Inter_Diffr

Inter_Diffr - 1 Interference and diffraction •...

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Unformatted text preview: 1 Interference and diffraction • Interference and diffraction phenomena are both results of the superposition of EM waves from sources that are coherent to each other • Interference: combined effects from discrete sources • Diffraction: combined effects from a continuous distribution of sources Interference from two coherent sources - Young’s experiment : At point Q , the distance traveled for EM wave from source A is different from that from source B , ∆ : path difference. phase difference δ = π 2 period travel of time extra × = T 1 c 2 ⋅ ⋅ ∆ π = λ ∆ π ⋅ 2 • Phase difference of 0, 2 π , 4 π , … are equivalent as far as interference is concerned 2 Assuming sources A and B gives out EM waves of the same intensity, so that E T = E A + E B = E cos ω t + E cos( ω t+ δ ) Using complex notation, T E ~ = E 0 e i ω t + E 0 e i( ω + δ ) = E 0 e i ω t (1 + e i δ ) = E 0 e i ω t e i δ/2 (e-i δ/2 + e i δ/2 ) ∴ E T = 2 E cos ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ 2 δ cos ( ω t + 2 δ ) Since I T ∝ < E T 2 > and I ∝ < E 2 > , ∴ I T = 4 I 0 cos 2 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ 2 δ = 0 when δ = π , 3 π , 5 π … = 4I when δ = 0, 2 π , 4 π … If the screen is located far from the sources, ∆ ≅ d sin θ ≅ d tan θ = L dx ∴ I T = 4 I 0 cos 2 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ L dx λ π = 0 when x = d 2 L λ , d 2 L 3 λ , d 2 L 5 λ … = 4I when x = 0, d L λ , d L 2 λ … ⇒ The well-known Young’s interference bright and dark fringes on the screen 3 • What happen if the two sources are not completely coherent with respect to each other? • A wave is perfectly coherent if the phase has a definite value by any delay on the wave itself • A perfectly coherent wave must be monochromatic , i.e. a perfect sinusoidal wave with no beginning and ending • An EM wave with a finite duration must consist of a band of frequency components e.g. A wave with finite duration of 2 ∆ t : E(t) = t i e ω −- ∆ t < t < ∆ t = otherwise This can be written as an integral over the frequency: ( ) ∫ ∞ ∞ − − = ω ω π ω d e E t E t i ) ( ~ 2 1 The spectral distribution E( ω ) is given by the Fourier transform of E(t) : ( ) ∫ ∫ ∆ ∆ − − ∞ ∞ − = = t t t i t i dt e dt e t E E ) ( ) ( ~ ω ω ω ω = t 2 t t sin ∆ ∆ ω ω ∆ ω ω ⋅ − − ) ( ] ) [( 4 The power spectrum I ( ω ) is proportional to the square of ( ) ω E ~ : ( ) ( ) 2 ~ ω ω E I ∝ The power spectrum of our e.g. look like this: Since ] ) [( t sin ∆ ω ω − = 0 when ω = ω ± t ∆ π ∴ The bandwidth of this spectral distribution, ∆ω , is given by ∆ω∆ t = π • This ‘uncertainty principle’ tell us that a finite wave duration requires a non-zero spread of frequency • Atoms emit light by jumping down from an excited state. Finite excited state lifetime ⇒ finite pulse duration ⇒ frequency spread • A collection of uncorrelated atoms gives an incoherent source 5 Coherent length l c : average length of the light wave within...
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Inter_Diffr - 1 Interference and diffraction •...

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