Lecture_2 - Fourier Analysis and Interferometers Phys 331,...

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Fourier Analysis and Interferometers Phys 331, Lecture 2, October 11, 2011
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Fourier Analysis Fourier series for periodic functions: Hecht, Pg 304 Fourier’s Theorem: a function f(x), having a spatial period λ , can be synthesized by a sum of harmonic functions whose wavelengths are integral of submultiples of λ . 0 00 ( ) cos( ) sin( ) 2 mm A f x A mkx B mkx ∞∞ = + ∑∑ 0 2 ( )cos( ) m A f x mkx dx λ = 0 2 ( )sin( ) m B f x mkx dx = k=2 π / λ Determine A and B through Fourier analysis.
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Hecht, Pg 303, Fig. 7.26 Superposition of 6 harmonic waves Hecht, Pg 305, Fig. 7.27 Approximation of a Sawtooth Wave Decomposing Periodic Functions frequency spectrum f(x) f(x) Saw-tooth function
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Decomposing a Square Wave Decomposing Periodic Functions +1 f(x) -1 λ /2 - λ /2 0 00 ( ) cos( ) sin( ) 2 mm A f x A mkx B mkx ∞∞ = + ∑∑ Odd function: f(x)=-f(-x) -> A m =0. 0 2 ( )sin( ) m B f x mkx dx λ =
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Pg 306, Fig. 7.29 Decomposing a Square Wave Sum of 2 sines Sum of 3 sines “ODD” Square Wave 0 2 (1 cos ) m m A Bm m π = = 0 λ /2 −λ /2 Decomposing Periodic Functions
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“EVEN” Square Wave 0 2 4 sin 2 / ; 2 2/ 0 m
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Lecture_2 - Fourier Analysis and Interferometers Phys 331,...

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