Analysis of the Square Beam Energy Efficiency of a Homogenizer Near the Target for Laser Shock Peening
 Author: Kim Taeshin, Hwang Seungjin, Hong Kyung Hee, Yu Tae Jun
 Publish: Current Optics and Photonics Volume 20, Issue3, p407~412, 25 June 2016

ABSTRACT
We analyzed through numerical simulations the properties of a square beam homogenizer near the target for laser shock peening. The efficiency was calculated near the target by considering the plasma threshold of the metals. We defined the depth of focus of the square beam homogenizer with a given efficiency near the target. Then, we found the relationship between the depth of focus for the laser shock peening and four main parameters of the square beam homogenizer: the plasma threshold of the metal, the number of lenslets in the arraylens, the focal length of the condenser lens and the input beam size.

KEYWORD
Beam shaping , Beam homogenizer , Depth of focus , Laser shock peening

I. INTRODUCTION
Laser shock peening (LSP) is a cold metalworking process that uses high power laser pulses to generate plasma shock waves by hitting a metal surface [1]. Laserinduced plasma shock waves produce residual compressive stress causing plastic deformation in the target [4]. The LSP can improve metal properties such as surface hardness, abrasion durability, corrosion resistance, and fatigue strength [16]. In general, the conventional laser beam shape is close to circular. The circular beam leads to nonirradiated areas and overlapped areas. The two types of areas have disadvantages with regard to the efficiency of the LSP [7, 8]. A square beam is required to reduce the area and increase the efficiency [1, 711]. In addition, the LSP becomes inhomogeneous owing to the nonuniform intensity distribution within the laser beam. A uniform beam distribution is also required to homogenize the plasma area peened by a laser beam. Therefore, a square beam homogenizer is required to enhance the efficiency of the LSP and to improve the peening quality simultaneously.
The working distance (WD) of a square beam homogenizer is important in the view of the industrial application of the LSP. The long WD, which has various height targets, makes peening possible without any unnecessary movement in the beam homogenizing system. It is related to the depth of focus (DOF) of the square beam homogenizer. We need to analyze the DOF of a square beam homogenizer in order to maximize the WD and increase the efficiency of the LSP. However, the DOF in the case of the LSP (LSPDOF) is different from the conventional DOF considered in most optical systems, which implies the tolerance of placement of the image plane in relation to the lens, as LSPDOF must include the concept of plasma threshold. There are two perspectives about WD: spatial uniformity and efficiency. The LSPDOF means WD for the industrial application of LSP in terms of efficiency.
In this study, we defined the LSPDOF using the efficiency of the LSP, which is defined as the ratio of the energy in the peening area to the energy in an input beam irradiated area. We found the relationship between the LSPDOF and four parameters of the square beam homogenizer: the plasma threshold intensity (
I_{th} ) of the metal, the number (N_{AL} ) of lenslets in the arraylens, the focal length (f_{C} ) of the condenser lens and the input beam size (D_{in} ).II. SIMULATION METHOD
To simultaneously reform the beam shape and distribution, we selected a multiaperture beam homogenizer, which consists of two positive arraylenses and a condenser lens [12, 13] (Fig. 1). The first arraylens, with square lenslet, divides the incident beam into several square beamlets. These beamlets are relayed by the second arraylens, which has the same specifications as the first arraylens. The condenser lens superposes all the beamlets, and the square beam is generated at the target plane.
The final square image size is given as [12, 13]
where
p is the pitch size of the arraylens,d _{12} is the distance between the two arraylenses, andf _{1},f _{2},f_{C} are the focal lengths of each of the lenses (Fig. 1). The final image sizeD is obtained from the magnifying power , wheref_{eq} is the equivalent focal length of two arraylenses .The beam homogenizer was configured in CODE V, a lens design program. With the illumination analysis tool of CODE V, we obtained the 201 beam distributions near the target from −1.0 cm to 1.0 cm at intervals of 0.01 cm (Fig. 1). Then, the plasma threshold was applied to the given beam distributions. The circular and nonuniform incident beam had a 1.3 cm diameter at full width half maximum (FWHM) (Fig. 1), a 532 nm wavelength, a 10 ns pulse width, and an energy of 6.9
J per one pulse. Figure 2(a) illustrates the configuration of the beam homogenizer for preliminary simulation. Table 1 shows the parameters of the square beam homogenizer for the preliminary simulation.The square beam homogenizer should generate a maximum square image size
D =0.26 cm in order to peen Alloy 22, which has the 10 GW/cm^{2} ofI_{th} [1] with the incident beam. Figure 2(b) shows the LSP region among the irradiated input beam region. The beam distributions at positions −0.4, −0.3, −0.2, 0.0, 0.2, and 0.4 cm from the target are shown in Figure 2(c). The white zone indicates the area over the plasma threshold. With the given beam distributions near the target, the efficiency is obtained by considering theI_{th} as one of the properties of the square uniform beam for the LSP. The efficiency is the ratio of the energy (E_{LSP} ) in the plasma area to the input energy (E_{in} ) on the target. The efficiency is defined aswhere
I (x ,y ,z ,t ) is the intensity distribution near the target and the temporal shape of the input pulse was supposed to be rectangular.Figure 2(c) also shows the energy efficiency near the target. The maximum point of the energy efficiency graph (
η_{M} ) is not located on the target position but −0.23 cm away from the target. We set the minimum energy efficiency as 0.5 similar to FWHM in order to define the LSPDOF from this graph which has arbitrary shape. According to this definition, the LSPDOF is 0.7 cm.III. SIMULATION RESULTS and DISCUSSION
Metal has a specific plasma thresholds intensity (
I_{th} ). The number of lenslets (N_{AL} ) in the arraylens is related to the uniformity of the square beam [12, 13]. The focal length (f_{C} ) of the condenser lens and the input beam size (D_{in} ) are the parameters in a conventional DOF. For these reasons, we selected these four main parameters in order to find the relationship with the LSPDOF.In the first simulation, the different
I_{th} of three types of metals were applied to the beam distributions previously determined in the preliminary simulation (Table 1.). The three types of metals are Alloy 22, Ti6AL4V, and 316 L SS, and theseI_{th} are 10, 5, and 2.5 GW/cm^{2}, respectively [1]. When the square uniform beam (D =0.26 cm) is radiated at different metals, the efficiency is shown in Fig. 3.Figure 3 shows the efficiency near the target for the
I_{th} . When theI_{th} are 10, 5, and 2.5 GW/cm^{2}, positions of theη_{M} are −0.23, −0.10 and −0.12 cm from the target, respectively; the LSPDOF are 0.7, 1.8, and 5.6 cm respectively. As theI_{th} decreases, the position of theη_{M} approaches the target plane. This, in turn, causes an increase in the efficiency and elongates the LSPDOF. The overall efficiency level ascends whenI_{th} descends. However, for a lowI_{th} , it is not always optimal to peen the target with a small square beam size. If the efficiency fulfills the minimum required value in the LSP industrial field, the square image size on the target can be increased in order to peen the target efficiently.he second simulation was to change the number of lenslets in the arraylens (
N_{AL} ) and it was performed twice differently according to each parameter,f _{2} andd _{12}. When theN_{AL} is equal to 5×5, 7×7, and 11×11,f _{2} is changed along with theN_{AL} in the first case and thed _{12} is changed along with theN_{AL} in the second case. The other parameters are the same as parameters of the preliminary simulation except for theN_{AL} ,f _{2} andd _{12} forD =0.26 cm. Table 2 lists values of main parameterN_{AL} , two dependent parametersf _{2},d _{12} and LSPDOF about the two cases.In the first case (Fig. 4(a)), when the
N_{AL} is equal to 5×5, 7×7, and 11×11 the positions of theη_{M} are −0.30, −0.23 and −0.21 cm respectively; the LSPDOF is 0.9, 0.7, and 0.6 cm respectively. In the second case (Fig. 4(b)), when theN_{AL} is equal to 5×5, 7×7, and 11×11 the positions of theη_{M} are −0.33, −0.23 and −0.22 cm respectively; the LSPDOF values are 0.9, 0.7, and 0.6 cm respectively. There is a similarity between the two cases as shown in Fig. 4(a) and (b). It means that the adjustment of interval (d _{12}) is more convenient than the changing of the lens (f _{2}). It is known that the uniformity of the square beam at the target plane becomes better when theN_{AL} rises [12, 13], but the LSPDOF shortens in all the cases. There is a tradeoff between the beam uniformity and the LSPDOF. When theN_{AL} is increased, the maximum point of the energy efficiency is also increased and the slope of the energy efficiency graph becomes steeper.The purpose of the third simulation was to change the focal length of the condenser lens (
f_{C} ) and it was carried out two times differently according to each parameter,f _{2} andd _{12}. When thef_{C} values are 8.2, 10.0, and 12.0 cm,f _{2} is changed along withf_{C} in the first case andd _{12} is changed along withf_{C} in the second case. The other parameters are the same as parameters of the preliminary simulation except for thef_{C} ,f _{2} andd _{12} forD =0.26 cm. Table 3 lists values of main parameterf_{C} two dependent parametersf _{2},d _{12} and LSPDOF about the two cases.In the first case (Fig. 5(a)), when the
f_{C} values are 8.2, 10.0, and 12.0 cm the positions of theη_{M} are −0.23, −0.19, and −0.13 cm, respectively; the LSPDOF values are 0.7, 0.7, and 0.8 cm, respectively. In the second case (Fig. 5(b)), when thef_{C} values are 8.2, 10.0, and 12.0 cm the positions of theη_{M} are −0.23, −0.17, and −0.11 cm, respectively; the LSPDOF values are 0.7, 0.7, and 1.1 cm, respectively. There are slight differences between the two cases about the positions of theη_{M} . The LSPDOF in the second case is longer than the result of the first case. In this simulation, adjusting the interval (d _{12}) is not only convenient but also advantageous to the LSPDOF. When thef_{C} values increase, the LSPDOF becomes longer and theη_{M} shifts to the right owing to the influence of the spherical aberration. We need to analyze input beam size additionally to understand the spherical aberration and the LSPDOF.In the fourth simulation, the input beam size (
D_{in} ) was changed. The parameters of the square beam homogenizer were equal to the conditions in the preliminary simulation except for theD_{in} . Figure 6(a) shows the efficiency for the three cases of the changes inD_{in} .When the
D_{in} values are 2.5, 1.8 and 1.0 cm, the positions of theη_{M} are −0.23, −0.13, and 0.23, respectively; the LSPDOF values are 0.7, 1.0, and 2.6 cm respectively. When theD_{in} is smaller, the position of theη_{M} passes through the target plane and the LSPDOF becomes longer. Figure 6(b) shows the concept of the changing beam waist; when the different height of the incident beamlets is going to the condenser lens. Because the influence of the spherical aberration is diminished when theD_{in} is small, the slope of the irradiated input beam is gentle and the position of the beam waist shifts to the right. The smallD_{in} is related to long conventional DOF. However, smallD_{in} is not always a good condition for the LSP. When theD_{in} is small, the number of beamlets that pass through the arraylens is small as well. Decreasing of the number of beamlets will diminish the uniformity of the square beam at the target plane [12, 13]. Research about the uniformity of the square beam needs to be studied as further work.IV. CONCLUSION
We have designed a multiaperture square beam homogenizer to reform the shape and distribution of the input beam simultaneously. The LSPDOF was defined with the efficiency near the target. A long LSPDOF is required, as it is important to have a long WD in application field of the LSP. When the square beam homogenizer was applied to the LSP, we found that low
I_{th} , smallN_{AL} , longf_{C} and smallD_{in} were advantageous in obtaining an elongated LSPDOF. Thef _{2} andd _{12} were selected as the dependent parameters. Modifyingd _{12} is convenient and beneficial to obtain long LSPDOF more than varyingf _{2} whenf_{C} is the main parameter. In addition, we have calculated and analyzed the maximum efficiency position of the LSP according to change the four parameters. Understanding these characteristics is useful to design a square beam homogenizer for the LSP. Further work in this subject would involve an analysis of the uniformity of the beam and its effect on the peening quality

[FIG. 1.] The parameters of the beam homogenizer and simulation concept.

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[FIG. 2.] (a) The simulation configuration of the beam homogenizer for Alloy 22, which has 10 GW/cm2 plasma threshold (b) The LSP region in the irradiated input beam region (c) The efficiency near the target and the beam distributions at each defocus point ？0.4, ？0.3, ？0.2, 0.0, 0.2, and 0.4 cm from the target.

[TABLE 1.] The parameters of the square beam homogenizer for the first simulation (D=0.26 cm)

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[FIG. 3.] The efficiency graph of metals having different Ith.

[TABLE 2.] The values of two parameters of the square beam homogenizer and the LSPDOF about the two cases when the NAL is changed

[FIG. 4.] The graph shows the efficiency curve versus defocus in the fixed case of (a) f2 and (b) d12. The NAL is 5×5 (red), 7×7 (green), and 11×11 (blue) respectively.

[TABLE 3.] The values of two parameters of the square beam homogenizer and the LSPDOF about the two cases when the fC is changed

[FIG. 5.] The graph shows the efficiency curve versus defocus in the fixed case of (a) f2 and (b) d12 . The fC is 8.2 (red), 10.0 (green), and 12.0 (blue) cm respectively.

[FIG. 6.] (a) The efficiency graph of the changing in Din, (b) The changing of the irradiated input beam region due to the change in Din.