Ch7. Interference and Diffraction 1. Spatial and temporal coherence 2. Cross terms and fringes 3. Examples 4/7/2016 The Temporal Coherence Time and the Spatial Coherence Length tc The temporal coherence time is the time over which the beam wavefronts remain equally spaced. Or, equivalently, over which the beam remains sinusoidal. Lc The spatial coherence length is the distance over which the beam wavefronts remain flat. 4/7/2016 Spatial and Temporal Coherence Spatial and Temporal Incoherence 4/7/2016 Spatial Coherence; Temporal Incoherence Temporal Coherence; Spatial Incoherence Spatial and Temporal Coherence: Lasers Emit Temporally Coherent Light The coherence time is given by: t c 1/ Dv where Dn is the light bandwidth (the width of the spectrum). Sunlight is temporally very incoherent because its bandwidth is very large (the entire visible spectrum). 4/7/2016 The Spatial Coherence The van Cittert-Zernike Theorem states that the spatial coherence area Ac is given by: D c 2 d 2 2 where d is the diameter of the light source and D is the distance away. Basically, wavefronts smooth out as they propagate away from the source. Starlight is spatially very coherent because stars are very far away. 4/7/2016 The Irradiance Revisited The most general plane-wave electric field is: E r , t Re E0 exp i (k r t ) where the amplitude is both complex and a vector: E0 E0 x , E0 y , E0 z The irradiance is: c c * * * * I E0 E0 E E E E E E 0 x 0 x 0 y 0 y 0 z 0 z 2 2 4/7/2016 Polarization Dependence Because the irradiance is given by: c c * E0 x E0 x* E0 y E0 y * E0 z E0 z * I E0 E0 2 2 combining two waves of different polarizations is different from combining waves of the same polarization. Different polarizations (say x and y): c E0 x E0 x* E0 y E0 y * I1 I 2 I 2 Same polarizations (say x and x, so we'll omit the x-subscripts): I Therefore: 4/7/2016 c * * * E1 E1 2 Re E1 E2 E2 E2 2 I I1 c Re E1 E2 I 2 * Cross term! Spatial Crossed Terms x k k cos zˆ k sin xˆ k k cos zˆ k sin xˆ k z k r k cos z k sin x k r k cos z k sin x k I 2 I 0 c Re E0 exp[i(t k r )]E0* exp[ i(t k r )] Cross term is proportional to: Re E0 exp i ( t kz cos kx sin E0 exp i ( t kz cos kx sin Re exp 2ikx sin cos(2kx sin ) 4/7/2016 * Fringes (in position) Examples: Fresnel's Biprism A prism with an apex angle of about 179° refracts the left half of the beam to the right and the right half of the beam to the left. Fringe pattern observed by interfering two beams created by Fresnel's biprism 4/7/2016 Angle Dependence of Fringes The fringe spacing, : Large angle: 2 /(2k sin ) /(2sin ) sin /(2) 0.5 m / 200 m Small angle: 1/ 400 rad 0.15 0.1 mm is about the minimum fringe spacing seen by eye 4/7/2016 A Misaligned Michelson Interferometer If the input beam is a plane wave, crossing beams maps delay onto position. Re E0 exp i ( t kz cos kx sin E0 exp i ( t kz cos kx sin Re exp 2ikx sin cos(2kx sin ) 4/7/2016 * Fringes (in position) A Misaligned Michelson Interferometer Now, suppose an object is placed in one arm. In addition to the usual spatial factor, one beam will have a spatially varying phase, exp[if(x,y)]. Now the cross term becomes: Re{ exp[if(x,y)] exp[-2ikxsin]} Distorted fringes (in position) 4/7/2016 A Misaligned Michelson Interferometer Placing an object in one arm of a misaligned Michelson interferometer will distort the spatial fringes. 4/7/2016 Mach-Zehnder Interferometer The Mach-Zehnder interferometer is usually operated “misaligned” and with something of interest in one arm. 4/7/2016 Newton's Rings 4/7/2016 Quiz Q: He-Ne lasers can have coherence times as long as about a second. This is amazing; how many cycles that have to be locked in the beam? 4/7/2016 Newton's Rings Get constructive interference when an integral number of half wavelengths occur between the two surfaces (that is, when an integral number of full wavelengths occur between the path of the transmitted beam and the twice reflected beam). This effect also causes the colors in bubbles and oil films on puddles. 4/7/2016 Combining a Beam with a Delayed Replica of Itself Has “Fringes” The irradiance is given by: I I1 c Re E1 E2 I 2 * Suppose the two beams are E0exp(it) and E0exp[i(t-t)], that is, a beam and itself delayed by some time t: I 2 I 0 c Re E0 exp[it ] E0* exp[i (t t )] 2 I 0 c Re E0 exp[it ] 2 2 I 0 c E0 cos[t ] 2 Fringes (in delay) I I 2 I 0 2 I 0 cos[t ] 4/7/2016 - t The Michelson Interferometer The Michelson Interferometer splits a beam into two and then recombines them at the same beam splitter. Fringes (in delay) 4/7/2016 - The Michelson Interferometer I out I 1 I 2 c Re E0 exp i (t kz kL1 ) E0 exp i (t kz kL2 ) I I 2 I Re exp ik ( L2 L1 ) * since I I1 I 2 (c 0 / 2) E0 2 2 I 1 cos(k DL) Fringes (in delay) 4/7/2016 -

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# Ch7. Interference and Diffraction 1. Spatial and temporal coherence 3. Examples