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Chapter 7: Interference of light

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2.Sports. Obstruction of an opponent, resulting in penalty. in·ter·fer·ence constructivedestructive 3.Physics.Superposition of two or more waves, resulting in a new wave pattern. 1.Life. Hindrance or imposition in the concerns of others.

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HeNe laser

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Radio City Rockettes, New York, NY

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J.R. Stroop "Studies of interference in serial verbal reactions" Journal of Experimental Psychology 18: (1935). rood blauw oranje paars oranje blauw groen rood blauw paars groen rood oranje blauw rood groen paars oranje rood blauw groen rood blauw paars oranje blauw rood groen paars oranje rood blauw

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Peacock

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Soap bubbles

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2-beam interference propagation distance from source of disturbance initial phase (at t =0) from superposition principle:

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- Electric fields are rapidly varying ( ~ Hz) - Quickly averages to 0 -Instead of measuring E directly, measure radiant power density = irradiance, E e (W/m 2 ) = time average of the square of the electric field amplitude -Note: to avoid confusion, Pedotti 3 now uses the symbol I instead of E e Measuring interference

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Irradiance at point P I1I1 I2I2 I 12 I =++ - when E 1 and E 2 are parallel, maximum interference - when orthogonal, dot product = 0; no interference

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The interference term I 12 dot product of electric fields: simplify by introducing constant phases: use trigonometry: 2cosAcosB = cos(A+B) + cos(B-A) and consider again the time average: kills it

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The interference term I 12 simplify by introducing : to yield the interference term of the irradiance:

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Irradiance formula if E 1 E 2, then -where is the phase difference -for parallel electric fields

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Interference mutually incoherent beams (very short coherence time) mutually coherent beams (long coherence time) constructive interference destructive interference maximum when cos = 1 minimum when cos = -1 = ( 2m ) = ( 2m+1 )

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Interference fringes maximum when I 1 = I 2 = I = 4 !?!

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Interference in time and space Youngs experiment wavefront division Michelson interferometer amplitude division

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The double slit experiment (first performed in 1803)

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Double slit experiment with electrons

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Criteria for light and dark bands conditions for interference: - approximate arc S 1 Q to be a straight line - optical path difference = a sin constructive destructive m = 0, 1, 2, 3, …

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Interference from 1 source: reflection Fresnels mirrors Lloyds mirror part of the wavefront is reflected off each mirror part of the wavefront is reflected; part goes direct to the screen

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Fresnels mirrors as solar collectors

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part of the incident light is refracted downward and part upward Interference from 1 source: refraction Fresnels biprism

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Interference via amplitude division - thin films - oil slicks - soap bubbles - dielectric coatings - feathers - insect wings - shells - fish - …

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Interference intermezzo

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The Dancing Couple-1663-Jan Steen

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Anatomy of a soap bubble

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Soap bubble interference

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optical path difference: = n f ( AB + BC ) = n f (2 t ) Thin film interference: normal incidence

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= m : constructive interference = (m + ½) : destructive interference where m = 0,1,2,… Thin film interference: non-normal incidence optical path difference: = n f ( AB + BC ) – n 0 ( AD ) = 2 n f t cos t

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Keep in mind the phase Simple version: phase of reflected beam shifted by if n 2 > n 1 0 if n 1 > n 2 Correct version: use Fresnel equations! hard reflection soft reflection

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Summary of phase shifts on reflection TE modeTM mode air glass external reflection n 1 < n 2 TE modeTM mode air glass internal reflection n 1 > n 2 n1n1 n2n2 n1n1 n2n2

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Back to the bubbles

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How thick here (red band)? t n>1 180 o phase change 0 o phase change Constructive interference for 2t ~ (m + ½) At first red bandm = 0 t ~ ¼ (700 nm) Colors indicate bubble thickness

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Bright: Colored monochromatic stripes occur at (1/4) for visible colors White: Multiple, overlapping interferences (higher order) Dark:Super thin; destructive interference for all wavelengths (no reflected light) pop! Dark, white, and bright bands

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Multiple beam interference r, t : external reflection r, t: internal reflection Note: thickness t ! geometric series where is the phase difference

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Multiple beam interference Introduce Stokes relations: r=-r and tt=1-r 2 and simplify to get: Irradiance:

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Working through the math, youll arrive at: Multiple beam interference where I i is the irradiance of the incident beam Likewise for transmission leads to:

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Fabry-Perot interferometer (1897) d This simulation was performed for the two sodium lines described above, with reflectivity and the separation of the mirrors increasing from 100 microns to 400 microns. simulation of two sodium lines: 1 = m 2 = m mirror reflectivity r = 0.9 mirror separation: m

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Fabry-Perot interferometer where F is the coefficient of finesse: see chapter 8

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Fabry-Perot interferometer: fringe profiles Michelson - transmission maxima occur when = 2 m - as r approaches 1 (i.e. as F increases), the fringes become very narrow - see Chapter 8 for more on Fabry-Perot: fringe contrast, FWHM, finesse, free spectral range

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Constructive reflection 2d = (m+1/2)λm=0, 1, 2, 3... Destructive reflection 2d = mλ m=0, 1, 2, 3... Fringes of equal thickness

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Newtons rings pattern depends on contact point: goal is concentric rings white-light illumination

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Constructive reflection 2d = mλ m=0, 1, 2, 3... Destructive reflection 2d = (m+1/2)λm=0, 1, 2, 3... Oil slick on pavement

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Glass: n = 1.5 MgF 2 coating: n = 1.38 To make an AR coating for = 550 nm, how thick should the MgF 2 layer be? Thin film coatings: anti-reflective

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Broadband anti-reflective films

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thin layers with a high refractive index n 1,interleaved with thicker layers with a lower refractive index n 2 path lengths l A and l B differ by exactly one wavelength each film has optical path length /4: all reflected beams in phase ultra-high reflectivity: % or better over a narrow wavelength range Multilayer mirrors

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Anodized titanium

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Natural multi-layer reflectors

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Exercises You are encouraged to solve all problems in the textbook (Pedrotti 3 ). The following may be covered in the werkcollege on 5 October 2011: Chapter 7: 1, 2, 7, 9, 15, 16, 24

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