One of the main application of x-rayFTH lies in the field of imaging magnetic samples with structure sizes between 50 nm and 1 µm. Magnetic contrast with x-rays is usually provided by the x-ray magnetic circular dichroism (XMCD). This effect describes a change of the resonant atomic scattering factor near absorption edges of a magnetized material in dependence on the helicity of the incident circularly polarized photons. Fundamentally, theXMCDgoes back to a difference in the transition probability of a core electron to the first unoccupied spin-polarized states above the Fermi level when absorbing a left or right circularly polarized photon. The spin-polarization of the valence band, e.g. the 3dband in case of the elements Fe, Co, and Ni, is caused by the Stoner energy splitting of the spin-up and spin-down states which in fact is the origin of the material’s magnetization. A very thorough explanation of theXMCDeffect can be found in the book of Stöhr and Siegmann [Sto06]. Here, the influence on the radiation is again described in terms of absorption and phase shift. The definition of the refractive index (Eq. 2.4) for right (n+) and left (n−) circularly polarized light is therefore modified for the influence of theXMCDeffect in the way:
n±(ω) = 1−(δ0±(εk·m)∆δ) + i(β0±(εk·m)∆β). (2.42) Here, δ0 andβ0 denote the optical constants for unpolarized or linearly polarized light, where theXMCDeffect is not present. The sign of the corrections for circular polarization
±∆δ and ±∆β to these constants depends on the helicity of the incident radiation. Similar to the usual optical constants, the correction constants are strongly dependent on the
1 It is very common to describe optical imaging system via transfer functions [Goo05]. As most imaging systems, e.g. microscopes and photo-cameras, rely on incoherent imaging and, thus, on the transport of intensity, usually the optical transfer function (OTF) is used which is the normalized auto-correlation of the amplitude transfer function. The modulus of theOTFis called themodulation transfer function (MTF).
photon energy (eV)
780 790 800
(10
-3 ) 1
-1
-2
5
0
-5 0
(10
-3 )
Figure 2.5: The optical constantsδ0 andβ0 of Co in the vicinity of the L2and L3 absorp-tion edges and the dichroic correcabsorp-tions∆δ and∆β for magnetized media. The figure was reproduced from Ref. [Mer04].
photon energy (Fig. 2.5). The strength of the XMCDeffect is additionally given by the orientation of the light’s propagation direction εk = k/|k| (which is usually identified with the z-direction in this thesis) and the sample’s magnetization direction m. The orientations are connected with the scalar product, i.e. the effect is maximized for parallel or anti-parallel orientation and vanishes if the magnetization is perpendicular to the light propagation. Using Eq. 2.42 it is straight forward to obtain the transmission function for magnetic samples, following the consideration for Eq. 2.9:
t±(x, y)'1−µ±(x, y)−iφ±(x, y), (2.43)
with the helicity dependent constants:
µ±(x, y) =µ0(x, y)±∆µ(x, y) (2.44)
= 2π λ
Zd
0
[β0(x, y, z)±(εz·m(x, y, z))∆β(x, y, z)] dz , (2.45)
φ±(x, y) =φ0(x, y)±∆φ(x, y) (2.46)
= 2π λ
Zd
0
[δ0(x, y, z)±(εz·m(x, y, z))∆δ(x, y, z)] dz . (2.47) The constantsµ0(x, y) andφ0(x, y) cover non-magnetic contributions, e.g. the height or material topology, whereas∆µ(x, y) and ∆φ(x, y) provide the magnetic morphology of the sample. Usually the magnetic signal is weak compared to the non-magnetic variation.
For this reason, it is favorable to remove all non-magnetic contributions by recording holograms with both positive and negative helicity of the light. The images showing magnetic contrast are then reconstructed from the difference of both holograms:
p= 1
2F−1{(F {bρ+o−})∗F {bρ+o−} −(F {bρ+o+})∗F {bρ+o+}} (2.48) p= 1
2 o∗−∗o−−o∗+∗o++b∗ρ∗∗o−+bo∗−∗ρ−b∗ρ∗∗o+−bo∗+∗ρ , (2.49) where o+ and o− characterize the sample’s exit wave for positive and negative helicity of the light, respectively. The reference exit waveρ shall be independent on the photon polarization.1 Focusing on the image reconstruction formed by the reference–object cross-correlation and putting in the transmission function yields by analogy with Eq. 2.24:
pi(x, y) = 1
2{b∗(1−µ−(x, y)−iφ−(x, y))−b∗(1−µ+(x, y)−iφ+(x, y))} (2.50)
= 1
2{b∗(µ+(x, y) + iφ+(x, y)−µ−(x, y)−iφ−(x, y))} (2.51)
=b∗(∆µ(x, y) + i∆φ(x, y)). (2.52)
And analogously, the twin image:
p∗i(x, y) =b(∆µ(x, y)−i∆φ(x, y)). (2.53)
1 The case of using a “magnetic” reference, i.e. a reference wave that is subjected to theXMCDeffect, is of special interest when imaging with linearly polarized light, but still searching for magnetic contrast [Sac12].
The reconstruction of the difference hologram, thus, outputs an image containing purely magnetic information without any cross-talk from the sample topography. The magnetic contrast is observed either in the real part or the imaginary part of the reconstructed image:1
Repi(x, y) =|b|∆µ(x, y), (2.54)
Impi(x, y) =|b|∆φ(x, y). (2.55)
The magnitude|pi(x, y)|=|b|p∆µ2(x, y) +∆φ2(x, y) is not an appropriate channel as the sign of ∆µ and∆φreflecting the magnetization orientation is lost. The same holds for the phase argpi(x, y) as it only reflects the sign of the magnetization and—under experimental conditions including noise—becomes imprecise or even undefined for positions with low or zero magnitude. In the real and imaginary part, the reconstructed intensity values are proportional to the z-component of the material’s magnetization.
Nevertheless using the real or imaginary part for magnetic contrast still has a number of drawbacks: (i) Since the dichroic parts of optical constants are heavily changing with the photon energy (Fig. 2.5), the magnetic contrast found either in the real or imaginary part is changing as well or may even vanish [Sch07a]. (ii) Real part and imaginary part are sensitive to an exactly known position of q=0 in the hologram in order to avoid any artificial phase modulation in the reconstruction. In a digital hologram, the position must be known with sub-pixel accuracy. (iii) If the illumination of the sample is not a plane wave, the reference wave will have a certain phase differenceβ [Sta08], i.e.b=|b|exp(iβ).
The reconstruction will be affected by this phase shift and the observed magnetic contrast in the real and imaginary part is altered.
The way to analyze magnetic FTH images proposed in this work is to use a signed magnitude—a property which is insensitive to the implications above stated. The value is calculated either by rotating the reconstructed wave by an angle −α in the complex plane in a way that the real part shows the maximum contrast:
M{pi(x, y)}= Re{pi(x, y) exp(−iα)} (2.56)
=|b|(∆µ(x, y) cosα+∆φ(x, y) sinα) (2.57) or by calculating the magnitude signed by the sign of the real part:
M{pi(x, y)}= sgn∆µ(x, y)|pi(x, y)| (2.58)
=|b| ∆µ(x, y)
|∆µ(x, y)| q
∆µ2(x, y) +∆φ2(x, y). (2.59) In the ideal case, the angle α in Eq. 2.57 is simply given by the phase of the reconstructed image at a certain position (x, y), i.e. α(x, y) = argpi(x, y). In most cases, it is sufficient
1 It is assumed thatbhas only a small phase difference, i.e.|b| 'Reb.
i
1 -1
-i
i
1 -1
-i
1 -1
(a) (b) (c)
(d)
Figure 2.6: Illustration of the signed magnitude. (a)Model sample consisting of one up and one down magnetized region (domain). (b)Reconstructed exit wave values presented in the complex plane. Magnetic contrast is observed in the real and in the imaginary part of the reconstruction. The non-zero phase shift can be caused by (i) energy dependence of the optical constants, (ii) imprecise centering of the hologram, or (iii) a non-plane-wave illumination of the sample. The real and the imaginary parts have opposite signs, but equal magnitude in oppositely magnetized regions. The signed magnitude is calculated(c)by either rotating the reconstructed image by an angleαin the complex plane or(d)by using the magnitude together with the sign of the real part.
to choose a constant angle (α(x, y) =α0) for the whole reconstructed image. In the second procedure (Eq. 2.59), the signed magnitude is always calculated locally.1 Taking the real part as the reference for choosing the sign of the magnitude is motivated by the fact, that∆β retains its sign over the L2 orL3 absorption edge whereas ∆δ flips its sign where
∆β reaches its maximum magnitude (Fig. 2.5). It is important to note that the linearity between reconstructed intensity andz-component of the magnetization is not affected by the signed magnitude procedure.
If the illumination function features a certain phase difference β between the positions of the object and the reference source, the reconstructed image (Eq. 2.52) is affected by an additional phase term:
pi(x, y) =|b|exp(−iβ)(∆µ(x, y) + i∆φ(x, y)). (2.60) This phase difference can be removed by the signed magnitude procedure as well. In the case of procedure Eq. 2.57, one has to additionally rotate by the angleβ, i.e. to multiply with exp(iβ). In Eq. 2.59, the phase term is already effectively removed by calculating the magnitude ofpi(x, y).
In principle, the signed magnitude also removes phase gradients in the reconstructed image due to an improper centering of the hologram. But in practice, it turns out that it is more favorable to first remove the phase gradient as described in Ref. [SN09] and then take the signed magnitude using a constant angleα0 for the whole reconstructed image.
1 In practice, “locally” refers to a single pixel in the digital reconstruction matrix.
In order to promote a better understanding of the signed magnitude contrast, a special, but experimentally important case of sample system shall be considered in the following.
The sample shall consist of a thin-film, laterally uniform magnetic layer. Thus, the sample does not exhibit any element or topological contrast. Also the magnitude of the magnetization shall be considered as constant with the direction of magnetization mainly pointing out-of-plane, i.e. either parallel or anti-parallel to the k-vector of the light with normal incidence to the film surface. At the boundary between oppositely magnetized domains, the magnetization direction gradually rotates by 180° over the width of the so-called domain wall. In this model case, the dichroic part of the constants in Eq. 2.45 and Eq. 2.47 and the magnetic contrast in Eq. 2.52 is solely varying with the cosine of the angle γ(x, y) enclosed by the magnetization and the light propagation vector.
pi(x, y) =b∗2πd
λ (∆β+ i∆δ) cosγ(x, y). (2.61)
When only considering the oppositely magnetized domains (cosγ=±1) and neglecting the domain walls, one obtains:
pi(x, y) =±b∗2πd
λ (∆β+ i∆δ). (2.62)
In Fig. 2.6 this situation is illustrated by considering a model sample consisting of two oppositely magnetized domains denoted by black and white color and by up and down arrows. The reconstructed complex exit wave values presented in the complex plane have equal magnitude but opposite sign. The strongest contrast is found when using the values for the magnitude, but provided with different sign in order to distinguish the differently magnetized regions.
The advantage of the signed-magnitude contrast representation becomes particularly apparent when monitoring the reconstructed magnetic information using different photon energies which is shown in Fig. 2.7. The energy-dependent values for ∆δ and ∆β have been taken from Ref. [Mer04] (Fig. 2.5). The calculation of the reconstructed magnetic contrast of the model sample reveals that the contrast vanishes in the real or imaginary part of the reconstructed exit wave for certain energies and in case of the imaginary part even switches the sign. On the other hand, the signed magnitude always gives the best combined contrast.
Experimentally, the new signed magnitude procedure was first applied in Ref. [Pfa11].
In the next chapter all images presented have been processed in this way.
779.0 eV 780.0 eV 781.1 eV 781.8 eV 783.8 eV
i i i i i
-i -i -i -i
-1 -1 -1 -1 -1
-i
1 1 1 1 1
Im
Re
Figure 2.7: Calculated magnetic contrast as observed in the imaginary part, in the real part and in the signed magnitude of the reconstruction at different photon energies around an absorption edge, in this case the Co L3-edge. The values for the optical constants were taken from Ref. [Mer04] as presented in Fig. 2.5. The model sample is the same as used in Fig. 2.6. The first image row shows the reconstructed values for two oppositely magnetized regions in the complex plane. The images beneath illustrate the reconstruction contrast yielded in the different channels. At every energy, the signed magnitude gives the highest contrast with consistent sign.