest (<15 μm). In this time range, the distribution acquires Gaussian shape as the delay increases and is the most suitable for absorbed heat evaluation. The final decay step (τ₄ = 190 μs) describes the overall cooling of the observed volume and the transport of heat outside the observed area (>15 μm).

While global analysis is insightful for elucidating the dominant features of the data, it is not based on any underlying physical model of energy transport in the material. To obtain a physical picture of the observations, we follow it up by constructing a heat diffusion model intended to simulate the experimental results. Given the initial temperature distribution in the sample, the heat transfer is described by the following partial differential equation (Eq. 2).

Solving the Eq. 2 yields time-dependent temperature distribution in the sample. This, in turn, can be used to calculate the resulting refractive index change [31]:

Thermal conduction described by Eq. (2) is the major process through which the heat is transported out of the affected zone. It should be noted that even in a solid phase thermal equilibration is significantly affected by the fact that thermal conductivity (${\lambda }_i\left(T\right)$) and heat capacity ($C\left(T\right)$) are both temperature dependent. The data given in [32] was used to calculate the amount of heat necessary to raise the temperature of a mass unit, while the data for heat conductivity was taken from [33].

The heat sources in our case are a) laser excitation and b) resolidification of material melted upon excitation. Both of these sources can be accounted for in the initial condition, if the analysis of the data starts at sufficiently late delay times when all the material is solid and the effects of laser pulse are over. The only conceivable sink of heat could be black body radiation that removes thermal energy from the excited area in the form of infrared photons. Assuming the worst-case scenario, when the temperature of the entire visibly affected volume (5x5x50 μm) is equal to the melting temperature of fused silica, the radiative loss of heat during the first 5 μ s does not exceed 2% of the calculated deposited energy (see below) and can safely be neglected in the modeling. Therefore the simulation was done assuming that there are no heat sources or sinks.

Equation (2) was solved using the split-step method. The initial conditions for the simulation were found by estimating the heat distribution in three dimensions from a single early delay (5 μs) phase shift map. The property of radialsymmetry was used to obtain the distribution in three dimensions by employing rotation and normalization operation on the two-dimensional experimental refractive index data. This produced a spatial distribution of refraction index change, which could be recalculated to temperature distribution using the temperature dependence of the refractive index, determined by Waxler et al. [31]. Data at delay times, when the maximum temperature was above 650°C, were excluded from the modeling because: a) due to the induction of stress the volumetric expansion may become significant at higher temperatures and b) no temperature dependent refractive index data is available for these temperatures. The earliest delay time from which the data could be used for the simulation was 5 μs. The phase maps recorded at this delay was selected for the calculation of the initial condition. We chose to stop the simulation at 40 μs, where the signal-to-noise ratio of the data was still acceptable. The solution of Eq. (2) over this time range produces a spatial temperature distribution as a function of time that allows the calculation of experimentally observed refractive index data using Eq. (3).

 

Fig. 5 Temporal response to 7.61 μJ incident energy excitation. Left - comparison between experiment and model - phase dependence at 3 different positions of the phase map. Right - comparison between experiment and model at 3 different delays, vertical and horizontal cross-sections.

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Fig. 6 The thermal energies evaluated at different time delays in fused quartz, for three pump energies.

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The model was varied to describe experimental data by changing a single free parameter - the value of absorbed energy. This value was estimated from the entire set of refractive index changes at different delay times, thereby increasing the accuracy of estimation, and, at the same time, testing the validity of time-dependent modeling assumptions. The absorbed energy was estimated to be 4.2±0.25, 5.7 ±0.3, 7.2±0.34 μJ for the 5.86, 7.61, 9.37 μJ incident energies respectively. The error bars were evaluated at two standard deviations from the mean. The comparison between the experimentally observed and measured refractive index dynamics is presented in Fig. 5.

Close agreement between the simulation and experiment is obtained, confirming that thermal energy is conserved during the investigated time window. This allows us to expect that the absorbed energy can be estimated using a single phase map, as long as the assumptions of the model are valid (in practice, at all excitation energies lower than the ones included in time-dependent simulation). Such estimations are important because they allow us to determine the relationship between the absorbed and incident energies. This is done by measuring a series of refractive index spatial profiles at a fixed delay with different excitation energies (i.e. the data shown in Fig. 3). When this procedure is applied to all time points of a kinetic series as if they were independent, the absorbed energy estimate was obtained for different delay times. The same approach as described above, used for the determination of initial condition (i.e. 1 - refractive index rotation and normalization to convert two-dimensional data into three dimensions, 2 - recalculation of refractive index change into heat energy). The data is plotted for three incident energies in Fig. 6. It can be seen that obtained energy values are randomly distributed around the average value and no clear time dependence is observed, validating the single-shot evaluation approach.

 

Fig. 7 The absorbed thermal energy as a fraction of incident energy, measured at different time delays in fused quartz, for three pump energies. Red bars - extrapolated thermal energy absorption threshold.

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Using the fixed-delay and variable energy approach of evaluation we have also investigated the response of material to different incident energies. The results are shown in Fig. 7. On a logarithmic energy axis, the data points seem to be scattered around a straight line, therefore the data was fitted to a logarithmic function (Eq. 4):

This allowed us to extrapolate how the material would respond to lower incident energies. Assuming that such extrapolation is valid, there should be energy deposition even below the damage or ablation threshold. The energy fraction (damage criterion) required to trigger irreversible damage (apparent modification of the surface) is in the range of 10 %, while the energy fraction needed to start ablation is closer to 15 %. In our experiments, the highest absorbed energy fraction seems to reach approximately 80% of incident energy. As shown in the graphs, the error-bars of the evaluation are determined by the signal to noise ratio in phase contrast data. When the signal is low (due to late probe delays or low pump intensity) the error bars increase. Typically, the standard error in the thermal lens dissipation experiments ranges between 10% and 21% (st. error), however for the lowest energies it reaches 41%.

4. Conclusions

We have applied time-resolved digital holography for the assessment of absorbed energy fraction and direct visualization of the dynamics of thermal lens induced in fused quartz by a single femtosecond laser pulse. Time-dependent evolution of induced phase change was directly measured in the range of 1 ns to 110 μ s, enabling us to analyze multiple thermal relaxation processes, as well as providing the estimate of the residual energy deposited in the sample. The fraction of laser pulse energy absorbed during the nonlinear interaction at fluences above the damage threshold was determined to be in the range of 30-80%. The data shows that the deposited energy remains constant during the investigated time range, suggesting that no additional dissipation effects are taking place. It was found that the absorbed energy depends logarithmically on incident energy. This trend predicts the presence of absorbed heat even the below damage threshold. The energy fraction (damage criterion) required to trigger the irreversible damage process in fused silica is estimated to be 10%, while the energy fraction needed to start the ablation is 15 %.