CN112114393B - Parabolic rotating surface array integrator and design method thereof - Google Patents

Parabolic rotating surface array integrator and design method thereof Download PDF

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CN112114393B
CN112114393B CN202010791084.0A CN202010791084A CN112114393B CN 112114393 B CN112114393 B CN 112114393B CN 202010791084 A CN202010791084 A CN 202010791084A CN 112114393 B CN112114393 B CN 112114393B
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CN112114393A (en
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唐运海
陈宝华
吴泉英
范君柳
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Suzhou University of Science and Technology
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    • G—PHYSICS
    • G02—OPTICS
    • G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
    • G02B5/00—Optical elements other than lenses
    • G02B5/08—Mirrors
    • G02B5/10—Mirrors with curved faces
    • G—PHYSICS
    • G02—OPTICS
    • G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
    • G02B27/00—Optical systems or apparatus not provided for by any of the groups G02B1/00 - G02B26/00, G02B30/00
    • G02B27/0012—Optical design, e.g. procedures, algorithms, optimisation routines
    • G—PHYSICS
    • G02—OPTICS
    • G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
    • G02B27/00—Optical systems or apparatus not provided for by any of the groups G02B1/00 - G02B26/00, G02B30/00
    • G02B27/09—Beam shaping, e.g. changing the cross-sectional area, not otherwise provided for
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Abstract

The invention discloses a parabolic rotating surface array integrator and a design method thereof. The reflecting surface of the integrator is formed by alternately arranging a series of concave parabolic rotating surfaces and convex parabolic rotating surfaces, and the curvatures of the rotating surfaces are kept continuous. Each parabolic rotating surface reflects incident light to a focusing light spot to form light spots with the same size and shape. The incident light with uneven light intensity distribution is divided and reflected by a series of parabolic rotating surfaces and superposed on the focusing light spots to form the focusing light spots with regular shapes and even light intensity distribution. The parabolic rotating area array integrator lens provided by the invention can convert incident light with uneven light intensity distribution into focused light spots with sharp boundaries and even light intensity. The parabolic rotating surface array integrator mirror provided by the invention has no abrupt change in surface curvature, is beneficial to single-point diamond lathe processing, and has high production efficiency.

Description

Parabolic rotating surface array integrator and design method thereof
Technical Field
The present invention relates to a device for shaping a laser beam, and more particularly, to an optical element for focusing and converting a parallel laser beam into a uniform spot.
Background
The superiority of laser in material modification, quenching and cutting is receiving increasing attention. If the uniformity of the light intensity of the output beam of the laser is not high, the material processing degree in the laser action area is inconsistent. The integrator mirror (or integrator) can convert the laser beam with non-uniform light intensity distribution into the beam with uniform light intensity distribution. A diffraction type optical integrator is proposed in a document 1 (a diffraction optical element design algorithm [ J ] optics report, 2007,27 (9): 1682-1686) for beam shaping), but the large-area diffraction integrator has high requirements on processing technology and great preparation difficulty. Reference 2 (ZL 01278581.4), which does not relate to the profile characteristics of the reflection bands and to the methods involved, discloses a focusing integrating mirror consisting of a multi-band reflecting curved surface. Reference 3 (ZL 200520127504) discloses a reflection optical system composed of three mirrors for obtaining a laser beam equalizing device of a large area uniform square spot. The device needs more lenses, needs a complex adjusting device and has higher cost. Reference 4 (royal courage, chenhong, te ye, ye te ye, design of a belt integrator for high power laser processing [ J ]. University of beijing university of industry (3): 334-336.) discloses a design method of a belt integrator, which is designed by replacing a curved surface with a plurality of reflecting slopes in a direction perpendicular to a rotation direction of a spherical or aspherical (paraboloidal, ellipsoidal, hyperboloid, etc.) reflecting condenser. The curvature of each reflecting inclined plane on the reflecting surface designed by the method is suddenly changed, and the processing is difficult.
Disclosure of Invention
The invention aims to overcome the defects and designs an integrating mirror with continuous surface curvature, which can convert parallel laser beams with uneven illumination distribution into focused light spots with even illumination.
The invention provides a parabolic rotating area array integrator mirror, which is shown in figure 1 and is characterized in that: the reflecting surface of the integrator is formed by alternately connecting a plurality of concave parabolic rotating surfaces and convex parabolic rotating surfaces. All concave and convex parabolic surfaces of rotation are tangent at the boundary.
The generatrix of all the parabolic rotating surfaces is a part of a parabola. The plane where the generatrix is located is intersected with the focusing light spot of the integrating mirror to form a series of focal plane line segments. The focus of the parabola is positioned at the intersection of the two end points of the generatrix and the connecting line of the two end points of the focal plane line segment. The rotating shafts of all the parabolic rotating surfaces are positioned at the intersection line of the generatrix on the boundary of the parabolic rotating surfaces and the plane where the focal plane line segment on the boundary of the focusing light spot of the integrating mirror is positioned.
As shown in fig. 2, the generatrix of the concave parabolic rotation surface at the two boundaries corresponds to the focal plane line segment on the boundary on the opposite side of the focal spot of the integrator. The two generatrixes and the corresponding focal plane line segments form two intersected planes respectively. The axis of rotation of the concave parabolic surface of rotation is located at the intersection of the two intersecting planes. The revolution axis of the concave parabolic rotation surface is located between the parabolic rotation surface and the focal spot of the integrator mirror.
As shown in fig. 2, the plane of the concave parabolic rotation surface where the generatrix intersects with the focus spot of the integrator mirror to form a corresponding focal plane line segment. The focus of the parabola where the bus is located at the intersection point of the connecting line formed by connecting the two end points of the bus and the two end points of the corresponding focal plane line segment in a non-lateral mode. The focal point is located between the generatrix and the focal plane line segment.
As shown in fig. 3, the generatrix of the convex parabolic rotation surface at two boundaries corresponds to the focal plane line segment on the same side boundary of the focal spot of the integrator. The two generatrixes and the corresponding focal plane line segments form two intersected planes respectively. The axis of rotation of the convex parabolic surface of rotation is located at the intersection of the two intersecting planes. The revolution axis of the convex parabolic rotating surface is located on the same side of the parabolic rotating surface as the focal spot.
As shown in fig. 3, the plane of the convex parabolic rotation surface where the generatrix is located intersects with the focal spot of the integrator mirror to form a corresponding focal plane line segment. The focus of the parabola where the bus is located at the intersection point of the connecting line formed by connecting the two end points of the bus and the two end points of the corresponding focal plane line segment on the same side. The focus is positioned at the same side of the generatrix and the focal plane line segment.
The invention provides a parabolic rotating area array integrating mirror, which comprises the following steps:
(1) Determining a generatrix equation
The coordinate plane yoz of the Cartesian coordinate system is intersected with the reflecting surface of the integrator to form a series of parabolic segments connected end to end, and B is set as 0 B 1 ，B 1 B 2 ，B 2 B 3 For each parabolic segment, as shown in FIG. 4, the focus of each segment is F 01 ，F 12 ，F 23 The test is that a focal plane line segment formed by the intersection of a coordinate plane yoz and a focusing light spot of an integrating mirror is CD.
The parabolic equation expressed in focal coordinates is:
(y-y F ) 2 ＝4z F (z+c) (1)
wherein c is a parabolic parameter, y F And z F Respectively the y-coordinate and the z-coordinate of the parabolic focus F.
Let B 0 B 1 Is a parabolic segment at the edge of the integrator mirror and is a concave parabola, the parabolic segment B 0 B 1 Focal point F of 01 Is located on line segment B 0 D and B 1 C at the intersection of C, and point B 0 And B 1 On a parabola, there are
Figure GDA0003928024410000031
Wherein c is 01 Is a parabolic parameter, y F01 And z F01 Respectively a parabolic focus F 01 Y and z coordinates of (c), y B0 And z B0 Are parabolic end points B respectively 0 Y and z coordinates of (2), y B1 And z B1 Are parabolic end points B respectively 1 Y and z coordinates of (2), y C And z C Y and z coordinates, y, of the focal plane line segment CD end point C, respectively D And z D Respectively, the y coordinate and the z coordinate of the end point D of the focal plane line segment CD.
Due to B 0 B 1 Is a parabolic segment at the edge of the integrator mirror, B 0 The point coordinates may be set to a known determined value, i.e. y B0 And z B0 Are known. The width of the paraboloid in the z direction can be set to be a fixed value L,0<L<L CD Therefore has z B1 ＝z B0 + L. In the above equation 4 there are 4 free variables y F01 、z F01 、y B1 And c 01 And the variable y F01 、z F01 、y B1 Is positive, so can solve to y F01 、z F01 、y B1 And c 01 The unique solution of (a). Thereby obtaining the parabolic segment B 0 B 1 The equation of (c).
Second segment parabolic segment B 1 B 2 Is a convex parabola with a focus F 12 Is located on line segment B 1 C and B 2 D at the intersection of the extension lines, and point B 1 And B 2 In a parabolic segment B 1 B 2 Above, there are
Figure GDA0003928024410000032
Wherein c is 12 Is a parabolic parameter, y F12 And z F12 Are parabolic foci F respectively 12 Y coordinate ofAnd z-coordinate, y B1 And z B1 Are parabolic end points B respectively 1 Y and z coordinates of (2), y B2 And z B2 Are parabolic end points B respectively 2 Y-coordinate and z-coordinate. I.e. y B1 And z B1 Is known, z B2 ＝z B1 + L. In the above equation 4 there are 4 free variables y F12 、z F12 、y B2 And c 12 . And the variable y F12 、z F12 And y B2 Positive values. So can solve y F12 、z F12 、y B2 And c 12 The unique solution of (a). Thereby obtaining the parabolic segment B 1 B 2 The equation of (c).
The focal point coordinate (y) of the parabolic segment on the whole integral mirror can be obtained by solving in sequence F ,z F ) Equation and displacement parameter c. Thus obtaining a series of generatrix equations of the parabolic rotating surface.
(2) Determining the rotation axis and the rotation radius
B 'is a series of arc line segments formed by the intersection of the coordinate plane xoy and the reflecting surface of the integrator' 0 B' 1 ，B' 1 B' 2 ，B' 2 B' 3 At the same time, as shown in fig. 5, the center of each arc segment is O 01 ，O 12 ，O 23 The creation, the focal plane line segment formed by the intersection of the coordinate plane xoy and the focusing light spot of the integrator is C 'D'.
(x-x O ) 2 +(y-y O ) 2 ＝R 2 (4)
Wherein x Ox And y Ox Respectively are the y coordinate and the z coordinate of the circle center O of the circular arc, and R is the radius of the circular arc.
B 'is provided' 0 B' 1 Is a parabolic segment at the edge of the integrating mirror and is a concave arc line, and the arc line segment B' 0 B' 1 Center of circle O 01 Is located at line segment B' 0 D 'and B' 1 C 'at the intersection point, and point B' 0 And B' 1 On the circular arc line, there are
Figure GDA0003928024410000041
Wherein R is B'0B'1 Is the radius of rotation of the surface of rotation, x O01 And y O01 Are respectively arc line focus O 01 X and y coordinates of (2), x B'0 And y B'0 Are respectively arc line end points B' 0 X and y coordinates of (2), x B'1 And y B'1 Are respectively arc line end points B' 1 X-coordinate and y-coordinate. x is the number of C' And y C' X and y coordinates, x, of the end C ' of the focal plane segment C ' D ', respectively D' And y D' Respectively, the x coordinate and the y coordinate of the endpoint D ' of the focal plane line segment C ' D '.
Due to B' 0 B' 1 Is a circular arc segment at the edge of the integrating mirror, B' 0 The point coordinates may be set to a known determined value, x B'0 And y B'0 Are known. The width of the circular arc line can be set to be a fixed value L',0<L'<L C'D' Therefore, has x B'1 ＝x B'0 + L'. In the above equation 4 there are 4 free variables x O01 、y O01 、y B'1 And R B'0B'1 And is positive. So can solve x O01 、y O01 、y B'1 And R B'0B'1 The unique solution of (a). Thereby obtaining the arc wire segment B' 0 B' 1 The position and radius of rotation of the axis of rotation of (a).
Second segment parabolic segment B' 1 B' 2 Is a convex parabola with a center O 12 Is located at line segment B' 1 C 'and B' 2 D 'is at the intersection of the extension lines, and point B' 1 And B' 2 At arc line segment B' 1 B' 2 Above, there are
Figure GDA0003928024410000051
Wherein R is B'1B'2 Is the radius of rotation of the surface of rotation, x O12 And y O12 Respectively being arc line focus O 12 X and y coordinates of (2), x B'1 And y B'1 Are respectively arc line end points B' 1 X and y coordinates of (2), x B'2 And y B'2 Are respectively arc line end points B' 2 X-coordinate and y-coordinate.
Due to circular arc segment B' 1 B' 2 And B' 0 B' 1 Tangent, B' 1 The point coordinates have already been obtained. The width of the circular arc line is a fixed value L', i.e. x B'1 And y B'1 Is known as x B'2 ＝x B'1 + L'. In the above equation 4 there are 4 free variables x O12 、y O12 、y B'2 And R B'1B'2 And is positive. So can solve x O12 、y O12 、y B'2 And R B'1B'2 Is determined. Thereby obtaining the position and radius of rotation of the axis of rotation of the region.
Solving in turn can yield the position (x) of a series of rotation axes on the whole integrating mirror O ,y O ) And a radius of rotation R.
(3) The parabola generated in the step (1) is used as the generatrix of the parabolic rotating surface, and the rotating shaft and the rotating radius generated in the step (2) are used for obtaining integral mirror surface-shaped coordinates (x, y, z) according to the following formula,
Figure GDA0003928024410000052
wherein x p And y p The initial positioning coordinate is the grid discrete point coordinate of the projection of the reflecting surface of the integrator on the xoy coordinate plane or the discrete point coordinate on the Archimedes spiral. y is F ，z F And c is the parabolic focus coordinate and parabolic displacement parameter in step (1). x is the number of O ，y O And R is the rotation axis coordinate and the rotation radius in the step (2).
And converting the integral mirror surface shape data into path data of a single-point diamond lathe, and processing the path data into an integral mirror.
Each parabolic rotating surface on the integrating mirror can reflect the incident light of the corresponding area to form a light spot with the same shape and size at the focused light spot. Although the light intensity distribution of the light spots formed by reflection of each parabolic rotating surface is different, the parabolic rotating surface array divides and reflects incident light, and then the reflected light spots are mutually superposed to finally form a focused light spot with relatively uniform light intensity distribution. Theoretically, regardless of the intensity distribution of the incident light, if the incident light is divided into an infinite number of regions and then reflected and superimposed on the focused light spot, a light spot with uniform intensity distribution will be formed.
Drawings
FIG. 1 is an oblique view of an integrator mirror element according to the present invention;
FIG. 2 is a schematic structural view of a concave parabolic rotating surface in the shape of the reflecting surface of the integrator;
FIG. 3 is a schematic view of a convex parabolic rotating surface structure of the reflecting surface shape of the integrator in the present invention;
FIG. 4 is a schematic view of a sectional light path between the reflecting surface of the integrator and the yoz plane in the present invention;
FIG. 5 is a schematic view of a sectional light path between the reflecting surface of the integrator and the xoy coordinate plane in the present invention;
fig. 6 is a three-dimensional network diagram of the integrator obtained in example 1 after the reflection surface of the integrator is rotated by 45 °;
fig. 7 is a sectional line of the reflection surface of the integrator mirror obtained in this example 1 at the xoz surface;
FIG. 8 is a simulated optical path diagram of the integrator mirror of this embodiment 1;
FIG. 9 is a diagram showing the distribution of incident light intensity in the present example 1;
FIG. 10 is the light intensity distribution at the spot focused by the integrator mirror in this embodiment 1;
FIG. 11 is a longitudinal graph of the light intensity distribution at the focused spot of the integrator mirror in this embodiment 1;
FIG. 12 is a transverse intensity distribution curve of the focused light spot of the integrator mirror in this embodiment 1;
fig. 13 is a three-dimensional network diagram of the integrator mirror with a reflection surface rotated by 45 ° obtained in example 2;
FIG. 14 is the light intensity distribution at the spot focused by the integrator mirror in this embodiment 2;
FIG. 15 is a longitudinal graph showing the light intensity distribution at the focal spot of the integrator mirror in this embodiment 2;
FIG. 16 is a transverse intensity distribution curve of the spot focused by the integrator mirror in this embodiment 2;
Detailed Description
The technical solution of the present invention is further described with reference to the accompanying drawings and examples.
Example 1
In the embodiment, the reflecting surface of the integrator is an oblique cutting surface of a cylinder with the radius of 25mm, and as shown in fig. 1, the whole reflecting surface is divided into a series of square lattices with the side length of 7.07mm, and each lattice corresponds to a parabolic rotating surface. The reflecting light path of the integrator is shown in fig. 6, parallel light beams enter the reflecting surface of the integrator from top to bottom, and the focusing light spot of the integrator is positioned in the right horizontal direction and is 300mm away from the center of the cylinder. The focusing light spot is a rectangular light spot with the width of 1mm and the length of 10 mm.
Parabolic end B in equation set (2) 0 Y coordinate of (a), y B0 ＝0，z B0 =0; the width of the paraboloid in the z direction is a fixed value L =5mm, so z B1 =5; coordinates y of focal plane line segment CD end point C point C = -300 and z C Coordinate y of point d =0, D = -300 and z D ＝10。
Get y by solution F01 、z F01 、y B1 C, and c, thereby obtaining the parabolic segment B 0 B 1 Equation (c) of (a), i.e. y can be obtained B1 And z B1 ，z B2 ＝z B1 + L. And then can be solved to y in the equation set (3) F12 、z F12 、y B2 C, and c, thereby obtaining the parabolic segment B 1 B 2 The equation of (c).
And solving the equations of the parabolic segments on the whole integral mirror in sequence to obtain a series of bus equations of the parabolic rotating surfaces.
Endpoint B 'of circular arc line in equation set (5)' 0 Coordinate x of B'00 ＝0，y B'00 =0, the width of the circular arc line is L' =5mm B'1 And (5). Coordinates x of end point C 'of focal plane line segment C' D C' ＝0，y C' Coordinate x of = -300,D D' ＝10，y D' ＝-300。
Thereby obtaining the circular arc segment B' 0 B' 1 The position and radius of rotation of the rotating shaft of (1) may be obtained as a circular arc line end point B' 1 Coordinate x of B'1 And y B'1 The value of (c). Arc line terminal B' 2 X coordinate x of B'2 ＝x B'1 + L'. Thereby obtaining the position and radius of rotation of the axis of rotation of the region.
The sequential solution can result in the positions and rotation radii of a series of rotation axes across the entire integrating mirror.
And obtaining the reflecting surface shape of the parabolic rotating surface array integrator according to the formula (7).
Fig. 6 is a three-dimensional mesh surface diagram of the integrated mirror surface obtained in this example after rotating 45 °, and it can be seen that the integrated mirror surface area is substantially in a band shape.
Fig. 7 is a sectional view of the obtained integrated mirror shape and xoz plane in this embodiment, and it can be seen from the partial position in the figure that: the integral mirror surface shape provided by the invention changes continuously, and the whole surface shape has no curvature mutation. Compared with the prior art, the free-form surface of the curvature change connecting line can reduce the processing difficulty and improve the processing efficiency and the processing precision.
Fitting and inputting the reflecting surface of the integrator into optical simulation software, and simulating by using a light ray tracing method to obtain a focusing light spot light intensity distribution diagram. Fig. 8 is a reflected light path diagram of the integrating mirror in the present embodiment, and fig. 9 is an incident light intensity distribution diagram. It can be seen that: the incident beam has a Gaussian light intensity distribution. FIG. 10 is a diagram showing the light intensity distribution at the spot focused by the integrating mirror in this embodiment. The shape of the focusing spot is a rectangle with the length of 10mm and the width of 1mm, fig. 11 is a longitudinal curve graph of the light intensity distribution at the focusing spot of the integrating mirror in the embodiment, and it can be seen that the light intensity at the edge of the focusing spot is reduced extremely rapidly, and the light intensity outside the focusing spot is almost zero. FIG. 12 is a transverse intensity distribution curve of the spot focused by the integrating mirror in the present embodiment.
Example 2
In the embodiment, the light spot focused by the integrating mirror is a rectangular light spot with the length and the width of 10mm, and other parameters are all the same as those in the embodiment 1. The reflecting surface calculation procedure is also the same as in embodiment 1.
Fig. 13 is a three-dimensional mesh surface diagram of the integrated mirror surface obtained in this example after rotating 45 °, and it can be seen that the integrated mirror surface area is continuously and alternately distributed with concave surfaces and convex surfaces.
The simulated optical path of the integrating mirror in this embodiment is the same as that in embodiment 1, and the light intensity of the incident light beam is the same as that in embodiment 1, that is, the light intensity is gaussian distribution in fig. 9. FIG. 14 is a diagram showing the light intensity distribution at the spot focused by the integrating mirror in this embodiment. The shape of the focusing light spot is a square light spot with the length of 10mm and the width of 10mm, fig. 15 is a longitudinal curve graph of the light intensity distribution at the focusing light spot of the integrating mirror in the embodiment, and it can be seen that the light intensity at the edge of the focusing light spot is almost linearly reduced, and the light intensity outside the focusing light spot is almost zero. FIG. 16 is a transverse intensity distribution curve of the spot focused by the integrator mirror in this embodiment. The variation of the light intensity inside the focusing light spot is about 5% of the light intensity of the whole focusing light spot.

Claims (2)

1. A parabolic rotating area array integrator mirror, characterized in that: the reflecting surface of the integrating mirror is formed by alternately connecting a plurality of concave parabolic rotating surfaces and convex parabolic rotating surfaces; the adjacent concave parabolic rotating surface and the convex parabolic rotating surface are tangent at the boundary; the generatrix of all the parabolic rotating surfaces is a part of a parabola, the plane where the generatrix of all the parabolic rotating surfaces is located is intersected with the focusing light spot of the integrating mirror to form a series of focal plane line segments, and the focus of all the parabolas is located at the intersection point of the two ends of the generatrix of the parabolic rotating surfaces where the parabolas are located and the connecting line of the two ends of the focal plane line segments; the rotating shafts of all the parabolic rotating surfaces are positioned at the intersection line of the generatrix on the boundary of the parabolic rotating surface where the rotating shafts are positioned and the plane where the focal plane line segment on the boundary of the focusing light spot of the integrating mirror is positioned.
2. The parabolic rotating area array integrator mirror of claim 1, wherein the design comprises the steps of:
(1) Determining a generatrix equation
The coordinate plane yoz of Cartesian coordinate system is intersected with the reflecting surface of the integrator to form a series of parabolic segments connected end to end, and B is set as 0 B 1 ，B 1 B 2 ，B 2 B 3 At each parabolic section, the focus is F 01 ，F 12 ，F 23 The creation is that a focal plane line segment formed by the intersection of a coordinate plane yoz and a focusing light spot of an integrating mirror is CD;
let B 0 B 1 Is a parabolic segment at the edge of the integrator mirror and is a concave parabola, the parabolic segment B 0 B 1 Focal point F of 01 Is located on line segment B 0 D and B 1 C at the intersection point, and point B 0 And B 1 On the parabola, there is
Figure FDA0003928024400000011
Wherein c is 01 Is a parabolic parameter, y F01 And z F01 Are parabolic foci F respectively 01 Y and z coordinates of (2), y B0 And z B0 Are parabolic end points B respectively 0 Y and z coordinates of (2), y B1 And z B1 Are parabolic end points B respectively 1 Y and z coordinates of (2), y C And z C Y and z coordinates, y, of the focal plane line segment CD end point C, respectively D And z D Respectively the y coordinate and the z coordinate of the end point D of the focal plane line segment CD;
due to B 0 B 1 Is a parabolic segment at the edge of the integrator mirror, B 0 The point coordinates may be set to a known determined value, i.e. y B0 And z B0 To be known, the width of the paraboloid in the z direction is set to a fixed value L,0<L<L CD Z is therefore B1 ＝z B0 + L, 4 free variables y in the 4 equations of formula (I) above F01 、z F01 、y B1 And c 01 And the variable y F01 、z F01 、y B1 Is positive, so can solve to y F01 、z F01 、y B1 And c 01 Thereby obtaining the parabolic segment B 0 B 1 The equation of (c);
second segment parabolic segment B 1 B 2 Is a convex parabola with a focus F 12 Is located on line segment B 1 C and B 2 D on the intersection of the extension lines, and point B 1 And B 2 In a parabolic segment B 1 B 2 Above, there are
Figure FDA0003928024400000021
Wherein c is 12 Is a parabolic parameter, y F12 And z F12 Respectively a parabolic focus F 12 Y and z coordinates of (c), y B1 And z B1 Are parabolic end points B respectively 1 Y and z coordinates of (c), y B2 And z B2 Are parabolic end points B respectively 2 Y and z coordinates of (a), i.e. y B1 And z B1 Is known, z B2 ＝z B1 + L, 4 free variables y in the 4 equations of the above formula (II) F12 、z F12 、y B2 And c 12 And the variable y F12 、z F12 And y B2 Is positive, so can solve to y F12 、z F12 、y B2 And c 12 Thereby obtaining the parabolic segment B 1 B 2 The equation of (c);
the focal point coordinate (y) of the parabolic segment on the whole integral mirror can be obtained by solving in sequence F ,z F ) Equation and displacement parameter c, namely obtaining a bus equation of a series of parabolic rotating surfaces;
(2) Determining the rotation axis and the radius of rotation
B 'is a series of arc line segments formed by the intersection of the coordinate plane xoy and the reflecting surface of the integrator' 0 B' 1 ，B' 1 B' 2 ，B' 2 B' 3 In every section of arc segment, the circle center is O 01 ，O 12 ，O 23 A focal plane line segment formed by intersection of the coordinate plane xoy and the focusing light spot of the integral mirror is C 'D';
b 'is provided' 0 B' 1 Is a parabolic segment at the edge of the integrating mirror and is a concave arc line, and the arc line segment is B' 0 B' 1 Center of circle O of 01 Is located at line segment B' 0 D 'and B' 1 C 'at the intersection point, and point B' 0 And B' 1 On the circular arc line, there are
Figure FDA0003928024400000031
Wherein R is B'0B'1 Is the radius of rotation of the surface of rotation, x O01 And y O01 Are respectively arc line focus O 01 X and y coordinates of (2), x B'0 And y B'0 Are respectively arc line end points B' 0 X and y coordinates of (2), x B'1 And y B'1 Are respectively arc line end points B' 1 X and y coordinates of (2), x C' And y C' X and y coordinates, x, of the end C ' of the focal plane segment C ' D ', respectively D' And y D' Respectively an x coordinate and a y coordinate of an end point D ' of the focal plane line segment C ' D ';
due to B' 0 B' 1 Is a circular arc segment at the edge of the integrating mirror, B' 0 The point coordinates are set to a known determined value, i.e. x B'0 And y B'0 Setting the width of the circular arc line as a fixed value L',0<L'<L C'D' Therefore has x B'1 ＝x B'0 + L', 4 free variables x in the 4 equations of the above formula (III) O01 、y O01 、y B'1 And R B'0B'1 And is positive, so x can be solved O01 、y O01 、y B'1 And R B'0B'1 Thereby obtaining the circular arc line segment B' 0 B' 1 The position and radius of rotation of the rotating shaft of (a);
second segment parabolic segment B' 1 B' 2 Is a convex parabola with the center O 12 Is located at line segment B' 1 C 'and B' 2 D 'is at the intersection of the extension lines, and point B' 1 And B' 2 In a circular arc line segment B' 1 B' 2 Above, there are
Figure FDA0003928024400000032
Wherein R is B'1B'2 Is the radius of rotation of the surface of rotation, x O12 And y O12 Respectively being arc line focus O 12 X and y coordinates of (2), x B'1 And y B'1 Are respectively arc line end points B' 1 X and y coordinates of (2), x B'2 And y B'2 Respectively being an end point of a circular arc lineB' 2 The x-coordinate and the y-coordinate of (c),
due to circular arc segment B' 1 B' 2 Tangent to B '0B '1, B ' 1 The point coordinates are obtained, the width of the circular arc line is a fixed value L', namely x B'1 And y B'1 Is known as x B'2 ＝x B'1 + L', 4 free variables x in the 4 equations of the above formula (IV) O12 、y O12 、y B'2 And R B'1B'2 And is positive, so x can be solved O12 、y O12 、y B'2 And R B'1B'2 Thereby obtaining the position and the radius of rotation of the rotation axis of the corresponding region;
solving in turn can yield the position (x) of a series of rotation axes on the whole integrating mirror O ,y O ) And a radius of rotation R;
(3) The parabola generated in the step (1) is used as the generatrix of the parabolic rotating surface, and the rotating shaft and the rotating radius generated in the step (2) are used for obtaining integral mirror surface-shaped coordinates (x, y, z) according to the following formula,
Figure FDA0003928024400000041
wherein x p And y p Is an initial positioning coordinate whose value is a grid discrete point coordinate of the projection of the reflecting surface of the integrator on the xoy coordinate plane or a discrete point coordinate on the Archimedes spiral, y F ，z F And c is the parabolic focus coordinate and parabolic displacement parameter, x, in step (1) O ，y O And R is the rotation axis coordinate and the rotation radius in the step (2).
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