Process Optimization of Wire-Based Laser Metal Deposition of Titanium

Titanium is an astonishing material, which has a lot of outstanding properties. The strength-to-density ratio is outstanding, and due to its affinity to oxygen, it develops a passivating layer. This layer makes it resistant against most corrosive environments and it can be used in aggressive atmospheres. Thereby, titanium meets all requirements for being an ideal material for several turbomachinery components [1,2]. In aero-engines, it is mainly used in the compressor part as it is in the TFE 731 or the CFM56-5C2 [3]. Additionally, the jet nozzle and parts of the case are made out of titanium [3].

In aerospace engineering, the high toughness and low density are the main reasons for the usage of titanium alloys. Therefore, the parts have an extreme Buy-to-Fly ratio up to 95% in the worst case, depending on the part's complexity. Because of the high strength, resistance and low thermal conductivity, the machining of titanium is often laborious and expensive.

To overcome especially the limitations of the classic, conventional subtractive process chain, a different approach might be taken. Therefore, the Fraunhofer IPT introduced an alternative process chain, where a near-net-shape part can be produced by additive manufacturing (AM). Afterward, this part can be postprocessed even by water jet milling or surfaced finished using electrochemical machining [4].

Additionally, using AM processes to repair turbine blades is also possible. This perspective gains a high interest, taking into account that by now repair is a major factor on the turbomachinery market [5].

The following paper will focus on the improvement of the near net-shape-process as the crucial first step in an alternative production chain.

Additive manufacturing processes by melting material directly onto the work piece are for instance used to generate near-net-shape parts. As a form of energy supply for these kinds of processes, a laser beam, an electronic beam, or also an electric arc is suitable [6].

When processing titanium alloys by additive manufacturing, three major challenges have to be coped: oxidation, distortion, and the adjustment of the microstructure.

The high reactivity of titanium leads to a reaction of the material with oxygen. Therefore, titanium can burn or even explode, especially when the surface-to-volume ratio is large [2]. Besides, just a few ppm amount of oxygen, hydrogen, or nitrogen will cause a significant deformation of the crystal structure. This causes a decrease of the properties of the material [1,7]. Oxygen has a solubility of 30 wt %, but already 0.05 wt % is sufficient in pure titanium to increase the brittleness [8]. The phenomenon is more pronounced by oxygen being dissolved in the melt pool. However, temperatures of 200 °C are already sufficient to dissolve oxygen in solid titanium. According to the DVS (German welding society), oxidation at temperatures above 450 °C has no tolerable effects [9]. Therefore, for laser metal deposition (LMD), the melt pool as well as the heated titanium material has to be covered with shielding gas.

The low thermal conductivity of titanium alloys combined with the thermal expansion coefficient causes high inner tensions between the substrate and the bead. The high melting point of the material enforces this challenge. Additionally, the high-temperature strength avoids the relieving of the tensions during the cooling down phase. These inner tensions cause a distortion of the part. A hardening effect based on lattice distortion occurs, which can be compared with martensitic hardening of steel, enforcing the inner tensions.

The last challenge is the adjustment of the microstructure. The microstructure depends on the thermomechanical history of the process. Titanium alloys have two different phases: the alpha phase (hexagonal) and the beta phase (body-centered cubic). The hexagonal structure of the alpha phase is anisotropic. The plastic deformability is higher in the alpha phase than in the beta phase. However, the ductility of the alpha phase as well as the total strength is lower. Duplex alloys like Ti6Al4V allow an adaption of material properties in a wide range by adjusting the phase type and the microstructure through thermal treatment [1,2]. Ti6Al4V, called titanium grade 5, is by far the most used titanium alloy for industrial applications [2]. More than 50% of the industrial used titanium is made from this alloy [2].

Therefore, titanium alloys are challenging for all additive manufacturing techniques. The most advanced technique nowadays is selective laser melting. It is the AM process with the highest accuracy and process stability on the market. Airbus showed that it is possible to reduce part weight about 20%, if these parts were realized by selective laser melting due to an optimized geometry.¹ Instead of a laser beam, Arcam uses an electron beam in its powder-based AM system for a selective material melting process. The vacuum inside the working chamber is a general necessity for the electron beam process, but also prevents oxidation [10].

In comparison with buildup parts out of a powder bed, a different class of AM processes exists. For these processes, the material is not preplaced in a separated process step, but instead is fed directly into the energy beam, e.g., the laser or electron beam, and into the melting pool. The material is typically provided in wire or powder form. Feeding the material into the melting pool gives the process the ability to be used on three-dimensional substrates and complex shape geometries. This property allows the usage for repair applications. Therefore, the LMD process was chosen as the most promising additive manufacturing technique. It was decided to use the wire-based LMD process for reasons explained in the state of the art.

For wire as well as powder, laser is the preferred energy source due to the high accuracy of the heat input and the controllability. For the manufacturing of larger parts made out of titanium alloys, Norsk Titanium has reached a technological readiness level 8 for a plasma based process [11]. In comparison with that approach Waam uses an electronic arc welding process [12]. Both investigations showed that the toughness level as well as the fatigue test is comparable with normally processed titanium. Sciaky realized an additive manufacturing process where titanium alloys can be processed using an electron beam and wire. It is reported that this process is used by Lockheed Martin [13].

These processes have been already integrated in separate machines. Inside these machines, the process chamber is usually filled with argon or there is a vacuum as it is necessary for the electron beam process. While the inert environment prevents oxidation, there are several disadvantages. For instance, the flooding or vacuuming of the chamber is time and energy consuming. The setup time also increases drastically. If technical problems occur during additive manufacturing, switching to a milling process to recondition the surface will also be time consuming. To avoid these issues and enhance the ability to integrate additive processes into subtractive process chains, hybrid machines exist.

Examples for hybrid machines are the LASERTEC 65 (DMG-Mori), Hybrid HSTM 1000 (Hamuel Reichenbach Mill-Turn), Integrex i-400AM (Mazak), Replicator and WFL Millturn Technologies [14]. Also, for more than 15 years, the controlled metal buildup system was built up by the Fraunhofer IPT, where a wire-based LMD process is combined with a milling process [15].

Laser metal deposition can be performed with the use of powder, wire, or both [16]. A disadvantage using wire is the physical bonding between the melting pool and the machine system through the wire. The machine system has to be accurate to prevent collision with the substrate. Especially the process stability is challenging [17]. The wire is usually fed into the laser beam laterally. Alternatively, systems exist where the wire is perpendicular to the work piece while one or several laser beams are directed on the melting pool sideways. This configuration is nearly symmetric, making the process independent from any orientation. As a disadvantage, the size of the melt pool usually is bigger, causing a larger heat-affected zone. As will be discussed later, this will make local shielding more difficult or can even prevent it.

To enlarge the process stability while using wire, Mok et al. investigated different parameter sets. They changed the laser power, wire speed, and wire feed orientation. To assure that the wire will be melted, they used a large laser spot of 4 × 7 mm² [18,19]. To avoid oxidation, the process chamber was flooded with argon as it was done by Brandl and coworkers [20–23] and Åkerfeldt et al. [6]. Brandl and coworkers used a robot system instead of a machine system, although robots are known for being less accurate [20–23]. The laser spot had a 4 mm diameter [22]. In another publication, Brandl and coworkers compared an arc process with a laser process [20]. The tensile tests showed similar results. It can be assumed that the heat input was comparable between the processes due to the huge spot size. All literature references used a large laser spot to stabilize the process. Due to the caused large heat-affected zone, a local shielding was not possible.

Powder-based processes, on the other hand, also have some disadvantages. Not all of the powder sticks to the melt pool. Thereby, an overspray exists. The amount of the overspray is related to the shape of the nozzle and process optimization [24,25]. The powder might be a hazard to the environment and is a loss in process efficiency. When highly reactive material is processed, powder has an inherent disadvantage. The high surface-to-volume ratio enlarges the risk of oxidation and contamination humidity [26]. Regardless of these disadvantages, Kelbassa was able to realize the repair of a turbine blade manufactured out of Ti6Al4V. Thereby, he faced two main challenges: distortion and oxidation. The distortion reached up to 1 mm. To avoid oxidation, he used a special shielding gas concept [27].

For the realization of the alternative process chain, the parts have to be cleaned after a powder based process. This is necessary to prevent a contamination of the environment as well as to prevent damages on the machinery of the production chain.

Concluding, for an alternative production chain for additive manufacturing of titanium, lateral wire-based laser metal deposition (LMD-W) with local shielding gas concept seems to be the most promising process. This concept can be integrated into process chains more easily and allows the buildup as well as the repair of complex shaped bodies.

To avoid the flooding of the whole process chamber with argon, a local shielding is necessary. Due to the high affinity to oxygen, the local shielding of additive manufacturing of titanium alloys is challenging. There are two ways to address this challenge. The first one is to enlarge the shielded area as far as possible. The second one is to reduce the heated area, which has to be covered by shielding gas, by a proper process design.

The improvement of the shielding gas was presented by the authors in a previous study. In the mentioned study, it was shown that local shielding is possible [28].

This publication will focus on a simulation approach to optimize the process to minimize the heat-affected zone and the melt pool.

During the LMD-W process, the wire, the laser beam, and the surface of the substrate are related through the melt pool shown in Fig. 1. To assure a stable process, the same amount of material brought into the melt pool by the wire has to solidify at the end of the melt pool. Thereby, the welding speed on the surface and the wire speed have a certain ratio. If the speed of the wire is lower than the welding speed, droplets will form. The ratio therefore has to be higher than one.

The size of the laser spot has an influence on the melt pool as well as on the heat-affected zone. The size of the melt pool again determines the size of the bead that forms during the solidification. The melt pool has to be at least large enough to melt the wire. The wire diameter applied in this study is 1.2 mm. The stability can be increased by a larger melt pool, allowing a displacement of the wire due to feeding problems or tolerances of the wire and the machine system. As a consequence, the size of the heat-affected zone will rise.

To investigate the stable regime of processes, process maps are a common approach. Process maps show process features as a function of important parameters. Process maps can be generated either through a series of experiments or simulations [29].

The temperature distribution in the workpiece and thereby the melt pool characteristics can be simulated with an analytical, semi–analytical, or numerical approach. A literature review on thermal simulation of welding process is given by Mackwood and Crafer [30].

Numerical approaches can incorporate a large amount of effects for example the effects caused by the dynamics between the gas or the liquid phase or the Marangoni convection. Thereby, numerical approaches are able to accurately describe the size of the melt pool and the dimensions of the beads [31,32]. Analytical approaches usually neglect several physical effects. Thereby the computing capacity is much lower. As a drawback, analytical approaches normally are not able to consider complicated intensity distributions [33]. Semi-analytical approaches are a trade of between both.

The laser beam was generated by a 4.5 kW diode laser by Laserline. The optic was by Precitec. The intensity distribution was measured with a focus monitor from Primes.

In total, 11 planes of the intensity distribution were measured with a plane distance of 1 mm. The interpolated distribution (left) as well as the distribution in two measured planes (right) can be seen in Fig. 2.

For each measured plane in the laser beam, henceforth will be called laser level, the temperature field was simulated by a semi-analytical approach. The number of the level is corresponding to the distance in millimeters in z-axis direction, shown in Fig. 2 (left-hand side).

Each point of intersection in the mesh, which was interpolated based on the measured data, was taken as one point source with a scaled intensity. The parameters of titanium at room temperature were used for the simulation.

Figure 3 shows two examples of the plots generated by the simulation. The temperature fields are analyzed in a depth of z = 0.1 mm. Important values of the temperature field are indicated by the following isotherms:

• 200 °C—The temperature where the oxidation process begins.

• 450 °C—The temperature where the oxidation gets critical.

• 1700 °C—The temperature which can be interpreted as a rough estimation of the melt pool.

To generate process maps, the feed rate was varied from 25 mm/min to 2000 mm/min by steps of 25 mm/min and the absorbed laser power was changed from 5 to 400 W by steps of 5 W. From each of those 6400 temperature fields, which were calculated for each laser level, several characteristics were used for further analysis.

These characteristics are the width, length, and the total area of the areas contained by the isotherms.

The width of the area plays an important role in the process. It is obvious that in a continuous process the heat-affected zone perpendicular to the additive manufactured track is proportional to the width of the area above a certain temperature. In addition, the width of the area above 1700 °C, as an estimation of the melt pool, has to be at least as large as the wire to guarantee a stable process.

In Fig. 4, the diagrams show the width of the melt pool as a function of the effective power and the feed rate based on the 6400 simulated temperature fields. Thereby, each pixel in the diagram represents the results of one simulation. The process diagrams caused by the intensity distribution of the laser levels 1 and 7 are shown exemplary.

The necessary width of a melting pool on the surface is at least as large as the wire diameter. Therefore, the width of 1.2 mm as well as the width of 1.5 mm is marked in white to show the threshold of a stable process.

The change of the laser intensity distribution by defocusing has a minor effect on the temperature field compared to the speed and the power. By comparing the diagrams, it is obvious that the process window is narrower for a defocused laser beam. The power that is necessary to generate a molten pool width of 1.2 mm is lower for the defocused beam (laser level 1). This is because the size of the laser beam is larger, heating up a larger area of the surface.

In the center of the beam at laser level 7 where the intensity profile is more top-hat shaped, the dependency of the feed rate and effective power is nearly linear.

When comparing laser level 7, shown in Fig. 4, with laser level 5, shown in Fig. 5, nearly no difference can be seen. This result confirms that in the Rayleigh length the intensity distribution is negligible. For further investigation, the laser level 5 was chosen for three reasons:

First, the range of accepted parameters is larger than in the lower area outside of the Rayleigh length, which reaches until the fifth laser level.

Second, compared to the laser level 7 the laser level 5 is beneficial for the wire feeding as will be explained in the following chapter.

Third, it is estimated that the homogenous profile will result in a more stable process. For the laser level 1, the size of the heated area may be smaller, but the temperature increases in the center due to the inhomogeneous distribution.

If phase changing and the behavior of molten material are taken into account, the results will change slightly. It can be estimated that although the model predicts a minor dependency of defocusing, the real process results show a higher one.

As explained before, further investigations will be undertaken for laser level 5, which is shown in Fig. 5. There, the overview of the width of the area above 1700 °C, 450 °C, and 200 °C is displayed. The white lines again indicate the stable process parameters, where the width of the melt pool is estimated to be between 1.2 and 1.5 mm.

It is evident that the heat-affected zone increases with the increasing of the power. From the diagram, it also can be seen that higher speeds cause smaller heat-affected zones at the same power. The width of the area above 450 °C and 200 °C increases slower than the 1700 °C. As an example, for a speed above 1000 mm/min the width of an area above 450 °C can be minimized below 2.4 mm for the estimated melt pool width of 1.2 mm.

To minimize the width of the heat-affected zone, as a conclusion, the speed should be as high as possible. The effective laser power to generate the melt pool has to be between 80 W and 300 W depending on the speed. The power should be as low as possible to guarantee a stable process.

While the heat-affected zone perpendicular to the process is mainly depending on the width of the area, the area on the work piece that has to be covered with shielding gas is also depending on the total area respectively the length of the area above certain temperatures. For higher speeds, this length as well as the area will be increased. To confirm that the zone also is minimized, Fig. 6 shows the area above 450 °C in laser plane 5.

Based on the simulations, different area sizes can be chosen. For the evaluation, the following process window was determined. It can be seen that the area above 450 °C is not as linearly dependent as the width of the area above 1700 °C. Thereby with increasing feed rate, the area grows slower than the width of the area for a stable process indicated again by the white lines. Therefore, the speed should not be increased too much. The speed should be between 1000 mm/min and 1500 mm/min. Thereby, the size of the 450 °C area can be minimized to 14 mm². The estimated process window will be in the dotted contour line displayed in Fig. 6.

Concluding, the feed rate for the practical evaluation of the simulations was set between 1000 and 1500 mm/min. The absolute laser power is estimated to be in between 1246 W and 1786 W. Afterward, the power was iteratively minimized by a trial and error approach.

After the speed, the power, and the defocusing were defined, the wire setup was analyzed. As described before for the process, a noncoaxial wire supply was used. The wire was fed into the laser beam laterally. The projection of the laser on the wire can be used as a rough estimation of the size of the melt pool.

The intensity distribution of the laser beam on the wire depends on the angle between wire and substrate, the displacement of the wire in x and y direction, as well as the laser level, in which the wire is injected into the laser beam.

For the simulations, the wire to beam angle was set to 30, 45, and 60 deg. Displacements in x and y up to 0.6 mm in both directions were considered. The simulations were performed for each of the eleven laser levels of the measured intensity distribution. Moreover, a series of simulation were done with a change of the wire to beam angle from 15 deg to 75 deg in steps of one degree without displacement. Some of the projected intensity profiles on the wire are shown in Fig. 7.

To determine the ideal parameters, several aspects have to be considered. For example, the intensity distribution should be as homogeneous as possible. A contact between not molten wire and substrate should be avoided and the projected area should be as small and round as possible. Thereby, the tensions on the melt pool are minimized to avoid a dripping effect of the molten wire. Additionally the tolerance against displacement of the wire has to be as high as possible. Based on the knowledge gained in 20 years developing and investigating wire-based additive laser processes, the simulations were examined.

The conclusion is that the laser level 5 results in the most appropriate intensity distribution for the wire setup. Thereby, a part of the beam is able to directly access the substrate. This will stabilize the melt pool on the substrate, where the intensity distribution on the wire is nearly top hat.

It is assumed that the configuration will allow a displacement of the wire by ±0.1 mm in x, y, and z direction. The wire-laser angle has to be set to 30 deg to achieve a homogeneous intensity distribution on the surface of the wire as well as to minimize the melt pool size. A large melt pool size with a concave surface is known to be error prone and therefore should be avoided.

These results defined a rough process window and defined optimal machine parameters, which finally were validated.

The process investigations were done on a five-axis machine. The accuracy of the machine system is more than 8 μm and it can reach speeds up to 20,000 mm/min. It more than fulfills the requirements of the designed process. The wire feeding was optimized to meet the restriction determined in the previous chapter. The optimization of the gas flow as well as the distortion effects has already been described in a previous publication [28].

The wire, which was used for the investigation, had a diameter of 1.2 mm. The wire material actually fulfills not only the requirements of titanium grade 5, but also of titanium grade 23. This material has the same composition, but finer tolerances especially for oxygen.

The optimized parameters were adapted to the process by a trial and error approach. Thereby, it was possible to find a series of stable parameters, which were used to buildup parts with a low waviness and a nearly top-hat plateau in each layer. In the end, already 1100 W laser power was sufficient. The feed rate was set to 1500 mm/min. Figure 8 shows a demonstrator part, which was buildup with this setup. The outer shape of the demonstrator is the style of a near-net-shape-profile of a turbine blade. The buildup demonstrator has a thickness of about 13 mm and length of 106 mm. The height is 9 mm.

Additional to the demonstrator, specimens were buildup for metallurgical analysis and for tensile tests to identify if the process is capable of achieving proper mechanical strength and the right microstructure.

As it was emphasized before, the microstructure is important for the processing of titanium alloys. Titanium grade 5 contains alpha and beta-phases. These phases can form three different microstructures: lamellar, globular and bimodal. To adjust the microstructure, thermal as well as mechanical processing is typically necessary. The mechanical processing has to generate slips. These slips, which are induced for example by forging, are necessary to induce recrystallization.

When Ti6Al4V is heated up above beta-transus as it is done by the melting or in the heat-affected zone, the material transforms. If the temperature is lowered slowly, the alpha phase grows as lamellas in one direction forming the so-called alpha colonies. These are separated by beta phases [1].

When the cooling is done faster, the direction of the growing alpha colonies tends to be not synchronized. Instead, another alpha lamellar nucleates generates perpendicular to the first by the stress, which is induced due to the growth. This forms the so-called “basket weave” structure or Widmanstatten structure [2].

This is a special type of lamellar structure with a high fatigue strength, but lower ductility. The lattice distortion, which is also associated with high cooling rates, is known for hardening effects. Both phenomena result in a higher resistance limit but potentially brittle structure [2].

The fatigue limit does not only have a dependency on the kind of microstructure, but also on the characteristic size. Fine lamellar microstructure can exceed the crack propagation resistance of globular microstructure. Cracks will be deflected by the fine structure. Coarse lamellar microstructures on the other hand have a low crack resistance value [1,2].

Titanium grade 5 parts that were manufactured by wire-based LMD also showed a Widmanstatten structure, as can be seen in Fig. 9. The cross section was prepared perpendicular to the processing direction. The pale lines indicate the heat-affected zone of each bead. In this zone, high tensions exist that prevent the etching. It can be assumed that the direction of alpha colonies partly is oriented by the high tension during the solidification. Thereby, the material will not be anisotropic.

This also is enforced by the grain growth during the heat treatment that occurs during the thermal input by processing the upper layers. Thereby the grains of the buildup parts are already very coarse and deformed in the buildup direction. For turbomachine applications, this might be beneficial to the higher thermal strength of oriented grains.

One specimen was built up to extract four tensile specimens out of it. The mean yield strength of the tested specimen is 874 MPa, and the ultimate strength is 934 MPa. The lowest of the values of yield strength is 854 MPa, and the lowest ultimate strength is 912 MPa [28]. The values are similar to the ones found in the literature for additive manufacturing [18–20]. The norm DIN 17896 specifies the yield strength as 865 MPa and the ultimate strength as 930 MPa. Therefore, the process is capable of fulfilling the requirements.

The grains of the parts are coarse, and an anisotropy exists after additive manufacturing. The anisotropy is caused by the microstructure, the grain direction and the residual stresses in the work piece. If both effects can be neglected or are beneficial, the process can be used without a postweld thermo-mechanical treatment to buildup parts. This is the case for turbine blades or comparable parts being under high unidirectional load at high temperatures.

The influences of different thermal treatments without additional mechanical process steps were discussed by the authors in Ref. [28]. In the following, these results shall be reflected according to their effect on the buildup of turbomachinery components.

The cross section of the specimen and the microstructure after two different heat treatments are shown in Fig. 10. The left side shows the results of a thermal treatment below beta-transus. The right side shows the treatment above. The actual temperature-time-cycle can be found in Ref. [28].

By a thermal treatment below beta-transus, the grains are deformed but do not recrystallize. The microstructure changes. However, a globular structure, which was intended, was not achieved. Although the alpha phase did appear, a bimodal structure was not generated. Thereby, the thermal treatment has to be adapted or a mechanical processing is necessary.

When the material has to be more ductile to avoid brittleness at an impact, as it is especially necessary in the first stator for instance, a thermal treatment below beta-transus seems suitable.

Above beta-transus, a recrystallization appears. The recrystallization is more significant in the lower part of the specimen. This indicates that here a higher amount of compressive tensions exists. The microstructure is finer than the one directly achieved after the LMD processing.

In the case that the material of a part of a turbine should be shock absorbing and ductile, the grains should not be orientated and have to be finer. Thereby they can deform during high load. This might be achieved with thermal treatment above beta-transus combined with a mechanical processing like HIPing or peening for thin layers [26].

In this paper, the thermal optimization, the adaption of the wire configuration, the optimization of the layer buildup, and the impact of different thermal treatments were presented. The following results were found:

To avoid negative thermal effects and oxidation, the laser spot size should be as large as the estimated melt pool and, therefore, the bead size.

The laser projection onto the wire is important for the process stability at higher feed rates. The wire angle was set to 30 deg and a slight defocusing was configured. By choosing a small laser spot, large melt pool sizes were avoided.

Finally, the results of potential thermal treatments suggest adapting the thermal treatment to the necessities of the parts for turbomachinery components.

Titanium alloys are materials with high potential for the manufacturing of several turbomachinery components. To gain this potential, the processing of the material has to be more efficient than it is today. Additive manufacturing is a key technology to overcome restrictions of classic production chain. Laser metal deposition with wire (LMD-w) seems capable to tab the potential, but the necessary technological readiness level is not given yet.

The authors will go on developing and optimizing the wire-based laser metal deposition for titanium grade 5. The next steps are going to be the optimization of the wire supply. This will increase the possible wire speed and thereby the buildup rates. Further challenges are going to be the distortion of the material as well as the effect on the postprocessing steps such as peening.

This project has received funding from the European Union's Horizon 2020 research and innovation program under the grant agreement No. H2020–FoF-2016- 723917-OPENHYBRID.² The dissemination of the project herein reflects only the author's view and the Commission is not responsible for any use that may be made of the information it contains.

• H2020 Industrial Leadership (H2020 - FoF - 2016 - 723).

c_p =

heat capacity, J/kgK

k_x,k_y =

count variable of the intensity matrix

n_y,n_x =

number of elements of the intensity matrix

P_L =

laser power, W

r =

radius, m

T_Enviroment =

temperature of the environment, ° C

κ =

thermal diffusivity, m²/s

μ =

geometric efficiency factor

$μkxky$ =

intensity factor at the point x, y on the surface

v =

feed rate, m/s

ρ =

density, kg/m³

$Δxkx,Δyky$ =

distance to center of point source