By Peter W. Hawkes

*Advances in Imaging & Electron Physics* merges long-running serials-*Advances in Electronics & Electron Physics* and *Advances in Optical & Electron Microscopy*. The sequence positive aspects prolonged articles at the physics of electron units (especially semiconductor devices), particle optics at low and high energies, microlithography, snapshot technology and electronic picture processing, electromagnetic wave propagation, electron microscopy, and the computing tools utilized in these kind of domains.

- Contributions from top professionals
- Informs and updates on the entire most modern advancements within the field

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**Extra info for Advances in imaging and electron physics**

**Sample text**

Its deﬁnition is FRFT: Fa ðuÞ ¼ OFa ðf ðtÞÞ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ZN j 2 1 À j cot a j cot a:u2 2 eÀj:csc a:ut e2:cot a:t f ðtÞdt: (3) ¼ e 2p ÀN 41 Linear Canonical Transform When a s Np, N is an integer, F2Np ðuÞ ¼ OF2Np ðf ðtÞÞ ¼ f ðuÞ; Fð2Nþ1Þp ðuÞ ð2Nþ1Þp ¼ OF ðf ðtÞÞ ¼ f ðÀuÞ: It satisﬁes the additivity property as follows: À Á OFa OFb ðf ðtÞÞ ¼ OFb OFa ðf ðtÞÞ ¼ O aþb F ðf ðtÞÞ: (4) (5) It is easy to see that, when a ¼ 0, p/2, p, and 3p/2, the FRFT is reduced to the identity operation, the FT, time-reverse operation, and the IFT, respectively.

D;Àb;Àc;aÞ OF b2 c2 d2 #" : a1 b1 c1 d1 # : (10) ða;b;c;dÞ OF ðf ðtÞÞ ¼ f ðtÞ: (11) ¼ In particular, a2 That is, the inverse of the LCT with parameters {a, b, c, d} is the LCT with parameters {d, Àb, Àc, a}. Since the additivity property of LCT can be represented by the multiplication of two 2 Â 2 matrices, as in Eq. " # t t LCT a b : (12) ! f c d f to represent the LCT with parameters {a, b, c, d}. This is called the metaplectic representation (Folland, 1989). For simpliﬁcation, we call it the ABCD matrix.

6 Testing the Correct Liquid Thickness in the Electron Microscope Once the vacuum level is all right, the gun valve is carefully opened and the specimen is imaged at low magniﬁcation (5,000–20,000 x). The ﬁrst step is to ﬁnd the lateral (x,y) position of the window at low magniﬁcation. Once light comes through, it needs to be checked if the liquid layer is sufﬁciently thin. This can be done by monitoring the transmitted current. , 2010). The number of electrons N scattered into an angle larger than the opening angle b of the ADF detector can be calculated by the following equation (Reimer & Kohl 2008): t tsðbÞNA N ¼ N0 1 À exp À ; (1) ¼ N0 1 À exp À A l where N0 is the number of incoming electrons, t the sample thickness, l the mean free path length, s(b) the partial elastic cross section, NA Avogadro’s number, and A the atomic weight.