MIT2_71S09_uquiz2_sol

MIT2_71S09_uquiz2_sol - 2.71 Optics 1. Solution: Quiz 2...

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Unformatted text preview: 2.71 Optics 1. Solution: Quiz 2 Solutions Spring `09 z g1 (x, z) = a1 exp i2 sin + cos , 6 6 x z g2 (x, z) = a2 exp i2 - sin + cos , 3 3 x At z = 0, a1 = 1 a2 = 1 2 x 1 x 3 2 I(x) = | 1 (x, 0) + g2 (x, 0)|2 = a1 ei2 2 + a2 e-i2 2 g 2a1 a2 x 1 3 2 2 = (a + a2 ) 1 + 2 cos 2 + 1 2 a + a2 2 2 1 21 1 1 x 1+ 3 2 = 1+ 1+ cos 2 1 4 2 1+ 4 x 5 4 2 = 1 + cos 2 where = 4 5 1+ 3 m= 4 2 period contrast = 5 1+ 3 5 4 5 9 9 1 I(0) = 1 + = = =2 4 4 5 5 4 4 After phase-shifting g1 by /2, we get x 5 4 5 1 I(0) = = 1 I(x) = 1 + cos 2 + 4 5 2 4 4 2. (a) Solution: x 1 gt (x) = 1 + cos 2 2 1 1 i2 x 1 -i2 x + e = 1 + e 2 2 2 1 1 1 1 1 Gt (u) = (u) + u - + u+ 2 2 2 x gP P - (x ) Gt f 1 1 f 1 f = (x ) + x - + x + 2 4 4 f 1m 5cm = = 5mm 10m Fourier: Scaling: 1 (b) Solution: To preserve all the input's details to the outputs, all diffracted orders must pass the pupil mask. a > 1cm. (c) Solution: Yes. If the illumination is ei2 then: gin (x) = gillum (x)gt (x) 1 i2 x 1 i2( + 1 )x 1 i2( - 1 )x = e + e + e 2 2 2 1 1 1 1 1 Gin (u) = u - + u- - + u- + 2 4 4 1 1 f 1 f gP P - (x ) (x - f ) + x - f - + x - f + 2 4 4 So all diffracted orders shift by f at the pupil plane. The original limit amin = 1cm is not enough to accommodate all orders. A pupil mask that is symmetric around the optical axis must be extended to f + f : x 2 (d) Solution: The glass plate shifts the 0th order by : 1 1 1 gP P - (x ) = (x ) + (x - 5) + (x + 5) 2 4 4 1 1 1 gP P + (x ) = - (x ) + (x - 5) + (x + 5) 4 2 4 phase shift by no change Fourier: Scaling: 1 1 1 GP P + (u) = - + ei2u5mm + e-i2u5mm 2 4 4 x x x 1 1 i2 10m 1 -i2 10m gout (x ) = GP P + =- + e + e f 2 4 4 1 1 x 1 x = - + cos 2 = - 1 - cos 2 2 2 10m 2 10m 2 2 1 x Iout (x ) = gout (x ) = 1 - cos 2 4 10m 2 1 2x 2x 2 = 2 sin 2 = sin 2 4 10m 10m 3 MIT OpenCourseWare http://ocw.mit.edu Spring 2009 2.71 / 2.710 Optics For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. ...
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