wk7b - Today’s summary • Multiple beam interferometers:...

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Unformatted text preview: Today’s summary • Multiple beam interferometers: Fabry-Perot resonators – Stokes relationships – Transmission and reflection coefficients for a dielectric slab – Optical resonance • Principles of lasers • Coherence: spatial / temporal IT 2.71/2.710 Optics 0/20/04 wk7-b-1 Fabry-Perot interferometers IT 2.71/2.710 Optics 0/20/04 wk7-b-2 Relation between r , r ’ and t , t ’ a ar at air a ’ a ’ t ’ a ’ r ’ glass air glass Proof : algebraic from the Fresnel coefficients or using the property of preservation of the preservation of the field properties upon time reversal 1 2 = ′ + − = ′ t t r r r field properties upon time reversal Stokes relationships IT 2.71/2.710 Optics 0/20/04 wk7-b-3 Proof using time reversal a ar at air glass att ’ ar at air IT 2.71/2.710 Optics 0/20/04 wk7-b-4 glass atr ’ art ar 2 ⇔ ( ) ( ) 1 2 2 = ′ + ⇒ = ′ + ′ − = ⇒ = ′ + t t r a t t r a r r t r r a Fabry-Perot Interferometer incident reflected transmitted L Resonance condition: reflected wave = 0 ⇔ all reflected waves interfere destructively n m L 2 λ = wavelength in free space refractive index IT 2.71/2.710 Optics 0/20/04 wk7-b-5 Calculation of the reflected wave incoming a reflected ar transmitted at e i δ reflected at e i 2 δ r ’ transmitted at e i δ t ’ reflected at e i 3 δ ( r ’ ) 2 transmitted at e i 2 δ r ’ t ’ reflected at e i 4 δ ( r ’ ) 3 transmitted at e i 3 δ ( r ’ t ’ ) 2 reflected at e i 5 δ ( r ’ ) 4 transmitted at e i 4 δ ( r ’ ) 3 t ’ λ π δ nL 2 = air, n =1 glass, n air, n =1 IT 2.71/2.710 Optics 0/20/04 wk7-b-6 Calculation of the reflected wave ( ) { } ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ′ − ′ ′ + = + ′ + ′ + ′ ′ + = δ δ δ δ δ 2 2 2 4 4 2 2 2 reflected e 1 1 e e e 1 e i i i i i r r t t r a r r r t t r a a " 1 2 = ′ + − = ′ t t r r r Use Stokes relationships ( ) δ δ 2 2 2 reflected e 1 e 1 i i r r a a − − = IT 2.71/2.710 Optics 0/20/04 wk7-b-7 Transmission & reflection coefficients ( ) δ δ 2 2 2 reflected e 1 e 1 i i r r a a − − = δ 2 2 d transmitte e 1 i r t t a a − ′ = 2 r R ≡ ( ) ( ) ( ) δ δ δ 2 2 2 2 d transmitte 2 2 2 2 reflected sin 4 1 1 t coefficien on transmissi sin 4 1 sin 4 t coefficien reflection R R R a a R R R a a + − − = = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − = = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ IT 2.71/2.710 Optics 0/20/04 wk7-b-8 IT 2.71/2.710 Optics 0/20/04 wk7-b-9 Transmission & reflection vs path R =0.95 R =0.5 IT 2.71/2.710 Optics 0/20/04 wk7-b-10 Fabry-Perot terminology nL mc 2 ( ) nL c m 2 1 + ( ) nL c m 2 2 + frequency ν band width free spectral range resonance frequencies Fabry-Perot terminology IT 2.71/2.710 Optics 0/20/04 wk7-b-11 ∆ν FWHM FWHM Bandwidth is inversely proportional to the finesse finesse F (or quality factor quality factor ) of the cavity ∆ν FSR R R F − ≡ 1 π nLF c 2 FWHM = ∆ν F FSR FWHM ν ν ∆ = ∆ ( ) ( ) ( ) finesse range spectral free bandwidth = Spectroscopy using Fabry-Perot cavity...
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This note was uploaded on 04/17/2011 for the course ME 525 taught by Professor Lucth during the Spring '11 term at Purdue University-West Lafayette.

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wk7b - Today’s summary • Multiple beam interferometers:...

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