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Unformatted text preview: MASSACHUSETTS INSTITUTE OF TECHNOLOGY 2.710 Optics Spring ’09 Solutions to Problem Set #6 Due Wednesday, Apr. 15, 2009 Problem 1: Grating with tilted plane wave illumination 1. a) In this problem, one–dimensional geometry along the x –axis is considered. The Fresnel diffraction pattern, the field just behind the grating illuminated by the plane wave, is m x 2 π g + ( x, z = 0) = g t ( x ) g − ( x, z = 0) = exp i sin 2 π exp i θx . (1) 2 Λ λ Note that the transmission function can be expanded as ∞ m x m 2 π g t ( x ) = exp i sin 2 π = J q exp iq x . (2) 2 Λ 2 Λ q = −∞ Using eq. (2), we can rewrite eq. (1) as ∞ m 2 π qλ g + ( x, z = 0) = J q exp i θ + x . (3) 2 λ Λ q = −∞ qλ Since exp i 2 λ π θ + Λ x represents a tilted plane wave whose propagation angle is θ + qλ/ Λ, eq. (3) implies that the transmitted field just behind the grating is consisted of a infinite number of plane waves, where q denotes diffraction order and the amplitude of the diffraction order q is J q ( m/ 2). The propagation direction of the zero–order is identical as one of the incident tilted plane wave. 1.b) The field behind the grating is identical to eq. (1). When the observation plane is in the far–zone, the Fraunhofer diffraction pattern is g ( x , z ) = g + ( x, z = 0) exp − i 2 π ( x x ) d x. (4) λz Note that we neglected the scaling factor and phase term because the scaling factor change overall magnitude of diffraction pattern and the phase term does not contribute to intensity. Substituting eq. (2) into (4), we obtain the field distribution of the Fraun hofer diffraction as ∞ g ( x , z ) = J q m exp i 2 π θ + qλ exp − i 2 π ( xx ) d x 2 λ Λ λz q = −∞ ∞ m q θ x = J q exp i 2 π + x exp − i 2 π x d x 2 Λ λ λz q = −∞ ∞ m x q θ = J q δ − + . (5) 2 λz Λ λ q = −∞ 1 (a) on–axis plane wave (b) tilted plane wave Figure 1: The whole diffraction patterns rotate by θ as the incident plane wave rotates The intensity of the Fraunhofer diffraction pattern is ∞ I ( x , z ) =  g ( x , z )  2 = J 2 m δ x − q + θ . (6) q 2 λz Λ λ q = −∞ In the far–region, we should observe a infinite number of diffraction orders. The in tensity of the diffraction order is proportional to J q 2 ( m/ 2) and the offset between two neighboring diffraction orders is ( λz ) / Λ. The zeroth order is located at x = zθ . 1.c) In both cases (Fresnel and Fraunhofer diffraction), the diffraction patterns of the grating probed by a onaxis and tilted plane waves are identical except the angular shift by the incident angle θ , as shown in Fig. 1. 2 Problem 2: Grating spherical wave illumination 2.a) Using the same approach as in Prob. 1, we obtain z i 2 π x 2 + y 2 1 x e λ g + ( x, y, z = 0) = g t ( x, y ) g − ( x, y, z = 0) = 1 + m cos 2 π exp iπ ....
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 Spring '09
 SeBaekOh
 Fourier Series, Diffraction, Fraunhofer diffraction, Fraunhofer, Fraunhofer diffraction pattern

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