JP7564522B2 - METHOD FOR DESIGNING MIRROR AND ASTAGITABER CONTROL MIRROR HAVING REFLECTING SURFACE WHEREIN THE DESIGN EQUATION IN THE METHOD APPLY - Google Patents

Description

The present invention relates to a method for designing a mirror that is manufactured by forming a reflecting surface on the hollow inner or outer peripheral surface of a substrate, and to an astigmatism-controlled mirror that has a reflecting surface for which the design formula in the design method holds true.

A synchrotron radiation soft X-ray beam has the characteristic that its characteristics differ 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 the diffraction gratings used focus the soft X-rays only in the spectroscopic direction, resulting in "astigmatism," where the light source position is different in the spectroscopic direction and in directions where the light is not focused.

Conventionally, optical systems that handle beams with different characteristics in the vertical and horizontal directions have used a two-stage focusing optical system that arranges two mirrors to handle each of the vertical and horizontal directions, and sets the light source points independently in the vertical and horizontal directions to make the focusing points coincident. Specifically, known methods include arranging two elliptical mirrors vertically and horizontally, arranging two bent mirrors (mechanically bent cylindrical mirrors) to create an approximate shape, and arranging a bent mirror and a sagittal cylinder mirror facing each other in the horizontal direction. However, such a method of combining two mirrors results in a complex mechanism and an additional chamber, which increases costs and makes adjustments more difficult.

One possible single mirror that can eliminate astigmatism is the toroidal mirror (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, in principle, the focused light size increases.

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 obtain 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.

However, this AO mirror is a mirror that 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 because the reflecting surface is approximated to an axially symmetric shape, this approximation creates a limit to the reduction in 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.

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 A. Wagner-Gentner, U.U. Graf, M. Philipp, D. Rabanus, "A simple method to design astigmatic off-axis mirrors", Infrared Physics & Technology 50, 2007, p42-p46

In view of the above situation, the present invention aims to provide a method for designing a mirror that can be used to create a single mirror that allows the light source position and focusing position to be set independently in the vertical and horizontal directions, thereby allowing free conversion of astigmatism, and that can also accommodate beams in the X-ray range by reducing the focusing size, has a simple design formula, is applicable in a wide range of applications, and can be used suitably as an optical system that handles beams with different characteristics in the vertical and horizontal directions.

In light of this current situation, the inventors conducted extensive research and discovered that a method for expressing the properties of a beam with astigmatism in geometrical optics can be provided by newly defining a "source ray" and a "focus ray" in the vertical and horizontal directions, respectively, and by assuming that all incident light rays passing through the reflective surface of the mirror pass through the "source rays" in the vertical and horizontal directions, and that all outgoing light rays emitted from the reflective surface of the mirror pass through the "focus rays" in the vertical and horizontal directions, and by applying Fermat's principle, which states that the "optical path length" from the light source position to the focus position is constant, it is possible to provide a design method for a mirror that can solve the above problems, which led to the completion of the present invention.

That is, the present invention includes the following inventions.
(1) A method for designing a mirror produced by forming a reflective surface on a hollow inner or outer peripheral surface of a substrate, the method including: setting a central axis of the inner or outer peripheral surface of the substrate as a z-axis, and a cross section perpendicular to the z-axis as an xy plane; an incident beam has a light source in the x-axis direction at a position displaced by L _1x along the z-axis from a predetermined position on the z-axis, and a light source in the y-axis direction at a position displaced by L _1y along the z-axis from the predetermined position on the z-axis; an outgoing beam is focused at a position displaced by L 2x along the z-axis from the predetermined position on the z-axis in the x-axis direction, and a light source in the y-axis direction at a position displaced by L _2x along the z-axis from the predetermined position on the z-axis in the y-axis direction; It is assumed that the light is focused at a position displaced by _2y from the mirror, all incident light rays passing through the mirror pass through a first light source line passing through the position of the light source in the x-axis direction and extending in a direction perpendicular to both the x-axis direction and the z-axis direction, and a second light source line passing through the position of the light source in the y-axis direction and extending in a direction perpendicular to both the y-axis direction and the z-axis direction, and all outgoing light rays emitted from the mirror pass through a first focused light line passing through the focusing position in the x-axis direction and extending in a direction perpendicular to both the x-axis direction and the z-axis direction, and a second focused light line passing through the focusing position in the y-axis direction and extending in a direction perpendicular to both the y-axis direction and the z-axis direction, and an arbitrary point on the reflecting surface of the mirror is denoted as M, and the coordinates of the intersection of the first light source line and the incident light ray at point M and the intersection of the second light source line and the incident light ray at point M are defined as L1x, L2x, L3x, L4x, L5x, L6x, L7x, L8x, L9x, L1x, L1x, L2x, L3x, L4x, L5x, L6x, L7x, L8x, L9x, L1x, L1x, L1x, L2x, L3x, L4x, L5x, L6x, L7x, L8x, L9x, L1x, L1x, L1x, L1x, L2x, L2x, L3x, L4x, L4x, L5x, L6x, L7x, _L8x , L1 ... and expressing the coordinates of an intersection point between the outgoing light from point M and the first condensed light and an intersection point between the outgoing light from point M and the second condensed light using _L2x , _{L2y ,} and designing the mirror using a design equation for the reflecting surface derived based on these coordinates and that the optical path length from the light source position to the condensed light position is constant for any point on the reflecting surface in each of the x-axis direction and the y-axis direction.

(2) The first light source line and the second light source line are respectively a straight line S _x extending in the y-axis direction and a straight line S _y extending in the x-axis direction, the first condensed light line and the second condensed light line are respectively a straight line F _x extending in the y-axis direction and a straight line F _y extending in the x-axis direction, the incident length from the light source position in the x-axis direction to point M is calculated as the distance from one of two intersection points between the incident light and the equiphase surface A 1x that is closer to the second light source line S _y to point M, with an arc having an intersection point P _y0 between the second light source line S _y and the z-axis as its center and passing through the intersection point P _x0 between the first light source line S x and the z-axis as its center, being rotated around the first light source line S _x as an equiphase surface A _1x , and the exit length from point M to the condensed light position in the x-axis direction is calculated as the distance from the one of two intersection points between the incident light and the equiphase surface A _1x that is closer to the second light source line S _y to point M, and the exit length from point M to the condensed light position in the x-axis direction to point M ... The incident length from the light source position in _{the y} -axis direction to point _M is calculated by rotating an arc having a center at an intersection point P x0 of the first light source line S _x and the z-axis, passing through an intersection point Q _x0 of the first light source line F _x and the z-axis, and rotating the arc around the first light source line F x to obtain an equiphase surface A _2x , and calculating the distance from one of the two intersection points between the output light beam and the equiphase surface A _2x that is closer to the second light source line F _y to point M. The incident length from the light source position in the y-axis direction to point M is calculated by rotating an arc having a center at an intersection point P _x0 of the first light source line _{S x and} the z-axis, passing through an intersection point P _y0 of the second light source line S _y and the z-axis to obtain an equiphase surface A _1y , and calculating the distance from one of the two intersection points between the input light beam and the equiphase surface A _1y that is closer to the first light source line S a distance from the intersection point closer to _x to point M, and an emission length from point M to the focusing position in the y-axis direction is calculated by rotating an arc having an intersection point _Qx0 of the first focusing line _Fx and the z-axis as its center, passing through an intersection point _Qy0 of the second light source line _Fy and the z- _axis and extending in a direction perpendicular to the y- _axis , around the second focusing line Fy as an _axis to obtain an equiphase surface _A2y , and calculating an optical path length that is a sum of an incident length and an exit length for focusing in each of the x-axis and y-axis directions.

(3) The distance from one of the two intersection points between the incident light ray and the equiphase surface _A1x that is closer to the second light source line S _y to point M is obtained by calculating the distance from an intersection point P _x between the incident light ray and the first light source line S _x to point M, and adding or subtracting the distance from the intersection point P _x to the arc that defines the equiphase surface A _1x to the distance. The distance from one of the two intersection points between the exit light ray and the equiphase surface A _2x that is closer to the second condenser line F _y to point M is obtained by calculating the distance from an intersection point Q _x between the exit light ray and the first condenser line F _x to point _{M, and adding or subtracting the distance from the intersection point Q x to} the arc that defines the equiphase surface A _2x to the distance. a distance from a point of intersection closer to _x to point M is obtained by determining a distance from a point of intersection _Py between the incident light beam and the second light source line S _y to point M, and adding or subtracting a distance from the point of intersection _Py to the arc defining the equiphase surface A _1y to that distance; and a distance from a point of intersection closer to a first focusing line _Fx of two points of intersection between the exit light beam and the equiphase surface A _2y to point M is obtained by determining a distance from a point of intersection _Qy between the exit light beam and the second focusing line _Fy to point M, and adding or subtracting a distance from the point of intersection _Qy to the arc defining the equiphase surface A _2y to that distance.

(4) The method for designing a mirror according to any one of (1) to (3), wherein the design formula is the following formula obtained by weighting a first formula f _x (x, y, z) = 0 derived from the fact that the optical path length from the light source point to the focusing point is constant in the x-axis direction, and a second formula f _y (x, y, z) = 0 derived from the fact that the optical path length from the light source point to the focusing point is constant in the y-axis direction:

(5) An astigmatism controlled mirror having a reflecting surface for which the design formula in the mirror design method according to any one of (1) to (4) holds, wherein the values of L _1x and L _1y are different and the values of L _2x and L _2y are the same, and an outgoing beam that is focused to one point can be obtained from an incoming beam having astigmatism.

(6) An astigmatism controlled mirror having a reflecting surface for which the design formula in the mirror design method according to any one of (1) to (4) holds, the values of L _1x and L _1y are the same, and the values of L _2x and L _2y are different, and an outgoing beam having astigmatism is obtained from an incident beam diverging from a single point.

The mirror design method of the present invention allows the light source position and focusing position to be set independently in the vertical and horizontal directions on a single mirror, making it possible to fabricate a mirror that allows free conversion of astigmatism. It also makes it possible to reduce the focusing size and accommodate beams in the X-ray range. Furthermore, the design formula is simple, the range of applications is wide, and it is possible to fabricate a mirror that can be suitably used as an optical system that handles beams with different characteristics in the vertical and horizontal directions.

FIG. 2 is a conceptual diagram showing the “light source ray” and the “light collecting ray” in the design direction of the mirror according to the present invention. 1 is a conceptual diagram showing an equiphase surface A _1x in the vicinity of an intersection point P _x on a sagittal light source line S _x . FIG. 13 is a conceptual diagram showing an equiphase surface A _1y in the vicinity of an intersection point P _y on a meridional light source line S _y . FIG. 2 is a schematic diagram of a mirror in which a reflective surface is formed on the hollow inner or outer peripheral surface of a substrate. FIG. 13 is a diagram showing an example of focusing a beam using a rotating mirror having a conic section as its ridgeline. FIG. 13 is a diagram showing an example of diffusing a beam using a rotating mirror having a conic section as its ridgeline. FIG. 2 is an explanatory diagram for explaining a ray tracing calculation method based on geometric optics. FIG. 2 is an explanatory diagram for explaining a diffraction integral calculation method based on wave optics. FIG. 2 is a schematic diagram showing an optical system arrangement of mirrors according to the first embodiment. 1A shows the radial distribution of the reflecting surface of Example 1, where FIG. 1A is a two-dimensional distribution of radius r(z, φ) when the vertical axis is set to the z coordinate and the horizontal axis is set to the φ coordinate. FIG. 1B is a radial profile of the reflecting surface at φ=0° (z-axis direction (longitudinal direction)) indicated by the dashed line in FIG. 1A. FIG. 1C is a radial profile of the reflecting surface at z=0 (circumferential direction) indicated by the dashed line in FIG. 1A and 1B show simulation results of the focusing performance of Example 1, in which FIG. 1A shows the result of calculating the variation of light rays on the focusing surface based on geometric optics, and FIG. 1B shows a two-dimensional intensity distribution on the focusing surface calculated based on wave optics. FIG. 11 is a schematic diagram showing an optical system arrangement of mirrors according to a second embodiment. 13A shows the radial distribution of the reflecting surface of Example 2, where FIG. 13A is a two-dimensional distribution of radius r(z, φ) when the vertical axis is set to the z coordinate and the horizontal axis is set to the φ coordinate. FIG. 13B is a radial profile of the reflecting surface at φ=0° (z-axis direction (longitudinal direction)) indicated by the dashed line in FIG. 13A. FIG. 13C is a radial profile of the reflecting surface at z=0 (circumferential direction) indicated by the dashed line in FIG. 13 shows the results of a simulation of the focusing performance of Example 2, in which (a) shows the results of calculating the variation of light rays in the x-axis direction focusing surface based on geometric optics, (b) shows a two-dimensional intensity distribution in the x-axis direction focusing surface calculated based on wave optics, (c) shows the results of calculating the variation of light rays in the y-axis direction focusing surface based on geometric optics, and (d) shows a two-dimensional intensity distribution in the y-axis direction focusing surface calculated based on wave optics.

The mirror design method of the present invention is a method for designing a mirror that is manufactured by forming a reflective surface on the hollow inner or outer peripheral surface of a substrate. Below, the mirror design method of the present invention will be explained with reference to a representative embodiment.

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 focusing (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 focusing 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 focusing point. In this invention, the idea of newly defining "light source line" and "focusing 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.

FIG. 1 is a conceptual diagram showing "light source line" and "light collecting line". The central axis of the inner or outer peripheral surface of the substrate is the z axis, and the cross section perpendicular to the z axis is the xy plane. The incident beam has a light source in the x axis direction (horizontal direction in this example) at a position displaced by L _1x along the z axis from a predetermined position on the z axis, and a light source in the y axis direction (vertical direction in this example) at a position displaced by L _1y along the z axis from the predetermined position on the z axis. In addition, the outgoing beam is collected at a position displaced by L _2x along the z axis from the predetermined position on the z axis in the x axis direction (horizontal direction), and is collected at a position displaced by L _2y along the z axis from the predetermined position on the z axis in the y axis direction (vertical direction).

All incident light rays passing through the mirror are considered to pass through a first light source line (S x ) that passes through the position of the light source in the x-axis direction (horizontal direction) and extends in a direction perpendicular to both the x-axis direction (horizontal direction) and the z-axis direction, and a second light source line (S _y ) that passes through the position of the light _{source in the y-axis direction (vertical direction) and extends in a direction perpendicular to both the y-axis direction (vertical direction) and the z-axis direction. The first light source line (S x )} and the second light source line (S _y ) are defined in this manner.

Furthermore, all of the emitted light rays emitted from the mirror are considered to pass through a first focusing line (F x ) that passes through the focusing position in the x-axis direction (horizontal direction) and extends in a direction perpendicular to both the x-axis direction (horizontal direction) and the z-axis direction, and a second focusing line (F _y ) that passes through the focusing position in the y-axis direction (vertical direction) and extends in a direction perpendicular to both the y-axis direction (vertical direction) and the z-axis direction. The first focusing line (F _x ) and the second focusing line (F _y ) are defined in _this manner.

In this example, the first light source line (S _x ), the second light source line (S _y ), the first condensed light line (F _x ), and the second condensed light line (F _y ) are each straight lines, but they may be curved lines. FIG. 1 shows the case where L _1x > L _1y > 0 and L _2x > L _2y > 0, but the magnitude relationship between L _1x and L _1y may be reversed, or the magnitude relationship between L _2x and L _2y may be reversed, and these constants may be negative. When L _1x or L _1y has a negative value, the incident beam is reflected by the reflecting surface of the mirror while being condensed toward the downstream. When L _2x or L _2y has a negative value, the outgoing beam has a wavefront that diverges from a position upstream of the mirror.

By defining the "light source line" and the "focus line" as described above, it is possible to define the incident light ray and the outgoing light ray passing through any point on the reflecting surface of the mirror. That is, assuming that any point on the reflecting surface of the mirror is M(x, y, z), the coordinates of the intersection (P _{x ) between the first light source line (S x )} and the incident light ray to the point M, and the intersection (P _y ) between the second light source line (S _y ) and the incident light ray to the point M can be expressed by the following formulas (1) and (2) using the above-mentioned displacements L _1x and L _1y . Similarly, the coordinates of the intersection (Q _x ) between the outgoing light ray from the point M and the first focus line (F _{x )} , and the intersection (Q _y ) between the outgoing light ray from the point M and the second focus line (F y ) can be expressed by the following formulas (3) and (4) using the above-mentioned displacements L _2x and L _2y .

Then, based on _the fact that the optical path length (the sum of the incident length and the exit length ₎ from the light source position to the focusing position is constant for any point on the reflecting surface in each of the coordinates _Px , _Py , Qx, and Qy, and in the x-axis direction and the y-axis direction, respectively, a design equation for the reflecting surface can be derived.

In this embodiment, the distance between each of the intersection points _Px , Py, _Qx , and _Qy on the light _source line and the condenser line and an arbitrary point M(x, y, z) on the reflecting surface is not directly determined as the incident length or the exit length, but the coordinates of the intersection points of the light source line and the condenser line defined by straight lines are used, and the optical path length is compensated as follows to obtain a more accurate design formula.

(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 distance 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.

First, on the incident side, consider the equiphase surface in the vicinity of the intersection point P _x on the light source line S _x described above. On the light source line S _x , a wavefront converging toward the light source line S _y should be observed. Although it is not possible to strictly define the phase on S _x under such assumptions, here, the intersection point between S _y and the z axis is set as P _y0 , and a phase distribution according to the distance from P _y0 exists on S _x , that is, the beam before entering the mirror (reflection surface) has a wavefront that converges to the light source line S _y in the y-axis direction (vertical direction). Based on this idea, as shown in FIG. 2, an arc B 1x that is centered on the intersection point P _y0 between the second light source line _{S y} and the z axis and extends in a direction perpendicular to the x axis through the intersection point P _x0 between the first light source line S x and the z _axis is rotated around the first light source line S _x to form an equiphase surface A _1x . It is more accurate to determine the incident length from the light source position in the x-axis direction to point M as the distance from one of two intersection points between the incident light ray and the equiphase surface _A1x that is closer to the second light source line _Sy to point M.

Here, the distance from the intersection point closer to the second light source line S _y of the two intersection points between the incident light ray and the equiphase surface A _1x to point M on the reflecting surface of the mirror is calculated by first calculating the distance from the intersection point P _x between the incident light ray and the first light source line S _x to point M, and then adding or subtracting (in the example shown in the figure, subtracting) the distance from the intersection point P _x to the arc B _1x that defines the equiphase surface A _1x , that is, the distance between P _{x and} H _1x , with H _1x being the foot of the perpendicular line from P _x to the arc B _1x . That is, the distance between H _1x and M in the following formula (5) is set as the incidence length in the x-axis direction. The reason why this formula is an approximation is that there is no guarantee that the point H _1x exists on the straight line P _x M. However, it is of course possible to calculate the distance using a formula other than this approximation. In this example, as described above, the foot of the perpendicular line from _Px to the arc _B1x is defined as _H1x , and the distance between Px _{and H1x} is added/subtracted to obtain an approximation. However, instead of the perpendicular line to the arc _B1x , the distance may be calculated more accurately by using the distance from _Px to one of the two intersection points between the incident ray and the equiphase surface _A1x that is closer to the second light source line S _y .

Next, also on the incident side, consider an equiphase surface in the vicinity of the intersection point P _y on the light source line S _y . On the light source line S _y , a wavefront diverging from the light source line S _x should be observed. Although it is not possible to strictly define the phase on S _y under such assumption, here, the intersection point of S _x and the z axis is set as P _x0 , and a phase distribution according to the distance from P _x0 exists on S _y , that is, the beam before entering the mirror (reflection surface) has a wavefront diverging from the light source line S _x in the x-axis direction. Based on this idea, as shown in FIG. 3, an arc B 1y having a center at the intersection point _{P x0} of the first light source line S _x and the z axis and passing through the intersection point P _y0 of the second light source line S _y and the z axis and extending in a direction perpendicular to the y axis is rotated around the second light source line S _y as an equiphase surface A _1y . The incident length from the light source position in the y-axis direction to point M is determined as the distance from one of two intersection points between the incident light ray and the equiphase surface _A1y that is closer to the first light source line _Sx to point M on the reflecting surface of the mirror.

The distance from the intersection point closer to the first light source line _Sx of the two intersection points between the incident light ray and the equiphase surface _A1y to point M is calculated by first calculating the distance from the intersection point _Py between the incident light ray and the second light source line _Sy to point M, and then adding or subtracting (adding in this example) to the distance, the distance from the intersection point _Py to the arc _B1y that defines the equiphase surface _A1y , that is, the distance between Py _{and H1y} , where _H1y is the foot of the perpendicular line from _Py to the arc _B1y . That is, the distance between _{H1y and} M in the following formula (6) is determined as the incident length in the y-axis direction.

As for the exit side, similar to the entrance side, consider the equiphase surface near the intersection point _Qx on the first focusing line _Fx , and the equiphase surface near the intersection point _Qy on the second focusing line _Fy . In the first focusing line _Fx , a wavefront diverging from the second focusing line _Fy should be observed. Although it is not possible to strictly define the phase on _Fx under such assumptions, here, the intersection point of _Fy and the optical axis z is set as _Qy0 , and it is assumed that there is a phase distribution on _Fx according to the distance from _Qy0 . In addition, in the second focusing line _Fy , a wavefront converging toward the first focusing line _Fx should be observed. Although it is not possible to strictly define the phase on _Fy under such assumptions, here, the intersection point of _Fx and the exit optical axis z is set as _Qx0 , and it is assumed that there is a phase distribution on _Fy according to the distance from _Qx0 .

Based on these considerations, a more accurate exit length is obtained, as in the case of the incident side. Specifically, although not shown in the figures, compensation is performed by adding or subtracting the distance from the intersection _Qx to the arc _B2x that defines the equiphase surface, that is, the distance between _H2x and _{Qx, where H2x is the foot of the perpendicular line from Qx to the arc B2x , and the} distance from the intersection _Qy to the arc _B2y that defines the equiphase surface, that is, the distance between _Qy and _H2y, where H2y is the foot of the perpendicular line from _Qy to _the arc _B2y , to obtain a more accurate exit length in the x-axis direction and the y-axis direction as shown in the following formulas (7) and (8).

(Calculation of optical path length)
Using the incident length and exit length thus obtained, the optical path length for focusing in each of the x-axis and y-axis directions is calculated. First, the incident length f _1x (x, y, z) when focusing on focusing in the x-axis direction is calculated from equation (5) using the following equations (9) to (11).

Similarly, the exit length f _2x (x, y, z) focusing on the light collection in the x-axis direction is expressed by the following equations (12) to (14) from equation (7.9).

Then, when the reference optical path length from the light source point to the focusing point in the x-axis direction is set to _Lx , the conditional equation required for focusing in the x-axis direction is derived as the following equation (15).

Next, the condition equation required for focusing in the y-axis direction is derived in a similar manner. That is, the incident length f _1y (x, y, z) for focusing in the y-axis direction is obtained from equation (6) as shown in the following equation (16).

Similarly, the exit length f _2y (x, y, z) in terms of light collection in the y-axis direction is expressed by the following equation (17) from equation (8).

Then, when the reference optical path length from the light source point to the focusing point in the y-axis direction is set to _Ly , the conditional expression required for focusing in the y-axis direction is derived as the following expression (18).

Ideally, a set of points (x, y, z) that simultaneously satisfies the focusing condition in the x-axis direction of equation (15) and the focusing condition in the y-axis direction of equation (18) represents the desired shape of the reflecting surface of the mirror, but if the solution of such simultaneous equations is used as a design formula, the solution can only exist under special conditions such as " _L1x = _L1y and _L2x = _L2y . " In order to obtain a design formula that represents a more generalized shape of the reflecting surface that can be established under other conditions, the inventors weighted equations (15) and (18) and used a new formula f(x, y, z) shown in equation (19) as the design formula.

(Design formula)
That is, the design formula is f(x,y,z)=0 (formula (15)), which is the first formula (formula for the x-axis direction focusing condition) derived from the fact that the optical path length from the light source point to the focusing point is constant in the x-axis direction, and _fy (x,y,z)=0 (formula (18)), which is the second _formula (formula for the y-axis direction focusing condition) derived from the fact that the optical path length from the light source point to the focusing point is constant in the y-axis direction, weighted by α and β as shown in the following (19): f(x,y,z)=0. α is a weighting coefficient for focusing in the x-axis direction, and β is a weighting coefficient for focusing in the y-axis direction.

By substituting equations (9) through (18) into equation (19), the design equation for the reflecting surface is derived as shown in equation (20).

As can be seen from equation (20), it can be confirmed that an equation with good symmetry in the x-axis and y-axis directions has been obtained. In the derivation so far, we have assumed that " _L1x > _L1y > 0 and _L2x > _L2y >0", but even if this assumption is not made, that is, even if the magnitude relationship is reversed or each setting value takes a negative value, the same equation (design equation) shown in equation (20) can be derived. However, all of the four constants _L1x , _L1y , _L2x , and _L2y are positive or negative values and cannot be 0.

Furthermore, since the reflecting surface is made from the hollow inner or outer peripheral surface of the substrate, a specific design formula can be obtained by setting a reference point and substituting the coordinates thereof to set the constant term “αL _x + βL _y ” in the above formula (20).

A schematic diagram of a mirror in which a reflective surface is formed on the hollow inner or outer peripheral surface of a substrate is shown in Figure 4. The definition of the coordinate system is the same as in Figure 1. Reference points M _0x (r ₀ , 0, 0) and M _0y (0, r ₀ , 0) are set, where r ₀ is a constant representing the reference radius. First, assuming that M _0x (r ₀ , 0, 0) satisfies equation (20), which is an implicit function expression of the collecting mirror shape, the following equation (21) is obtained by substituting the coordinates of the reference point M _0x (r ₀ , 0, 0) into equation (20).

In order for this equation (21) to hold regardless of the setting of the weighting coefficient α, L _x and _Ly must satisfy equations (22) and (23), respectively.

Similarly, by substituting the coordinates of the reference point M _0y into equation (20), equation (24) is obtained. From equation (24), the constraints on L _x and L _y are similarly derived as equations (25) and (26).

The values of _Lx and _Ly to be obtained are obtained by taking a weighted arithmetic average of the values derived from the reference points _M0x and _M0y using the weighting coefficient α for the light concentration in the x-axis direction and the weighting coefficient β for the light concentration in the y-axis direction. That is, _Lx and _Ly are expressed by the following equations (27) and (28).

By substituting equations (27) and (28) into equation (20), the design equation for the mirror reflecting surface is derived as the equation shown below in equation (29).

(Examples of mirrors that can be designed)
In setting the condition of formula (29), by setting the values of L _1x and L _1y to different values and the values of L _2x and L _2y to the same value (the same value), it is possible to design an astigmatism control mirror having a reflecting surface that can obtain an outgoing beam that is focused to one point from an incident beam having astigmatism. Conversely, by setting the values of L _1x and L _1y to the same value and the values of L _2x and L _2y to different values, it is possible to design an astigmatism control mirror having a reflecting surface that can obtain an outgoing beam having astigmatism from an incident beam that diverges from one point.

In addition, by using the design formula (formula (29)), it is possible to design a mirror with a reflecting surface where the light source and light collection positions coincide in both the x-axis direction and the y-axis direction. For example, by substituting _L1x = _L1y = _L1 > 0 and _L2x = _L2y = _L2 > 0 into the design formula (formula (29)), the formula for a rotational ellipsoidal mirror can be obtained as shown in the following formula (30).

Furthermore, by substituting L _1x =L _1y =L ₁ <0 and L _2x =L _2y =L ₂ >0 into the design equation (equation (29)), the equation for the rotating hyperbolic mirror can be obtained as shown in the following equation (31).

Furthermore, by setting _L1 to positive or negative infinity in the above equation (30) or equation (31), the equation for the parabolic mirror of revolution can be obtained as shown in the following equation (32).

(Design limit)
First, as described above, none of L _1x , L _1y , L _2x and L _2y is 0. Furthermore, for example, when the light-collecting action of the mirror is limited to only the vertical direction, that is, when L _1x = +∞ and L _2x = +∞, L _1y > 0, L _1y > 0 and α = 0 are given, equation (29) takes the form of the following equation (33).

Equation (33) is clearly the equation for an elliptical cylindrical surface. It can be seen that two elliptical cylindrical mirrors are arranged to sandwich the optical axis from the vertical direction, indicating a design failure. The reason for the failure is that the design did not have positive focusing performance in both the vertical and horizontal directions. The positive focusing performance mentioned here is the ability to "change the beam in a more convergent direction." This condition is expressed by the following equation (34), where the entrance length is _L1 and the exit length is _L2 .

Equation (34) indicates that if the curvature of the wavefront diverging from upstream is defined as positive, then the curvature decreases after reflection by a mirror. In this case, the shape of the mirror becomes concave. Figure 5 shows an example of focusing a beam using a rotating mirror with a conic section as its ridge. All of these satisfy equation (34). In order to design a mirror that uses a hollow inner surface, it is a necessary and sufficient condition that equation (34) be satisfied in both the vertical and horizontal directions.

It is also possible for a conic section to increase the curvature of a wavefront by utilizing its convex profile. Figure 6 shows a list of examples of increasing the curvature of a wavefront using a rotating conic section. All of these examples satisfy the following equation (35). When a mirror that increases the curvature of the wavefront in both the vertical and horizontal directions is designed using this method, it becomes a "mirror that uses a cylindrical outer surface."

When the curvature of the wavefront does not change, that is, when 1/ _L1 = -1/ _L2 holds, the ridgeline of the mirror will have a linear profile with no curvature, and the design will fail. In summary, the conditions for the design equation for the hollow inner surface mirror to hold are 1/ _L1x > -1/ _L2x and 1/ _L1y > -1/ _L2y , and the conditions for the design equation for the outer surface mirror to hold are 1/ _L1x < -1/ _L2x and 1/ _L1y < -1/ _L2y .

(Cross-Section Profile)
The cross-sectional profile of the reflecting surface can be confirmed by substituting x = 0 or y = 0 into the design equation (equation (29)). For example, when x = 0, L _1x > 0, L _1y > 0, L _2x > 0, and L _2y > 0, the intersection line (cross-sectional profile) between the yz plane and the mirror reflecting surface is expressed by the following equations (36) and (37). Equation (36) represents an elliptical function with the light source point of the y-direction focused light and the focusing point of the y-direction focused light as its foci. C in the equation is a constant term that represents the reference optical path length.

Although the embodiment of the present invention has been described above, the present invention is not limited to these examples, and can of course be embodied in various forms within the scope of the present invention. In this embodiment, the light source line and the condenser line are straight lines, and the distance between the straight line and the equiphase surface in its vicinity is compensated for, but such compensation is not necessarily required. It is also preferable to use arc lines or other curves as the light source line and the condenser line, and to obtain the result 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 at a different position. Coordinate transformation may of course be performed.

Next, as design examples of the astigmatism control mirror according to the present invention described above, two types of mirrors were designed: a mirror intended to eliminate astigmatism (Example 1) and a mirror intended to add astigmatism (Example 2). The performance of each mirror was confirmed by simulation using both geometric optics and wave optics. The results are described below.

(Simulation method)
Considering an actual use example of a hollow mirror, we calculated the focusing performance under partial illumination conditions where the soft X-ray beam illuminates only a part of the entire circumference (360°) of the mirror (reflecting surface) as shown in Figure 7. The installation condition was horizontal (x-axis direction) deflection. In the ray tracing calculation based on geometric optics, a group of rays passing through the light source line in the x-axis and y-axis directions shown in Figure 1 is defined and made to enter the reflecting surface of the mirror. 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.

The normal vector n(x, y, z) at each position on the mirror's reflecting surface is a unit vector parallel to the gradient vector of the function f(x, y, z) defined in equation (29) (equation (38)). As shown in Figure 7, the incident light ray 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 ray at the focusing surface is evaluated.

In diffraction integral calculations based on wave optics, the centers of curvature in the x-axis and y-axis directions of the wavefront of the beam incident on the reflecting surface of the mirror are made to coincide with the light source Sx that focuses the light in the x-axis direction _and the light source _Sy that focuses the light in the y-axis direction, respectively. Figure 8 is a schematic diagram. It is assumed that a line light source with no thickness exists, and that the beam is incident with uniform intensity over the entire effective area of the reflecting surface. The wave field U _M (x _M , y _M , z _M ) of the beam at point M (x _M , y _M , z _M ) on the reflecting surface is expressed by the following equations (39) and (40).

In equation (39), λ is an arbitrary constant representing the wavelength of the beam, and _I0 is an arbitrary constant representing the incident intensity. The complex wave field U _M (x _M , y _M , z _M ) on the reflecting surface of the mirror is propagated to coordinate Q (x _Q , y _Q , z _Q ) on the focusing surface according to the following equations (41) and (42).

In equation (41), dS represents a small area on the reflecting surface, and θ( _xM , _yM , _zM ) represents the oblique incidence angle at each position on the reflecting surface. The output is the intensity distribution, which is the square of the absolute value of the complex wave field _UQ ( _xQ , _yQ , _zQ ) on Q (equation (43)). By the above procedure, the intensity distribution on the focusing surface is calculated based on wave optics.

(Mirror Design of Example 1)
Table 1 shows a list of constants used in the mirror design of Example 1. The incident length is different in the x-axis direction (horizontal direction) and the y-axis direction (vertical direction), and the exit length is the same in the x-axis direction and the y-axis direction. The reference radius was set to 5 mm so that the oblique incidence angle was about 10 mrad. The arrangement of the light source line, the condensed light line, and the mirror (reflecting surface) is shown in Figure 9.

To obtain the shape of the mirror's reflecting surface, the implicit function shown in the design equation (29) is solved. A solution set for equation (29) is obtained using a cylindrical coordinate system. Here, the cylindrical coordinate system (z, φ, r) and the Cartesian coordinate system (x, y, z) are assumed to have the corresponding relationship shown in the following equation (44).

(Radius distribution of the reflecting surface in Example 1)
The calculated radius distribution of the reflecting surface of the mirror is shown in Fig. 10. Fig. 10(a) shows a two-dimensional distribution of radius r(z, φ) when the vertical axis is set to the z coordinate and the horizontal axis is set to the φ coordinate. The reflecting surface of the mirror in Example 1 has a hollow shape with different diameters in the y-axis direction (vertical direction) and the x-axis direction (horizontal direction).

Figure 10(b) shows the radial profile of the reflecting surface at φ=0° (z-axis direction (longitudinal direction)) indicated by the dashed line in Figure 10(a). This is an elliptical function corresponding to the focusing in the x-axis direction (horizontal direction). Figure 10(c) shows the radial profile of the reflecting surface at z=0 (circumferential direction) indicated by the dashed line in Figure 10(a). It can be seen that there are two peaks in the circumferential direction due to the crushing of the mirror.

(Simulation results of light collection performance in Example 1)
Next, the results of a simulation of the focusing performance are shown in Fig. 11. Fig. 11(a) shows the results of calculating the dispersion of light rays on the focusing surface based on geometric optics. It can be seen that all light rays are concentrated in an area of 1 nm or less both vertically and horizontally.

Figure 11(b) shows the two-dimensional intensity distribution at the focal plane calculated based on wave optics. The photon energy was set to 300 eV. The beam was focused to an area of 430 nm (x-axis direction (horizontal direction)) x 170 nm (y-axis direction (vertical direction)) (FWHM).

(Mirror Design of Example 2)
Table 2 shows a list of constants used in the mirror design of Example 2. The incident length is the same in the x-axis direction (horizontal direction) and the y-axis direction (vertical direction), while the exit length has different values in the x-axis direction and the y-axis direction. The reference radius _r0 was set so that the oblique incidence angle on the reflecting surface was about 10 mrad. The arrangement of the light source line, the focusing line, and the mirror (reflecting surface) is shown in Figure 12. As in Example 1, a solution set of equation (29) is obtained using a cylindrical coordinate system.

(Radius distribution of the reflecting surface in Example 2)
The calculated radius distribution of the reflecting surface of the mirror is shown in Figure 13. As with the mirror in Example 1, it can be seen that the mirror has a hollow shape with different diameters in the y-axis direction (vertical direction) and the x-axis direction (horizontal direction), and that a two-peak distribution is formed in the circumferential direction due to the crushing of the mirror.

(Simulation results of light collection performance in Example 2)
The results of a simulation of the focusing performance are shown in Fig. 14. Fig. 14(a) shows the results of calculating the dispersion of light rays on the focusing surface (z = _L2x ) in the x-axis direction (horizontal direction) based on geometric optics. It can be seen that all light rays are concentrated in an area with a width of 1 nm or less in the x-axis direction (horizontal direction).

Figure 14(b) shows the two-dimensional intensity distribution at the focusing surface in the x-axis direction (horizontal direction) calculated based on wave optics. The photon energy was set to 300 eV. The beam is focused to an area 52 μm (FWHM) wide in the x-axis direction (horizontal direction).

14(c) and (d) show the simulation results of the focusing performance on the focusing surface (z = _L2y ) in the y-axis direction (vertical direction). The focusing width in the y-axis direction (vertical direction) was 130 nm in geometric optics and 7.4 μm (FWHM) in wave optics.

Claims (6)

A method for designing a mirror produced by forming a reflective surface on a hollow inner or outer peripheral surface of a substrate, comprising the steps of:
A central axis of the inner circumferential surface or the outer circumferential surface of the base material is defined as a z-axis, and a cross section perpendicular to the z-axis is defined as an xy-plane;
The incident beam has an x-axis direction light source at a position displaced by L _1x along the z-axis direction from a predetermined position on the z-axis, and has a y-axis direction light source at a position displaced by L _1y along the z-axis direction from the predetermined position on the z-axis;
the emitted beam is focused in the x-axis direction at a position displaced by L _2x along the z-axis from the predetermined position on the z-axis, and in the y-axis direction at a position displaced by L _2y along the z-axis from the predetermined position on the z-axis;
All incident light rays passing through the mirror pass through a first light source line passing through a position of the light source in an x-axis direction and extending in a direction perpendicular to both the x-axis direction and the z-axis direction, and a second light source line passing through a position of the light source in a y-axis direction and extending in a direction perpendicular to both the y-axis direction and the z-axis direction;
All of the emitted light rays emitted from the mirror pass through a first focusing line that passes through the focusing position in the x-axis direction and extends in a direction perpendicular to both the x-axis direction and the z-axis direction, and a second focusing line that passes through the focusing position in the y-axis direction and extends in a direction perpendicular to both the y-axis direction and the z-axis direction;
An arbitrary point on the reflecting surface of the mirror is denoted as M, and the coordinates of the intersection of the first light source line and the incident light to point M and the intersection of the second light source line and the incident light to point M are expressed using the L _1x and L _1y . In addition, the coordinates of the intersection of the outgoing light from point M and the first condensed light and the intersection of the outgoing light from point M and the second condensed light are expressed using the L _2x and L _2y .
A method for designing a mirror, comprising the steps of: designing a 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 in both the x-axis direction and the y-axis direction. The first light source line and the second light source line are respectively defined as a straight line S _x extending in the y-axis direction and a straight line S _y extending in the x-axis direction,
The first collecting line and the second collecting line are respectively a straight line F _x extending in the y-axis direction and a straight line F _y extending in the x-axis direction,
The incident length from the light source position in the x-axis direction to point M is determined as the distance from one of two intersections between the _incident light _ray and the equiphase surface _A1x that is closer to the second light source line S _y out of two intersections between the incident light ray and _{the equiphase surface A1x , the} incident length being determined as the distance _from the ...
The emission length from point M to the focusing position in the x-axis direction is determined as the distance from one of two intersections between the emission _light and the equiphase surface _A2x , which is closer to the second focusing line _Fy , to point _M , when an arc extending in a direction perpendicular to the x-axis and centered at an intersection Qy0 between the second focusing line Fy and the z-axis and passing through an intersection Qx0 between the first focusing line _Fx and the z-axis is rotated about the first focusing line _Fx as an _axis to obtain an equiphase surface _A2x .
The incident length from the light source position in the y-axis direction to point M is determined as the distance from one of two intersections between the incident light ray and the equiphase surface _A1y that is closer to the first light source line _Sx , to point M, where an arc extending in a direction perpendicular to the y-axis is centered at an intersection _Px0 between the first light source line _Sx and the z-axis, passes through an intersection _Py0 between the second light source line Sy and the z- _axis , and is rotated about the second light source line _Sy as an _axis to form an equiphase surface A1y;
The emission length from point M to the focusing position in the y-axis direction is determined as the distance from one of two intersections between the emission light ray and the equiphase surface _A2y that is closer to the first focusing line _Fx , to point _M , where an arc that is centered at an intersection _Qx0 between the first focusing line _Fx and the z-axis, passes through an intersection Qy0 between the second light source line _Fy and the z-axis, and extends in a direction perpendicular to the y _- axis, is rotated about the second focusing line _Fy as an axis to obtain an equiphase surface A2y.
As a result, the optical path length, which is the sum of the incident length and the exit length, is calculated for each of the light collection in the x-axis direction and the y-axis direction.
The method for designing a mirror according to claim 1. the distance from one of the two intersection points between the incident light ray and the equiphase surface _A1x closer to the second light source line S _y to point M is obtained by calculating the distance from an intersection point P _{x between the incident light ray and the first light source line S x to} point M, and adding or subtracting the distance from the intersection point P _x to the arc defining the equiphase surface _A1x to the distance;
the distance from one of the two intersection points between the output light beam and the equiphase surface _A2x closer to the second condenser line _Fy to point M is obtained by calculating the distance from an intersection point _Qx between the output light beam and the first condenser line _Fx to point M, and adding or subtracting the distance from the intersection point _Qx to the arc defining the equiphase surface _A2x to the distance;
the distance from one of the two intersection points between the incident light ray and the equiphase surface _A1y closer to the first light source line _Sx to point M is obtained by calculating the distance from an intersection point _Py between the incident light ray and the second light source line _Sy to point M, and adding or subtracting the distance from the intersection point _Py to the arc defining the equiphase surface _A1y to the distance;
The distance from one of the two intersection points between the output light beam and the equiphase surface _A2y closer to the first condensing line _Fx to point M is obtained by calculating the distance from an intersection point _Qy between the output light beam and the second condensing line _Fy to point M, and adding or subtracting the distance from the intersection point _Qy to the arc that defines the equiphase surface _A2y to the calculated distance.
3. The method for designing a mirror according to claim 2. The design formula is
The method for designing a mirror according to any one of claims 1 to 3, comprising the following formula, which is a weighted version of a first formula f _x (x, y, z) = 0 derived from the optical path length from the light source point to the focusing point being constant in the x-axis direction, and a second formula f _y (x, y, z) = 0 derived from the optical path length from the light source point to the focusing point being constant in the y-axis direction:

A mirror having a reflecting surface for which the design formula in the mirror design method according to any one of claims 1 to 4 is satisfied,
The values of L _1x and L _1y are different, and the values of L _2x and L _2y are the same;
An astigmatism control mirror that can obtain an output beam that is focused to a single point from an input beam that has astigmatism. A mirror having a reflecting surface for which the design formula in the mirror design method according to any one of claims 1 to 4 is satisfied,
The values of L _1x and L _1y are the same, and the values of L _2x and L _2y are different;
An astigmatism control mirror that produces an astigmatism-bearing output beam from an input beam diverging from a single point.

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