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45. 9) where r = [(x − X)2 + (y − Y)2 + Z 2 ]1/2 . If the distance from the origin to the evaluation point P2 is r = [X 2 + Y 2 + Z 2 ]1/2 , then the distance between a point in the aperture at (x, y, 0) and a point in the diffracted field at (X, Y, Z) can be written as r = R 1 + (x2 + y2 )/R2 − 2(xX + yY)/R2 1/2 . 10) In the far field, because the aperture is located near the origin of the coordinate system and R x2 , y2 , the second term is small compared with unity and the other term, so the distance r can be approximated as r∼ = R 1 − 2(xX + yY)/R2 1/2 .

34) we can express d as a function of Λ and Λ, but number of algebra steps will yield a simple equation for optimum blaze depth as a function of the diffraction angle θd : d = λ Λ d= . 37) Even at this optimum grating depth, there is still a significant loss of efficiency. This enhanced scalar efficiency is lower because of the reduced fill factor at the grating. This broadens the sinc envelope in the far field by a factor of Λ/Λ . The efficiency loss is therefore equal to (Λ /Λ) (Ref. [2]). 45. 0, but as the ratio drops, the efficiency loss predicted by an extended scalar theory becomes significant.

Note that the expansion for the reflected wave is simply a plane-wave expansion, with each component representing a diffraction order of the reflected wave. The reason the basis vectors in this expansion are plane waves is because the modes of a wave propagating in unbounded free space are in fact plane waves. However, this is not the case when waves are subject to inhomogeneities or are confined in one or more dimensions. Such is the case in region II. Because the diffracting structure is assumed to have periodicity Λ in the x direction, αm satisfies the Bragg condition: αm = α0 + 2πm .

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