METHOD FOR DESIGNING MIRROR AND ASTAGITABER CONTROL MIRROR HAVING REFLECTING SURFACE WHEREIN THE DESIGN EQUATION IN THE METHOD APPLY

Abstract

To provide a method for designing a single mirror that allows individual setting of a light source position and a condensation position in the vertical direction and the horizontal direction, and allows free conversion of astigmatism according to the setting.SOLUTION: In a method for designing a mirror, an arbitrary point on a reflection surface is taken as M; the coordinates of the intersection of a sagittal light source ray and an incident ray on the point and the intersection of a meridional light source ray and the incident ray on the point are represented by using L1s, L1m; the coordinates of the intersection of an emission ray from the same point and a sagittal condensed ray and the intersection of the emission ray from the point and a meridional condensed ray are represented by using L2s, L2m; and a mirror is designed by using a design formula of the reflection surface derived based on these coordinates and the fact that a light path length from a light source position to a condensation position is constant in regard to arbitrary points on the reflection surface for condensation in a sagittal direction and condensation in a meridional direction.SELECTED DRAWING: Figure 1

Description

[Technical field]

[0001] The present invention relates to a method for designing a mirror produced by forming a reflecting surface on the surface of a plate material, and to an astigmatism control mirror equipped with a reflecting surface for which a design equation in the design method holds true. [Background technology]

[0002] Synchrotron radiation soft X-ray beams have different characteristics in the vertical and horizontal directions. The beam size tends to be smaller in the vertical direction than in the horizontal direction. The coherent width is larger in the vertical direction than in the horizontal direction. Furthermore, in spectroscopic systems using diffraction gratings, which are widely used in soft X-ray beamlines, the divergence angle of the beam in the vertical direction increases. In addition, the spectrometers containing diffraction gratings used focus soft X-rays only in the spectral direction, resulting in "astigmatism," where the light source position is different in the spectral direction and in non-focusing directions.

[0003] Conventionally, as an optical system that handles beams with different characteristics in the vertical and horizontal directions, a two-stage focusing optical system that arranges two mirrors that handle each direction, and sets the light source point independently in the vertical and horizontal directions to make the focusing point coincident, has been used. Specifically, a method of arranging two elliptical cylindrical mirrors vertically and horizontally, a method of establishing an approximate shape by arranging two bent mirrors (mechanically bent cylindrical mirrors), and a method of arranging a bent mirror and a sagittal cylindrical mirror facing each other in the horizontal direction are known. However, such a method of combining two mirrors makes the mechanism complicated, and the number of chambers increases, which increases costs and makes adjustment difficult.

[0004] A toroidal mirror is a single mirror that may be able to eliminate astigmatism (Non-Patent Document 1). However, a toroidal mirror is an approximation of an existing ellipsoidal mirror, and is easily manufactured by setting a uniform radius of curvature in both the long and short directions of the reflecting surface. Even if it can eliminate astigmatism, it has the drawback that the focused size increases in principle.

[0005] Astigmatic off-axis mirrors (AO mirrors) have also been proposed as mirrors that can reduce the focused size compared to toroidal mirrors and can set independent light sources and focusing points vertically and horizontally (Non-Patent Document 2). This mirror is shaped to find a curved surface that smoothly connects different conic sections in the long and short directions, based on the principle that an elliptical curve is used as the ridgeline of the reflecting surface to focus a beam diverging from one point to another, a parabola is used to parallelize a beam diverging from one point, and a hyperbola is used to convert a beam focused toward one point into a beam focused toward another point.

[0006] However, this AO mirror is defined by rotating the longitudinal conic section profile around the straight line (major axis) connecting the foci of the transverse conic section to obtain a curved surface, and since the reflecting surface is approximated to an axially symmetric shape, this approximation creates a limit to the reduction of the focused size. This is not a problem for beams in the terahertz range with long wavelengths, but it cannot be used for beams in the X-ray range. In addition, the design formula is very complicated, including multiple coordinate transformations, and the parameters are also complicated, making it difficult to understand and use. [Prior art documents] [Non-patent literature]

[0007] [Non-Patent Document 1] William A. Rense, T. Violett, "Method of Increasing the Speed ​​of a Grazing-Incidence Spectrograph", JOURNAL OF THE OPTICAL SOCIETY OF AMERICA, Vol. 49, No. 2, February 1959, p139-p141 [Non-Patent Document 2] A. Wagner-Gentner, UU Graf, M. Philipp, D. Rabanus, "A simple method to design astigmatic off-axis mirrors", Infrared Physics & Technology 50, 2007, p42-p46 Summary of the Invention [Problem to be solved by the invention]

[0008] In view of the above-mentioned circumstances, the present invention seeks to provide a method for designing a mirror that can be used to fabricate a single mirror in which the light source position and the focusing position can be set independently in the vertical and horizontal directions, thereby enabling free conversion of astigmatism, and which can also accommodate beams in the X-ray region by keeping the focusing size small, has a simple design formula, has a wide range of applications, and can be suitably used as an optical system for handling beams whose characteristics differ between the vertical and horizontal directions. [Means for solving the problem]

[0009] In view of the current situation, the inventors conducted extensive research and found that, as a method of expressing the properties of a beam having astigmatism in terms of geometrical optics, a "source ray" and a "focusing ray" can be newly defined for focusing in the sagittal direction and focusing in the meridional direction, respectively, and that all incident light rays passing through the reflective surface of the mirror pass through each of the "source rays" in the vertical and horizontal directions, and all outgoing light rays emitted from the reflective surface of the mirror pass through the "focusing rays" in the vertical and horizontal directions. By applying Fermat's principle, in which the "optical path length" from the light source position to the focusing position is constant, it is possible to provide a design method for a mirror that can solve the above-mentioned problems, and thus completed the present invention.

[0010] That is, the present invention includes the following inventions. (1) A method for designing a mirror made by forming a reflecting surface on the surface of a plate material, in which the optical axis of the incident beam on the mirror is z 1 axis, and the cross section perpendicular to it is x 1 y 1 The optical axis of the beam emitted from the mirror is z 2 axis, and the cross section perpendicular to it is x 2 y 2 Let the plane be x 1 Axis and x 2 The axis is parallel to the sagittal direction of the reflecting surface, and the incident beam is z 1 On-axis z 1 Axis and z 2 Intersection point M on the reflecting surface with the axis 0 From z 1 Along the axial direction L 1s A light source for sagittal focusing is provided at the displaced position, and the z 1 The intersection point M on the axis 0 From z 1 Along the axial direction L 1m At the displaced position, a light source for focusing in the meridional direction is provided, and the output beam is focused in the sagittal direction at the z 2 The intersection point M on the axis 0 From z 2 Along the axial direction L 2s The light is focused at a displaced position, and the z 2 The intersection point M on the axis 0 From z2 Along the axial direction L 2m The light is focused at a displaced position, and all incident rays via the mirror pass through the source position in the sagittal focus at x 1 Axis and z 1 A sagittal light source line extending in a direction perpendicular to both axes and a meridional light source line passing through the position of the light source in the focusing direction are 1 Axis and z 1 All the exit rays that pass through the meridional light source rays extending in a direction perpendicular to both axes and are emitted from the mirror pass through the focusing position in the sagittal focusing direction x 2 Axis and z 2 A sagittal focusing line extending in a direction perpendicular to both axes and a meridional focusing line passing through the focusing position in the y direction 2 Axis and z 2 Let M be an arbitrary point on the reflecting surface of the mirror. The coordinates of the intersection of the sagittal light source line and the incident light beam to point M, and the intersection of the meridional light source line and the incident light beam to point M are expressed as the coordinates of the intersection of the sagittal light source line and the incident light beam to point M. 1s , L 1m The coordinates of the intersection of the outgoing ray from the same point M and the sagittal condensed ray and the intersection of the outgoing ray from the same point M and the meridional condensed ray are expressed as 2s , L 2m and designing the mirror using a design equation for the reflecting surface derived based on these coordinates and the fact that the optical path length from the light source position to the focusing position is constant for any point on the reflecting surface for focusing in the sagittal direction and focusing in the meridional direction.

[0011] (2) The sagittal light source line and the meridional light source line are respectively expressed as y 1 Straight line S extending in the axial direction s , x 1 Straight line S extending in the axial direction m The sagittal and meridional focusing rays are respectively defined as y 2 Straight line F extending in the axial direction s , x 2 Straight line F extending in the axial direction mThe incident length from the light source position to point M for the sagittal direction focusing is the meridional light source line S m and z 1 Intersection point P with axis m0 The sagittal light source line S s and z 1 Intersection point P with axis s0 Through x 1 The arc extending perpendicular to the axis is called the sagittal light source line S s The rotational arc surface rotated around the axis is called equiphase surface A. 1s The incident light and the equiphase surface A 1s The meridional light ray S is one of the two intersection points with m The exit length from point M to the focusing position for the sagittal focusing is calculated as the distance from the intersection point on the side closest to the meridional focusing line F m and z 2 Intersection point Q with axis m0 The sagittal focusing ray F s and z 2 Intersection point Q with axis s0 Through x 2 The circular arc extending perpendicular to the axis is called the sagittal focusing ray F s The rotational arc surface rotated around the axis is called equiphase surface A. 2s The outgoing light ray and the equiphase surface A 2s Of the two intersections with the meridional condensing ray F m The incident length from the light source position to point M for the light collection in the meridional direction is calculated as the distance from the intersection point on the side closer to the sagittal light source line S s and z 1 Intersection point P with axis s0 The meridional light source line S m and z 1 Intersection point P with axis m0 Through y 1 The arc extending perpendicular to the axis is called the meridional light ray S m The rotational arc surface rotated around the axis is called equiphase surface A. 1m The incident light and the equiphase surface A 1m Of the two intersections with the sagittal light source line S sThe exit length from point M to the focusing position for the sagittal focusing line F s and z 2 Intersection point Q with axis s0 The meridional focusing ray F m and z 2 Intersection point Q with axis m0 Through y 2 The circular arc extending perpendicular to the axis is called the meridional condensing ray F m The rotational arc surface rotated around the axis is called equiphase surface A. 2m The outgoing light ray and the equiphase surface A 2m Of the two intersections with the sagittal focusing ray F s The method for designing a mirror according to (1), further comprising the steps of: determining the distance from the intersection point closer to point M to point M; and calculating the optical path lengths for the focusing in the sagittal direction and the focusing in the meridional direction.

[0012] (3) The incident light and equiphase surface A 1s The meridional light ray S is one of the two intersection points with m The distance from the intersection point closest to point M is the distance between the incident ray and the sagittal light source line S s Intersection point P s The distance from the point M to the intersection point P is calculated. s From the equiphase surface A 1s The distance to the arc that defines the exit ray is added or subtracted to obtain the equiphase surface A 2s Of the two intersections with the meridional condensing ray F m The distance from the intersection point closest to point M is the distance between the exit ray and the sagittal focusing ray F s Intersection point Q s The distance from the point M to the intersection point Q is calculated. s From the equiphase surface A 2s The distance to the arc that defines the incident light is added or subtracted to obtain the equiphase surface A 1m Of the two intersections with the sagittal light source line S s The distance from the intersection point closer to point M is the distance between the incident ray and the meridional light source line S m Intersection point Pm The distance from the point M to the intersection point P is calculated. m From the equiphase surface A 1m The distance to the arc that defines the exit ray is added or subtracted to obtain the equiphase surface A 2m Of the two intersections with the sagittal focusing ray F s The distance from the intersection point closer to point M is the distance between the exit ray and the meridional focusing ray F m Intersection point Q m The distance from the point M to the intersection point Q is calculated. m From the equiphase surface A 2m The method for designing a mirror according to (2), further comprising the steps of: adding or subtracting the distance to the arc defining said mirror;

[0013] (4) z 1 Axis and z 2 Intersection point M on the reflecting surface of the axis 0 The surface in contact with the reflecting surface is a uv plane, and the M 0 The direction of the normal passing through the z axis is the w axis, and the v axis is the z axis. 1 axis and z 2 The u axis is perpendicular to both the v axis and the w axis, and the intersection point M 0 the origin, the uv plane and the optical axis z 1 The oblique incidence angle between 0 A rectangular coordinate system is defined based on the mirror, and the coordinates are expressed as x 1 y 1 z 1 Coordinate system, x relative to the optical axis of the output beam 2 y 2 z 2 and expressing the design equation in a uvw coordinate system.

[0014] (5) The first formula f is derived from the fact that the optical path length from the light source point to the focusing point is constant for focusing in the sagittal direction. s The second formula f is derived from (u,v,w)=0 and the fact that the optical path length from the light source point to the focusing point is constant for the focusing in the meridional direction.m 5. The method for designing a mirror according to claim 4, comprising the following formula (1) with (u,v,w)=0 weighted:

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[0015] (6) A mirror having a reflecting surface for which the design formula according to any one of (1) to (5) is satisfied, 1s and L 1m The values ​​of L are different. 2s and L 2m An astigmatism control mirror, which has the same value of and can obtain a focused output beam from an input beam with astigmatism.

[0016] (7) A mirror having a reflecting surface for which the design formula according to any one of (1) to (5) is satisfied, 1s and L 1m The values ​​of L 2s and L 2m An astigmatism controlled mirror has different values ​​of , resulting in an astigmatism-bearing output beam from an input beam diverging from a single point.

[0017] (8) A mirror having a reflecting surface for which the design formula according to any one of (1) to (5) is satisfied, 1m and L 2m The value of L is positive or negative infinity, and 1s and L 2s is a given value (where L 1s +L 2s ≠0) and has light-gathering performance only in the sagittal direction. Effect of the Invention

[0018] According to the mirror design method of the present invention, a single mirror can be manufactured in which the light source position and the focusing position can be set independently in the vertical and horizontal directions, thereby enabling free conversion of astigmatism. Also, the focusing size can be kept small to accommodate beams in the X-ray region. Furthermore, the design formula is simple, the range of applications is wide, and a mirror that can be suitably used as an optical system that handles beams with different characteristics in the vertical and horizontal directions can be manufactured. [Brief description of the drawings]

[0019] [Figure 1] FIG. 2 is a conceptual diagram of a mirror designed by the design method according to the present invention. [Diagram 2] A conceptual diagram showing a coordinate system based on the incident beam. [Diagram 3] FIG. 1 is a conceptual diagram showing a coordinate system based on an emitted beam. [Figure 4] 1 is a conceptual diagram showing an equiphase surface A1s in the vicinity of an intersection point Ps on a sagittal light source line Ss. [Diagram 5] A conceptual diagram showing an equiphase surface A1m near the intersection point Pm on the meridional light source line Sm. [Figure 6] FIG. 2 is a schematic diagram showing an optical system arrangement of mirrors in the first embodiment. [Figure 7] 1 shows the calculated shape of the mirror reflecting surface in Example 1, where (a) shows the distribution of height w(u,v) relative to the mirror origin when the horizontal axis is set to the u coordinate and the vertical axis is set to the v coordinate, (b) shows the short-side cross-sectional profile indicated by the dashed line in (a), and (c) shows the long-side cross-sectional profile indicated by the dashed line in (a). [Figure 8] 1 shows the results of a simulation of the focusing performance of Example 1, in which (a) shows the result of calculating the dispersion of light rays on the focusing surface based on geometric optics, and (b) shows the intensity distribution on the focusing surface calculated based on wave optics assuming soft X-rays of 300 eV. [Figure 9] FIG. 11 is a schematic diagram showing an optical system arrangement of mirrors according to a second embodiment. [Figure 10]1 shows the calculated shape of the mirror reflecting surface in Example 2, where (a) shows the distribution of height w(u,v) relative to the mirror origin when the horizontal axis is set to the u coordinate and the vertical axis is set to the v coordinate, (b) shows the short-side cross-sectional profile indicated by the dashed line in (a), and (c) shows the long-side cross-sectional profile indicated by the dashed line in (a). [Figure 11] 1 shows the results of a simulation of the focusing performance of Example 2, in which (a) shows the result of calculating the dispersion of light rays on the focusing surface based on geometric optics, and (b) shows the intensity distribution on the focusing surface calculated based on wave optics assuming soft X-rays of 300 eV. [Figure 12] FIG. 11 is a schematic diagram showing an optical system arrangement of mirrors according to a third embodiment. [Figure 13] 1 shows the calculated shape of the mirror reflecting surface in Example 3, where (a) shows the distribution of height w(u,v) relative to the mirror origin when the horizontal axis is set to the u coordinate and the vertical axis is set to the v coordinate, (b) shows the short-side cross-sectional profile indicated by the dashed line in (a), and (c) shows the long-side cross-sectional profile indicated by the dashed line in (a). [Figure 14] 1 shows the results of a simulation of the focusing performance of Example 3, in which (a) shows the results of calculating the variation of light rays on the focusing surface based on geometric optics on the vertical (meridional) focusing surface, (b) shows the intensity distribution on the focusing surface calculated assuming a 300 eV soft X-ray on the same focusing surface, (c) shows the results of calculating the variation of light rays on the focusing surface based on geometric optics on the horizontal (sagittal) focusing surface, and (d) shows the intensity distribution on the focusing surface calculated assuming a 300 eV soft X-ray on the same focusing surface. [Figure 15] FIG. 13 is a schematic diagram showing an optical system arrangement of mirrors according to a fourth embodiment. [Figure 16] 1 shows the calculated shape of the mirror reflecting surface in Example 4, where (a) shows the distribution of height w(u,v) relative to the mirror origin when the horizontal axis is set to the u coordinate and the vertical axis is set to the v coordinate, (b) shows the short-side cross-sectional profile indicated by the dashed line in (a), and (c) shows the long-side cross-sectional profile indicated by the dashed line in (a). [Figure 17] FIG. 13 is a diagram showing the difference between the mirror shape of Example 4 and a truncated cone. [Figure 18]13 shows the results of a simulation of the focusing performance in the sagittal direction in Example 4, where (a) shows the result of calculating the dispersion of light rays on the focusing surface based on geometric optics, and (b) shows the intensity distribution on the focusing surface calculated based on wave optics assuming soft X-rays of 300 eV. [Figure 19] FIG. 4 is a graph showing the height distribution of the reflecting surface of each mirror in Example 1 and Comparative Examples 1 and 2. [Figure 20] FIG. 13 is a graph showing the results of subtracting the mirror height distribution of Example 1 from the mirror height distribution of Comparative Example 2. [Figure 21] 1 shows the results of a simulation of focusing performance, in which (a) is the ray variation of Comparative Example 1 (toroidal mirror), (b) is the two-dimensional intensity distribution at 300 eV calculated based on the wave optics of Comparative Example 1, (c) is the ray variation of Comparative Example 2 (astigmatic off-axis mirror), (d) is the two-dimensional intensity distribution at 300 eV calculated based on the wave optics of Comparative Example 2, (e) is the ray variation of Example 1 (astigmatism control mirror according to the present invention), and (f) is the two-dimensional intensity distribution at 300 eV calculated based on the wave optics of Example 1. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0020] The method for designing a mirror according to the present invention is a method for designing a mirror produced by forming a reflective surface on the surface of a plate material. Hereinafter, the method for designing a mirror according to the present invention will be described with reference to typical embodiments.

[0021] The present invention aims to freely convert astigmatism, and designs mirrors with higher accuracy based on Fermat's principle, which states that "light follows the path with the shortest optical distance." When limited to converging (or diffusing) mirrors, Fermat's principle can be translated into the expression that "for any point on the mirror surface (reflecting surface), the sum of the distance from the light source point and the distance to the converging point is constant." When the incident beam or outgoing beam has astigmatism, the law of constant optical path length cannot be immediately applied. This is because, as the name suggests, an astigmatized beam does not have a single light source point or converging point. In this invention, the idea of ​​newly defining "light source line" and "converging line" has been conceived, and a design method has been realized by making it possible to express the properties of an astigmatized beam in terms of geometrical optics.

[0022] FIG. 1 is a conceptual diagram showing the "source ray" and the "focus ray." For sagittal focusing, the incident beam is focused along the optical axis z of the incident light. 1 z on 1 Axis and z 2 Intersection point M, which is the intersection point on the reflecting surface with the axis 0 From L 1s The light source is located at a displaced position, and the emitted beam is parallel to the optical axis z 2 Above the intersection point M 0 From along the optical axis, L 2s The beam is focused at the displaced position. For focusing in the meridional direction, the incident beam is focused on the optical axis z of the incident beam. 1 Above the intersection point M 0 From L 1m The light source is located at a displaced position, and the emitted beam is parallel to the optical axis z 2 Above the intersection point M 0 From L 2m Assume that the light is focused at a displaced position.

[0023] Then, all the incident light rays passing through the mirror pass through the position of the light source in the sagittal direction and are aligned along the optical axis z of the incident light. 1 The direction perpendicular to both the sagittal and axial directions (y 1 A sagittal light source line (S s), and the sagittal direction (x 1 A meridional light line (S m ) is considered to pass through the sagittal light source ray (S s ), meridional light ray (S m )

[0024] In addition, all of the outgoing light rays emitted from the mirror pass through the focusing position in the sagittal direction and are aligned along the optical axis z of the outgoing light. 2 The direction perpendicular to both the sagittal and axial directions (y 2 A sagittal focusing ray (F s ), and the sagittal direction (x 2 A meridional condensing ray (F m ) is considered to pass through the sagittal focusing ray (F s ), meridional condensation ray (F m )

[0025] In this example, the sagittal light source line (S s ), meridional light ray (S m ), sagittal focusing ray (F s ), meridional condensation ray (F m ) are straight lines, but they may be curved lines. 1s >L 1m >0 and L 2s >L 2m Although the case of >0 is shown, these constants can also be negative. 1s or L 1m If L is negative, the incident beam is reflected by the mirror's reflective surface on its way downstream. 2s or L 2m When is negative, the exit beam has a wavefront that appears to diverge from a position upstream of the mirror.

[0026] By defining the "source ray" and "collected ray" as above, it is possible to define the incident ray and the outgoing ray that pass through any point on the reflecting surface of the mirror. That is, let M be any point on the reflecting surface of the mirror, and the sagittal source ray (S s ) and the intersection point (P s ), and meridional light rays (S m ) and the intersection point (P m ) by dividing the coordinates of the displacement L 1s , L 1m Similarly, the outgoing ray from point M and the sagittal convergent ray (F s ) and the intersection point (Q s ), and the outgoing ray from point M and the meridional focusing ray (F m ) and the intersection point (Q m ) by dividing the coordinates of the displacement L 2s , L 2m This can be expressed by the formula:

[0027] And these P s , P m , Q s , Q m Based on the fact that the optical path length (the sum of the entrance length and exit length) from the light source position to the focusing position is constant for each coordinate of the sagittal direction focusing and the meridional direction focusing for any point on the reflecting surface, a design equation for the reflecting surface can be derived.

[0028] Any point M on the reflecting surface of a mirror can be expressed as M(u,v,w) by defining a uvw Cartesian coordinate system based on the mirror. In other words, the intersection point M on the reflecting surface of the incident light and the outgoing light 0 The surface in contact with the reflecting surface is a uv plane, and the M 0 The direction of the normal line passing through is defined as the w axis, the v axis is defined as the direction perpendicular to both the incident optical axis and the outgoing optical axis, and the u axis is defined as the direction perpendicular to both the v axis and the w axis. 0 the origin, the uv plane and the optical axis z 1 The oblique incidence angle between 0 It was decided.

[0029] However, in Fig. 1, the sagittal light source line (S s ) and meridional light rays (S m ) is not the u axis but the incident beam optical axis z 1 Similarly, the sagittal focusing ray (F s ) and meridional focusing ray (F m ) is the optical axis of the emitted beam z 2 , and is orthogonal to the uvw coordinate system. Although it is possible to directly calculate the optical path length from the light source line and the condensed line which are set obliquely with respect to the uvw coordinate system, this is complicated. Therefore, in this embodiment, the optical path length is calculated after conversion to a coordinate system based on each of the incident beam optical axis and the exit beam optical axis, and then substituted into the design equation for the astigmatism control mirror.

[0030] The transformation into a coordinate system based on the optical axis of the incident beam is as follows. Figure 2 shows a schematic diagram of the coordinate system after transformation. If the optical axis of the incident beam is z 1 axis, and the cross section perpendicular to it is x 1 y 1 Let the mirror be a plane. 1 ,y 1 ,z 1 ) are given by equation (2).

[0031]

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[0032] Incident ray passing through point M and sagittal source ray S s Intersection point P s , and also the incident ray and meridional light ray S m Intersection point P m The coordinates of 1 y 1 z 1 In the coordinate system, the displacement L 1s , L 1m Using these, it can be expressed by the following equations (3) and (4).

[0033]

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[0034] Similarly, the transformation into a coordinate system based on the optical axis of the outgoing beam is as follows. A schematic diagram of the transformed coordinate system is shown in Figure 3. If the optical axis of the outgoing beam is z 2 axis, and the cross section perpendicular to it is x 2 y 2 Let the mirror be a plane. 2 ,y 2 ,z 2 ) are given by equation (5).

[0035]

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[0036] Emitting ray and sagittal focusing ray F passing through point M s Intersection point Q s The coordinates of the exit ray and the meridional focusing ray F m Intersection point Q m The coordinates of are x 2 y 2 z 2 In the coordinate system, the displacement L 2s , L 2m Using these, it can be expressed by the following equations (6) and (7).

[0037]

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[0038] And, as mentioned above, these P s , P m , Q s , Q m Based on the fact that the optical path length (the sum of the entrance length and exit length) from the light source position to the focusing position is constant for each coordinate of the sagittal direction focusing and the meridional direction focusing for any point on the reflecting surface, a design equation for the reflecting surface is derived.

[0039] In this embodiment, the intersection points P s , P m , Q s, Q m and an arbitrary point M on the reflecting surface is not taken as the incident length or exit length as it is, but the following optical path length compensation is performed using the coordinates of the above-mentioned intersection of the source line and the focusing line defined by straight lines to obtain a more accurate design formula.

[0040] (Optical path length compensation) Consider Fermat's principle when a normal light source point and a focal point can be defined. The equiphase surface near the light source point is a sphere centered on the light source point, and the equiphase surface near the focal point is a sphere centered on the focal point. Keeping in mind that light rays are always perpendicular to the equiphase surface, the law of constant optical path length can be rephrased as the optical path of a light ray connecting an arbitrary point on a specific equiphase surface near the light source point and a corresponding point on a specific equiphase surface near the focal point is constant. Even when the incident beam contains astigmatism as in the present invention, a more accurate design formula can be derived by performing compensation taking into account the equiphase surface.

[0041] First, on the incident side, the sagittal light source line S s The above intersection point P s Consider the equiphase surface in the vicinity of the sagittal source line S s Now, the meridional light source ray S m Under this assumption, a wavefront converging toward the sagittal source line S s Although it is not possible to define the phase above precisely, we use S m and z 1 The intersection point with the axis is P m0 Tooki, S s Above is P. m0 There is a phase distribution according to the distance from the mirror, i.e., the beam before entering the mirror (reflecting surface) is y 1 In the axial direction, the meridional light source line S m Based on this idea, as shown in Figure 4, the meridional light source line S m and z 1 Intersection point P with axis m0 The sagittal light source line S s and z 1 Intersection point P with axis s0 Through x1 Arc B extending perpendicular to the axis 1s , the sagittal light source line S s The rotational arc surface formed by rotating around the axis is called the equiphase surface A. 1s The incident length from the light source position in the sagittal direction to point M is the incident light ray and the equiphase surface A 1s The meridional light ray S is one of the two intersection points with m It is more accurate to calculate the distance from the intersection point closer to point M.

[0042] Here, the incident light and the equiphase surface A 1s The distance from the intersection point of the incident light ray and the sagittal light source line S s Intersection point P s The distance from to point M is calculated, and the intersection point P s From the equiphase surface A 1s The arc B defines 1s The distance to, i.e., P s From arc B 1s The foot of the perpendicular line to H 1s As P s H 1s The distance between the incident length f is calculated by adding or subtracting (subtracting in this example). 1s is expressed by equation (8). The reason why this equation is an approximation is that point H 1s is the line P s However, it is of course possible to use a calculation other than this approximation. s From arc B 1s The foot of the perpendicular line to H 1s As P s H 1s The distance between arcs B and C is approximated by adding / subtracting the distance between arcs B and C. 1s P s From the above, the incident light and the equiphase surface A 1s The meridional light ray S is one of the two intersection points with m Alternatively, the distance to the intersection point closer to the center of the intersection may be used for more accurate calculation.

[0043]

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[0044] To this t' 1x (Equation (10)), t' 1y By introducing (Equation (11)), it can be transformed into the following Equation (9).

[0045]

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[0046] Next, on the same incident side, the meridional light source ray S m The above intersection point P m Consider the equiphase surface in the vicinity of the meridional light ray S m Now, the sagittal light source ray S s Under this assumption, we should observe a wavefront diverging from S m Although it is not possible to define the phase above precisely, we use S s and z 1 The intersection point with the axis is P s0 Tooki, S m Above is P. s0 There is a phase distribution according to the distance from the mirror, i.e., the beam before entering the mirror (reflecting surface) is x 1 In the axial direction, the sagittal light source line S s Based on this concept, as shown in Figure 5, the sagittal light source line S s and z 1 Intersection point P with axis s0 The meridional light source line S m and z 1 Intersection point P with axis m0 Through y 1 Arc B extending perpendicular to the axis 1m , the meridional light source ray S m The rotational arc surface formed by rotating around the axis is called the equiphase surface A. 1m Let y 1 The incident length from the light source position in the axial direction to point M is the incident light ray and the corresponding equiphase surface A 1mOf the two intersections with the sagittal light source line S s It is calculated as the distance from the intersection point closest to point M on the mirror's reflecting surface.

[0047] Incident light and equiphase surface A 1m The distance from the intersection point of the incident ray and the meridional light source line S m Intersection point P m The distance from to point M is calculated, and the intersection point P m From the equiphase surface A 1m The arc B defines 1m The distance to, i.e., P m From arc B 1m The foot of the perpendicular line to H 1m As P m H 1m The distance between the incident length f is calculated by adding or subtracting (adding in this example). 1m is expressed by equation (12).

[0048]

number

[0049] To this t' 1x , t' 1y By introducing the above, it can be transformed into the following equation (13).

[0050]

number

[0051] On the exit side, as on the entrance side, the sagittal focusing ray F s Above intersection point Q s The corresponding equiphase surface and meridional focusing line F m Above intersection point Q m Consider the equiphase surfaces in the vicinity of the sagittal focusing ray F s So, the meridional ray F m Under this assumption, we should observe a wavefront emanating from F sAlthough it is not possible to define the phase above precisely, here we use F m and the output optical axis z 2 The intersection point of Q m0 Toki, F s Above is Q m0 It is assumed that there is a phase distribution according to the distance from the m Now, the sagittal focusing ray F s Under this assumption, we should observe a wavefront converging toward F m Although it is not possible to define the above topology precisely, here we use F s and the output optical axis z 2 The intersection point of Q s0 Tooki, F m Above is Q s0 It is assumed that there exists a phase distribution according to the distance from .

[0052] Based on these considerations, a more accurate exit length is calculated, as in the case of the incident side. Specifically, although not shown in the figure, the intersection point Q s Arc B that defines the equiphase surface from 2s The distance to, i.e., Q s From arc B 2s The foot of the perpendicular line to H 2s H 2s Q s Distance between and intersection point Q m Arc B that defines the equiphase surface from 2m The distance to, i.e., Q m From arc B 2m The foot of the perpendicular line to H 2m Q m H 2m The compensation is performed by adding or subtracting the distance between the two points, and the exit length of the focused light in the sagittal direction f 2s , the exit length of the focused light in the meridional direction f 2m For each of these, a more accurate exit length can be obtained.

[0053]

number

[0054]

number

[0055] f 2s t' 2x , t' 2y By introducing the above, the equations can be transformed into the following equations (16) to (18).

[0056]

number

[0057] f 2m t' 2x , t' 2y By introducing the above, it can be transformed into the following equation (19).

[0058]

number

[0059] (Calculation of optical path length) Using the entrance length and exit length thus obtained, the optical path length for focusing in each direction, sagittal direction and meridional direction, is calculated. The reference optical path length from the light source point to the focusing point in the sagittal direction is L s =L 1s +L 2s Then, the condition required for focusing in the sagittal direction is derived as follows:

[0060]

number

[0061] Similarly, the reference optical path length from the light source point to the focal point in the meridional direction is L m =L 1m +L 2m Then, the condition required for focusing in the meridional direction is derived as follows:

[0062]

number

[0063] Ideally, the set of points (u,v,w) that simultaneously satisfy the sagittal focusing condition in equation (20) and the meridional focusing condition in equation (21) will be the desired shape of the reflecting surface of the mirror. 1s =L 1m And L 2s =L 2m In order to obtain a design equation that expresses a more generalized shape of the reflecting surface that can be applied under other conditions, the inventors weighted the equations (20) and (21) to obtain a new design equation, f(u,v,w)=0, shown in equation (22).

[0064] (design formula) That is, the design formula is the first formula (the formula for the sagittal focusing condition) f, which is derived from the fact that the optical path length from the light source point to the focusing point is constant for sagittal focusing. s (u,v,w)=0 (Equation (20)) and the second equation (the meridional focusing condition equation) f, which is derived from the fact that the optical path length from the light source point to the focusing point is constant for focusing in the meridional direction, m The equation (21) (u,v,w)=0 is weighted using α and β to give the following equation (22), f(u,v,w)=0, where α is a weighting coefficient for focusing in the meridional direction, and β is a weighting coefficient for focusing in the sagittal direction.

[0065]

number

[0066] Equation (22) is the mirror design equation. 1x ,t' 1y ,t' 2x ,t' 2y If we rewrite it based on the uvw coordinate system, we get the following equations (23) to (26).

[0067]

number

[0068] As can be seen from equation (22), we can confirm that we have obtained an equation with good symmetry for the sagittal and meridional focusing directions. 1s >L 1m >0 and L 2s >L 2m > 0' has been assumed, but even if this assumption is not made, that is, even if the magnitude relationship is reversed or each set value takes a negative value, the same equation (design equation) shown in formula (22) can be derived. However, L 1m , L 1s , L 2m , L 2s All four constants above can be positive or negative and cannot be zero.

[0069] (Examples of mirrors that can be designed) In the condition setting of equation (22), L 1s and L 1m Set the value of to different values ​​and L 2s and L 2m By setting the values ​​of to the same value, it is possible to design an astigmatism control mirror with a reflecting surface that can obtain a focused output beam from an astigmatism-bearing input beam. Conversely, L 1s and L 1m Set the value of to the same value and set L 2s and L 2m By setting the value of L to different values, it is possible to design an astigmatism-controlled mirror with a reflecting surface that can obtain an astigmatism-bearing output beam from an incident beam diverging from a single point. 1m and L 2m Set the value of to positive or negative infinity and L 1s and L 2s Set the value to a given value (where L 1s +L 2s ≠ 0), it is possible to design an astigmatism control mirror that has light-collecting performance only in the sagittal direction.

[0070] In addition, by using design equation (22), it is possible to design a mirror with a reflecting surface where the light source and the light focusing position of both the sagittal direction and the meridional direction are the same. For example, if L 1s =L 1m =L 1 , and L 2s =L 2m =L 2 By substituting, we obtain the following equation (27).

[0071]

number

[0072] The reflecting surface designed by Eq. (27) is L 1 , L 2 Depending on whether it is positive or negative and its magnitude relationship, the shape will be one of the following: ellipsoid of revolution, hyperboloid of revolution, paraboloid of revolution, or plane. Table 1 shows the classification. What is particularly noteworthy is that not only concave mirrors, but also convex mirrors and planes can be designed using the same formula.

[0073] [Table 1]

[0074] In addition, when the light-collecting effect of the mirror is limited to the meridional direction, that is, L 1s = +∞, and L 2s When = +∞, equation (22) is simplified to the following equation (28).

[0075]

number

[0076] Equation (28) represents an ellipse, a parabola, a hyperbola, or a straight cylinder. As with equation (27), L 1m and L 2m Depending on whether the value is positive or negative and the magnitude relationship, different shapes such as concave, flat, or convex can be expressed.

[0077] In this way, it can be seen that the design formula for the planar astigmatism control mirror of the present invention is a highly versatile (widely applicable) design formula that encompasses existing ellipsoidal mirrors, hyperboloidal mirrors, paraboloidal mirrors, one-dimensional ellipsoidal mirrors (one type of KB mirror), one-dimensional hyperboloidal mirrors, one-dimensional parabolic mirrors, and planar mirrors.

[0078] Although the embodiment of the present invention has been described above, the present invention is not limited to these examples, and can be embodied in various forms without departing from the spirit of the present invention. In this embodiment, the light source line and the condenser line are treated as straight lines, and the distance between the straight line and the equiphase surface in the vicinity thereof is compensated for, but such compensation is not necessarily required. It is also preferable to use an arc line or other curved line as the light source line and the condenser line, and to obtain the line without compensation or using a compensation method or approximation method other than the above compensation. The position of the origin of the design equation for the reflecting surface may be different. Coordinate transformation may of course be performed. EXAMPLES

[0079] Next, as design examples of the astigmatism control mirror according to the present invention described above, four mirrors were designed: a mirror intended to eliminate astigmatism (Examples 1 and 2), a mirror intended to add astigmatism (Example 3), and a mirror intended to focus light only in the sagittal direction (Example 4). The performance of each mirror was confirmed by simulation using both geometric optics and wave optics, and a comparison was made between the mirror of Example 1 and a conventional mirror. The results will be described below.

[0080] (Simulation method) The mirror was set to reflect the beam vertically upward. In other words, the long side (meridional) direction of the mirror (reflecting surface) is responsible for vertical focusing, and the short side (sagittal) direction is responsible for horizontal focusing. In ray tracing calculations based on geometric optics, a group of rays passing through the light source line for each focusing in the meridional and sagittal directions shown in Figures 1 to 3 is defined, and made to enter the reflecting surface of the mirror. At this time, the thickness of the light source line, i.e., the size of the light source, is set to 0. The light rays are emitted uniformly over the entire effective range of the reflecting surface.

[0081] The normal vector n(x,y,z) at each position on the mirror's reflecting surface can be calculated from the gradient vector of the function f(u,v,w) defined in equation (22) as a numerical solution that is a unit vector parallel to the gradient vector (equation (29)). As shown in Figure 1, the incident light is reflected symmetrically with respect to the normal vector of the mirror's reflecting surface and propagates to the focusing surface. In this way, the variation of the light on the focusing surface is evaluated.

[0082]

number

[0083] In the diffraction integral calculation based on wave optics, first, lattice points are arranged at equal intervals in the u and v directions on the reflecting surface of the mirror, and the coordinates M(u,v,w) of each point are then calculated based on the coordinate system M(x 1 ,y 1 ,z 1 ) is converted into the wavefront x of the beam incident on the reflecting surface of the mirror. 1 ,y 1 The center of curvature in the sagittal direction is the light source S s , a light source S focused in the meridional direction m Let us assume a line source with no thickness, and the beam is incident on the entire effective area of ​​the reflecting surface with uniform intensity. x 1 y 1 z 1 A point M(x 1 ,y 1 ,z 1 ) wave field U M (x 1 ,y 1 ,z 1 ) becomes the following equations (30) and (31).

[0084]

number

[0085] In equation (30), λ is an arbitrary constant representing the wavelength of the beam, I 0 is an arbitrary constant that represents the incident intensity. x 1 y 1 z 1 The complex wave field U on the reflecting surface of the mirror expressed in the coordinate system M (x 1 ,y 1 ,z 1 ) into x as shown in equation (32). 2 y 2 z 2 Convert to the U coordinate system. M (x 2 ,y 2 ,z 2 )

[0086]

number

[0087] Furthermore, according to equations (33) and (34), the wave field U M (x 2 ,y 2 ,z 2 ) to x 2 y 2 z 2 Point Q(x Q ,y Q ,z Q )

[0088]

number

[0089] In equation (33), dS represents the infinitesimal area on the reflecting surface, and θ(x 2 ,y 2 ,z 2 ) represents the oblique incidence angle at each position on the reflecting surface. The output is the complex wave field U on Q. Q (x Q ,y Q ,z Q) (Equation (35)). By the above procedure, the intensity distribution on the focusing plane is calculated based on wave optics.

[0090]

number

[0091] (Mirror Design of Example 1) Table 2 shows a list of constants used in the mirror design of Example 1. The entrance length has different positive values ​​in the vertical and horizontal directions, and the exit length has the same positive value in the vertical and horizontal directions. A schematic diagram of the optical system layout is shown in Figure 6.

[0092] [Table 2]

[0093] (Shape of the reflecting surface in Example 1) The calculated shape of the reflective surface of the mirror is shown in Figure 7. Figure 7(a) shows the distribution of height w(u,v) relative to the mirror origin when the horizontal axis is set to the u coordinate and the vertical axis is set to the v coordinate. Figure 7(b) shows the short-side cross-sectional profile indicated by the dashed line in Figure 7(a), and Figure 7(c) shows the long-side cross-sectional profile indicated by the dashed line in Figure 7(a). The reflective surface of the mirror in Example 1 is a concave surface with different curvatures in the long-side and short-side directions.

[0094] (Simulation results of light collection performance in Example 1) The results of the light-collecting performance simulation are shown in Figure 8. Figure 8(a) shows the results of calculating the light ray dispersion on the focusing surface based on geometric optics. It was confirmed that all the light rays were concentrated in an area of ​​10 nm or less in both the horizontal and vertical directions.

[0095] Figure 8(b) shows the intensity distribution at the focal plane calculated based on wave optics assuming a 300 eV soft X-ray. The beam is focused to an area of ​​160 nm horizontally × 440 nm vertically (FWHM). The spot size of the focused beam is smaller because the horizontal numerical aperture is larger.

[0096] (Mirror Design of Example 2) Table 3 shows a list of constants used in the mirror design of Example 2. The vertical (meridional) focusing incidence length has a positive value, while the horizontal (sagittal) focusing incidence length has a negative value. In other words, the incident beam has a source point upstream of the mirror in the vertical direction, while it is focused toward a point downstream of the mirror in the horizontal direction. The exit length has the same positive value in the vertical and horizontal directions. A schematic diagram of the optical system layout is shown in Figure 9. This mirror is intended to eliminate extreme astigmatism, where the positive and negative curvatures are reversed.

[0097] [Table 3]

[0098] (Shape of the reflecting surface in Example 2) The calculated shape of the reflective surface of the mirror is shown in Figure 10. Figure 10(a) shows the distribution of height w(u,v) relative to the mirror origin when the horizontal axis is set to the u coordinate and the vertical axis is set to the v coordinate. Figure 10(b) shows the short-side cross-sectional profile indicated by the dashed line in Figure 10(a), and Figure 10(c) shows the long-side cross-sectional profile indicated by the dashed line in Figure 10(a). The reflective surface of the mirror in Example 2 is saddle-shaped, with a concave profile in the long-side direction and a convex profile in the short-side direction.

[0099] (Simulation results of light collection performance in Example 2) The results of the light-collecting performance simulation are shown in Fig. 11. Fig. 11(a) shows the light beam dispersion on the focusing surface calculated based on geometric optics. It was confirmed that all the light beams were concentrated in an area of ​​10 nm or less in both the horizontal and vertical directions.

[0100] Moreover, Fig. 11(b) shows the intensity distribution on the focusing surface, calculated assuming a 300 eV soft X-ray. The beam was focused in an area of ​​160 nm horizontally × 320 nm vertically (FWHM). This simulation showed that the mirror of Example 2 can eliminate the aberration of the incident beam with extreme astigmatism, where one side is focused and the other side is diverged.

[0101] (Mirror Design of Example 3) Table 4 shows a list of constants used in the mirror design of Example 3. The entrance length has the same positive value in the vertical and horizontal directions, and the exit length has different positive values ​​in the vertical and horizontal directions. A schematic diagram of the optical system arrangement is shown in Figure 12.

[0102] [Table 4]

[0103] (Shape of the reflecting surface in Example 3) The calculated shape of the reflective surface of the mirror is shown in Figure 13. Figure 13(a) shows the distribution of height w(u,v) relative to the mirror origin when the horizontal axis is set to the u coordinate and the vertical axis is set to the v coordinate. Figure 13(b) shows the short-side cross-sectional profile indicated by the dashed line in Figure 13(a), and Figure 13(c) shows the long-side cross-sectional profile indicated by the dashed line in Figure 13(a). The reflective surface of the mirror in Example 3 is a concave surface with different curvatures in the long-side and short-side directions.

[0104] (Simulation results of light collection performance in Example 3) The simulation results of the light-collecting performance are shown in Fig. 14. Fig. 14(a) shows the vertical light-collecting surface (z 2 =L 2m It was confirmed that all the light rays were concentrated in a vertically wide area of ​​60 nm.

[0105] Figure 14(b) shows the intensity distribution in the vertical focusing plane for a 300 eV soft X-ray beam, calculated based on wave optics. The beam is focused to a width of FWHM 39 μm. Similarly, the intensity distribution in the horizontal focusing plane (z 2 =L 2s The focusing performance at the horizontal focusing point was evaluated by simulation. The results are shown in Figure 14(c) and (d). The focusing width of the horizontal focusing was 40 nm in the geometric optics and 7.7 μm (FWHM) in the wave optics.

[0106] (Mirror Design of Example 4) Table 5 shows a list of constants used in the mirror design of Example 4. For vertical focusing, the entrance length and exit length have positive infinity. This indicates that the mirror has no focusing performance in the meridional direction. In contrast, for horizontal focusing, the entrance length and exit length are both positive. A schematic diagram of the optical system layout is shown in Figure 15.

[0107] [Table 5]

[0108] (Shape of the reflecting surface in Example 4) The calculated shape of the reflective surface of the mirror is shown in Figure 16. Figure 16(a) shows the distribution of height w(u,v) relative to the mirror origin when the horizontal axis is set to the u coordinate and the vertical axis is set to the v coordinate. Figure 16t(b) shows the short-side cross-sectional profile indicated by the dashed line in Figure 16(a), and Figure 16(c) shows the long-side cross-sectional profile indicated by the dashed line in Figure 16(a). It can be seen from Figure 16(c) that the reflective surface of the mirror of Example 4 has a completely linear profile in the long-side direction.

[0109] The difference between the mirror shape and a truncated cone is shown in Figure 17. The RMS value of the difference was 77 nm. It can be seen that the shape of a mirror that has focusing performance only in the sagittal direction can be well approximated by a truncated cone.

[0110] (Simulation results of light collection performance in Example 4) Next, the simulation results of the light-collecting performance are shown in FIG. 18. FIG. 18(a) shows the vertical light-collecting surface (z 2 =L 2s It was confirmed that all the light rays were concentrated in an area with a width of 10 nm or less.

[0111] Figure 18(b) shows the intensity distribution in the vertical focusing plane for a 300 eV soft X-ray beam, calculated based on wave optics. The beam was focused to an area with a width of 770 nm (FWHM).

[0112] (Comparison with various mirrors) Next, a comparison is made with an existing toroidal mirror (Comparative Example 1) and an astigmatic off-axis mirror (Comparative Example 2) that are set to the same conditions as the design conditions of Example 1 shown in Table 2 and are intended to eliminate astigmatism.

[0113] (Comparison of reflective surface shapes) The shapes of the reflecting surfaces of the mirrors of Example 1 and Comparative Examples 1 and 2 are shown in Fig. 19. As can be seen from Fig. 19(b) and (c), the shapes of these mirrors are almost the same in Comparative Example 2 and Example 1. However, as can be seen from Fig. 20, which shows the result of subtracting the height distribution of the mirror of Example 1 from the height distribution of the mirror of Comparative Example 2, there is a difference in the order of μm between their shapes.

[0114] (Comparison of light collection performance) The simulation results of the focusing performance are shown in Figure 21. The variation of the light beam shown in the left column of Figure 21 was 2 mm for Comparative Example 1 (toroidal mirror), 20 μm for Comparative Example 2 (astigmatic off-axis mirror), and 2 nm for Example 1 (astigmatism control mirror according to the present invention).

[0115] The results of calculations based on wave optics of the focusing performance for a 300 eV soft X-ray beam are shown in the right column of Figure 21. The toroidal mirror of Comparative Example 1 no longer focuses light. The astigmatic off-axis mirror of Comparative Example 2 focuses light, but has a main peak that is widely spread in the horizontal direction. The astigmatism control mirror of Example 1 focused light to the diffraction limit size in both the meridional and sagittal directions. Of the three mirrors, only Example 1 (astigmatism control mirror) enables diffraction-limited focusing in the soft X-ray region.

Claims

1. A method for designing a mirror produced by forming a reflective surface on a plate material, comprising the steps of: The optical axis of the incident beam on the mirror is z １ axis, and the cross section perpendicular to it is x １ y １ The plane is The optical axis of the beam emitted from the mirror is z ２ axis, and the cross section perpendicular to it is x ２ y ２ The plane is x １ Axis and x ２ Let the axis be parallel to the sagittal direction of the reflecting surface, If the incident beam is z １ On-axis z １ Axis and z ２ Intersection point M on the reflecting surface with the axis ０ From Z １ Along the axial direction L １ｓ A light source for sagittal focusing at a displaced position, and １ The intersection point M on the axis ０ From Z １ Along the axial direction L １ｍ A light source for focusing in the meridional direction is provided at the displaced position. The output beam is focused in the sagittal direction at the z ２ The intersection point M on the axis ０ From Z ２ Along the axial direction L ２ｓ The light is focused at a displaced position, and the z ２ The intersection point M on the axis ０ From Z ２ Along the axial direction L ２ｍ The light is focused at the displaced position, All incident rays through the mirror pass through the position of the light source in the sagittal focus at x １ Axis and z １ A sagittal light source line extending in a direction perpendicular to both axes and a y line passing through the position of the light source in the convergence in the meridional direction １ Axis and z １ Passing through meridional light rays extending in a direction perpendicular to both axes, All the rays emitted from the mirror pass through the focusing position in the sagittal focusing direction x ２ Axis and z ２ A sagittal focusing line extending in a direction perpendicular to both axes and a meridional focusing line passing through the focusing position in the y direction ２ Axis and z ２ It is assumed that the light passes through a meridional condensing ray extending in a direction perpendicular to both axes, Let M be an arbitrary point on the reflecting surface of the mirror. The coordinates of the intersection of the sagittal light source line and the incident light beam to the point M, and the intersection of the meridional light source line and the incident light beam to the point M are expressed as the coordinates of the intersection of the sagittal light source line and the incident light beam to the point M. １ｓ , L １ｍ and the coordinates of the intersection of the outgoing ray from the same point M with the sagittal condensed ray and the intersection of the outgoing ray from the same point M with the meridional condensed ray are expressed as ２ｓ , L ２ｍ It is expressed using A method for designing a mirror, characterized in that a design equation that represents the reflective surface is obtained based on these coordinates and the fact that the optical path length from the light source position to the focusing position is constant for any point on the reflective surface for both sagittal and meridional focusing.

2. The sagittal light source line and the meridional light source line are respectively defined as y １ A straight line S extending in the axial direction ｓ , x １ A straight line S extending in the axial direction ｍ year, The sagittal condensed ray and the meridional condensed ray are respectively defined as y ２ A straight line F extending in the axial direction ｓ , x ２ A straight line F extending in the axial direction ｍ year, The incident length from the light source position to point M for the sagittal direction focusing is the meridional light source line S ｍ and z １ Intersection point P with axis ｍ０ The sagittal light source line S ｓ and z １ Intersection point P with axis ｓ０ Through x １ The arc extending perpendicular to the axis is called the sagittal light source line S ｓ The rotational arc surface rotated around the axis is called equiphase surface A. １ｓ The incident light and the equiphase surface A １ｓ Among the two intersection points with the meridional light source ray S ｍ The distance from the intersection point closest to point M is calculated. The exit length from point M to the focusing position in the sagittal direction is the meridional focusing line F ｍ and z ２ Intersection point Q with axis ｍ０ The sagittal focusing line F ｓ and z ２ Intersection point Q with axis ｓ０ Through x ２ The arc extending perpendicular to the axis is called the sagittal focusing line F ｓ The rotational arc surface rotated around the axis is called equiphase surface A. ２ｓ The outgoing light ray and the equiphase surface A ２ｓ Of the two intersections with the meridional condensing ray F ｍ The distance from the intersection point closest to point M is calculated. The incident length from the light source position to point M for the light collection in the meridional direction is the sagittal light source line S ｓ and z １ Intersection point P with axis ｓ０ The meridional light source line S ｍ and z １ Intersection point P with axis ｍ０ Through y １ The arc extending in a direction perpendicular to the axis is called the meridional light source line S ｍ The rotational arc surface rotated around the axis is called equiphase surface A. １ｍ The incident light and the equiphase surface A １ｍ Of the two intersections with the sagittal light source line S ｓ The distance from the intersection point closest to point M is calculated. The exit length from point M to the focusing position in the meridional direction is the same as that of the sagittal focusing line F ｓ and z ２ Intersection point Q with axis ｓ０ The meridional focusing line F ｍ and z ２ Intersection point Q with axis ｍ０ Through y ２ The arc extending perpendicular to the axis is called the meridional focusing line F. ｍ The rotational arc surface rotated around the axis is called equiphase surface A. ２ｍ The outgoing light ray and the equiphase surface A ２ｍ Of the two intersections with the sagittal focusing ray F ｓ The distance from the intersection point closest to point M is calculated. Thereby, the optical path length is calculated for the sagittal direction focusing and the meridional direction focusing. The method for designing a mirror according to claim 1.

3. The incident light and the equiphase surface A １ｓ Among the two intersection points with the meridional light source ray S ｍ The distance from the intersection point closest to the incident light ray to the sagittal light source line S ｓ Intersection point P ｓ The distance from the point M to the intersection point P is calculated. ｓ From the equiphase surface A １ｓ Add or subtract the distance to the arc that defines The emitted light beam and the equiphase surface A ２ｓ Of the two intersections with the meridional condensing ray F ｍ The distance from the intersection point closest to the point M is the distance between the exiting ray and the sagittal focusing ray F ｓ Intersection point Q ｓ The distance from the point M to the intersection point Q is calculated. ｓ From the equiphase surface A ２ｓ Add or subtract the distance to the arc that defines The incident light and the equiphase surface A １ｍ Of the two intersections with the sagittal light source line S ｓ The distance from the intersection point closer to point M is the distance between the incident ray and the meridional light source line S ｍ Intersection point P ｍ The distance from the point M to the intersection point P is calculated. ｍ From the equiphase surface A １ｍ Add or subtract the distance to the arc that defines The emitted light beam and the equiphase surface A ２ｍ Of the two intersections with the sagittal focusing ray F ｓ The distance from the intersection point closest to the point M is the distance between the exit ray and the meridional focusing ray F ｍ Intersection point Q ｍ The distance from the point M to the intersection point Q is calculated. ｍ From the equiphase surface A ２ｍ Add or subtract the distance to the arc that defines 3. The method for designing a mirror according to claim 2.

4. Z １ Axis and z ２ Intersection point M on the reflecting surface of the axis ０ The surface in contact with the reflecting surface is a uv plane. The M in the uv plane ０ The direction of the normal line passing through is the w axis, v axis to z １ axis and z ２ the u-axis is a direction perpendicular to both the v-axis and the w-axis, Intersection M ０ is the origin, the uv plane and the optical axis z １ The oblique incidence angle between ０ A Cartesian coordinate system based on the mirror is defined as follows: The coordinates are expressed as x with respect to the optical axis of the incident beam. １ y １ Z １ Coordinate system, x relative to the optical axis of the outgoing beam ２ y ２ Z ２ Transform each into a coordinate system, The design equation is expressed in a uvw coordinate system. The method for designing a mirror according to any one of claims 1 to 3.

5. The design formula is The first formula f is derived from the fact that the optical path length from the light source point to the focusing point is constant for the focusing in the sagittal direction. ｓ The second equation f is derived from the fact that (u, v, w) = 0 and the optical path length from the light source point to the light collection point is constant for the light collection in the meridional direction. ｍ 5. The method for designing a mirror according to claim 4, wherein the design is based on the following equation (1) with (u, v, w)=0 being weighted: [0010]

6. The optical axis of the beam incident on the mirror is the z 1 axis, and the cross section perpendicular to the z 1 axis is the x 1 y 1 plane, The optical axis of the beam emitted from the mirror is defined as the z 2 axis, and the cross section perpendicular to the z 2 axis is defined as the x 2 y 2 plane. Let the x 1 and x 2 axes be parallel to the sagittal direction of the reflecting surface, The incident beam has a light source for focusing in the sagittal direction at a position displaced by L 1s along the z 1 axis direction from an intersection M 0 on the z 1 axis of the z 1 axis and the z 2 axis on the reflection surface, and has a light source for focusing in the meridional direction at a position displaced by L 1m along the z 1 axis direction from the intersection M 0 on the z 1 axis; The output beam is focused at a position displaced by L 2s along the z 2 -axis direction from the intersection point M 0 on the z 2 -axis in the sagittal direction, and is focused at a position displaced by L 2m along the z 2 -axis direction from the intersection point M 0 on the z 2 -axis in the meridional direction, All incident light rays passing through the mirror pass through a sagittal light source line that passes through the position of the light source in the sagittal focus and extends in a direction perpendicular to both the x 1 axis and the z 1 axis, and a meridional light source line that passes through the position of the light source in the meridional focus and extends in a direction perpendicular to both the y 1 axis and the z 1 axis; All light beams emitted from the mirror pass through a sagittal focusing line that passes through the focusing position in the sagittal direction and extends in a direction perpendicular to both the x 2 axis and the z 2 axis, and a meridional focusing line that passes through the focusing position in the meridional direction and extends in a direction perpendicular to both the y 2 axis and the z 2 axis; If a surface including an intersection point M 0 of the incident light and the outgoing light on the reflecting surface is defined as a uv plane, a normal direction of the uv plane passing through M 0 is defined as a w axis, a v axis is defined as a direction perpendicular to both the incident optical axis and the outgoing optical axis, a u axis is defined as a direction perpendicular to both the v axis and the w axis, the intersection point M 0 is defined as the origin, and an oblique incidence angle between the uv plane and the optical axis z 1 is defined as θ 0 , A reflecting surface represented by the following formula (22) (wherein t' 1x , t' 1y , t' 2x , and t' 2y are the following formulas (23) to (26), respectively): Said L １ｓ and L １ｍ The values ​​of L are different. ２ｓ and L ２ｍ and An astigmatism control mirror that can obtain an output beam that is focused to a single point from an input beam that has astigmatism. [0025] [0030]

7. The optical axis of the beam incident on the mirror is the z 1 axis, and the cross section perpendicular to the z 1 axis is the x 1 y 1 plane, The optical axis of the beam emitted from the mirror is defined as the z 2 axis, and the cross section perpendicular to the z 2 axis is defined as the x 2 y 2 plane. Let the x 1 and x 2 axes be parallel to the sagittal direction of the reflecting surface, The incident beam has a light source for focusing in the sagittal direction at a position displaced by L 1s along the z 1 axis direction from an intersection M 0 on the z 1 axis of the z 1 axis and the z 2 axis on the reflection surface, and has a light source for focusing in the meridional direction at a position displaced by L 1m along the z 1 axis direction from the intersection M 0 on the z 1 axis; The output beam is focused at a position displaced by L 2s along the z 2 -axis direction from the intersection point M 0 on the z 2 -axis in the sagittal direction, and is focused at a position displaced by L 2m along the z 2 -axis direction from the intersection point M 0 on the z 2 -axis in the meridional direction, All incident light rays passing through the mirror pass through a sagittal light source line that passes through the position of the light source in the sagittal focus and extends in a direction perpendicular to both the x 1 axis and the z 1 axis, and a meridional light source line that passes through the position of the light source in the meridional focus and extends in a direction perpendicular to both the y 1 axis and the z 1 axis; All light beams emitted from the mirror pass through a sagittal focusing line that passes through the focusing position in the sagittal direction and extends in a direction perpendicular to both the x 2 axis and the z 2 axis, and a meridional focusing line that passes through the focusing position in the meridional direction and extends in a direction perpendicular to both the y 2 axis and the z 2 axis; If a surface including an intersection point M 0 of the incident light and the outgoing light on the reflecting surface is defined as a uv plane, a normal direction of the uv plane passing through M 0 is defined as a w axis, a v axis is defined as a direction perpendicular to both the incident optical axis and the outgoing optical axis, a u axis is defined as a direction perpendicular to both the v axis and the w axis, the intersection point M 0 is defined as the origin, and an oblique incidence angle between the uv plane and the optical axis z 1 is defined as θ 0 , A reflecting surface represented by the following formula (22) (wherein t' 1x , t' 1y , t' 2x , and t' 2y are the following formulas (23) to (26), respectively): Said L １ｓ and L １ｍ The values ​​of L ２ｓ and L ２ｍ having reflective surfaces with different values ​​of An astigmatism control mirror that produces an astigmatism-bearing output beam from an input beam diverging from a single point. [0045] [0050]

8. The optical axis of the beam incident on the mirror is the z 1 axis, and the cross section perpendicular to the z 1 axis is the x 1 y 1 plane, The optical axis of the beam emitted from the mirror is defined as the z 2 axis, and the cross section perpendicular to the z 2 axis is defined as the x 2 y 2 plane. Let the x 1 and x 2 axes be parallel to the sagittal direction of the reflecting surface, The incident beam has a light source for focusing in the sagittal direction at a position displaced by L 1s along the z 1 axis direction from an intersection M 0 on the z 1 axis of the z 1 axis and the z 2 axis on the reflection surface, and has a light source for focusing in the meridional direction at a position displaced by L 1m along the z 1 axis direction from the intersection M 0 on the z 1 axis; The output beam is focused at a position displaced by L 2s along the z 2 -axis direction from the intersection point M 0 on the z 2 -axis in the sagittal direction, and is focused at a position displaced by L 2m along the z 2 -axis direction from the intersection point M 0 on the z 2 -axis in the meridional direction, All incident light rays passing through the mirror pass through a sagittal light source line that passes through the position of the light source in the sagittal focus and extends in a direction perpendicular to both the x 1 axis and the z 1 axis, and a meridional light source line that passes through the position of the light source in the meridional focus and extends in a direction perpendicular to both the y 1 axis and the z 1 axis; All light beams emitted from the mirror pass through a sagittal focusing line that passes through the focusing position in the sagittal direction and extends in a direction perpendicular to both the x 2 axis and the z 2 axis, and a meridional focusing line that passes through the focusing position in the meridional direction and extends in a direction perpendicular to both the y 2 axis and the z 2 axis; If a surface including an intersection M 0 of the incident light and the outgoing light on the reflecting surface is defined as a uv plane, a normal direction of the uv plane passing through M 0 is defined as a w axis, a v axis is defined as a direction perpendicular to both the incident optical axis and the outgoing optical axis, a u axis is defined as a direction perpendicular to both the v axis and the w axis, the intersection M 0 is defined as the origin, and an oblique incidence angle between the uv plane and the optical axis z 1 is defined as θ 0 , A reflecting surface represented by the following formula (22) (wherein t' 1x , t' 1y , t' 2x , and t' 2y are the following formulas (23) to (26), respectively): Said L １ｍ and L ２ｍ The value of L is positive or negative infinity, and １ｓ and L ２ｓ is a predetermined value (where L １ｓ +L ２ｓ ≠0), An astigmatism controlled mirror that has focusing performance only in the sagittal direction. [006] [0070]

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