exceed N/4 (orM/4 in the y-direction). In particular, this ratio is independent of the wavelength, sample–detector distance and detector pixel size, although for a particular wavelength and a particular detector a proper distance has to be chosen to achieve the optimum ratio in the experiment.
where the complex ˜ψ(Q, ω) describe the amplitude and phase of each spectral component.
As polarization effects should be omitted for now ˜ψis treated as a scalar. The intensity of the light field at the positionQ is given by:
I(Q) =hψ∗(Q, t)ψ(Q, t)i, (2.84)
where the angle brackets denote an averaging over times much longer than the temporal fluctuations of the electromagnetic field.
It is now assumed that the light reaches Q through two point-like apertures at the positionsP1 and P2 (Fig. 2.12). The light field inQ is then expressed by a superposition of the light field at the positionsP1 andP2, each suitably propagated and delayed:
ψ(Q, t) =K1ψ(P1, t−t1) +K2ψ(P2, t−t2) . (2.85) K1 andK2 are complex constants called the propagators. Basically, they are dependent on the size of the pinholes and on the distances (ri) between the pointsPi and Q. The time delaysti are readily calculated from these distances asti =ri/c. At this point, it shall already be noted that this arrangement of two pinholes and a screen (detector) will not only serve as a model for deriving the coherence properties of the light field as done in the following, but this model is already suitable for also obtaining the influence of partial coherence on the optical system’sPSF.
Using Eq. 2.84, the intensity of the light atQ becomes:
I(Q) =K1∗K1hψ∗(P1, t−t1)ψ(P1, t−t1)i+K2∗K2hψ∗(P2, t−t2)ψ(P2, t−t2)i +K1∗K2hψ∗(P1, t−t1)ψ(P2, t−t2)i
+K2∗K1hψ∗(P2, t−t2)ψ(P1, t−t1)i (2.86) The first two terms can be identified by the intensities (Ii(Q)) each pinhole would separately generate at the positionQ. Additionally, for the cross-correlation terms the definition of themutual coherence function is introduced:
Γ12(τ) =hψ∗(P1, t+τ)ψ(P2, t)i, (2.87)
source
detector plane P
1
P
2
Q r
1
r
2
Figure 2.12: Geometry for deriv-ing the general interference law.
Reproduced from Ref. [Goo85].
withτ being the time delay between both beams inQ. Eq. 2.86 can be now written as:
I(Q) =I1(Q) +I2(Q) + 2|K1K2|ReΓ12(τ). (2.88) If both pinholes coincide (P1=P2) one obtains the self-coherence of the light field:
Γ11(τ) =hψ∗(P1, t+τ)ψ(P1, t)i, (2.89) and one further realizes that:
Γ11(0) =I1(P1) and I1(Q) =|K1|2I1(P1) =|K1|2Γ11(0). (2.90) It is thus common to normalize the mutual coherence function in such a way that:
γ12(τ) = Γ12(τ)
pΓ11(0)Γ22(0). (2.91)
The correlation functionγ12 is called the complex degree of coherence. Now Eq. 2.88 turns into the general interference law for stationary optical fields:
I(Q) =I1(Q) +I2(Q) + 2qI1(Q)I2(Q) Reγ12(τ). (2.92) Both, the temporal and spatial coherence properties are included in the single factorγ12(τ).
In order to point out the impact of γ12(τ) on the interference pattern, it is convenient to use a somewhat different expression. Let ¯ω be the center frequency of the light spectrum;
one can always write:
γ12(τ) =|γ12(τ)|exp{−i[¯ωτ −α12(τ)]}, (2.93) if
α12(τ) = ¯ωτ+ argγ12(τ), (2.94)
and the interference law becomes:
I(Q) =I1(Q) +I2(Q) + 2qI1(Q)I2(Q)|γ12(τ)|cos[¯ωτ −α12(τ)]. (2.95) When considering a sufficiently narrow spectrum of the light, the phase factor exp(−i¯ωτ) will produce the well known cosine-shaped interference fringes due to the time delay between the two beams. The remaining part |γ12(τ)|will limit the contrast between the maxima and minima of the interference pattern by an slowly varying envelope which is directly related to the spectrum of the radiation and to the spatial coherence between the points P1 and P2. Thus, by measuring the interference contrast commonly known as the visibility, it is possible to directly determine γ12(τ) experimentally. For a fully coherent beam |γ12(τ)|equals unity, for partially coherence it is smaller and it vanishes
for incoherent conditions.
In order to gain a better understanding ofγ12(τ) it is useful to separately discuss the temporal and spatial coherence properties.
2.7.2 Temporal coherence
For the analysis of the temporal coherence, the previously discussed arrangement is first reduced to the case were the position of the two pinholes coincides, but a certain time delay betweens the beams is introduced. This arrangement is known as the Michelson interferometer. In this case the interference law in Eq. 2.92 reduces to:
I(Q) = 2I1(Q) (1 + Reγ(τ)) (2.96)
whereγ(τ) is the normalized time-correlation function of the light field, called the normal-ized self-coherence function, and is given by
γ(τ) = Γ(τ)
Γ(0) with Γ(τ) =hψ∗(Q, t+τ)ψ(Q, t)i. (2.97) Via a Fourier transform both quantities are connected to spectral properties of the light field:
Γ(τ) =
∞
Z
−∞
S(ω) exp(−iωτ)dω and γ(τ) =
∞
Z
−∞
s(ω) exp(−iωτ)dω . (2.98) S(ω) is called thespectral density orpower spectrumof the light field,s(ω) is thenormalized spectral density. Eq. 2.98 is known as the Wiener-Khintchine theorem. Using Eq. 2.83 one also findsS(ω) =|ψ(ω)˜ |2. Thus, the spectral density can be also seen as a frequency-space analogue to the intensity and the interference law (Eq. 2.92) exists for the spectral density as well:
S(Q, ω) =S1(Q, ω) +S2(Q, ω) +qS1(Q, ω)S2(Q, ω)w12(ω), (2.99) where w12(ω) is called the spectral degree of coherence. It is the normalized version of the cross-spectral density given by the Fourier transform of the mutual coherence function:
w12= W12(ω)
pS1(Q, ω)S2(Q, ω) with W12=
∞
Z
−∞
Γ12(τ) exp(−iωτ)dω . (2.100) Although the following discussion will focus on narrow-band light, it shall be briefly commented also on interference of broad-band illumination as it is described by the spectral interference law in Eq. 2.99. While in a double slit experiment using monochromatic light, the intensity is modulated on the detector as expressed by Eq. 2.92, in the case of a white
illumination the spectrum of the light is affected. This effect is well known from, e.g., the dispersion of light at a diffraction grating. There have been attempts to use a white x-ray illumination together with an energy-dispersive detector for coherent diffraction experiments as well [Pie05].
Coming back to the case of a narrow light spectrum s(ω) with a center frequency ¯ω, one can transform Eq. 2.96 analogously to Eq. 2.95:
I(Q) = 2I1(Q){1 +|γ(τ)|cos[¯ωτ −α(τ)]}. (2.101) When scanning the delayτ between the two beams the intensity oscillates according to cos(¯ωτ). In addition the amplitude of the oscillation is modulated by|γ(τ)|which is given by the Fourier transform of the spectral density (Eq. 2.98). Depending on the shape of the spectrum, e.g. Gaussian, Lorentzian or rectangular, the envelope of the oscillation also changes its appearance. However, the width of the envelope is always on the order of τc'1/∆ν with∆ν being the frequency bandwidth of the light spectrum. τcis called the coherence time. From the coherence time the longitudinal coherence length is obtained as:
ll=cτc'¯λ2/∆λ . (2.102)
At a soft x-ray synchrotron beamline where the light is delivered with a spectral purity (E/∆E) of more than 1000 the longitudinal coherence lengths are on the order of 1 µm to 10 µm.
2.7.3 Spatial coherence
The spatial coherence properties of a light source are best investigated under quasi-monochromatic conditions. These conditions are fulfilled if the spectrum of the illumination is so narrow that the coherence time is much longer than the maximum time delay occurring in the experiment. In this case, the complex degree of coherence can be rewritten as:
γ12(τ)∼=µ12exp(i¯ωτ), (2.103)
with
µ12=γ12(0) = hψ∗(P1, t)ψ(P2, t)i
pI1(P1)I2(P2) , (2.104)
which is the normalized spatial correlation function of the light field, called thecomplex coherence factor. In general,µ12 depends on the nature of the source and its properties.
But for the case of an incoherent (chaotic) source which consists of an extended collection of independent radiators, the complex coherence factor is related to the intensity distribution
of the sourceI(u, v) by the van Cittert-Zernicke theorem:
µ(∆x,∆y) = exp(−iφ)RR∞−∞I(u, v) exphi2π¯λz(∆x u+∆y v)idudv RR∞
−∞I(u, v) dudv . (2.105)
The theorem states that for a chaotic source (i) the coherence factor depends only on the relative position (∆x,∆y) of the two points P1 and P2 in the sample plane and not on their absolute positions1 and (ii) the coherence factor is given by the normalized Fourier transform of the source’s spatial intensity distributionI(u, v). By analogy with the coherence time, one can also define a transversal coherence areaAtr at the distance z from the source which covers the transverse areaAS:2
Atr= λ¯2z2
AS . (2.106)
Experimentally, the coherence factor was determined only for very few synchrotron beamlines [Pat01; Sin08; Tra05; Tra07]. The spatial coherence strongly depends on the beamline slit settings defining the source size, on the photon energy [Len01], and on the overall geometry. The relative distance of two pointsP1 andP2 where the coherence factor falls below 0.5 is typically on the order of 20 µm for a synchrotron beamline operated at 1 keV to 2 keV.
2.7.4 Implications for holographic imaging
In the following, the principles of partial coherence as outlined in the previous sections shall be applied to the case ofFTH imaging. By analogy with Sec. 2.6, the influence of partial coherence on thePSF(Sec. 2.4) of the imaging system is investigated. Again, the reference source as well as the object are assumed to be δ-like transmission structures separated by the vectorr0. Following the general interference law (Eq. 2.92) the signal measured on the detector is then given by:
I(q) =I1{2 +γ12(τ) +γ12∗ (τ)}. (2.107) The intensity I1 is identified with the intensity of the light field propagated from the pinhole to the detector. For a δ-like pinhole the intensity is constant over the whole q-space. Due to the interference of the light emitted by the two pinholes, the intensity distribution is modulated according to the coherence properties of the light expressed in γ12. The occurring time delays τ depend on the position on the detector which is, in turn,
1 The phase factor exp(−iφ) arises due to a path length difference from the source to the pointsP1and P2 and can be neglected for large distances from the source.
2 In one dimension, it is possible to define the transversal coherence length as: ltr = ¯λz/lS wherelS
denotes the source size in this dimension.
represented by the momentum transferq. TheFTHreconstruction is obtained via a Fourier transform of the intensity pattern. As already outlined (Eq. 2.23), the cross-correlations γ12(τ) and γ12∗ (τ) form the image and the twin image, respectively, while the constant offset 2I1 results in the object’s auto-correlation. The PSF of the real image is thus given by:
hreal(r0;r0) =F−1{γ12(τ)}. (2.108)
Under certain conditions, in particular including a narrow spectrum of the light, one finds that the coherence factor γ12(τ) separates into a temporal and a spatial part as discussed in Secs. 2.7.2 and 2.7.3. This separation called reducibility of the coherence factor is in detail treated in Ref. [Goo85] and is assumed to be valid for the following calculation:
γ12(τ) =µ12γ(τ). (2.109)
In addition, it is assumed that µ12 only depends on the separation r0 of both pinholes, but not on their absolute positions, i.e. µ12 = µ(r0), as it is suggested by the van Cittert-Zernicke theorem (Eq. 2.105). Expanding the Fourier transform gives:
hreal(r0;r0) =µ(r0)
∞
Z Z
−∞
γ(τ) exp(i ¯qr0) d ¯q. (2.110) Of course, one has to be aware that τ depends on the position on the detector, i.e. on q:
¯
ωτ = ¯qr0. (2.111)
Quantities tagged by a bar again represent the properties of the light in the center the light spectrum. It appears to be favorable to use a coordinate system aligned to the vector r0, where eq denotes the unit vector parallel to r0, and e⊥ denotes the perpendicular direction:
¯
q= ¯qqeq+ ¯q⊥e⊥, r0 =rq0eq+r0⊥e⊥ and r0=r0eq, (2.112) which leads to:
hreal(r0q, r0⊥;r0) =µ(r0)
∞
Z Z
−∞
γ(τ) exp(i(¯qqrq0+ ¯q⊥r0⊥)) d¯qqd¯q⊥. (2.113)
As τ only depends on ¯qq the integration over ¯q⊥ trivially ends up in aδ-function δ(r0⊥).
The integration over ¯qq is changed to an integration over τ by using Eq. 2.111:
hreal(r0q, r0⊥;r0) =µ(r0)ω¯ r0δ(r0⊥)
∞
Z Z
−∞
γ(τ) expir0q r0ωτ¯
dτ . (2.114)
Using the Wiener-Khintchine theorem (Eq. 2.98) readily gives:
hreal(r0q, r0⊥;r0) =µ(r0) ω¯ r0 s
rq0 r0ω¯
δ(r⊥0 ). (2.115)
This result describes the FTH reconstruction of a δ-like object under partial coherence conditions. The spatial coherence is reflected by the factorµ(r0).1 A reduction of the spatial coherence only reduces the contrast of the reconstruction, but is not acting on the resolution.
The influence of the temporal coherence in Eq. 2.115 on the reconstruction is much more severe, as it also affects the spatial resolution. Along the direction of the vectorr0,
(a) (b)
0 0.2 0.4 0.6 0.8 1
0 2 4 6 8 10 12 14
position ( m)
normalized intensity
(c)
20 m
Figure 2.13: Influence of spectral broadening at anFELsource. The images in (a)and (b)were recorded using the same test object, but with different single pulses of the x-ray
source. As the spectrum of the the pulses is varying from shot to shot also the image qual-ity changes. Due to the broad spectrum of the radiation, the reconstruction is smeared out along the direction defined by the vector connecting object and reference aperture. The red arrows point in that direction. (c)Intensity slices through a single object dot for both reconstructions as indicated in panels (a) and (b). The data have been already published in [Pfa10a].
1 The factor ¯ω/r0 in Eq. 2.115 appears as a dimensional scaling factor. As the coherence factorγ12(τ) can be interpreted as the fringe visibility in the hologram, it is dimensionless. Its Fourier transform (from reciprocal space to direct space) will thus need to have the dimension of 1/m. However, the spectral densitys(ω) has the unit “per Hertz”, i.e. seconds.
the reconstruction is modulated with the spectrum of the light s(ω).1 When assuming a peak-like spectrum around the center frequency ¯ω with the width ∆ω, the object is still reconstructed at the position r0q = r0 but the reconstruction is broadened by the factor ∆r0 = ∆ωωr0 [Pfa10a]. At a synchrotron beamline with a typical energy resolution E/∆E=ω/∆ω better than 1000, the broadening is on the order of 1 nm to 5 nm for object–
reference spacings below 10 µm. This situation changes when performing experiments at non-monochromatized beamlines at FEL sources with a spectral bandwidth of up to 1 %.
In this case, the expected broadening of a few tens of nanometers is comparable to the resolution limits set by the reference aperture and the detector size.
Such an example is presented in Fig. 2.13 from experiments performed at the FEL
FLASH in Hamburg [Pfa10a]. The figure shows two reconstructions of the same object—a test object consisting of the number “13” written in a dot matrix.2 The images were recorded with different single pulses from the source. Due to the variation in the spectrum of the x-ray pulses [Ack07], the reconstructions differ from each other. The influence of the spectral width becomes visible by a one-dimensional broadening of the dots in the direction of the vector connecting reference and sample position. Note that this effect could be used for a single-shot compatible analysis of the wavelength spectrum of the source when the geometry of the sample is optimized for this purpose [Fle11].