Aby Joseph_Report 9 July 12.doc

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Project Report

Transmission characteristics of Al and Si filters in

soft x-ray/ VUV region

Summer Internship (15th May 15th July, 2012)

Submitted by

Aby Joseph

Undergraduate Student (3rd year)

B.Tech. Engineering Physics

IIT Delhi

Under the guidance of

Dr. M.H. Modi

X-ray Optics Section, Indus Synchrotrons Utilization Division

Raja Ramanna Centre for Advanced Technology

Indore - 452013

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Contents

1. Introduction 3

1.1. Synchrotron radiation 4

1.1.1.a.) Bending magnet radiation

b.) Undulators and wigglers

1.2. Advantages of synchrotron radiation sources 6

1.3. Indus-1 synchrotron radiation source 6

1.4. Optical properties x-rays 6

2. Experimental details 8

2.1. Reflectivity beamline at Indus-1

3. Methodology: Reflectivity and Transmission through Thin Films 9

3.1. Simulation results 12

3.1.1. Case I Transmission vs. angle of incidence 12

3.1.2.Case II Transmission vs. wavelength 13

4. Characterization of Si and Al transmission filters 15

4.1. Data analysis 15

4.1.1. Contribution of higher harmonics 16

4.1.2. Silicon filter 17

4.1.3. Aluminum filter 20

4.2. Summary 21

Appendix 22

References 23

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1. Introduction

Thin foils of different materials are used in x-ray region as an absorber and edge filters.

Absorbers of different thicknesses are primarily used to reduce the incident intensity as perthe detector capacity. Edge filters are used to selectively absorb high energy spectrum

coming from wide energy sources such as synschrotrons, laboratory based x-ray generator etc.

Aluminium is a very important filter material. It has a wide bandpass (170 800 ) and is

very good for visible light rejection. This filter provides visible light rejection in the range of

10-9. This filter is more durable and easier to process than many other filter materials.

Aluminium filters are used to obtain high resolution pictures of the sun in the soft x-ray

region of the electromagnetic spectrum. [1]

Thus Iit is important to characterize the performance of this filmfoils, especially near the

absorption edges (170 ). The primary aim of this project was to do data analysisanalyseon

the experimental transmission curve of Al and Si obtained from a wavelength scan of the

aluminium filters (1500 thick), under condition of normal incidence of in the soft x-ray

radiationregion (xxx- yyy). For the simulations, the important filter the parameters are

initially considered as: 1500 foil thickness, standard optical constants and zero roughness.

Based on the modifications required in the initial assumptions regarding the different

parameters of the filter we expected to characterize the filter and give possible reasons why

certain parameters (eg. thickness distribution between filter material and oxide layer) were

different from the expected values.

A thorough understanding of the methodology involved in calculating the reflectivity and

transmission through thin films was required before setting out on the data analysing sof the

transmission experimental data of filters. A detailed study into the basic equations leading up

toofthe Parratt formalism [2] and a its mathematical subsequent derivation of the same was

done undertaken. The simulation code written in MATLAB was used to understand the

behaviour of the reflectivity and transmission of thin films as a function of angle and

wavelength, their dependence on thickness of layer and roughness and the effect of oxide

layer on a transmission filter. Such studies have already been done before and are well

known, but were required nonetheless for a deeper understanding of the basic phenomena

involved.

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The project work is carried out on reflectivity beamline at Indus-1 synchrotron source. The

This report presnt work is divided into four parts: The first part covers basic aspects of

synchrotron radiation and the synchrotron radiation source at RRCAT Indus-1 NDUS 1. It

then talks of the usefulness of soft x-rays. The second part covers the details of the

experimental work. related to this project. Basic aspects of the beamline and other relevant

details regarding of transmission measurement of transmission through filters are discussed.

The third part discusses the methodology used in calculating the reflectivity and transmission

through a multilayer thin film. The basic equations leading up to the Parratt formalism [2] are

discussed and a complete derivation of the same is given in the Appendix. The fourth part

describes the data analysis on the experimental results obtained of transmission through

measurements of Si and Al filters . It describes the steps involved in matching the simulated

curve with the experimental one and including the interpretations of the final results.

1.1 Synchrotron Radiation

Synchrotron radiation refers to radiation emitted by charged particles travelling at relativistic

speeds in an applied magnetic field (which keeps them along curved paths). From a historical

perspective, synchrotron radiation was first observed from storage rings of electrons or

positive ions. These storage rings were circular and radiation was emitted due to the curved

path of the electrons. Modern synchrotrons are specifically designed to produce synchrotron

radiation with high brilliance. Third generation synchrotrons have a polygon-like structure

with straight sections and rounded corners. The straight sections contain periodic magnetic

structures (undulators and wigglers) which confine the electron beam to move in a sinusoidal

path with an effective propagation in a straight line (i.e. the small periodic displacements of

the electrons are perpendicular to the direction of propagation of the beam). The amplitude of

the oscillations depends on the strength of the magnetic field. The rounded corners contain

bending magnets which force the electrons to travel in a curved path, which again leads to

emission of radiation.

Three types of magnetic structures are used to produce synchrotron radiation: bending

magnets, undulators and wigglers.

1.1.1a Bending Magnet Radiation

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Accelerated charged particles emit electromagnetic radiation. Consider a charged particle

moving in a curved trajectory at relativistic speed. In the moving frame of reference of the

electron, the radiation emitted is isotropic (uniform in all directions). However, in the

laboratory frame of reference, the beam of radiation is highly collimated. This can be seen by

solving electron equation of motion in moving frame of reference of electron and Lorentz

transforming the results to the laboratory frame of reference.

The radiation emitted in the range -/4 < ` < /4 is beamed in the direction of motion of the

electron in the range -1/ < < 1/

Figure 1.1: The two pictures show the distribution of emitted radiation form an accelerated charged particle

(relativistic) in the frames S and S` respectively (images taken from [14]).

In synchrotrons, the relativistic speed of the charged particles will change the observed

frequency of radiation due to the Doppler effect by the Lorentz factor

1.1.1b Undulators and Wigglers

Undulators are periodic magnetic structures with relatively weak magnetic fields. The

periodicity causes the electron to experience harmonic oscillations resulting in small angular

excursions call undulations. The amplitude of these undulations is small and the resultant

radiation cone is narrow. The angular half width of the radiation cone is , where N is the

number of magnetic periods. Thus we have tightly confined electron beam with radiation of

small angular divergence and relatively low spectral width.

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Wigglers are different from undualtors in that they have stronger magnetic fields. The

oscillation amplitude is thus larger and the radiation cone is broader both in space and angle.

The radiation spectrum is similar to that of bending magnets, but characterized by larger

photon flux and shift to harder x-rays.

1.2 Advantages Properties of Synchrotron Radiation Sources

In earlier radiation sources (like x-ray tubes), the emission lines were not continuously

tunable and the x-rays were emitted in all directions. Special optics was needed to collect the

x-rays and redirect them into roughly collimated beam.

Synchrotron Radiation sources provide several advantages over other sources. High intensity

photon beam (high flux) is available and allows rapid experiments or use of weakly scattering

crystals. A broad spectrum of radiation with wide tunability (from microwaves to hard x-rays)

is available. We get a highly collimated beam with small angular divergence (high brilliance).

Both linear and circularly polarized light are available. Pulsed light emission with pulse

durations as low as tens of picoseconds allows resolution of processes on the same time scale.

1.2.13 Indus-1 Synchrotron Radiation Source

Indus-1 is a 450 MeV electron storage ring and is the first Indian synchrotron machine

(commissioned in early 1999). It provides a broad electromagnetic spectrum extending from

far infrared to soft x-ray region with a critical wavelength of 61 . It has made possible

research using the soft x-ray/vacuum ultraviolet radiation (10 < < 1000 ). Since the

wavelengths considered are relatively shorter, one can explore physical structure of materials

with much better resolution than that offered by visible light. Also, this wavelength range

covers many absorption thresholds of various elements. [5]

1.34 Optical properties X-rays

In x-ray region, The refractive index of all materials for wavelengths in the x-ray region is

very close to unity. Therefore,iIt is difficult to bend x-rays by using transmission optics as is

done in the case of visible light. Also, x-rays show high absorption which limits the efficient

use of transmission lenses.

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Because the refractive index is less than unity, it is possible to reflect the radiation completely

from the surface of a material below a certain critical angle of incidence known as critical

angle (with respect to the surface) (Snells law).

The value of critical angle increases with wavelength ( ) and it lies in the

range of a few degrees for soft x-rays (10 A < < 300 A). Above the critical angle, the

reflectivity drops rapidly and the fall is proportional to where q . Soft x-rays

are important due to the numerous atomic resonances of different materials lying in this

region, which makes it possible to exploit these wavelengths for elemental identification.

Important filters like the aluminium and silicon filter can also be characterized using this

range of wavelengths, which includes the absorption edges of both. [5]

In the x-ray region, the refractive index is close to unity and is defined as ,

where (delta) is the deviation of real part from unity, and imaginary part (beta) is

absorption index. The terms delta and beta are referred to as optical constants. In soft x-ray

region their typical values lie in the range ~10-2 10-3. The inner atomic electrons of atoms

contribute to the absorption of x-rays which gives a series of jumps in the absorption index at

energies given by the binding energies of these electrons.

2. Experimental Details

2.1 Reflectivity beamline at Indus-1

The reflectivity beamline comprised of pre and post focusing toroidal mirrors and a toroidal

grating monochromator. The beamline provides monochromatic radiation in 40-1000

wavelength region with high flux and moderate spectral resolution. The reflectivity beamline

in soft x-ray and vacuum ultraviolet region is installed on one of the bending magnets of

Indus-1. This beamline is used for characterization of optical elements (mirrors, gratings, thin

films, multilayers, detectors, absorption edge filters etc.) in the soft x-ray/ vacuum ultra violet

region in the wavelength range 40-1000 . The beamline optics consists of a toroidal pre-

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mirror, a toroidal grating monochromator, a toroidal post-mirror for focussing the beam on

the sample and a high precision reflectometer[6].

Monochromatic radiation from the TGM is comprised of higher harmonicsHigher diffraction

order content in a grazing incidence monochromator is generally quite high, especially for

low energy gratings. A filter wheel mechanism has been incorporated in the beamline just

after the exit slit of the monochromator to introduce an appropriate transmission filter in the

SR beam path. Various filters such as Al, Si, B Indium grown on Ni mesh of 87%

transmission Two filters (Al and Si) areare used to suppress the higher harmonic contents in

the beamline. [6]in the form of ~1000 1500 thick metal foils mounted on an 87%

transmitting Ni mesh [6].

For this project data from a wavelength scan on transmission filters in the above mentioned

beamline was used. The transmission through Al and Si filters was measured in the

wavelength range 100 360 . This range includes the absorption edge of both materials

(124 and 170 for Si and Al respectively). The step size taken to do the wavelength scan

on the filter () was 0.3 .

3. Methodology: Reflectivity and Transmission through Thin

Films

The aim of the first set of simulations was to plot the reflectivity and transmission of

multilayer films using the recursion formalism given by Parratt [2]. The first part of this

report describes the basic equations leading up to the final expressions of the recursion

formalism of Parratt. Next, the results of the simulations for 1.) Transmission vs. Angle and

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2.) Transmission vs. Wavelength are shown for different cases; and the inferences which can

be drawn from these graphs are discussed along with the results.

The starting point is the set of Fresnel formulas for refraction and transmission coefficients at

the boundary between two materials of refractive indices n1 and n2. The electromagnetic wave

is incident at an angle 1 (from normal) with an angle of refraction 2. The final expressions

used for the simulations consider only s-polarization.

(3.1)

(3.2)

Figure 3.1: Schematic of multilayer film showing reflected and transmitted amplitudes just above each interface.

Tn and Rn are the amplitudes of the electric field just above the nth interface.

The recursion relation of Parratt [2] is given by:

(3.3)

where,

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(z-component of wavevector in layer n)

is the angle of incidence (w.r.t surface) (air to layer 1)

z is the z-coordinate (i.e. zn is the cumulative sum of all layer thicknesses upto the n th

interface)

There are two approaches to calculate the reflectivity or transmission of N layers: a matrix

method and a recursion method. The literature like the classic paper of Parratt [2] and M.

Tolans book X-ray Scattering from Soft-Matter Thin Films [7] give only the final

expressions for the amplitude of the fields at each interface (after considering multiple

reflection and transmission from each interface). The book by Spiller [8] gives the starting

equations relating the amplitude of forward-running waves and backward-running waves

of the nth interface with that of the n-1th interface. Thus a recursion formalism can be derived

to get the expression for transmitted field at the bottom of the last interface. We have

attempted to derive the final expressions:

The basic equations relating the amplitude of the fields at the bottom two interfaces are given

below:

since there is no backscattered wave (from air medium) at the N+1th interface.

(3.4)

(3.5)

(3.6)

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For we consider the contribution of the wave which propagates upto the Nth

interface, gets reflected, and travels the same distance back. The phase term is therefore

instead of as given in Spiller [8]. (where,

and dN is the thickness of the nth layer).

(3.7)

(3.8)

The complete derivation is given in the Appendix.

Roughness

For considering the effect of roughness of the interfaces the reflection and transmission

coefficients at each interface are modified by multiplying by an exponential factor. The

literature [7] gives two sets of modification factors depending on the case considered. The set

of expressions which is commonly used is given below:

(3.9)

(3.10)

However, Tolan [7] states that while the above set of expressions is valid in the case of

reflectivity measurements, it gives . The reason is an approximation used

while averaging the phase factors in Eqs. 2.19 and 2.20 in [7]. A more rigorous derivation

leads to the following expressions:

(3.11)

(3.12)

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As can be seen, the transmission coefficient is now modified and . However,

it was observed that in the case of multilayer films, increasing the roughness () of certain

interfaces decreased the transmission while doing the same for certain other interfaces

increased the overall transmission. We believe that this is due to the interference of the

reflected waves from the different interfaces of the multilayer film. More systematic studies

will be done on this later.

In the simulation code, Eqs. (3.9) and (3.10) were used because it led to the intuitive

behaviour of a decrease in the overall transmission on increasing the roughness parameter ().

3.1 Simulation Results

Simulations of Fresnel reflection and transmission through multilayer films have been carried

out for various cases. Case-I looks at how the transmission (of soft x-rays) through thin film

depends on thickness of the film (transmission vs. angle). Case II helps us understand the

behaviour of transmission through thin film near the absorption edge of the material, through

a transmission vs. wavelength curve.

Simulations for the cases discussed above give a greater insight into the basic phenomena

involved in reflection and transmission through thin films. Varying certain parameters and

comparing the corresponding curves helps us obtain certain relationships and more

importantly helps in understanding the limited range of values for which these relationships

are valid. Also, information on certain parameters for performing experiments can be obtained

from such simulations (for e.g.: required angular resolution in an angle-scan). Detailedinferences based on these simulation results are discussed along with the same.

3.1.1 Case I Transmission vs. Angle of incidence

Pure Si film in airWavelength 200 , Thickness of film varied

Effect of thickness of film on transmission

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In Figure 3.2, the wavelength is fixed and the thickness of the film is varied. A thicker film

will have higher absorption and we see how the curve is lower for higher thicknesses. The

critical angle is given

approximately by . In

graphs of transmission vs.

angle, the transmission is

very low below the critical

angle and increases suddenly

at the critical angle and then

saturates towards some

value as the case of normalincidence is approached. As

shown in Figure 3.2, for

higher thicknesses the graph

is as expected. However, at lower thicknesses (eg. 100 ) the characteristic behaviour near

critical angle is not very apparent. A possible explanation is that at lower thicknesses there is

less absorption, and the field penetration is much more.

3.1.2 Case II Transmission vs. Wavelength

The experimental apparatus at Indus-I is such that it can give a wavelength-scan at a fixed

angle (normal incidence) for measurements of transmission of multilayer films. Two common

filter materials used in the lab are silicon and aluminium. For simulations, the range of

wavelengths taken was such that it included the absorption edge of both these materials (124

and 170 respectively).

13

0 10 20 30 40 50 60 70 800.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Transmission

Angle of incidence (degrees)

100 250 400 700 1000

Thickness of Si film

Figure. 3.2: Transmission curve for wavelength=200 .The thickness of silicon film is varied.

Silicon (case I) and Aluminium (case II)

(both with oxide layers on both sides)Wavelength range 50 200 Thicknesses Si film: 500

SiO2 layer: 50 on both sides


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As can be observed from Figures 3.8 3.13 the behaviour of the transmission curve directly

corresponds with the behaviour of the beta vs. wavelength curve (for silicon). Thus the

transmission is strongly dependant on the absorption (characterized by beta). These

simulations to study transmission behaviour of a filter helped in the data analysis of

wavelength scan of Al and Si filter which is discussed in the next section.

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120 160 200 240 280 320 360

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Transmission

Wavelength ()

515306090

Angle (w.r.t surface)

SiO2 (50 ) - Si (500 ) - SiO2 (50 )

Figure 3.8: Transmission vs. Wavelength for Si

120 160 200 240 280 320 360

-0.02

0.00

0.02

0.04

0.06

0.080.10

0.12

Delta/Beta

Wavelength ()

Delta (deviation from real part of refractive index)Beta (imaginary part of refractive index)

Figure 3.9: Optical constants for Si

120 160 200 240 280 320 360

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Figure 3.10: Optical constants for SiO2


Delta/Beta

Wavelength ()

120 160 200 240 280 320 360

0.0

0.1

0.2

0.3

0.40.5

0.6

0.7

0.8

Figure 3.11: Transmission vs. Wavelength for Al

Transmission

Wavelength ()

515306090

Angle (w.r.t surface)

Al2O3 (50 ) - Al (500 ) - Al2O3 (50 )

120 160 200 240 280 320 360

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

Figure 3.12: Optical constants for Al

Delta (deviation from real part of refractive index)

Beta (imaginary part of refractive index)

Delta/Beta

Wavelength ()

120 160 200 240 280 320 360

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

Fi ure 3.13: O tical constants for Al2O3


Delta/Beta

Wavelength ()


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4. Characterization of Si and Al transmission filters

As mentioned in Section 2.1, we obtained an experimental curve of transmission vs.

wavelength for Si (1200) and aluminium (1500) filter, for normal incidence angle light.The two films were deposited on a Ni mesh (87% transmission). The thickness of the

respective oxide layers on the two films is unknown. The range of wavelengths includes the

absorption edges of both Si and Al (124 and 170 respectively). The step size () taken

was 0.3 . Further information regarding the beamline is given in Section 2.1.

Cartoon

Our aim was to match simulation results considering similar structure with the experimentalcurve and try and explain the various variations. This would give us a better understanding of

the basic phenomena involved and a deeper insight into the true composition of the filter.

4.1 Data Analysis

For the simulations, we considered a film (Si and Al) of particular thickness surrounded by

oxide layer on both sides (SiO2 and Al2O3 respectively). The simulated curve was obtainedconsidering the thickness of the

film as given by the manufacturer

(1200 for Si and 1500 for Al

filter) and standard optical

constants (obtained from CXRO

website [13]). There was a large

discrepancy between the simulatedand experimental curves. Because

of the huge difference in amplitude

between the two curves (Figure

4.1), we suspected that the

thicknesses of the respective layers were very different from that initially assumed (SiO2 (0

) Si (1200 ) SiO2 (0 )). Also we did not consider the contribution of higher

harmonics. The simulation curve obtained from these preliminary considerations is given inFigure 4.1.

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120 140 160 180 200 220 240 260 280 300 320 340 360

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

SiO2 (0 ) - Si (1200 ) - SiO2 (0 )

Transmission

Wavelength ()

simulatedexperimental

Figure 4.1: Simulated transmission curve withoutconsidering the contribution of higher harmonics


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4.1.1 Contribution of Higher Harmonics

The absorption edge of Si is at 124 . Because of a small apparent edge at around 250

(roughly twice 124 ), we suspected a significant contribution of higher harmonics in the

experimental curve. Data of the fractional composition of the first three harmonics as a

function of wavelength (beamline specific) was taken from [11]. The data was interpolated in

the required range of wavelengths (115 360 ) with the required number of data points

(~500).

Higher orders of diffraction from the monochromator lead to contribution of higher

harmonics. Thus the transmission at a single wavelength actually consists of the

transmission at the wavelengths , /2, /3 and so on, with different fractional weightages

for each harmonic. The fractional composition of these higher harmonic wavelengths depends

on several factors, including the grating diffraction efficiency, detector responsivity, mirror

reflectivity etc. The fractional composition of higher harmonics was calculated from the SXR

analysis in different wavelengths coming from the TGM of the reflectivity beamline [11].

For example, in the case of silicon the transmission curves were calculated for the ranges: (I)

105 360 , (II) 52.5 180 and (III) 35 120 corresponding to the 1 st, 2nd and 3rd

harmonics respectively. The

three transmission curves were

then added after multiplying

with the corresponding

weighting factors (given as a

function of wavelength). The

final curve obtained is shown in

Figure 4.2.

As we can see from Figure 4.2,

after considering the

contribution of higher harmonics

the overall shape of the simulated transmission curve matches with that of the experimental

one. We now need to take care of the difference in amplitude between the simulated andexperimental curves.

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120 140 160 180 200 220 240 260 280 300 320 340 360

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Figure 4.2: Simulated transmission curve after consideringthe contribution of higher harmonics

SiO2 (0 ) - Si (1200 ) - SiO2 (0 )

Transmission

Wavelength ()



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4.1.2 Silicon filter

For matching the simulation data with the experimental one, various parameters of the filterwere changed in a step by step manner:

Thickness

Thickness of film and oxide layer:- It was observed that, in the case of Si, increasing the Si

film thickness had a significant

effect on the vertical extent of the

transmission curve near the

absorption edge. Whereas

increasing the oxide layer thickness

brought the transmission curve

down but had little effect on the

vertical extent of the same. (Figure

4.3).

The thickness distribution (between

oxide and film layers) was changed

keeping the total thickness of the filter around 1200 (for Si filter). It was observed that to

match the simulation with experiment, considerably thick oxide layers had to be incorporated.

The total thickness considered is around 1140 (Table 4.1 and Figure 4.4).

Optical constants

Imaginary part of refractive index (beta):- Proportional increases in the beta and thickness of

any layer led to an almost similar effect on the transmission curve. An increase in either of the

two means an increase in the total absorption (where is the imaginary part of

refractive index and x is the thickness of layer). The logarithm of total absorption is linearly

proportional to both beta and thickness of layer. However, a change in either of these two

parameters led to slightly different effects on the features on the transmission curve. These

features are predominantly due to the interference between the reflected waves from multipleinterfaces. Thus a change in the thickness of a layer would have a significant impact on the

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120 140 160 180 200 220 240 260 280 300 320 340 3600.0

0.1

0.2

0.3

0.4

0.5

0.60.7

0.8

0.9

1.0

Transmission

Wavelength

Figure 4.3: Effect of considering 100 oxide layeron both sides of 1200 Si film

100A oxide both side

116 118 120 122 124 126 128 130 132 1340.0

0.2

0.4

0.6

0.8

B

A


18/24

wavy features seen on the transmission curve; and the effect of a change in beta would be

somewhat different in this respect.

Deviation from real part of refractive index (delta):- In the case of Si (with thicknesses as

given in Table 4.1), it was observed that changing the delta values (even up to three times the

standard values) had very little effect on the transmission curve.

However, during preliminary investigations (when the thickness of the pure Si layer was

much larger ~1200 ) it was observed that increasing the delta by a significant amount (2-3

times original value) brought about wavy features onto the transmission curve. Since the

experimental curve did not show such features it was considered appropriate to smoothen

out the same in the simulated curves by increasing the roughness of the layers. However, as

mentioned previously, the thickness distribution was greatly modified (Table 4.1 and 4.2)

from that initially assumed. It was not seen necessary to make the above mentioned changes

to the delta values in this case

A brief digression regarding the quantification of roughness of interfaces in the case of

transmission is given below:

Roughness

The roughness of the interfaces is quantified by multiplying the reflection and transmission

coefficients at each interface by an exponential factor. While the factor to be used in the case

of reflectivity calculations is well known (Nevot Croce factor), it is not very clear which

factor to use in the case of transmission calculations. The literature [7] gives two sets of

expressions for quantifying roughness in the case of transmission calculations. More detailed

discussion on which expression was finally used in the above simulations is given in Section

3.

Effect of Ni mesh

The manufacturers of the filters (Luxel Corporation [12] ) have mentioned on their website

that the film was deposited on a Nickel mesh (of 87% transmission [6]). This Ni mesh is used

so that an aluminium film of larger size can be supported. The transmission through this film

increases with increase in size of pores of mesh. But a mesh with large pore size is difficult tomanufacture. It was seen that including the effect of mesh in the simulations (i.e. considering

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an overall decrease in the transmission by the given factor {.87}) led to better matching of

simulation and experimental curves in the case of both Si and Al.

Silicon filter (finalized parameters)

Thicknesses SiO2 (330) Si (480 ) SiO2 (330 )

Optical constants Standard values (unchanged)

Roughness (four interfaces) 5 10 10 5

Table 4.1: Finalized simulation parameters for Silicon filter

120 160 200 240 280 320 3600.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

Transmission

Wavelength ()


SiO2 (330 ) - Si (480 ) - SiO2 (330 )

Figure 4.4: Finalized simulated transmission curve matched with that of experiment for Silicon filter

4.1.3 Aluminium filter

The behaviour of the simulated transmission curve for the Aluminium filter was similar tothat of the Silicon filter with respect to changes in the several parameters discussed

previously. Certain differences between the two filters are given below:

Thickness

Thickness of film and oxide layer: - In the case of Aluminium, it was observed that the effect

of increasing the oxide layer thickness was different from that of Si. In this case, an increase

in the oxide layer thickness not only brought the transmission curve down, it also significantlyreduced the vertical extent of the curve (unlike the effect observed in case of Si).

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Finalizing the thickness distribution:- Due to the behaviour mentioned above fixing the

thicknesses of the film and oxide layer for Aluminium filter was a more dynamic process than

that of Silicon filter. The best possible combination of thicknesses is believed to be that given

in Table 4.2.

Aluminium filter (finalized parameters)

Thicknesses Al2O3 (70 ) Al (1450 ) Al2O3 (70 )

Optical constants Standard values (unchanged)

Roughness (four interfaces) 5 10 10 5

Table 4.2: Finalized simulation parameters for Aluminium filter

120 160 200 240 280 320 360-0.05

0.00

0.05

0.100.15

0.20

0.25

0.30

0.35

0.40

0.45

Transmission

Wavelength ()


Al2O3 (70 ) - Al (1450 ) - Al2O3 (70 )

Figure 4.5: Finalized simulated transmission curve matched with that of experiment for Aluminium

filter

As can be seen in the transmission curves of Figure 4.3 and 4.4, there is a small horizontal

shift (2.0 for Si and 2.7 for Al) between the simulated and experimental curve. This shift

is particularly apparent near the absorption edge. This small shift is observed in experimental

curves due to a small inevitable error in the calibration.

4.2 Summary

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The simulated and experimental curves match reasonably well for the range of wavelengths

from the absorption edge of the respective materials (Si 124 and Al 170 ) to about

200-220 for the two filters with parameters as given in Tables 4.1 and 4.2.

It was observed that there is significant contribution of higher harmonic content in the

experimental results of transmission though filters. Thus this factor was included in the

simulations. It was observed that there is significant amount of oxide layer in the case of

Silicon filter (330 on both sides), and a relatively less amount in the case of Aluminium

filter (70 on both sides). The total thickness, however, remained approximately same as

given by the manufacturer (1140 and 1590 for Si and Al filters respectively).

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Appendix

Figure 5.1: Schematic of multilayer film showing reflected and transmitted amplitudes just above each interface.

since there is no backscattered wave (from air medium) at the N+1th interface.

(5.1)

(5.2)

(5.3)

(5.4)

(5.5)

(5.6)

(5.7)

(5.8)

where, dN is the thickness of the nth layer.

Dividing Eq. (5.6) by ,

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(5.9)

where, and dN is the thickness of the nth layer).

However, the Parratt formalism [2] gives the following expression for :

(5.10)

where, and is the z-coordinate of the nth interface

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The two expressions (5.9 and 5.10) are equivalent and give the same results for transmission

calculations. The difference between the two is that in one the individual thicknesses of the

layers are considered, while in the other the cumulative sum of the layers upto the n th interface

is considered (i.e. z-coordinate). Both expressions were used to run simulations of

transmission through multilayer films and the results obtained were the same.

References

1. Forbes R. Powell, Optical Engineering, Vol. 29 No. 6, 614-624 (1990).

2. L.G. Parratt, Phys. Rev. 95, 359 (1954)

3. David Attwood, Soft X-ray and Extreme Ultraviolet Radiation, Cambridge

University Press.

4. Jens Als-Nielson, Elements of Modern X-ray Physics, Wiley (2nd edition, 2011)

5. Mohammed H.Modi, Surfaces and Interface Studies in X-ray Mirrors, PhD Thesis,

RRCAT Indore, India (2003)

6. R.V. Nandedkar, Current Science, Vol. 82, No. 3, 298-304 (2002)

7. M. Tolan, X-Ray Scattering from Soft-Matter Thin Films, Springer (1999)

8. E. Spiller, Soft X-Ray Optics, SPIE Optical Engineering Press (1994)

9. D. K. G. de Boer, Phys. Rev. B, Vol. 44 number 2, 498 (1991)

10. V. Holy, X-Ray Scattering from Thin Films, Springer Tracts in Modern Physics, Vol.

149, Springer (1998)

11. Mohammed H. Modi, Applied Optics, Vol. 51, No. 16, 3552-3557 (2012)

12. http://www.luxel.com

13. http://www.cxro.lbl.gov/

14. http://asd.gsfc.nasa.gov/Volker.Beckmann/school/download/Longair_Radiation2.pdf

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http://www.luxel.com/http://www.cxro.lbl.gov/http://asd.gsfc.nasa.gov/Volker.Beckmann/school/download/Longair_Radiation2.pdfhttp://www.luxel.com/http://www.cxro.lbl.gov/http://asd.gsfc.nasa.gov/Volker.Beckmann/school/download/Longair_Radiation2.pdf

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