Basic Properties of High-Dynamic Beam Shaping with Coherent Combining of High-Power Laser Beams for Materials Processing

Abstract

Lasers with average powers of several kilowatts have become an important tool for industrial applications. Temporal and spatial beam shaping was demonstrated to improve existing and enable novel applications. A very promising technology for both highly dynamic beam shaping and power scaling is the coherent combining of the beams of an array of high-power fundamental mode fibers. However, the limited number of fibers allows only limited spatial resolution of the common phase front. It is therefore favorable to work with plane or spherical common phase fronts, which generate a “point”, i.e., a diffraction pattern with a strong main lobe in the focal plane. By applying a tilt to the common phase front, points can be positioned in the focal plane with high spatial resolution. The Civan DBL 6–14 kW investigated in this work allows switching between positions of the points with 80 MHz. Sequences of points can be used to create arbitrary shapes. The time constants of points and shapes are very critical for this type of shape generation. The current paper analyzes the relevant time constants for setting points and creating shapes and relates them to time constants in laser processes. This is mandatory to deterministically influence laser processes.

1. Introduction

In the last decades, cw lasers with average powers of several kilowatts became an important tool in many industrial applications. The applications are taking advantage of efficient and reliable laser processes ranging from the hardening of materials over laser cutting and welding of virtually any material to additive manufacturing including directed energy deposition (DED), laser metal wire deposition, and powder bed fusion [1]. Unfortunately, the parameter window for reliable processes is often small. For example, the change of the feed rate strongly manifests itself in the change of the shape of the cutting front as reported in [2,3] or in the shape of the keyhole as seen in [4,5,6]. Therefore, actively influencing the process with temporal or spatial beam shaping has gained importance in the last few years. Beam shaping techniques such as wobbling, multi-spot beams, or ring-shaped intensity distributions allowed a significant reduction of process issues as described in [7,8,9,10]. Spatial and temporal shaping of the laser beam enabled very deep penetration welding [11] and welding of materials that are difficult to process such as copper [12]. Furthermore, beam shaping is an important tool to improve productivity and quality in laser powder bed fusion (LPBF) as is summarized in [13]. In-situ monitoring and modelling of the LPBF-process with shaped beams is given in [14]. A comprehensive overview of the methods and benefits of dynamic beam shaping including its use in industrial application can be found in [15]. However, these works were limited to specific processes and beam shaping technologies. Versatile and dynamic changes were not intended.

1.1. Ultra-Fast Dynamic Beam Lasers

A very promising technology for both realizing versatile, highly dynamic beam shaping and scaling of the average laser power is the coherent beam combining (CBC) of an array of collimated high-average power fundamental mode fiber lasers with optical phased array (OPA) technology. The active control of the optical phase of each fiber allows creating a combined beam with a common phase front with an arbitrary surface. CBC was already proposed to increase the average power of diode lasers in the 1980s, e.g., in [16]. The capability for power scaling of ultra-fast lasers was demonstrated since 2018 in [17,18]. Scaling of the average power of fundamental mode lasers was proposed in 2005 in [19]. In 2020, Civan presented the first cw high-average power CBC laser with OPA technology, called “Dynamic Beam Laser” (DBL) [20]. The stable and reliable operation of the DBL system was demonstrated, making it a promising new technology for industrial laser applications. Digital electro-optical controls allow the very fast adjustment of the overall optical phase with frequencies of up to 80 MHz. Recently, the scaling capability of this technology was demonstrated with a 100 kW DBL laser system [21]. This is the first time that highly dynamic CBC-laser systems at high average powers are available for industrial applications. The large additional number of laser materials processing parameters, which are introduced with dynamic CBC technology, makes deterministic beam shaping for improving processes a challenging task. The basic properties of high-dynamic beam shaping with the coherent combining of high-power laser beams for materials processing will therefore be described and for the first time analyzed in detail in the following.

1.2. Basic Properties of Dynamic CBC

An arbitrary, common phase front allows generating an arbitrary beam pattern in the focal plane of a focusing lens. In principle, any intensity pattern in the focal plane can be generated with freely formable phase fronts, as is realized, e.g., with spatial light modulators (SLM) in ultra-fast laser processing as described in [22]. Today, SLMs have about one million pixels per cm² of active area, allowing the phase front to be formed with very high spatial resolution. Unfortunately, CBC usually comprises only a few tens to a few hundred fibers. This limited number of coherently coupled fibers allows only limited spatial resolution of the common phase front. It is therefore advantageous to work with elementary plane or spherical common phase fronts. In the focus, such common phase fronts create a diffraction pattern with an almost diffraction limited main lobe in the center. The principle of CBC is sketched in Figure 1a. The beams of a bundle of N_x × N_y = N_Fibers fundamental mode fiber lasers are collimated and arranged to a beam array, representing a large, collimated virtual beam with a common phase front. The DBL 6–14 kW laser, which was used for this paper, features a 6 × 6 array of fundamental mode fibers, each with an M² ≅ 1.2, with a total maximum average power of 12 kW and a wavelength of λ = 1064 nm.

Figure 1b shows the resulting intensity distribution in the focal plane, when such a collimated, virtual beam with a plane common phase front, which is perpendicular to the beam propagation, is focused with a lens of focusing length f_Lens. The large main lobe in the center has the property of a fundamental mode beam. This main lobe is called “CBC-point” in the following. It contains about 60% of the total power of the fiber laser array. The side lobes are due to diffraction, which results from the square arrangement of the array of collimated fiber lasers and the gap between the fibers. The color scale of the normalized intensities of the laser beams in Figure 1a and the intensity distribution in the focal plane in Figure 1b ranges from 0 (blue) to 1 (red). It is noted that power scaling by increasing the number of fundamental mode fibers does not reduce the beam quality of the main lobe. The intensity distribution will be discussed in more detail in the following. The very fast control of the phase front with 80 MHz allows the fast creation of a point with a minimum time constant of 12.5 ns.

Because the average power of fundamental mode beams out of common laser architectures is either limited by thermal effects like in disk lasers or by non-linear effects in the fibers in fiber lasers [1]. CBC is the only technology that allows scaling of the average power of fundamental mode beams without physical limits.

By tilting the common phase front, the main lobe can be positioned in the focal plane. The tilt of the phase front can be adjusted with very high resolution, which is only limited by the accuracy of the relative phase shift generated in the electro-optic modulator, which is usually >8 bits. It is noted that by positioning the main lobe out of the center, the maximum intensity decreases at the expense of the side lobes. Applying a sequence of points at arbitrary positions allows creating arbitrary beam shapes. This technique requires considering the spatial overlapping of the diffraction patterns at the different positions of the main lobe. Furthermore, creating a shape as a sequence of positions of points results in clear dependencies of the defined time constants for the positioning of the points, their dwell time at one position, the build-up time of the complete shape, and the duration of one shape in a sequence of shapes.

In the following, the basic spatial and temporal properties of high-dynamic, high-power coherent laser beam combining are described in detail. Furthermore, the temporal properties of the resulting beam shapes are related to typical process time constants to give a guide on how to deterministically influence the laser process to improve existing processes and enable novel processing strategies.

1.3. Implication of Process Time Constants on Beam Shaping

Laser processes, which use the liquid phase of the material such as powder bed fusion, cutting, or keyhole welding, are highly dynamic, and usually typical time constants t_Proc with the corresponding typical frequencies f_Proc = 1/t_Proc are involved. The time constants of the process are determined, on the one hand, by convection and surface tension, as, for example, how long it takes to open or close a capillary or the emergence of pores. Recent work showed that the opening and closing time of a keyhole in steel is in the order of t_Proc = 1 ms [23]. In [24] it was seen in the keyhole welding of steel that the tip of the capillary could clearly follow an excitation with frequencies up to f_Proc = 100 Hz, i.e., within about t_Proc = 10 ms, while no clear correlation to the excitation was found for frequencies of f_Proc = 1 kHz and higher. In [25] it was measured that cutting off the tip of the keyhole, which might result in a pore, happened in about t_Proc = 4 μs. On the other hand, the time constants are determined by heat conduction, as for example during solidification of the liquid material. The thermal diffusion length [1] is an accurate physical quantity to describe the expansion of the temperature field into the material. The duration for a thermal diffusion length of 100 μm, which is in the order of a typical focus diameter, is about t_Proc = 100 μs in steel and about t_Proc = 20 μs in aluminum and copper.

Therefore, processing parameters should be well adjusted in both space and time for stable processes applying accurate and fast beam shaping. The process time constants divide any kind of beam shaping with CBC beams into the three distinct regimes “quasi-static beam shaping”, “dynamic beam shaping”, and “resonant beam shaping”.

(1)

Quasi-static beam shaping is achieved when the movement of the beam within a shape or a sequence of shapes is created in a time, which is much shorter than the relevant process time constant. In this case, the process cannot follow the movement of the laser beam or each single shape of the sequence and “sees” a temporally and spatially averaged beam shape. For the same example of keyhole welding of steel with a beam shape, which is created by a beam moving on a circle, if the circle is created in a time < 1 ms, the keyhole has about the width of the circle.

(2)

Dynamic beam shaping is achieved when the movement of the beam within a shape or a sequence of shapes takes place in a time that is much longer than the relevant process time constant. In this case, the process follows the movement of the laser beam or adapts to each shape of the sequence. For the example of keyhole welding of steel with a beam shape, which is created by a beam that is moving on a circle, if the circle is created in a time >1 ms, the width of the keyhole is about the width of the beam and follows the movement of the beam along the circumference of the circle. This situation is often called wobbling.

(3)

Resonant beam shaping is achieved when the movement of the beam within a shape or a sequence of shapes is created in a time, which is about the same as the relevant process time constant. In this case, the process might be excited by the movement of the laser beam or each single shape of the sequence.

Figure 2 basically illustrates the meaning of the three beam shaping regimes using the example of deep-penetration welding of steel 1.4301 with DBL beam shapes and a feedrate of 20 m/min. Rows (a) and (b) show the two triangle-like beam shapes involved. Rows (c–f) show single frames out of high-speed videos taken from the surface of the welded sample. The single frames were selected at the time steps noted below. The bright area represents the melt pool, and the dark triangle is the keyhole as marked in figure (f) of the top row. The triangle-like shapes were created within about 2 μs, i.e., in a time much shorter than 1 ms, which was the typical keyhole closing time measured for the arrangement used. In the top row (quasi-static) the shape was constantly applied. The very fast creation of the shape forces the keyhole to adapt to the average position of the laser beam, which is the triangle. This means that the process was influenced in the same manner as with a static shape. In the middle row (dynamic), the orientation of the triangle was switched from tip forward to tip backward, each 5 ms. This duration was clearly long enough that the keyhole had time to adapt to each orientation. The process can therefore be influenced by dynamically changing a quasi-static shape. In the bottom row (resonant), the triangles were switched every 0.5 ms, which was about the keyhole closing time. In the sequence of frames, it can be seen that the keyhole began to form without being able to fully adapt to the triangular shape. This resonant switching of the shapes, for example, strongly influenced the melt flow around the keyhole. A detailed explanation of the time constants can be found in Section 4.

These three temporal regimes of beam shaping should be kept in mind when the basic properties of CBC beam shaping are discussed in the following.

2. CBC Points

Figure 3 visualizes the principle of operation of a DBL laser. The set point, representing the center of the main lobe, can be set at an arbitrary position in the green grid. The calculated, corresponding CBC intensity distribution in the focal plane, i.e., the corresponding “CBC point”, is shown color-coded in the blue square, which is called the active area in the following. The active area contains >99.99% of the total laser power. In the example shown in Figure 3, the point was set at the origin of the coordinate system (marked with the gray arrow), yielding the intensity distribution shown in the blue active area, and corresponds to the intensity distribution shown in Figure 1b.

In addition, Figure 3 shows the corresponding coordinate systems with the origin in the center, the maximum for setting points x_set,max and y_set,max in the drawing area, and the maximum extensions x_active,max and y_active,max of the active area. The width of the active area is d_ActiveArea = 2 × x_active,max. Both the drawing grid and the active area are a square, and the behavior of the beam is identical in the x-direction and y-direction. Therefore, only the x-coordinate will be treated in the following. It is noted that x_active,max is about 2.75 times larger than x_set,max, both given by the focal length of the focusing lens, as will be explained later.

2.1. Intensity Distribution in the Focal Plane

For the simplest case of a set point set at x = y = 0 as shown in Figure 3, the resulting normalized CBC intensity distribution in the focus along the x-coordinate is shown in Figure 4. The intensity is normalized with the maximum intensity of the main lobe, I_max,0,0. The x-coordinate is normalized with the width of the active area, d_ActiveArea. It is noted that the intensity distribution is identical along the y-coordinate.

The main lobe has a width of d_MainLobe, contains about 60% of the total laser power, and has the properties of a fundamental mode beam. The height of the major side lobes is about 6% of I_max,0,0. The relative position of the side lobes is |x_MSL| = 0.36, which results from the definition of the active area, as described below.

For a collimated, almost fundamental mode beam with an M² = 1.2, the diameter of the main lobe can be approximated with [1]

$$d_{MainLobe}=1.27\times M^2\times {\lambda }_L\times {\displaystyle \frac{f_{Lens}}{D_{BL}}}$$

(1)

where λ_L is the wavelength of the laser, f_Lens is the focal length of the lens and the diameter of the beam on the focusing lens, and D_BL = D_CBC-beam is the diameter of the array of the collimated beams. For a diameter of D_CBC-beam = 18 mm of the collimated CBC beam, the diameter in μm of the main lobe can therefore be approximated with

$${\displaystyle \frac{d_{MainLobe}}{\mu m}}=91\times {\displaystyle \frac{f_{Lens}}{m}}$$

(2)

where f_Lens is the focal length of the focusing lens in meters. Assuming a Gaussian intensity distribution of this main lobe containing 60% of the total laser power, the maximum intensity in the center of the active area is about

$$I_{max,\mathrm{0,0}}={\displaystyle \frac{60\%·2\times P_{Laser}}{{\pi \times (0.5\times d_{MainLobe})}^2}}=1.5\times {\displaystyle \frac{P_{Laser}}{d_{MainLobe}^2}}$$

(3)

2.2. Setting Points at Arbitrary Positions

Figure 5 describes the change of the diffraction pattern, shown in the active area, and the position of the main lobe, when the set point is set out of the center of the coordinate system along the x-axis, shown in the drawing grid. This is achieved by adding an appropriate tilt to the common phase front. The top row shows the position of the set point (gray arrows) with the corresponding diffraction patterns in the active area. The bottom row shows the corresponding relative intensity along the normalized x-axis through the coordinate origin (blue lines). In (a), the point is set in the center of the coordinate system, as described above. When the point is set to the right of the origin as shown in (b), the side lobes move together with the main lobe. The intensity of side lobe on the right side decreases while the intensity in the side lobe on the left (yellow arrow) increases.

The maximum intensity of the lobes follows the dashed, orange envelope, which has a Gaussian shape. The 1/e² diameter of the envelope is given by Equation (1), when D_BL equals the diameter of the collimated beam of a single fiber. For a single beam diameter of d_SingleBeam = 2.6 mm, the 1/e² diameter can be approximated by

$${\displaystyle \frac{d_{Envelope}}{\mu m}}=610\times {\displaystyle \frac{f_{Lens}}{m}}$$

(4)

The width of the envelope can be used to define the width of the active area. In the present example, the width of the active area was arbitrarily chosen as

$${d_{ActiveArea}=1.55\times d}_{Envelope}$$

(5)

so that the active area contains > 99.99% of the total laser power. If the set point is set at x_set,max, as shown in Figure 5c, (marked with the gray arrow), the main lobe and the left side lobe have the same maximum intensity and are at the same absolute x-position. In fact, this defines the size of the drawing grid for the laser system considered here. Considering Equation (5) and the relative positions of the main side lobes of |x_MSL| = 0.36, the relative position x_set,max/d_ActiveArea = 0.18.

The maximum of the set points x_set,max = y_set,max in the drawing area, as well as the maximum extensions x_active,max = y_active,max of the active area, are therefore defined by the focal length of the focusing lens, f_Lens, and therefore also related to the diameter of the main lobe, d_MainLobe. Combining Equations (2), (4), and (5), the following Equations (6) and (7) allow calculating the respective absolute values in micrometers for a focal length given in meters.

$${\displaystyle \frac{x_{set,max}}{\mu m}}=172\times {\displaystyle \frac{f_{Lens}}{m}}=1.9\times {\displaystyle \frac{d_{MainLobe}}{\mu m}}$$

(6)

$${\displaystyle \frac{x_{active,max}}{\mu m}}=473\times {\displaystyle \frac{f_{Lens}}{m}}=5.2\times {\displaystyle \frac{d_{MainLobe}}{\mu m}}$$

(7)

Consequently, all geometric properties of the beam shape scale linearly with the used f_Lens. The maximum intensity at the position, where the main lobe and the side lobe have the same height, is then given by

$$I_{MainLobe}\left(x_{set,max}\right)=54\%\times I_{max,\mathrm{0,0}}$$

(8)

It is noted that the identical pattern is created if the point is set −x_set,max.

A point can be set at any arbitrary position in the green coordinate system by applying the appropriate tilt of the common phase front. Figure 6 shows the diffraction pattern in the active area for a point set at the position marked with the blue arrow. This results in the CBC point shown in the blue area (yellow arrow) with its corresponding intensity distribution.

Apart from the CBC point (top right in the active area), three clearly visible side lobes of different maximum intensity appear. Their maximum intensity and position depend on the x,y-position of the set point, x_set and y_set. Considering the Gaussian shape of the envelope and Equation (3), the peak intensity of the main lobe for an arbitrary set position can be approximated by

$$I_{MainLobe}(x_{set},y_{set})=1.5\times {\displaystyle \frac{P_{Laser}}{d_{MainLobe}^2}}\times e^{-{\displaystyle \frac{8\times (x_{set}^2+y_{set}^2)}{d_{Envelope}^2}}}$$

(9)

Table 1. Summary of the basic CBC beam properties for different focal lengths of the focusing lens and a CBC laser with an average power of 12 kW, a CBC beam diameter of 18 mm, and a single beam diameter of 2.7 mm.

fLens (m)	dMainLobe (μm)	Imax,0,0 (W/cm2)	xset,max (μm)	xactive,max (μm)	dActiveArea (μm)
0.75	68	3.9 × 108	130	356	712
1.0	91	2.2 × 108	173	474	949
1.5	136	9.8 × 107	259	712	1423
3.0	273	2.5 × 107	518	1423	2847

summarizes the values for d_MainLobe, I_max,0,0, x_set,max, x_active,max, and d_ActiveArea for a few typical f_Lens for a laser with 12 kW of average power.

The choice of the focal length of the focusing lens is crucial when designing the optical layout for using a CBC laser with a given application.

On the one hand, the focal length defines the size of the active area. Within this area, the process is influenced by the deposited laser energy, dependent on the positions of the points defined in the drawing area. On the other hand, the focal length defines the maximum intensity, which is achievable with a given laser power. The threshold for laser processes is, depending on the feedrate, correlated to the intensity of the beam on the surface of the workpiece, such as, for example, the threshold for the generation of a capillary for deep-penetration welding [1]. Therefore, great care must be taken that the intensity in the main lobe exceeds the threshold by the desired amount but the intensity in the side lobes is below the threshold or that the intensity in the side lobes is high enough if they should contribute to the process.

3. A Sequence of CBC Points Results in a Shape

3.1. The Principle of Shapes

As mentioned before, the basic principle of a CBC system with a relatively low number of coupled fiber lasers is to create one CBC point at a time. Therefore, every shape must be composed as a sequence of CBC points. This means that one point after another is set in the drawing grid. During the process, the laser system addresses these points in the same sequence as they were set. Therefore, each shape is created with of a total number of N_SP set points, which can be set at N_Pos different positions. A set point at the i-th position can be set more than once, i.e., N_C,i times (N_C,i ≥ 1), which means that at this position the identical diffraction pattern will be generated N_C,i-times. In this case, N_SP and N_Pos become different. The impact of this is explained below. The total number of points in a shape, N_SP, is therefore given by

$$N_{SP}={\sum }_{i=1}^{N_{Pos}}{N}_{C,i}$$

(10)

Figure 7 shows an example for a square shape with eleven set points (marked with gray arrows) at nine different positions (N_SP = 11 and N_Pos = 9). The position in the center (at i = 5 where x = y = 0) was set three times. The meaning of the diffraction pattern displayed in the active area is explained in Section 4.

The sequence of the points must be considered when designing processes. The current sequence is marked by the red arrow. The clever selection of the set points and considering the side lobes in the diffraction pattern allows to create virtually any shape.

3.2. Sequences of Shapes

The principle of operation described above allows to define sequences of arbitrary shapes. For example, such a sequence could consist of the shapes shown in Figure 8. The false color representation is normalized with the maximum value achieved in the active area and ranges from 0 (blue) to 1 (red).

Sequences of shapes are very promising for dynamically influencing the process with different shapes because each shape might have a completely different influence on process parameters such as keyhole size and keyhole shape, temperature gradients, or melt flow velocities.

3.3. Depth of Focus and Active Focus Steering in z-Direction

In the direction of beam propagation (z-coordinate), the depth of focus ∆z_Focus is about plus and minus the Rayleigh length of the main lobe, z_R,ML, which is given by [1]

$$z_{R,ML}={\displaystyle \frac{\pi \times {\left(0.5\times d_{MainLobe}\right)}^2}{M^2\times {\lambda }_L}}$$

(11)

Combining Equations (2) and (10), and with M² = 1.2 and λ_L = 1.064 μm, ∆z_Focus in mm can be approximated by

$${\displaystyle \frac{\Delta z_{Focus}}{mm}}=\pm 5.1\times {\left({\displaystyle \frac{f_{Lens}}{m}}\right)}^2$$

(12)

Furthermore, an additional adjustable spherical profile on the common phase front allows the active steering of the position of the focus in z-direction over a range, over which the peak of the main lobe in the center of the active area is ≥90% of I_max,0,0. The focus steering range is therefore defined to be

$$\Delta z_{Steering}=\pm {\displaystyle \frac{1}{e}}\times z_{R,ML}\times {\displaystyle \frac{d_{Envelope}}{d_{MainLobe}}}$$

(13)

For the diameter of the CBC beam of D_CBC-beam = 18 mm and of the collimated beam of a single fiber of D_SingleFibre = 2.7 mm with M² = 1.2 and λ_L = 1.064 μm, an active focus steering range ∆z_Steering is achieved, which is about

$${\displaystyle \frac{\Delta z_{Steering}}{mm}}=\pm 12.8\times {\left({\displaystyle \frac{f_{Lens}}{m}}\right)}^2$$

(14)

Table 2. Summary of the depth of focus and the active steering range ∆z_Steering for the DBL 6–14 kW laser for a few typical focal lengths.

fLens (m)	Depth of Focus ∆zFocus (mm)	Active Focus Steering ∆zSteering (mm)
0.75	±2.9	±7.2
1.0	±5.1	±12.8
1.5	±11.5	±28.9
3.0	±45.9	±115.4

gives a summary of the depth of focus and the active steering range ∆z_Steering for a few typical focal lengths for the DBL 6–14 kW laser, as described with Equations (12) and (14).

4. Time Constants

It is important to understand that—due to the principle of operation of a CBC system as discussed here—the shapes, which are shown in Figure 8, do not actually exist. Each shape shown in the active area represents the summed-up values of the energy per unit area of all set points of the shape. To determine the energy per unit area, it is necessary to define the dwell time, t_dwell, at each position, taking into account the complete diffraction pattern for the respective CBC point. Furthermore, this is where the process time constants described in the introduction must be considered to decide whether the resulting beam shaping is quasi-static, dynamic, or resonant. The different beam-shaping time constants are therefore critical and will be discussed in detail in the following.

The principle of a CBC system as discussed in this paper requires us to define a few decisive durations or frequencies. This includes both the timing of the re-positioning of the CBC points and the change between shapes in a sequence of shapes. In the present DBL example, this was performed by defining the total number of set positions, N_Pos, the total number of set points, N_SP, in the shape, the shape refresh frequency, f_SR, and the shape duration, t_SD. As mentioned above, the shortest possible time for the re-positioning of a point is t_SP,min = 12.5 ns for the laser investigated here. Figure 9 shows an example shape with the corresponding parameters.

These four shape parameter values define all frequencies and time constants involved in CBC dynamic beam shaping.

4.1. Number of Set Positions and Number of Set Points

The example shape in Figure 9 consists of N_SP = 11 set points at N_Pos = 9 positions. The distinction between N_Pos and N_SP is necessary because the generation of CBC points takes place at a fixed frequency. Therefore, increasing the number of set points at one position (N_C,i ≥ 1) allows us to increase the dwell time at this position (see next section).

4.2. Shape Refresh Frequency

The shape refresh frequency f_SR = 500 kHz defines with what frequency the complete shape is refreshed. This means that it takes the shape refresh time t_SR = 1/f_SR = 2 μs to generate all N_SP points of the shape. The lifetime of one single point, t_SP = 181.8 ns, within the shape results from the shape time and the total number of set points t_SP = t_SR/N_SP.

The dwell time t_dwell,i at the ith position is increased by generating the point N_C,I times, i.e., t_Dwell,i = N_C,i × t_SP. Therefore, t_Dwell,i = t_SP = 181.8 ns for i ≠ 5 and t_Dwell,5 = 545.4 ns. Increasing the dwell time means that the locally deposited energy per unit area, Φ_i(x,y), in the active area is increased N_C,I times for the i-th intensity distribution I_i(x,y). This increase applies for the complete active area, i.e., for the main lobe and all side lobes.

$${\Phi }_i(x,y)=N_{C,i}\times t_{SP}\times I_i(x,y)=t_{Dwell,i}\times I_i(x,y)$$

(15)

It is noted again that increasing N_C,i does only increase the locally deposited energy per unit area but not the local intensity.

The shapes shown in Figure 7, Figure 8 and Figure 9 represent the sum over all set positions, N_Pos, of the energy per unit area deposited in the active area, given by

$$\Phi (x,y)={\sum }_{i=1}^{N_{Pos}}{\Phi }_i(x,y)$$

(16)

It is noted that Equation (16) just applies if the feedrate v_F is low enough that the diffraction pattern does not move more than about 10% of the diameter of the main lobe during the shape refresh time, i.e., v_F < 0.1^.d_MainLobe/t_SR.

4.3. Shape Duration

The shape duration t_SD = 10 ms defines how long the shape is persistent until the next—or the same—shape is applied. The shape duration frequency is therefore f_SD = 1/t_SD = 100 Hz. During the shape duration, the shape is drawn N_S = 5000 times, where N_S = t_SD ^. f_SR. It is recommended to choose the above numbers so that N_S becomes an integer to guarantee that the shape is drawn completely until the end of the shape duration.

Each shape in a sequence of shapes has its own effective total number of positions, number of set points, shape frequency, and shape duration. This means that the dwell time is given for all positions within one shape, but it is adapted for each shape within a sequence, according to the respective number of positions, points, and the respective shape frequency. The total time, t_Seq, for completing a sequence of N_Shapes is the sum of the durations of each shape given by

$$t_{Seq}={\sum }_{j=1}^{N_{Shapes}}{t}_{S{D}_j}$$

(17)

It is noted that all frequencies should in principle be integer divisors of the maximum OPA modulation frequency (80 MHz in this example), and therefore all durations should be integer multiples of the shortest possible duration of a point (12.5 ns). This means that the odd number of set points chosen for the example in Figure 9 together with the values for the shape refresh frequency results in a maximum deviation of 6.8 ns to the perfect synchronization.

The duration of each shape and the total time for the complete sequence must be synchronized with the process, e.g., with the position on the workpiece and the total processing time. This is process specific as well as application specific and must be carefully performed by the user of such a laser to achieve deterministic process results.

5. Summary of the Relevant Shape Parameters

The relevant time constants and frequencies, which result from the system constraints and from the user-defined settings, are summarized in this section.

5.1. System Constraints

In

Table 3. Summary of the decisive system constraints for CBC lasers. The values for the limits apply to the example laser in this paper. The arrow at the item indicates that this is a calculated value.

Item	Symbol	Example Limits DBL 6–14 kW
Maximum OPA modulation frequency	fOPA,max	80 MHz
⇨ Shortest possible lifetime of one point	tSP,min = 1/fOPA,max	12.5 ns
Maximum total number of points in a shape	NSP,max	1024
Maximum number of shapes in a sequence	NShapes,max	14

, the decisive system constraints for CBC lasers are given. The limiting values in the column to the right apply to the DBL 6–14 kW laser used for this work.

5.2. Definable and Resulting Shape Parameters

Table 4. Summary of the defined (bold) and resulting (italic) time constants and frequencies. The values in the right column refer to the example given in Figure 9. All positions were set one time except for the position i = 5 at x = y = 0, which was set three times. An arrow at the item indicates that this is a calculated value.

Item	Symbol	Example 7
Total number of positions in the shape	NPos	9
Number of set points at the ith position	NC,i	1 (i ≠ 5) 3 (i = 5)
Total number of points in the shape	$N_{SP}={\sum }_{i=1}^{N_{Pos}}{N}_{C,i}$	11
Shape refresh frequency	fSR	500 kHz
⇨ Duration for drawing the complete shape	tSR = 1/fSR	2 μs
⇨ Lifetime of one single CBC point	tSP = tSR/NSP	181.8 ns
⇨ Point frequency in the shape	fP = 1/tSP = NSP/tSR = NSP × fSR	5.5 MHz
⇨ Dwell time at the ith position	tDwell,i = NC,i × tSP	181.8 ns (i ≠ 5) 545.4 ns (i = 5)
Duration of the shape (persistence)	tSD	10 ms
⇨ Shape duration frequency	fSD = 1/tSD	100 Hz
⇨ The shape is drawn NS times per duration	NS = tSD × fSR	5000

gives a summary of the definable and resulting time constants and frequencies for a shape. The example values in the right column refer to the example given in Figure 9. In this example, all positions were set one time except for the position i = 5 at x = y = 0, which was set three times. The user-defined values are written in bold; the resulting values are written italic.

It is mandatory that all the values are checked and compared to the limiting system values when designing new shapes and sequences for novel processes and applications.

5.3. Comparison with Process Time Constants

The values resulting from the definition of the parameters of the shape must be compared with the relevant time constants to determine the process regime of the planned process, as described in the Introduction. The comparison is summarized in

Table 5. Meaning of the defined and resulting time constants and frequencies with respect to quasi-static, dynamic, and resonant beam shaping, as defined in the Introduction. The references (a) and (b) refer to the example described in the text.

Quasi-Static Beam Shaping
Shape Parameter	Frequency	Time	Resulting Effect in the Process
Shape refresh	fSR ≫ fProc	tSR ≪ tProc (a)	The process adapts to the shape (required for each shape in a sequence)
Shape duration	fSD ≫ fProc	tSD ≪ tProc	The process adapts to an average of all shapes in the sequence (“average shape”)
Dynamic beam shaping
Shape parameter	Frequency	Time	Resulting effect in the process
Shape refresh	fSR ≪ fProc	tSR ≫ tProc	The process follows the diffraction pattern (“moving beam”, “wobbling”)
Shape duration	fSD ≪ fProc	tSD ≫ tProc (b)	The process adapts to each shape in a sequence (the shapes must be quasi-static)
Resonant beam shaping
Shape parameter	Frequency	Time	Resulting effect in the process
Shape refresh	fSR ≈ fProc	tSR ≈ tProc	The beam excites the process resonantly (“resonant stirring”)
Shape duration	fSD ≈ fProc	tSD ≈ tProc	The shape sequence excites the process resonantly (“resonant shaking”)

.

The example given in Figure 2 in the Introduction and the following example illustrate the meaning of Table 5. In [23,24], a typical process time constant of t_Proc = 1 ms for the opening and closing of the keyhole in steel was reported. This time constant was compared with the parameters taken for the shape shown in Figure 9.

• The shape refresh time of t_SR = 2 μs is much shorter than the process time constant of t_Proc = 1 ms (t_SR ≪ t_Proc^(a)). Therefore, this is a quasi-static beam shape. The keyhole cannot follow the movement of the CBC point and adapts its shape to the beam shape.
• The shape duration of t_SD =10 ms is much larger than the process time constant of t_Proc = 1 ms (t_SD ≫ t_Proc^(b)). This means that the process has time to dynamically adapt to each shape in the sequence of quasi-static shapes.

Conversely, taking into account the restrictions and dependencies summarized in Table 3, Table 4 and Table 5 allows the correct shape parameters to be determined if a specific effect on the process is to be achieved.

6. Conclusions

Coherent beam combining with OPA technology adds an enormous number of degrees of freedom to laser processing and applications. However, due to the limited number of combined fiber lasers, it is advantageous to create one CBC point at a time and to create shapes with a sequence of CBC points. Designing dynamic beam shapes to optimize processes therefore requires careful consideration of all shape parameters. The resulting temporal and spatial properties of both the sequence of CBC points and the created shapes must be related to the temporal and spatial process requirements, particularly the local intensity on the surface of the sample, the locally deposited energy, and the process time constants. Typical values for many laser processes can be found in textbooks, such as, e.g., in [1].

Careful selection of the spatial and temporal settings of the beam shaping parameters and comparison with the process requirements guarantees a successful and deterministic process design using high-dynamic beam shaping. An example for successful processing with a sequence of shapes with a DBL is presented in reference [26].

A detailed treatment of beam shaping with DBL lasers for industrial applications, including experimental work, is therefore very important and planned for future publications. This will include concepts for future improvements regarding both CBC technology and process. An important feature for CBC lasers would be to be able to manage the intensity of the side lobes, which might be achieved with the dynamic spatial filtering of the beam. Furthermore, the active control of processes with dynamic beam lasers, including online AI optimization, would bring laser processing to a new level.