1. Introduction

Recent experimental results in laser material processing of glass with femtosecond laser indicate that a tilted pulse front may influence the light matter interaction. A femtosecond laser source with 500 fs pulse duration and a wavelength λ = 1030 nm is coupled into a scanner system [1] and focused by a telecentric objective. The laser’s repetition rate is f_rep = 5 MHz at an average power of 0.5 W. The processing of borosilicate glass is effected by moving the samples at a stage velocity v = 1 mm/s. The large repetition rate and low scanning velocity are necessary to provide heat accumulation and thus a temperature gradient in the focal region. The thermal gradient in the focal region results in anisotropic material properties enabling the ponderomotive force induced by a pulse front tilt (PFT). The scanner is fixed at a certain angle θ. Therefore, a line is generated within the sample without any movement of the scanner. Microscope images of the processing results for two scan angles θ = ± 2.9° and two opposing feed directions are presented in Fig. 1. The periodical structures which arise for each line exhibit a width of about 50 µm equal to the visible heat affected zone. The focal diameter is about 1 µm in diameter and the pulse separation is about 0.2 nm. Therefore, the quasi-periodic structure in the heat affected zone of 10-100 µm in feeding direction is not a result of individual pulses but a result of thermal gradient, pulse front tilt and the dynamics of the absorptivity of the material, which are under on-going investigation. A comparison of the lines from Fig. 1(a) shows a difference of the structures depending on the feed direction of the sample, similar to the results of Kazansky et al. [2–4]. When carrying out the same experiment for an opposite scan angle θ₂ = + 2.9°, the process leads to different results as shown in Fig. 1(b). In this case, the difference of the processing results for both feed directions is significant less than for θ₁ = −2.9°. The comparison of Figs. 1(a) and 1(b) shows clearly a dependency of the resulting structures from different pulse properties at different scan angles. The asymmetry with respect to θ = 0° is caused by a constant PFT present in the raw beam of the laser. The constant PFT of the raw beam is overlaid with a variable PFT which depends on the focusing lens and the scan angle. For this reason, pulse deformations due to dispersive optical elements of a focusing lens are discussed and investigated for different field angles.

 

Fig. 1 Microscope images of laser processed glass for two different scan angles θ₁ = −2.9° (a) and θ₂ = + 2.9° (b) and opposing feed directions at v = 1 mm/s.

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If the laser processing is performed without pulse overlap (e.g. by using a repetition rate 100 kHz), the structured dots are not depending on the scan angle. Therefore, both a varying pulse-front tilt and anisotropic materials properties due to a thermal gradient are necessary to obtain the quill-writing effect [3].

The origins of propagation time differences (PTD) or pulse front tilt (PFT) are dispersive effects occurring when ultra-short laser pulses propagate through optical elements. Furthermore, pulse broadening due to group velocity dispersion (GVD) may occur in dispersive optics. Both effects have been studied in detail for single lenses and axially parallel incidence of light by Bor [5]. When utilizing scanner systems, the light path through the focusing lens depends on the scan angle θ as visualized in Fig. 2. Thus, the pulse deformation and the temporal pulse expansion are a function of the scan angle. The impact of variable field angles on the PTD has not been discussed in literature, yet, though a detailed examination is essential to understand the scan angle dependent effects of glass processing presented above.

 

Fig. 2 Principle of pulse deformations induced by a laser scanner system.

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2. Simulation methods

The effects of PTD and pulse broadening are being simulated utilizing methods based on ray tracing as well as wave optics, using the software packages Zemax [6] and VirtualLab [7], respectively.

The phase front of a laser pulse propagates with the phase velocity while the pulse front propagates with the group velocity. These different speeds cause a time difference between the phase and pulse front called propagation time difference (PTD). The PTD can be calculated using [5]

Pulse broadening is caused by group velocity dispersion (GVD), which appears for a nonzero second derivative of n with respect to λ. The group velocities for upper and lower wavelength windows within a pulse differ. Thus, the pulse duration increases. This effect can be calculated adopting the model [5]

2.1 Ray tracing analysis with Zemax

For the ray tracing analysis, an f-θ-lens is modeled in Zemax [6]. A set of 21 rays, homogeneously distributed along the y-direction of the entrance pupil with diameter d, is traced through the optical system. The path length l_i of each ray in lens i depends on the ray’s pupil coordinate ρ normalized with respect to d, and the angle of incidence. For each lens, the Sellmeier model for the refractive index n(λ) is extracted from the glass catalog and the derivative of n with respect to λ is calculated. Inserting these values into Eq. (1) and summing ΔT for the lenses yields the total PTD for the 21 rays. Finally, the PTD is plotted versus the normalized pupil coordinate ρ. Based on the same method, the pulse broadening is calculated. Instead of the first, the second derivative of n with respect to λ is computed. Equation (2) yields the pulse broadening in dependence of the bandwidth Δλ. In the following, we consider Gaussian shaped bandwidth-limited pulses with a time bandwidth-product of 0.441.

2.2 Wave optical analysis with VirtualLab

The wave optical analysis is carried out with VirtualLab [7]. The evaluation method is visualized in the flow chart in Fig. 3.

 

Fig. 3 Flow chart of the PTD calculation in case of the wave optical simulation.

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First, the f-θ-lens is modeled in the same way as in Zemax. As laser pulse, we use a Gaussian pulse with 100 fs duration, 1 µm wavelength and a plane wave as spatial field distribution. Both amplitude and phase information of the electric field are calculated in a certain observation plane (z = z₀) close behind the f-θ-lens. The propagation from the source to the observation plane is subdivided into three sections: source to first surface of the lens system, the lens system itself, and last surface of the lens system to the observation plane. For each section, the program decides on its own whether the Fresnel, Fraunhofer, far field, or geometric optics operator is to be used.

The resulting field data in the observation plane, spectral phase ϕ(x, y, λ) and spectral field amplitude A(x, y, λ), are analyzed as follows. A cross section of the phase along the y-axis ϕ(x = 0, y, λ₀) at the center wavelength λ₀ is extracted. Because the phase features phase shifts, the data is unwrapped with respect to an offset of 2 π. A multiplication of ϕ(y) with λ/(2πc) yields the time t_ϕ(y) when the phase front crosses the observation plane.

The spectral amplitude A(x, y, λ) is extracted along the same cross section as the spectral phase. The data A(y, λ) is transformed into the time domain utilizing an inverse Fourier transformation. Thus, the time dependent squared field amplitude A²(y, t) is obtained. In accordance with the input beam diameter, we define the threshold for the lateral expansion of the pulse in y-direction as 50% of the highest squared amplitude. The two threshold points y_min and y_max correspond to the normalized pupil coordinates ρ_y = ± 1 of the ray tracing, respectively. Within these borders, Gaussian shaped curves of the form

Bor [5] defines the PTD as the distance between phase and pulse front, measured perpendicular to the phase front, multiplied by 1/c. Here, we define the PTD as the difference between time t_p(y) from Eq. (3) when the pulse front passes the observation plane (z = z₀) and time t_ϕ(y) when the phase front passes,

Pulse and phase front have slightly different sampling rates. Fitting parabolas to the data and subtracting the parabolas simplified the calculation of the PTD. Because the phase can only be determined relatively, we set the PTD equal zero for the marginal ray (ρ_y = + 1). In fact, the PTD is not zero if the lens thickness is nonzero for the marginal rays. However, due the phase’s relative definition, this does not affect the pulse duration in the focal region.

To calculate the pulse broadening due to group velocity dispersion, we use the squared field amplitude A²(y, t). The pulse duration coincides with the full width half maximum (FWHM) of the fitted Gaussian curves in Eq. (3).

3. Simulation results

The PTD and pulse broadening of the telecentric f-θ-lens shown in Fig. 4 were investigated for two scan angles, 0° and 6°. A scan angle θ = 0° corresponds to an incident beam parallel to the optical axis. The input field diameter was d = 5 mm and the focal length 80 mm. The duration of the Gaussian pulse was τ₀ = 100 fs (FWHM) at a central wavelength of λ = 1 µm resulting in a bandwidth of Δλ = 14.7 nm.

 

Fig. 4 Schematic of the f-θ-lens with a focal length of 80 mm. Two scan angles were investigated: 0° and 6°.

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3.1 Ray tracing results

The resulting PTD calculated with ray tracing is illustrated in Fig. 5(a). The absolute delay between phase and pulse front is in the order of 2 ps. In case of θ = 0°, the PTD is symmetric around the chief ray (ρ_y = 0). The PTD is parabolic and has its maximal relative delay of 8.5 fs at ρ_y = 0. For a scan angle of 6°, the PTD is asymmetric which means that the pulse front is tilted. The pulse front for ρ_y = −1 is delayed about 94 fs with respect to the pulse front for ρ_y = + 1. An illustrative reason for this tilt is that one marginal ray (ρ_y = −1, lower ray in Fig. 4) propagates a longer distance in the dispersive lenses than the other marginal ray (ρ_y = + 1, upper ray).

 

Fig. 5 Ray tracing results for the f-θ-lens obtained with Zemax. (a) PTD for a scan angle of 0° and 6°. For θ = 6° the pulse front is tilted. (b) Pulse broadening for a scan angle of 0° and 6°. The initial pulse duration is 100 fs.

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The pulse broadening Δτ predicted by ray tracing is in the order of 70 fs assuming an initial pulse duration τ₀ = 100 fs. It features the same dependence on the pupil coordinate ρ_y as the PTD (compare Figs. 5(a) and 5(b)). The relative increase in pulse duration is about 70%. The bandwidth Δλ and therefore the pulse broadening are reciprocal to the initial pulse duration τ₀.

3.2 Wave optical results

The resulting spatio-temporal intensity distribution of the wave optical simulation is shown in Fig. 6(a). The pulses for a scan angle of 0° and 6° are separated in space. Due to a spherical phase front and PTD, the center of the pulse reaches the observation plane later than the outer parts. It can be clearly seen from the data that the pulse is tilted for θ = 6°. The extracted phase and pulse fronts for the two scan angles are shown in Fig. 6(b). In case of normal incidence (θ = 0°), the pulse front is delayed about 10 fs for y = 0 mm. For a scan angle of θ = 6° the pulse front is clearly deformed indicating a pulse front tilt. The maximum PTD of 115 fs appears at y = 5 mm.

 

Fig. 6 Wave optical results obtained with VirtualLab. (a) Spatio-temporal intensity distribution along the y-axis directly behind the f-θ-lens. The pulse for θ = 6° (top) is tilted. (b) Comparison of phase and pulse fronts to calculate the PTD for θ = 0° and θ = 6°.The maximal PTDs are 10 fs and 115 fs, respectively.

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4. Discussion

The main simulation results of the f-θ-lens are listed in Table 1. The PTD results obtained by ray tracing and wave optical analysis are in good agreement.

Table 1. Comparison of the Main Simulation Results of the f-θ-Lens

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The results in Figs. 7(a) and 7(b) show the same dependence on the pupil coordinate ρ_y. Quantitatively, the PTD calculated wave optically with VirtualLab is approximately 20% larger than the result obtained using Zemax. One reason for this deviation might be the threshold definition of the pulse’s lateral border as 50% of the highest squared amplitude.

 

Fig. 7 Comparison of the PTD (a-b) and of the pulse duration (c-d) calculated with Zemax and VirtualLab for the two scan angles. The initial pulse duration was 100 fs.

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The ray optical and wave optical results for the pulse broadening presented in Figs. 7(c) and 7(d) have the same dependence on the pupil coordinate. But there is an offset of approximately 15 fs and 13 fs for θ = 0° and θ = 6°, respectively.

Since the PTD is constant along one ray behind the lens system [5], the pulse front will be horseshoe-shaped in the focal region [8]. The pulse broadening due to PTD in the focal region approximately matches the PTD directly behind the lens system. In addition, the pulse broadening due to GVD must be taken into account. As a rough approximation, we predict that the pulse duration in the focus is equal to the initial pulse duration plus the PTD plus the pulse broadening. Thus, the focal pulse durations are assumed to be approximately 190 fs for θ = 0° and 290 fs for θ = 6°.

A comparison of the experimental results with the simulation results is possible only in a qualitative way. Previous studies show that laser processing of transparent dielectrics reacts very sensitive on tilted pulse fronts and pulse broadening [2–4]. So, the change of pulse properties which is predicted by the theoretical examination could be a reasonable explanation for processing results which depend on the scan angle. For a quantitative investigation the simulation of the high NA objective used for the process is required. Furthermore, measurements of the pulse front tilt depending on the scan angle are planned which provides results directly comparable to the simulation.

5. Conclusion

Since the requirements concerning processing speed and pulse duration are increasing over time, it will be necessary to design optical systems which provide homogeneous pulse properties over the whole scan field. The simulation of scan angle depending pulse deformations in focusing lenses is an important step to find methods of compensation. First results obtained by ray tracing as well as wave optical simulations show a significant change of the pulse shape for different scan angles. Even within a small scan range of 6° a PTD variation of about 100 fs occurs. Further investigation will show if well-established ray tracing tools are appropriate for design and optimization of PTD and PFT corrected optical systems for laser scanners.