Design of an X-ray transfer beamline for the Soft X-ray project at MAX IV

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TVE-F19003

Examensarbete 15 hp Juni 2019

Design of an X-ray transfer

beamline for the Soft X-ray project at MAX IV

Milad Emadi Johnny Ljung

Patric Beas Petersson

Sofia Tynelius

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Teknisk- naturvetenskaplig fakultet UTH-enheten

Besöksadress:

Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0

Postadress:

Box 536 751 21 Uppsala

Telefon:

018 – 471 30 03

Telefax:

018 – 471 30 00

Hemsida:

http://www.teknat.uu.se/student

Abstract

Design of an X-ray transfer beamline for the Soft X-ray project at MAX IV

Milad Emadi, Johnny Ljung, Patric Beas Petersson, Sofia Tynelius

At the MAX-IV lab in Lund, there is a current goal to build a new soft X-Ray laser. The beam will be generated from a free-electron laser (FEL), which is an instrument consisting of high-speed electrons. The electrons move through alternating magnetic fields, causing the beam to become monochromatic. After the FEL, the X- rays will enter a beamline consisting of different optical

components, such as mirrors, gratings and slits. This project investigated the necessary parameter values of the components, in order for the new X-Ray laser to focus the beam enough. The project consisted of a theoretical part and a simulation part.

The use of so-called Kirkpatrick-Baez mirrors enables the beam to be very focused. The best focus achieved was 7.23um*10.87um for ''Pink beamline'' and the intensity at the end was 71.5%, which meant that only 30% of the rays were lost. For the monochromatic beamline, a loss of intensity is inevitable. With a pair of KB-

mirrors, this beam was focused to be 6.95um*9.80um. The energy spread is ranging from 6.198 eV to 0.3442 eV. The analytical calculations for the spot size matched well with the simulations.

The pink beamline which was built in Ray satisfied the criterias of a spot size and intensity loss. The monochromatic beamline did fullfil the criterias of spot size and narrowing the energy

spread. A loss of intensity will for this beamline be inevitable.

Studying the misalignment effect showed that the components were most sensitive for vertical misalignment. The most sensitive

parameters were the curvature of the mirrors.

Examinator: Martin Sjödin Ämnesgranskare: Maria Strömme

Handledare: Vitaliy Goryashko, Peter Salén

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Sammanfattning

En frielektronlaser är ett instrument som består av elektroner med en hastighet nära ljushastigheten.

Lasern fungerar så att många magneter med alternerande magnetfält får laserstrålen att endast bestå av en våglängd. I det här projektet undersöktes en laser av röntgenstrålar i spektrumet

”mjuk röntgenstrålning”, vilket täcker vådlängder mellan 1nm och 5nm. Målet var att med hjälp av ett optiskt system av speglar och gitter fokusera strålarna så mycket som möjligt. Projektet var indelat i två delar; en teoretisk del och en simuleringsdel. För den teoretiska delen användes en matrismetod för att ta fram en systemmatris för hela strålgången. Varje optisk komponent har en överföringsmatris som beskriver hur strålens höjd och vinkel ändras efter komponenten.

Matrismetoden går ut på att multiplicera ihop matriserna för alla strålgångens komponenter, i omvänd ordning. Den resulterande systemmatrisen kan sedan användas för att bestämma strålens diameter. I simuleringsdelen användes programmet Ray-UI för att simulera varje del av strålgången. I programmet kan användaren till exempel se strålens storlek, hur många strålar som reflekteras mot en spegel och stålens utbredning. Olika parametrar för de optiska kompo- nenterna testades för att hitta den bästa fokuseringen av strålen. Användningen av så kallade Kirkpatrick-Baez-speglar (KB-speglar) gör att strålen kan fokuseras väldigt mycket. Fokuserin- gen blev som bäst 5.36 µm i bredd och 6.94 µm i höjd för ”Pink beamline” och intensiteten i slutet var 81.3%, vilket innebar att endast cirka 20% av strålarna gått förlorade. För ”Monochro- matic beamline” blev spridningen i fotonenergin kraftigt reducerad vilket är syftet med denna strålgång. Med hjälp av ett par KB-speglar fokuseras denna stråle till 6.95µm*9.80µm. Ener- gispridningen går från 6.198 eV till 0.3442 eV. Resultaten från simuleringen stämde väl överens med de teoretiska resultaten.

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1 Background

1.1 Introduction

At MAX-Lab in Lund, research on X-Ray science has been ongoing for over 40 years. In 2016, the Max-IV facility was opened and there are plans on building a Soft X-Ray beamline. Just like certain colors in the visible spectrum correspond to certain wavelengths of light, soft X-rays are defined as electromagnetic radiation or photons that have the wavelength 1 − 5 nm, which is equal to having a photon energy of 250 − 1240 eV [1]. The interval in the electromagnetic spectrum next to the soft X-rays with higher photon energies correspond to hard X-rays, whereas the interval with lower photon energies corresponds to extreme-ultraviolet light, often shortened to EUV. While these are not the focus of this study, the processing of photons in these intervals is usually done in a very similar way, and as such this study may be useful in these areas as well [2]. The current goal however is to build a new soft X-Ray laser at MAX-IV in Lund. This project will investigate the necessary parameter values of the optical components in the new beamline, in order for the new X-Ray laser to focus the beam enough.

A free-electron laser (FEL) is an instrument containing a laser consisting of high-speed electrons, moving close to the speed of light. The electrons move freely in an undulator, which consists of several magnets with alternating magnetic fields. When passing the undulator, the electron beam oscillates slightly which causes the beam to radiate energy in the form of light. This radiation interferes constructively so that the emitted light is nearly monochromatic. The FEL is a kind of laser with a wide span of frequencies, ranging from microwaves to X-Rays. To produce FEL X-Rays, the undulator must be made long. The radiation from the undulator then interacts with the electron beam, which results in a very powerful X-Ray, since millions of electrons radiate in phase [3].

After photons with soft X-ray levels of energy have been generated from the FEL, they will enter a beamline consisting of different optical components in order to attain the desired effects at the end of the beamline. For example, one purpose of the FEL structure is to be able to take a picture of sorts on an atomic or molecular scale, something that can be of great use in understanding the structure of biomolecules, which may be useful in improving current and developing new medicine. When designing this beamline, or the optical components it contains which for the majority of the time are mirrors, there are many different parameters that need to be taken into consideration, e.g. distances between mirrors and mirror sizes.

This study will use ray optics when modelling the beamline. While newer theories such as wave optics exist and can be used, ray optics can still describe the path a ray takes from a starting point to an ending point through different optical components. It is also easier due to the fact that this study only focuses on the reflection, transmission and absorption of light, rather than other more complicated phenomena such as diffraction. Thus, more extensive tools for describing the behaviour of light are unnecessary as they are simply more convoluted while not adding any real depth to the results [4].

The project will consist of a theoretical part and a simulation part. The goal is to design an X-ray transfer beamline for transporting and focusing the pulses from the FEL onto the experimental sample. The transfer is done by a line-up of optical tools such as mirrors, slits and gratings. By the end of the project there will be a theoretical model for the X-ray transfer beamline with an accompanying simulation to compare the analytical and experimental results.

An important goal is to find an analytical solution to transfer a generated FEL pulse without changing the original pulse outside acceptable tolerances. The other part will be to create a

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functional simulation to be able to effectively and accurately analyze how changes in parameters affect the transfer beamline.

The goal for the theoretical part, which is an analytical model of ray propagation, is to model an optical pulse as a Gaussian distribution of rays, to be able to perform ray tracing using the ray transfer matrix method. For the simulation part, the goals are to model ray tracing in the program Ray-UI and build the model of a so-called ”pink” beam beamline and a beamline with a monochromator. Sketches of the bealine are shown in figure 1.

Figure 1: Sketch of the beamlines, taken from [5].

In the pink beamline, the goal is to use mirrors to direct the original beam into having different properties, e.g. regulating the intensity, spot size, or resolution at the sample at the end of the beamline. In the monochromating beamline, the goal is to get a very small spread in energy (and therefore wavelength) of the beam, a spread which is unavoidable in the FEL prior, and as such a grating and a slit is used to diffract the different wavelengths and extract a certain wavelength through a slit.

In both parts of the project, one concluding goal is to do a misalignment study of the parameters.

At the end, the theoretical and simulational results are compared for the pink beamline, whereas the monochromator is left in the simulational stage.

1.2 Geometrics and Kirkpatrick-Baez mirrors

In this project, many different types of geometrics are used. One specific geometrical shape that will be used in the mirrors is that of an ellipse. The shape of an ellipse can be defined in many different ways, usually through the parameters short half-axis and long half-axis. When dealing with incoming rays onto an elliptical surface, this is not the most useful way to describe the ellipse. Instead an ellipse’s entrance and exit arm lengths are defined, as can be seen in figure 2. Given any point on the ellipse’s boundary, the entrance and exit arm length is simply the lengths between the point and the two focal points of the ellipse if they were to be projected onto the long half-axis. These two parameters can be used instead to yield the same ellipse as the one described by short and long half axis, but in this project the entrance arm will naturally be the path the ray takes toward the mirror (from focal point 1), and the exit arm the path

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the ray takes away from the mirror (toward focal point 2). Another important property of an ellipse is the radius of curvature. In contrast to the simplest case, the circle, an ellipse has a varying radius of curvature. For that reason we define a local radius of curvature depending on the entrance and exit arm lengths (figure2), as seen in equation9.

Figure 2: A picture showing the entrance and exit arm length of a given ellipse, where it can be seen that any ray passing through one focal point will also pass through the other.

The first mirror is said to be an ellipsoidal mirror. However, this mirror and any other mirror used in this project that is set as an ellipse in any focal plane, has either their entrance or exit arm length set to a very large number to correspond to infinity. This results in the focal point related to the aforementioned arm length can be seen as ”infinitely” far away, and the rays on this side of the reflection as parallel. A beam of parallel rays is called a collimated beam. A mirror with these properties, with one focal point infinitely far away, is equivalent to a parabolic mirror.

As such mirrors in the shape of a paraboloid reflect all incoming rays to become parallel, so that regardless of angular spread when the beam hits the mirror, one ends up with a collimated beam afterwards. Or, in the opposite manner, if collimated rays hit a paraboloidal mirror, they will all be focused on the same spot. This will prove to be an essential property for both the simulations and analytical solutions. In this report, these two shapes of mirrors will be used interchangably, since any mentioned ellipse will actually in reality be a parabola.

A certain pair of mirrors, Kirkpatrick-Baez mirrors or KB mirrors for short, is an integral part of the optical components used. The KB mirrors consist of two plane-cylindrical ellipsoidal mirrors, placed orthogonally to each other. The beam hits the first mirror and becomes focused in one plane, then moves on to the second one which focuses in the other plane, this seen in figure 3. The two mirrors will have different focal points in such a way that both focal points align in one spot, which is the spot whose properties are of interest and subject of optimising, and where the sample will be placed.

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Figure 3: The setup of a Kirkpatrick-Baez (KB) mirror pair.

1.3 Transfer matrix method

When a ray passes through an optical system consisting of several optical components, it is possible to determine changes in position and direction by using a matrix method. Each optical component has its own matrix, which describes the change in height and angle of the ray when passing the component. The matrix method requires the ray to be paraxial, which means that the angle between the ray and the optical axis is small throughout the system. The paraxial approximation leads to the following approximations,

sin(θ) ≈ θ, tan(θ) ≈ θ, cos(θ) ≈ 1.

(1) The purpose of the ray transfer matrix method is to get a system matrix that describes the whole system with just one matrix. This system matrix is obtained by multiplying all the matrices for the individual components in reverse order. The angle of the ray will be changed with each refraction and reflection, while the height of the ray, i.e. the position above the optical axis, will be changed during translation between the components. The matrix method is valid since the changes in height and direction can be describes by linear equations when assuming paraxial approximation.

The ray can be described by two coordinates, namely the height above the optical axis, y, and the angle relative to the optical axis, α. A simple translation of a ray through a homogeneous medium is shown in figure 4. Before the translation, the coordinates describing the ray are (y₀, α0), which is the initial height and angle. After the translation, the angle is unchanged. The height however, is changed by L·tan(α₀), as shown in figure4. Using the paraxial approximation gives:

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α₁ = α₀ y1= y0+ Lα0,

(2) where L is the axial progress of the ray. Rewriting these two equations in matrix form gives

y1

α1

=

1 L 0 1

y0

α0

. (3)

The 2x2 matrix in this expression is the ray-transfer matrix M, which in this case shows how translation affects the ray.

Figure 4: Translation of a ray through homogeneous medium.

In a similar way, it is possible to determine a transfer matrix for reflection at a spherical surface, like the one shown in figure 5. Since the mirror is concave, R is considered negative. There is a sign convention for the angles as well, also shown in figure 5. For rays pointing upward, the angle is considered positive, and for rays pointing downward, the angle is considered negative.

From figure 5, the following expressions can be deduced,

α = θ + φ = θ + y

−R α⁰ = θ⁰− φ = θ⁰− y

−R,

(4) where the paraxial approximation has been used for sin(φ). The law of reflection, θ = θ⁰, together with equation 4gives:

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α⁰ = θ + y

R = α +2y

R. (5)

Since the reflection takes place in one point of the mirror, the height of the ray is unchanged by the reflection. Thus, the matrix equation describing the reflection is

y⁰ α⁰

=

1 0

2

R 1

y α

, (6)

where the 2x2 matrix is the transfer matrix for reflection against a spherical surface.

Figure 5: Reflection of a ray at a spherical surface.

When the surface is elliptical instead of spherical, the transfer matrix looks the same with the exception that the radius R is replaced with the radius of curvature ρ in element M₂₁, since not all normal vectors go through the center of the ellipse. An expression for ρ can be determined by looking at reflection againt a mirror as well as translation before and after the mirror. The system matrix for this will be

1 Len

0 1

1 0

2 ρ 1

1 Lex

0 1

=

"

1 +²_ρ L_en(1 + ²_ρ) + L_ex

2 ρ

2Len

ρ + 1

#

, (7)

where L_en and L_ex are the entrance and exit arm, respectively. Multiplying this system matrix with the initial parameter vector r₀ =

y0

α0

gives

y₁ = (1 + 2

ρ)y₀+ (L_en(1 +2

ρ) + L_ex)α₀ α1= 2

ρy0+ (2Len

ρ + 1)α0.

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This system of two translations and one reflection does not change the height of the beam, meaning that y₁ = y₀ = 0. Using this and equation 8gives

2

ρ = − 1 L_en− 1

L_ex. (9)

Thus, the reflection matrix for an elliptical surface can be expressed with the entrance and exit arm lengths, as

M =

1 0

−_L¹

en −_L¹

ex 1

. (10)

It is also important to note that for an off-axis mirror the local radius of curvature ρ will be adjusted by an angle ρ = ρ₀· cos(θ) in the horizontal direction and ρ = _cos(θ)^ρ⁰ in the vertical direction, where θ is the rotation of the matrix off the optical axis.

When considering two dimensions, the initial parameters will be written as r₀ =





 x₀ β0

y₀ α₀





 . The 2x2 transfer matrix will then be replaced by a similar 4x4 matrix, describing the change of the ray in both directions. A translation matrix in two dimensions will therefore look like

M =







1 L 0 0

0 1 0 0

0 0 1 L

0 0 0 1







, (11)

where the upper left corner describes one direction and the lower right the other [6].

Now assume the ray passes through an optical system with three components. The coordinates for the height and angle will be changed according to

y₁ α1

= M₁·

y₀ α0

y2

α₂

= M₂·

y1

α₁

y3

α3

= M3·

y2

α2

,

(12) at the first, second and third component. Combining all these equations yields

y₃ α3

= M₃· M₂· M₁·

y₀ α0

(13)

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and thus the complete system matrix can be written as M = M₃· M₂· M₁. It is important to multiply the matrices in reverse order, since matrix multiplication is not commutative. Generally, for a system with N components, the system equation can be written as

y_f α_f

= M_N · M_{N −1}· ... · M₂· M₁·

y0

α₀

(14) and the system matrix can thus be expressed by M_N· M_{N −1}· ... · M₂· M₁. [7]

1.4 Superposition of rays

In this study the beam is modelled as a superposition of rays. Suppose that an arbitrary ray has initial parameters y₀, x₀, α₀ and beta₀, where y₀ and x₀ are the coordinates for the initial position of the ray in the yx − plane and α₀, β₀ are the initial beam divergences with respect to the y and x axes. Now suppose that a Gaussian distribution of rays is formed by generating 1000 random starting parameters within fixed intervals, each belonging to a separate ray. This gives a beam of 1000 unique rays that can all be traced through the beamline system by using equation14, which provides a means of examining how the beam propagates through the system.

The rays that the beam is comprised of follow a Gaussian distribution formula,

f (y₀, α₀) = 1 2πσ_yσ_αe⁻

(y−y0)2 2σ2y e⁻

(α−α0)2

2σ2α , (15)

which is analogous for the x₀ and β₀ parameters. The probability to find all rays within the entire coordinate system is 100%, which gives the probability equation,

Z ∞

−∞

Z ∞

−∞

f (y0, α0)dy0dα0= 1. (16)

Since the variables y₀and α₀are independent in equation15, it is possible to write the probability equations separately,

Z ∞

−∞

f (y0)dy0= 1, (17)

Z ∞

−∞

f (α0)dα0 = 1. (18)

This allows a much simpler formulation of the expectation values of the initial parameters, which provide the root-mean-square (RMS) beam size and angle of divergence,

hy²₀i = Z

y²₀f (y0, α0)dy0dα0 = σ²_y. (19) The equation for α is analogous and gives hα²₀i = σ_α². The matrix for the initial position can now be written as

σ0 =

hy²₀i hy₀α0i hy₀α0i hα²₀i

=σ_y² 0 0 σ_α²

. (20)

Now examining a ray travelling a distance L from the initial position gives the new position y₁ = y₀+ Lα₀ according to equation2. It is possible to examine the change in expectation value if the position changes from y = y₀ to y = y₀+ Lα₀, according to

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hy²₁i = Z ∞

−∞

(y₀+ Lα₀)²f (y₀, α₀)dy₀dα₀

= Z ∞

−∞

y₀²f (y0)dy0

Z ∞

−∞

f (α0)dα0

+ 2L Z ∞

−∞

y0f (y0)dy0

Z ∞

−∞

α0f (α0)dα0+ L²

Z ∞

−∞

α²₀f (α₀)dα₀ Z ∞

−∞

f (y₀)dy₀,

(21) where the first term is σ_y², because R∞

−∞f (α₀)dα₀ = 1 due to the probability distribution, the second term is just 0 because of the multiplication between an even and an odd function inte- grated to infinity and the final term is L²σ²_α, becauseR∞

−∞f (y₀)dy₀ = 1 again due to probability distribution mentioned earlier. This gives a new result for the ray through drift

hy²₁i = σ²_y+ L²σ_α², (22)

which can be used to determine the ray propagation of the entire beam. Repeating this process for the angle α allows the formulation of a new law of propagation for the beamline - instead of always employing the result in equation22, the result can be expressed by the following matrix equation:

σ = M σ0M^T =

M₁₁² σ_y²+ M₁₂²σ²_α M₁₁M₁₂σ_y²+ M₁₂M₂₂σ²_α M11M12σ_y²+ M12M22σ_α² M₂₁² σ_y²+ M₂₂² σ_α²

, (23)

where M is the transfer matrix for the system. The top left entry of the matrix corresponds to the result in equation22and the bottom right entry corresponds to the analogous result for the angle. The cross-terms describe how off-focus the ray is in the examined position, but this is not examined in this study and therefore no further explanation is provided. This new matrix equation provides a very convenient way of computing the propagation of the entire beam.

1.5 Ray-UI

The software used to perform the ray tracing is called Ray-UI and is developed by the German research centre Helmholtz-Zentrum Berlin (HZB). Ray-UI allows the user to simulate a beamline with the possibility to add components such as slits and gratings, and can also simulate Gaussian distributed beams. The user also has the possibility to look at non-ideal mirrors by adding misalignments and misorientations to each mirror. The parameters for each component are accessed by just clicking on the component [8].

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Figure 6: The user interface of Ray-UI.

As shown in figure 6, the beamline is at the top with all components and image planes. Down to the left are all parameters for each component. Down to the right is the simulation of all the rays displayed, accompanied by distributions for where all the rays impinge the component. The width of this distribution will give the spot size at that point. The footprint, i.e. the picture showing where each ray impinges the mirror, will show rays in certain colours as seen in figure 6, stating what happened to the rays during the hit. A green colour indicates that the ray has been reflected on the component, a red colour indicates that the ray has been absorbed in the component and a grey colour indicates that the ray has never impinged the component. Except for footprint, the user can also access the intensity of the ray on the component and look at the beam properties. The beam properties display the energy distribution and the horizontal and vertical beam divergence.

Image planes can be added before or after components to see what the beam looks like in that very position, the footprints on each component only show how the when the beam impinges it.

In Ray-UI, the components are added to the beamline and assigned parameters which is either found in the papers about the Athos-FEL or tuned so that there is no intensity drop between the mirrors.

When the beamline is created, results or data for every component such as intensity before and after (number of rays transmitted or absorbed) and the distance after preceeding object can be seen and selected in the row at the top. An image plane placed at the end will give the result of the spot size, and also how the energy varies at that point. When constructing this beamline, the same components were used as in the ATHOS soft X-Ray FEL which is under construction in Switzerland and the second FEL at the Paul Scherrer Institut (PSI). In 2016 the PSI opened a hard X-Ray FEL called Aramis, which cover the FEL properties from 5 to 10 kev. This new ATHOS soft X-Ray FEL is planned to be finished in 2020 [5].

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1.6 Criterias of final beam

After dicussions with the supervisors, the following criterias of the beamlines have been taken into consideration. The beam should be focused so that the spot size is between 3 to 10 µm. The intensity for the Pink beamline should be as high as possible, the minimum allowed intensity at the end is 50%. For the monochromatic beamline, the energy spread should be reduced so that the resolving factor (equation (24)) is between 3000 and 4000. The resolving power is calculated as follows,

R = δλ λ = E

δE (24)

The total photon energy divided by the energy spread [4].

2 Execution

2.1 Analytical Method

2.1.1 Matrix Method

The pink beamline consists of four mirrors and five translations, one between each of the mirrors, meaning that there are nine matrices needed to express the whole system. The first mirror is an ellipsoid and the second is plane. For the plane mirror, the same matrix as for a spherical surface, described by equation 6, may be used, if the radius is set to infinity. Therefore, the matrices for these are

R1 =







1 0 0 0

2

c1 1 0 0

0 0 1 0

0 0 _c²

2 1







and R₂ =







1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1





 ,

where c₁ and c₂ are the radius of curvature, which can be expressed by the mirror’s entrance and exit arm length according to equation 9. Since the KB-mirrors are cylindrical in one direction and elliptical in the other, they only focus the ray in one direction each. The mirror is therefore an ellipsoidal with infinitely large radius of curvature in one direction. Thus, the reflection matrices for the first and second KB-mirrors can be expressed as

R3 =







1 0 0 0

0 1 0 0

0 0 1 0

0 0 _ρ²

1 1







and R₄ =







1 0 0 0

2

ρ2 1 0 0

0 0 1 0

0 0 0 1





 ,

where ρ is the radius of curvature for each mirror. In total, there are five translation matrices in the pink beamline, one between each mirror. They all have the same structure,

T_N =







1 LN 0 0

0 1 0 0

0 0 1 L_N

0 0 0 1





 ,

(16)

but with different lengths L. The resulting system matrix is then

M = T₅· R₄· T₄· R₃· T₃· R₂· T₂· R₁· T₁ (25) This is a 4x4 matrix and its elements are

M₁₁= 1 +^2(L⁴_ρ^+L5)

1 +^2(L⁴_c^+L⁵⁾

1 + (1 + ^2(L⁴_ρ^+L⁵⁾

1 )^2(L²_c^+L³⁾

1

M₁₂= L4+ L5+ L2(1 +^2(L⁴_ρ^+L5)

1 ) + L3(1 +^2(L⁴_ρ^+L5)

1 ) + L1(^2(L⁴_c^+L⁵⁾

1 + (1 +^2(L⁴_ρ^+L5)

1 )(1 +^2(L²_c^+L³⁾

1 ))

M₂₁= _ρ²

1 +_c²

1 + _c⁴

1ρ1(L2+ L3) M₂₂= 1 +^2L_ρ²

1 +^2L_ρ³

1 + L₁(_ρ²

1 +_c²

1 +_c⁴

1ρ1(L₂+ L₃)) M₃₃= 1 +^2L_ρ⁵

2 +^2L_c⁵

2 + (1 +^2L_ρ⁵

2 )^2(L²^+L_c³^+L⁴⁾

2 )

M₃₄= L5+ (1 + ^2L_ρ⁵

2 )(L2+ L3+ L4) + L1(^2L_c⁵

2 + (1 +^2L_ρ⁵

2 )^2(L²^+L_c³^+L⁴⁾

2 )

M₄₃= _ρ²

2 +_c²

2 + _c⁴

2ρ2(L2+ L3+ L4) M₄₄= 1 +_ρ²

2(L2+ L3+ L4) + L1(_ρ²

2 +_c²

2 +_c⁴

2ρ2(L2+ L3+ L4))

M₁₃ = M₁₄ = M₂₃ = M₂₄ = M₃₁ = M₃₂ = M₄₁ = M₄₂ = 0.

This system matrix is then used in conjunction with equation 23 to calculate the beam propagation through the system.

2.1.2 Misalignment - Spot Size and Central ray

It is interesting to know how misalignment in different parameters of the beamline affects the final result of the focusing (spot size), as well as the position of the central beam after passing through the system. This is studied by implementing the aforementioned transfer matrix method in Matlab using typical values for parameters and using equation 23 to calculate the beam propagation. The parameters can then be varied in fixed intervals around the typical values and the change in spot size with regards to the variation can be examined. The result can be plotted as spot size on the y-axis and the varying parameter size on the x-axis to clearly show the dependence.

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2.2 Simulation in Ray-UI

2.2.1 Light Source

In order to be able to start simulating any beamline, a source of a beam is necessary. In reality, the X-ray photons are generated by accelerating electrons and sending them through an undulator as has been previously stated. In Ray-UI, it is possible to have two types of undulators as the source of rays, however, the source parameters are already available. Therefore, a different type of source is chosen, specifically a point source.

The point source is used in both the pink beam and the monochromator beamline, and its parameters can be seen in figure 7and 8.

Figure 7: The UI of Ray-UI showcasing the point source and its relevant parameters.

The number of outgoing rays is set to 20000, which is a number that is not overly important for the simulation. This number is simply chosen as a large enough number so that the rays can accurately enough represent a Gaussian distribution, while not being too large to cause unnecessary long computational time. When the photons exit the FEL into the beamline, they are expected to have a certain spread in space, which can be shown to have the distribution of a Gaussian. The rays also have a certain angular divergence, also having a Gaussian distribution.

These divergences are first introduced in the setting of source width, in horisontal (x-) direction,

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and height in vertical (y-) direction. The one-sigma, set to be the same for both directions, is chosen to be 0.088 mm [4].

’

Figure 8: The UI of Ray-UI showcasing the point source’s footprint and distribution.

One thing to note here is that while the width and height is set to 0.088 mm, this is different from what the footprint suggests, stating a width of about 0.2 mm. This is due to the fact that the footprint doesn’t use the standard deviation, one-sigma, as its width but instead the full width at half maximum, or FWHM. The FWHM for a Gaussian is related to one-sigma through equation26

F W HM = 2

√

2ln2 σ ≈ 2.355 σ, (26)

which explains the difference between the parameter set and the shown value on the footprints.

This is something to keep in mind when five-sigma values and similar is used.

In addition to the horizontal and vertical width on the source, an angular divergence in these two directions is set. The divergence is that of a Gaussian, and is set to 0.0053 mrad [4].

The energy of the photons are as previously stated set to 250-1240 eV. For most of the simulations though, 1240 eV is used. Some exceptions to this however, are included in the study and will be discussed later on. The photon energy is also set to have a certain spread, so called band-width , with the width set to ±0.5% [4].

All of the above mentioned parameters and settings to the point source is constant throughout all simulations. Since the rays are all distributed randomly in the software within these boundaries however, two simulations will not be the same. Usually this leads to values differing a few percent, but sometimes one receives some bad runs of the simulation with drastically lower values, for

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reasons unknown. This strengthens the need for multiple runs of the same simulation, something to keep in mind.

After being reflected and collimated by the paraboloidal mirror, the beam lack angular spread but no longer travels straight, instead it has a certain angle from its original path. While this is not an obstacle in a simulation, when constructing the actual beamline, for practical reasons one needs to offset this by adding a simple plane mirror, whose purpose is to offset this angle without affecting the beam in any other way such as focusing it further. The drawback of doing this is of course losing out on a part of the total intensity of the beam due to absorption. This is however necessary for the beamline and doesn’t affect the intensity considerably.

2.2.2 Pink Beamline

The ultimate goal of the beamline for the so-called pink beam is to focus the beam of soft X-rays onto a very small spot without too much of a reduction in intensity. For example, the size of the focusing spot was should be as small as possible, at least below 10 µm. The intensity on the spot needs to be as high as possible to not have wasted the energy of the photons that have been carefully and intricately generated in the FEL prior.

After having set the point source, the next step was to create the general structure of the rest of the optical components in the beamlines. For the pink beamline, an ellipsoidal mirror, a plane mirror, and two more ellipsoidal mirrors (the KB-pair) were introduced in this order, following the point source. Hereafter they will be referred to as EM (ellipsoidal mirror, corresponding to M1 in figure 1), PM (plane mirror, corresponding to M3 in figure 1), KB1 and KB2 (KB-pair, corresponding to M4 in figure 1). In front of every mirror, image planes were put to be able to view the cross-section of the incoming beam before it were to impinge each mirror. At the end of the beamline after the two KB-mirrors another image plane was set, once again as a means to view the beam, or in this case the spot, where it in reality is supposed to hit the sample.

In accordance with the Athos SwissFEL [5], most of the distances between the mirrors were then set. More specifically, the distance point source-EM was set to 55 m, EM-PM to 10 m, and PM-KB1 to 85 m. The effect of different incidence angles on the mirrors was investigated as well as different settings to achieve a small focus spot size. In the same manner, the so-called grazing angle for all mirrors was set to 1.5^◦, since with larger angles the amount of reflected rays is so low that the simulations simply yield nothing. The grazing angle is derived from 90^◦ subtracted with the normal angle of incidence[2]. The dimensions of the mirrors, while they later on were to be modified and studied, were in this stage set large enough not to lose out on any incoming rays. 100 mm in width and length was sufficient.

As the beam of rays propagates through space, it widens due to the divergence of the rays, reducing intensity. To lessen this effect, or optimally counteract it completely, a parabolic mirror may be used.

2.2.2.1 KB-mirrors and Spot Size The final step in the pink beam beamline is to actually focus the beam down to an acceptable size. This is commonly done with a KB-pair of mirrors.

With the general idea of the structure of the beamline complete, the next more complicated part is to adjust several parameters to achieve the desired results. The first KB-mirror is set to have an infinite entrance arm since a collimated beam was now received. The second mirror has the exact same parameters except for having an azimuthal angle of 90^◦, due to the fact that it needs to be orthogonal. It also has an exit arm slightly shorter than the first, since they need to allign

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at one point, the focal point. An image plane was added to the focal point, where both exit arms end. The distance between the two KB-mirrors is set to 500 mm, the focal point is set to be at 1500mm from the second KB-mirror. With these parameters a high intensity is preserved and a small spot size can be obtained.

2.2.2.2 Intensity: materials, angles, and mirror sizes It is imperative that a high intensity is preserved throughout the beamline, and to reduce the drop in intensity on each optical component the main things to consider are the grazing angle, materials and coatings, the variance between photon wavelengths, and the necessary mirror sizes. One of the first things to consider is the grazing angle of the beam on each of the mirrors. With light of this high energy levels, a very small gazing angle is necessary not to lose most of the incoming rays to absorption or transmission of the mirror. Grazing incidence angles between 0.5 and 1.5 were studied. The larger angles are desired in case of several beamline branches in order to create sufficient offset between e.g. the endstations and other components. The smaller angles produce higher transmission because of the higher reflectivity but leads to an increase in mirror size.

The substrate, i.e. the base material, of the mirrors are silicon. The material which is chosen as coating on the silicon substrate is B4C (Boron Carbide), because of its ability to absorb FEL radiation without mirror damage. When the intensity is high, the risk of damaging mirrors are high, therefore a coating with this material is suitable. The thickness of the coating is set to 50 nm [5].

While a photon energy of 1240 eV is usually the setting in all simulations, an analysis of the reflectance of the materials Silicon and B4C is done in a program called XOP, to simply plot a graph over the reflectance as a function of photon energy. Another parameter here is the angle, which will change this graph. As such, several graphs of this study were necessary, each displaying the wavelength (energy-) dependence of light onto a certain material at a certain angle. Since an optimisation of the entire beamline is the goal, the minimum mirror sizes were also examined in detail after finding a working basic setup for the pink beamline.

2.2.2.3 Misalignment - Intensity The distance between each mirror is usually not very important for the simulation. However, each distance involving an ellipsoidal mirror (i.e. every distance), needs to be taken into consideration when setting the focus of the curved mirrors.

The misalignment study was comprehensive and included varying 6 parameters for all the mirrors in the pink beamline. The 6 parameters are horizontal, vertical and longitudinal misalignment and horizontal, vertical and longitudinal misorientation. Misalignment is a variation in position while misorientation is an angular variation. Each parameter was increased one at a time until the intensity in the focal point was decreased to 0%. The size of the mirrors were determined in two ways. The first way was to calculate the length by approximating the mirror length as the hypotenuse of a right-angled triangle, which is exact for the plane mirror, and accurate even for the elliptical mirrors due to the large entrance or exit arm length of the mirrors. The cathetus is known from the image plane right before the mirror and calculating the five-sigma width of the spot shown. The angle is the grazing angle which is known. This size of the mirrors is here called the calculated mirror dimensions. The other way was to compare the size of the mirrors to the intensity between each mirror in such a way that no beams were lost between mirrors, here called the compared dimensions. The only intensity drops should come from absorption, and not from mirrors being undersized.

Design of an X-ray transfer beamline for the Soft X-ray project at MAX IV

Examensarbete 15 hp Juni 2019

Design of an X-ray transfer

beamline for the Soft X-ray project at MAX IV

Milad Emadi Johnny Ljung

Patric Beas Petersson

Sofia Tynelius

Abstract

Design of an X-ray transfer beamline for the Soft X-ray project at MAX IV

Sammanfattning

Contents

1 Background

2 Execution