## 1 Introduction  
Laser drilling is a process where material is removed by a focused laser beam to produce usually a hole with a circular cross-section. [PERSON] et al. extensively covered the basics of laser drilling in a review [1]. It is used for a variety of applications, e.g., the production of cooling channels and lubrication holes for turbine components and engine parts, the fabrication of orifices for nozzles and controlled leaks, microvia hole drilling in circuit interconnects, and biomedical applications such as needles, catheters, and sensors [2; 3]. Also, other electrical components such as hard disks, displays, computer peripherals, and telecommunication devices utilize laser drilling on a micro scale [3]. The four most used processes for laser drilling are single pulse, percussion (using multiple pulses in the same spot), trepenning (moving the laser beam along the circumference of the hole), and helical drilling (trepanning while simultaneously adjusting the focal plane). Laser drilling is a relatively well-understood process that has been studied for decades. There are certain aspects of drilling that have been of special interest. Challenges related to producing high precision and high aspect ratio holes have been investigated [4]. Aspect ratios greater than 50 have been produced with high quality, usually by employing helical or trepenning drilling [5; 6; 7]. In addition to high precision and aspect ratios, high-speed drilling and micro drilling are utilized, especially in aerospace applications [8]. Burr formation and elimination are also important issues, studied e.g., by [PERSON] et al. [9]. Their investigations showed which laser parameters cause burr formation and how to minimize it. [PERSON] et al. investigated the effect of laser parameters on hole taper and circularity in percussion drilling of steels [10]. [PERSON] and [PERSON] investigated similar hole characteristics and drilling optimization in a case of titanium alloy material [11]. Their studies provide guidance on how to optimize processing parameters to produce smaller holes with less tapering, and better circularity. Hole quality properties, including circularity and taper, are also important because of the need to control the shape of the drilled hole [12]. [PERSON] et al. developed a new type of optical systems that enable control of the hole taper to produce either positive or negative taper [13; 14]. Some effort has been expended to combine electro-discharge machining with laser drilling to produce high quality holes without the recast layer and heat affected zones, which can be problematic in laser drilling [15; 16; 17]. More specialized studies include [PERSON] et al. 's examination of curved drilling using inner hole reflections [18]. They were able to drill a 50-um diameter hole with a curvature of about 3\({}^{\circ}\), beginning at a depth of 400-600 um. On a more fundamental level, [PERSON] et al. investigated the hole characterization and formation [19]. They developed a numerical model to simulate the drilled hole geometry for AISI (American Iron and Steel Institute) 304 stainless steel. [PERSON] and [PERSON] investigated the drilling of tool steels by using a nanosecond diode-pumped Nd:YAG (neodymium doped yttrium-aluminum-garnet) laser [20]. With a laser wavelength of 1064 nm, the maximum material removal rate obtained was about 1 um per pulse. [PERSON] et al. studied single pulse drilling with a millisecond-pulsed Nd:YAG laser [21]. They determined particle size distribution, the angle of trajectory, molten layer thickness, and the temporal variation of melt ejection. Drilling depth estimation can be difficult unless a vast experimental data bank is available. [PERSON] et al. monitored the plasma plume from the side, and estimated the depth online through image acquisition and analysis [22]. Their results indicate that the cumulative plasma size correlates with the drilling depth, with a high degree of confidence.