## Abstract

We demonstrate time-resolved tomography with 200 fs resolution for the three-dimensional analysis of the non-linear dynamics of ultrafast laser-matter interaction inside the volume of transparent materials. We reconstruct as an example the three-dimensional spatial distribution of the transient extinction coefficient induced by focusing higher-order Bessel-Gaussian-beams into Gorilla glass. This approach can be employed to gaseous, liquid and transparent solid state matter which interact with laser light.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Ultrashort pulse lasers have become a unique tool for a huge variety of applications, ranging from investigations of nonlinear propagation effects for atmospheric analysis [1,2], spectroscopic investigations in gases [3, 4], analysis of the interaction with water with a focus on medical applications [5,6] to interactions with solids like crystals, polymers or glasses, where permanent material modifications are locally induced [7–9]. The reason is, apart from the short interaction time that yields a high temporal resolution, the high intensities that lead to nonlinear interaction processes which enable interactions with practically all materials, whether absorbing or transparent, gaseous, liquid or solid.

The interaction between matter and ultrashort laser pulses is a notoriously complex process involving transient states and dynamics that span time scales from microseconds down to the femtosecond regime [7,10–12]. At its heart, different ionization mechanisms [13] and the interplay between radiation and plasma give rise to highly nonlinear behavior [14,15]. In the aftermath of the energy deposition itself, the presence of non-equilibrium material states [7, 11, 16–18] and their various competing dissipation channels present serious challenges to a comprehensive numerical simulation of the overall process and the resulting permanent material modifications. Moreover, the wide range of possible objects of investigation show multifaceted differences in their interaction with ultrashort laser pulses. Thus, in-situ observation of these transient processes is essential for their understanding and tailoring. Typically, this is achieved by pump-probe measurements yielding the required temporal resolution in the femtosecond regime. However, these techniques typically provide only two-dimensional spatial information due to observation in- or perpendicular to the laser beam direction [5, 6, 12, 19, 20]. Complex three-dimensional structural dynamics are therefore hard to analyze. Such complex dynamics occur e.g. in ultrafast laser processing, where advanced focusing strategies based on asymmetric beam shaping are used. For example, special beam geometries are applied in glass processing to induce cracks with a predominant direction [21,22] or to generate hollow structures [23,24]. Another application example is the 3D additive manufacturing using short laser pulses, where micro- or nanoobjects are realized via two-photon polymerization. In this case, the final size of the structure is dependent on complicated three-dimensional effects, like thermal and chemical interactions as well as chain-scission of organic and inorganic materials [25,26].

In this paper, we demonstrate a tomographic approach to obtain the full three-dimensional information of the laser-matter interaction, in particular the change in the extinction, with high spatio-temporal resolution and the possibility to reveal details of potentially obscured or even hollow structures. However, this technique can be also adapted to polarization, refractive index, phase difference or dark field sensitive measurements. For example, we analyze the interaction of ultrashort higher-order Bessel-Gaussian beams with glass. Such intensity distributions/ beam shaping approaches show promising prospects for material modifications and processing, e.g. the precise cutting of glass [24,27–29]. Already before the last section, where the limitations of our setup will be discussed in detail, we would like to emphasize at this point that multiple shots are needed for reconstruction purposes and thus, need reproducible conditions. This implements that identical laser pulses and comparable material reaction are required, common to all pump-probe setups.

## 2. Materials and methods

For the time-resolved three-dimensional analysis of the transient state evolution with sufficient temporal and spatial resolution a transverse pump-probe microscope [19,20], recording cross-sectional images, would have to be set up, acquiring images in different projection directions which have to be tomographically reconstructed [30]. Instead, in the approach presented here, only one projection is sufficient; the experimental setup is sketched in Fig. 1.

Single laser pulses at a wavelength of *λ* = 1026 nm and Δ*τ* = 200 fs pulse duration are emitted from a PHAROS-SP system from Light Conversion and split by a beam splitter. One beam can be stretched in time to the duration of interest e.g. 7.5 ps here, and serves as pump pulse, whereas the other (probe beam), transmits the interaction area transversely. This probe beam is frequency doubled to *λ* = 513 nm to distinguish it from parasitic background radiation from the pump beam and samples the laser-matter interaction in a time window of 200 fs (pulse duration) at a given point in time defined by an adjustable delay line. The starting point *t* = 0 ps is defined by the first occurrence of significant signal drop in the shadowgraphic image. For instance, we measure here after a delay time of 7.5 ps, i.e. at the end of the pump pulse. However, it is possible to sample at any specific point in time, during the generation of free electrons or their subsequent relaxation.

A telescope with 20× magnification images the interaction area onto a CCD camera where shadowgraphic images are recorded. A bandpass filter in front of the CCD is used to avoid overexposure due to the broadband plasma radiation and scattered pump light. As glass sample Corning Gorilla^{®} Glass 3 is chosen and moved after every shot to a pristine position.

The setup is completed by implementing digital-holographic beam shaping techniques into the pump light path. Holograms displayed on a reflective liquid-crystal-on-silicon based spatial light modulator (SLM, Hamamatsu LCoS X13138-03 with 1280 × 1024 pixels, illumination angle < 5 °) allow the flexible generation of Bessel-Gaussian beams of desired order with varying orientation. A subsequent 4*f*-setup with 20× telescopic demagnification scales the shaped optical fields to the required micrometer dimensions and focuses them into the glass volume. Figure 2 illustrates the employed phase-only transmission functions *T*(*x*, *y*) displayed by the SLM for higher-order Bessel-Gaussian beam shaping.

A radial symmetric blazed phase grating (Fresnel-axicon-type) with transmission according to *T*_{ax} (*r*) = exp (*ιβr*) [31] acts as carrier into which superpositions of azimuthal phase components ${T}_{\text{az}}(\varphi )=\text{exp}\left\{\iota \text{arg}\left[{\sum}_{m}\text{exp}(\iota {l}_{m}\varphi )\right]\right\}$, *l _{m}* ∈ ℤ are multiplexed in order to obtain Bessel-Gaussian beams of higher order [32], see Figs. 2(a), (c) and (d). The slope

*β*of these digital axicons is connected to “real” axicons of angle

*α*and refractive index

*n*via

*β*≈ 2π (

*n*− 1)

*α*/

*λ*[31]. The simulated propagation behavior of the generated beams in vacuum with corresponding transverse beam profiles is shown for three examples in Fig. 3 where we consider two superpositions Figs. 3(a), (b) and one pure higher-order Bessel-Gaussian beam Fig. 3(c). For the first two examples, the typical petal-like intensity features distributed point-symmetrically around the optical axis are clearly recognizable. Consequently, we adopt the appellation “petal-like” [32] for this class of beams. Common to all beam examples is the extreme aspect ratio in the focal zone of a few micrometer in transverse to several hundreds in longitudinal direction [24,28,33].

Using transverse pump-probe microscopy we gain access to the local extinction zone in the (*x*, *z*)-plane (*z*-axis corresponds to propagation direction, *y*-axis to observation direction), by recording shadowgraphic images *I _{S}* (

*x*,

*z*) and corresponding background images

*I*

_{0}(

*x*,

*z*).

From this information we calculate the optical depth *τ* [34](dimensionless) from

Here, all effects (e.g. absorption, free electrons, Bremsstrahlung, scattering effects, ...) that locally reduce the transmission of the probe pulse [20,34] are contributing to increased values of *τ*(*x*, *z*). Considering Lambert-Beer’s law and our coordinate system definition (for one projection) depicted in Fig. 4, the optical depth represents the integral over the local extinction coefficient *κ*(**r**) = *κ* (*x*, *y*, *z*) along *y*-direction: *τ* (*x*, *z*) = ∫ d*y* *κ*(*x*, *y*, *z*) (coordinate system Σ).

However, for volume rendering of a three-dimensional object, a finite number of two-dimensional projections from different projection angles are required [30]. State-of-the-art tomographic techniques are based on a rotational scan around the measuring object. In our case a rotational scan of probe illumination and observation system around the glass sample would be needed. This approach is tremendously challenging since the fs resolution would suffer from the smallest changes within the delay line. Moreover, this realization becomes complex from a technical and optical point of view. For example, a change of the observation direction causes wavefront aberrations due to the different angles of incidence on different sides of the cuboid glass sample, which needs to be corrected foremost. In addition, variations in the sampling time due to different propagation lengths through the glass would have to be taken into account.

Thus, an alternative concept to access the different projections is needed. One possibility is the rotation of the pump beam around a desired angle *θ*. Typical beam rotation concepts are based on Dove prism or telescopes consisting of cylindrical lenses [35], which are prone to adjustment errors. A much simpler approach is to make use of the SLM not only for beam shaping but also for rotation by varying the orientation of the displayed phase mask, see Fig. 2(b).

Using this approach easily allows to obtain the projections for different rotations around the *z*-axis. In the rotated system Σ′ [again, see Fig. 4], the corresponding projections *τ _{θ}* (

*x*,

_{θ}*z*) along

*y*are directly accessible

_{θ}*τ*(

_{θ}*x*,

_{θ}*z*) is known as the Radon transform of

*κ*(

*x*,

*y*,

*z*) [30].

Several techniques exist to numerically invert Eq. (1) and, thus, to completely reconstruct the spatial distribution of the extinction coefficient *κ* (*x*, *y*, *z*) [30]. We apply the filtered backprojection algorithm to the measured projections *τ* (*x _{θ}*,

*z*) to perform the inverse Radon transform [30] and render volumetric images.

## 3. Results and discussion

In order to demonstrate the efficacy of this approach we analyze the non-radial symmetric extinction zone caused by focusing the first-order petal-like Bessel-Gaussian beam [*E _{p}* = 40 μJ, see Fig. 3(a)] into the glass. The corresponding measured optical depths

*τ*(

_{θ}*x*,

_{θ}*z*) at a delay time of 7.5 ps for five different projection angles

*θ*are depicted in Figs. 5(a) – (e), as an example. Here, the beam orientation at

*θ*= 0 [Fig. 5(a)] equals the one used in the propagation simulation shown in Fig. 3(a). Similar to the beam’s simulated intensity distribution, the extinction zone is characterized by two parallel-running elongated areas of 300 μm length, transverse distance of ≈ 2 μm and maximum optical depth values of

*τ*(

*x*,

*z*= 0) ≈ 1.0. Beam rotation to

*θ*= π/4 and π/2, respectively, gradually leads to a single elongated projection exhibiting maximum

*τ*-values of ≈ 1.9 at

*z*= 0. A further increase of the rotation angle to

*θ*= 3π/4 and, finally, π restores the original optical depth distribution since

*τ*=

_{θ}*τ*

_{θ+π}.

For the reconstruction of the spatial extinction coefficient *κ*(**r**) using the backprojection algorithm the sinogram representation [30] is beneficial. Adapted to the problem at hand, we plot *τ _{θ}* (

*θ*,

*x*) for a specific

_{θ}*z*-value and include the complete measurement series where

*θ*was varied from 0 to π within 41 steps, see Fig. 6(a). In this representation, the aforementioned transition from a two-lobe extinction projection at

*θ*= 0 to a single, stronger one at

*θ*= π/2 and back to the original distribution at

*θ*= π can be seen more clearly. From this measured data we derive the transverse distribution of the extinction coefficient

*κ*(

*x*,

*y*) at the focal position

*z*= 0 using the inverse Radon transform. The result depicted in Fig. 6(b) clearly shows an extinction distribution exhibiting two distinct maxima of distance 2 μm distributed point-symmetrically around the optical axis. In this particular case highest reconstructed

*κ*-values were determined to about 0.7 μm

^{−1}.

To complete the reconstruction of *κ*(**r**) in all three dimensions we compute the inverse Radon transform for each *z*-position. This volume data allows to evaluate the measurement as a “flight” through the entire extinction distribution representing the material’s complete spatial extinction response caused by the pump pulse at a fixed temporal delay, see
Visualization 1.

To gain visual access to the complete 3D-data of *κ*(**r**) simultaneously, an isosurface representation can be extracted, which is depicted in Fig. 7 (also, see
Visualization 2). Here, five areas of equal extinction relative to the maximum value of *κ*_{max} = 0.7 μm^{−1} are combined to surfaces of equal color and transparency.

As a next step we analyze a more complex situation and investigate the measured optical depths *τ*(*x*, *z*) arising from focussing the petal-like Bessel-Gaussian beam of third order [*E _{p}* = 74 μJ, cf. Fig. 3(c)] into the sample. During the beam rotation from

*θ*= (0 . . . π) within 81 steps (due to the more complex energy distribution) the extinction distribution is subject to considerable alternating fluctuations and is, for example, altered from two to three elongated

*τ*-zones for

*θ*= 0 and

*θ*= π/6, respectively, see Figs. 8(a), (b). Between these projection angles, in a transitional area, a single noticeably broadened optical depth distribution can be found, see, e.g., Fig. 8(c), where

*θ*= π/12.

The corresponding sinogram *τ _{θ}* (

*θ*,

*x*) at a fixed propagation distance

_{θ}*z*= 0 [cf. Fig. 9(a)], expresses this process more clearly where a threefold repetition until

*θ*= π can be seen for symmetry reasons, since

*τ*=

_{θ}*τ*

_{θ+π/3}, for this particular beam. The reconstruction of the transverse extinction distribution

*κ*(

*x*,

*y*) shown in Fig. 9(b) reveals this symmetry and six petals with maximum

*κ*-values of 0.4 μm

^{−1}appear.

Additionally, the isosurface representation of the total *κ*(**r**)-distribution reveals their parallel-running elongated behavior with similar *κ*(**r**) - values along the entire propagation distance, see Fig. 10(a) and
Visualization 3.

To highlight the ability of our technique we focus the pure Bessel-Gaussian beam of third order [*E _{p}* = 74 μJ] into the glass. The beam’s caustic is characterized by an elongated ring-like distribution, see Fig. 3(c). Also in this case, the corresponding reconstructed extinction distribution, depicted in Fig. 10(b), accurately follows this intensity profile and an extinction zone of hollow cylindrical symmetry appears exhibiting ≈ 3 μm diameter, a few hundred μm length and maximum

*κ*-values of 0.1 μm

^{−1}. This example, emphasized in Visualization 4, clearly illustrates the ability to transfer tomographic imaging techniques onto ultrafast detection methods and reveals details of potentially obscured or even hollowed extinction distributions.

The spatial resolution *d*_{min} of these details is given by the diffraction limit determined by NA and wavelength of the employed observation microscope which corresponds to *d*_{min} = 800 nm in our particular case. Choosing *d*_{min} as desired resolution for the reconstruction of the *κ* (**r**)-signal with a corresponding effective diameter of *D*_{max} = 8 μm, see, e.g., Fig. 9, the number of required projections *N* within the interval *θ* = [0, π] for a sufficient sampling is estimated to *N* > π*d*_{min}/*D*_{max} ≈ 31 [30]. Further increasing of *N* will decrease the impact of statistical errors, such as, e.g., camera noise, and, thus, will yield smoothed reconstruction signals. However, the corresponding spatial resolution limit will not be affected. It is important to note that the measurement technique reported here is based on the reproducibility of the laser-matter interaction process as is the case for all pump-probe measurements. Thus, typical statistical phenomena like filamentation of intensive femtosecond laser pulses [15] can not be reconstructed clearly. This also applies to processes close too a certain threshold value, like the generation of free electrons. Small fluctuations of the laser intensity or material impurities would hinder a definite rendering. In addition, the sample used needs to be invariant for a sample translation and beam rotation. With respect to the technical limitations we have to consider Eq. (1) where we assume to measure optical depths as integrals along straight lines. This condition is fulfilled only in a small region around the focus where plane phase fronts are present and which can be identified in first approximation with the axial Rayleigh length *z*_{R}. The transverse (*x*, *y*)-extension of the extinction zone under investigation has to fit inside an area with edge length of ≈ 2*z*_{R}. Considering the high NA of 0.4 of the observation objective used in our setup we obtain a Rayleigh length of *z*_{R} ≈ 5 μm. Larger extensions have to be characterized with adapted objective lenses or the procedure described in [30] has to be applied where the rotational scan of the extinction zone is combined with a depth scan in *y _{θ}*-direction and an additional, depth-dependent filter function is required [30] for the backprojection algorithm.

Finally, we would like to pay attention to Fig. 5 where parallel to the dual extinction zone close to *x* = 0, further weaker elongated extinction zones are apparent (in fact, they appear in all measurements). One might assume that these lines are caused by the sidelobes which are typical for Bessel-Gaussian beams [cf. Fig. 3]. However, as these extinction lines alternate with areas of negative extinction (*τ*_{min} ≈ −0.2), which would indicate amplification, this effect is similar to diffraction of coherent sources on slits or edges, known as Gibbs phenomenon, see e.g. [37], and is further enhanced by the nonlinearity of the logarithm applied to the shadowgraph signals.

The measured overshoots have a strong influence on the reconstruction of *κ* as can be seen in Fig. 6(a) where a ring-like structure surrounds the actual signal. To circumvent this issue, deconvolution algorithms [38], known from astronomy and microscopy, respectively, could be applied to the measured data. A deconvolution with a simulated or measured point-spread function would damp the overshoots and yield a cleaned extinction signal.

## 4. Conclusions

To conclude, the presented procedure combines all features of classical tomographic measurements where arbitrary spatial extinction distributions are reconstructed with the ability to investigate the process with a high temporal resolution provided by pump-probe imaging techniques. Although we focused on the spatial properties within this work, the procedure allows to investigate the temporal dynamics of the three dimensional extinction coefficient *κ*(**r**) with a resolution of 200 fs, provided by state-of-the-art pump-probe microscopes. The procedure is not limited to investigate the spatio-temporal extinction behavior of the glass sample only but could be expanded with little effort, e.g. to characterize the three-dimensional energy distribution of the two beam interaction of externally refuelled optical filaments [2] or double coaxial beams [39] in air, where a complex spatio-temporal beam shaping is needed and multi-filamentation needs to be suppressed. Moreover, this approach can be implemented for various other pump-probe setups using interferometric approaches or polarization sensitive devices. This could unravel the complex three-dimensional laser-matter interaction and could pave the way towards precisely controlled two- und three-dimensional in-volume modifications.

## Funding

Federal Ministry of Education and Research (BMBF) (FKZ 13N13927, FKZ 13N13930); TRUMPF Laser- und Systemtechnik GmbH.

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