Accounting for laser beam characteristics in the design of freeform optics for laser material processing

2.1 Solution of the inverse heat conduction problem

A laser beam that is moving at constant feed speed v→ over the surface of a workpiece is inducing a temperature profile T that will after a while reach a steady state. This steady-state temperature profile can be calculated using the heat conduction equation [18]:

−∇(k(T)∇T)=s−v→ρ(T)cP(T)∇T

where k(T) refers to the heat conductivity, ρ(T) refers to the mass density, and c_P(T) refers to the specific heat at constant pressure. The absorbed intensity I is either included in the source term s (for volume absorption) or as surface flux in a Neumann boundary condition of the form −k n→ ∇T=A⋅I (for surface absorption) with the boundary normal n→ and the absorptivity A. Additionally, boundary conditions need to be specified for every (other) boundary of the workpiece.

This is the usual approach for calculating the temperature profile for a given intensity distribution, which in the given context is called the solution of the “direct” heat conduction problem. The corresponding “inverse” heat conduction problem is then defined as follows [19]: given the information of the temperature profile, the intensity distribution must be reconstructed. Although the direct heat conduction problem can be solved with standard solution strategies for partial differential equations [finite difference method (FDM) and finite element method (FEM)], the inverse heat conduction problem is known to be ill-posed and special regularization strategies must be implemented.

The presented solution strategy for the inverse heat conduction problem is based on the conjugate gradient method with adjoint problem [20], [21].

In this approach, the temperature is assumed to be prescribed on a set of N points, which are located somewhere within the workpiece or on its surface. It is then possible to qualify a given intensity distribution by comparing, at these points, the temperature it induces with the prescribed temperature through calculating the least-squares sum

S=∑i=1N(Yi−T(r→i))2

Here, Y_i is the prescribed temperature at point r→i and T(r→i) is the calculated one.

This least-squares sum is minimized iteratively. In each iteration, the intensity distribution is updated according to the instruction

In+1=In−βndn

Here, dⁿ is the actual update direction that is given as a combination of the previous update direction and the steepest decent direction given by the gradient of S with respect to the intensity distribution, i.e. ∇S_I.

dn=γndn−1+∇SIn

βⁿ and γⁿ are real numbers that are called step size and conjugation coefficient, respectively.

The computation of the different quantities – ∇SIn,βⁿ, γⁿ – is rather comprehensive and includes the solution of two further partial differential equations: the adjoint problem and the sensitivity problem.

In the conjugate gradient method with adjoint problem, a regularized solution is obtained by limiting the number of iterations. See Refs. [7], [8] for more information.

2.2 Spot size optimization

This algorithm, as described above, is applicable in a situation where the intensity distribution’s dimensions are well specified. However, for great dimensions, it might yield results that are not realizable as the total power needed is higher than the available maximum laser power. In such a case, the intensity’s dimensions must be decreased and the prescribed temperature profile must be adapted where necessary. Furthermore, in cases where the temperature profile is prescribed through a temperature-time-development, the feed speed must also be adjusted.

Here, it is suggested to adapt the intensity distribution’s dimensions iteratively according to

Dk=Dk−1⋅min(0.99,c⋅PmaxPactual)=Dk−1⋅f

where D is the size in the x- and y-directions, respectively, P_max is the maximum available laser power, P_actual is the needed power in the last iteration, and c is a factor (usually chosen between 1 and 2). This factor c accounts for the fact that the needed power does not scale linearly with the extent and shall slow down the convergence.

In the implementation, the spot size reduction is performed for an equally spaced intensity with a previously defined discretization size δ. Thus, D^k is chosen as

Dk=δ⋅floor (nDk−1⋅f−1)

with nDk−1 being the number of points in the previous iteration.

In cases where the temperature profile is prescribed as “within the irradiated area”, the points with prescribed temperature values can be adjusted via

pk=pk−1⋅Dk−1Dk

where, for p, the points’ x and y-coordinates must be inserted. An analogous relation holds for a possible adaptation of the feed speed.

2.3 Freeform optics design strategy

The freeform design algorithm [15] is based on calculating the intensity distribution generated by a given freeform optics at a target point (x_t, y_t) on the target surface by integrating the incoming light [11]:

E(xt, yt)=∫L(xt, yt, ϕt, θt)cosθtdΩt=∫L(xt, yt, ϕt, θt)cosθtsinθtdθtdϕt

Here, E(x_t, y_t) is the intensity at position (x_t, y_t) and L(x_t, y_t, ϕ_t, θ_t) is the radiance at position (x_t, y_t) propagating into the direction given by (ϕ_t, θ_t). The index t refers to coordinates in the target plane.

The integrals over the solid angles are then approximated by sums and a single ray is associated with every single entry. As the radiance stays constant, when a ray propagates through any optical systems (as long as Fresnel losses are neglected), it is then possible to insert the ray’s radiance in the source plane, i.e.

E(xt, yt)=∑j=1NraysL(xt,j, yt,j, ϕt,j, θt,j)cosθt,jsinθt,jΔθt,jΔϕt,j=∑j=1NraysL(xs,j, ys,j, ϕs,j, θs,j)cosθt,jsinθt,jΔθt,jΔϕt,j

Thus, the intensity on the target plane can be calculated by considering rays emitted from the target plane and traced backward through the system. If they intersect the source, the radiance is evaluated and the according term is added to the sum.

In this setup, the optical surfaces are assumed to be B-splines that have several advantages; the most important one is that they enable a continuous parameterization. The degrees of freedom are currently given by the control points’ z-components.

This calculation of the intensity distribution is then combined with a standard optimization strategy, which is, as before, based on a least-squares sum. This time, the current intensity distribution is compared to the target intensity distribution on a set of target points.

S=∑i=0Npoints(Eactual(xt,i, yt,i)−Etarget(xt,i, yt,i))2

2.4 Stabilization

This algorithm, as described in Ref. [15], yields freeform optics that are in principle satisfactory. However, the surfaces of the freeform optics exhibit many oscillations that are due to numerical issues. Furthermore, irregularities also appear in the obtained target intensity distribution, which arise from the impossibility to obtain sharp transitions between darker and brighter areas. On the one hand, this holds for edges at the boundary of the target; on the other hand, it also influences edges (i.e. sharp lines) in the inner part of the target distribution.

Therefore, the algorithm is extended to stabilize it. There are two main extensions:

a. Refinement: Some refinement strategy has been implemented that iteratively increases the number of degrees of freedom of the freeform surfaces. At the beginning, an initial design is calculated with only ≤100 degrees of freedom. When the optimization converges, the number of degrees of freedom is increased by a factor that usually lies between 1 and 4. With this, the optics is recalculated and the procedure is repeated several (usually up to five) times leading to ≥1000 degrees of freedom in the final design.

b. Smoothing of target distribution: In the second extension, the target intensity distribution is adapted to account for the finite étendue of the laser beam. This assures that the target intensity distribution is achievable, at least in principle.

The following considerations are based on the concept of étendue, which is defined differentially as [22]

dU=n2dA cos θ dΩ

Here, a ray bundle with solid angle dΩ is considered to pass through a differential surface area dA, whereas the angle between the bundle’s propagation direction and the surface normal is given by θ. The surrounding material is supposed to have the index of refraction given by n. The étendue can thus be depicted as a measure for the light’s extension in space and angle, i.e. the volume it occupies in phase-space.

It has been proven that the étendue stays constant when the light propagates through an optical system (no scattering assumed). Thus, there will also be some impact on the achievable target intensity distribution, which especially results in the broadening of sharp edges. An estimation for the minimum width of any edge can be obtained by equalizing the étendue of the source to that in the target plane. Supposing that both source and target are surrounded by air, it holds

Us=Ut ⇔∫​dAs cos θsdΩs=∫​dAt cos θtdΩt 

As only an approximation for the width of the edge shall be provided, it is possible to assume that the emittance angle of a ray does not depend on its position on the source and that, furthermore, the maximum angle under which rays impact on the target plane does not depend on the target position. It is then possible to derive

As(sin θs,max)2=At(sin θt,max)2

where θ_S,max is the source divergence angle, A_s is the source size, and θ_t,max is the maximum angle under which rays impact on the target plane. Then, A_t is the minimal area on which the light can be concentrated. The minimal width of the edge can thus be determined by calculating

dt≈sin θs,maxsin θt,maxAs

Before starting the optimization, the target intensity distribution is smoothed until every edge is approximately broadened to that size. With this, the algorithm proves to converge more reliably.

3.1 Derivation of the intensity distribution

As stated in Section 1, the aim is to calculate an intensity distribution for a laser thin-film processing application that induces a stepped temperature distribution of 400°C and 1500°C, respectively, within the (rectangular) area irradiated by the laser beam. To this end, a steel sheet with dimensions of 4×4×1 mm³ is assumed over whose z-surface the laser beam is initially moving with ν_y=−500 mm/s. As the penetration depth of light in the visible and infrared wavelength range in steel is very low, the absorption of the laser radiation is simulated as a surface flux, i.e. through a Neumann boundary condition. Furthermore, the boundary condition T=20°C is applied to the boundary at y=−2 mm and the other boundaries are supposed to be adiabatic.

Start dimensions of 2×2 mm² are chosen for the intensity distribution and the procedure from Section 2.2 is applied to compute an intensity distribution for which the total power lies below 800 W. Additionally, as the prescribed temperature profile is given as a temperature-time-profile, the feed speed is adapted to the spot size so that the total irradiance time stays constant.

This approach then yields an intensity distribution with an extent of 1.2×1.2 mm², as shown in Figure 1. The total power is 778 W, which lies well below 800 W. It consists for each temperature step of three thin lines with high-intensity values at the edges of the corresponding area. The rest of the intensity distribution is rather flat.

Figure 1:

Calculated intensity distribution.

There are small oscillations visible at the sharp edges of the intensity distribution, which are due to mapping issues of the N points with prescribed temperature to the FDM mesh vertices. However, they can be neglected as the intensity distribution will be smoothed in Section 3.2 anyway.

The temperature distribution that is calculated (for a smaller volume) using this intensity distribution is shown in Figure 2. The two different temperature steps are clearly distinguishable and well-defined and the temperature is very homogeneous in each step. This temperature distribution has been obtained using the commercial FEM software ANSYS [23] and thus provides a validation for the computed intensity distribution.

Figure 2:

Temperature profile that has been obtained with the intensity distribution in Figure 1.

3.2 Freeform optics design

The intensity distribution shown in Figure 1 has a rather small extent and very sharp features. The geometry that is assumed for the experimental setup is depicted in Figure 3. The laser beam with a super-Gaussian-like intensity distribution (it is taken from measurements) is incident on a freeform mirror, which changes the propagation direction by 90°. The working distance is assumed to be 100 mm. Figure 3 also shows the measured input intensity and the (smoothed) target intensity.

Figure 3:

Geometrical setup for the freeform optics design showing also the input and prescribed intensity distributions.

According to the approximate relation given in Section 2.4, the minimum achievable edge width can be estimated to be 0.3 mm. As this is a significant fraction of the intensity’s general size, it is not possible to neglect effects due to the finite divergence angle. Thus, it is not possible to apply an algorithm assuming zero étendue light.

Therefore, the algorithm described in Section 2.3 is implemented and the target intensity is smoothed as shown in Figure 3. As an initial guess, a parabolic surface is assumed with 49 degrees of freedom. The target intensity distribution is prescribed on 151×151 equally spaced points on the target plane. The source intensity measurement is interpolated on 400×400 positions and the angular spectrum of the source is assumed to be Gaussian with a divergence angle of 2.44 mrad. The freeform mirror is refined three times leading to 1225 degrees of freedom in the final setup.

The obtained intensity distribution is simulated using the ray tracing software ZEMAX [24] and the result is shown in Figure 4. It is very close to the prescribed (smoothed) intensity distribution in Figure 3, although there is, of course, some deviation to the initial intensity distribution in Figure 1 due to the finite étendue and the smoothing.

Figure 4:

Obtained intensity distribution.

The mirror design is shown in Figure 5 where the small overall curvature arises as a light beam with a diameter 26 mm is focused to a much smaller extent. It is furthermore obvious that the surface and the achieved intensity are quite smooth and that they show much less irregularities and oscillations as for similar situations in our previous publication [15].

Figure 5:

Freeform optics design scaled by a factor of 10 in the direction of the overall surface normal.

As a final check, it is possible to evaluate the temperature profile that is obtained with the intensity distribution in Figure 4. Figure 6 shows the result. It is obvious that the effect of the finite divergence angle is recognizable and that there is a difference between the temperature distributions in Figures 2 and 6, respectively. Most prominently, the sharp edges at the boundary of the laser spot as well as the edge between the lower and the upper temperature step are broadened. This had to be expected as the intensity distribution is broadened likewise by the finite étendue. However, the temperature profile is supposed to agree sufficiently with the requirement, as the different temperature steps are clearly distinguishable.

Figure 6:

Calculated temperature profile for the intensity distribution shown in Figure 4.