In practice, the illumination sources used by each form of ptychography exhibit a limited spatial and temporal coherence. The rarity of ideally coherent electron and X-ray sources has led to the theoretical and experimental examination of coherence effects in CP setups [19, 29]. Here, we re- examine the impact of partial coherence in CP, and introduce for the first time a thorough analysis of partial coherence for FP. We use our phase space model to show that in either optical setup, partially coherent illumination does not limit the ability to recover an exact sample amplitude and phase estimate. We conclude that while partial coherence impacts CP and FP performance differently, it remains a mathematical separable expression that can be removed by computational post-processing. This finding motivates future investigations to examine the potential for a variety of limited-coherence, high-throughput sources to improve the acquisition rates and noise performance for both CP and FP.
5.4.1
Partially coherent source description
To accurately model experimentally realistic optical sources, we must introduce a statistical measure of spatial coherence into our phase space descriptions of CP in Eq. (5.5) and FP in Eq. (5.11). We achieve this by treating the field emitted by the optical source,U(x0, t), as a temporally stationary stochastic process and examining its correlation across space and time: hU(x01, t1)U∗(x02, t2)i = ˜
Γ(x0
1, x02, τ). Here, ˜Γ is the light’s mutual coherence, τ =t2−t1 is a constant time difference, and the expectation value is performed over time. Furthermore, we model the source using the conjugate coordinatex0, since this is the plane where we expect to find the illumination source.
From the Weiner-Khinchine theorem, the cross-spectral density (CSD) of this stochastic process is defined through a Fourier transform of the mutual coherence, Γ(x01, x02, ω) =R ˜
Γ(x01, x02, τ)e−jωτdτ. The spectral density C(x0, ω) = Γ(x0, x0, ω) represents the intensity of light at location x0 at a certain frequency ω. We will assume our illumination sources are fully spatially incoherent within their photon-generating area, leading to a CSD function at source planeL,
ΓL(x01, x02, ω) =γ2C(x10, ω)δ(x01−x02), (5.13) whereCrepresents the geometric shape of the source intensity for each frequencyω(typically a circ- function in two dimensions),γis its spatial coherence cross section, andδis a Dirac delta function. For the remainder of this section, we will drop our interest in the spectral variableω for simplicity, assuming a notch filter is used in the experiment to effectively isolate a narrow spectrum from the source. Although not detailed here, effects of a spectrally broad (i.e., temporally incoherent) source are an important consideration and may be included through incoherent superposition of the following equations. The Van Cittert-Zernike theorem relates the CSD of the source, ΓL in Eq. (5.13), to the CSD a distancezaway, Γz:
Γz(∆x) =e−2jkqz
Z
C(x0)ejkz(x
0∆x)
dx0 ≈C˜(x0), (5.14) where a constant multiplier is neglected for simplicity, ∆x=x1−x2 and q=x21−x22. Assuming (x2
1−x22)/λz <<1 allows us to neglect the phase factor up front. With this assumption, we arrive at
an approximate scaled Fourier relationship between the shape of an incoherent illumination source,
C, and the CSD function, Γz, at any subsequent plane a large distancez from this source.
5.4.2
CP with partially coherent light
In conventional ptychography, the first distant plane the source’s light interacts with is the aperture plane A(x). Here, the light’s CSD function Γ`(x1−x2) is given by Eq. (5.14), with z = `. The aperturea(x) then modulates Γ`(x1−x2) before the light is focused by the lens to the sample plane.
As developed in [28], it is direct to show that the CSD at the sample plane, ΓS, is given by,
Γ˜aS(x01, x02) = Z
C(q)˜a(x01−q) ˜a∗(x02−q)dq. (5.15) Here, we used the variable replacementq=x0 for notational clarity.
We can update our original expression for the intensity at the detector g(p, x) in Eq. (5.4) to reflect our new partially coherent probe beam with a simple replacement. Instead of multiplying the sampleψwith coherent probe wave ˜a, we multiplyψwith the probe wave CSD in Eq. (5.15):
g(p, x) =
Z Z
ΓaS˜(x01, x02)ψ(x01−p)ψ∗(x02−p)exp [−jkx·(x01−x02)]dx01dx02. (5.16) Plugging Eq. (5.15) into Eq. (5.16) and performing several straightforward manipulations (outlined in Appendix C of [28]) produces the following mathematical description of the CP data matrix
g(p, x) in terms of the aperture’s WDF, the sample’s WDF, and the illumination source’s geometric shapeC:
g(p, x) =
Z Z Z
C(q)Wψ(x0−p, u)W˜a(x0−q, x−u)dx0du dq. (5.17)
Partially coherent light alters CP’s data matrix with an additional convolution along the scan vari- ablep(Fig. 5.6(a)). The goal of ptychographic data post-processing under partially coherent illu- mination is to recover a complex description of the sample Wψ from data matrix g(p, x). This is achieved by deconvolving the effect of bothW˜a andC(q). This is identical to the coherent case, but with an additional (yet still separable) blurring term.
5.4.3
FP with partially coherent light
Unlike CP, FP uses anarray of spatially offset and partially coherent LEDs at its illumination plane. We now use p, our shift variable, to represent the distance from a given LED to the optical axis. The CSD of one LED may be expressed by modifying Eq. (5.13) to incorporate a spatial offset by
p: ΓL(x01, x02) =γ2C(x01−p)δ(x01−x02). This LED’s shifted source light first illuminates the sample at planeS(x). We can write the CSD at the sample plane, ΓS, in terms of ΓL as,
ΓS(ρ1−ρ2) = Z
C(x0−p)ejkx0(ρ1−ρ2)dx0= ˜C(ρ
1−ρ2)exp(−jkp(ρ1−ρ2)), (5.18) where (ρ1, ρ2) replace (x1, x2) as the spatial coordinates at the sample plane S(x), for notational clarity. As developed in [28], we may transform this CSD through the imaging lens to the detector plane to write,
ΓD(x1, x2) = Z Z
C(q)! q!
*
x! p!=
g0(p, x) C(q)! q!(a) Conventional ptychography !
(b) Fourier ptychography! x! p! g(p, x)
*
gF0(p, x) x! p! x! p!=
gF(p, x) p! p! blur! blur! 0! 1!Figure 5.6: Partially coherent light manifests itself as an additional convolution along the data matrix scan dimensionpfor both (a) CP and (b) FP. The convolution is one-dimensional, as indicated by the vertical bar. With matrices rotated by 90◦ with respect to one another, this convolution will
mix the data from each respective setup in a unique manner. For this simulation, we used the same setup parameters as for Fig. 5.2 and Fig. 5.4, but assumed each illumination sourceC(q) (i.e., LED) is a square with a width of 200µm(figure adapted from [28]).
where ΓD(x1, x2) is the CSD of partially coherent light at the detector. Finally, we may express the measured data matrix in FP in terms of the aperture WDF, sample WDF, and LED source geometry now with,
gF(p, x) =
Z Z Z
C(q)Wψ(q−u−p, x0)W˜a(u, x0−x)dx0du dq. (5.20)
Details of this last step are in Appendix D of [28]. Comparing Eq. (5.20) to Eq. (5.11)’s coherent description of FP, we see that partial coherence manifests itself again as an additional convolution along the data matrix p-dimension (Fig. 5.6(b)). Practically, this indicates that each FP image, captured from a different LED and compiled along p, will begin to look increasingly similar with increasingly incoherent illumination. In the limit of a completely incoherent source, spatial shifting will leave all image features nearly unchanged. Since this blur remains a separable function, it is still possible to deconvolve the effects of both C and Wa to obtain an accurate sample estimate
Wψ. Put simply, using a partially coherent source in a CP setup blurs together the sample’s spatial information within its recorded data matrix. In FP, using an array of partially coherent sources instead blurs the sample’s spatial frequency content, as Fig. 5.6 clearly depicts.
x! p! (a) CP! (b) FP! g(p, x) x! p! W-1(C, σ)! x! p! RMSEp = 0.065 x! p! RMSEF = 0.029 W-1(C, σ)! gF(p, x) Blurred: C = 100μm! Deconvolved!
(c) Ptychography and FP recovery error vs. LED size!
ψ(x' )! x' (mm)! -.7! .7! (d) Input data g0(p, x)! x! p! 0! 1!
Figure 5.7: Simulation of partially coherent effects produce blurred (a) CP and (b) FP data matrices of an example grating. A Wiener filter can approximately recover the coherent data matrix for each setup, from which an accurate sample reconstruction is direct. (c) Reconstruction error as a function of LED diameter (i.e., blur kernel width) increases for both CP and FP, although FP’s error is consistently lower. (d) The chirped grating sample and its coherent CP data matrix, for comparison (figure adapted from [28]).