Ray-tracing is the commonly used technique to calculate the absorption of light in laser deep-penetration welding or drilling. Since new lasers with high brilliance enable small capillaries with high aspect ratios, diffraction might become important. To examine the applicability of the ray-tracing method, we studied the total absorptance and the absorbed intensity of polarized beams in several capillary geometries. The ray-tracing results are compared with more sophisticated simulations based on physical optics. The comparison shows that the simple ray-tracing is applicable to calculate the total absorptance in triangular grooves and in conical capillaries but not in rectangular grooves. To calculate the distribution of the absorbed intensity ray-tracing fails due to the neglected interference, diffraction, and the effects of beam propagation in the capillaries with sub-wavelength diameter. If diffraction is avoided e.g. with beams smaller than the entrance pupil of the capillary or with very shallow capillaries, the distribution of the absorbed intensity calculated by ray-tracing corresponds to the local average of the interference pattern found by physical optics.
© 2012 OSA
Laser micro processing applications such as micro cutting, welding and drilling have become more and more important in the recent years. Although these applications have been experimentally and theoretically studied for several decades, no comprehensive model exists to describe the laser-material interaction up to now. Generally, to find optimized processing parameters, extensive experimental parametric studies are carried out.
Because the laser acts as a thermal tool, the heating of a workpiece is the first and most elementary process to be understood. The heat source is determined by the distribution of absorbed intensity. Normally it is assumed that the light absorption can be described by the Fresnel equations . Since a significant amount of power will be reflected from a local surface, multiple reflections have to be taken into account in the calculation.
The local absorptivity on the walls of a capillary depends on the index of refraction, the angle of incidence, the polarization  and the roughness . In general, the capillary walls are at an elevated temperature (e.g. evaporation) during processing. To calculate the absorbed intensity, the refractive index of the material at these high temperatures has to be known. Unfortunately, at high metal temperatures, there is little experimental data available. In most studies the refractive index is taken as constant by assuming that the consideration of the temperature dependence would not lead to qualitative changes. The importance of the polarization of the incident beam is also known from experimental investigations [4–6]. Tailored polarization distributions can positively influence the absorption. Linear polarization might be more efficient in cutting  and drilling [5,6] under some conditions. Nevertheless because of their easier utilization, direction-independent symmetric polarization states are applied in most of the cases. The commonly used processing beams are circularly or stochastically polarized. More recently, however, radially and azimuthally polarized beams have gained considerable attention. By using these beams, significant improvements of drilling  and cutting velocity  have been reported.
The absorptance and the distribution of the absorbed intensity which are determined by the local absorptivity have great influence on the efficiency of laser processing. To measure the absorptance (the ratio of the absorbed power to the incident power) of a bore hole, M. Schneider et al.  used an integrating sphere and confirmed that the aspect ratio of the hole affects the number of laser beam reflections as well as the absorptance. In this case, only the total absorptance was determined. The experimental comparison of laser-drilled blind holes shows that the distribution of the locally absorbed intensity is of high importance for the produced hole shape . But the distribution of the absorbed intensity inside a processing capillary has not been measured so far to the best of our knowledge.
Meanwhile, numerous investigations applied the ray-tracing method to calculate the absorbed intensity [10–13]. These calculations typically show high peak intensities at the bottom of the capillary [12,13]. The reason is that geometrical beams with finite power can be focused to a virtually infinitely small spot. Another reason why geometrical optics is expected to fail follows from diffraction  at the entrance pupil of a capillary. The Fresnel number for an electromagnetic wave passing through an aperture is defined by F = a2/dλ. Here a is the characteristic size (e.g. radius) of the aperture, d the distance from the aperture, and λ the wavelength of the light. Geometrical optics is only applicable if F >> 1. However, this condition will not be satisfied in the case of laser processing of micro components such as fuel injection nozzles with small diameters (about 100 μm) and high aspect ratios (more than 10) [14,15]. In addition the F is defined for an incident plane wave and an infinitely thin aperture. For small boreholes or welding keyholes this condition is not fulfilled even at short distances from the entrance. Thus, it is believed that diffraction becomes important.
Physical optics is a more sophisticated method that can be used to calculate the absorbed intensity. Indeed, the solution of the Maxwell's equations produces very accurate results including diffraction. But these calculations are much more involved, require considerable computer memory and are very time-consuming. Therefore only two dimensional (2D) models are commonly considered. For the realistic 3D case the advantage of the ray-tracing method becomes evident because of the reduced calculation effort. Even for comparatively large 3D model systems with several multiple reflections the calculation time is noncritical if an efficient algorithm is used. Furthermore, ray-tracing can easily be extended e.g. to diffuse reflection  by distributing the reflection angles.
Hence, for this kind of calculations, one has to make the choice between the fast and efficient ray-tracing method or the inherently more accurate but very involved solution of the Maxwell's equation. In this paper our aim is to identify under which conditions ray-tracing will give satisfactory results to help making more educated decisions on which of the methods should be applied to a given situation. To this end we calculated and compared the light absorption inside small capillaries with the two methods. Firstly we implemented an own algorithm for the ray-tracing calculations. The second approach was the numerical solution of the Maxwell's equations using the Finite Element Method (FEM). From the various available FEM software packages we chose to use the RF(Radio Frequency) module of COMSOL. Simple 2D capillary models were used in Cartesian and axial coordinates. The incident beam was a plane wave for the Cartesian model. For the axial model the simplest fully symmetric ring-mode was used. The calculated total absorptance as well as the distribution of the absorbed intensity was compared between both methods.
To calculate the power absorption inside 2D capillaries three model geometries with different incident fields were studied as shown in Fig. 1 .
The analyzed capillaries in the Cartesian coordinate system are grooves with a rectangular and a triangular shape (Fig. 1, case A and B). A plane wave-front with TE or TM polarization is assumed at the groove entrance. The corresponding direction of polarization of the electric field vector is parallel to the y (TE) or x (TM) axis. In the axially symmetric coordinate, as plotted in Fig. 1 case C, cone-shaped capillaries are analyzed. For this model the stationary solution of the electric field does not allow to consider linearly or circularly polarized incident waves. We therefore choose the simplest field with axial polarization states, i.e. the azimuthally and the radially polarized fundamental ring mode with an intensity distribution given by
All calculations were performed with CrNi steel as the processed material and 1030 nm for the incident wavelength. The index of refraction was found to be 2.59-4.87i by a best fit of the Fresnel equations to experimentally determined data of the absorptivity at TE and TM polarization for several incident angles as shown in Fig. 2(a) . The measurement of the absorptivity was performed using a standardized calorimetric setup according to ISO 11551:2003 as depicted in Fig. 3 . A CrNi sample was irradiated for a short but well-known time by a linear polarized cw laser beam at 1030 nm. With the known specific heat of the sample the amount of absorbed energy follows from the extrapolation of the exponential cooling curve.
Since the material properties and especially the refractive index change with temperature, the absorptivity in a given interaction zone also depends on the temperature of the sample which may vary between room temperature and evaporation temperature. The main findings discussed in the following, i.e. the influence of diffraction and interference, do not significantly depend on the absolute value of the given refractive index.
The graph on the right in Fig. 2 shows the variation of the normalized absorbed intensity taking into account the projection of the incident beam with varying angle of incidence. For TM polarization the absorptivity reaches its maximum value at the Brewster angle with 80°, followed by a steep decrease between 83° and 90° (near to grazing incidence). As discussed below this steep change strongly influences the distribution of the absorbed intensity inside a capillary.
3. Calculation methods
The two fundamentally different methods “ray-tracing” and “physical optics” mentioned above were applied to study the light absorption in capillaries. A self-developed program was used for ray-tracing. The physical optics calculations were performed with the commercial software COMSOL RF module. The two methods were compared mainly with respect to the total absorptance and the distribution of the absorbed intensity. In the following we first describe the two calculation methods.
3.1 Ray-tracing method
To calculate the total absorptance and the distribution of the absorbed intensity by means of geometrical optics the main step is to trace the propagation of independent rays. Fig. 4 shows the flow chart of the ray-tracing program. First, a capillary model is defined in a 3D Cartesian system as shown in Fig. 1(case A and B). As the ray-tracing algorithm was implemented to calculate the beam propagation also in much more complex cavities, the capillaries are specified as a list of elements with a triangular shape as commonly used for this kind of modeling.
For the calculation numerous rays are generated at random positions at the capillary entrance pupil to represent the incident beam. The power Pj|j = 0 of each ray is proportional to the local beam intensity. As multiple wall reflections are taken into account, the subscript j denotes the number of reflections of the considered ray. According to the propagation of the ray, the triangle element at which the ray hits the wall of the capillary can be calculated geometrically and the angle of incidence, the absorptivity A and the reflection vector are simultaneously obtained. The difference between the incident power Pj and the absorbed power APj is the power of the reflected ray Pj+1 for this reflection event. At least 7 millions of rays with a maximum of 40 multiple reflections were traced inside the capillary in our calculations.
At the end of the simulation loops the absorbed power of each triangle Pabs_triangle is calculated. The absorbed intensity of one triangle is defined by
3.2 Physical optics
The solution of the Maxwell's equations was performed to study the influence of diffraction (caused by the entrance aperture of the investigated cavities) to the distribution of the electromagnetic field inside the capillary taking into account the corresponding boundary conditions. For this investigation the RF module of the software COMSOL was used since its basic function is the calculation of electromagnetic wave propagation by numerically solving Maxwell's equations with the finite element method.
In order to reduce the numerical errors the mesh size of an element has to be chosen smaller than λ/5. Due to the exploited symmetries the incoming beam was defined by a line source. This boundary was given by an oscillating electric field and transparent for the wave reflected from the capillary. The surface impedance boundary condition  was used for the metal surface. Reflections at the boundary of the calculation area have been suppressed by impedance matching.
The calculation returns the x and z components of the absorbed intensity in the material of the walls of the capillary i.e. Iabs,x(x, z) and Iabs,z(x, z). The absorbed intensity on the walls is therefore determined by
4. Rectangular groove
Figure 5 shows the comparison of the total absorptance calculated by means of the ray-tracing and the physical optics methods. Here the aspect ratio of the rectangular groove is defined as the ratio of the depth to the width of the groove. The total absorptance comprises the absorptance on the side wall and the bottom of the groove.
For very small aspect ratios the groove is shallow (nearly a plane). In this case the Fresnel number is large and diffraction is negligible . The incident power is mainly absorbed at the bottom of the groove. For these shallow grooves the total absorptance calculated by the two methods is similar. With increasing aspect ratio, however, the total absorptance obtained from physical optics increase significantly due to diffraction at the pupil of the groove-entrance. The diffracted wave is reflected several times at the walls before and after being reflected at the bottom of the capillary. The absorption at the bottom of the capillary therefore decreases with increasing aspect ratio. This behavior is not observed with ray-tracing as it is purely due to the diffraction effect. That is why the two methods lead to increasingly different results with increasing aspect ratio. Thus, ray-tracing completely fails for this geometry with small Fresnel number.
According to Fig. 5(b) the difference between ray-tracing and physical optics is even more pronounced for the TM polarization. At an aspect ratio of 30 the simulation based on physical optics predicts a total absorptance of nearly 100%, whereas the absorptance calculated by ray-tracing is limited to below 30%. This is due to the steep decrease of the absorptivity beyond the Brewster angle mentioned in Fig. 2(a). Even a small variation of the angle of incidence caused by the diffraction at the entrance pupil will lead to a significant increase of the absorptivity at the groove walls. The comparison of the results obtained with physical optics in the Figs. 5(a) and 5(b) shows that the total absorptance strongly depends on the polarization as the absorptance differs by more than a factor of two at high aspect ratios.
Figure 6 shows the distribution of the absorbed intensity Iabs inside the rectangular groove. Iabs is normalized by the maximum intensity Imax of the incident beam. Since the absorptivity of the perpendicularly incident rays as considered by the ray-tracing method is the same for the TE and TM polarization, only one chart is shown here (Fig. 6(a)). With ray-tracing the absorbed intensity distributes uniformly on the bottom and no power is absorbed on the walls. According to physical optics the diffracted wave undergoes several reflections on the walls and the bottom, which is clearly seen from Figs. 6(b) and 6(c). For TE polarization the normalized absorbed intensity is distributed quite uniformly on the wall with a value of about 0.5%. In contrast the TM polarized beam leads to a value of about 5.2% near the entrance pupil, again showing the significant influence of the polarization.
5. Triangular groove
The results obtained for the triangular groove show a completely different behavior. Figure 7 shows the calculated total absorptance with varying aspect ratio for a linearly polarized incident plane-wave. The ray-tracing results agree notably well with those obtained with physical optics. At an aspect ratio of 0.1 the total absorptance is about 0.283. For aspect ratios higher than about 5 the total absorptance approaches 100%. As seen from the moderate deviations the ray-tracing method is applicable to study the total absorptance of the triangular groove.
Since for the TE polarization the absorptivity decreases with increasing angle of incidence (Fig. 2(a)) the total absorptance is found to decrease slightly at first with increasing aspect ratio. As the first angle of incidence reaches 39°, the number of multiple reflections starts to increase leading to a corresponding increase of the total absorptance. For the TM polarized light the total absorptance is higher and increases more quickly as the number of reflections increases stepwise at the angles of incidence of 39° and 58°, which is expressed in the two steps of total absorptance shown in Fig. 7.
The distribution of the absorbed intensity along the wall of the triangular groove depicted in Fig. 8 however shows that the influence of diffraction and interference is nevertheless not negligible. The ordinate shows the position on the wall measured from the groove entrance (not the vertical depth).
From Figs. 8(a) and 8(d) it is interesting to note that the distribution of absorbed intensity obtained from physical optics oscillates around the values calculated by means of ray-tracing. This oscillating behavior is due to the interference which is not considered by ray-tracing. For small aspect ratios the diffraction can be neglected and the ray-tracing results equal the local average of the interference pattern found by physical optics. Furthermore, due to the effect of multiple reflections the absorbed intensity increases at deeper positions inside the groove.
According to Fig. 8(b) the latter behavior is also observed by the ray-tracing in the groove with an aspect ratio of 10. After the multiple reflections the rays reach the bottom of the groove where the highest intensity is absorbed. At a position below 90 µm above the bottom the width between the two walls is smaller than the wavelength (1030 nm in our calculation). In the ray-tracing method the rays are propagated through a hole no matter how small it is. With physical optics, however, the incident wave cannot enter into the hole if its width is significantly smaller than the wavelength . Hence the results obtained with physical optics are different in that the intensity decreases to 0 at the position of 96 μm where the distance between the two walls is 400nm. The difference becomes even more pronounced for the groove with an aspect ratio of 30 as shown in Fig. 8(c). Hence, in the case of TE polarization the ray-tracing method is not suitable to describe the intensity near the bottom of the high-aspect ratio grooves.
Due to diffraction and its impact on the Fresnel absorption in the case of TM polarization it can clearly be seen from Fig. 8(e) that the ray-tracing results deviate too much from physical optics already at the aspect ratio of 10. The perpendicularly incident rays first reach the wall with an angle of incidence of 87°. The corresponding absorptivity is close to 0 (Fig. 2(a)). Hence, with ray-tracing the ray power will be absorbed later after multiple reflections deeper down the wall as we have discussed in the case of TE polarization. In contrast, most of the power is absorbed in the upper part of the groove according to physical optics. Again the difference of the two methods is more pronounced with increasing aspect ratio, Fig. 8(f). Consequently ray-tracing cannot be applied to calculate the distribution of absorbed intensity in triangular grooves with high aspect ratios.
6. Conical capillary
To simulate a more realistic situation of laser drilling we extended the calculations to a conical capillary. The incident fields were polarized either in azimuthal or radial direction. The intensity distribution is given by Eq. (1). It was found that the variation of the total absorptance with increasing aspect ratio obtained with this model is similar to the results discussed above with the (Cartesian) triangular groove (Fig. 7). Hence ray-tracing is suitable to calculate the total absorptance in capillaries with triangular cross-section. Differences are again noticed in the distribution of the absorbed intensity along the wall of the conical capillary, as plotted in Fig. 9 .
For the case of azimuthal polarization and a small aspect ratio (Fig. 9(a)) the influence of diffraction is negligible. But the calculation with physical optics clearly shows interference effect. Again the two calculation models agree well if the interference pattern obtained by physical optics is averaged over the interference period. The distribution of the absorbed intensity follows the doughnut-like intensity distribution of the incident beam.
With an aspect ratio of 15, the two methods start to differ near the bottom of the capillary in the case of azimuthally polarized light (Fig. 9(b)). The reasons are the same as for the case of TE polarization in the (Cartesian) triangular groove. The discrepancy is even more pronounced for the radial polarization (Fig. 9(c)). Again the explanation is the impact of diffraction on the Fresnel absorption with its steep decrease of the absorbed intensity at near-to-grazing incident angles.
Our analysis above reveals that the entrance pupil is the main cause for diffraction. From this one may conclude that the incident intensity at the edge of the entrance pupil is likely to influence the diffraction effect and with this the applicability of ray-tracing. To verify this hypothesis the waist radius ω0 of the radially polarized incident beam was varied from 2 µm to 20 µm. The aspect ratio of the conical capillary was 5 and the radius of entrance was 10 µm. To better visualize the effect of diffraction on the distribution of the absorbed intensity the ray-tracing results are compared with the average values obtained by physical optics smoothed over the period of the interference pattern. Figure 10 shows the resulting distribution of the absorbed intensity with the different beam radii.
In Fig. 10(a), the 2 μm waist radius is smaller than the radius of the entrance. The beam intensity at the edge of the entrance pupil is almost zero. The ray-tracing results agree well with the smoothed physical optics results, which means that diffraction can be neglected.
By increasing the waist radius to 10 μm and subsequently to 20 µm, the beam intensity at the edge of the entrance pupil is significantly increased. As this increases the influence of diffraction the validity of ray-tracing is reduced.
To evaluate the applicability of ray-tracing for the calculation of the absorption of light in a capillary we have compared this method with the simulations based on physical optics. The total absorptance and the distribution of the absorbed intensity were determined for different 2D Cartesian and conical axisymmetric geometries. The results show that ray-tracing is appropriate to calculate the total absorptance in the triangular grooves and in the conical capillaries. In all other cases ray-tracing fails due to the neglected effects.
The effect of interference should be considered since it has significant influence on the distribution pattern of the absorbed intensity. If diffraction is avoided for geometries with small aspect ratios or with beams waist smaller than the entrance pupil of the capillary the distribution of the absorbed intensity obtained by ray-tracing, however, agree well with the average of the physical optics results smoothed over the interference pattern.
The diffraction at the entrance pupil gains importance with increasing aspect ratios. The influence of diffraction effects are determined by the angle of incidence (hence the inclination of the wall), the direction of polarization, and the beam intensity at the edge of the entrance pupil:
- (1) Ray-tracing is not applicable to calculate the total absorptance and the distribution of absorbed intensity in rectangular grooves due to the steep walls.
- (2) In the case of TM or radial polarization, the distribution of the absorbed intensity is much more sensitive to diffraction because of the steep change of absorptivity at large angles of incidence beyond the Brewster angle.
- (3) If the local intensity of the incident beam near the edge of the entrance pupil is not negligible, the diffraction is pronounced.
Ray-tracing also fails to calculate the distribution of absorbed intensity in the case of beam propagation inside of the capillaries with sub-wavelength diameter. This shape should be avoided for calculations by means of ray-tracing.
In conclusion, ray-tracing can lead to reasonable results under appropriate conditions but in most cases the deviation from physical optics are not negligible. For the future the interference effects could also be considered in ray-tracing by taking into account the phase of each ray.
This work is supported by the National Natural Science Foundation of China, with Grant No. 51205210 and the Deutsche Forschungsgemeinschaft (DFG), with Grant No. GR 3172/7-1.
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