JP2022108208A - Mirror design method and astigmatism control mirror including reflection surface in which design formula in the design method is established - Google Patents

Abstract

To manufacture a single mirror that allows free conversion of astigmatism by individually setting a light source position and a condensation position in the vertical direction and the horizontal direction, can be applied to a beam in an X-ray region by further reducing the condensation size, and can be suitably used as an optical system that deals with a beam with different characteristics in the vertical direction and the horizontal direction, and to provide a design method.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 first light source ray and an incident ray on the M point and the intersection of a second light source ray and the incident ray on the M point are represented by using L1x, L1y; the coordinates of the intersection of an emission ray from the M point and a first condensed ray and the intersection of the emission ray from the M point and a second condensed ray are represented by using L2x, L2y; and a design formula of the reflection surface is used which is 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 an x-axis direction and a y-axis direction.SELECTED DRAWING: Figure 1

Description

The present invention relates to a method of designing a mirror manufactured by forming a reflecting surface on a hollow inner peripheral surface or an outer peripheral surface of a base material, and an astigmatism control mirror having a reflecting surface that satisfies the design formula in the designing method.

Synchrotron soft X-ray beams are characterized by different characteristics in the vertical direction and in the horizontal direction. The beam size tends to be smaller in the vertical direction than in the horizontal direction. The coherence width is larger in the vertical direction than in the horizontal direction. Furthermore, spectroscopy systems using diffraction gratings, which are prevalent in soft X-ray beamlines, increase the vertical divergence angle of the beam. In addition, since the spectroscope including the diffraction grating used condenses soft X-rays only in the spectroscopic direction, different "astigmatism" occurs between the light source position in the spectroscopic direction and the non-condensed direction.

Conventionally, as a technique for an optical system that handles beams with different characteristics in the vertical direction and the horizontal direction, for example, a two-stage condensing optical system with two mirrors that handle each direction in the vertical direction and the horizontal direction is used. , a technique is used in which the light source points are set independently in the vertical direction and the horizontal direction, and the condensing points are matched. Specifically, two elliptical cylinder mirrors are arranged vertically and horizontally, two bent mirrors (mechanically bent cylindrical mirrors) are arranged to establish an approximate shape, and a bent mirror and a sagittal cylinder mirror are used. is known as a method of arranging two sheets of . However, in the case of such a method of combining two mirrors, the mechanism becomes complicated, and the number of chambers increases, resulting in an increase in cost and difficulty in adjustment.

A toroidal mirror is one of the possibilities for eliminating astigmatism with a single mirror (Non-Patent Document 1). However, the toroidal mirror is an approximation of the existing spheroidal mirror, and it is a mirror that is easy to manufacture by setting a uniform radius of curvature in each of the longitudinal and lateral directions of the reflecting surface, eliminating astigmatism. Even if it can be done, there is a drawback that the focused light size increases in principle.

An astigmatic off-axis mirror (AO mirror) has also been proposed as a mirror that can make the light collection size smaller than a toroidal mirror and that can set independent vertical and horizontal light sources and light collection points (Non-Patent Document 2). . This mirror has an elliptic curve to focus a beam diverging from one point to another point, a parabola to collimate a beam diverging from one point, and a different beam to another point. Based on the principle that each hyperbola is applied as the ridgeline of the reflecting surface in order to convert it into a beam condensing toward a point, different conic curves are set in the longitudinal direction and the lateral direction, and they are smoothly connected. It has a shape that requires a curved surface.

However, this AO mirror is a mirror defined by rotating the longitudinal conic profile about a straight line (major axis) connecting the focal points of the transverse conic sections to obtain a curved surface. is approximated to an axially symmetrical shape, there is a limit to the suppression of the focused light size due to the approximation. There is no problem with long-wavelength beams in the terahertz region, but they cannot be used with beams in the X-ray region. In addition, the design formula is extremely complicated, including several coordinate transformations, and the parameters are also complex, difficult to understand, and difficult to 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

Therefore, in view of the above-mentioned situation, the present invention aims to solve the problem by using a single mirror that can set the light source position and the light condensing position independently in the vertical direction and the horizontal direction. Aberrations can be freely converted, and the size of the focused beam can be kept smaller to support beams in the X-ray region. The design formula is simple, and the range of applications is wide. Vertical and horizontal directions are possible. The object of the present invention is to provide a method of designing a mirror that can fabricate a mirror that can be suitably used as an optical system that handles beams with different characteristics.

In view of the current situation, the present inventors have made intensive studies and found that as a method of geometrically expressing the properties of a beam with astigmatism, a "light source line" and a "condensed line" are used in the vertical and horizontal directions, respectively. , all incident rays passing through the reflective surface of the mirror pass through vertical and horizontal "source rays", and all outgoing rays emitted from the reflective surface of the mirror pass through the vertical and horizontal By applying Fermat's principle that the "optical path length" from the light source position to the condensed position is constant, it is possible to provide a mirror design method that can solve the above problems. , have completed the present invention.

That is, the present invention includes the following inventions.
(1) A method for designing a mirror manufactured by forming a reflecting surface on a hollow inner peripheral surface or an outer peripheral surface of a base material, wherein the central axis of the inner peripheral surface or the outer peripheral surface of the base material is the z-axis. , the incident beam has a light source in the x-axis direction at a position displaced L _1x along the z-axis direction from a predetermined position on the z-axis, and a light source in the x-axis direction from the predetermined position on the z-axis to the z-axis having a light source in the y-axis direction at a position displaced L _1y along the direction, and the emitted beam is condensed at a position displaced L _2x along the z-axis direction from the predetermined position on the z-axis in the x-axis direction; In addition, all the incident light rays that are condensed in the y-axis direction from the predetermined position on the z-axis to a position displaced by L _2y along the z-axis direction and pass through the mirrors pass through the position of the light source in the x-axis direction and the x A first light source line extending in a direction orthogonal to both the axial direction and the z-axis direction, and a second light source line extending in a direction orthogonal to both the y-axis direction and the z-axis direction through the position of the light source in the y-axis direction. a first converging ray in which all the output rays that pass through the light source line and are emitted from the mirror extend in a direction orthogonal to both the x-axis direction and the z-axis direction through the converging position in the x-axis direction; M , the coordinates of the intersection of the first light source line and the light ray incident on the point M, and the coordinates of the intersection of the second light source line and the light ray incident on the point M, using the L _1x and L _1y , and , the coordinates of the intersection of the outgoing light beam from the point M and the first condensed light beam, and the coordinates of the intersection point of the outgoing light beam from the point M and the second condensed light beam using the L _2x and L _2y , Using a reflecting surface design formula derived based on these coordinates and the fact that the optical path length from the light source position to the condensing position is constant with respect to any point on the reflecting surface in the x-axis direction and the y-axis direction. A method of designing a mirror, characterized in that the mirror is designed by

(2) A straight line S _x extending in the _y -axis direction and a straight line Sy extending in the x-axis direction are used as the first light source line and the second light source line, respectively, and the first converging line and the second converging line Let the light rays be a straight line F _x extending in the _y -axis direction and a straight line F _y extending in the x-axis direction. and extending in a direction orthogonal to the _x -axis through the intersection point _Px0 of the first light source line S _x and the z-axis is rotated about the first light source line S _x as an axis. With the arc surface of rotation as the equal phase surface A _1x , the distance from the two intersections of the incident light beam and the equal phase surface A _1x on the side closer to the second light source line _Sy to the point M, The emission length from the point M to the condensed position in the _x -axis direction is centered on the intersection _Qy0 of the second condensed light Fy and the _z -axis and the intersection of the first condensed light Fx and the z-axis. A circular arc passing through Q _x0 and extending in a direction perpendicular to the _x -axis is rotated about the first condensed ray F _x as an arc plane A _2x . The distance from the point of intersection on the side closer to the second condensed ray F _y out of the two points of intersection with to the M point, and the incident length from the light source position in the y-axis direction to the M point is the first light source line The _{second light source line Sy} is the equiphase surface _A1y , which is rotated around the axis, and from two intersections of the incident light beam and the equiphase surface _A1y , the intersection on the side closer to the first light source line _Sx to point M and the emission length from the point M to the condensed position in the y-axis direction is centered on the intersection point _Qx0 between the first condensed light Fx and the _z -axis, and the second light source line _Fy and A circular arc extending in a direction perpendicular to the _y -axis through the intersection point _Qy0 with the z-axis is rotated around the second condensed light beam Fy as an equiphase plane _A2y , and the emitted light beam and the It is obtained as the distance from the two intersections with the equiphase surface A _2y that is closer to the first condensed ray F _x to the point M, and the incident The method of designing a mirror according to (1), wherein the optical path length is calculated as the sum of the length and the exit length.

(3) The distance from the two intersections of the incident light ray and the _equiphase plane A _1x on the side closer to the second light source line Sy to point M is the distance between the incident light ray and the first light source line S Obtaining the distance from the intersection point P _x with _x to the M point, and adding or subtracting the distance from the intersection point P _x to the arc defining the equal phase plane A _1x to the distance, Of the two intersections of the output light beam and the _equiphase plane A _2x , the distance from the intersection on the side closer to the second condensed ray _Fy to point M is The distance from the intersection point _Qx to the M point is obtained, and the distance from the intersection point _Qx to the arc defining the equal phase surface _A2x is obtained by adding or subtracting to the distance, and the incident ray and the _equiphase surface _A1y , the distance from the point of intersection closer to the first light source line _Sx to the point M is the intersection point _Py to the M point, and add or subtract the distance from the intersection point P _y to the arc defining the equal phase surface A _1y to the distance, and obtain the same phase as the emitted ray Of the two intersections with the surface A _2y , the distance from the intersection closer to the first condensed ray _Fx to the point M is the distance from the intersection _{Qy of the emitted ray and the second condensed ray Fy to} the M The mirror design method according to (2), wherein the distance to the point is obtained, and the distance from the intersection point _{Qy to the arc defining the equal phase surface A2y is} added or subtracted to the distance. .

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

(5) A mirror having a reflecting surface satisfying the design formula according to any one of (1) to (4), wherein the values of L _1x and L _1y are different, and the values of L _2x and L _2y is matched to provide a converging output beam from an astigmatic input beam.

(6) A mirror having a reflecting surface that satisfies the design formula according to any one of (1) to (4), wherein the values of L _1x and L _1y match, and the values of L _2x and L _2y Astigmatism control mirrors with different values to provide an astigmatic output beam from an input beam diverging from a single point.

According to the method of designing a mirror according to the present invention, a single mirror is used, and the light source position and the light condensing position can be set independently in the vertical direction and the horizontal direction, thereby freely converting astigmatism. possible mirrors can be made. In addition, it is possible to cope with beams in the X-ray region by suppressing the size of the focused light. Furthermore, the design formula is simple, the range of applications is wide, and a mirror can be manufactured that can be suitably used as an optical system for handling beams with different characteristics in the vertical direction and in the horizontal direction.

FIG. 4 is a conceptual diagram showing a “light source line” and a “condensed light line” in the design direction of the mirror according to the present invention; FIG. 4 is a conceptual diagram showing an _{equiphase plane A 1x in} the vicinity of an intersection point Px on the sagittal light source line _Sx . FIG. 2 is a conceptual diagram showing an equiphase surface A _1y near an intersection point P _y on a meridional light source line _Sy ; FIG. 3 is a schematic diagram of a mirror in which a reflecting surface is formed on the hollow inner peripheral surface or the outer peripheral surface of a base material; FIG. 4 is a diagram showing an example of condensing a beam using a rotator mirror having a conic curve as a ridgeline; FIG. 4 is a diagram showing an example of diffusing a beam using a rotating body mirror having a conic curve as a ridgeline; Explanatory drawing explaining the ray tracing calculation method based on geometric optics. Explanatory drawing explaining the diffraction integral calculation method based on wave optics. 4 is a schematic diagram showing the optical system arrangement of the mirror of Example 1. FIG. Shows the radius distribution of the reflecting surface of Example 1, (a) is a two-dimensional distribution of the radius r (z, φ) when the vertical axis is set to the z coordinate and the horizontal axis is set to the φ coordinate, (b) is (a ), the radial profile of the reflective surface at φ = 0° (z-axis direction (longitudinal direction)) indicated by a dashed line in (c) is the reflection at z = 0 (circumferential direction) indicated by a dashed line in (a) Fig. 3 shows a radial profile of a face; Shows the simulation results of the light collection performance of Example 1, (a) is the result of calculating the variation of light rays on the light collection surface based on geometric optics, and (b) is the result on the light collection surface calculated based on wave optics. The figure which shows two-dimensional intensity distribution. FIG. 8 is a schematic diagram showing the optical system arrangement of the mirrors of Example 2; Shows the radius distribution of the reflecting surface of Example 2, (a) is a two-dimensional distribution of the radius r (z, φ) when the vertical axis is set to the z coordinate and the horizontal axis is set to the φ coordinate, (b) is (a ), the radial profile of the reflective surface at φ = 0° (z-axis direction (longitudinal direction)) indicated by a dashed line in (c) is the reflection at z = 0 (circumferential direction) indicated by a dashed line in (a) Fig. 3 shows a radial profile of a face; The simulation results of the light collection performance of Example 2 are shown, (a) is the result of calculating the variation of light rays on the light collection surface in the x-axis direction based on geometric optics, and (b) is the x calculated based on wave optics. Two-dimensional intensity distribution on the axial light-collecting plane, (c) is the result of calculation of the light ray dispersion on the y-axis direction light-collecting plane based on geometrical optics, and (d) is the y-axis direction calculated based on wave optics. The figure which shows the two-dimensional intensity distribution in a condensing surface.

The method of designing a mirror of the present invention is a method of designing a mirror manufactured by forming a reflecting surface on a hollow inner peripheral surface or an outer peripheral surface of a base material. The mirror design method according to the present invention will be described below with reference to typical embodiments.

The present invention aims at free conversion of astigmatism, and designs mirrors with higher precision based on Fermat's principle that ``light travels along the path with the shortest optical distance''. Fermat's principle is that when limited to a condensing (or diffusing) mirror, "For any point on the mirror surface (reflecting surface), the sum of the distance from the light source point and the distance to the condensing point is constant." It is possible to convert to a representation. If the incoming or outgoing beam has astigmatism, the law of constant optical path length cannot be readily applied. This is because an astigmatic beam, as the name suggests, does not have a single source point or focal point. In the present invention, we conceived of newly defining "light source line" and "condensed light", and this is a design method realized by making it possible to express the properties of a beam with astigmatism in geometrical optics. be.

FIG. 1 is a conceptual diagram showing a "light source line" and a "condensed line". Let the central axis of the inner or outer peripheral surface of the base material be the z-axis, and the cross section perpendicular to it be the _xy plane. Having a light source in the axial direction (horizontal direction in this example) and having a light source in the y-axis direction (vertical direction in this example) at a position displaced by _L1y along the z-axis direction from the predetermined position on the z-axis do. In addition, the emitted beam is condensed at a position displaced by _L2x along the z-axis from a predetermined position on the z-axis in the x-axis direction (horizontal direction), and is converged on the z-axis in the y-axis direction (vertical direction). Assume that the light is condensed at a position displaced by L _2y along the z-axis direction from a predetermined position.

All incident light rays passing through the mirror pass through the position of the light source in the x-axis direction (horizontal direction) and extend in a direction orthogonal to both the x-axis direction (horizontal direction) and the z-axis direction. a line (S _x ) and a second light source line (S _y ) passing through the position of the light source in the y-axis direction (vertical direction) and extending in a direction orthogonal to both the y-axis direction (vertical direction) and the z-axis direction; considered to pass through Thus, the first light source line (S _x ) and the second light source line (S _y ) are defined.

Furthermore, all outgoing light rays emitted from the mirror pass through the condensed position in the x-axis direction (horizontal direction) and extend in a direction orthogonal to both the x-axis direction (horizontal direction) and the z-axis direction. and a second condensed _ray ( F _y ). Thus, the first condensed ray (F _x ) and the second condensed ray (F _y ) are defined.

In this example, the first light source line (S _x ), the second light source line (S _y ), the first condensed line (F _x ), and the second condensed line (F _y ) are straight lines. may be curved. FIG. ₁ shows a case _where L _1x >L _1y >0 and L _2x >L _2y > _{0 .} Inversion of the magnitude relationship is also included, and it is possible for these constants to take negative values. If L _1x or L _1y take negative values, the incident beam will be reflected by the reflective surface of the mirror on the way to focusing downstream. If L _2x or L _2y takes negative values, the exit beam has a wavefront that appears to have diverged from a position upstream of the mirror.

By defining the "source ray" and "concentrated ray" as above, for any point on the reflecting surface of the mirror, we can define the incoming and outgoing rays passing through that point. That is, an arbitrary point on the reflecting surface of the mirror is M(x, y, z), the intersection point (P _x ) between the first light source line (S _x ) and the incident light ray to point M, and the second Each coordinate of the intersection point (P _y ) between the light source line (S _y ) and the light incident on the point M can be expressed by the following equations (1) and (2) using the displacements L _1x and L _1y . can. Similarly, the intersection point (Q _x ) between the outgoing light beam from the point M and the first condensed light beam (F _x ), and the intersection point ₍ Q _y ) can be expressed by the following equations (3) and (4) using the displacements L _2x and L _2y described above.

_Then , the _{optical path length} (incident A design formula for the reflecting surface can be derived based on the fact that the sum of the length and the output length) is constant.

In this embodiment, the distance between each intersection P _x , P _y , Q _x , Q _y on the light source line and condensed line and an arbitrary point M (x, y, z) on the reflecting surface is the incident length Alternatively, the following optical path length compensation is performed so as to obtain a more accurate design formula by using the coordinates of the intersection of the light source line and the condensed light defined by straight lines instead of using the emission length.

(Compensation for optical path length)
Let us consider Fermat's principle when we can define a normal light source point and a light focus point. The equiphase surface near the light source point is a spherical surface centered on the light source point, and the equiphase surface near the condensing point is a spherical surface centered on the condensing point. Bearing in mind that rays are always orthogonal to the equiphase surface, the law of constant optical path length means that any point on a particular equiphase surface near the source point and the corresponding point near the focal point In other words, the optical length of a ray connecting points on a specific equiphase plane is constant. Even when the incident beam includes astigmatism, as in the present invention, a more accurate design formula can be derived by performing compensation in consideration of the equal phase surface.

First, on the incident side, the near _equiphase surface corresponding to the above-described intersection point Px on the light source line _Sx is considered. At the source line _Sx , a wavefront converging towards the source line _Sy should be observed. Although it is not possible to strictly define the phase on _Sx based on this assumption, here the intersection of Sy and the _z -axis is defined as _{Py0, and the distance from Py0 on Sx} is It is assumed that a beam having a corresponding phase distribution, that is, a beam before being incident on a mirror (reflection surface) has a wavefront converging on the light source line Sy in the _y -axis direction (vertical direction). _Based on this _idea , as shown in FIG. 2, _{an x} A rotating arc surface formed by rotating the arc _B1x extending in the direction orthogonal to the axis about the first light source line _Sx is assumed to be an equal phase surface _A1x . The incident length from the light source position in the x-axis direction to point M is the distance from the two intersection points of the incident light beam and the equivalent phase surface _A1x , which is closer to the second light source line _Sy , to point M. Asking is more accurate.

Here, the distance from the two intersections of the incident light beam and the _equiphase plane A _1x , which is closer to the second light source line Sy, to point M on the reflecting surface of the mirror is The distance from the intersection point _Px with the first light source line _Sx to point M is obtained, and the distance from the intersection point _Px to the arc _{B1x defining the equal phase plane A1x ,} that is, The distance between P _x H _1x is obtained by adding or subtracting (subtracting in the example of this figure) the foot of the perpendicular extending from P _x to arc B ₁ x as H ₁ x. That is, let the distance between _H1xM in the following formula (5) be the incident length in the x-axis direction. The reason why this expression is an approximation is that there is no guarantee that the point _H1x exists on the straight line _PxM . However, of course, it may be determined by calculation other than such an approximation formula. In this example, as described above, the foot of the perpendicular line _{extending from Px to the arc B1x is H1x, and the distance between PxH1x is added / subtracted} to obtain an approximate _value . The distance from _Px to the two intersections of the incident ray and the equivalent phase plane _A1x , which is closer to the second light source line _Sy , than the perpendicular line obtained by can be

Subsequently, similarly on the incident side, the near _equiphase surface corresponding to the above-described intersection point _Py on the light source line Sy is considered. A wavefront diverging from the light source line _Sx should be observed on the light source line _Sy . Although it is not possible to strictly define the phase on Sy based on this assumption, here the intersection of _Sx and _z -axis is defined as _Px0 , and the distance from _Px0 on _Sy is It is assumed that there is a corresponding phase distribution, that is, the beam before entering the mirror (reflection surface) has a wavefront diverging from the light source line _Sx in the x-axis direction. _Based on this _idea , as shown in _FIG . 3, _y A rotating arc surface formed by rotating the arc _B1y extending in the direction perpendicular to the axis about the second light source line _{Sy is assumed to be an equal phase surface A1y .} The incident length from the light source position in the y-axis direction to point M is the distance between the two intersections of the incident light beam and the equivalent phase surface _A1y , which is closer to the first light source line _Sx , on the reflecting surface of the mirror. It is obtained as the distance to the M point.

Of the two intersections between the incident light ray and the _equiphase surface _A1y , the distance from the intersection on the side closer to the first light source line _Sx to the point M is The distance from P _y to point M is obtained, and the distance from the intersection point P _y to the arc B _{1 y} defining the _equiphase surface A 1 y, that is, the perpendicular drawn from P _y to the arc B ₁ y is obtained by adding or subtracting (adding in this example) the distance between P _y and H ₁ y, with H _{1 y} being the foot of H 1 y. That is, let the distance between H _1y M in the following formula (6) be the incident length in the y-axis direction.

On the output side, similarly to the incident side, an equiphase surface in the vicinity corresponding to the intersection point Qx on the first condensed beam _Fx and an _equiphase surface in the vicinity corresponding to the intersection point _Qy on the second condensed beam _Fy Consider the isophase surfaces, respectively. In the first focused beam _Fx , a wavefront diverging from the second focused beam _Fy should be observed. Although it is not possible to strictly define the phase on Fx based on this assumption, here the intersection of _Fy and the optical axis _z is defined as _Qy0 , and the distance from _Qy0 on _Fx is It is assumed that there is a corresponding phase distribution. Also, in the second focused beam _Fy , a wavefront _converging towards the first focused beam Fx should be observed. Although it is not possible to strictly define the phase on Fy based on this assumption, here, the intersection of _Fx and the output optical axis _z is defined as _Qx0 , and the distance from _Qx0 on _Fy is It is assumed that there exists a phase distribution corresponding to

Based on these considerations, a more accurate exit length will be obtained in the same way as for the incident side. Specifically, although illustration is omitted, the distance from the intersection point _{Qx to the arc B2x defining} the _equiphase plane, that is, the foot of the perpendicular from Qx to the arc _B2x is set to _H2x . The distance between H _2x Q _x and the distance from the intersection point Q _y to the arc B _2y defining the equiphase surface, that is, between Q _y and H _2y where the foot of the perpendicular from Q _y to the arc B _2y is H _2y Compensation is performed by addition or subtraction using the distance, and more accurate emission lengths in the x-axis direction and the y-axis direction can be obtained as in the following equations (7) and (8).

(Calculation of optical path length)
Using the incident length and the exit length obtained in this manner, the optical path length for converging light in each direction of the x-axis direction and the y-axis direction is calculated. First, the incident length f _1x (x, y, z) when condensing light in the x-axis direction is calculated from the equation (5) as shown in the following equations (9) to (11).

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

Then, if the reference optical path length from the light source point to the condensing point in the _x -axis direction is set as Lx, the conditional expression necessary for condensing in the x-axis direction is derived as shown in the following equation (15). .

Next, a conditional expression necessary for condensing light in the y-axis direction is also derived by the same procedure. That is, the incident length f _1y (x, y, z) focusing on the condensed light in the y-axis direction is obtained from the equation (6) by the following equation (16).

Similarly, the emission length f _2y (x, y, z) focusing on the light collection in the y-axis direction is expressed by the following formula (17) based on the formula (8).

Then, when the reference optical path length from the light source point to the condensing point in the _y -axis direction is set to Ly, the conditional expression necessary for condensing in the y-axis direction is derived as shown in the following equation (18). .

Ideally, a set of points (x, y, z) that simultaneously satisfy the x-axis direction condensing condition of equation (15) and the y-axis direction condensing condition of equation (18) is the desired mirror. As for the shape of the reflecting surface, if the solution of such simultaneous equations is used as the design formula, the solution can only exist under special conditions such as "L _1x =L _1y and L _2x =L _2y ". In order to obtain a more general design equation representing the shape of the reflective surface that can also hold under other conditions, the inventors weighted equations (15) and (18) to give equation (19). A new formula f(x, y, z) was used as the design formula.

(design formula)
That is, the design formula is f _x (x, y , z) = 0 (equation (15)) and the second equation (equation of the y-axis direction condensing condition) derived from the fact that the optical path length from the light source point to the condensing point is constant in the y-axis direction. A certain f _y (x, y, z) = 0 (formula (18)) is weighted using α and β as shown in (19) below with the formula f (x, y, z) = 0 be. α is a weighting factor for condensed light in the x-axis direction, and β is a weighting factor for condensed light in the y-axis direction.

Here, by substituting the equations (9) to (18) into the equation (19), the design equation of the reflecting surface is derived as the equation shown in the equation (20).

As can be seen from the equation (20), it can be confirmed that an equation having good symmetry with respect to the x-axis direction and the y-axis direction was obtained. In the derivation so far, we have assumed "L _1x > L _1y > 0 and L _2x > L _2y >0". However, the same equation (design equation) shown in Equation (20) is derived. However, the four constants L _1x , L _1y , L _2x , and L _2y are all positive or negative values and cannot be 0.

Furthermore, the specific design formula is to set a reference point because the reflecting surface is made from the hollow inner peripheral surface or outer peripheral surface of the base material, and substitute the coordinates to obtain the above formula (20). It is obtained by setting the constant term “αL _x +βL _y ”.

FIG. 4 shows a schematic diagram of a mirror in which a reflecting surface is formed on the hollow inner peripheral surface or outer peripheral surface of a base material. The definition of the coordinate system is the same as in FIG. Set reference points M _0x (r ₀ , 0, 0) and M _0y (0, r ₀ , 0). where _r0 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 shape of the collector mirror, the reference point M _0x (r ₀ , 0, 0) is added to Equation (20). ), the following equation (21) is obtained.

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

Similarly, by substituting the coordinates of the reference point _M0y into the equation (20), the equation (24) is obtained. From this equation (24), restrictions on L _x and _Ly are similarly derived as in equations (25) and (26).

L _x and L _y to be obtained are weighted values derived from the reference points M _0x and M _0y using the weighting coefficient α for the light collection in the x-axis direction and the weighting coefficient β for the light collection in the y-axis direction. weighted arithmetic mean. That is, L _x and _Ly are represented 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 equation (29) below.

(Examples of mirrors that can be designed)
In the condition setting 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 match (the same value), it has astigmatism. An astigmatism control mirror can be designed with a reflective surface that provides a converging output beam from an incident beam. Conversely, by setting the values of L _1x and L _1y to match and the values of L _2x and L _2y to different values, an astigmatic output beam can be obtained from an incident beam diverging from a single point. An astigmatism control mirror can be designed with the resulting reflective surface.

Also, by using the design formula (Formula (29)), it is possible to design a mirror having a reflecting surface in which the light source and light condensing positions in both the x-axis direction and the y-axis direction match. For example, by substituting L _1x =L _1y =L ₁ >0 and L _2x =L _2y =L ₂ >0 into the design formula (formula (29)), a spheroid is obtained as shown in the following formula (30). We can obtain the equation of the surface mirror.

By substituting L _1x =L _1y =L ₁ <0 and L _2x =L _2y =L ₂ >0 into the design formula (formula (29)), the rotational twin We can get the formula for the curved mirror.

Furthermore, by setting L1 to positive or negative infinity in the above equation (30) or (31), the following equation (32) for _a parabolic mirror of revolution can be obtained. .

(design limit)
First, as described above, none of L _1x , L _1y , L _2x and L _2y is zero. Further, for example, when the condensing action of the mirror is limited to the vertical direction only, that is, when L _1x =+∞ and L _2x =+∞, L _1y >0, L _1y >0, and α=0 are given. , Eq. (29) becomes the following Eq. (33).

Equation (33) is clearly the equation for the elliptic cylinder surface. It can be read that the two cylindric mirrors are arranged so as to sandwich the optical axis from the vertical direction, indicating a failure of the design. The reason for the failure is that it did not have positive condensing performance in both the vertical direction and the horizontal direction. Positive focussing performance, as discussed herein, is the ability to "turn the beam in a more convergent direction." This condition is expressed by the following equation (34) where L ₁ is the incident length and L ₂ is the outgoing length.

Equation (34) shows that the curvature of the wavefront emanating from the upstream is defined as positive and the curvature decreases through reflection by the mirrors. At this time, the shape of the mirror becomes concave. FIG. 5 shows an example of condensing a beam using a rotating body mirror having a conic curve as a ridge line. All of these satisfy formula (34). In order to design a mirror that utilizes the inner surface of a hollow mold, it is a necessary and sufficient condition that Eq. (34) be satisfied in both the vertical and horizontal directions.

A conic section can also take advantage of its convex profile to increase the curvature of the wavefront. FIG. 6 shows a list of examples of increasing the curvature of the wavefront using a conic section of revolution. All of these examples satisfy the following formula (35). If a mirror that increases the curvature of the wavefront in both vertical and horizontal directions is designed using this method, it becomes a "mirror using a columnar outer surface".

When the curvature of the wavefront does not change, that is, when 1/L ₁ =−1/L ₂ holds, the ridgeline of the mirror becomes a straight profile with no curvature and the design fails. In summary, satisfying 1/L _1x >−1/L _2x and 1/L _1y >−1/L _2y is the condition for the design formula of the hollow inner peripheral mirror to hold, and 1/L _1x <−1 /L _2x and 1/L _1y <−1/L _2y are the conditions for the design formula of the outer peripheral mirror to hold.

(cross-sectional profile)
By substituting x=0 or y=0 into the design formula (formula (29)), the cross-sectional profile of the reflecting surface can be confirmed. For example, when x = 0, L _1x > 0, L _1y > 0, L _2x > 0, and L _2y > 0, the line of intersection (cross-sectional profile) between the yz plane and the mirror reflection surface is given by the following equation (36), It is expressed by the formula (37). Equation (36) expresses an elliptic function whose focal points are the light source point of the y-direction condensed light and the y-direction condensed light condensed point. C in the formula is a constant term representing the reference optical path length.

Although the embodiments of the present invention have been described above, the present invention is by no means limited to such embodiments, and can of course be embodied in various forms without departing from the gist of the present invention. In the present embodiment, the light source line and the condensed light are straight lines, and the distance between the straight lines and the equiphase planes in the vicinity thereof is compensated. However, such compensation is not necessarily required. It is also preferable to obtain arc lines and other curves as light source lines and condensed rays without compensation or by a compensation method other than the above compensation or an approximation method. The position of the origin of the design formula for the reflecting surface may be different. Needless to say, the coordinates may be transformed.

Next, as design examples of the astigmatism control mirror according to the present invention described above, a mirror for the purpose of eliminating astigmatism (Example 1) and a mirror for the purpose of adding astigmatism (Example 2) ) were designed, and the performance of each mirror was confirmed by simulation using both geometrical optics and wave optics.

(Simulation method)
Considering the actual use of a hollow mirror, as shown in Fig. 7, the soft X-ray beam illuminates only part of the entire circumference (360°) of the mirror (reflecting surface) under partial illumination conditions. The light collection performance was calculated. The installation condition was horizontal (x-axis direction) deflection. In the ray tracing calculation based on geometrical optics, a group of rays passing through the light source lines in the x-axis direction and the y-axis direction shown in FIG. 1 is defined and made incident on the reflecting surface of the mirror. The thickness of the light source line, that is, the size of the light source is assumed to be zero. To emit light uniformly over the entire effective range of a reflective surface.

The normal vector n(x, y, z) at each position on the reflecting surface of the mirror is a unit vector parallel to the gradient vector of the function f(x, y, z) defined by equation (29). There is (equation (38)). As shown in FIG. 7, the incident light ray is symmetrically reflected to the normal vector of the reflecting surface of the mirror and propagates to the collecting surface. In this way, the dispersion of light rays on the condensing surface is evaluated.

In the diffraction integration calculation based on wave optics, the center of curvature in the x-axis direction and the y-axis direction of the wavefront of the beam incident on the reflecting surface of the mirror is defined as the x-axis direction light source S _{x and the} y-axis direction light source S Match _Sy . FIG. 8 is a schematic diagram thereof. A linear light source with no thickness is assumed, and the beam is assumed to be incident on the entire effective area of the reflective surface with uniform intensity. The wave field U _M (x _M , y _M , z _M ) of the beam at the point M (x _M , y _M , z _M ) on the reflecting surface is given by the following equations (39) and (40).

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 converted to coordinates Q (x _Q , y _Q , z _Q ).

In equation (41), dS represents a minute 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 that is the square of the absolute value of the complex wave field U _Q (x _Q , y _Q , z _Q ) over Q (equation (43)). By the above procedure, the intensity distribution on the condensing plane 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 incident angle was about 10 mrad. FIG. 9 shows the arrangement of light source lines, condensed lines, and mirrors (reflecting surfaces).

To obtain the reflecting surface shape of the mirror, the implicit function shown in the design formula of equation (29) is solved. We obtain the solution set of equation (29) using a cylindrical coordinate system. Here, it is assumed that the cylindrical coordinate system (z, φ, r) and the Cartesian coordinate system (x, y, z) have the correspondence of the following equation (44).

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

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

(Simulation result of light collection performance of Example 1)
Next, FIG. 11 shows the simulation result of the light condensing performance. FIG. 11(a) shows the result of calculating the variation of light rays on the condensing surface based on geometrical optics. It can be confirmed that all light rays are concentrated in a region of 1 nm or less both vertically and horizontally.

Also, FIG. 11(b) shows a two-dimensional intensity distribution on the condensing plane calculated based on wave optics. Photon energy was set at 300 eV. The beam is focused on an area of 430 nm (x-axis direction (horizontal direction))×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. While the incident length is the same in the x-axis direction (horizontal direction) and the y-axis direction (vertical direction), 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 incident angle to the reflecting surface was about 10 mrad. FIG. 12 shows the arrangement of light source lines, condensed lines, and mirrors (reflecting surfaces). Similar to Example 1, a cylindrical coordinate system is used to obtain the set of solutions of Equation (29).

(Radius distribution of reflecting surface of Example 2)
FIG. 13 shows the calculated radius distribution of the reflecting surface of the mirror. Similar to the mirror of Example 1, it has a crushed hollow shape with different diameters in the y-axis direction (vertical direction) and in the x-axis direction (horizontal direction). You can read what's on it.

(Simulation result of light collection performance of Example 2)
FIG. 14 shows the simulation results of light collection performance. FIG. 14(a) shows the results of calculation of light ray variations on the x-axis direction (horizontal direction) condensing plane (z=L _2x ) based on geometrical optics. It can be confirmed that all rays are concentrated in a region with a width of 1 nm or less in the x-axis direction (horizontal direction).

FIG. 14(b) shows a two-dimensional intensity distribution in the x-axis direction (horizontal direction) condensing plane calculated based on wave optics. Photon energy was set at 300 eV. The beam is focused on a region with a width of 52 μm (FWHM) in the x-axis direction (horizontal direction).

Similarly, FIGS. 14(c) and 14(d) show simulation results of light collection performance on the y-axis direction (vertical direction) light collection plane (z=L _2y ). The light collection width in the y-axis direction (vertical direction) was 130 nm in geometrical optics and 7.4 μm (FWHM) in wave optics.

Claims (6)

A method for designing a mirror manufactured by forming a reflecting surface on a hollow inner peripheral surface or an outer peripheral surface of a base material, comprising:
Let the central axis of the inner peripheral surface or the outer peripheral surface of the base material be the z-axis, and the cross section orthogonal to this be the xy plane,
The incident beam has a light source in the x-axis direction at a position displaced _L1x along the z-axis direction from a predetermined position on the z-axis, and a position displaced _L1y along the z-axis direction from the predetermined position on the z-axis has a light source in the y-axis direction,
The emitted beam is condensed at a position displaced L _2x along the z-axis direction from the predetermined position on the z-axis in the x-axis direction, and is L along the z-axis direction from the predetermined position on the z-axis in the y-axis direction. Condensed at a position displaced by _2y ,
All incident light rays passing through the mirror have a first light source line passing through the position of the light source in the x-axis direction and extending in a direction orthogonal to both the x-axis direction and the z-axis direction, and the light source line in the y-axis direction. passing through a second light source line extending in a direction orthogonal to both the y-axis direction and the z-axis direction through a position;
A first converging ray extending in a direction orthogonal to both the x-axis direction and the z-axis direction through the converging position in the x-axis direction, and the above-mentioned Suppose that a second condensed ray passes through the condensed position and extends in a direction orthogonal to both the y-axis direction and the z-axis direction,
Let M be an arbitrary point on the reflecting surface of the mirror, and let the coordinates of the intersection of the first light source line and the light ray incident on the point M and the intersection of the second light source line and the light ray incident on the point M be Expressed using the L _1x and L _1y , and the coordinates of the intersection of the outgoing ray from the point M and the first condensed ray, and the intersection of the outgoing ray from the M point and the second condensed ray , expressed using the L _2x and L _2y ,
Using a reflecting surface design formula derived based on these coordinates and that the optical path length from the light source position to the condensing position is constant with respect to any point on the reflecting surface in the x-axis direction and the y-axis direction. A method of designing a mirror, characterized in that the mirror is designed by The first light source line and the second light source line are a straight line S _x extending in the _y -axis direction and a straight line Sy extending in the x-axis direction, respectively;
The first condensed light and the second condensed light are a straight line F _x extending in the y-axis direction and a straight line F _y extending in the x-axis direction, respectively;
The incident length from the light source position in the x-axis direction to point M is centered at the intersection point _Py0 between the second light source line Sy and the z-axis, and the intersection point P between the first light source line _Sx and the _z -axis. A circular arc passing through _x0 and extending in a direction perpendicular to the _x -axis is rotated around the first light source line Sx as an equiphase surface A _1x , and the incident light beam and the equiphase surface A _1x is the distance from the closest side of the second light source line _Sy to the point M,
The emission length from the point M to the condensed position in the _x -axis direction is centered on the intersection _Qy0 of the second condensed light Fy and the _z -axis and the intersection of the first condensed light Fx and the z-axis. A circular arc passing through Q _x0 and extending in a direction perpendicular to the _x -axis is rotated about the first condensed ray F _x as an arc plane A _2x . is the distance from the point of intersection closer to the second condensed ray F _y to the point M,
The incident length from the light source position in the y-axis direction to point M is centered at the intersection point P _x0 between the first light source line S _x and the z-axis, and the intersection point P between the second light source line _Sy and the z-axis. A circular arc passing through _y0 and extending in a direction perpendicular to the _y -axis is rotated around the second light source line _Sy , and the circular arc surface is defined as an equal phase plane _A1y . is the distance from the point of intersection closer to the first light source line S _x to the point M,
The emission length from the point M to the condensed position in the y-axis direction is centered on the intersection point _Qx0 between the first condensed light beam Fx and the _z -axis and the intersection point between the second light source line _Fy and the z-axis. A circular arc passing through Q _y0 and extending in a direction perpendicular to the y-axis is rotated about the second condensed light beam F _y as an _{equiphase surface A 2y .} is the distance from the intersection on the side closer to the first condensed ray F _x to the point M,
With this, the optical path length, which is the sum of the incident length and the outgoing length, is calculated for each of the light collections in the x-axis direction and the y-axis direction.
The method of designing a mirror according to claim 1 . Of the two intersections of the incident light ray and the _equiphase plane A _1x , the distance from the intersection closer to the second light source line Sy to point M is the distance between the incident light ray and the first light source line _Sx . Obtaining the distance from the intersection point P _x to the M point, and adding or subtracting the distance from the intersection point P _x to the arc defining the equal phase plane A _1x to the distance,
Of the two intersections of the output light beam and the _equiphase plane A _2x , the distance from the intersection on the side closer to the second condensed ray _Fy to point M is Obtaining the distance from the intersection point Q _x to the M point, and adding or subtracting the distance from the intersection point Q _x to the arc defining the equal phase surface A _2x to the distance,
Of the two intersections of the incident light ray and the equiphase surface _A1y , the distance from the intersection on the side closer to the first light source line _Sx to point M is the distance between the incident light ray and the second light source line _Sy Obtaining the distance from the intersection point P _y to the M point, and adding or subtracting the distance from the intersection point P _y to the arc defining the equal phase plane A _1y to the distance,
Of the two intersections of the output light beam and the _equiphase surface _A2y , the distance from the intersection on the side closer to the first condensed light Fx to point M is the distance between the output light beam and the second condensed light _Fy . Obtaining the distance from the intersection point Q _y to the M point, and adding or subtracting the distance from the intersection point Q _y to the arc defining the equiphase surface A _2y to the distance,
3. The method of designing a mirror according to claim 2. The design formula is
The first formula f _x (x, y, z) = 0 derived from the fact that the optical path length from the light source point to the condensing point is constant in the x-axis direction, and the light condensing from the light source point in the y-axis direction The second formula f _y (x, y, z) = 0 derived from the fact that the optical path length to the point is constant, is weighted by the following formula, according to any one of claims 1 to 3 How the mirror is designed as described.

A mirror having a reflecting surface that satisfies the design formula according to any one of claims 1 to 4,
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 provides a converging output beam from an astigmatic input beam. A mirror having a reflecting surface that satisfies the design formula according to any one of claims 1 to 4,
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 provides an astigmatic output beam from an input beam that diverges from a single point.

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