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===The phase contrast transfer function===
The phase [[contrast transfer function]] (CTF) is a function of limiting apertures and [[Aberration in optical systems|aberrations]] in the imaging lenses of a microscope. It describes their effect on the phase of the exit wave ''φ<sub>e</sub>('''x''','''u''')'' and propagates it to the image wave. Following ''Williams and Carter'',<ref>{{cite book |title=Transmission electron microscopy: A textbook for materials science |last=Williams |first=David B. |author2=Carter, C. Barry |year=1996 |publisher=Plenum Press |location=New York |isbn=978-0-306-45324-3 |url-access=registration |url=https://archive.org/details/transmissionelec0002will }}</ref> if we assume the WPOA holds (thin sample) the CTF becomes
: <math>CTF(u)=A(u)E(u)\sin(\chi(u))</math>
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