Light in Bulk Matter - 8-11 keV 穿透式X 光顯微鏡系統中Zone plate 傾斜之研究

The refractive index is defined as:

v

n c  











0 (1.1)

, where v is the phase velocity of light in the medium, c is the speed of light in vacuum,



is the permittivity,  is the permeability,



₀ and



₀ are the permittivity and permeability of free space (



_r 



₀ and



_r 



₀). In fact, the refractive index is frequency dependent. The

dependence of n on wavelength/ frequency of light is the effect of dispersion.

When light waves impinge on a medium, they will interact with the atoms in the medium. Each atom can be thought as a classic forced oscillator being driven by the time-varying electric force

) (t

E

q

F

_E  _e .

3 An electron which is displaced by a small distance x from equilibrium suffers the restoring force

x

m



_s² and damping force

m_e dx , where



_s is the resonance frequency, and x is the displacement between the negative cloud and the positive nucleus. If the directions of the restoring force and damping force of the electron are opposite to the direction of the driving force, by Newton’s second law:

If the frequency of the electric field is



, the light wave can be expressed as

) exp(

)

(

t E

₀

j t

E





, and we can anticipate that the electron will oscillate at the same frequency.

We assume the solution of displacement x is:

) 1.3 The Dispersion Equation and Complex Refractive Index

The density of dipole moments P is:

P





₀

 E



q

xN

(1.4) where N is the density of electrons (number of electrons per unit volume). By eq. (1.3), we can rewrite eq. (1.4) as:

Eq. (1.1) is known as the dispersion equation. We can see that, the refractive index is a function of the frequency.

If there are N molecules per unit volume, each with fj oscillators having resonant frequency, where j=1, 2, 3, …, and the dispersion equation can be expressed as:

, where fi terms are weighting factors, and they satisfy the requirement:



f

i ^1

(1.9)

f

i terms are known as transition probabilities or oscillator strengths.

If ² is much larger than

Since the frequencies of X-rays are large, the refractive indices are approximately unity. Besides,



is usually positive, and the real part of refractive index is slightly less than1. By Snell’s law, the refractive angle will be larger than the incident angle. Therefore, X-ray can’t be focused by ordinary lenses, and conventional refractive optics is not suitable for X-ray.

The imaginary part of the refractive index, , governs the absorption. Consider a light wave the wave vector,

2 2 ( 1 ) ( 1 )

k . As a result, the optical field will become:

The first term is the original wave in vacuum, the second term is due to the phase retardation, and the third term is the attenuation. The intensity of the wave is:

The amount of phase retardation is:

6 

 k

₀

 r

(1.16) , and the absorbed energy is:

E

₀²(1

e

^²^k^^r) (1.17)

Chapter 2 Transmission X-ray Microscope (TXM) at NSRRC

Transmission X-ray microscope (TXM) is one of the X-ray microscopes.

NSRRC’s TXM was designed and installed in September 2004 by NSRRC and Xradia, it was the first zone plate based hard X-ray microscope operated at 8-11 keV. The resolution of this TXM is about 60 nm in hard X-ray region.

My experiment is performed in this end station. The structure, optical components and principles of the TXM system will be introduced in this chapter.

2.1 The Structure of TXM

Fig. 2-1 The structure of TXM system. The zone plate can be regarded as a X-ray lens.

The structure of TXM system is shown in Fig. 2-1. X-ray come out from the synchrotron, pass through the energy filter and the condenser. Photons with energy from 8 to 11 keV are filtered through the energy filter. After passing through the condenser, X-rays become convergent, hollow cone, and focused on the sample. The zone plate which is put behind the sample acts as an objective lens in microscope. In the scintillated system, the scintillator converts X-ray to visible light, and the CCD gets the image. By Fourier theory, the optical wave distribution can be calculated, if we know the original wave function, the distances between each element, and the transparencies of the sample and zone plate. Those elements in TXM system will be introduced in the following sections.

2.1.1 The Light Source of TXM System

The Taiwan Light Source (TLS) is a 1.5GeV synchrotron, constructed in October 1993 in Hsin Chu, Taiwan, also and it was the first third generation synchrotron in Asia. The NSRRC’s TXM is in beam line BL01B end station.

Synchrotron is a kind of cyclic particle accelerator. The electric and the magnetic fields are carefully synchronized with the traveling particle beam in the synchrotron. The electric force accelerates the particles. The particles circulate, because the magnetic force B



is perpendicular to the direction of velocity vof the particles. In order to enhance the intensity of X-ray, the insertion devices (IDs) are inserted into a straight section of the storage ring. Insertion devices are composed of arrays of magnets, and they generate spatial periodic magnetic field. When charged particles pass through an insertion device, the magnetic fields will generate transverse acceleration of the particles, which emit the synchrotron radiation in the projection direction. With the help of IDs, the synchrotron radiation is highly-brilliant, forward-directed and quasi-monochromatic. In beam line BL01B at NSRRC, an insertion device, superconducting wavelength shifter (SWLS) was installed at the downstream of the straight section.

The beam line BL01B is about 30 meters long from the front to the end station, it was composed of two optical components, the focusing mirror(FM), and the double crystal monochromator (DCM). The FM is used to focus X-ray to the sample, and the DCM which consists of two Ge(111) crystals can give the resolving power up to 1000.

The photon beams filtered through the DCM are monochromatic with energies tunable from 5keV to 20keV with an expect photon flux about

mA photons

/ sec/ 200 10

7 

¹¹ (an photon flux of

5  10

¹¹photons

/ sec/ 300

mA has been measured) with an average energy resolution (

E / E

) of 110^³. The beam spot is about 1

mm

0.3

mm

at the end of the beam line, the emittance in horizontal and vertical directions are different.

A shutter and an ion chamber are placed between the X-ray source and the TXM system. Shutter can protect the optical system from…, and the ion chamber is used to measure the X-ray flux.

2.1.2 Condenser

The condenser in TXM system is a 15 cm-long, circularly symmetric single

reflection glass capillary. The reflection angle is about 0.5mrad respect to the propagation direction, and the measured reflection angle from the condenser is from 0.87 mrad. to 1.13mrad. The total internal reflection gives the reflection up to 90% for 8keV at 0.5 mrad. The limitation of the critical angle for the condenser is about 4 mrad. for the roughness is about 10nm. The shape of the capillary is hyperbolic type and made by the capillary puller in Xradia Inc. Fig. 2-2 shows the structure of the condenser.

Fig. 2-2 The optical layout of the condenser.

The diameters of entrance and exit are about 300 μm and 200 μm, respectively.

Since the foot print of the X-ray beam is about 1

mm

0.3

mm

(horizontal  vertical) on the entrance of the condenser, the condenser will make flux loss in the horizontal.

As shown in Fig. 2-2, there is a gold bead of 100 μm which blocks X-ray at the end of the condenser, so the output of the condenser is the hollow cone beam. The beam comes out from the condenser will focus at the sample position. There is a low resolution CCD camera which can be placed in the beam path in the microscope to monitor the shape and flux of the focal spot.

2.1.3 Zone Plate

In TXM system, the zone plate is made of gold on the silicon nitride membrane by electron beam lithography, it acts as a focal lens. The distance between the zone plate and the sample is equal to the focal length of zone plate, 27mm. And the zone plate of 75 to 85  diameter is illuminated by the hollow cone beam from the m condenser. A zone plate consists of many concentric rings. Light passing through the zone plate will diffract around the rings. And the zone plate uses diffraction to focus light. The principle of zone plate will be explained later.

The zone plate has a diffraction-limited resolution:

δm=K1λ/NA=2K1dr/m (2.1) , where m is the diffraction order, and dr is the width of the outmost ring,

61

resolution of first order is about 60nm. By the way, the aspect ratio of zone plate is defined as:

Aspect ratio≡

dr h (2.2)

where h is the thickness of zone plate.

The optimal thickness of zone plate is about 1.8μm to achieve a phase shift of π at 8 keV (the aspect ratio is about 36). Since the achievable aspect ratio for gold zone plate was about 18 when TXM is installed, the fabrication of an ideal zone plate which can achieve a phase shift of πis still a big challenge. In order to achieve phase shift ofπ/2 at 8keV, the thickness of gold has to be about 890nm, the calculation is shown below:

By eq. (1.10), the complex refraction index can be expressed as n=1-δ+iβ. For gold, δ ≈4.2*10^-5 at 8 keV (Ref [2])

The aspect ratio is about 18. To fabricate a zone plate of high aspect ratio is difficult.

2.1.4 Scintillated System

The scintillated system in TXM is composed of the scintillator and the CCD detector, the scintillated system is shown in Fig. 2-3.

Fig. 2-3 The internal structure of scintillated system.

The scintillator is able to convert X-ray to the visible light. The scintillator is made of CsI with the quantum efficiency up to 40% at 8keV. There is a visible-light 20 objective lens behind the scintillator, the object lens is used to magnify the image on the scintillator. And the magnified image will be reflected to a CCD by a mirror with 45 degree. The CCD is PIXIS 1024, which has 1024 1024 pixels with readout rate of 20Mbit/sec.

There is a flight tube filled with helium between the zone plate and the

scintillated system. Since the flux of X-ray decays fast in the air, the helium is purged into the flight tube to suppress the flux decay. If the whole system can be installed in vacuum, the flux of X-ray will be maintained very well.

2.2 Wave Propagation

By Fourier optics theory, optical waves are described as a weight sum of plane waves with different spatial frequencies. We can calculate the propagation of optical waves through various optical systems. Fourier optics uses the spatial frequency domain as the conjugate of the spatial domain, and Fourier transform (FT) theory is often used as a tool to solve the problems of wave propagation.

In general, a homogeneous optical wave can be expressed as a weighted superposition of elementary plane wave solutions:

 

^









 U f

f

e

^j ^f ^x ^f ^y

df

y

x

U ( , ) ( , )

²^⁽ ^x ^y ⁾ (2.3)

, where f_x and

f are the spatial frequency, and

U

(

f

_x,

f

_y) represents a plane wave with spatial frequency of (

f

_x,

f

_y). The plane wave spectrum representation of the optical field is the basic foundation of Fourier optics. We can see that, the above equation is an inverse Fourier transform of the plane wave

U

(

f

_x,

f

_y).

respectively, as shown in Fig. 2-4. The Heygens-Fresnel principle can be stated as:

 

^

Fig. 2-4 Diffraction Geometry The optical field at P0=(x,y) is expressed as:

The above equation is seen to be a convolution, where p(x,y)=

exp( ) exp[ (

₂ ₂

) / 2 ]

is the propagator in real space. Another form of the result can be written as:

)

We refer to the forms of the result of eq. (2.7) and (2.9), as the Fresnel diffraction integral. By convolution theorem,

)}

By the results of eq. (2.7) and (2.10), we can calculate the wave function of the propagated wave, as long as we know the original wave function and the propagation distance.

2.3 Principle of Zone Plate

In section 2.1, the zone plate and other optical components in TXM system have been introduced. Zone plates were created by Augustin-Jean Fresnel, so they are also known as Fresnel zone plates. A zone plate consists of many concentric

ring-shaped zones, known as Fresnel zones. Alternate zones are open the others block the incident beam (amplitude zone plate), or advance/retard the phase of incident beam (phase zone plate).

2.3.1 Amplitude Zone Plate

(a) (b)

Fig. 2-5 (a) Binary zone plate, it focuses light at many points. (b) Sinusoidal zone plate, it has one focal point.

There are two types of amplitude zone plates in Fig. 2-5, the binary zone plate and the sinusoidal zone plate. The transmittance of a binary zone plate must be 1 or 0.

That is, the rings of binary zone plate are completely opaque. But the transmittance of a sinusoidal zone plate varies gradually in a sinusoidal manner. Besides, a binary zone plate has many focal points, but a sinusoidal zone plate has only one focal point.

Light passing through zone plate will diffract around the rings. Unlike optical or reflective type lenses, zone plates use diffraction of X-ray to focus light rather than

using refraction. If the optical waves are all in phase, the focal spot will total constructively interfere. As a result, we can see a focus spot behind the zone plate.

Fig. 2-6 A zone plate obstructs the blue paths. Only the optical waves come from red paths can pass through the zone plate. The phase differences of the waves are equal to multiples of 2π at focal point.

The radiuses of the Fresnel zones are related to the focal length of zone plate and the wavelength of incident beam. In Fig. 2-6, the blue lines and red lines are optical paths. The optical path length difference (OPD) between any two paths of the same color is a multiple of the wavelength, so the phase difference is a multiple of 2 π. That is, the optical waves from the paths of the same color are all in phase.

However, the optical waves from the paths of different colors are out of phase (the OPD is

/ 2

). As shown in Fig. 3-1, the zone plate obstructs the blue paths, the optical waves come from red paths will constructively interfere at the focal point. If we remove the red lines, let blue lines pass, it won’t change the wave distribution on the image plane. Zone plates produce equivalent diffraction patterns no matter whether the central disk is opaque or transparent.

Since the OPD between two nearby zones is 

/ 2

, the radius of the zones are required to be (Ref [3]):

4

2 2

 ⁿ

r_n

 

(2.11)

, where r_n is the radius of the nth zone. If f  n



, eq. (2.11) can be simply expressed as:

In this approximation, all zones have the same areas and contribute equally to the irradiance at the focal point.

2.3.2 Phase Zone Plate

Unlike the amplitude zone plates, the phase zone plates change the phase of incident beam. This is done by adding or subtracting the optical path length in the zones, using refractive material of appropriate thickness. The transmission function of a pure phase zone plate can be expressed as (Ref [1]):

2 sin( )]} wavelength, and Cm=1/m is the weight of each diffraction number. The term 1/2 represents the non-diffraction zero order background, and m=1, 3, 5,… represent for the first, the third, the fifth order, etc. The absolute value of T is 1, the phase zone plate doesn’t change the intensity of incident beam, it only changes the phase term.

By the transmission function, we can see that, the cross-section of zone plate is approximately rectangular shape.

There are many diffraction orders behind a phase zone plate, and they focus at different focal points, as shown in Fig. 2-7. The focal length is inverse proportional to the diffraction order, focal length of m th order is fm=f/m. As a result, the phase zone plate can be regarded as a lens with many focal points.

Fig. 2-7 Light waves diffract around zone plate, and there will be many diffraction orders behind the zone plate (It only shows the zero order, first order, and the third order in this picture). The focal length of each order is inversed proportional to the diffraction order.

As a result, the zone plate can be regarded as a lens with many focal points.

Besides, the focal length f, numerical aperture (NA), and the diffraction-limit

resolution δ of zone plate are related to the radius and the width of the outmost ring, r and dr, the diffraction order m, and the wavelength



. The following equations show the relationships (Ref[1]):

fm=f/m=

mλ

2rdr (2.14)

NA=

dr m



(2.15)

δm=

m dr

= 2K NA

K₁



₁

(2.16)

, where

0 . 3  K

₁

 0 . 61

, depending on the illumination condition. In TXM system, the resolution is given by δm=1.22dr/m.

By the equations, we can see that the higher the diffraction order, the shorter the focal length, the larger the numerical aperture, and the better the resolution.

Chapter 3 The Simulation of Wave Propagation and

Affections Due to Zone Plate Tip / Tilt in TXM

The structure of TXM and the optical components in TXM system is introduced in chapter 2. There are some simulation results in this chapter. A Matlab program is designed to calculate the optical fields in TXM system, and it also can calculate the transmission function of a tipped / tilted zone plate. There are some simulated images acquired at different tip / tilt angles, the images will be compared and the relationship between the image quality and the tip / tilt angle will be discussed.

In this chapter, the calculation of X-ray optical fields in TXM system is based on the paraxial approximation in scalar wave field theory (Ref. [4]). The

approximation requires that the propagation distances are much larger than the wavelength and that the sample and zone plate can be described by thin element transmission functions. Besides, quasi-monochromatic object illumination conditions are assumed. We set the wave length of X-ray to be 1.5 angstrom (the photon energy is about 8.27keV) in our calculation and simulation.

3.1 Wave Propagation in TXM System

3.1.1 Calculations of Optical Fields

The optical fields in TXM system can be calculated by Fourier optics theory which has been introduced in section 2.2.

Fig. 3-1 Optical wave propagates from the condenser to the image plane. The

19 transparencies of the condenser, the sample and the zone plate.

A part of the TXM system (from the condenser to the image plane of the zone plate), and the optical fields at many positions are shown in Fig. 3-1. U1 is the optical field at the output of the condenser, it’s convergent, hollow cone beam. The wave function U1 can be calculated by multiplying a spherical wave U0 by the transparency T condenser. That is,

T_condenser ^bead ^condenser

,

We can calculate the optical field U3 by multiple U2 and the transparency of the sample T sample.

3.1.2 Wave Distributions

By eq. (3.1-11), if the distances z1, z2, z3, the optical field U1, and the

transparencies of sample and zone plate are know, the optical field at each position in Fig. 3-1 can be calculated. The calculations were implemented by using a Matlab code, and there is an example in this section. In this example, the wavelength of X-ray is 1.5 Å (the energy of a photon is about 8keV), the focal length of zone plate

f=27mm, z1=f=27mm, z2= f

9

10

, z3=10f, R_condenser 30.5



m, R_bead 27.5



(a) (b)

(e) (f)

(g) (h)

(i) (j)

(k) (l)

(m) (n)

(o) (p)

Fig. 3-2 (a) The amplitude of U1, | U1|. (b) The phase of U1. (c) The amplitude of U2 ,

|U2|. (d) The magnified version of (c). (e) The transparency of the sample,

Tsample. (f) The magnified version of (f). (g) The amplitude of U3 , | U3| (h) The magnified version. (i) The amplitude of U4 , | U4|. (j) The phase of U4. (k) The imaginary part of the transparency of a phase zone plate, Im{TZP}. The focal length of this zone plate is 27 mm at 8keV. (l) The magnified version of (k). (m) The amplitude of U5 , | U5|. (n) The phase of U5. (o) The amplitude of U6 , | U6|.

(p) The magnified version of (o).

The transparency of the sample, Tsample, is shown in Fig. 3-2 (e) and (f), it is reversal “NCTU”. Fig. 3-2 (k), (l) show the imaginary part of the phase zone plate with 27mm focal length at 8 keV, we can see that the zone plate is composed of many concentric rings.

Since the optical field U1 at z=0 is a convergent spherical wave with curvature R=z1, after it propagates for a distance z1, the beam will be focused to be a small spot on the sample; and the optical field at z= z1 is U2, as shown is Fig. 3-2 (c) and (d). The sample filters some X-rays out, so the shape of U3 is like the sample. After the beam propagates for z2, the wave distribution of U4 is blurred, as shown is Fig. 3-2 (i). The

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