By Alison Pohl
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Extra resources for Oxford English for Careers: Technology 2: Teacher's Resource Book by Alison Pohl (2008-08-14)
When we let the imaginary part go to zero we find that all of the integrals Rk are well defined. We could also get the integrals by taking the derivatives of the integral R0(γ) with respect to γ. 79) 0 We can use the same sort of reasoning on these integrals. 80) However, unlike Rk these integrals do not vanish when k is odd. 83) −∞ This has the stationary point at ξ = 0. To obtain the first term in the method of stationary phase, we argued that the major contribution to this integral came from the region around ξ = 0.
In order to understand the behavior of H(x,γ) near such points we need to include cubic terms in the Taylor series expansion of the phase near the stationary point, and this analysis is based on the Airy integral. We will not give any further discussion of the Airy integral or caustics since when discussing beam shaping we do not present any examples where caustics occur in the classical sense of the word. All of the problems we analyze lead to lenses whose phase functions do not have inflection points.
102) ∞ 0 This reduces to the previous problem when α = 0. When α is nonzero, we can analyze this problem in a similar fashion. In this case, we see that we will get oscillations on both sides of x = 0, but the oscillations will be bigger on the side where the stationary phase solutions predict that V is bigger. 103) −∞ and f(ξ) has a discontinuity at ξ*, then we need to break the integral up into two parts. 104) ξ* Suppose that the function q(x,ξ) has a stationary point at ξ(x), and that ξ(x*) = ξ*.