By Gregory J. Gbur
The 1st textbook on mathematical equipment targeting innovations for optical technology and engineering. perfect for higher department undergraduates and graduates. robust emphasis is put on connecting mathematical options to optical structures. Essay difficulties in response to learn guides and diverse routines enhance the relationship among the speculation and its functions.
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Extra resources for Mathematical Methods for Optical Physics and Engineering
23. A body-centered cubic lattice has primitive vectors 1 a1 = a(−ˆx + yˆ + zˆ ), 2 1 a2 = a(ˆx − yˆ + zˆ ), 2 1 a3 = a(ˆx + yˆ − zˆ ). 2 Show that the reciprocal lattice is a face-centered cubic lattice, with primitive vectors 1 a1 = α(ˆy + zˆ ), 2 1 a2 = α(ˆx + zˆ ), 2 1 a3 = α(ˆx + yˆ ). 2 24. Let a1 , a2 , and a3 represent a set of primitive vectors, and g1 , g2 , and g3 represent the reciprocal lattice. Demonstrate using vector algebra identities that the reciprocal of the reciprocal lattice vectors are equal to the original vectors a1 , a2 , and a3 .
12 Derivation of the von Laue condition. Instead of considering the crystal as various families of planes, von Laue looked at the X-rays scattered by individual units. Let us suppose that an X-ray is incident upon the crystal with wavevector ki and is scattered with wavevector kr , where |ki | = |kr | = k. The phase of a plane wave with wavevector k may be written as k · r. For two atoms separated by a lattice vector R, the relative phase difference between them is (ki − kr ) · R, as illustrated in Fig.
In anticipation of future discussion, we take some time to formally deﬁne a linear vector space and some other related spaces. 1 (Linear vector space) A linear vector space S is deﬁned as a set of elements called vectors, which satisfy the following ten properties related to elements |x , |y and |z : 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. |x + |y ∈ S (completeness with respect to addition), (|x + |y ) + |z = |x + (|y + |z ) (associativity), |x + |0 = |x (existence of zero), ∃ |y such that |x + |y = |0 (existence of negative element), |x + |y = |y + |x (commutativity), a |x ∈ S (completeness with respect to scalar multiplication), a(b |x ) = (ab) |x (associativity of scalar multiplication), (a + b) |x = a |x + b |x (ﬁrst distribution rule), a(|x + |y ) = a |x + a |y (second distribution rule), 1 |x = |x (existence of unit element).
Mathematical Methods for Optical Physics and Engineering by Gregory J. Gbur