Baohua Chena, Quanying Wua,*, Yunhai Tanga, Junliu Fana, Xiaoyi Chenb, Yi Sunc
a Jiangsu Key Laboratory of Micro and Nano Heat Fluid Flow Technology and Energy Application, School of Physical Science and Technology,
Suzhou University of Science and Technology, Suzhou 215009, China
b Suzhou Mason Optical Co., Ltd., Suzhou, Jiangsu 215028, China
c Soochow Mason Optics Co., Ltd., Suzhou, Jiangsu 215028, China
Annular laser beam
Optical system
Integrating mirror
Intensity uniformity
mirror and a parabolic cylindrical mirror. The parameters of the mirrors are obtained in accordance with the design requirements of the annular beam. The rotation equation of the parabolic
cylindrical mirror is derived with the same annular beam diameter, while the apex angle of the
conical mirror changes. The uniformity of the annular beam intensity is improved by changing the
parabolic cylindrical mirror into a concave–convex parabolic cylindrical integrating mirror,
which is designed on the basis of the principles of surface division and beam superposition. The
mirrors are processed by single-point diamond turning. An experimental facility is built to analyze
the size and uniformity of the beam intensity distribution. The annular beam-width error is less
than 3%, and the uniformity is 89%. The surface of the concave–convex parabolic cylindrical
integrating mirror is smooth and continuous. The experimental data correspond to the theoretical
design.
1. Introduction
Laser beam shaping and modulation have an important role in fiber optic communications, laser cutting, and laser welding [1,2]. Industrial thin-walled pipe welding is usually completed with a focused laser beam spot combined with automated machinery [3,4]. The welding effect of this method is poor and inefficient due to the low accuracy of the stroke trajectory of automated machinery and the nonuniform intensity distribution of the focused beam. Therefore, new optical systems are proposed to solve these problems by directly shaping the beam into an annular beam [5–8]. Most of the optical systems used for annular beam shaping are transmissive [9–11], consisting of a conical lens and a focusing lens. Nevertheless, limited by the conical lens polishing process, the tip of the lens center is prone to rounding, resulting in a nonuniform center beam and reducing its quality. Transmissive systems with a lens film layer cannot support high-power laser beams for a long time and induce optical system length redundancy and other problems, affecting the final welding efficiency and accuracy. The mirrors of the reflective optical system can be processed by ultra-precision single-point diamond turning (SPDT) with high efficiency and good precision, and the reflectivity is 98% after gold plating on the metal surface [12]. However, such optical systems still use the same vertical angle of the conical mirror, resulting in a structure in which the position of the focusing mirror cannot be freely changed and the freedom of design is limited [13,14]. When the incident beam is Gaussian, the annular-shaped intensity distribution is not uniform. The problem of thermal deformation cannot be solved in the welding process for large annular weld gap.
In this study, a reflective optical system is designed to address the problems of limited degrees of freedom of reflective optical systems and nonuniform focused annular beam based on conical and parabolic mirrors. A parabolic rotation matrix is derived for any conical mirror vertical angle to increase the design freedom of the optical system. Then, a concave–convex parabolic cylindrical integrating mirror is designed to increase the annular ring width of the focused annular beam and optimize its intensity distribution to form an annular beam with uniform intensity distribution.
2. Design method
2.1. Initial structure of the optical system
The optical system consists of a conical mirror M1 and a parabolic cylindrical mirror M2, as shown in Fig. 1. It is designed on the basis of the annular beam's diameter Ø, the working distance Z1, and the beam's size H. The meridional plane coordinate system is defined by the axis X and the optical axis Z. A circular parallel laser beam is incident onto M1, and its propagation direction is turned around 90◦ and then reflected on M2. Finally, the entire beam converges on the focal point F. Given that the focal point F is offset from the optical axis Z, a focused annular beam is formed in the focal plane with a radius equal to the offset distance. To summarize, the coordinates of the focus F are determined by the working distance Z1 and the annular beam's diameter Ø, and the size of M1 is also affected by the incident beam's size H. Optical system parameters can be obtained from the initial conditions.
M1's reflective surface is formed by a conical line rotating around the optical axis Z, and the conical line equation L(x, z) in the meridional plane is defined as follows:
The apex angle a of M1 is 90◦, and its bottom diameter can be set in accordance with the incident laser size H.
M2's reflective surface is formed by the parabola rotating around the optical axis Z, and its symmetry axis is the axis X. The parabola P(x, z) in the meridional plane is defined as follows:
where f is the focal length of the parabola, l is the distance between the parabola vertex S and the Z axis, and the focal point F coordinates are F(XF, ZF). If XF is equal to –d and ZF is equal to zero, the radius of the focused annular beam is d. The focal length f is the unknown parameter in Eq. (2). The edge point t is located on P(x, z), its Z coordinate is –Z1, and its X coordinate is equal to the radius r, the value of which is reasonably set by the optical system's size. Finally, the focal length f can be calculated by substituting t(r, –Z1) into Eq. (2).
2.2. Optical system with conical mirror apex angle changed
The reflected beam on M1 changes from 1 to 2 when the apex angle of M1 is α′ , as shown in Fig. 2. The symmetry axis X′ of the parabola should be parallel to the reflected light 2 to keep focusing and the focus position unchanged. In fact, the parabola P(x, z) is rotated around the focus F by a certain angle θ to obtain a new parabola P′ (x′ , z′ ), and the angle θ is equal to 90º–α′ . where T is a point on the parabola P(x, z) before the rotation, and the vector to the focus F is FT̅→ = (x − XF,z − ZF). T′ is the rotated point of T, and the vector to the focus F is FT̅→′ = (x′ − XF,z′ − ZF). The position of point T′ (x′ , z′ ) can be calculated using the following formula:
Fig. 1. The annular beam optical system consists of a conical mirror M1 and a parabolic cylindrical mirror M2.
Fig. 2. Optical system with conical mirror apex angle changed. The solid blue line represents the beam propagation process when the apex angle is α, and the dotted line represents the propagation process when the apex angle is α′.
where γ is the angle between the vector FT̅→and the axis X, and▕ FT̅→▕ is the modulus of the vector FT̅→. The above formula is simplified as follows:
where the coordinates of T′ and T are transformed into each other by the rotation matrix Tθ, so the parabolic P′ (x′ , z′ ) equation is as follows:
The optical systems are designed using conical mirrors with three types of apex angles of α = 90◦, α′ > 90◦, and α′′ < 90◦, as shown in Fig. 3. Given that the position of M2 is changed when the apex angle of M1 changes, the optical system can be designed by selecting the optimal apex angle of M1 on the basis of actual conditions, such as the working space and mechanical structure.
The annular laser beam radius can be determined by the coordinates of the focus F in the above design methods. The F coordinates are F(–d, 0), and the upper and lower beams obtained by ray tracing simulation meet first and then propagate to the focal plane, as shown in Fig. 1. When the F coordinates are F(0, 0), the entire laser beam on M2 converges into a focused point. When the F coordinates
Fig. 3. Optical systems with different conical mirror apex angles. (a) the apex angle α = 90◦.(b) the apex angle α′ > 90◦.(c) the apex angle α′′ < 90◦.
are F(d, 0), the laser beams propagate directly to the focal plane without overlapping. Although the annular beam has the same size as F (–d, 0), its intensity distribution and practical use are different.
Fig. 4(a) shows the annular beam received by the detector viewer when the F coordinates are F(–d, 0), and Fig. 4(b) shows the intensity distribution curve of the annular beam. The peak of intensity is at the outside edge, and its distribution diminishes monotonically from the outside to the interior. It is appropriate for internal welding between components in the application field of laser welding in Fig. 4(c).
Fig. 5(a) shows the annular beam received by the detector viewer when the F coordinates are F(d, 0). Fig. 5(b) indicates that the peak of intensity is at the inner edge, and its distribution is opposite to that in Fig. 4(b). As shown in Fig. 5(c), it is suitable for external welding of components in laser welding.
2.3. Design of a uniform annular laser beam
Beam uniformity σ can be measured by the ratio of the difference between the maximum and minimum intensity and the average intensity, as shown in Formula (7). Fig. 4 and Fig. 5 show the annular laser beam intensity distribution of the focal plane is not uniformly designed by the above method.
As shown in Fig. 6, M2 is changed into a concave–convex parabolic cylindrical integrating mirror to improve the uniformity of the beam intensity [15,16]. The surface of M1 is separated into areas 1, 2, 3. on the basis of the annular ring width CD, and set the width of each section along the Z-axis as Z11, Z12, Z13.
where the laser beam is reflected on the concave mirror in areas 1 and 3, then converges at focal points F1 and F3, and eventually reaches CD. The beam in area 2 is reflected on the convex mirror and travels in the opposite direction along the virtual focus F2, eventually reaching CD as well, and the area 2 width is less than the CD width.
The intensity of the Gaussian laser beam incident on areas 1, 2, and 3 is monotonically decreasing. Its intensity steadily drops from point D to point C reflected to CD by the concave parabolic mirror above area 1 and increases by the convex parabolic mirror above area 2. As a result, the annular-focused beam intensity at CD becomes uniform by the concave–convex surface.
When the apex angle of the conical mirror is α′′ , the concave parabolic equation Pn1(Xn1, Zn1) with F1 (XF1, ZF1) as the focus can be defined as follows:
where points A and B are located on Pn1(Xn1, Zn1), and F1 is the intersection of lines AD and BC. The coordinates of A(XA, ZA), C(XC, ZC), and D(XD, ZD) are calculated from the initial condition. The ZB coordinate in B(XB, ZB) is equal to ZA+Z11. The value of XB, the coordinate of F1, and the focal length fn1 in Eq. (8) can be solved using the following equations:
Fig. 4. Annular beam intensity distribution at F(-d, 0) of the focal plane. (a) The annular beam received by the 10 × 10 mm detector viewer. The place marked by the circle shows that the beam intensity on the left is low, while that on the right is high. (b) Intensity distribution curve. (c) Internal welding of tubular parts. It shows the beam path is applicable to internal welding of tubular parts.
Fig. 5. Annular beam intensity distribution at F(d, 0) of the focal plane. (a) The annular beam received by the 10 × 10 mm detector viewer. The place marked by the circle shows that the beam intensity on the left is high, while that on the right is low. (b) Intensity distribution curve. (c) external welding of tubular parts. It shows the beam path is suitable for external welding of tubular parts.
Fig. 6. Design of the concave–convex parabolic cylindrical integrating mirror. (a) The path diagram of the laser beam on the integrating mirror.It shows that the incident laser beam is divided into areas 1,2,3 by the integrating mirror and then superimposed on CD. (b) Integrating mirror design schematic.
Fig. 7. (a) Uniform annular laser optical system. Surface 1 represents a conical mirror and surface 2 represents a concave–convex parabolic cylindrical integrating mirror. (b) The Uniform annular beam received by the 10 × 10 mm detector viewer. (c) Intensity distribution curve.The dashed circle mark display that the annular ring width is near a rectangle.
Similarly, the convex parabolic equation Pn2(Xn2, Zn2) with F2 (XF2, ZF2) as the focus can be defined as follows:
where points A(XA, ZA), C(XC, ZC), and D(XD, ZD) are the known coordinates, and the value of ZE in E(XE, ZE) is equal to ZB+Z12. Combined with Eq. (9), the coordinate of the focal point F2 and focal length fn2 can be calculated in Eq. (10). This can ensure continuous smoothness at the junction points of concave and convex surfaces, such as B and E, and satisfy the following constraints:
A concave–convex parabolic cylindrical integrating mirror is achieved on the basis of the above method, as shown in Fig. 7(a). Fig. 7(a) depicts the uniform annular laser beam optical system, where surface 1 represents a conical mirror and surface 2 represents a concave–convex parabolic cylindrical integrating mirror. The radiant intensity received by the detector viewer is depicted in Fig. 7(b). The distribution curve of the annular ring width is near a rectangle in Fig. 7(c). The uniformity is more than 80%, and its value will be higher as divided regions increase.
3. Experiment
The design parameters of the optical system are provided in Table 1, with the outer diameter d′ of the focal plane uniform annular laser beam being 12 mm and the inner diameter d′′ being 6 mm. The incident beam's diameter H is 20 mm, and the radius size r of the left side of the concave–convex parabolic cylindrical integrating mirror is 35 mm. The working distance Z1 is 150 mm, and the uniformity of the annular beam intensity is greater than 85%. The optical system parameters are calculated by MATALB using Eqs. (1)– (10), as listed in Table 2 and Table 3. The conical mirror's size H′ is 28 mm, and its apex angle α′′ is 86◦. The coordinates of points C and D are (3, 0) and (6, 0), respectively, and the rotation angle θ of each parabolic mirror is 4◦.
Fig. 8(a) shows the curve of integrating mirror. The width of each area is 2 mm, which is much smaller than their focal length. Therefore, the overall curve is not directly see a pattern similar to waves, but rather a straight line. Point G and point J are adjacent points at the concave-convex junction. The difference between their X values is 2 µm, and the difference between their Z values is 5 µm. There is no jumping point, so the whole curve is smooth. Fig. 8(b) shows the incremental change rate of Z value with X value on the curve. In the concave area from point A to point B, the change rate increases gradually. In the convex area from point B to point E, the change rate gradually decreases, so the whole change rate is an obvious broken line chart.
The material of the mirrors is oxygen-free copper, and their surfaces are rotationally symmetrical and readily manufactured using SPDT technology, as shown in Fig. 9(a). The tip error of the processed conical mirror can be regulated below 1 µm, the apex angle error is less than 0.001◦. Compared with glass polishing, it takes less time to achieve 5 nm roughness by SPDT. Fig. 9(b) shows the optical system with the uniform annular beam focused on the left white screen. The optical mounts and components are all coaxial, and the distance between the white light screen and the parabolic mirror is 150 mm.
The white screen is replaced with a CCD camera with a target surface size of 2/3 in. and a pixel size of 4.5 µm. The annular laser beam received by the detector surface is shown in Fig. 10(a). There are speckles and stray light surrounding the annular beam due to the external light source and exposure noise. The curve of the intensity distribution is shown in Fig. 10(b). The annular laser beam width occupies 686 pixels corresponding to 3.09 mm, and the error is 3% compared with the theoretical value. The mean intensity of the curve is 222.4 W/m2 . The intensity of the high-energy point is 230.6 W/m2 , while the intensity of the low-energy point is 205.3 W/ m2 . The uniformity σ is as follows:
4. Conclusions
An optical system for generating annular beams using a conical mirror and a parabolic cylindrical mirror is designed in this study. The rotation equation of the parabolic cylindrical mirror is deduced to improve the design freedom. The concave–convex parabolic cylindrical integrating mirror is designed on the basis of the principles of surface division and beam superposition. As a result, this method can construct an annular beam using a minimal number of mirrors. The uniformity of the beam intensity has also been improved and meets the application fields of higher precision. The experimental result shows that the diameter error of the annular beam is less than 3%, and the uniformity reaches 89%.
Fig. 8. (a) The curve of the integrating mirror. Concave areas are indicated by blue lines and convex areas by red lines. The area width is much smaller than the focal length, so the whole curve looks like a straight line. (b) Incremental change rate of Z value with X value on the curve.
Fig. 9. Experimental optical system. (a) conical mirror and the concave–convex parabolic cylindrical integrating mirror. (b) Annular laser beam experimental device.
Fig. 10. (a) Annular laser beam on the detector surface of CCD. (b) Intensity distribution curve. The intensity of the high-energy point is 230.6 W/ m2 , while the intensity of the low-energy point is 205.3 W/m2 , the difference is only 25 W/m2 .
Funding
National Natural Science Foundation of China (NSFC) (61875145, 11804243); Natural Science. The Jiangsu Key Disciplines of the Fourteenth Five-Year Plan (Grant No. 2021135). Natural Science Foundation of the Jiangsu Higher Education Institutions of China (17KJA140001); the Jiangsu Province Key Laboratory (KJS1710). Suzhou Industry Prospect and Key Core Technology Project (SYC2022145).
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Data availability
No data was used for the research described in the article.
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