Open Access
Add to BibSonomyAbstract
Ablation of silicon and metals is investigated using a 1064 nm pulsed fiber laser, with pulse energy up to 0.5 mJ, peak powers up to 10 kW, and pulse widths from 10 to 250 ns. A simple thermal model is employed to explain the dependence of scribe depth and shape on pulse energy or peak power. We demonstrate that pulses of high peak powers have shallow penetration depths, while longer pulses with lower peak powers have a higher material removal rate with deeper scribes. The key parameter that enables such variation of performance with changes in peak pulse power or peak irradiance on the material surface is the nonlinear increase of the absorption coefficient of silicon or metals as its temperature increases.
©2010 Optical Society of America
1. Introduction
Lasers generating nanosecond pulses, ranging from one to several hundred nanoseconds, have been used in material processing applications that range from ablation, cutting, and drilling, to marking [1–3]. As is often the case, the type of optical pulses used in ablation experiments are defined by the type of laser used, as well as the technology employed for pulsing and extracting energy out of the laser system [4,5]. A wide array of results have been published regarding ablation using a variety of lasers, each with its own pulse shape, pulse energy, pulse repetition frequency (PRF), wavelength, and average power [6–9].
The procedure of optimizing a laser machining process, such as scribing, drilling or cutting, involves changing some the laser’s parameters, such as pulse energy or power, while observing the corresponding change to the material and process speed. This procedure inherently is limited by the performance of the laser itself, whereby optimizing one of the laser’s parameters in order to improve the machining operation, invariably changes other parameters as well.
The most common pulsed laser in use today is of the diode-pumped solid-state (DPSS) type. In such a system, pulsing is defined by an intra-cavity Q-switch element that is electronically controlled. In this case, pulse duration, pulse frequency and pulse energy are governed by a carefully tuned and balanced electronic switching system and consequently, adjusting one of the pulse parameters often impacts other parameters. For example, changing PRF or pulse width would unavoidably lead to changes in pulse peak power or pulse shape. It is thereby difficult to perform a concise study to evaluate and discern the effects of the pulse shape, peak power, and pulse energy on the process at hand.
In this study, a MOPA (master-oscillator power-amplifier) fiber laser is used to evaluate the effect of peak pulse power and pulse shape on the ablation process of hard materials such as silicon, metal, or thin film materials [10,11]. We will also demonstrate that in the nanosecond pulsed regime the nonlinear thermal absorption coefficient of silicon and metals has a dominating contribution on the ablation process.
2a. Pulsed fiber laser configuration
The experimental configuration uses a pulsed fiber laser, Model MOPA-M-1μm-10W, by Multiwave Photonics (see Fig. 1 ). This laser is configured to emit pulses with well controlled pulse widths, pulse energy and peak power, ranging from 10 ns to 250 ns, pulse energy up to 0.5 mJ, peak powers up to 10 kW and PRF from single shot up to 500 kHz.
Fig. 1 (a) Schematic of the fiber laser used in these experiments. The seed laser and amplifier chain is controlled to form pulses of preconfigured peak power and energy. (b) Power distribution of the delivered beam to target. The M2 for this beam is 1.15 and is measured to be constant throughout the experiment. (c) MOPA-M-1μm-10W laser used in this experiment.
The laser system contains a modulated seed laser, followed by a series of fiber amplifiers, ending with an optically isolated fiber delivery end, and a collimated beam of M2 = 1.15. This beam propagation factor is also measured to be stable and uniform as PRF, average power, pulse width or peak power are adjusted.
This system also has the feature that pulse parameters are controlled to remain constant as other parameters are adjusted. For example, pulse width can be controlled as peak power is held constant, or alternately, pulse peak power and pulse width can be adjusted without change of the emitted pulse energy. Furthermore, these parameters are designed to remain constant as the PRF is adjusted, up to the maximum average power of the laser. These capabilities are important for discerning the various thermal ablation processes that are driven by pulse energy or peak power, as well as increasing process throughput as PRF and scanning speed are increased.
2b. Experimental configuration
The system of Fig. 1 is installed in a marking station for delivery of the optical beam to the material surface (see Fig. 2 ). The optical beam is delivered via a scanner head and an imaging lens that produces a focused spot of about 20 μm in diameter on the surface. The Rayleigh Range is calculated to be about 300 μm. The scanning range addressed by each scribing operation is 25 mm x 25 mm, although the system is capable to address a much wider scanning range.
Fig. 2 Schematic of the marking system used to deliver the laser’s output to target. The focused spot is about 20 μm in diameter and has a calculated Rayleigh Range of 300 μm. The beam is focused at the top surface of the work piece.
The operation of the laser and scanner is controlled by an external controller. This controller also defines gating and pulse width selection for the laser via TTL control signals. An external trigger is also provided for control of the laser’s PRF, which can be adjusted at will, from single-shot to 500 kHz.
3. Operating the laser at constant pulse energy
The MOPA laser can be operated in a variety of pulse configurations. In particular, pulse widths, average power, peak power and PRF can be adjusted at will to produce a multitude of pulse shapes and pulse energies. Of these varieties of pulses, a group of pulses is chosen for our experiments that have the same pulse energy. This mode of operation is shown in Fig. 3 .
Fig. 3 The laser of Fig. 1 was operated at pulse widths of 11.6, 20, 31.6, 50.6, 98.6, 150.5, 201.5 and 243 ns, while its pulse energy is maintained at about 120 μJ. These measurements correspond to average power of 12 W and pulse repetition frequency (PRF) of 100 kHz. These pulse widths are simply referred to in this paper as 10, 20, 30, 50, 100, 150, 200 and 250 ns. The figure on the right shows the corresponding shape of these pulses at 100 kHz. The 250 ns pulse shape is not shown, although the 200 and 250 ns pulses are quite similar in shape. The 75 ns pulse is included for reference only.
Figure 3 shows a group of superimposed pulses, with different peak powers and pulse widths, but having the same pulse energy, and operated with the same PRF. Due to the nature of the leading edge commonly observed in fiber lasers, particularly with high energy pulses more than 200μJ, pulse width measurement is recorded at the 10% point from the base for pulses that are 100 ns and longer.
The MOPA system of Fig. 1 also has no observed spontaneous emissions between pulse emissions, i.e. all the pulse energy measured in Fig. 3 represents all the emitted power of the system. Often, spontaneous emissions drain the energy from the gain medium and compromise the energy of short pulses, 10 to 30 ns.
The laser is operated at PRF from 10 to 100 kHz, average power up to 11 W, peak powers up to 10 kW, and pulse energy up to 540 μJ. In particular, the delivery system is calibrated to produce 10 W of average power on target. This translates to about 10 kW of peak power (for 10 ns pulses) and 100 μJ of pulse energy, at the PRF of 100 kHz. At the silicon surface, the focused spot is about 20 μm, which leads to 3.2 GW/cm2 peak irradiance, and 32 J/cm2 of fluence. In contrast, the 250 ns pulse of Fig. 3 has peak power that is 20 times less; hence its irradiance level is about 0.15 GW/cm2, but with the same 32 J/cm2 fluence.
Ablation, drilling and scribing experiments were conducted using mono-crystalline silicon, multi-crystalline silicon, aluminum and alumina. The analysis presented here is applied to silicon and aluminum, although the concept and the observed effects can be applied to other materials since the dominant process here is thermally-based, with the critical parameters being pulse peak power, pulse energy, and the material’s thermal conductivity, absorption coefficient and boiling temperature.
4. Scribing silicon
Multiple scribed lines are written on single and multi-crystalline silicon using pulses that are 10, 20, 30, 50, 100, 150, 200 and 250 ns wide, all with 100μJ of energy. The depths and widths of these scribes are measured accordingly for each pulse width for comparison to our theoretical model. Figure 4 shows sample scribes using 10, 30, 50, 150 and 200ns in multi-crystalline silicon. The wafer was 200 μm thick and was cleaved to enable the SEM pictures of Fig. 4 and the corresponding width, depth and shape measurements.
Fig. 4 Successive lines are scribed on the surface using 10 W average power, but with different pulse widths ranging from 10 to 250 ns. The system was operated at constant pulse energy (100 μJ) and 20 μm spot diameter at the silicon surface. PRF is 100 kHz, scan rate is 1 m/sec, and pulse overlap is 66%. The SEM photos shown here are for scribed multi-crystalline silicon using pulse widths of 10, 30, 50, 150 and 200 ns, respectively.
Other common beam parameters for Fig. 4 scribes are 10 W average power at the surface, 20 μm spot diameter, 100 kHz PRF, 1 m/s scan speed and 66% pulse overlap. All the pulses have the same pulse energy; consequently, pulse peak powers are proportionately higher for shorter pulses. As seen in Fig. 4, shorter pulses with higher peak powers have shallower penetration depths. This shorter and higher peak power pulse also causes the dispersion of the silicon to a wider area as the material sublimates and is ejected from the surface. As the ejected material cools down, it returns back to solid in the form of small particles that settle to the neighboring areas. In contrast, larger ejected particles are formed when a wider and lower peak power pulse is applied.
The ablation process consequently is driven by the high peak power, which causes the fast sublimation of the material and a stronger shock wave that disperses the material. The intensity of the shock wave generated as the material is ejected manifests itself in the size of the ridge that is formed about the scribe, and the size of the area covered by the debris after it settles down. Short pulses have very low material accumulation at the ridge, while longer pulses show accumulated debris near the scribe rim.
5. Analytical model
In order for ablation to occur, the material that is close to the surface has to be rapidly heated past its vaporization temperature within the early portion of an incoming pulse [2]. Pulse energy is absorbed causing the temperature and pressure of the underlying material to be raised beyond their critical sublimation values, causing the material close to the surface to explode. An ablated and ionized plume results, leaving behind a well-defined ablation pit [2].
A detailed modeling of the above process is relatively complex due to a rich array of mechanisms that are involved. However, for long enough pulses of sufficient irradiance levels (for example > 10 ns and irradiance levels > 108 W/cm2) the thermal process plays an important role.
To simulate the ablation process, we assume an input laser beam that has a Gaussian transverse intensity profile, with a peak at center, and is incident on the surface of a material with relatively high optical absorption. We will also assume that this beam has a temporal pulse profile p(t), that is 10ns or longer.
The temperature change of a high absorption material due to an incident Gaussian laser pulse can be approximated by3
where ΔT is the temperature rise, and W is the beam’s (1/e) field radius, τ is the pulse width, Imax is the peak intensity, κ is the material’s diffusivity, and K is the material’s conductivity. The fraction of the pulse energy that is absorbed by the irradiated material is represented by the factor γ. For high absorption materials, γ is equal to 1- R, where R is the Fresnel energy reflectivity for the material, for example γ = 0.625 for silicon. In what follows, we will assume that surface reflectivity is constant since the refractive index change with temperature and intensity is relatively small. The pulse temporal profile is represented by the function p(t), so that the laser intensity is . For simplicity, we will also assume that p(t) is a square-shaped pulse. This assumption is reasonable for the pulse shapes shown in Fig. 3.Equation (1) assumes that the optical absorption coefficient is high enough such that the pulse energy is absorbed mostly at the surface. This assumption is accurate for metals. Silicon, however, has a relatively low absorption coefficient at room temperature, but increases very rapidly as the temperature rises since it varies as T4 [12–14]. For example, at 300 K, the absorption coefficient α = 10/cm. At 1000 K it rises to 1200/cm. The 50 to 75% spatial overlap of consecutive optical pulses applied to the silicon surface raise the initial temperature of the substrate and condition the surface temperature to be well above the room temperature as consecutive pulses are applied. Hence, it is reasonable to assume that the absorption coefficient is already high enough to absorb most of the energy at the silicon surface.
Equation (1) is used to calculate the surface temperature of silicon, as a function of depth z, and radial distance, r, from the peak of the input beam after the energy of one pulse is absorbed. The surface temperature profile for silicon (where K = 145.3 W/K kg, κ = 0.907x10−4 m2/s, and γ = 0.625) is shown in Fig. 5 .
Fig. 5 The surface temperature distribution of the silicon material as a function of radial distance for various incident pulse widths, 10 to 250 ns. Pulse energy is 100 μJ for all pulses. 10 ns pulses have 10 kW peak powers, while longer pulses have proportionately lower peak powers, as shown in Fig. 3. Note that this temperature rise, which is due to the absorption of a single incident pulse is significantly higher than the boiling temperature of silicon.
Also drawn in Fig. 5 is a horizontal line representing the boiling temperature for silicon (melting point = 1,687 K, and boiling temperature = 2628 K). It is worthwhile to note that on axis, the surface temperature for a 10 ns pulse is significantly higher than that for 250 ns pulse, which in turn is also significantly higher than the boiling temperature of silicon. Such high temperatures would sublimate the silicon material. Since the energy of the shock wave is also proportional to the temperature of the expanding silicon gas, it follows that the shock wave intensity would be proportional to the incident peak irradiance level. Consequently, the dominant effects here are thermal evaporative and shock wave-related.
It is also reasonable to conclude that the surface temperature is directly affected by the incident peak irradiance level. The higher the peak irradiance, the higher the rate of absorption and the localization of the absorbed energy close to the surface, which highlights the behavior outlined above in Fig. 4. Additionally, this process is also affected by pulse shape. Additionally, the material’s temperature rise would be balanced by its thermal conductivity, affecting the size of the ablated surface area. Consequently, control of the laser’s peak power and repeatability of the pulse shape is significant for uniform scribing and marking of hard materials since ablation depends on the rate of energy accumulation.
6. Ablation depth vs. pulse width
Equation (1) is further employed to estimate the ablation depth of the scribe for aluminum and for silicon. For simplicity, we plot the temperature rise vs. the depth z for different pulse widths at the center of the beam (i.e. r = 0). To simplify, the pulse shape in the temporal domain is assumed to be of a square shape. The intersection of the curves with the boiling temperature (horizontal line) is assumed to correspond to the depth to which the material is ablated. To match the calculation to the experimental configuration, the beam diameter is set at 20 μm, and the pulse energy at 100 μJ. Figure 6 shows the temperature profile vs. depth for aluminum for different pulse lengths.
Fig. 6 Temperature rise of aluminum vs. sample’s depth for different incoming pulse widths. The horizontal black line represents boiling temperature of aluminum. The intersection of the curves with the boiling temperature line represents the ablation depth.
We can derive an analytical expression for the ablation depth as a function of radial distance r. In this case, we take advantage of the mean-value theorem [15], which states that for a continuous function f(t), there exists a point a < tc < b that satisfies the relationship,
Applying Eq. (2) to Eq. (1), we can derive an expression for the ablation depth h as a function of radial position r with respect to the center of the beam,
where tc = βτ, and ΔTB, is the boiling temperature for the sample minus the initial temperature, which is normally room temperature at 300 K. The parameter β is found numerically to be 0.5 and gives results accurate to better than 5% for ablation depths over a pulse width range of 5 ns to 300 ns.The above equations are derived for a single pulse. If a train of pulses are employed, then one has to add the ablated depth caused by the previous pulse a distance S away from the current pulse. Hence, for a scanned beam, where pulses overlap spatially, the effective ablation depth is,
where S is the spatial separation between neighboring pulses and is equal to the scan speed in m/s times the inverse of the PRF, the period between pulses.This model is applied to aluminum samples (K = 2.387 W/cm K, and κ = 0.84 cm2/s, Tboil = 2720 K) and is compared to the experimental measurements for pulses ranging from 10 ns to 150 ns. These results are shown in Fig. 7 .
Fig. 7 Ablation depth vs. pulse width for a single trace (solid curve) and measured depths (diamonds) for aluminum. Pulse energy is held constant at 100 μJ and beam diameter is 20 μm. Pulses are 10 μm apart. The error bars on the experimental measurements are ± 1 μm, which corresponds to the surface roughness and irregularity of the bottom of the trench. This irregularity is due to filamentation and recasting of the liquid aluminum.
In this case the scan speed was 1 m/s and the PRF was 100 kHz, resulting in pulse spatial separation S = 10 μm. The energy per pulse was held constant at 100 μJ for all pulse widths. The absorbed energy for each pulse is assumed to be 70% of the total incident energy. The samples’ initial temperature is assumed to be 300 K. While the model’s prediction of the pit’s depth is in good agreement with experimental data, it under estimates the pit’s depth for pulse widths shorter than 20 ns., The model employed here is purely thermal in nature, and become inadequate to fully account for the ablation process for pulses shorter than 10ns [16].
The higher thermal conductivity of metals causes heat to be conducted to other areas on the surface, raising its temperature closer to melting. A longer and lower peak power pulse also leads to more melting and recasting of the material as successive pulses are absorbed. It is thereby suggested that pulses with higher peak powers (shorter than 50 ns) are more appropriate for metal surface ablation and texturing applications.
Fiber lasers have a characteristic leading edge, particularly when the pulse is longer than 150 ns. The longer tail contributes thermal energy, without having the peak power needed for material sublimation. The energy in the tail section of the pulse raises the temperature of the surface closer to its melting temperature. On the other hand, the limited tail-50 ns pulses shown in Fig. 3 have peak powers resulting in substantially higher effective temperatures than the boiling temperature of aluminum. The absence of this tail limits the amount of heating of the substrate during pulse absorption, leading to less melting and recasting.
The ablation pit’s radius can be found from Eq. (3) by setting the depth h(r) to zero for the perimeter of the pit. The resulting solution for r = ro, is given by,
Equation (5) represents the ablation radius ro as a function of pulse width. A correlation between the experimental and theoretical values for scribe width (or ablation pit diameter) is shown in Fig. 8 for aluminum (left figure) and crystalline silicon (right figure). The predicted pit sizes are larger than the actual measurement data for aluminum due to vapor condensation. A SEM inspection of the pits created in aluminum, shows very clearly that there is a significant material condensation along the pit’s edge which refills the hole created by ablation, resulting in a reduction of the pit’s diameter for aluminum material. The pulse energy was set at 100μJ and beam diameter was 20μm for all cases.
Fig. 8 Ablation pit diameter vs. pulse width as defined by Eq. (5) (solid red curve) and measured diameter (blue x and diamond) for aluminum (on the left) and crystalline silicon (right). .The error bars on the experimental measurements are ± 1.5 μm, corresponding to the surface roughness and irregularity at the bottom of the pit. This roughness, measured by a non-contact optical surface profiler, is due to filamentation and recasting of the liquid silicon as it is ejected.
Note that Eq. (5) predicts a quadratic pit shape, as shown in Fig. 9 , which is rather unexpected since one intuitively expects a Gaussian shape following the pulse’s spatial Gaussian profile. This is shown by matching the measurements from the SEM pictures of Fig. 4 and a quadratic curve fit according to Eq. (5). A Gaussian curve is seen to give a poor fit to the SEM profile.
Fig. 9 (a) Surface profile measurement of an ablation channel formed on multi-crystalline silicon sample using 150 ns pulses, and (b) A quadratic curve fit to the depth profile (blue x) according to Eq. (4) (red curve) and a Gaussian curve fit (dotted brown curve).
8. Ablation of silicon
At room temperature, metals have a much higher optical absorption coefficient than silicon. At room temperature, the optical absorption coefficient of silicon at 1060nm is about 10 /cm, which implies that silicon ablation, should be ineffective at 1060nm. However, as the temperature is increased, the population of the conduction band of silicon increases, causing the optical absorption coefficient to increase rapidly, which explains why silicon ablation at 1060nm is effective at all. The optical absorption coefficient of silicon at 1060nm increases with temperature according to the equation [12],
where αo is the absorption coefficient at ambient temperature To, assumed to be at room temperature, 300 K. Thus, the absorption coefficient becomes 1235 /cm at 1000 K. The thermal properties of silicon are strongly temperature dependent too. For example, the temperature dependence of conductivity and diffusivity for silicon follow the formulas [17],These formulas indicate that silicon’s room temperature conductivity and diffusivity at room temperature (K = 1.45 W/cm K, and κ = 0.9 cm2/s) become K = 0.32 W/cm K, and κ = 0.16 cm2/s at a temperature of 1000 K. This indicates the importance of heating the silicon material between pulses. The heat equation becomes difficult to solve when the temperature dependence of the conductivity and diffusivity are included. However, a simple approximation to fit the theoretical calculation to the observed experimental results is to fix these parameters is to fix the silicon temperature at higher than room temperature.
Adopting this approach, we show in Fig. 10 a comparison of experimental data for the pit depth for ablated crystalline silicon as a function of pulse width for fixed pulse energy of 100 μJ, a beam diameter at the sample of 20 μm. The scan speed was 1 m/s and the pulse rep rate was 100 kHz, resulting in a pulse separation of 10 μm. The ambient temperature used for generating the curves using Eq. (3) along with Eq. (7) and Eq. (8) was 550 K and γ = 0.625. . The corresponding comparison for pit diameter is shown in Fig. 8. The reduction in the pit diameter for shorter pulse widths seems to be caused by material condensation at the pit’s edges.
Fig. 10 Ablation pit’s depth vs. pulse width for mono-crystalline silicon. The error bars on the experimental measurements are estimated to be ± 0.5 μm, corresponding to the surface roughness at the bottom, as well as the resolution of the detection microscope.
The ablation depth measurement shows significantly shallower ablation depth with short and high peak power pulses. As our thermal model predicts, the depth of generated pit is also dependent on pulse peak power. The higher thermal conductivity of metals contributes to recasting of the material on the adjacent ridges which distorts the formed scribed pit, particularly when the deposited power to the surface is high, such as when the pulse overlap is increased to 75%.
9. Conclusion
In conclusion, we have developed a model that predicts the ablation depth and pit size after exposure of silicon, metals, and other hard materials to nanosecond pulses from a fiber laser. This model shows good agreement with experimental measurements on silicon, as well as predicts a comparable behavior for metals. The model also predicts a quadratic shape for the ablated pit in silicon substrates which is consistent with measurements gathered from SEM cross-sections and surface profiling measurements.
The theoretical model calculates the temperature of the material in the presence of thermal diffusion and using an input beam of Gaussian spatial profile. For simplicity, a square temporal pulse profile is used. The process of ablation is explained in terms of surface temperature increases due to absorption of the incoming pulse energy. Agreement is demonstrated between theory and experiment for ablation depth values, scribe widths and channel shape. Since the model is basically thermal in nature, we expect it to be limited to pulse widths where thermal processes dominate such as pulse widths longer than 10 ns.
The material used for these tests are multi-crystalline silicon, mono-crystalline silicon and aluminum, with pulse widths that range from 10 ns to 250 ns. The fundamental property that defines ablation behavior in this simple analytical model is the temperature-dependent absorption coefficient for silicon. We demonstrated that pulses with high peak powers have shallow penetration depths, while longer pulses, with lower peak powers, have a higher material removal rate with deeper scribes as ablation is accompanied with pre-heating of the surface, melting of the material, and a shock wave that ejects the melted material.
This model and experimental demonstration also shows that material ablation benefits from a leading peak pulse shape often observed in fiber lasers. The MOPA-type fiber lasers, as shown in Fig. 1, with pulse widths of more than 150 ns and pulse energy of more than 0.3 J are noted for their leading edge. Consequently, the approximation used here of a square pulse could be adapted to explain the ablation behavior if the shape is broken down into successive pulses of varying widths and peak powers, and subsequently interpreting the effects in terms of the thermal model defined above.
Acknowledgement
We would like to acknowledge the support of Paulo Guerreiro, João Sousa and Martin Berendt of Multiwave Photonics, Niels Schilling and Jan Rabe of Fraunhofer IWS, M. Aljassim of NREL, and Zemetrics, Inc.
References and links
1. J. C. Miller, and R. F. Hagland, Jr., eds., “Laser Ablation and Desorption,” Academic Press, San Diego, CA (1998).
2. R. E. Russo, “Laser Ablation,” Focal Point 46(9), 14A (1995).
3. F. John, Ready, “Effects of High-Power Laser Radiation”, pp 79 (Academic Press, 1971).
4. D. Breitling, A. Ruf, and F. Dausinger, “Fundamental aspects in machining of metals with short and ultrashort laser pulses,” Proc. SPIE 5339, 49–61 (2004). [CrossRef]
5. X. Zeng, X. I. Mao, R. Greif, and R. Russo, “Experimental investigation of ablation efficiency and plasma expansion during femtosecond and nanosecond laser ablation of silicon,” Appl. Phys., A Mater. Sci. Process. 80(2), 237–241 (2005). [CrossRef]
6. H. Herfurth, R. Patwa, T. Lauterborn, S. Heinemann, and H. Pantsar, “Micromachining with tailored Nanosecond Pulses,” in Proceedings of SPIE 6796, 67961G–1 (2007).
7. J. König, S. Nolte, and A. Tünnermann, “Plasma evolution during metal ablation with ultrashort laser pulses,” Opt. Express 13(26), 10597–10607 (2005). [CrossRef] [PubMed]
8. A. Schoonderbeek, C. A. Biesheuvel, R. M. Hofstra, K.-J. Boller, and J. Meijer, “The influence of the pulse length on the drilling of metals with an excimer laser,” J. Laser Appl. 16(2), 85–91 (2004). [CrossRef]
9. P. Deladurantaye, D. Gay, A. Cournoyer, V. Roy, B. Labranche, M. Levesque, and Y. Taillon, “Material micromachining using a pulsed fiber laser platform with fine temporal nanosecond pulse shaping capability,” in Proceedings of SPIE 7195, 71951S–1 (2009).
10. S. T. Hendow, J. Sousa, N. Schilling, and J. Rabe, “Pulsed MOPA Fiber Laser Optimized for Processing Silicon and Thin-Film Materials,” 5th Int. Workshop on Fiber Lasers, Dresden, Germany, Sept. 30-Oct.1, 2009.
11. S.T. Hendow, S.A. Shakir, and J.M. Sousa, “MOPA fiber laser with controlled pulse width and peak power for optimizing micromachining applications,” Photonics West 2010, Paper 7584–42, January 27, 2010.
12. J. Paul Chernek and Jay A Orson, “A simple thermal response model for a p-doped silicon substrate irradiated by 1.06 and 1.32 micron lasers,” Laser-Induced Damage in Optical Materials, Proc. SPIE 4679, 186 (2002). [CrossRef]
13. D. E. Ackley, A. P. De Fonzo, and J. Tauc, “Optical Absorption Below the Gap in Crystalline and Amorphous Silicon at High Temperatures,” Physics of Semiconductors: Proceedings of the 13th International Conference, Rome, 1976.
14. K. Dieter Schroder, “Semiconductor Material and Device Characterization,” Third Edition, pp 613, IEEE Press, John Wiley & Sons Inc. 2006.
15. A. Ganino, Korn and Theresa M. Korn, “Mathematical Handbook for Scientists and Engineers,” pp. 119 (Dover Edition, 2000).
16. B. Luther-Davies, A. V. Rode, N. R. Madsen, and E. G. Gamaly, “Picosecond high-repition-rate pulsed laser ablation of dielectrics: the effect of energy accumulation between pulses,” Opt. Eng. 44(5), 051102 (2005). [CrossRef]
17. P. P. Pronko, S. K. Dutta, D. Du, and R. K. Singh, “Thermophysical effects in laser processing of materials with picosecond and femtosecond pulses,” J. Appl. Phys. 78(10), 6233 (1995). [CrossRef]
