research papers
Aberrationfree aspherical lens shape for shortening the focal distance of an already convergent beam
^{a}Diamond Light Source Ltd, Chilton, Didcot, Oxfordshire OX11 0DE, UK
^{*}Correspondence email: john.sutter@diamond.ac.uk
The shapes of single lens surfaces capable of focusing divergent and collimated beams without aberration have already been calculated. However, nanofocusing compound refractive lenses (CRLs) require many consecutive lens surfaces. Here a theoretical example of an Xray nanofocusing CRL with 48 consecutive surfaces is studied. The surfaces on the downstream end of this CRL accept Xrays that are already converging toward a focus, and refract them toward a new focal point that is closer to the surface. This case, so far missing from the literature, is treated here. The ideal surface for aberrationfree focusing of a convergent incident beam is found by analytical computation and by ray tracing to be one sheet of a Cartesian oval. An `Xray approximation' of the Cartesian oval is worked out for the case of small change in index of refraction across the lens surface. The paraxial approximation of this surface is described. These results will assist the development of largeaperture CRLs for nanofocusing.
Keywords: Xray; lens; oval; nanofocusing; aberration.
1. Introduction
Compound refractive lenses (CRLs) have been used to focus Xray beams since Snigirev et al. (1996) demonstrated that the extremely weak refraction of Xrays by a single lens surface could be reinforced by lining up a series of lenses. As the Xray focal spot size has been brought down below 1 µm, more lenses have been necessary to achieve the very short focal lengths required. Because the absorption of Xrays in the lens material is generally significant, it thus becomes critical to design the CRL with the shortest length possible for the given focal length in order to minimize the thickness of the refractive material through which the Xrays must pass.
In recent years, designs for novel nanofocusing lenses have been proposed. Xray refractive lenses will deliver ideally focused beam of nanometer size if the following conditions can be satisfied:
(1) The lens material should not introduce unwanted scattering.
(2) The fabrication process should not introduce shape errors or roughness above a certain threshold.
(3) Absorption should be minimized in order to increase the lens effective aperture.
(4) The lens designs should not introduce geometrical aberrations.
In response to the first three of these conditions, planar microfabrication methods including electron beam lithography, silicon etch and LIGA have successfully been used to fabricate planar parabolic CRLs, singleelement parabolic kinoform lenses (Aristov et al., 2000) and singleelement elliptical kinoform lenses (EvansLutterodt et al., 2003) from silicon. However, the best focus that can be obtained is strongly dependent on material and Xray energy. A collimating–focusing pair of elliptical silicon kinoform lenses with a focal length of 75 mm has successfully focused 8 keV photons from an undulator source of 45±5 µm full width at halfmaximum (FWHM) into a spot of 225 nm FWHM (Alianelli et al., 2011). On the other hand, although silicon lenses can be fabricated with very high accuracy, they are too absorbing to deliver a focused beam below 100 nm at energy values below 12 keV, unless kinoform lenses with extremely small sidewalls can be manufactured. As a result, diamond has come to be viewed as a useful material for CRLs. Diamond, along with beryllium and boron, is one of the ideal candidates to make Xray lenses due to good refractive power, low absorption and excellent thermal properties. Planar refractive lenses made from diamond were demonstrated by Nöhammer et al. (2003). Fox et al. (2014) have focused an Xray beam of 15 keV to a 230 nm spot using a microcrystalline diamond lens, and an Xray beam of 11 keV to a 210 nm spot using a nanocrystalline diamond lens. Designs of diamond CRLs proposed by Alianelli et al. (2016) would potentially be capable of focusing down to 50 nm beam sizes. Many technical problems remain to be solved in the machining of diamond; however, our current aim is to provide an ideal lens design to be used when the technological issues are overcome. We assume that technology in both reductive and additive techniques will advance in the coming decade and that Xray refractive lenses with details of several tens of nanometers will be fabricated. This will make Xray refractive optics more competitive than they are today for the ultrashort focal lengths. When that happens, lens designs that do not introduce aberrations will be crucial.
The aim of this paper is to define a nanofocusing CRL with the largest possible aperture that can be achieved without introducing aberrations to the focus. The determination of the ideal shape of each lens surface becomes more critical as the desired aperture grows. Suzuki (2004) states that the ideal lens surface for focusing a plane wave is an ellipsoid, although in fact this is true only if the index of refraction increases as the Xrays cross the surface [see Sanchez del Rio & Alianelli (2012) and references therein, as well as §2.4 of this paper]. The same author also proposes the use of two ellipsoidal lenses for pointtopoint focusing, but a nanofocusing Xray CRL requires a much larger number of lenses because of the small refractive power and the short focal length. EvansLutterodt et al. (2007), in their demonstration of the ability of kinoform lenses to exceed the numerical aperture set by the critical angle, used Fermat's theorem to calculate the ideal shapes of their four lenses, but explicitly described the shape of only the first lens (an ellipse). Sanchez del Rio & Alianelli (2012) pointed out the general answer, known for centuries, that the ideal shape of a lens surface for focusing a point source to a point image is not a conic section (a curve described by a seconddegree polynomial). Rather, it is a Cartesian oval, which is a type of quartic curve (i.e. a curve described by a fourthdegree, or quartic, polynomial). An array of sections of Cartesian ovals is therefore one possible solution to the task of designing an Xray lens for singledigit nanometer focusing. Previous authors in Xray optics have not calculated analytical solutions (which do exist), but instead relied on numerical calculations of the roots without asking how well conditioned the quartic polynomial is; that is, how stable the roots of the polynomial are against small changes in its coefficients. However, in this paper it will be shown that finite numerical precision can cause errors in the calculation of the Cartesian oval when the change in across the lens surface becomes very small, as is usually the case with Xrays. Moreover, it has not been made explicit in the literature when it is reasonable to approximate the ideal Cartesian oval with various conic sections (ellipses, hyperbolas or parabolas). As a result, for the sake of rigor, the authors have considered it worthwhile to find the analytical solutions explicitly. This has not been done before in any recent papers. Finally, Alianelli et al. (2015) state that no analytical solution exists for a lens surface that accepts an incident beam converging to a point and that focuses this beam to another point closer to the lens surface. This paper will concentrate on that very case and will show that in fact such a lens surface can be described by a Cartesian oval. The existence of such solutions removes the necessity of using pairs of lens surfaces of which the first slightly focuses the beam and the second collimates the beam again.
Schroer & Lengeler (2005) proposed the construction of `adiabatically' focusing CRLs, in which the aperture of each lens follows the width of the Xray beam as the beam converges to its focus. Fig. 1 displays a schematic of an adiabatic CRL. A potential example of such a CRL, which would be made from diamond, is given in Table 1. This example treats Xrays of energy 15 keV, for which diamond has an index of refraction where = 3.23 × 10^{−6}. The very small difference between the index of refraction of diamond and that of vacuum is typical for Xray lenses. Calculations of surfaces 2, 24 and 48 of Table 1 will be demonstrated in the following treatment. The loss of numerical precision when using the exact analytical solutions of the Cartesian oval at small will be avoided by an approximation of the quartic Cartesian oval equation to lowest order in . The cubic equation resulting from this `Xray approximation' will be shown to be numerically stable. The approximate cubic and the exact quartic equation will be shown to agree when the latter is numerically tractable. As in Sanchez del Rio & Alianelli (2012), conic approximations will be made to the ideal surface in the paraxial case. The results of this paper agree with theirs in showing that Cartesian ovals, even when calculated by the cubic `Xray' approximation, introduce no detectable aberrations, and that elliptical or hyperbolic lens surfaces introduce less aberration than the usually used parabolic surfaces. However, this paper will also demonstrate that at sufficiently high apertures even the elliptical or hyperbolic approximation will produce visible tails in the focal spot. This is especially true for surface 48, the final surface and the one with the smallest focal length, where the elliptical and hyperbolic approximation produces tails in the focus at an aperture only slightly larger than that given in Table 1.

At the end of this treatment, the focal spot profiles calculated by ray tracing for the compound refractive lens (CRL) of Table 1 will be compared with the diffraction broadening that inevitably results from the limited aperture. The absorption in the lens material limits the passage of Xrays through a CRL to an effective aperture A_{eff} that is smaller than the geometrical aperture. This in turn restricts the numerical aperture (NA) of a CRL of N surfaces to a value A_{eff}/(2q_{2N}), where q_{2N} is the distance from the last (Nth) surface to the final focus. Lengeler et al. (1999) derive a FWHM of 0.75λ/(2NA) for the Airy disk at the focal spot. This yields the diffraction broadening and hence the spatial of the CRL. Formulas for the effective aperture of a CRL in which all lens surfaces are identical have been derived by Lengeler et al. (1998, 1999), and Schroer & Lengeler (2005) have derived formulas for the effective aperture of an adiabatically focusing CRL. Very recently Kohn (2017) has reexamined the calculation of the effective aperture, surveying the various definitions appearing in the literature and distinguishing carefully between onedimensionally and twodimensionally focusing CRLs. In this paper the effective aperture and the numerical aperture will be estimated numerically by ray tracing, taking full account of the absorption to which each ray is subjected along the path from the source to the focus. For simplicity, the lens surfaces will all be assumed to be onedimensionally focusing parabolic cylinders. It will be shown that the parabola is an adequate approximation to the ideal Cartesian oval within the effective aperture of the CRL in Table 1.
2. Principles
2.1. Definitions and derivation of ideal lens surface
The first task of this article is to calculate the exact surface y(x) of a lens that bends an already convergent bundle of rays into a new bundle of rays converging toward a closer focus. The required quantities are labelled and defined in Fig. 2. According to Snell's Law taken for the rays at an arbitrary point P,
Let the coordinate vector of P be . Fig. 2 shows that
It is also seen in Fig. 2 that
Substitution of equations (1)–(4) into (5) yields the firstorder ordinary differential equation
One can rearrange this to find an expression for the surface slope , which will be useful for design calculations once a solution for y(x) has been obtained,
This is a nonlinear differential equation, but nevertheless it can be solved by noticing that the numerators of each fraction of equation (6) are the derivatives of that fraction's denominator. A simple variable substitution thus presents itself,
As the initial condition, one may set y(x = 0) = 0 as shown in Fig. 2. In that case, V_{1}(x = 0) = q_{1}^{ 2} and V_{2}(x = 0) = q_{2}^{ 2}. Integration of both sides of equation (6) starting from x = 0 can then be written
where s is a dummy variable. Now, = dV_{1} and = dV_{2}, allowing equation (10) to be rewritten in the very simple form
The integrals on both sides of equation (11) are elementary and yield the result
Equation (12) describes a Cartesian oval with the two foci F_{1} and F_{2} shown in Fig. 2. Note that the distances of any point on y(x) from F_{1} and F_{2} are r_{1} = {x^{2} + [q_{1} + y(x)]^{2}}^{1/2} and r_{2} = {x^{2} + [q_{2} + y(x)]^{2}}^{1/2}, respectively. Equation (12) can then be written in the standard form for a Cartesian oval (Weisstein, 2016),
The case = 0 may be of some interest, and is physically achievable for the problem of this article () if . In this case, equation (12) can be squared and rearranged into
and = < 0. This is a circle of radius centred at , where
2.2. Closedform solutions of ideal lens surface
2.2.1. Derivation of algebraic equation
By adding q_{1} to both sides of equation (12) and then squaring the equation, one obtains
Rearranging this equation to put the radical alone on one side, then squaring it again, yields
Equation (17) is a quartic polynomial equation in both x and y. It can be written as a quadratic equation in x^{2}, since no odd powers of x appear in it, and by using the quadratic formula a closedform expression of x(y) can be calculated. However, the inversion of this function to obtain y(x) is difficult, and y(x) would be far more useful for design calculations of the lens surface. It was thus decided to solve equation (17) for y(x) explicitly. Writing equation (17) in powers of y yields
The calculation of y(x) therefore amounts to finding the roots of the quartic polynomial equation (18) for any x. As a quartic polynomial equation with real coefficients, equation (18) is guaranteed to have four solutions, of which
– all four may be real, or
– two may be real, while the other two are complex and conjugates of each other, or
– all four may be complex, forming two pairs of complex conjugates.
No more than one of these roots can satisfy the original equation of the Cartesian oval, equation (12). To be physically significant, that root must be real. If equation (18) produces no real root that satisfies equation (12) when calculated at some particular x, the ideal lens surface does not exist at that x. This raises the possibility that the ideal lens surface may be bounded; that is, it has a maximum achievable aperture.
2.2.2. Calculation of roots of quartic polynomial equation
Analytical procedures for calculating the roots of cubic and quartic equations were worked out in the 16th century. Nonetheless, explicit solutions of such equations are published so rarely that a detailed description of the method will be useful for the reader. Note that the procedure for quartic equations includes the determination of one root of a cubic equation. Many standard mathematical texts explain the solution of cubic and quartic equations; Weisstein (2016) has been followed closely here.
The first step in solving a general quartic equation ay^{4} + by^{3} + cy^{2} + dy + e = 0 is the application of a coordinate transformation to a new variable given by
and the division of both sides of the general equation by a such that one obtains a `depressed quartic', that is, a quartic with no cubic term, in . This has the form = 0, where
By substituting the coefficients of equation (18) into equations (19) and (20), one obtains the coordinate transformation,
and the depressed equation in , which has the following coefficients,
The strategy now is to add to both sides of the depressed equation a quantity , where u is a real quantity that will be determined shortly. Knowing that = , one obtains from the depressed equation the following,
The left side of equation (25) is thus a perfect square. Notice that the right side of equation (25) is a quadratic equation. Therefore it too will be a perfect square if u can be chosen to make its two roots equal; that is, if its discriminant D equals zero,
Equation (26) is known as the `resolvent cubic'. As a cubic equation with real coefficients, it is guaranteed to have three roots, of which either one or all will be real. The analytical solution of a general cubic equation u^{3} + fu^{2} + gu + h = 0 begins with the calculation of two quantities A and B, of which the general formula is shown on the left, and the value for the resolvent cubic is shown on the right,
The discriminant of a general cubic equation is D_{c} = A^{3} + B^{2}. The next step depends on the value of D_{c}.
(i) . One root of the cubic equation is real and the other two are complex conjugates. The real root is (1/3) f + S + T, where S = and T = . For the resolvent cubic, the real root u_{1} is
(ii) D_{c} = 0. All roots of the cubic equation are real and at least two are equal. S and T in equation (28) are then equal. Thus, for the resolvent cubic,
(iii) . All roots of the cubic equation are real and unequal. In this case, an angle is defined such that
One of the real roots of the general cubic equation is then given by
which for the resolvent cubic yields
Note that in all these cases one may write
Substitution of u_{1} into equation (25) then yields a perfect square on both sides,
If , then equation (33) falls into two cases. In the first, one simply equates the square root of both sides,
Equation (34) is a quadratic equation in . Its two solutions are
In the second case, one equates the square root of the left side of equation (33) with the negative of the square root of the right side, so that
Equation (36), like equation (34), is a quadratic equation in . Its two solutions are
and are the four solutions of the depressed quartic. The explicit expressions in P, Q and U_{1} were calculated assuming that > 0; however, the same expressions are also valid if < 0. The only change is that the explicit expression for would appear like that for in equation (37), and the explicit equation for would appear like that for in equation (35). Because this does not affect the solution, it will not be mentioned further.
The solutions of the original quartic equation (18) are easily obtained from equations (35) and (37) by using equation (21),
In principle, any of these roots could be the one that satisfies the original equation for the Cartesian oval, equation (12). The simplest way to find this root is to evaluate equations (38) and (39) at x = 0. The root that equals zero there is the correct one. Equation (18) will have one and only one root equal to zero at x = 0, because then the constant (y^{0}) term vanishes but the linear (y^{1}) term does not.
If u_{1} = p, the formulas for the roots become somewhat simpler,
These roots must also be checked to determine which one fulfills equation (12) for the Cartesian oval.
If , then equation (33) can have a real solution only if both of the squared factors are equal to zero. This would require that = 0 and = 0 simultaneously. If that is not possible for the values of p, q and u_{1} calculated above, then no real solution exists.
Fig. 3 shows a set of examples of the exact solutions in equations (38) and (39) for various values of . The solutions clearly appear in two disconnected sheets, one inner and one outer. However, only one of these sheets satisfies the original lens equation (12). If > 1, it is the inner sheet; if < 1, it is the outer sheet. It is evident that as the outer sheet becomes very much larger than the inner sheet. How large the outer sheet becomes at x = 0 can be estimated as follows. First, define the following quantities from equations (22)–(24),
Substitution of these values into equation (27) yields A = 0 and B = 0. Thus the discriminant D_{c} of the resolvent cubic is zero, and from equation (29) one can define U_{10} = = (1/3)P_{0}. Because P_{0} < 0, U_{10} > P_{0} and equations (38) and (39) apply. Letting = and recalling the initial assumption q_{1} > q_{2}, one finds three solutions that remain bounded while the fourth solution y_{II  }(x = 0) diverges as . This sensitivity of the fourth root on the exact value of makes the quartic equation (18) illconditioned. Therefore the numerical evaluation of equations (38) and (39) is very sensitive to roundoff errors caused by the limited precision in cases in which is very small, even though the equations themselves remain theoretically exact. Such cases are not only common but normal in Xray optics, for which = generally has a magnitude on the order of 10^{−5} or less. An approximation that can capture the three bounded roots of equation (18) with high accuracy while ignoring the divergent root is therefore justified.
2.3. The `Xray approximation' to the ideal lens surface
Equation (18) can be rewritten as a quadratic equation in x^{2} simply by rearranging terms. The quadratic formula can then be applied to determine an equation for x^{2} in terms of y,
The `±' accounts for the two roots of any quadratic equation. However, if the plus sign is chosen, the resulting equation cannot be satisfied by the condition y(x = 0) = 0 as is required. Therefore only the minus sign yields a useful set of solutions for the lens surface.
Equation (45) is exact. However, the radical can be expanded by using the binomial theorem if
[For the Xray case where ≃ 1, this condition reduces approximately to .] The binomial theorem yields
Hence 1  (1 + z)^{1/2} ≃ (1/2)z+(1/8)z^{2}(1/16)z^{3} plus higherorder terms that will be discussed later. Note that this has no term in z^{0}. Substituting this into equation (45) and summing terms with the same power of on the righthand side, one obtains the approximate equation
For the linear term on the righthand side, one obtains
As , this quantity becomes very small as the terms in the numerator almost cancel out. Equation (48) therefore becomes subject to numerical errors caused by limited precision. However, remembering that in the Xray case = where is much less than 1, one can make a power series expansion of C_{1} in . The lowest term of this power series is
For the quadratic term on the righthand side of equation (47), one obtains
Like C_{1}, C_{2} also approaches zero as . The lowest term of the power series expansion of C_{2} in terms of is
For the cubic term on the righthand side of equation (47), one obtains
whose power series in terms of is found simply by setting = 0,
Finally one can calculate the lowest term in the power series expansion of ,
Substituting equations (49), (51), (53) and (54) into equation (47) and keeping only the lowestorder terms in yields
It is justified to keep all terms up to cubic on the righthand side of equation (55) because all are multiplied by the same power of . The neglected higherorder terms Y_{n} () on the righthand side of equation (55), which arise from the binomial expansion in equation (46), are given to lowest order in by
In the Xray approximation, these terms diminish rapidly with increasing n. Therefore the fourthorder term is already much less than the cubic term included in the righthand side of equation (55), and higherorder terms are smaller still. This justifies the neglect of terms beyond the cubic in equation (55).
In standard form, equation (55) is
This equation can be solved analytically. When x = 0 the roots are trivial: 0, q_{1} and q_{2}. The solutions for general x can be determined by the same methods used to calculate the resolvent cubic of the exact equation. The discriminant of equation (57) is D_{XR} = A_{XR}^{3} + B_{XR}^{2}, where
Notice that, since and , because (q_{1}^{2}  q_{1}q_{2} + q_{2}^{2}) = (q_{1}  q_{2} )^{2} + q_{1}q_{2} > 0. From these expressions, one obtains
Now one needs to determine the sign of D_{XR} at any given x. Notice that D_{XR} depends quadratically on x^{2}. Therefore one can use the quadratic formula to find the values x_{D0} at which D_{XR} = 0,
where B_{XR0} = B_{XR}(x = 0). From this one finds that
where D_{XR0} = D_{XR}(x = 0). Inspection of equation (60) shows that and that therefore , which proves that x_{D0 + }^{2} and x_{D0  }^{2} have opposite signs. Equation (61) shows that, if , and , since . Likewise, if , and . Only the positive squared x can yield real values of x; the negative squared x is discarded. There are thus two values of x at which the discriminant D_{XR}(x) = 0:
(i) .
(ii) .
The positive coefficient of the x^{4} term in equation (60) shows that D_{XR}(x) increases with increasing x^{2}. Therefore, the solutions of equation (57) are as follows:
(a) , all values of . Here the discriminant D_{XR} of equation (57) is negative. In this case the three roots of equation (57) are all real and unequal. The calculation of the roots begins with the calculation of an angle such that
The roots are then
One can now check these roots at x = 0, defining = . By using the trigonometric identity = , one can show that if = , then = B_{XR0}/(  A_{XR})^{3/2}, thus satisfying equation (65). One can also use the common trigonometric identity = 1 to find that, for as assumed here, = . Thus y_{XR1}(x = 0) = 0, y_{XR2}(x = 0) =  q_{1} and y_{XR3}(x = 0) =  q_{2}, as expected. Note that y_{XR1}(x) is the solution for the shape of the lens.
(b) = x_{D0}, (assuming ). The discriminant = 0. Therefore the roots are all real and two of them are equal. Substitution of equation (63) into equation (59) shows that in this case =  (  A_{XR})^{3/2}. Therefore, according to equation (65), = arccos(  1) = . Using equations (66)–(68), one finds the roots
Therefore, in this case, y_{XR1}(x) and y_{XR3}(x) together form the inner sheet of the Cartesian oval. Since we know that y_{XR1}(x) is the desired lens surface, this is consistent with the exact quartic equation.
(c) = x_{D0}, (assuming ). The discriminant = 0. Therefore the roots are all real and two of them are equal. Substitution of equation (64) into equation (59) shows that in this case = + (  A_{XR})^{3/2}. Therefore, according to equation (65), = arccos( + 1) = 0. Using equations (66)–(68), one finds the roots
Therefore, in this case, y_{XR2}(x) and y_{XR3}(x) form the inner sheet of the Cartesian oval, and y_{XR1}(x) must be on the Cartesian oval's outer sheet.
(d) , (assuming ). The discriminant D_{XR} is now positive [see equation (60)]. Therefore only one real root exists. (The other two are complex and hence not physically significant.) This root must join up with y_{XR2}(x) in equation (70). The real root is given by the expression
As , = , which does indeed join up with equation (70) as expected. Note that this does not form part of the solution to the lens surface, but it is included here for completeness.
(e) , (assuming ). Again, as the discriminant D_{XR} is positive, only one real root exists. This root must join up with y_{XR1}(x) in equation (72),
As , = , which does indeed join up with equation (72) as expected. This does form part of the solution to the lens surface.
Examples of the cubic Xray approximation of the Cartesian oval are shown in Fig. 4. The outputs displayed in these graphs were calculated using MATLAB (MathWorks, 2004) in the default double precision. At = 10^{−4}, the exact equation for the Cartesian oval is still well conditioned enough to deliver stable output, and the Xray approximation already agrees well with it. It is at values of below this that the usefulness of the Xray approximation becomes obvious. Attempts to use the exact formula result in inconsistent output, while the output of the Xray approximation remains stable.
Fig. 5 displays a series of SHADOW3 raytracing simulations (Sanchez del Rio et al., 2011) of the lens surfaces determined by using the Xray approximation for the six cases in Fig. 4. All of the 500000 rays originate from a twodimensional Gaussian source of 1 µm height and width. This is much smaller than a normal synchrotron electron beam source, but was chosen to keep down aberrations that appear when the source size becomes comparable with the lens aperture. The rays are randomly sampled in angle over a uniform distribution of horizontal width 1.6 µrad and vertical width 1.6 µrad. These widths were chosen in order to just exactly cover the full aperture of the lens surface. Two optical elements are used in each simulation. The first optical element is used solely to turn the divergent rays from the source point into a convergent beam. It is a perfectly reflecting, ideally shaped ellipsoidal mirror located 23.10287 m from the source point. This mirror is set to a central grazing incidence angle of 3 mrad. It is shaped so that its source point coincides with the original source of the rays and its image point lies 23.10287 m downstream. The second optical element is the lens surface. It is located 0.010 m downstream from the first optical element. The image point lies 8.6912 m downstream. The basic shape is a plane surface of aperture 0.07425 mm horizontal × 0.07425 mm vertical. To this plane is added a spline interpolation generated by the SHADOW3 utility PRESURFACE from a 501 × 501 mesh of points calculated by MATLAB from the Xray approximation of this section. The index of refraction is taken as constant in the medium upstream from the lens surface and in the medium downstream from the lens surface. Absorption is neglected in both media. The displayed plots are all taken at the final focal point. In all six simulations, the distribution of rays in the image fits well to Gaussians of FWHM very close to 0.886 µm, the geometrical demagnified source size, in both height and width. A calculation of the spot size using the SHADOW3 utility RAY_PROP on 17 frames over a range within ±0.8 m from the image point at 8.6912 m showed that this point was indeed, as required, the point at which the rays converged (see Fig. 6). It is therefore demonstrated that the Xray approximation can indeed generate lens surfaces that focus convergent beam.
2.4. The paraxial approximation to a conic section
If the incident rays deviate from the central line x = 0 by only a small amount, the calculations of the ideal lens surface and of the lens surface in the Xray approximation both show that the value y(x) of the lens surface will also be small. In this case, one can assume that the cubic term in equation (47) is much smaller than the quadratic term, thus leading to the condition
for which the cubic term in equation (47) can be neglected. [Recall that = and that C_{2} and C_{3} are defined in equations (50) and (52), respectively.] In the Xray approximation, equation (77) reduces to the simple condition
If equation (77) (for the general case) or equation (78) (for the Xray approximation) is fulfilled, then the paraxial approximation is valid. Equation (47) then reduces to
and equation (57) for the Xray approximation reduces to
The solutions y(x) of these equations are conic sections. The type of conic section depends on the sign of the quadratic term y^{2}. Beginning with general values of , one can complete the square of equation (79) for two cases:
(i) . The paraxial approximation to the ideal lens surface is the ellipse
(ii) . The paraxial approximation to the ideal lens surface is the hyperbola
In the Xray approximation, one can complete the square of equation (80) for two cases:
(i) . The paraxial approximation to the lens surface is the ellipse
(ii) . The paraxial approximation to the lens surface is the hyperbola
In the limit and , equations (83) and (84) approach the conic sections calculated by Sanchez del Rio & Alianelli (2012).
An even stricter paraxial approximation is obtained if, in addition to the condition given in equations (77) or (78), one demands that the quadratic y^{2} term in equations (79) or (80) be much smaller than the linear y term. For general , this imposes the additional requirement
which in the Xray approximation becomes
[One can see that equation (86) is in fact more stringent than equation (78) by showing that 1/(q_{1} + q_{2}) < (q_{1}+q_{2})/q_{1}q_{2} for positive q_{1} and q_{2}.] If the conditions of equations (85) or (86) are fulfilled, the lens surface may be approximated as a parabola. For general , the lens surface is then approximately
and in the Xray approximation the lens surface is approximately
where F is the geometrical focal length. Double differentiation of equation (88) yields the well known relationship between the radius R and the focal length F of a single lens surface in the Xray approximation, F = .
3. Testing the paraxial approximation
Figs. 7 and 8 demonstrate how, in the paraxial approximation, the best conic section (ellipse for , hyperbola for ) and the best parabola deviate from the Xray approximation to the ideal Cartesian oval for = , the value for diamond at 15 keV. Surfaces 2, 24 and 48 were selected from Table 1 to demonstrate that the conic section approximations fail at decreasing apertures as the curvature of the surface increases. As mentioned by previous authors, the parabola deviates from the Xray approximation at much smaller apertures than does the best ellipse or hyperbola. Each plot's horizontal axis is scaled to make visible the aperture at which even the best ellipse or hyperbola begins to deviate from the Xray approximation. Thus, for surface 2, which has an aperture of 74.25 µm, one would expect the parabolic approximation to be sufficient because it matches the Xray approximation well out to < 2500 µm. For surface 24, which has an aperture of 59.52 µm, the parabolic approximation could still be sufficient, but, as the parabola only matches the Xray approximation out to < 200 µm, one might prefer to give this surface an elliptical or hyperbolic shape. For surface 48, which has an aperture of 46.760 µm, even the elliptical/hyperbolic approximation begins to fail at the edges; therefore, this surface must follow the ideal curve. As a result, surface 48 was chosen for the SHADOW ray traces of Fig. 9. To emphasize the improvement offered by the Xray approximation over the ellipse/hyperbola, the aperture of surface 48 was slightly widened to 63.600 µm, at which Figs. 7 and 8 show that the ellipse or hyperbola fails severely at the edges. A value = was chosen because of the limited precision given to the index of refraction input in SHADOW. In each simulation, 500000 rays were randomly selected from a Gaussian source of root mean square width 0.1 µm (FWHM 0.23548 µm) and uniform angular distribution. Although the chosen size of the source is much smaller than the electron beam sizes of real synchrotron storage rings, it is applied here to approximate a true point source, eliminating aberrations that would appear in the focal spot if the source size were comparable with the lens surface's aperture. Two optical elements were created. The first was a purely theoretical spherical mirror designed to reflect all rays from the source at normal incidence. This element exists only to produce the necessary convergent beam for the second element, which is the lens surface itself. The second element is situated 12.227 mm upstream from the focus of the spherical mirror. It is simulated with a plane figure to which a spline file generated by the SHADOW utility PRESURFACE is added. MATLAB was used to calculate a cylinder for onedimensional focusing with 501 points over a width of 46.760 µm in the nonfocusing direction and 681 points over a width of 63.600 µm in the focusing direction. The rays in the calculated profiles were sorted into 250 bins according to their position. Figs. 9(a) and 9(c) show the beam profiles generated at the nominal focus 10.999 mm downstream from surface 48, comparing them with the original source. The profiles generated by the ellipse or hyperbola are slightly but noticeably lower at the peak and have slightly larger tails than those generated by the Xray approximation to the ideal curve. The profiles generated by the parabola show a loss of about 50% of the peak intensity and correspondingly severe tails. Depth of focus plots showing the variation of the beam size versus the distance along the beam direction from the nominal focus were generated by the SHADOW3 utility RAY_PROP and are displayed in Figs. 9(b) and 9(d). The Xray approximation yields a lens surface that minimizes the beam width at the nominal focus, as required. The FWHM of the beam profile at this minimum is 0.210 µm, which matches the geometrically demagnified source size. The ellipse/hyperbola shifts the minimum of the beam width closer to the lens surface by 10–20 µm, and this minimum is still not quite as small as that achieved by the Xray approximation. The parabola shifts the minimum of the beam width by about 60–70 µm from the nominal focus, and this minimum is considerably larger than that achieved by either the Xray approximation or the ellipse/hyperbola. These results again demonstrate that the Xray approximation can generate lens surfaces that focus convergent beam better than the approximate conic sections can do.
4. Diffraction broadening
SHADOW was used to calculate the effective aperture of the CRL in Table 1. 500000 rays of 15 keV energy were created from a point source. They were uniformly distributed in angle so that the entire geometrical aperture of the first lens surface (0.075 mm × 0.075 mm) was illuminated. Each lens surface was assumed to be a parabolic cylinder, focusing in the vertical direction only. All of the lens surfaces except the first were taken to be unbounded so that their limited geometrical apertures would not cut off any of the rays propagating inside the CRL. The lens material is diamond, which for 15 keV Xrays has an index of refraction differing from 1 in its real part by −3.23 × 10^{−6}. The μ is 0.282629 mm^{−1}. The calculated intensity distribution on the last surface (number 48) and the calculated angular distribution of the intensity converging onto the final focus are displayed in Figs. 10(a) and 10(b), respectively. The effective aperture is the FWHM of the plot in Fig. 10(a), 26.47 µm. The corresponding numerical aperture is half the FWHM of the angular plot in Fig. 10(b), 1.204 mrad. The diffraction broadening therefore amounts to 0.75λ/(2NA) = 25.74 nm, which is much less than the focal spot widths in Fig. 9. Moreover, within the FWHM effective aperture in Fig. 10(a), a parabola is still a sufficiently good approximation to the ideal shape of the final lens surface, as shown in Figs. 7(e) and 7(f).
5. Conclusions
The immediate goal of this paper was to prove that an analytical solution, namely a Cartesian oval, exists for a lens surface that is to refocus an incident beam converging to a point into a new beam converging to a point closer to the lens surface. This result serves the longterm goal of designing aberrationfree aspherical CRLs that will in future produce Xray beam spots of 50 nm width and, further on, even 10 nm width. Numerical difficulties that arose in the analytical calculation of the Cartesian oval when the change in
across the lens surface is small, as is usual for Xray optics, were overcome by a cubic approximation that was numerically stable. The focusing performance of lens surfaces following the cubic `Xray' approximation was compared with that of lens surfaces shaped either as ellipses or hyperbolas, or as parabolas, as previous authors have suggested. Elliptical or hyperbolic lens surfaces yield stronger peaks and lower tails at the focus than do parabolic lens surfaces, but surfaces that follow the cubic Xray approximation provide better focal profiles than either. Examples taken from a proposed adiabatically focusing lens, in which the radius of curvature and the aperture of the lens surfaces both decrease along the beam direction, indicate that the advantages of the Xray approximation over conic sections are most apparent in the final, most strongly curved, lenses.Acknowledgements
L. A. would like to thank M. Sanchez del Rio (ESRF) for the many illuminating discussions on optics and on numerical methods in SHADOW.
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