Optimized IR synchrotron beamline design

Moreno, T.

doi:10.1107/S1600577515010978

research papers

JOURNAL OF
SYNCHROTRON
RADIATION

ISSN: 1600-5775

Volume 22| Part 5| September 2015| Pages 1163-1169

https://doi.org/10.1107/S1600577515010978

Optimized IR synchrotron beamline design

Thierry Moreno ^a ^*

^aExperimental Division, Synchrotron SOLEIL, L'Orme des Merisiers, Saint-Aubain, BP 48, 91192 Gif sur Yvettes Cedex, France
^*Correspondence e-mail: thierry.moreno@synchrotron-soleil.fr

Edited by S. Svensson, Uppsala University, Sweden (Received 14 January 2015; accepted 6 June 2015; online 22 July 2015)

Synchrotron infrared beamlines are powerful tools on which to perform spectroscopy on microscopic length scales but require working with large bending-magnet source apertures in order to provide intense photon beams to the experiments. Many infrared beamlines use a single toroidal-shaped mirror to focus the source emission which generates, for large apertures, beams with significant geometrical aberrations resulting from the shape of the source and the beamline optics. In this paper, an optical layout optimized for synchrotron infrared beamlines, that removes almost totally the geometrical aberrations of the source, is presented and analyzed. This layout is already operational on the IR beamline of the Brazilian synchrotron. An infrared beamline design based on a SOLEIL bending-magnet source is given as an example, which could be useful for future IR beamline improvements at this facility.

Keywords: infrared synchrotron beamline; bending magnet; optical aberrations; optical path.

1. Introduction

The intense and transversally coherent radiation delivered by infrared (IR) synchrotron beamlines (Duncan & Williams, 1983 ; Schweizer et al., 1985 ) makes the use of bending-magnet sources very attractive to perform micro-spectroscopy but requires highly collimated beams in order to match to the spectroscopic techniques, which mainly couple micro-scale mapping (Carr et al., 1998 ; Dumas & Tobin, 2003 ) with Michelson interferometry (Jackson & Mantsch, 1995 ). Before collimation, the large source emission is first focused, usually using a single toroidal-shaped mirror, to create a secondary source at the exit of the front-end. This secondary source contains geometrical aberrations produced by the circular shape of the bending-magnet source that cannot be correctly removed by the toroidal mirror. These aberrations grow as the square of the horizontal beamline aperture and make it difficult to properly collimate the beam afterwards. This paper presents an optical design, consisting of two front-end mirrors, a cylindrical one focusing the horizontal emission of the IR source, and a cone-shaped one focusing the vertical emission. The geometrical aberrations produced by the source and the optics are analyzed using the optical path method (Noda et al., 1974 ; Howells, 1992 ) and the best mirror configuration, removing source aberrations, is obtained. This method is then applied to the design of an IR beamline example based on a SOLEIL (Source Optimisée de Lumière à l'Energie Intermédiaire du LURE) bending-magnet source (Level et al., 2002 ). In this study, only the synchrotron radiation mode of IR emission is considered, but, since our optical layout removes the aberrations from all positions of the source trajectory, in particular at the entrance of the bending magnet, it also suits the edge radiation mode (Bosch et al., 1996 ; Roy et al., 2000 ). Although IR synchrotron radiation is almost fully transversely coherent, optical beam properties such as intensities, sizes and divergences are here accurately described using a ray-tracing approach (Moreno & Idir, 2001 ).

2. Method

Synchrotron radiation emission occurs when relativistic electrons are accelerated radially by a magnetic field from a bending magnet, thus following a circular trajectory. The photon source size results from the convolution of three terms: the geometrical aberrations produced by the circular shape of the electron trajectory (synchrotron radiation mode), the electron beam size and the diffraction limit (Hecht & Zajac, 1987 ). When the geometrical aberrations are small in comparison with the two other terms, for example, for small beamline apertures or for very large wavelengths as in the far IR, the horizontal and vertical profiles of the photon source are Gaussian, whereas for large extraction apertures the geometrical aberrations prevail and the profiles maintain the aberrations of the source shape. In this case, the horizontal profile of the source emission resembles the one obtained from reflection through a cylindrical mirror and retains the aberrations resulting from the circular shape of the bending-magnet source. In the vertical direction, the profile is Lorentzian and is formed by the overlapping of all the vertical emissions produced at each point in the electron trajectory. In order to remove these aberrations resulting from the source geometry, two dedicated mirrors are used to focus separately the horizontal and vertical components of the source emission. This approach has been widely used and for a very long time in the soft and hard X-ray domains to design focusing optics (Kirkpatrick & Baez, 1948 ) and monochromators (Chen et al., 1990 ).

2.1. Horizontal focusing mirror

Horizontally, the circular emission of the synchrotron radiation mode produces, for large horizontal beamline apertures, a beam profile containing coma aberrations (Hecht & Zajac, 1987) similar to the ones resulting from the reflection of a point source through a cylindrical mirror. So therefore, by using a cylindrical mirror in tangential reflection with a coma aberrated source, one can produce a point-like aberration-free image. The main parameters of this mirror such as its position inside the front-end, orientation and radius of curvature will be determined through the optical path method (Noda et al., 1974; Howells, 1992). Fig. 1 shows the top view scheme of the cylindrical source (bending-magnet electron trajectory) and the cylindrical horizontal focusing mirror. The method minimizes the path distance between two rays, one emitted tangentially from the center of the circular source and reaching the center of the mirror and the image plane, and a second emitted with an angle α relatively to the central ray and reaching the same image position. The path difference is expressed as a Taylor expansion of the position w or the angle α (Fig. 1) with constant coefficients depending on the distances p and q between the center of the mirror and the center of the source and the image, respectively, the mirror grazing angle θ and the radii of curvature of the mirror R and the source ρ (Fig. 1).

Figure 1
Horizontal source aberration scheme. A cylindrical mirror is used to remove the horizontal aberrations generated by the synchrotron radiation source (circular trajectory). The mirror position, orientation and radius of curvature are determined by minimizing the optical path difference $[\Delta F_{\rm{H}}]$ = $[(\,p'+d+q')-(\,p-d+q).]$

Since the optical path method is defined for a point light source, we consider here the source point resulting from the intersection of the two rays (Fig. 1) and the optical path difference $[\Delta F_{\rm{H}}]$ = $[(\,p'+d+q')]$ − ( p-d+q) which depends on d = $[\rho\tan({{\alpha}/{2}})]$ (see Appendix A).

As the parameter d depends on α, the best choice is to express the path difference $[\Delta F_{\rm{H}}]$ as a Taylor expansion of α, using the same notation as in Noda et al. (1974) and Howells (1992):

$[\Delta F_{\rm{H}} = C_{10} \alpha + C_{20} \alpha^2 + C_{30} \alpha^3 +C_{40} \alpha^4 + \ldots,\eqno(1)]$

where C₁₀, C₂₀, C₃₀ and C₄₀ are the longitudinal tilt, longitudinal defocus, coma and the spherical aberration coefficients, respectively.

The coefficient C₁₀ (tilt) is zero due to Fermat's principal (Hecht & Zajac, 1987) which imposes C₁₀ = $[\partial\Delta F_{\rm{H}}/\partial\alpha\vert_{\alpha=0}]$ . The C₂₀ (defocus) coefficient is

$[C_{20} = {{p^2} \over {2}}\left[\,{{1} \over {p}} + {{1} \over {q}} - {{2} \over {R \sin(\theta)}}\right],\eqno(2)]$

where p, q, θ and R are the distances of the mirror to the central position of the circular source and the image plane, the grazing angle and the radius of curvature of the mirror, respectively (Fig. 1).

The full expressions of the C₃₀ (coma) and C₄₀ (spherical aberration) coefficients are quite complicated (see Appendix A) but can be simplified under certain conditions. When C₂₀ = 0 (the lens equation), C₃₀ simplifies to

$[C_{30} = {{p^3} \over {4 \tan(\theta)}} \left(\,{{1} \over {p^2}} - {{1} \over {q^2}}\right)- {{\rho} \over {6}}.\eqno(3)]$

When $[\theta \not = {{\pi}/{2}}]$ , C₃₀ is zero for

$[q = {p} \big/ \left[1 - {{2 \rho \tan(\theta)}\over{3 p}}\right]^{1/2}.\eqno(4)]$

Relation (4) generates two solutions, according to whether the curvatures of the mirror and the source trajectory are oriented in the same way ( $[\rho\,\,\gt\,\,0]$ and $[q\,\,\gt\,\,p]$ ) like in Fig. 1 or opposite ( $[\rho\,\,\lt\,\,0]$ and $[q\,\,\lt\,\,p]$ ). For C₂₀ = C₃₀ = 0, C₄₀ simplifies to

$[C_{40} = {{p^4} \over {32 \tan^2 (\theta)}} \left(\,{{1} \over {p}}+{{1} \over {q}}\right) \left[{{p} \over {q}} \left({{3} \over {q}}-{{1} \over {p}}\right)^2 - \tan^2 (\theta)\left(\,{{1} \over {p}}-{{1} \over {q}}\right)^2 \right].\eqno(5)]$

Combining equations (4) and (5), the grazing angle can be removed,

$[f(u) = \sqrt{u} (3u-1)-{{3 p} \over {2 |\rho|}}(1-u^2)(1-u) = 0,\eqno(6)]$

with u = p/q.

The set of equations (2), (4) and (6) allows the determination of the best mirror configuration, removing the horizontal bending-magnet aberrations up to coefficient C₄₀ included. As for (4), equation (6) gives two solutions depending of the source and mirror curvature orientation, u $[\lt]$ 1 if the source and the mirror curvatures are oriented in the same way and u $[\gt]$ 1 otherwise.

2.2. Vertical focusing mirror

In the vertical direction, the photon source size is made up of the overlapping of the radiation emitted all along the electronic trajectory. The vertical aberrations can be almost totally removed using a cone-shaped mirror. Fig. 2 shows a scheme of this mirror, oriented with its axis of curvature perpendicular to the source emission. Horizontally, the direction of the light emitted at any position from the source is centered tangentially to the electronic trajectory, so each source point of the trajectory is linked to the horizontal angle of emission α. By applying the lens equation to each ray emitted tangentially to the source, the local radius of curvature at the position where the ray reaches the mirror can be determined and hence the shape of the mirror can be deduced. Relation (7) gives the Taylor expansion of the mirror radius obtained as a function of the position x along the mirror (Fig. 2),

$[\eqalignno{ R(x) = {}& {{2pq} \over {\sin(\theta)(\,p+q)}} \bigg\{1- {{q \rho} \over {p^2 (\,p+q)}} x \cr& +\ {{1} \over {2p^2}} \bigg[1+{{p \rho^2 (q-p)} \over {p^2 (\,p+q)^2}} \bigg] x^2+\ldots \bigg\},&(7)}]$

which corresponds to a cone-shaped mirror, where R(x) is the local radius of curvature at the position x along the mirror, p and q are the source and image distances to the center of the mirror, ρ the bending-magnet radius and θ the mirror grazing angle. Since the mirror section is circular, the same method as the one presented in §2.1, but using a point light source ( $[\rho]$ = 0), can be applied to optimize the mirror position and orientation. Fig. 3 shows the side view scheme of the vertical mirror configuration. The optical path difference (Noda et al., 1974; Howells, 1992) is given as a function of the angle φ between the two rays (Fig. 3):

$[\eqalignno{ \Delta F_{\rm{V}} & = (\,p'+q')-(\,p+q) \cr& = \ C_{10} \varphi+C_{20} \varphi^2+C_{30} \varphi^3+C_{40} \varphi^4 + \ldots,&(8)}]$

where C₁₀, C₂₀, C₃₀ and C₄₀ are the longitudinal tilt and defocus, coma and the spherical aberration coefficients of the optical path difference $[\Delta F_{\rm{V}}]$ , respectively.

Figure 2
Cone-shaped mirror evaluation. The vertical aberrations generated by the synchrotron radiation source are removed with a cone-shaped mirror. The mirror cone section is determined by applying the lens equation to each ray emitted tangentially at the source.

Figure 3
Vertical source aberration scheme. The mirror position, orientation and central radius of curvature are determined by minimizing the optical path difference $[\Delta F_{\rm{V}}]$ = $[(\,p'+q')-(\,p+q)]$ .

The aberration coefficients C₂₀, C₃₀ and C₄₀ are given by equations (2), (3) and (5) for a point light source ( $[\rho]$ = 0) and with the parameters p, q, R and θ of the vertical focusing mirror (Fig. 3).

For C₂₀ = 0, the coefficient C₃₀ vanishes either for q = p or θ = $[{{\pi}/{2}}]$ (Gauss illumination).

For C₂₀ = C₃₀ = 0, the coefficient C₄₀ becomes

$[\eqalign{ C_{40} & = {{p} \over {4\tan^2(\theta)}_{\vphantom{\Big|}}} \quad{\rm{for}}\quad q = p \quad {\rm{and}} \cr C_{40} & = -{{p^4} \over {32}} \bigg({{1} \over {p}}+{{1} \over {q}}\bigg) \bigg({{1} \over {p^2}} -{{1} \over {q^2}}\bigg)\quad{\rm{for}}\quad \theta = {{\pi}/{2}}.}\eqno(9)]$

To summarize, the aberrations resulting from the vertical focusing mirror are removed up to the C₃₀ coefficient, for either p = q or θ = $[\pi/2]$ , and up to coefficient C₄₀ if p = q and θ = $[\pi/2]$ . In practice, $[\theta \not = \pi/2]$ and the best position for the vertical focusing mirror is half the distance from the source to the image plane (p = q).

For very large horizontal apertures, the intersection between the peripheral vertical beam emission and the mirror becomes non-circular and thus the vertical aberrations are less efficiently removed at the mirror borders.

2.3. Focused beam sizes

The beam size on the image plane $[\sigma_{\rm{tot}}]$ results from the convolution of the geometrical beamline aberrations $[\sigma_{\rm{geo}}]$ and the optical magnification M of both the electron beam size $[\sigma_{\rm{e}}]$ and the diffraction-limited size $[\sigma_{\rm{dlim}}]$ (when defined at the source location). Assuming Gaussian distributions, the horizontal and vertical beam size on the image plane is given by

$[\sigma_{\rm{tot}} = \left[\sigma_{\rm{geo}}^2 +\left(M \sigma_{\rm{dlim}}\right)^2 \,\,+\,\, \left(M \sigma_{\rm{e}}\right)^2\right]^{1/2},\eqno(10)]$

where M = q/p is the magnification of the horizontal or the vertical focusing mirror. The horizontal and vertical electron beam sizes $[\sigma_{{\rm{e}}X}]$ and $[\sigma_{{\rm{e}}Z}]$ are determined by the accelerator design. The diffraction-limited size $[\sigma_{\rm{dlim}}]$ is related to the natural radial opening angle $[2\theta_{\rm{n}}]$ of coherent emission of the bending magnet and depends on the working wavelength λ through the following equation (Bosch et al., 1996) (for a fringe contrast of 50%):

$[\sigma_{\rm{dlim}} = {{\lambda} \over {4 \pi 2\theta_{\rm{n}}}} \quad{\rm{and}}\quad \theta_{\rm{n}}\,({\rm{rad}}) = \left({{3 \lambda} \over {4 \pi \rho}}\right)^{{1}/{3}}.\eqno(11)]$

This natural opening angle $[2\theta_{\rm{n}}]$ also determines the coherent photon divergence of the bending-magnet emission.

The geometrical aberrations $[\sigma_{{\rm{geo}}X}]$ and $[\sigma_{{\rm{geo}}Z}]$ produced by the optical layout are obtained by integrating the ray aberrations $[\Delta y]$ = $[(q/p)\,\partial\Delta F/\partial\alpha]$ over the full horizontal and vertical source divergences (Howells, 1992).

3. Application

The method described in §2 is now applied to the design of an IR beamline using a SOLEIL bending-magnet source. The beamline has an aperture of 80 × 40 mrad (H × V) and a front-end distance of 12 m. The optical properties of the beamline are calculated by ray tracing at wavelengths of λ = 1, 10 and 100 µm. Table 1 gives the electronic parameters of the source while Table 2 gives, from equation (11), the natural opening angle $[2\theta_{\rm{n}}]$ , the limit of diffraction size at the three wavelengths and the Gaussian source size resulting from the convolution of the electron source and the limit of diffraction. In the horizontal direction, the position, orientation and radius of curvature of the cylindrical mirror is obtained by solving equations (2), (4) and (6). Fig. 4 shows, for the three wavelengths, the horizontal RMS beam size focused at p + q = 12 m (focal plane) as a function of the parameter u = p/q. As expected, two positions minimize the horizontal beam size according to the relative orientation of the mirror curvature with respect to the one of the source: u $[\lt]$ 1 for opposite curvatures and u $[\gt]$ 1 for curvatures with the same sign.

Table 1
Source parameters

Electron energy, intensity	E₀ = 2.734 GeV, I₀ = 0.5 A
Electron beam size	$[\sigma_{{\rm{e}}X}\times \sigma_{{\rm{e}}Z}]$ = 63.1 × 32.4 µm RMS (H × V)
Bending magnet	B = 1.72 T, ρ = 5.28 m
Photon divergence	$[\Delta'_X\times \Delta'_Z]$ = 80 × 40 mrad (H × V)

Table 2
Natural opening angle, limit of diffraction and photon source size (H × V)

λ (µm)	$[2\theta_{\rm{n}}]$ (mrad) RMS	$[\sigma_{\rm{dlim}}]$ (µm) RMS	$[\sigma_{\rm{e}} \times \sigma_{\rm{dlim}}]$ (µm) RMS
1	7.1	11.7	64.2 × 34.5
10	15.4	51.7	81.6 × 61.0
100	33.1	229.0	237.5 × 231.3

Figure 4
Horizontal beam size on the focal plane (12 m from the source) as a function of u = p/q.

From relation (6), the two configurations minimizing the geometrical aberrations are u = 0.521 and u = 2.022. By taking into account the electron beam size and the limit of diffraction, Fig. 4 shows that both minima are shifted to higher values of u due to the convolution of equation (10). Because of the horizontal magnification of the optical system, the minimum horizontal beam size is obtained in the region u $[\gt]$ 1. This is the solution we choose for the optical design of our example. With a horizontal beamline aperture of 80 mrad, the best configuration allowing the beam to pass through the front end (see Fig. 5) is to choose u = 1.9, which defines the parameters of the horizontal correcting mirror: p = 7862 mm, q = 4138 mm, θ = 80.27° and R = 5501.3 mm. In this configuration, the curvatures of the source and the mirror are opposed. The vertical focusing mirror is placed at the same distance to the source and to the image plane (p = q = 6 m) in order to remove the vertical C₂₀ and C₃₀ aberration coefficients. There is no gain in resolution by increasing, according to equation (9), the mirror grazing angle to reduce the C₄₀ aberration, because the limit of diffraction at the working wavelengths is far larger than the spherical aberration of the vertical focusing mirror at the defined position. The best configuration is to set its grazing angle at 45° to facilitate the alignment of the front-end optics. With a grazing angle of $[\theta]$ = 45° (upward reflection), the best cone-shaped profile R(x) = R₀(1+a x) is obtained from equation (7) for R₀ = 8485.3 mm and a = −7.333 × 10⁻⁵ mm⁻¹ where x is the coordinate along the axis of curvature of the mirror.

Figure 5
Optical scheme of the SOLEIL IR beamline example. The positions of the mirrors are given according to optical path distances from the source.

Fig. 5 and Table 3 give the main parameters of our IR beamline example, where mirror positions are given as optical path distances from the source. The beamline consists of two optical stages: the focusing stage inside the front end including the first M1 mirror (using a horizontal 4 mm slot to remove the hard X-ray components of the bending-magnet emission), the cone-shaped mirror M4 and the cylindrical mirror M5, both focusing the 80 × 40 mrad (H × V) bending-magnet emission on a focal plane located at 12 m from the source; and the collimating stage, made up of two parabolic shape mirrors (M7 and M9) collimating the beam to the experimental setup. These two mirrors are positioned in order to produce a collimated beam with a square shape at λ = 100 µm (far IR).

Table 3
Parameters of the SOLEIL IR beamline example

Element	Optical path distances	Geometry	Length × width at λ = 100 µm
Diaphragm	1.0 m	80 mm × 40 mm (H × V)
M1 mirror	1.3 m	Plane, slot of 4 mm	77 mm × 103 mm
		θ = 45°, upward
M2 mirror	2.3 m	Plane	115 mm × 159 mm
		θ = 45°, downward
M3 mirror	5.7 m	Plane	315 mm × 450 mm
		θ = 45°, downward
M4 mirror	6.0 m	Cone shaped, vertical focus	331 mm × 474 mm
		θ = 45°, upward
M5 mirror	7.862 m	Cylinder, horizontal focus	626 mm × 162 mm
		θ = 80.27°, sideward
M6 mirror	9.862 m	Plane	324 mm × 84 mm
		θ = 80.27°, sideward
CVD window	11.95 m		Ø 20 mm
Focal plane	12.0 m
M7 mirror	12.5 m	Parabola, vertically collimating	106 mm × 20 mm
		θ = 45°, downward
M8 mirror	13.3 m	Plane	72 mm × 75 mm
		θ = 45°, upward
M9 mirror	14 m	Parabola, horizontally collimating	75 mm × 110 mm
		θ = 45°, sideward left

Fig. 6 shows the ray-tracing simulation of the beam image at 12 m (focal plane) and 16 m (2 m after the last M9 mirror) from the source, respectively. Table 4 gives the corresponding beam sizes values and Table 5 the divergences and intensities delivered by the IR beamline. Since ray tracing propagates the light incohenrently, contrary to beam propagation codes like SRW (Chubar & Elleaume, 1998 ), it does not take into account IR diffractions resulting from mirror slots (M1) nor mirrors borders. Nevertheless, it is an accurate method to evaluate both geometrical aberrations resulting from bending-magnet sources and optical surfaces and beam properties such as intensities (outside diffractive areas), sizes and divergences. Thus, ray tracing is an efficient way to design and optimize IR synchrotron beamlines.

Table 4
Beam sizes (H × V) at the focal plane and 2 m after the last mirror

	Size (µm FWHM)	Size (mm at 99%)
λ (µm)	At the focal plane	2 m after M9
1	90 × 79	75 × 22
10	136 × 160	75 × 47
100	440 × 580	75 × 78

Table 5
Beam divergences (H × V) and intensities after the last mirror

λ (µm)	Divergence (µrad FWHM)	Flux (photons s⁻¹ 0.1% BW⁻¹)
1	180 × 40	2.3 × 10¹⁴
10	275 × 80	1.3 × 10¹⁴
100	874 × 290	5.7 × 10¹³

Figure 6
Image of the beam calculated by ray tracing: (a), (b), (c) Beam image at λ = 1, 10, 100 µm on a screen of 0.8 mm × 0.8 mm at the focal plane (12 m from the source). (d), (e), (f) Beam image at λ = 1, 10, 100 µm on a screen of 100 mm × 100 mm located 2 m after M9 (16 m from the source).

4. Conclusions

In this paper, an optical layout adapted to IR synchrotron beamlines is presented and analyzed. This layout consists of two shape-optimized mirrors, focusing separately the vertical and the horizontal emission of the bending-magnet source. It has been optimized using an analytical formulation based on the optical path method (Noda et al., 1974; Howells, 1992). It removes almost totally the optical aberrations generated by the beamline components and provides high collimated beams for large horizontal beamline apertures. This optical layout is already operational on the LNLS (Laboratórios Nacionais de Luz Síncrotron) IR beamline (Moreno et al., 2013 ), the main conclusions being that it is efficient, easy to align and inexpensive to produce. A similar optical layout has been proposed and accepted for the future IR beamline at the ALS (Advanced Light Source). An IR beamline design using a SOLEIL bending-magnet source and working at λ = 1, 10 and 100 µm is given as an example, which could be useful for future IR beamline improvements at SOLEIL.

APPENDIX A

Minimization of the horizontal bending-magnet aberrations

The optical path method is applied to the difference $[\Delta F]$ between the two paths (p′ + d) + q′ and (p − d) + q as defined in the scheme of Fig. 1, corresponding to the tangential reflection through a cylindrical mirror of a circular light source. $[\Delta F]$ is usually expressed as a Taylor expansion of the position w along the mirror (see Fig. 1):

$[\Delta F = A_{10} w+A_{20} w^2+A_{30} w^3+A_{40} w^4+\ldots,\eqno(12)]$

where A₁₀, A₂₀, A₃₀ and A₄₀ are the longitudinal tilt, defocus, coma and the spherical aberration coefficients of $[\Delta F]$ , respectively, themselves a function of the angle α between the two rays through the parameter d = $[\rho\tan(\alpha/2)]$ . For a cylindrical mirror, these aberration coefficients are well known and defined as (Howells, 1992):

$[\left. A_{10}={{ \partial\Delta F }\over{ \partial w }} \right\vert_{w\,=\,0}\eqno(13)]$

is null due to Fermat's principal;

$[A_{20}={{T_1+T_2}\over{2}},\eqno(14)]$

$[A_{30} = {{\cos(\theta)}\over{2}} \left(\,{{T_1}\over{p-d}}-{{T_2}\over{q}}\right),\eqno(15)]$

$[\eqalignno{ A_{40} = {}& -{{\sin(\theta)}\over{4R^3}} + {{1}\over{8R^2}} \left(\,{{1}\over{p-d}}+{{1}\over{q}}\right) \cr& +{{\cos^2(\theta)}\over{2}} \left[{{T_1}\over{(\,p-d)^2}}+{{T_2}\over{q^2}}\right] - {{1}\over{8}} \left(\,{{T_1^{\,2}}\over{p-d}}+{{T_2^{\,2}}\over{q}}\right),&(16)}]$

with

$[T_1 = {{\sin^2(\theta)}\over{p-d}}-{{\sin(\theta)}\over{R}}, \quad T_2 ={{\sin^2(\theta)}\over{q}}-{{\sin(\theta)}\over{R}}.]$

The tricky development of d = $[({{\rho}/{2}})\alpha - ({{\rho}/{24}})\alpha^3 + \ldots]$ inside A₂₀, A₃₀ and A₄₀ and α inside w allows us to develop $[\Delta F]$ as a function of constant coefficients of α, as defined by relation (1), where C₁₀, C₂₀, C₃₀ and C₄₀ are now independent of α and w.

Acknowledgements

The author thanks Paul Dumas for the very active and fruitful collaboration in the development of IR beamlines and Yves Petroff, Raul Freitas and Harry Westfalh Jr for their confidence in our optical layout and the very nice results obtained on their LNLS IR beamline. The author also thanks Michael Martin and Hans A. Bechtel for having accepted our design for the future ALS IR beamline.

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JOURNAL OF
SYNCHROTRON
RADIATION

ISSN: 1600-5775

Volume 22| Part 5| September 2015| Pages 1163-1169

https://doi.org/10.1107/S1600577515010978

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research papers\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

Optimized IR synchrotron beamline design

1. Introduction

2. Method

2.1. Horizontal focusing mirror

2.2. Vertical focusing mirror

2.3. Focused beam sizes

3. Application

4. Conclusions

APPENDIX A

Minimization of the horizontal bending-magnet aberrations

Acknowledgements

References

research papers