{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture07-FourierTransforms-OpticalDiffraction

# Lecture07-FourierTransforms-OpticalDiffraction - EMSE 312...

This preview shows pages 1–9. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: EMSE 312 – Diffraction Principles Lecture 7 Fourier Transforms Optical Diffraction (cont.) EMSE 312 DIFFRACTION PRINCIPLES EMSE 312 – Diffraction Principles Fourier Analysis ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 f exp i exp i 2 1 1 exp i3 exp i3 3 3 1 1 exp i5 exp i5 5 5 etc. θ θ θ π π θ θ π π θ θ π π = + + − − − − + + − a n n 1 2 3 4 5 6-6-5-4-3-2-1 EMSE 312 – Diffraction Principles Fourier Series – Fourier Transforms • Fourier Coefficients using ( ) ( ) ( ) ( ) ( ) ( ) ( ) n 2 2 2 2 1 a f e x p i n d 2 1 2 f x exp 2 inkx dx 2 1 f x exp 2 inkx dx a k π π λ λ λ λ θ θ θ π π π π λ π λ + − + − + − = − = ⋅ − = − = ∫ ∫ ∫ x 2 θ π λ = 2 d d x π θ λ = ngular Space eal Space EMSE 312 – Diffraction Principles Fourier Transforms • Basic premise: – Take a Fourier Series and “stretch” the wavelength to infinity – The Fourier Coefficient becomes a continuous function as the discrete values for 1/n become infinitely close to each other – Continuous function in “k”-space (reciprocal space) ( ) n a a k → ¡ ( ) ( ) ( ) ( ) ( ) n 2 2 1 a f e x p i n d 2 1 f x exp 2 inkx dx a k π π λ λ θ θ θ π π λ + − + − = − = − = ∫ ∫ a n n 1 2 3 4 5 6-6 -5 -4 -3 -2 -1 EMSE 312 – Diffraction Principles Fourier Transforms • The Fourier Transform of a single slit: ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) { } ( ) ( ) { } ( ) a / 2 a / 2 a / 2 a / 2 f k f x exp 2 ikx dx 1 exp 2 ikx dx 1 1 exp 2 ikx exp ika exp ika 2 ik 2 ik 1 1 1 exp ika exp ika sin ka k 2 i k π π π π π π π π π π π π +∞ + −∞ − − = − = ⋅ − = − − = − − − = ⋅ − − = ⋅ ∫ ∫ ¡ a 0; x a / 2 f (x) 1; a / 2 x a / 2 x a / 2 < − ⎧ ⎫ ⎪ ⎪ = − ≤ < ⎨ ⎬ ⎪ ⎪ ≥ ⎩ ⎭ x 1 a 2 − a 2 Transmission function EMSE 312 – Diffraction Principles Fourier Transforms • The Fourier Transform of a single slit: • • for or • for or • for or • for or ( ) sin ka π = ( ) ( ) ( ) k k x sin x 1 a lim sin ka lim sin ka a lim a k ka x π π π π → → → ⋅ = ⋅ = ⋅ = a 0; x a / 2 f (x) 1; a / 2 x a / 2 a a / 2 < − ⎧ ⎫ ⎪ ⎪ = − ≤ < ⎨ ⎬ ⎪ ⎪ ≥ ⎩ ⎭ x 1 a 2 − a 2 Transmission function ka n 2 π π = ⋅ k 2 n / a = ( ) sin ka 1 π = ( ) sin ka π = ( ) sin ka 1 π = − ka n 2 / 2 π π π = ⋅ + ka n 2 π π π = ⋅ + ka n 2 3 / 2 π π π = ⋅ + ( ) k 2 n 1 / 2 / a = + ( ) k 2 n 1 / a = + ( ) k 2 n 3 / 2 / a = + EMSE 312 – Diffraction Principles Fourier Transforms • The Fourier Transform of a single slit: • • for or • for or ( ) 2 sin ka π = ( ) ( ) ( ) ( ) 2 2 2 2 1 1 I k f k sin ka sin ka k k π π π π ⎛ ⎞ = = ⋅ = ⎜ ⎟ ⎝ ⎠ ¡ ¡ a 0; x a / 2 f (x) 1; a / 2 x a / 2 a a / 2 < − ⎧ ⎫ ⎪ ⎪ = − ≤ < ⎨ ⎬ ⎪ ⎪ ≥ ⎩ ⎭ x 1 a 2 − a 2 Transmission function ka n π π = ⋅ k n / a = ( ) 2 sin ka 1 π = ka (n 1/ 2) π π = + ( ) k n 1 / 2 / a = + EMSE 312 – Diffraction Principles Fourier Transforms...
View Full Document

{[ snackBarMessage ]}

### Page1 / 33

Lecture07-FourierTransforms-OpticalDiffraction - EMSE 312...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online