MSc Physics
Advanced Matter and Energy Physics
Master Thesis
Shadowgraphy of Metallic Micro-Droplets in
Plasma Sources of Extreme Ultraviolet Light
By:
Jim Cornelis Visschers
10452591
June 2017 60 ECTs from 01-07-2016 until 30-06-2017Supervisor:
dr. O.O. Versolato
Examiners:
prof. dr. W. Ubachs
prof. dr. P.C.M. Planken
June 26, 2017
Contents
1 Introduction 2
2 Theory 3
2.1 Imaging . . . 3
2.1.1 Image Resolution . . . 3
2.1.2 Diffraction limited imaging . . . 3
2.1.3 Image Blurring . . . 4
2.2 Light Source Coherence . . . 6
2.2.1 Temporal Coherence . . . 6
2.2.2 Spatial Coherence . . . 6
2.3 Coherence Effects on Shadowgraphy . . . 7
2.3.1 Speckle . . . 7 2.3.2 Fresnel Diffraction . . . 7 2.4 Shadowgraphy . . . 8 2.4.1 Focused Shadowgraphy . . . 10 2.5 K¨ohler Illumination . . . 11 2.6 2D-FFT . . . 12 3 Experimental Setup 13 3.1 Droplet Generator . . . 13 3.1.1 Illumination Scheme . . . 13 3.2 Illumination Sources . . . 13
3.2.1 Dye Based Light Sources . . . 14
3.2.2 Extended White Light Source . . . 15
3.2.3 Flash Lamp . . . 16
3.3 Evaluation Methods . . . 16
3.3.1 Specke . . . 17
3.3.2 Diffraction . . . 17
3.3.3 Feature Visibility . . . 18
4 Results & Discussion 19 4.1 Speckle Reduction . . . 21
4.2 Diffraction Reduction . . . 21
4.2.1 Spatial Coherence . . . 22
4.2.2 Temporal Coherence . . . 24
4.3 Feature Visibility . . . 24
5 Conclusion & Outlook 27
Abstract
This thesis presents the improvement of a shadowgraphy setup in a laser produced plasma (LPP) source of extreme ultraviolet light (EUV). A light source derived from a λ = 560 nm pulsed dye laser is introduced and employed in an incoherent fashion. The resulting shadowgraphs show little to none coherence effects such as speckle and diffraction.
1
Introduction
Shadowgraphy is an imaging method that records the shadow of a backlit obscuration or refractors. It is able to record changes in the refractive index of transparent media and is used to, for instance, visualize shockwaves of supersonic projectiles and explosions. When compared to typical imaging methods, one advantage of shadowgraphy imaging is that a less intense illumination source is required since light is directed towards the observation plane or camera and does not need to scatter from an object. The shadowgraph method was first understood by Robert Hooke in the 17thcentury [1]. He linked the observation of hot air
convection form candles to refractive index changes in the air itself. Since then, shadowgraphy and a related concept known as schlieren have been used in numerous experiments studying supersonic flow and shockwave dynamics in transparent media. A remarkable highlight of these imaging methods occurred in 1993, when shock waves of full sized jet fighters flying at 1.1 Mach were photographed using the schlieren technique [2]. More recently, the shadowgraphy and schlieren imaging methods have been used to make direct, time-resolved observations in laser-matter interaction experiments [3] ranging from probing plasma opacity [4] to the study of shockwave formation after laser ablation [5]. In laser produced plasma (LPP) sources of extreme ultraviolet (EUV) light, high-power lasers shoot at metallic micro-droplets to create a plasma which radiates at the desired ultraviolet bandwith [6]. Tin droplets are used because highly charged tin ions radiate 13.5 nm [7]. A double laser pulse system is used for efficient EUV production; a low energy prepulse deforms a tin droplet into a specific target shape after which a high energy main pulse turns the target into a plasma radiating EUV light [8]. By changing the prepulse parameters and the timing between the pre- and main pulse, different target shapes can be produced for main pulse irradiation.
A high resolution shadowgraphy setup is needed [9, 10] in order to investigate droplet dynamics such as droplet expansion, ligamentation, rim destabilization and hole formation [11]. However, despite the fact that various experiments use shadowgraphy as an investigation tool, little has been written on the specifics of the imaging method itself. A considerable amount of literature addresses the better known optical imaging methods such as microscopy, but on the whole, the shadowgraphy method remains sparsely investigated; as a result, improvement of a shadowgraphy system is not straightforward. Technical challenges concerning high resolution shadowgraphy in LPP sources of EUV light are abundant: Optical light from the plasma quickly saturates the observation cameras and the high droplet velocity after laser impact requires an imaging system with short temporal resolution. Furthermore, droplets are only a couple tens of micrometers large and the complete LPP system is in a vacuum making short distance microscopy impossible. Using a nanosecond pulsed light source on the same axis as a long distance microscope and the deforming tin droplet, high resolution shadowgraphs can be made. In order to evaluate the imaging effects caused by the size and spectral width of different light sources several test targets are used.
This thesis will improve a shadowgraphy setup on an LPP source of EUV light by modifying the illuminating source. Furthermore, light source coherence properties and their effect on shadowgraph image quality is investigated. First, relevant theory will be discussed, describing aspects ranging from imaging resolution to Fresnel diffraction of an edge to homogeneous illumination methods for long distance microscopes. Next follows the description of the experimental setups consisting of light sources, light waveguides, test targets and their evaluation methods. Results of the different systems are presented and remaining technical challenges are discussed and an outlook is given. Finally conclusions are presented.
Figure 1: (a) Computer generated Airy disk (Wikipedia). (b) Two, resolvable Airy patterns according to the Rayleigh Criterion. The visibility of the local minimum between the two Airy patterns, shown in the blue profile 0.144 from Eq. (4). (c) Airy pattern with fitted Gaussian curve.
2
Theory
2.1
Imaging
2.1.1 Image Resolution
Overall image resolution describes the amount of detail a digital image holds. Image resolution can be measured by looking at arrays of line pairs with increasing density. The more densely packed line pairs can be identified on an image, the higher the image’s resolution. The overall image resolution is dictated by a convolution of the three factors: sensor resolution, temporal resolution and spatial resolution.
Sensor resolution is determined by the number and size of pixels on the light capturing device, a light sensor or photographic plate. The smaller the pixel size and the larger number of pixels, the higher the sensor resolution. Pixel resolution is genreally taken as the system resolutoin in photography, although this is not correct since both temporal and spatial resolution are generally limiting optical imaging systems instead of sensor resolution. The maximum achievable sensor resolution is dictated by the Nyquist-Shannon sampling theorem [12]. Moir´e patterns appear on a digital image when the Nyquist criterion is approached.
An image’s temporal resolution is determined by the effective shutter speed of the imaging setup. Sensor or CCD readout speed, mechanical speed of shutters and flash duration are factors that determine temporal resolution. If the imaged object moves significantly during the exposure time of a picture, the picture will be blurred. Blurring obscures small image features that might lead to new insights. However, short temporal resolution requires an intense illumination source to adequately illuminate the picture.
Spatial resolution refers to the ability of an imaging setup to distinguish between small details of an object regardless of the sensor recording the image. An imaging system’s resolution is either limited by aberration or by diffraction, both cause image blurring. Aberration-limited systems can be improved with time and money: aligning paths and better aberration corrected optics will improve image quality. Imaging systems that are aberration free are fundamentally limited by the wave-like behavior of light, i.e. diffraction. The best resolution an imaging system is able to achieve is determined by its corresponding diffraction limit. 2.1.2 Diffraction limited imaging
The diffraction limit dictates a fundamental spatial resolution limit to an imaging setup. The diffraction size limit is set by the numerical aperture (NA) of the imaging objective and the wavelength of the light used to
image and is given by:
d = λ
2NA. (1)
Here, d is the smallest distinguishable feature, λ the wavelength of the illuminated light and NA the numerical aperture of the system. From Eq. (1) it can be concluded that shorter wavelength light used for imaging is able to resolve smaller features. This analogy holds for any optical system and, for instance, also describes the most tightly possible focused laser beam size. The best focused spot of light through a circular aperture, or smallest resolvable circular object is described by an Airy disk. An Airy disk stems from a Bessel function of the first kind and order one (J1(x)) [13] and is given by:
Airy(x) = 2J1(x) x
2
. (2)
for illustrative purposes a computer generated 2D Airy pattern is shown in Fig. 1a. A generally accepted criterion for the minimal distance between two resolvable airy patterns is when the first minimum of the first airy pattern coincides with the maximum of the second airy pattern. Or as the Rayleigh Criterion states:
θ = 1.220λ
D. (3)
Where θ is the minimal angular spacing between two just-resolvable Airy disks, or angular resolution, λ the wavelength of the light used and D the lens diameter. The scaling factor of 1.220 is an approximation for the first minimum in a J1(x). Fig. 1b shows two Airy patterns at the Rayleigh criterion. The visibility of
the center minimum in the combined profile is defined as: Visibility ≡ IMax− IMin
IMax+ IMin
. (4)
For the profile in Fig. 1b the visibility is 0.144. IMax and IMin are the maximum and minimum intensity
levels of the patterns area of interest. An Airy pattern can be approximated with a Gaussian Function as shown in Fig. 1c. The Gaussian fit has a R2 = 0.999 ± 10−3 and therefore it can be said that a Gaussian
describes an Airy pattern well. 2.1.3 Image Blurring
Image blurring or image sharpness describes the spatial intensity response of an imaging system to a step function in intensity. Measurement of the edge response and line spread funcion (LSF) give insight in how images are blurred by an imaging setup [14]. Image blurring can be fundamentally caused by the diffraction limit as covered in the previous section, but can also arise due to aberrations in the imaging system. Optical blurring reduces image contrast. Fig. 2a shows the edge response of an imaging system with Gaussian LSF. The response is described by an error function. The first derivative of the error function is a Gaussian function of equal width and is called the Line Spread Function (LSF). The σ of the LSF is equal to width of the error function. The broader the LSF width, the worse the resolution of the imaging setup as it is subject to Gaussian blur. Other sources report the 10% to 90% level of the edge response or the full-width at half maximum of the LSF as measures for image blurring [14], in this thesis, σLSF is maintained as blur
indicator because it is a single parameter found both in the edge response and in the LSF. An error function at position x0and with standard deviation σ is given by:
Erf(¯x) =2 π Z x¯ 0 e−t2dt (5) where ¯ x =x − x√ 0 2σ (6)
Figure 2: (a) Intensity step function (black) with the corresponding Gaussian blurred profile (red) and first derivative of the blurred profile, or LSF (blue). (b) Two Gaussian blurred edges separated by distance xd = 1 caused by a obscuring line of thickness 1. σLSF= 1 and the visibility of the profile is 0.24. (c) Theoretical visibility of increasingly dense line pairs per millimeter for different LSF widths σ.
Due to image blurring the visibility of densely packed line pairs can be diminished. The profile of two error functions separated from each other by distance xd centered around 0 is visible in Fig. 2b and given by
Profile(x) = 0.5 Erf −x −xd 2 √ 2σ1 + Erf x −xd 2 √ 2σ2 + 2 . (7)
The visibility of the pattern in Eq. (7) as defined by Eq. (4) is given by
Visibility(xd) = 1 + 0.5 Erf −xd 2√2σ1 + Erf −xd 2√2σ2 + 2 1 − 0.5 Erf −xd 2√2σ1 + Erf −xd 2√2σ2 + 2 (8)
In general one can assume σ1= σ2 as they are caused by the same optical system. Fig. 2b shows the effect
of Gaussian blurring on line visibility. When the width of the LSF is known a theoretical linepair visibility can be calculated. Line pair density in LP/mm versus visibility is then given by
Visibility(LP/mm) = Erf 125 √ 2 σLP/mm ! 2 − Erf 125 √ 2 σLP/mm ! . (9)
Fig. 2c shows the visibility of increasingly dense line pairs for various LSF widths according to Eq. (9). In practical situations, a series of line pairs that are Gaussian blurred result in a periodical profile. The visibility
of the periodical profile is given by:
Visibility = Amplitude y0
, (10)
as maximum and minimum values of a waveform are given by y0± Amplitude.
2.2
Light Source Coherence
Coherence is a light property and discribes phase correlations within a beam of light. When the phase correlations exist over time, phase evolution can be well described and the light is temporally coherent. When the light beam exhibits phase correlations over a large area within the beam the light is spatially coherent. Coherence is strongly related with interference effects which arise when coherent parts of the light beam interfere with each other. Both temporal and spatial correlation arise from the mutual coherence function ˜ γ12(τ ) = D ˜E 1(t + τ ) ˜E2∗(t) E T r D | ˜E1|2 E D | ˜E2|2 E . (11)
Where τ denotes a time difference and E1,2(∗) the (complex conjugate) of the electric field at positions 1 and 2. Different applications require light sources with different coherence properties. Narrow-bandwidth lasers are examples of very coherent light sources, while floodlights illuminating stadiums are examples of very incoherent light sources.
2.2.1 Temporal Coherence
Temporally coherent light has a very predicable phase evolution at a fixed position. If we evaluate the coherence at time delay τ at the same position, ˜γ12(τ ) = ˜γ11(τ ) it is found that temporal coherence is
described by: ˜ γtemp(τ ) = D ˜E(t + τ ) ˜E∗(t)E T | ˜E|2 . (12)
The coherence time is the time where ˜γtemp(τcoh.) = 1/e. The decay of the coherence function is linked to
the spectrum of the light as different frequencies will disturb the phase evolution of the light. The coherence time can be expressed as
τcoh=
1
π∆ν (13)
where ∆ν is the full-width at half maximum of the spectrum. The coherence time can be expressed in light propagation length by
Lcoh= cτcoh. (14)
The coherence length describes the distance between two points in a propagating light beam over which coherence is lost.
2.2.2 Spatial Coherence
Spatially coherent light has correlations and fixed phase relationships across a certain area in the beam profile. Evaluating the mutual coherence function as given in Eq. (11) at different positions for τ = 0, the degree of spatial coherence is given by:
˜ γspat(0) = D ˜E1(t) ˜E∗ 2(t) E T r D | ˜E1|2 E D | ˜E2|2 E . (15)
As a a source decreases in size, its spatial coherence will increase. The degree of spatial coherence can be expressed as a coherence area over which the spatial coherence deceases to 1/e [15]
Acoh=
λ2 0h2
D . (16)
Where λ0is the central wavelength of the light, h is the distance between the light source and the observation
plane and D the area of the source. The h2/D dependence indicates that the spatial coherence of light
increases the further it propagates from its source.
2.3
Coherence Effects on Shadowgraphy
Some forms of imaging, like optical coherence tomography, require light sources that are coherent to a certain degree. Shadowgraphy on the other hand, works best under complete incoherent illumination as coherence leads to two distinct imaging artifacts: speckle [16] and diffraction [13].
2.3.1 Speckle
An imaging phenomena that arises due to both spatial and temporal coherence is speckle [16]. Speckle arises when a coherent beam scatters from a rough surface. Many scattering areas reflect in a dephased but still coherent manner. Interference between the scattering areas at the point of observation leads to a granular speckle pattern. In imaging, speckle is unwanted as the background is not uniformly illuminated, visibility of objects is dicreased, especially when the size of the objet is of the order of the granular speckle size size. Speckle can be suppressed by interfering uncorrelated speckle patterns. Speckle contrast is defined as the intensity standard deviation in the pattern divided by the average intensity of the pattern. For M uncorrelated speckle patterns in the resulting speckle will have a contrast of
C =σ¯I I =
1 √
M. (17)
There are three ways to introduce uncorrelated speckle patterns in order to reduce the overall speckle con-trast. The first is to add different polarizations to the speckle creating light. With complete depolarization, the contrast of the speckle pattern is 1/√2 [16]. The second method to reduce speckle is by adding multiple angles of illumination. If the angles are sufficiently separated, speckle patterns will be uncorrelated once again. This method effectively decreases the spatial coherence of the illuminating light. The third method to decrease speckle is to change the wavelength of the illumination, creating substantial phase differences between reflected light of different wavelengths leads to uncorrelated speckle patterns. As the more wave-lengths are introduced in the temporal coherence of the illuminating light decreases. The required separation frequency in to achieve uncorrelated speckle patterns is:
∆ν ≈ c 2σz
. (18)
Where σz is the surface roughness of the optical components used. In summary it can be said that coherent
light sources generate speckle which can be reduced by increasing the size and wavelength distribution of the source, in other words decrease its temporal and spatial coherence.
2.3.2 Fresnel Diffraction
Diffraction is the second imaging phenomenon that arises whe light waves interfere with each other and themselves [13]. Diffraction of light can be described mathematically with two models: Fraunhofer and Fresnel. The former describes far-field diffraction effects, the latter near-field. Whether Fresnel or Fraunhofer diffraction should be considered depends on the Fresnel Number F given by
F = √a
where g is the distance between the diffracting aperture of size a and the observation screen. When F < 1 Fraunhofer diffraction patterns will form. When F > 1, the size of the aperture is large compared to√λg and Fresnel diffraction is visible. For a sufficiently large aperture, as is the for F > 1, one can imagine light passing through the aperture’s center and reaching the observation field undiffracted. Therefore Fresnel diffraction can be regarded as edge diffraction. As Fresnel diffraction is relevant in shadowgraphy it is evaluated and Fraunhofer diffraction is not regarded in this thesis. To evaluate Fresnel diffraction the so-called Fresnel integrals have to be examined:
S(x) = Z x 0 cos(πt2/2)dt (20) C(x) = Z x 0 sin(πt2/2)dt (21)
The Fresnel integrals are unfortunately not analytically solvable. However, the oscillating functions S(x) and C(x) can be plotted as a parametric plot in the complex plane as
˜
B(w) = S(w) + iC(w) = Z x
0
eiπt2/2dt. (22)
The result is shown in Fig. 3a and is called the Cornu spiral. The Cornu spiral is used to determine the shape of the Fresnel diffraction. The light intensity as position P behind a diffracting aperture in the Fresnel regime is given by:
Ip= I0 4| ˜B12(u)| 2| ˜B 12(v)|2, (23) where ˜B12(w) is given by ˜ B12(w) = [S(w) + iC(w)]ww21.
w1 and w2 are dimensionless numbers relating the position of the point of observation with respect to the
diffracting edges in the diffraction plane. The distance to a diffraction edge w represents a point on the Cornu spiral. The distance between two diffracting edges on the same plane is given by ˜B12(w) and the
length of the vector determines the illumination intensity. The dimensionless numbers u and v represent the distances along the x and y axes in the aperture plane and are given by:
u ≡ y 2(ρ0+ r0) λρ0r0) 1/2 ; v ≡ z 2(ρ0+ r0 λρ0r0 1/2 (24) the position of the diffraction edges are given by y, z and ρ0, r0are the distances from the light source to the
diffraction object and from the diffraction object to the observation point respectively. For diffraction from a horizontal knife edge, or semi-infinite opaque screen v2 = u2− ∞ and u1 = −∞ [13]. The illumination
pattern is shown in Fig. 3b. It is worth mentioning that at the position of the edge, the illumination intensity is 0.25 for the diffraction pattern, whereas it would be 0.5 for a Gaussian blurred system as discussed in a previous section. Depending on the edge effects visible in an image, threshold levels for edge and size determination should be adjusted accordingly.
2.4
Shadowgraphy
“Between microskopie and tel´escopie come stioscopie and ombroscopie - schlieren and shadow-graph techniques in English. All of a close family, these optical techniques have the same parents (...) and use the same optical stuff. Though there are hundreds of books on microscopes and telescopes, however, there is no modern book on schlieren and shadowgraph methods, hence the justification for this book.” - G.S. Settles (2001), Schlieren and shadowgraph Techniques, Visu-alizing phenomena in Transparent Media, Springer-Verlag [17]
Figure 3: (a)Cornu Spiral for −5 < w < 5. Increasingly negative values for w spiral around the coordinates (-0.5;-0.5) and increasingly positive values around (0.5;(-0.5;-0.5). (b) Fresnel diffraction of a knife edge. Note that at the actual edge, the illumination intensity is already 0.25 times the background illumination.
Figure 4: Simple shadowgraphy setup [17] with an illumination source of size D. S denotes the object of which a shadowgram is made on the screen. A circle of confusion -or half shadow- arises from the exte nded size of the illumination source.
Figure 5: So-called “focused” shadowgraphy setup (from [17]). This setup allows shadowgraph images with small distance g without becoming invasive to the test area.
Schlieren and shadowgraphy are imaging methods to visualize inhomogeneities in transparent media as mentioned in the introduction. The simplest observation of a shadowgraph is the visible hot air convection shadow from a candle on a white wall illuminated by a different candle. The result is visible due to a slight difference in refractive index between the hot air from the candle and the cold, ambient air in the room. Light that is diffracted more by a hot air section leaves a shadow on the wall. However, visualizations of these slight differences in density and temperature in transparent media is not the purpose of shadowgraphy in this thesis due to the fact that LPP sources of EUV light are contained in a vacuum. Therefore refraction theory and its place in shadowgraphy will not be covered in this thesis. The purpose of shadowgraphy is to image droplet deformation with high temporal resolution under low illumination intensity conditions compared to other optical imaging methods with similar temporal resolutions. Fig. 4 shows a simple shadowgraphy setup. This setup uses a divergent light source of size D illuminating an object of size S and observes the shadowgram at plane M . At the edge of the shadow the light source will only be partially obscured by the object causing blurr of the shadow. This blur is called the circle of confusion or half-shadow. The circle of confusion can be reduced by reducing the source size until diffraction effects due to spatial coherence become dominant over the circle of confusion [17]. The smallest light source diameter before diffraction effects become visible is given by:
Dmin= 1.33
p
λ0h(h − g)/g. (25)
Where h and g are the distances shown in Fig. 4. The smallest resolvable feature in a shadowgraph image is given by
δmin= 1.33
p
λ0g(h − g)/h. (26)
There are many different variations on schlieren and shadowgraphy imaging techniques, one particular in-teresting case is “focused” shadowgraphy.
2.4.1 Focused Shadowgraphy
For practical purposes it can be useful to image a shadowgraph at position M to a screen at position M∗. This method is called “focused” shadowgraphy since a focusing lens is necessary in order to image aforementioned planes. Fig. 5 illustrates a focused shadowgraphy setup using a point light source and a collimating lens in front of the test area. A big advantage of this focused shadowgraphy is that object magnification are independent from the distance between the imaging plane M and the object S as well as from the divergence of the light. A drawback of this imaging setup is that the numerical aperture (NA) of the objective will limit the resolution. Furthermore, due to the added imaging optics, the system will now exhibit a finite depth-of-field (DOF). The imaged plane M now has a thickness δM wherein everything appears in-focus on the imaged screen M∗. The expression for the depth of field of an optical imaging system is given by [18]:
DOF = 4cf
2NA(f − h∗)h∗
−4f2NA2
In Eq. (27): NA is the numerical aperture of the imaging system, f is the effective focal length of the imaging system, c is largest acceptable size of the circle of confusion and h∗ is the distance from the object in focus to the imaging system. A trade off between resolution and DOF becomes apparent. According to Eq. (1) a higher NA leads to a higher resolution and according to Eq.(27) a higher NA leads to a smaller DOF. The shadowgraphy system will still be subject to diffraction artifacts when the light source is small enough although now, the expression for the minimal source size is
Dmin= f
p
λ0/g. (28)
f is the focus length of the collimating or collector lens. For focused shadowgraphy systems that minimize g the DOF and size of the imaged object determine Dmin. In a similar fashion, the smallest resolvable feature
is given by
δmin= 1.33
p
λ0g. (29)
2.5
K¨
ohler Illumination
K¨ohler illumination is an illumination method used in optical microscopy [19]. This method of illumination uniformly illuminates the object plane and eliminates all structure from the illuminating source. Homoge-neous illumination is achieved by using a lens system that places the illumination source and imaged object into opposite conjugal planes as shown in Fig. 6. When the object or sample is in focus, the illumination source is completely defocussed and vice versa. An optimal K¨ohler illumination setup maximizes the NA of the light source and matches the NA of the projected spot with the NA of the objective. K¨ohler illumination maximizes the resolution and illumination efficiency of an optical imaging setup. In K¨ohler illumination sizes
Figure 6: Schematic representation of K¨ohler illumination in a traditional microscope. The indicated light paths show the conjugal planes of the lamp filament and the sample. Light from the lamp filament passes through the sample completely defocused and no source structure is seen in the sample image planes.(From: Wikipedia)
and distances between the light source, collector and condenser lenses and sample are calculable and given by the following expressions:
d1= FLCollector (DSource+ DLCondenser) DLCondenser , (30) d2= d1FLCollector d1− FLCollector , (31) d3= d2FLCondenser d2− FLCondenser . (32)
Where d1, d2 and d3 are the distances between the light source and collector, collector and condenser, and
component Y . The input numerical aperture is set by the size of the collector lens and the distance d1while
the numerical aperture of the spot created on the object/sample is determined by the size of the condenser lens and d3. After K¨ohler illumination is achieved, an imaging objective and camera are used to magnify
and record the image.
2.6
2D-FFT
A two-dimensional fast Fourier transform (2D-FFT) can be used to analyze the speckle content of shad-owgraphy background illumination. Just as with the normal one dimensional fast Fourier transform, the 2D-FFT decomposes an image into its frequency components. Structure or inhomogeneities visible The 2D Fourier transform is given by:
F (u, v) = Z ∞
−∞
Z ∞
−∞
f (x, y)e−i2π(ux+vy)dxdy. (33)
Where f (x, y) is the two dimensional signal and u, v are frequency components in the x, y direction respec-tively. Since images are discrete signals, fast Fourier transforms are taken. The 2D-FFT result of a digital image is on itself a digital image as well. The coordinates in the 2D-FFT represent the direction and fre-quency of the decomposing waveform over the image while the intensity of image represents the amplitude of the waveform. The image’s pixels represent the decomposing frequency space while the intensity serves the purpose of the Fourier coefficients. Low frequency components are shown near to the origin. The origin itself represents a non-oscillating, DC component of the image. The highest frequencies that can be probed are given by the Nyquist criterion, the smallest frequencies visible depend on the size of the decomposed picture. The largest frequency that is used in a 2D-FFT for a given the largest dimension l of the decomposed picture is 2j where j is the largest integer for which the inequality 2j
3
Experimental Setup
3.1
Droplet Generator
The experimental LPP source considered in this thesis is called the droplet generator (DG) and is used for studying laser-droplet interactions [9]. A schematic overview of the DG setup is shown in Fig. 7. Different 10 Hz Nd:YAG laser systems are used to ablate and deform tin droplets. A number of measurement are performed simultaneously on the DG setup in order to study different ongoing processes: Spectrometers capture the emission spectra from charged tin ions in the optical to EUV spectrum, faraday cups measure ions coming from the interaction zone, and quartz micro balances track deposition rates of tin in the vacuum chamber. Two shadowgraphy setups are used to observe droplet deformation and propulsion dynamics. While a 30◦ front view captures the deformation parallel to laser impact, a 90◦ side view observes the
defor-mation and velocity kick of the droplet orthogonal to the laser beam. Images are taken with a K2Distamax objective and an AVT MANTA camera. The optical system has a maximal numerical aperture of 0.083 and a magnification of 3.49. To avoid overexposure of the camera from plasma lightradiation, an optical bandpass filter is placed between the objective and the camera to filter out plasma light. The band pass filter transmits the shadowgraphy illumination wavelength and has a width of ±14 nm around its center frequency of. By scanning the delay of the pulsed shadowgraphy illumination light ∆t after laser impact, a stroboscopic movie is made of the deforming tin droplets. An 850 nm pulsed laser diode has been used as a shadowgraphy light source in published work regarding droplet propulsion and deformation studies [9]. According to Eq. (1) the diffraction limit using the λ = 850 nm light source is 5.12 µm. This thesis replaces the λ = 850 nm for a λ = 560 nm dye based illumination source for the use of shadowgraphy in the DG LPP source of EUV light. A 560 nm light source would be fundamentally decrease the diffraction limit to 3.37 µm and therefore has the potential to make higher resolution shadowgraphs. The novel light source is able to produce pulses of 5 ns which provide sufficient light for illumination and does not limit the temporal reso-lution of the imaging system. However, due to its coherence properties, the dye based light source exhibits other unwanted features like speckle and diffraction. These coherence effects are investigated and reduced for use in droplet deformation studies.
3.1.1 Illumination Scheme
To use the λ = 560 nm dye light source as an illumination source for shadowgraphy in the DG experiment light from the dye source is transported via waveguides from to the DG table as shown in FIg. 7. Two different waveguides are available for light transportation: A 17 m multimode fiber with a 600 µm core and a 15 m PMMA waveguide 3 mm in diameter. The µm fiber is bought from stock while PMMA waveguide is cut from a bundle and hand polished to a roughness of 1 µm. The PMMA waveguide is expected to have higher losses which are compensated for by coupling more light into the waveguide. A K¨ohler illumination setup is used to homogeneously illuminate the shadowgraph background. A f = 30 mm lens is used at a distance d1= 32 mm. The condenser lens (f = 200 mm) is placed at a distance of d2= 240 mm. The droplet
is an approximate d3 = 32 mm away from the condenser lens. Unfortunately, this setup is not an ideal
K¨ohler type setup as the output NA of the light beam does not match the NA of the objective. However, this illumination scheme is able to homogeneously illuminate shadowgraphs regardless of which waveguide is used for transportation of the light.
3.2
Illumination Sources
For use shadowgraphy on the DG experiment the performance of a λ = 560 nm dye based light source is investigated. Since the coherence properties of the dye light source influence its performance other light sources are used to study the general effects of coherence on shadowgraphy images. The light sources investigating coherence effects do not have to be applicable on the DG for tin droplet observation. Apart from various ways of using the dye based light source, an extended white light source and a white light flashlamp are used as illumination sources for the shadowgraphy setup.
10 Hz Nd:YAG 1064 nm λ/2 TFP PMT He Ne Delay generator CC D filt er CCD filter BD lens mirror mirror trigger λ/4 Pulsed Dye Laser System PMMA
PMMA
Figure 7: Schematic top view of the 10 Hz Droplet Generator Experiment. Droplets inside the vacuum chamber fall down through a sheet focus of a 632.8 nm He-Ne laser. A PhotoMultiplier Tube (PMT) collects the scattered light and serves as a trigger for the rest of the setup. Various Nd:YAG laser systems can hit the droplet and create a quickly expanding tin plasma. Front (30◦) and side (90◦) view shadowgraphs are taken at a time ∆t after laser impact. The dye light source is indicated as light source for the shadowgraphy setup. The plasma light emits continuum radiation visible on the shadowgraphs. An optical bandpass filter transmitting the light source wavelength is used to block excess tin plasma light.
3.2.1 Dye Based Light Sources
To improve the resolution in the DG shadowgraphy system a dye based light source is introduced as a light source. The light source is based on a PDL-2 dye laser from 1987. Rhodamine 6G dye dissolved in ethanol is used as it is able to operate at λ = 560 nm. Fig. 9 shows the schematics of the PDL-2 dye laser. The laser consists of three dye cells that are pumped by the second harmonic of a Nd:YAG laser (Surelite II). The first dye cell is placed inside a laser cavity where the back-end mirror is replaced by a dispersive prism and rotatable grating. A particular wavelength is reflected back into the oscillator dye cell. By varying the angle of the grating with respect to the incident light a lasing wavelength selection can be made. The second and third cell amplify the oscillator output. The spectrum of the dye laser is shown in Fig. 10a. The temporal coherence length for this operation mode derived from the spectral width is minimally 253 µm. The width of the peak is equal to the spectrometer resolution of 0.4 nm and therefore the indicated temporal coherence length is a lowest estimation. The light from the dye laser is coupled into two waveguides which transport
Figure 8: Schematic overview of the imaging system. The distances d1, d2, d3 are the distances from K¨ohler illumi-nation as described previously. Measurements with the extended white light source are preformed by replacing the the condenser lens with an aperture.
Figure 9: (a) Schematic overview of a PDL-2 dye laser. The dye laser is pumped with the second harmonic from an unseeded 10 Hz pulsed Nd:YAG laser giving pulses of 10 ns at λ =532 nm. A λ/2 plate is used in combination with a polarizing beamcube to attenuate power going into the dye laser. The power is adjusted to a level where the shadowgraph background is sufficiently illuminated. The dye laser is built up from three dye cells (orange), one in the oscillator, one that serves as a pre-amplifier and a main amplifier. The oscillator consists of an output coupler (OC) the dye cell, a prism disperser (PD) and a rotateable grating (Gr). The grating can be rotated in order to tune the laser wavelength.
the light to the DG experiment. Apart from the lasing configuration of the PDL-2, two different modes of operation are used which can be applied to the DG experiment for deformation and propulsion studies. The second method of using the dye based light source is by blocking the prism disperser and wavelength selection grating from light coming from the oscillator. The light is blocked using a white paper and an ill defined cavity is created between the output coupler and the piece of paper. In this mode of operation there is no wavelength selection and light coming from the oscillator into the pre-amplifier will consist of amplified spontaneous emission (ASE). The amplified spontaneous emission is amplified further in the amplification stages and used as illumination source. The spectrum of the ASE mode of operation is shown in Fig.10b. The spectrum is 4 nm wide and the corersponding temporal coherence length is 25.3µm. The final way the PDL-2 is used as a light source is by only illuminating the preamplifier with the pump light. The main amplifier is removed and the fiber coupling optics are placed directly next to the pre-amplifier. This method allows both ASE and fluorescence to be coupled into the waveguides. The spectrum of the single cell configuration of the PDL-2 is shown in Fig. 10. The temporal coherence length is calculated to be 20µm as the spectrum is slightly broader than for the ASE mode of operation. However this is an upper estimate as fluorescence with undefined phase is also used for illumination of the shadowgraphs. The three different operation modes are used to evaluate light source temporal coherence effects on shadowgraphy images.
3.2.2 Extended White Light Source
An extended white light source is used to study the effects of various degrees of spatial coherence on shad-owgraphy. The light source consists of a halogen lamp emitting a black body spectrum shown in Fig. 10d. Due to its spectrum and extended size the halogen light source is completely incoherent both temporally and spatially. Spatial coherence is introduced by varying the size of an aperture at the location of the condenser lens in Fig. 8. Given the assumption that coherence influences image resolution in a negative fashion, it is expected that shadowgraphs from the extended source with largest aperture size have the highest resolution.
Figure 10: Spectra of the different light sources (red) shown together with the filter transmission function in the imaging objective (black). Light reaching the CCD is the convolution of the two. (a) shows the spectrum of the lasing dye light source. (b) Shows Amplified Spontaneous Emission (ASE) of the dye light source and (c) shows the spectrum of the single cell operation mode. (d), (e) are spectra of the extended white ligst source and the white light flash lamp exiting a waveguide. The spectra of the latter two light sources are highly comparable.
3.2.3 Flash Lamp
A third illumination source is a flash lamp. As light from the dye light source has to be transported via optical waveguides to the imaging setup spatial coherence is introduced to the due to the finite size of the waveguides. Waveguides are not used for measurements performed with the extended white light source as spatial coherence is varies over these measurements. In order to make a comparison between the experiments with the extended light source and the dye laser a flashlamp is used. The flashlamp has an equal spectrum to the extended light source as shown in Fig. 10 and is coupled into the waveguide. The spatial coherence effects of the waveguides can now be investigated as this method has the same temporal coherence as the extended white light source and the same spatial coherence as the dye laser.
3.3
Evaluation Methods
A number of targets are used in order to study the performances of the different light sources and the effects of coherence on shadowgraphic images. Three observables have been of importance for the shadowgraphy application on the DG experiment.
2. Diffraction Effects
3. Visibility of small features (Resolution)
These three observables were evaluated by imaging various imaging objects and analysis methods. 3.3.1 Specke
Since speckle is observed in the background of the shadowgraphy image its investigation requires no imaged object. A homogeneous background is desired and therefore speckle is unwanted. The degree of speckle is defined by its contrast in Eq. (17) and its typical size is investigated with the 2D-FFT of the shadowgraphs. As speckle has a typical size, depending on the coarseness of the elements by which it is generated, a clear frequency band is expected to appear in the 2D-FFT when speckle is apparent in a shadowgraphy image. The normalized 2D-FFT of images exhibiting speckle allows comparison of speckle contrast and investigation of typical speckle size in the shadowgraph image across different operational modes of the dye laser.
Figure 11: Schematic overview of the knife edge normalization and analysis method. (a) Not normalized, out of focus knife edge profile. The first and final 50 pixels are taken as low and high normalization levels and are given the value of 0 and 1 respectively. (b) 21 individually normalized knife edge contours from the same image (black) are averaged. The averaged contour (green) shows diffraction features hardly seen on the individual contour level due to noise. (c) Heat map representation of averaged knife edge contours scanned through the focus of the objective.
3.3.2 Diffraction
Diffraction is investigated by imaging a knife edge. Where, depending on the coherence properties of the illumination source, the edge will exhibit either Gaussian blurring shown in Fig. 2a or Fresnel diffraction as shown in Fig. 3b. Two parameters are extracted from knife edge measurements, the width of the Line Spread Function (LSF) and the maximum intensity from the diffraction pattern. The σLSFindicates the sharpness
of the imaged knife edge and the maximum modulation intensity indicates how much diffraction is apparent in the image. During a measurement, the camera and objective are translated over 1 mm in 50 µm steps along the z-axis indicated in Fig. 8. In this manner the knife edge is scanned through the focus of the imaging objective. The development of the PSF width and maximum modulation are studied, the position where the width of the LSF is smallest for the knife edge is defined as the focus. For each position of the objective and camera 10±2 shadowgraphy images are taken. The uncertainty in images arises from the increasing exposure time in order to fully illuminate an image as the aperture is decreased in size. From each individual image a dirt-free, square window measuering 201 pixels in width and height is selected for image evaluation. First, 21 knife edge profiles are tested for overexposure and subsequently normalized. Profiles with a maximum gray scale of 255 are overexposed and are therefore omitted. Fig. 11a shows the normalization levels of a typical knife edge profile. After normalization, the width of the LSF is measured by fitting a Gaussian to the
first derivative of the profile and the maximum intensity is saved. The normalized profiles, LSF widths and maximum intensities from the same camera position are averaged. The averaged profiles for each position of the objective are used to produce a profile-through-focus-heatmap shown in 11c. The averaged values of the LSF width and maximum modulation intensity are used as indicators for sharpness and diffraction. 3.3.3 Feature Visibility
To asses the visibility of small features, a test target consisting of increasingly dense linepairs is used. Profiles of these linepair targets resemble periodical waves and the visibility of the pattern is given determined by the ratio between the amplitude and offset of the sinusoidal waveform and given in Eq. 10. Figure 12 shows a shadowgraphic image of a grid with 100 lines per mm (LP/mm), the profile and the resulting sinusoidal fit. The visibility of the pattern is determined to be 0.732. Every image is rotationally corrected before
Figure 12: (a) Shadowgraph image of resolution target showing 100 line-pairs per millimeter using an extended white light source. (b) Corresponding profile of 100 pixels. The half-period of the fitted sine is 2.7229 ± 3 × 10−4 which corresponds to an image-to-pixel size of 1.83µm per pixel, or a magnification of 3.5. According to Eq. (9) the visibility of these line pairs is 0.723.
image evaluation. For the case illustated in Fig. 12 a clockwise rotation of 0.1 degrees has been applied. In practice, corrective rotations larger than 0.5 degrees are seldom applied.
4
Results & Discussion
Fig. 13 shows shadowgraph images of deforming droplets using two different illumination sources. Fig. 13a shows a shadowgraph using a λ = 850 nm pulsed laser diode as illumination source. Background structure from the diode bars is visible. The diffraction limit given by Eq. (1) is 5.12 µm. The background structure and diffraction limit constrain the resolution of this image and as a result, droplet breakup phenomena are not clearly visible. Fig. 13b shows improved resolution shadowgraphs using a λ = 560 nm ASE dye based source. Due to the decrease in wavelength the diffraction limit is lowered to 3.37 µm. Background structure from the diode bars is eliminated by this method of illumination as light is transported through waveguides. However, the dye based light source introduces two distinct features which lower the resolution of the images made with the shadowgraphy setup: speckle and diffraction. Speckle, similarly to the diode bar structure, is visible in the background of the shadowgraph images and is a coherence feature of the light source. Speckle grains have a typical size and target features of a similar or smaller size will be hard to distinguish from the background. Therefore, speckle needs to be suppressed. Diffraction is also caused by the coherence properties of the illuminating light. Diffraction patterns are visible around the edge of the tin target and are able to mask small features along the rim of the tin target or droplet. In Fig. 13b the onset of ligamentation after 2.8µs is not visible due to diffraction. Both speckle and diffraction are suppressed to further increase the resolution of the shadowgraphy system.
Figure 13: Shadowgraph images of deforming tin targets a certain time after prepulse impact. (a) Shadowgraphs with 850 nm diode bar illumination source, (b) shadowgraphs with λ = 560 nm dye based ASE light source. For both illumination sources the droplet undergoes the breakup phenomenon called ligamentation.
Figure 14: Shadowgraph backgrounds from different illumination sources exhibiting various degrees of speckle. (a) shows the background of the dye light source in lasing mode with Csp= 0.48, (b) in ASE and (c) in single cell mode with Csp= 0.03 and Csp= 0.02 respectively. (d) is also illuminated with the single cell dye light source, but here a PMMA waveguide with a diameter of 3 mm is used to transport the light from the light source to the imaging setup. The background has a speckle contrast of Csp = 0.01. The 2D-FFT illustrates how speckle artifacts have a typical size distribution. For each illumination setup 45◦ profiles are taken corresponding to frequencies of pτ2
x+ τy2 −1 illustrating a clear cutoff frequency. The 2D-FFT images are normalized on to the DC component in the origin.
4.1
Speckle Reduction
Speckle is the first feature of the λ = 560 nm dye light source that is investigated and reduced. Background speckle on an image masks the imaged object and is therefore unwanted. Speckle is a result of both the spatial and temporal coherence properties of a light source. Supression of speckle in the shadowgraphy setup is achieved by first decreasing the temporal coherence of the dye light source and second by decreasing the spatial coherence of the dye light source. Fig. 14 shows the shadowgraph backgrounds for different operation modes of the dye light source. The lasing operation shown in Fig. 14 has the highest temporal coherence and is unfit for imaging deforming tin droplets in LPP sources of EUV light. The speckle contrast given by Eq. (17) is 0.48 for this particular background. From the 2D-FFT it becomes apparent that a typical speckle grain has a frequency of 0.05 µm−1 or, in other words a typical size of 20 µm. Temporal coherence is decreased with the ASE and single cell modes of the dye light source shown in Fig. 14b and Fig. 14c. Speckle contrast is reduced to 3% and 2% for these illumination sources respectively. The cutoff frequency in the 2D-FFT remains unchanged at 0.1 pixel−1 meaning that the typical size of the granularity does not change
with decreasing temporal coherence, only its relative intensity does. The 2D-FFT of Fig. 14d operation mode shows higher frequency rings. These rings arise due to the diffraction patterns of dirt particles on the optical components of the imaging objective. The dirt particles are also present in the lasing and ASE illumination although the speckle contrast overshadow the dirt diffraction for these modes of illumination. In order to reduce the speckle pattern even further, down to a contrast of 1%, the spatial coherence of the dye light source is decreased. Fig. 14d shows the background and 2D-FFT of the single cell operation mode of the dye light source, where the light is transported via a 3 mm PMMA waveguide instead of a 600 µm fiber. The 45◦ 2D-FFT profile decays fastest and no cutoff frequency is visible for the homogeneous illuminated image.
4.2
Diffraction Reduction
With reduced background speckle, diffraction effects become apparent and limit the shadowgraph image resolution. As previously mentioned, diffraction is an unwanted artifact in shadowgraphic images of tin targets in LPP sources of EUV light as it masks target rim features. Fig. 15 shows images of undisturbed droplets with and without diffraction. Fig. 15a uses the ASE dye based light source and Fig. 15b the single cell operation mode. Decreasing the temporal coherence of the dye light source decreases the diffraction surrounding the droplets only slightly. Since different wavelength contributions manifest in differently shaped Fresnel diffraction profiles according to Eq. (24) a sufficiently broad spectrum is able to wash out diffraction effects. Since the wavelength distribution of the dye light source is very localized it will not be able to wash out diffraction effects by only decreasing the temporal coherence. Furthermore, due to the optical filter, temporal coherence cannot be reduced to a level where diffraction is completely washed out by temporal incoherence. The unequal diffraction on either side of the droplet shown in Fig. 15 is due to misalignment of the light source to which the imaging setup is very sensitive. Misalignment effects are visible within a 100 µm window in the object plane. One result of the misalignment sensitivity is that during studies of droplet deformation, the droplet moves out of the alignment window. Therefore misalignment effects remain present in the stroboscopic movies of deforming droplets. To eradicate diffraction around an in focus droplet, as shown in Fig. 15c the spatial coherence of the light source is decreased. The optical component that determines the spatial coherence of the illumination light is the waveguide [20]. The coherence properties of the light from a fiber depend on the number of modes that are able to propagate through the fiber and and how far these modes are delayed with respect to each other. Fig. 15b and Fig. 15c use the same light source but Fig. 15b uses the 600 µm fiber while Fig. 15c uses the 3 mm diameter PMMA waveguide. The increased diameter of the waveguide allows more modes to propagate through the fiber. Since the diffraction pattern is decreased in Fig. 15c it can be concluded that the number of modes in the fiber waveguide were limiting the spatial incoherence and caused diffraction. Diffraction effects are studied using the previously described variety of (in)coherent sources.
Figure 15: Images of Undisturbed tin droplets in the DG experiment. (a) uses the ASE dye light source and the 600 µm fiber, (b) uses the single cell dye light source and also the 600 µm fiber, (c) uses the single cell dye light source and the 3 mm PMMA waveguide. The decreasing trend in diffraction is visible in the horizontal profiles through the middle of the droplet. Maximum modulation on the left and right side of the droplet are not equally high due to misalignment effects discussed in the text.
4.2.1 Spatial Coherence
Diffraction due to spatial coherence is tested using the extended white light source where spatial coherence is increased by closing down an aperture. The fact that a white light source is used is irrelevant because of the optical filter in the imaging objective. The illumination setup is similar to the setup shown in Fig. 8 where the aperture is placed at the location of the condenser lens. For nine different sizes of the aperture diffraction from a knife edge is measured. Diameters of the aperture range from completely open, 50 mm down to 6.2 mm almost completely closed. Fig. 16 shows the in and 300 µm out of focus profiles of the knife edge for the largest, middle and smallest aperture size. Along with the two patterns, the diffraction pattern development through focus is shown in a heatmap. To minimize misalignment effects present in the knife edge measurements, the shadowgraphy system is aligned using a alignment grid ensuring equal diffraction on both sides of the gird. The knife edge profiles are evaluated only in an aligned region of the shadowgraph. A complete overview of the LSF width and maximum modulation behaviour through focus for every aperture size is given in Fig. 17. Especially out of focus images with a small aperture show Fresnel diffraction. It can be observed that the LSF width in focus is not influenced by the size of the illumination source. However, Fig. 17a shows that source size does influence the LSF width for out of focus objects. A bigger source increases the LSF width analogous to the circle of confusion explained in Fig 4. Regardless of
Figure 16: Knife edge profiles in and 300 µm out of focus from an extended, white light illumination source. Profiles with an aperture size of (a) 50.0 mm, (b) 19.8 mm and (c) 6.2 mm. The in and out of focus profiles are shifted with respect to each other not to overlap. Increasing diffraction effects are visible for smaller aperture sizes. The heatmaps illustrate the full development of the knife edge profile in a through focus measurement.
distance to focus, smaller sources do increase diffraction effects as shown in Fig. 17b. According to Eq. (28) the minimum source size before diffraction effects appear is 52 mm, for f = d3 λ0= 560 nm and g = 0.25 µm.
Figure 17: (a) LSF width and (b) maximum modulation as a function of position to the focus in knife edge diffraction measurements. Different size light sources are used as illumination to vary the spatial coherence. It is observed that the width of the LSF for out of focus knife edges increases with source size.The maximal diffraction overshoot decreases with increasing source size.
4.2.2 Temporal Coherence
Diffraction effects caused by temporal coherence in the dye light source is investigated. Knife edge diffraction is observed for both the single cell dye light source and a white light flashlamp. In both cases light is transported via the optical waveguides and therefore both methods have equal spatial coherence. Identical knife edge measurements are performed and the LSF width and maximum modulation overview for the different light sources using different waveguides is shown in Fig. 18. The LSF widths of the two light sources in Fig. 18a, when using the same waveguide, are identical throughout the focus scan. In focus the difference in LSF width between the two waveguides is negligable. The 3mm PMMA waveguide shows a steeper increasing LSF width compared to the 600µm fiber when the knife edge is placed out of focus. These results are comparable to the LSF behaviour for different sized extended light sources shown in Fig. 17a. comparable to the LSF widths for different aperture sizes. Fig. 18b shows the behavior of the maximum modulation for the four different illumination methods. The small fiber has an overall higher maximum modulation than the PMMA waveguide which is one again in line with the observations from the extended light source. However, the maximum diffraction modulation caused by the dye light source is higher than the modulation caused by the white light flashlamp when they are both coupled into the 600 µm fiber. The difference in diffraction is caused by the difference in temporal coherence properties of the two illumination sources. The difference in diffraction modulation between the different light sources is reduced when the PMMA waveguide is used. The larger size of the source compensates the remaining temporal coherence of the dye light.
4.3
Feature Visibility
The linepair visibility of three illumination sources is measured. The extended white light source with open aperture theoretically gives the highest resolution as it is the most incoherent illumination source. The linepair visibility of the single cell dye light source using both available waveguides is also investigated as these configurations are applicable to the LPP source of EUV light. Fig. 19 shows the linepair visibility of the three sources. To minimize misalignment artifact the alignment grid is again used to align the imaging setup. The visibility for all three illumination sources decrease as line pairs become more and more densely packed. It can be seen that the 600 µm waveguide has the lowest resolution and the extended source size with open aperture the highest. Because of the small coherence of the extended source the linepair visibility should be explained by the LSF width as proposed in Eq. (9). The knife edge measurement using the extended source indicates σ = 1.7 µm and the comparison between the expected theoretical values and observed values is shown. The results deviate from the theoretical values which can be explained by remaining misalignment
Figure 18: (a) LSF width and (b) maximum modulation as a function of position to the focus in knife edge diffraction measurements. The single cell dye laser and flashlamp sources are used in combination with both available waveguides.
of the light source, being slightly out of focus and onset of the Nyquist criterion at the high end of linepair density. Furthermore astigmatism can cause these slight deviations as the σLSF is measured with vertical
profiles and linepair visibility with horizontal profiles.
Figure 19: Decreasing visibility of increasing linepairs per millimeter density for different illumination sources. The extended white light source has the highest resolution as it is least effected by coherence effects. Theoretical linepair visibility as predicted by Eq (9) using σ = 1.7 µm from Fig. 17a.
Figure 20: Shadowgraph series of deforming tin droplet using the 3 mm PMMA waveguide to transport the light from the single cell dye light source to the DG setup. A homogeneous background and little diffraction result in shadowgraphs with the ability to clearly resolve ligamentation and droplet breakup behavior.
5
Conclusion & Outlook
This thesis demonstrates the improvement of the shadowgraphy system in the DG LPP source of EUV light by implementing a new light source. The novel illumination setup consists of a single dye cell coupled into a 3mm PMMA waveguide which transports the light to the shadowgraphy setup. It has been demonstrated that in shadowgraphy, image resolution and artifacts depend on illumination source characteristics linked to coherence properties of the illuminating light. Decreasing the wavelength of the illuminating light decreases the diffraction limit and therefore, the smallest object that can theoretically be observed. The presented light source produces ns light pulses with low temporal and spatial coherence with a wavelength of λ = 560 nm, resulting in images with a homogeneous background and little diffraction. Furthermore a more general in-vestigation on the effects of temporally and spatially coherent light sources on shadowgraphy images has been presented. The improvement of the imaging method is visible in the comparison of Fig. 13 and Fig. 20. Phenomena such as droplet ligamentation and breakup can be studied in more detail leading to new insights in fluid dynamics. However, some aspects of the shadowgraphy setup still cause undesirable effects that could be improved.
First, misalignment effects apparent in Fig. 15 are present when investigating droplet deformation and propul-sion. Misalignment effects become visible in a window of 100 µm and therefore these effects remain present in the current shadowgraphy system. In order to minimize the misalignment effects of the knife edge and visibility target measurements, the imaging setup was aligned aligning on the alignment grid and shadow-graph images were only evaluated in aligned sections of the shadowgrams. For droplet deformation studies however, the grid alignment procedure is not possible and the illumination light has to be aligned to the droplet itself. As the droplet is propelled and deformed, it flies out of the aligned shadowgraph area and therefore misalignment effects will still be apparent in shadowgraphs investigating droplet propulsion and deformation dynamics. Increasing the source size decreases the misalignment effects and further improve-ment of the shadowgraphy imaging setup can be achieved by further increasing the source size.
Secondly, edge definition remains ambiguous. A proper threshold definition is useful for droplet and and droplet-fragment size determination. As mentioned before, Gaussian Blurring and Fresnel diffraction give different intensity levels at the physical edge location. As the shadowgraphy setup suffers from a combina-tion of both the proper threshold level is between 0.25 and 0.5 times the background intensity. There is no experimental way of determining what the appropriate threshold level is since the exact position of the knife edge is unknown in the shadowgraph. A threshold value can be found by simulating the system in an optical simulation toolbox that allows partial coherent imaging.
The results shown in Fig. 17 seem to indicate a trade-off between out-of-focus edge sharpness and the diffrac-tion effects for the choice of source size. The larger the source size the less the diffracdiffrac-tion; however, addidiffrac-tional blurring is observed caused by the circle of confusion when the object is out of focus. However, the apparent trade-off can be used to establish a more precise vollumetric understanding of the deforming droplet. For this purpose, two image evaluation methods can be employed to achieve three dimensional picture using the shadowgraphy images. The first consists of employing diffraction imaging: as objects are further out of focus focus, diffraction patterns will appear different on the shadowgraphs and the position can be calculated using back propagation [21]. A certain degree of coherence is required for this method as it uses diffraction patterns in order to determine the position of the out of focus object. The second method determines the position of out-of-focus object by using the circle of confusion [22]. In this case, the size of the circle of confusion as well as its intensity compared to the background intensity allow determination of the size and distance to focus of the object. It is noteworthy that the choice of volumetric visualisation method in this procedure depends solely on the source size and thereby surpasses the previously mentioned trade-off.
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