CN112114393A - Parabolic rotating surface array integrator and design method thereof - Google Patents
Parabolic rotating surface array integrator and design method thereof Download PDFInfo
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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
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 integrating 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 provided in document 1 (Ringyong, Wajia Sheng, Wuke difficultly, a diffraction optical element design algorithm [ J ] optics report, 2007,27(9): 1682-. Reference 2(ZL01278581.4) discloses a focus shaping integrating mirror composed of multi-band reflecting curved surfaces, and does not refer to the surface shape characteristics of the reflecting bands and the related methods. Reference 3(ZL200520127504) 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, a design of a belt integrator for high-power laser processing [ J ]. university of beijing university of industry (3): 334-. 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 mirror is formed by alternately connecting a plurality of concave parabolic rotating surfaces and convex parabolic rotating surfaces. All concave parabolic surfaces of rotation 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 rotating surface is located between the parabolic rotating surface and the focusing light 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 rotating shaft of the convex parabolic rotating surface is positioned on the same side of the parabolic rotating surface and the focal spot.
As shown in fig. 3, the plane of the convex parabolic rotating surface where the generatrix is located intersects with the focus light spot of the integrator 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 as0B1，B1B2，B2B3.., as shown in FIG. 4, the focus of each parabolic segment is F01，F12，F23,., the focal plane line segment formed by the intersection of the coordinate plane yoz and the focal spot of the integrator mirror is CD.
The parabolic equation expressed in focal coordinates is:
(y-yF)2＝4zF(z+c) (1)
wherein c is a parabolic parameter, yFAnd zFRespectively the y-coordinate and the z-coordinate of the parabolic focus F.
Let B0B1Is a parabolic segment at the edge of the integrator mirror and is a concave parabola, the parabolic segment B0B1Focal point F of01Is located on line segment B0D and B1C at the intersection point, and point B0And B1On the parabola, there is
Wherein c is01Is a parabolic parameter, yF01And zF01Respectively a parabolic focus F01Y and z coordinates of (2), yB0And zB0Are parabolic end points B respectively0Y and z coordinates of (2), yB1And zB1Are parabolic end points B respectively1Y and z coordinates of (2), yCAnd zCY and z coordinates, y, of the focal plane line segment CD end point C, respectivelyDAnd zDRespectively the y-coordinate and the z-coordinate of the end point D of the focal plane line segment CD.
Due to B0B1Is a parabolic segment at the edge of the integrator mirror, B0The point coordinates may be set to a known determined value, i.e. yB0And zB0Are known. The width of the paraboloid in the z direction can be set to beA fixed value L, 0<L<LCDTherefore has zB1＝zB0+ L. In the above equation 4 there are 4 free variables yF01、zF01、yB1C, and the variable yF01、zF01、yB1Is positive, so can solve to yF01、zF01、yB1And c01Is determined. Thereby obtaining the parabolic segment B0B1The equation of (c).
Second segment parabolic segment B1B2Is a convex parabola with a focus F12Is located on line segment B1C and B2D at the intersection of the extension lines, and point B1And B2In a parabolic segment B1B2Above, there are
Wherein c is12Is a parabolic parameter, yF12And zF12Respectively a parabolic focus F12Y and z coordinates of (2), yB1And zB1Are parabolic end points B respectively1Y and z coordinates of (2), yB2And zB2Are parabolic end points B respectively2Y-coordinate and z-coordinate. I.e. yB1And zB1Is known, zB2＝zB1+ L. In the above equation 4 there are 4 free variables yF12、zF12、yB2And cc12. And the variable yF12、zF12And yB2Positive values. So can solve yF12、zF12、yB2And c. Thereby obtaining the parabolic segment B1B2The equation of (c).
The focal point coordinate (y) of the parabolic segment on the whole integral mirror can be obtained by solving in sequenceF，zF) Equations and displacement parameters c. Thus obtaining a series of generatrix equations of the parabolic rotating surface.
(2) Determining the rotation axis and the radius of rotation
The coordinate plane xoy is intersected with the reflecting surface of the integrator to form a series of arc line segmentsIs set to B'0B′1，B′1B′2，B′2B′3.., as shown in FIG. 5, the focus of each parabolic segment is O01，O12，O23,., the focal plane line segment formed by the intersection of the coordinate plane xoy and the focusing light spot of the integrator mirror is C 'D'.
(x-xO)2+(y-yO)2＝R2 (4)
Wherein xOxAnd yOxRespectively 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.
Is B'0B′1Is a parabolic segment at the edge of the integrating mirror and is a concave arc line, and the arc line segment B'0B′1Center of circle O of01Is located at line segment B'0D 'and B'1C 'at the intersection point, and point B'0And B'1On the circular arc line, there are
Wherein R isB′0B′1Is the radius of rotation of the surface of revolution, xO01And yO01Respectively being arc line focus O01X and y coordinates of (2), xB′0And yB′0Are respectively arc line end points B'0X and y coordinates of (2), xB′1And yB′1Are respectively arc line end points B'1X-coordinate and y-coordinate. x is the number ofC′And yC′X and y coordinates, x, of the end C ' of the focal plane segment C ' D ', respectivelyD′And yD′Respectively, the x coordinate and the y coordinate of the endpoint D ' of the focal plane line segment C ' D '.
Due to B'0B′1Is a circular arc segment at the edge of the integrating mirror, B'0The point coordinates may be set to a known determined value, xB′0And yB′0Are known. The width of the circular arc line can be set to be a fixed value L ', 0 < L' < LC′D′Therefore, has xB′1＝xB′0+ L'. In the above equation 4 are4 free variables xO01、yO01、yB′1And RB′0B′1And is positive. So can solve xO01、yO01、yB′1And RB′0B′1Is determined. Thereby obtaining the arc wire segment B'0B′1The position and radius of rotation of the axis of rotation of (a).
Second segment parabolic segment B'1B′2Is a convex parabola with a center O12Is located at line segment B'1C 'and B'2D 'is at the intersection of the extension lines, and point B'1And B'2At arc line segment B'1B′2Above, there are
Wherein R isB'1B'2Is the radius of rotation of the surface of revolution, xO12And yO12Respectively being arc line focus O12X and y coordinates of (2), xB'1And yB'1Are respectively arc line end points B'1X and y coordinates of (2), xB'2And yB'2Are respectively arc line end points B'2X-coordinate and y-coordinate.
Due to circular arc segment B'1B'2And B'0B'1Tangent, B'1The point coordinates have already been obtained. The width of the circular arc line is a fixed value L', i.e. xB'1And yB'1Is known as xB'2＝xB'1+ L'. In the above equation 4 there are 4 free variables xO12、yO12、yB'2And RB'1B'2And is positive. So can solve xO12、yO12、yB'2And RB'1B'2Is determined. Thereby obtaining the position and radius of rotation of the axis of rotation of the region.
Solving in turn can result in the position (x) of a series of rotation axes on the whole integrating mirrorO,yO) 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,
wherein xpAnd ypThe 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 isF，zFAnd c is the parabolic focus coordinate and parabolic displacement parameter in step (1). x is the number ofO，yOAnd 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 light spots with the same shape and size at the focused light spots. 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 structural view of a convex parabolic rotating surface of the reflecting surface 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 mirror with a reflection surface rotated by 45 ° obtained in example 1;
fig. 7 is a sectional view of the reflecting surface of the integrator obtained in example 1 at the xoz plane;
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 embodiment 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 spot focused by 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 embodiments.
Example 1
In this embodiment, the reflecting surface of the integrator is a chamfered surface of a cylinder with a radius of 25mm, and as shown in fig. 1, the whole reflecting surface is divided into a series of square lattices with a 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)0Y coordinate of (a), yB0＝0，z B00; since the width of the paraboloid in the z direction is fixed to 5mm, z is set toB1(ii) 5; coordinate y of focal plane line segment CD endpoint C pointC-300 and z C0, coordinate y of point DD-300 and zD＝10。
Get y by solutionF01、zF01、yB1C, and c, thereby obtaining the parabolic segment B0B1Equation (c) of (a), i.e. y can be obtainedB1And zB1，zB2＝zB1+ L. And then can be solved to y in the equation set (3)F12、zF12、yB2C, and c, thereby obtaining the parabolic segment B1B2The 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.
Equation set (5) middle circular arc endpoint B'0Coordinate x ofB'00＝0，y B'000, 5mm width of the circular arc line, xB'15. Coordinates x of end point C 'of focal plane line segment C' DC'＝0，yC'Coordinate x of-300, DD'＝10，yD'＝-300。
Thereby obtaining the arc wire segment B'0B'1The position and radius of rotation of the rotating shaft of (1) may be obtained as a circular arc line end point B'1Coordinate x ofB'1And yB'1The value of (c). Arc line terminal B'2X coordinate x ofB'2＝xB'1+ L'. Thereby obtaining the position and radius of rotation of the axis of rotation of the region.
And sequentially solving to obtain the positions and the rotating radii of a series of rotating shafts on the whole 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 integrator mirror shape obtained in this embodiment and xoz, which is partially shown in the figure: the integral mirror surface shape provided by the invention continuously changes, 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 integrator mirror in this 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 light path of the integrating mirror in this embodiment is the same as that in embodiment 1, and the incident beam intensity 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 (7)
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, and all the concave parabolic rotating surfaces and the convex parabolic rotating surfaces are tangent at the boundary.
2. The concave parabolic rotating surface and the convex parabolic rotating surface of claim 1, wherein: 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 focal point of the parabola is located at the intersection of the connecting line of the two end points of the generatrix and the two end points of the focal plane line segments, and the rotating shafts of all the parabolic rotating surfaces are located at the intersection of the generatrix on the boundary of the parabolic rotating surfaces and the plane where the focal plane line segments on the boundary of the focusing light spot of the integrating mirror are located.
3. The concave parabolic surface of rotation of claim 1, wherein: the generatrix of two boundaries of the concave parabolic rotating surface corresponds to the focal plane line segment on the boundary of the different side of the focusing light spot of the integrating mirror, the two generatrixes respectively form two intersecting planes with the corresponding focal plane line segment, the rotating shaft of the concave parabolic rotating surface is positioned at the intersection line of the two intersecting planes, and the rotating shaft of the concave parabolic rotating surface is positioned between the parabolic rotating surface and the focusing light spot of the integrating mirror.
4. The concave parabolic surface of rotation of claim 1, wherein: the plane of the generatrix of the concave parabolic rotating surface is intersected with the focusing light spot of the integrating mirror to form a corresponding focal plane line segment, the focus of the parabola with the generatrix is positioned at the intersection point of connecting lines formed by connecting two end points of the generatrix and the opposite sides of two end points of the corresponding focal plane line segment, and the focus is positioned between the generatrix and the focal plane line segment.
5. The convex parabolic rotating surface of claim 1, wherein: the generatrix of two boundaries of the convex parabolic rotating surface corresponds to the focal plane line segment on the same side boundary of the focusing light spot of the integrating mirror, the two generatrixes respectively form two intersecting planes with the corresponding focal plane line segment, the rotating shaft of the convex parabolic rotating surface is positioned at the intersection of the two intersecting planes, and the rotating shaft of the convex parabolic rotating surface is positioned at the same side of the parabolic rotating surface and the focal light spot.
6. The convex parabolic rotating surface of claim 1, wherein: the plane of the bus of the convex parabolic rotating surface is intersected with the focusing light spot of the integrating mirror to form a corresponding focal plane line segment, the focus of the parabola with the bus is located at the intersection point of a connecting line formed by connecting two end points of the bus and two end points of the corresponding focal plane line segment on the same side, and the focus is located on the same side of the bus and the focal plane line segment.
7. The parabolic belt integrator of claim 1, wherein the design comprises the steps of:
(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 as0B1，B1B2，B2B3.., the focus of each parabolic segment is F01，F12，F23,., the focal plane line segment formed by the intersection of the coordinate plane yoz and the focusing light spot of the integrator mirror is CD;
let B0B1Is a parabolic segment at the edge of the integrator mirror and is a concave parabola, the parabolic segment B0B1Focal point F of01Is located on line segment B0D and B1C at the intersection point, and point B0And B1On the parabola, there is
Wherein c is01Is a parabolic parameter, yF01And zF01Respectively a parabolic focus F01Y and z coordinates of (2), yB0And zB0Are parabolic end points B respectively0Y and z coordinates of (2), yB1And zB1Are parabolic end points B respectively1Y and z coordinates of (2), yCAnd zCAre respectively cokeY and z coordinates of the surface segment CD end C, yDAnd zDRespectively a y coordinate and a z coordinate of an end point D of the focal plane line segment CD;
due to B0B1Is a parabolic segment at the edge of the integrator mirror, B0The point coordinates may be set to a known determined value, i.e. yB0And zB0As is known, the width of the paraboloid in the z direction can be set to a fixed value L, 0<L<LCDZ is thereforeB1＝zB0+ L, there are 4 free variables y in the 4 equations aboveF01、zF01、yB1C, and the variable yF01、zF01、yB1Is positive, so can solve to yF01、zF01、yB1And c01Thereby obtaining the parabolic segment B0B1The equation of (c);
second segment parabolic segment B1B2Is a convex parabola with a focus F12Is located on line segment B1C and B2D at the intersection of the extension lines, and point B1And B2In a parabolic segment B1B2Above, there are
Wherein c is12Is a parabolic parameter, yF12And zF12Respectively a parabolic focus F12Y and z coordinates of (2), yB1And zB1Are parabolic end points B respectively1Y and z coordinates of (2), yB2And zB2Are parabolic end points B respectively2Y and z coordinates of (i.e. y)B1And zB1Is known, zB2＝zB1+ L, there are 4 free variables y in the 4 equations aboveF12、zF12、yB2And c12And the variable yF12、zF12And yB2Is positive, so can solve to yF12、zF12、yB2C, and c, thereby obtaining the parabolic segment B1B2The equation of (c);
the focal point coordinate (y) of the parabolic segment on the whole integral mirror can be obtained by solving in sequenceF,zF) 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'0B'1，B'1B'2，B'2B'3.., the focus of each parabolic segment is O01，O12，O23,., a focal plane line segment formed by the intersection of the coordinate plane xoy and the focusing light spot of the integrator is C 'D';
is B'0B'1Is a parabolic segment at the edge of the integrating mirror and is a concave arc line, and the arc line segment B'0B'1Center of circle O of01Is located at line segment B'0D 'and B'1C 'at the intersection point, and point B'0And B'1On the circular arc line, there are
Wherein R isB'0B'1Is the radius of rotation of the surface of revolution, xO01And yO01Respectively being arc line focus O01X and y coordinates of (2), xB'0And yB'0Are respectively arc line end points B'0X and y coordinates of (2), xB'1And yB'1Are respectively arc line end points B'1X and y coordinates of (2), xC'And yC'X and y coordinates, x, of the end C ' of the focal plane segment C ' D ', respectivelyD'And yD'Respectively representing the x coordinate and the y coordinate of an endpoint D ' of the focal plane line segment C ' D ';
due to B'0B'1Is a circular arc segment at the edge of the integrating mirror, B'0The point coordinates may be set to a known determined value, xB'0And yB'0As is known, the width of the circular arc line can be set to a fixed value L', 0<L'<LC'D'Therefore, has xB'1＝xB'0+ L', there are 4 free variables x in the above equation 4O01、yO01、yB'1And RB'0B'1And is positive, so x can be solvedO01、yO01、yB'1And RB'0B'1Is obtained from the circular arc segment B'0B'1The position and radius of rotation of the rotating shaft of (a);
second segment parabolic segment B'1B'2Is a convex parabola with a center O12Is located at line segment B'1C 'and B'2D 'is at the intersection of the extension lines, and point B'1And B'2At arc line segment B'1B'2Above, there are
Wherein R isB'1B'2Is the radius of rotation of the surface of revolution, xO12And yO12Respectively being arc line focus O12X and y coordinates of (2), xB'1And yB'1Are respectively arc line end points B'1X and y coordinates of (2), xB'2And yB'2Are respectively arc line end points B'2The x-coordinate and the y-coordinate of (c),
due to circular arc segment B'1B'2And B'0B'1Tangent, B'1The point coordinates have been obtained, the width of the circular arc line being a fixed value L', i.e. xB'1And yB'1Is known as xB'2＝xB'1+ L', there are 4 free variables x in the above equation 4O12、yO12、yB'2And RB'1B'2And is positive, so x can be solvedO12、yO12、yB'2And RB'1B'2Thereby obtaining the position and radius of rotation of the axis of rotation of the region;
solving in turn can result in the position (x) of a series of rotation axes on the whole integrating mirrorO,yO) 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,
wherein xpAnd ypIs 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, yF，zFAnd c is the parabolic focus coordinate and parabolic displacement parameter, x, in step (1)O，yOAnd R is the rotation axis coordinate and the rotation radius in the step (2).
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