Supplementary material: Correcting lateral chromatic aberrations in
non-monochromatic X-ray microscopy
Ken Vidar Falch,1 Carsten Detlefs,2 Marco Di Michiel,2 Irina Snigireva,2 Anatoly Snigirev,3 and Ragnvald H.
Mathiesen1
1)
Norwegian University of Science and Technology, Department of physics, Hgskoleringen 1, 7491 Trondheim,
Norway
2)
European Synchrotron Radiation Facility, 71 Avenue des Martyrs, 38000 Grenoble,
France
3)
Immanuel Kant Baltic Federal University, 238300 Kaliningrad, Russia
in Figure S1. From these, the MP and MF were calculated as
CALCULATION OF RELATIVE MAGNIFICATION WITH
LONG LENS RAY TRACE MATRICES
Let MP and MF be the raytransfer matrices representing the parallel and the focused illumination cases, respectively. Their corresponding magnifications, MP and
MF , respectively, were taken to be the (1, 1) elements of
the two matrices. i.e MF = MF 1,1 and MP = MP 1,1 .
The ray transfer matrices that were used was a product of three basic component matrices. The free space
propagation matrix,
#
"
1 d
,
(S1)
R (d) =
0 1
Q (f ) =
1
0
,
(S5)
Where ωOBJ and LOBJ is the ω and L of the objective
lens. Even though the condenser is a long lens situated
some distance upstream of the sample MF still uses the
thin lens matrix. The length of the condenser was taken
into account by calculating the focal length F , as indicated in Figure S1, using the formula
F =
(S2)
and the CRL-propagation matrix.
"
#
cos (ωL) sin(ωL)
ω
MCRL (ω, L) =
,
− sin (ωL) cos (ωL)

MP = cos (ωL) 
1−
q
F
− F1
pq
(S3)
1
p
+
1
q
−
1−
1
F
p
F
+
1
pqF ω 2
(S6)

(S7)

cos (ωL) − q sin (ωL) (cos (ωL) − q sin (ωL)) p +
− sin (ωL)
1
ω tan(ωL)
and subtracting the distance between the condenser exit
plane and the sample. The parameters of the condenser
were as in the experiment, but rather than using the
measured distance, the condenser was placed so that the
focal spot of the condenser is exactly in the center of the
objective lens at E0 . The calculation was repeated for
each photon energy. The full expressions for Mp and MF
are
where ω = √1f T , where f is the focal length of a single
lenslet and T is the spacing between lenslets as indicated
=
MF = R (q) MCRL (ωOBJ , LOBJ ) R (p) Q (g) .
#
− f1 1
"
(S4)
and
the thin lens matrix
"
MP = R (q) MCRL (ωOBJ , LOBJ ) R (p) ,
sin(ωL)
ω
+ q cos (ωL)
#
(S8)
− sin (ωL) p + cos (ωL)
and

MF = 
cos (ωL) − q sin (ωL) +
−pω cos(ωL)+pω q sin(ωL)−sin(ωL)−q cos(ωL)ω
ωg
− sin (ωL) −
− sin(ωL)p+cos(ωL)
g
− −pω
cos(ωL)+pω q sin(ωL)−sin(ωL)−q cos(ωL)ω
ω
− sin (ωL) p + cos (ωL)

 (. S9)
2
FIG. S1: Explanation of CRL parameters.