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Uppsala University
Bachelor of Science Degree thesis
Department of Physics and Astronomy
Division of Molecular and Condensed Matter Physics
A Numerical Study of the Wavelength
Dependence in an Off-Plane XUV
Monochromator
Supervisor:
Stefan Plogmaker
Jan-Erik Rubensson
Subject Reader:
Johan Söderström
Author:
Jonatan Fast
June 30, 2015
Abstract
In this thesis the work is based around the HELIOS (High Energy Laser Induced
Overtone Source) laboratory, a facility at the Division of Molecular and Condensed
Matter Physics at the Department of Physics and Astronomy, Uppsala University, Sweden. HELIOS generates XUV (extreme ultraviolet) photon pulses with energies between
15-70 eV and pulse lengths shorter than 50 fs. It is believed that the grating in the
monochromator of the lab introduces a chirp, meaning that the different constituent
energies of the pulse will be ordered in an non homogeneous way. Since the lab is striving to create as short pulses with as high temporal resolution as possible, learning more
about any effects that might change the pulse is of great interest. If the source of the
chirp effect is known then possible countermeasures can start being discussed. To investigate the pulse a numerical ray trace is performed for a XUV pulse passing through the
c
monochromator using the BeamFour
software. A pulse consisting of 49500 rays with
mean energy 40 eV and FWHM 0.5 eV was traced, but no clear dependence between
wavelength and the pulse length could be observed. This seems to indicate that there
either is no chirp present in the monochromator or that it is of such a small order that
higher certainty (more rays) will be required to observe it.
Sammanfattning
I detta examensarbete baseras arbetet på HELIOS-laboratoriet (High Energy Laser Induced Overtone Source), en anläggning vid avdelningen för molekyl- och kondenserade
materiens fysik, institutionen för fysik och astronomi, Uppsala universitet, Sverige. Vid
HELIOS genereras XUV (extremt ultravioletta) fotonpulser med energier mellan 1570 eV och pulslängder under 50 fs. Vid avdelningen tror man att gittret i laboratoriets
monokromator introducerar en så kallad chirp, vilket innebär att de olika energierna
som bygger upp pulsen blir uppdelade i en icke homogen ordning. Eftersom labbet
strävar efter att skapa så korta oc kompakta pulser som möjligt är det av stort intresse för avdelningen att lära sig mer om effekter som påverkar pulsens form. Om
utsträckningen av sådana effekter är välkänd så kan man börja diskutera åtgärder för
att motverka dessa, eller åtminstone vara medveten om dem. För att undersöka hur
pulsen beter sig så har en numerisk strålspårning utförts för en XUV-puls med hjälp
c
av programvaran BeamFour
. En puls innehållande 49500 strålar med genomsnittlig
energi 40 eV och halvvärdesbredd (FWHM) 0.5 eV spårades, men inget tydligt beroende
mellan våglängd och pulslängd kunde observeras. Detta tyder på att det antingen inte
uppstår någon chirp i monokromatorn, alternativt att den är av sådan liten storlek att
högre mätsäkerhet (fler strålar) kommer behövas för att observera den.
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Contents
1 Introduction
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2 Background: Theory
2.1 The diffraction grating . . . . . . .
2.1.1 Blazed reflection grating . .
2.1.2 Specular reflection on blazed
2.1.3 Grazing angle . . . . . . . .
2.1.4 Off-plane setup . . . . . . .
2.2 Monochromator . . . . . . . . . . .
2.3 Chirp . . . . . . . . . . . . . . . . .
2.4 High harmonic generation . . . . .
2.5 Normal distribution . . . . . . . . .
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gratings
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3 Background: HELIOS
4 Methodology for numerical ray
4.1 Optical elements . . . . . . .
4.2 The source . . . . . . . . . . .
4.3 Handling of data . . . . . . .
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tracing
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5 Results & Discussion
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6 Outlook
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7 Conclusions
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8 References
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References
8.1 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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9 Appendix
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2
1
Introduction
In this thesis the work is based around the HELIOS (High Energy Laser Induced Overtone
Source) laboratory, a facility at the Division of Molecular and Condensed Matter Physics at
the Department of Physics and Astronomy, Uppsala University, Sweden. HELIOS generates
XUV (extreme ultraviolet) photon pulses with energies between 15-70 eV and pulse length
shorter than 50 femtoseconds. Working with such short pulses allows the division to perform
time-resolved experiments with high temporal resolution, studying the properties of gases,
liquids and surfaces and to observe processes which would otherwise happen too quick to
observe.
One of the main components of HELIOS is its monochromator which selects a narrow band of
energies from the greater spectrum of generated XUV radiation, to be transmitted and used
for different experiments. To separate different energies the monochromator uses a diffraction
grating. It is well known that a pulse that is dispersed on a grating will be subject to a time
delay (increased pulselength). Another effect may also be introduced, refered to as a chirp.
A chirp means that the different constituent energies of the beam pulse will be divided and
ordered in a non homogeneous way, also resulting in a lowered peak power. To keep the pulse
length as short as possible the lab uses an off-plane grating to work around the time delaying
effect of the dispersion. Using an off-plane grating means that notably fewer grooves will be
illuminated by the beam and hence the dispersion and time delaying effects will be smaller.
It is not well known how this setup is going to affect a potential chirp, thus calling for the
need of this investigation.
The aim of this thesis has been to investigate how a pulse is affected when passing through the
monochromator in HELIOS, to see if it is subject to any spatial chirp. Since creating optimal
pulses is of great interest for the HELIOS lab, the purpose of this task is clear: to get more
knowledge about how the pulse is affected when propagating through the monochromator
may be of help in future experiments and development of the laboratory.
The background of this report is divided into two main parts, the first one (section 2) will
cover an overview of concepts that are central to the work done, while the second part (section
3) will give a short introduction to the HELIOS laboratory. In section 4 the method used
for the numerical investigation is explained and in section 5 the results will be presented and
discussed. Lastly, in section 6 and 7 an outlook for future work based on the results is given,
followed by conclusions on the thesis.
3
2
Background: Theory
In this chapter a review of concepts and techniques relevant for the thesis will be given.
The main focus will be on the principles of diffraction gratings, especially concerning the
blazed type. Apart from that, a general introduction about monochromators, high harmonic
generation (HHG), chirp and normal distribution is given. The review is only meant to serve
as an overview of the underlying concepts, if the reader feels a need to learn more details
about them he/she is encouraged to look into the references.
2.1
The diffraction grating
A diffraction grating is an optical dispersion device that splits incident light into different
directions, depending on the wavelength/energy of the incident light and the structure of
the grating. A grating can be created in many different ways but the basic principle is to
create some sort of periodic variation in it’s shape. This periodic variation in a grating will
periodically alter the incoming electromagnetic waves, causing them to interfere alternatingly destructively and constructively. This ends up creating a pattern of interference with
principal maxima and minima, these are called 0:th, ±1:st, ±2nd diffraction order and so on
(Fig. 1).[1]
Figure 1: Light interacting with a grating is diffracted into several beams, propagating in
different directions, referred to as different orders of diffraction
4
To describe the diffraction of a plane wave, the grating equation is used:
d(sin θm − sin θi ) = mλ
(1)
Where d is the period of variations in the grating (width of one groove), θm and θi is the angle
between the gratings normal and the exiting respectively incident beams, λ is the wavelength
of the incident light and m defines the diffraction order. It’s clear to see from this expression
that the zeroth order diffraction, m = 0, is not diffracted at all and independent of λ. [2]
2.1.1
Blazed reflection grating
As mentioned, there are many different types of gratings and they can work both by transmission or reflection. In the monochromator of HELIOS a reflection grating with a saw tooth
shape is used, a so-called blazed grating (Fig. 2). This is the most commonly used grating
today and it was first produced in 1910 by Robert Williams [2].
The profile of the saw tooths are governed by the blaze angle, δ. By choosing different blaze
angles the grating will maximize the reflection of different energies and as such one can choose
the angle that provides highest efficiency for the energy of interest. [3]
(a)
(b)
Figure 2: The blazed reflection grating with blaze angle δ and distance d between two grooves.
In 2b a ray of light is incident with angle θi and reflected with angl θr , with respect to the
surface normal
5
2.1.2
Specular reflection on blazed gratings
If you are not familiar with the word specular reflection, it simply refers to normal reflection
of light on a mirror-like surface. This is described by the law of reflection which states that
the angle of the incident light to the surface normal equals that of the reflected light to
the surface normal. When light is dispersed into several diffraction orders, the order that is
closest to specular reflection is always going to be the one with the highest intensity. For the
standard grating (Fig. 1) the zero order is always going to be the order that is specularly
reflected, i.e. θi = θm=0 . Since the zero order is actually undiffracted it is not of much interest
in diffraction experiments and preferably you would want the intensity on one of the higher
order diffractions. If one instead looks at the blazed grating, the zero:th order is occurring
when the incident angle with respect to the facet normal is equal to the reflected angle to
the facet normal, this means that θi 6= θr (Fig. 2b). Looking at the grating from afar (think
of it as a mirror since the grooves are so small) it is then clear to see that the zero order
diffraction is no longer going to be a specular reflection. In this way most of the photon
beam intensity can be shifted towards a higher diffraction order. With some conversion, the
law of reflection can be rewritten to apply for a blazed grating:
θi − θr = 2δ
(2)
Where δ is still the blaze angle while θi and θr are the angles the incident respectively exiting
beam makes with the grating surface normal (Fig. 2b). The order that gets specularly
reflected, which also is the order that will receive most of the beams intensity, is found to
when θi = θr . By inserting this into eq. 1 one can easily find which order, m this is for a
certain wavelength λ. [4]
2.1.3
Grazing angle
When dealing with XUV rays one will usually set up the experiment such that the incident
radiation hits the optical surfaces at grazing incidence, that is at an angle that is almost
parallel to the surface. The grazing angle is measured between the surface plane and the
incident radiation, in contrast to the usual praxis of measuring incident angle between the
beam and the surface normal. Assume a grating is set up as in figure 2a and 2b, with light
propagating from left to right, quasi perpendicularly to the grooves. If a beam is then incident
at grazing angle (Fig. 3b) it will be reflected over a significantly bigger area compared to
if it wasn’t (Fig. 3a). Illuminating the different optical components with grazing angle will
give higher reflectiviy which usually is to be preferred in a monochromator. However, there
is a downside to this, when shining light with a grazing angle on a grating this will make
the beam diffract over more grooves (since the area is bigger) which in turn will lengthen the
pulse.
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(b) Grazing incidence
(a) ”Normal” incidence
Figure 3: Light incident at a grazing angle, 3b, will be reflected by a bigger area compared to
3a
2.1.4
Off-plane setup
If one instead aligns the grating so that the incoming rays are quasi parallel to direction of
the grooves, the only thing that affects how many grooves are irradiated will be the diameter
of the beam. This setup is referred to as off-plane and is shown in figure (Fig. 4).
Figure 4: The off-plane grating, with altitude γ and azimuthal angles α respectively β.
The altitude, γ, is the angle that the incident light makes with the direction of the grooves.
Any ray incident with altitude γ will leave the surface with the same altitude. The angles
α and β are azimuthal angles, α being the azimuth of the incoming light and β that of the
diffracted light of order m and wavelength λ. The grating equation can be modified for an
off-plane grating to look like
dsinγ(sin α + sin β) = mλ
(3)
To achieve the highest efficiency from the off plane grating one wants the light to make a
specular reflection on the groove surfaces, that is α = β. Equation 2 then gives α = β = δ,
δ still being the blaze angle. This means that the highest efficiency will be achieved when
the grating is rotated around its off-axis in such a way that all the grooves appear as a plane
mirror for the incoming beam of photons. [5]
2d sin γ sin δ = mλ
7
(4)
2.2
Monochromator
The word monochromator stems from greek where "mono" means "single" and "chroma"
means "colour", and as the name suggests a monochromator is an optical device that idealy
selects a single energy (colour) from a wider spectrum of energies. In reality you can’t select
a single energy, but only a very narrow band of energies, centered around the energy you are
interested in. There are many different variants of monochromators, but the basic principle is
to use some sort of dispersive element (for example a prism or grating) to split the incoming
light into it’s consituent energies. The different energies will then be dispersed with varying
angles and by putting a slit after the dispersion only the energies that are dispersed onto the
slit will be transmitted (Fig. 5)
Figure 5: In this example of a simple monochromator, white light is dispersed and only the
energies corresponding to green light is transmitted through the slit. Which wavelengths that
hit the slit will depend on the angle of incidence and can be calculated with the diffraction
equation (eq. 1)
2.3
Chirp
Since the dispersion angle of a grating depends on the energy of incoming radiation, rays of
different energy may travel slightly different paths after being diffracted. This results in the
different energies becoming separated and arranged in a certain (non homogeneous) order,
so that for example lower energies would be in the front of the pulse and higher energies
in the back (Fig. 6). This effect is referred to as a chirp, and for continuous radiation or
longer pulses it will usually not have a big impact. But for shorter pulses, especially in the
femtosecond region where HELIOS operates, these effects may be very clear and result in a
reduced peak power. [1], [3]
8
In facilities where one wants to perform time resolved experiments with high temporal resolution, such as HELIOS, the chirp is unwanted. By using the off-plane setup the time delay
introduced by a grating can be greatly reduced since the light will be diffracged by a fewer
amount of grooves, as described in section 2.3.3. However it is not known how this setup is
going to effetc the pulse in terms of a chirp, hence the need for this investigation.
Figure 6: A pulse front containing a spectrum of homogeneously divided wavelengths is incident on a grating. The pulse is tilted and the different wavelengths will be structured into a
non-homogeneous order, a so called spatial chirp.
2.4
High harmonic generation
High harmonic generation (HHG) is a technique used to generate extreme ultraviolet (XUV)
radiation. It’s principle is based upon focusing a laser beam with high energy density into
a gas, which causes the gas to generate higher harmonics of the generating laser beam. The
generated XUV beams from this process will follow the same direction of propagation as the
generating laser but with a smaller divergence and shorter pulse length. [6]
Figure 7: a) The electric field tilts the Coulomb barrier, the electron escapes via tunneling
and gets accelerated by the electric field, b) the electric field changes direction and accelerates
the electron back towards the potential well, c) the electron is reabsorbed, emitting the gained
kinetic energy as a photon.
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The technique can be described as a three-step process; tunnel ionization, acceleration and
recombination. Assuming the electromagnetic field of the generating laser beam is strong
enough then it will bend the coulomb barrier of the gas molecules that is keeping the electrons
in their place. This will result in giving the electrons a certain probability to escape from
the barrier by tunneling, hence ionizing the molecule. When an electron escapes from its
molecule it will be accelerated away by the electric field of the laser, gaining momentum.
The field of the laser is rapidly alternating so the direction of the electric field will reverse
after half the period of the field, accelerating the electron back towards the molecule it just
escaped from. The electron may then recombine with the molecule and the energy it gained
as momentum when accelerated by the field is released as light. The photons released during
the recombination will then be higher harmonics of the generating beam, that is photons
with higher energy than the laser. The process will generate photons with energy up to a
certain cutoff energy. [1], [3]
2.5
Normal distribution
Also known as the Gaussian distribution, the normal distribution is a probability distribution of random numbers that is very common in nature. Many physical processes are well
described by a normal distribution. The idea is that the quantities of a normal distribution
will be equally distributed on each side of a mean value, that will ideally represent the value
that is most probable. Mathematically it can be described by the probability density:
f (x, µ, σ) = √
1
2πσ 2
e−
(x−µ)2
2σ 2
(5)
Where x is the variate (x-axis), µ is the mean value around which the quantities are distributed and σ is the standard deviation. The standard deviation is a measurement of how
much the values are deviating from the mean value. As can be seen in Fig. 8 about 64% of
the values will be within a distance of ±σ from the mean value. A common expression when
working with normal distributions is the full width at half maximum (FWHM). The FWHM
is defined as the width of the curve at half its maximum and calculated as:
√
F W HM = 2 2ln2σ ≈ 2.355σ
Figure 8: The normal distribution
10
(6)
3
Background: HELIOS
In following section a brief description of the HELIOS laboratory is given, with focus on the
structure of the monochromator. Below (Fig. 9) a schematic of the laboratory is shown.
Figure 9: 800 nm laser pulses (A) is focused in the argon gas chamber (B) to generate XUVradiation. The radiation then passes through the monochromator (C) and is diverted (D) into
one of the two end stations, the ultra-high vacuum (UHV) chamber (E) or the micro-focus
refocusing chamber (F)
To generate XUV radiation pulses, HELIOS uses the HHG technique described in section
2.1. It is achieved in this case by focusing laser pulses (A in Fig. 9) with a wavelenght of 800
nm into a gas cell containing Argon (B). When the laser hits the gas cell higher harmonic
photons will be generated with energies ranging between 15-70 eV, in the XUV-region. The
XUV photons will keep propagating in the same direction as the laser, but with a smaller
divergence.
After being focused into the gas cell, the beam is led into the monochromator chamber (C),
where it first passes through a 200 nm thick aluminium filter that blocks out the 800 nm light
from the generating laser but transmits the XUV. The XUV radiation is then reflected on
the first mirror (”PM1” in fig. 10) which has a parabolic shape and will collimate the beam so
that all rays travel more or less parallel to each other. The collimated beam is diffracted on
an off-plane blazed grating (”transl. stage” in Fig. 10) where different energies are diffracted
with slightly different angles, in a conical shape. Lastly, the diffracted beam hits another
parabolic mirror (”PM2” in Fig. 10) that serves to refocus the beam onto the exit slit. Since
the different energies of the beam is dispersed at the off-plane grating, the focus for different
energies will lie next to each other in a conical shape. A slit with variable size is placed at
the focus of PM2 and by rotating the grating one can control which energies are transmitted
through the slit.
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Figure 10: A picture of the monochromator in HELIOS. The XUV radiation enters from the
right, passing through the aluminium filter, becomes collimated by the first mirror (PM1),
diffracted at the transl. stage and finally refocused (PM2) onto the exit slit by PM2
After being monochromatized the beam will continue into the beam routing chamber (D)
where it will be reflected into one of the end stations. The beam routing chamber contains
two mirrors that are interchanged depending on which end station you want to use. A
parabolic mirror with focal lenght 600 mm is used to collimate the beam and lead it to the
micro-focus refocusing chamber. The 600 mm mirror can quickly be changed to one with
400 mm focal length that is placed at a distance of 550 mm from the slit, so the XUV beam
will not originate from the focus point of the mirror. This means that instead of collimating
the light, the 400 mm mirror will refocus the radiation at a distance of 1470 mm, inside the
ultra-high vacuum (UHV) chamber. The UHV chamber is used for surface science, and the
micro-focus refocusing chamber to study liquids, gases and clusters. Details about the end
stations are not of great relevance for this thesis and will be left out. [6]
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4
Methodology for numerical ray tracing
To numerically study how a pulse of light is affected when passing through HELIOS monochroc
mator, the ray tracing software Beam Four
was used. Beam Four is a free software (since
2015) distributed by Stellar Software. For more details about the program and how it’s
operated the reader is refered to their website. [7] The basic concept is to simulate an environment modelled after HELIOS monochromator and then irradiate it with a XUV pulse and
observe how it is affected when passing through the different optical elements. The pulse will
consist af a great number of rays and each ray will be numerically traced as it travels through
the simulated optical components. The procedure of ray tracing can hence be divided into
two parts, first simulating the relevant optical elements and second generating a source of
radiation that can be traced through the optics.
4.1
Optical elements
A simple picture of the optical elements that were added in Beam Four is show in Fig. 11.
Notice that this isn’t a proper schematic with all angles and distance properly scaled but
just meant to give a conceptual overview of the setup.
Figure 11: A scetch of the simulated elements. The figure is only meant to give a conceptual
view, the distances and angles are not proportional.
In Beam Four the monochromator was simulated by its four main parts, the first parabolic
mirror (M1 in Fig. 11), the grating (G), second parabolic mirror (M2) and the slit (S). M1 is
placed at a distance of 400 mm from the source in such a way that the focus of the beam and
the mirror coincide, making the reflected beams nearly parallel to each other. The mirror is
also tilted so that it makes a grazing angle with the incident beam of 3◦ . Next the grating
(G) is placed at a distance of 200 mm from M1 and tilted so that the beam and plane of the
grating makes a grazing angle of 3.5◦ . At this stage the pitch (rotation around z-axis) of the
grating must also be chosen so that the choosen energy/wavelength gets specularly reflected
and focused on the slit. To calculate the pitch for a certain photon energy, eq. 4 can be
rewritten as
mλ
δp = arcsin
(7)
2dsinγ
13
Where δp represents the pitch of the grating. Since all rays are parallel, and incident on an
off-plane grating, the altitude γ will equal the incident grazing angle, γ = 3.5◦ . For specular
reflection, the altitude (angle between the ray and the grating plane, in the direction of
propagation) of incoming rays will be the same for incoming and reflected rays. The azimuthal
angles α and β, on the other hand, can and will differ depending on the energy of the photons.
Subsequently the beams will hit the second mirror (M2) with a grazing angle of 3◦ but due
to the spread of azimuthal angle caused by the grating, different rays will hit the mirror at
different location and hence be focused to on the slit (S) at different locations, in the shape
of a cone. The slit is located 200 mm from the second mirror.
At the slit, an opening size can be defined to control how big a spectrum of wavelengths that
are allowed to pass through. 550 mm from the slit opening, (where the beam is focused and
then exits the monochromator) a third parabolic mirror is placed (M3). This mirror has a
focal length of 400 mm, which means that the source is not in the focus and hence instead
of being collimated it will be refocused, at a distance of about 1500 mm.
4.2
The source
For this thesis it was not of main interest how the XUV pulse is physically created, but rather
what happens to the radiation as it passes through the monochromator. To perform the ray
tracing it was of interest to start with a ray source that represents the radiation created in
the actual lab, but with some simplifications. The XUV pulse generated for the numerical
ray tracing consisted of rays distributed in a 2 dimensional Gaussian circular distribution, as
described in section 2.5. The divergence of the rays was choosen such that the whole grating
(G in Fig. 11) would be illuminated. For the energy of the rays a mean value and a FWHM
was chosen, then a normal distribution of energies was generated around those parameters.
Since the starting position of the rays will be distributed over this 2-dimensional circular
disk, the length of the pulse will be infintely short. In other words the generated pulse that
is used for the ray tracing will have a pulse length of zero when starting. This of course isn’t
achievable in reality but it should not affect the result, any chirp effect occurring should be
the same no matter what the pulse length. In addition with a zero pulse length it should
be easier to get a clear result, and also it will be very easy to see any changes in the pulse
length.
4.3
Handling of data
Each time a run is made in Beam Four a new group of rays will be generated (using the same
method) and then traced as they pass through the setup of optical elements. The software
will only handle around 1500 rays for each run, in order to get more accurate results a source
with much more than 1500 rays was required. To solve this the simulations were performed
many times, each time with a new separately generated source of rays. For every run the path
14
traveled to different objects was traced and the results were imported into excel documents
where all data could be combined to get a more reliable result.
c
The data was then processed with aid of Matlab
, where all the rays were divided into
different bins with different energy interval. For each bin the total path traveled for the
corresponding rays were plotted in a histogram, from which the average value and FWHM
for the corresponding energy intervals could be extracted. The average path for different
energies could then be compared to look for any patterns. The full code is attached in the
appendix.
5
Results & Discussion
The source of the XUV radiation was generated using a Gaussian distribution, both for the
position of the rays and for their corresponding energy values, as described in the previous
chapter. From estimating the width of the actual focus in HELIOS to be ∼ 0.3 mm the
starting position of the rays was generated with a FWHM of 0.2 mm to represent a similar
diameter. For the energy of the rays a mean of 40 eV (wavelength of 31 nm) and FWHM of
0.5 eV was choosen. Using eq. 7 the blaze angle of the grating could be calculated so that
the 31 nm wavelength would be focused on the slit, this resulted in setting the blaze angle
to δ = 5.828924◦ . Keeping in mind that such high precision is hard to achieve in actaul
experiments, it may still be used in theoretical work such as this. A slit size of 20 µm was
used and the groove width d = 2.5 · 10−5 m corresponding to a groove density of 4 · 105 m−1 ,
which of course is the same as used in HELIOS.
By studying the beam closely around the region of the final focus the focal point was estimated
to be at a distance of approximately 1.57m from the third mirror (M3). To determine the
focus of a realistic beam is always going involve some good judgement being applied, hence
the human error should be kept in mind. However, if the chosen focal point turns out to be
a little bit off, this shouldn’t have a big impact on the appearance of any possible spatial
chirp.
As previously mentioned, each time a simulation was run 1500 rays were traced at the same
time. A total of 33 runs each with separate generation of the XUV ray source was performed,
resulting in a total of 49500 rays for the final source. On the following pages (Fig. 12-21) are
presented the result of adding all these 49500 rays together to create one bigger pulse. The
distribution of total path traveled by rays, both to the slit and to the focal point is presented
for ten energy intervals of 0.05 eV centerad around the mean 40.00 eV. Normal distribution
curves are fitted to the data so that the mean value and standard deviation for each curve can
be found. Notice that the x-axes are not showing the total path, but how much it deviates
from 1 m and 3.1098 m for the position of the slit and focal point, respectively. The reason
for this is purely esthetic, since the distances are very close to the two mentioned ones it
would require long decimal numbers when writing them out on the axis. For each plot the
mean value, µ, and standard deviation, σ, is also displayed.
15
(b) µ = 2.204 · 10−5 m, σ = 1.329 · 10−5 m
(a) µ = −7.207 · 10−7 m, σ = 1.218 · 10−5 m
Figure 12: Rays within the energy interval 39.75 - 39.80 eV
(b) µ = 2.238 · 10−5 m, σ = 1.206 · 10−5 m
(a) µ = −8.213 · 10−7 m, σ = 1.142 · 10−5 m
Figure 13: Rays within the energy interval 39.80 - 39.85 eV
(b) µ = 2.193 · 10−5 m, σ = 1.255 · 10−5 m
(a) µ = −9.206 · 10−7 m, σ = 1.137 · 10−5 m
Figure 14: Rays within the energy interval 39.85 - 39.90 eV
16
(b) µ = 2.193 · 10−5 m, σ = 1.239 · 10−5 m
(a) µ = −1.118 · 10−6 m, σ = 1.22 · 10−5 m
Figure 15: Rays within the energy interval 39.90 - 39.95 eV
(b) µ = 2.143 · 10−5 m, σ = 1.212 · 10−5 m
(a) µ = −8.96 · 10−7 m, σ = 1.209 · 10−5 m
Figure 16: Rays within the energy interval 39.95-40.00 eV
(b) µ = 2.244 · 10−5 m, σ = 1.187 · 10−5 m
(a) µ = −9.747 · 10−7 m, σ = 1.167 · 10−5 m
Figure 17: Rays within the energy interval 40.00-40.05 eV
17
(b) µ = 2.218 · 10−5 m, σ = 1.195 · 10−5 m
(a) µ = −7.616 · 10−7 mσ = 1.233 · 10−5 m
Figure 18: Rays within the energy interval 40.05-40.10 eV
(b) µ = 2.172 · 10−5 m, σ = 1.212 · 10−5 m
(a) µ = −8.585 · 10−7 m, σ = 1.222 · 10−5 m
Figure 19: Rays within the energy interval 40.10-40.15 eV
(b) µ = 2.24 · 10−5 m, σ = 1.13 · 10−5 m
(a) µ = −6.411 · 10−7 m, σ = 1.204 · 10−5 m
Figure 20: Rays within the energy interval 40.15-40.20 eV
18
(b) µ = 2.175 · 10−5 m, σ = 1.196 · 10−5 m
(a) µ = −8.737 · 10−7 m, σ = 1.201 · 10−5 m
Figure 21: Rays within the energy interval 40.20-40.25 eV
The total path traveled to the slit (Fig. 12-21a) seems to follow a normal distribution, which
is to be expected since the rays were generated using the normal probability density (Eq. 5).
A normal distribution curve could be fitted to this data. However after the slit a majority
of the rays will be stopped and not so many will actually reach the focus. As a result the
data for total path traveled to focus (Fig. 12-21b) does not show the same clear normal
distribution and the curve fitted after these distributions are less reliable.
To investigate if there are any patterns of a chirp, the mean path to the slit for each of the
energy intervals was calculated and plotted (Fig. 22). No similar plot is presented for the
number of rays that reached the focus since they were too few to extract any reliable mean
from. If there is a chirp it should show as a dependence of energy to mean path, no clear such
dependence is vissible. One could speculate about a quadratic dependence with regards to
the energy if only looking at the plot, but it’s hard to think of a physical explanation for such
dependence. Either way more data would be required before any substantial assumptions
could be made. It is well possible that with more intervals a dependence would become
visible. However without generating more rays, dividing the current data into more intervals
would mean less rays per interval. This would decreases the reliability of the mean value
calculated from the fitted curves (Fig. 12-21). In order to see a possible pattern it would
hence be required to generate a lot more rays.
19
Figure 22: The mean path traveled to the slit for the different energy intervals, once again
as deviation from 1 m. The data was not deemed reliable enough to fit a curve, any pattern
could just as well be random.
Lastly two plots of the complete pulse that reaches the slit and the focus (Fig. 23 & 24) are
shown. By calculating their FWHM we get an estimate of how long the pulse is, and hence
how much it’s been prolonged in time when passing through the optical parts, since the start
pulse is of zero length. Using the data presented in Fig. 23 and 24 the FWHM was calculated
using eq. 6, yielding approximately 2.8·10−5 m at both the slit and at the focus. Inside the
lab the radiation can be asumed to travel at the speed of light in vacuum, c = 2.997924 · 108
ms−1 , and hence the length of the pulse, l, can easily be transformed into a time by dividing
the distance by speed of light t = cl .
Using the FWHM as pulse length in distance yields the pulse length in time ∆t ≈ 94 fs and
∆tf ocus = 9.497 · 10−14 s≈ 95 fs. The fact that the pulse length is more or less the same both
at the slit and at the focus is an indicator that at least parts of the simulations has worked
correctly. The pulse length is expected to be increased after being diffracted on the grating
and refocused on the slit. However after passing through the slit and being refocused to the
end stations the pulse length should more or less stay the same, as appears to be the case.
A pulse length of about 95 fs is deemed reasonable since the divergence of the beam is such
big that the whole grating is illumiated, which is seldom the case when running HELIOS in
reality.
20
Figure 23: The path distribution to the slit for the complete generated pulse of 49500 rays.
The fitted curve gives a mean value of µ = −8.229 · 10−7 m and a standard deviation of
σ = 1.196 · 10−5 m
Figure 24: The path distribution to focal point for the complete generated pulse of 49500
rays. The fitted curve gives a mean value of µ = 2.191 · 10−5 m and a standard deviation of
σ = 1.209 · 10−5 m
The aim of this thesis was to investigate how the pulse is affected when passing through
the monochromator and see if a chirp could be visible. From the results (Fig. 22) no clear
evidence for a spatial chirp could be found. It has been clear that a lot more rays should
21
be traced before any solid conclusions can be drawn. Comparing figures 12-21a with 12-21b
shows a clear difference in result depending on number of rays traced. To solve the problem
with having too few rays reaching the focal point the slit size should be made bigger to allow
more rays through.
Of course there is also the possibility that there is no chirp pattern to be found at all. No
ray trace looking for a chirp in an off-plane setup has to my knowledge been done before and
hence there are no data to compare with. The possibility that I’m looking for something
that isn’t there exists. My experience in working with ray tracing is also very limited and
working with Beam Four has been a new experience to me. Limited knowledge and lack of
time doesn’t add well together so the factor of human error should be kept in mind when
considering the work of this thesis.
6
Outlook
There is a lot more work that can be done based on what’s been covered so far in this thesis.
The first priority would be to generate sources with more rays, and for this another software
could be considered. Beam Four has several advantages but it proved to be lacking in ability
to trace huge numbers of rays.
A lot of continued work investigating the properties of the monochromator could be done
based on the optical elements that have been set up in the software. Things that could be
investigated but were left out of this thesis due to time restraint include looking into the
shape of the beam at its focal point, since learning how the beam focuses could be of help
to easier calibrate the lab when performing measurements. It would be of great interest to
look into how the focus would change if the hyperbolic focusing mirror in the beam routing
chamber was swapped for a toroidal mirror, could this increase the quality of the focus enough
to justify the cost and work of switching it?
7
Conclusions
Using the ray tracing software Beam Four a total of 49500 rays with energies normaly distributed around a mean of 40 eV with FWHM 0.5 eV were traced through the components of
HELIOSs monochromator. The data was treated in Matlab to try and find any pattern of dependence between the energy/wavelength of the rays and the total path they travel through
the optical components of the monochromator. No such pattern could be distinguished,
which would suggest that if there is a chirp introduced it is very small. To make more certain
conclusion about whether a chirp is introduced, and how big it is, a much greater number of
rays should be traced, for which another software could be considered.
22
8
References
References
[1] MALVIN CARL Teich and BEA Saleh. Fundamentals of photonics, 2nd. Canada, Wiley
Interscience, 2007.
[2] Eugene Hecht. Optics, 4th. International edition, Addison-Wesley, San Francisco, 2002.
[3] Stefan Plogmaker. Techniques and application of electron spectroscopy based on novel
x-ray sources. 2012.
[4] Thorlabs. Gratings tutorial, Blazed (Ruled) Gratings. [Online]. Available from: https:
//www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=26. [Accessed 18th may
2015].
[5] Luca Poletto, Fabio Frassetto, and Paolo Villoresi. Diffraction gratings for the selection
of ultrashort pulses in the extreme-ultraviolet. 2010.
[6] Stefan Plogmaker. Development of helios-a laboratory based on high-order harmonic
generation of xuv photons for time-resolved spectroscopy. 2012.
[7] Stellar
Software.
Stellar
Software.
[Online].
http://www.stellarsoftware.com/. [Accessed 18th may 2015].
8.1
Available
from:
Figures
Figure 4: Luca Poletto, Paolo Villoresi and Fabio Frassetto. Diffraction Gratings for the Selection of Ultrashort Pulses in the Extreme-Ultraviolet. Available from: http : //www.intechopen.com/books
in − solid − state − lasers − development − and − applications/dif f raction − gratings −
f or − the − selection − of − ultrashort − pulses − in − the − extreme − ultraviolet. [Accessed
27th May 2015]
Figure 7: Carsten Winter. High-order Harmonic Generation. Wilhelms-Universität. Available from http : //www.intechopen.com/books/advances − in − solid − state − lasers −
development − and − applications/dif f raction − gratings − f or − the − selection − of −
ultrashort − pulses − in − the − extreme − ultraviolet. [Accessed 27th May 2015].
Figure 8: Wikipedia user Ainali. Normal Distribution. Wikipedia. Available from: http :
//commons.wikimedia.org/wiki/F ile : Standardd eviationd iagram.svg. [Accessed 27th May
2015].
Figure 9: Helios laboratory, Division of Molecular and Condensed Matter Physics, Department of Physics and Astronomy, Uppsala University.
23
9
Appendix
The matlab code used to generate plots:
format long
path_to_focus=focusAG;
path_to_slit=slitAG;
wavelength=waveAG;
%find index for all the nonzero values in slit resp focus data
index_focus=[];
L=length(wavelength);
for i=1:L
if strcmp(’
’,path_to_focus(i))==0
index_focus=[index_focus, i];
end
end
index_slit=[];
for i=1:L
if strcmp(’
’,path_to_slit(i))==0
index_slit=[index_slit, i];
end
end
%extract the nonzero values, convert to numerical vectors
focus=path_to_focus(index_focus);
wave_focus=wavelength(index_focus);
focus=cellfun(@str2num, focus);
wave_focus=cellfun(@str2num, wave_focus);
slit=path_to_slit(index_slit);
wave_slit=wavelength(index_slit);
slit=cellfun(@str2num, slit);
wave_slit=cellfun(@str2num, wave_slit);
%convert wavelength to energy
energy_focus=1.2398419e-6./wave_focus;
energy_slit=1.2398419e-6./wave_slit;
%Place the rays into bins corresponding to 10 intervalls of 0.5eV centered
%at 40eV
Bin_focus=cell(10,1);
for j=1:10
Bin_focus{j}=find(energy_focus>(39.75+0.05*(j-1)) & energy_focus<(39.75+0.05*j));
24
end
Bin_slit=cell(10,1);
for k=1:10
Bin_slit{k}=find(energy_slit>(39.75+0.05*(k-1)) & energy_slit<(39.75+0.05*k));
end
numbins=40;
FWHM=zeros(10,1);
average=zeros(10,2);
b=slit-1;
c=focus-3.1098;
%Generate plots for the 10 bins of rays reaching the slit
for l=1:10
x=b(Bin_slit{l,1});
topedge=max(x);
botedge=min(x);
binedges = linspace(botedge, topedge, numbins+1);
[h, whichbin] = histc(x, binedges);
curvefit(binedges,h);
end
%Generate plots for the 10 bins of rays reaching the focus
for l=1:10
y=b(Bin_focus{l,1});
topedge=max(y);
botedge=min(y);
binedges = linspace(botedge, topedge, numbins+1);
[h, whichbin] = histc(y, binedges);
curvefit(binedges,h);
end
Where the function cruvefit(binedges,h) is given by:
function [fitresult, gof] = curvefit(binedges, h)
%CREATEFIT(BINEDGES,H)
% Create a fit.
%
% Data for ’Curve fit’ fit:
%
X Input : binedges
%
Y Output: h
% Output:
%
fitresult : a fit object representing the fit.
%
gof : structure with goodness-of fit info.
%
25
%
See also FIT, CFIT, SFIT.
%
Auto-generated by MATLAB on 23-May-2015 14:14:18
%% Fit: ’Curve fit’.
[xData, yData] = prepareCurveData( binedges, h);
% Set up fittype and options.
ft = fittype( ’gauss1’ );
opts = fitoptions( ’Method’, ’NonlinearLeastSquares’ );
opts.Display = ’Off’;
opts.Lower = [-Inf -Inf 0];
opts.StartPoint = [205 3.05228250016555e-07 5.21158403134744e-06];
% Fit model to data.
[fitresult, gof] = fit( xData, yData, ft, opts)
% Plot fit with data.
figure( ’Name’, ’Curve fit’ );
h_1 = plot(fitresult, binedges,h, ’x’);
set(gca,’FontSize’,18,’FontWeight’,’bold’)
%axis([-3*10^-5 3*10^-5 0 400]);
legend( h_1,’Data’, ’Curve fit’, ’Location’, ’NorthEast’ );
% Label axes
xlabel(’Path traveled to slit, as deviation from 1m [m]’)
ylabel (’Number of rays’)
grid on
26