Coherence and Interference Prof Rick Trebino Georgia Tech
Coherence and Interference Prof. Rick Trebino Georgia Tech Coherence Temporal coherence Spatial coherence Interference Parallel polarizations interfere; perpendicular polarizations don't. The Michelson Interferometer Fringes in delay Measure of temporal coherence The Fourier transform spectrometer The Misaligned Michelson Interferometer Fringes in position Opals use interference between tiny structures to yield bright colors.
The temporal coherence time and the spatial coherence length The temporal coherence time is how long the beam remains sinusoidal at a single wavelength: Temporal Coherence Time, tc Wave-fronts tc The spatial coherence length is the transverse distance over which the beam wave-fronts remain flat: Spatial Coherence Length, xc Wave-fronts xc Since there are two transverse dimensions, we can define a spatial coherence area, Ac.
Spatial and Temporal Coherence Beams can be coherent or only partially coherent (indeed, even incoherent) in both space and time. Wave-fronts Spatial and Temporal Coherence: Temporal Coherence; Spatial Incoherence Spatial Coherence; Temporal Incoherence xc tc Spatial and Temporal Incoherence xc tc
How quickly will a light wave deviate from a perfect sine wave in time? Suppose the light wave has two frequencies: The two frequencies will become significantly out of phase with each other in a time, tc: E t So the phase will drift on a time scale of: ~ 2 p/Dw = 1/Dn where:
The coherence time is the reciprocal of the bandwidth. The largest frequency difference in the light wave will yield the shortest phase-drift time, which we call the coherence time: Sunlight and light bulbs are temporally very incoherent—and have very small coherences times (a few fs)—because their bandwidths are very large (the entire visible spectrum). Lasers can have much longer coherence times—as long as about a second, which is amazing; that's >1014 cycles!
The spatial coherence depends on the emitter size and its distance away. The van Cittert-Zernike Theorem states that the spatial coherence area Ac is given by: where d is the diameter of the light source and D is the distance away, and W = d 2/D 2 is the solid angle subtended by the source. Basically, wave-fronts smooth out as they propagate away from the source. Starlight is spatially very coherent because stars are very far away.
Irradiance of a sum of two waves Same polarizations Different polarizations Coherent addition Same colors Incoherent addition Different colors Interference only occurs when the waves have the same color and polarization.
The irradiance when combining a beam with a delayed replica of itself has fringes. The irradiance is given by: Suppose the two beams are E 0 exp(iwt) and E 0 exp[iw(t-t)], that is, a beam and itself delayed by some time t : Fringes (in delay) “Dark fringe” I “Bright fringe” t
Varying the delay on purpose Simply moving a mirror can vary the delay of a beam by many wavelengths. Input beam E(t) Mirror Output beam E(t–t) Translation stage Moving a mirror backward by a distance L yields a delay of: Do not forget the factor of 2! Light must travel the extra distance to the mirror—and back! Since light travels 300 µm per ps, 300 µm of mirror displacement yields a delay of 2 ps. Such delays can come about naturally, too.
We can also vary the delay using a mirror pair or corner cube. Mirror pairs involve two reflections and displace the return beam in space: But out-of-plane tilt yields a nonparallel return beam. E(t) Mirrors E(t–t) Input beam Output beam Translation stage Corner cubes involve three reflections and also displace the return beam in space. Even better, they always yield a parallel return beam: Hollow corner cubes avoid propagation through glass.
Input beam The Michelson Interferometer Mirror The Michelson Interferometer splits a beam into two and then recombines them at the same beam splitter. where DL = 2(L 2 – L 1) L 2 Output beam I 0 L 1 Beamsplitter Delay Mirror Fringes (in delay) “Dark fringe” Measure the wavelength of monochromatic light! l I “Bright fringe” DL = 2(L 2 – L 1)
Huge Michelson interferometers may someday detect gravity waves. Gravity waves (emitted by all massive objects) warp space-time ever so slightly. Relativity predicts them, but they’ve never been detected. Supernovae and colliding black holes emit gravity waves that may be detectable. Gravity waves are “quadrupole” waves, which stretch space in one direction and shrink it in another. They should cause one arm of a Michelson interferometer to stretch and the other to shrink, increasing DL. L 2 L 1 and L 2 ~ 4 km! L 1 Unfortunately, the distance difference (DL ~ 10 -16 cm) is less than the width of a nucleus! So such measurements are very difficult!
The LIGO project Cal. Tech LIGO The building containing an arm A small fraction of one arm of the Cal. Tech LIGO interferometer… Hanford LIGO The control center
The LIGO folks think big… The longer the interferometer arms, the better the sensitivity. So put one in space, of course.
Interference is easy when the light wave is a monochromatic plane wave. What if it’s not? For perfect sine waves, the two beams are either in phase or they’re not. What about a beam with a short coherence time? tc The beams could be in phase some of the time and out of phase at other times, varying rapidly. Remember that most optical measurements take a long time, so these variations will get averaged.
Adding a nonmonochromatic wave to a delayed replica of itself Delay = 0: Constructive interference for all times. Coherent. Bright fringe Delay = ½ period (<< tc): Destructive interference for all times. Coherent. Dark fringe |Delay| > tc: Incoherent addition. No fringes.
The Michelson Interferometer is a Fourier Transform Spectrometer L 2 Suppose the input beam is a pulse and is not monochromatic (but is spatially coherent): Þ L 1 Iout(t) = I(t) + I(t-t) + c e Re{E(t)E*(t-t)} Mirror Delay Now, Iout will vary in time, and detectors will integrate over the entire pulses. The measured output energy, Uout, is: where U is the energy per unit area (fluence) of the pulse in each arm. Notice that the 2 nd term is the Field Autocorrelation! Recall that the Fourier Transform of the Field Autocorrelation is the spectrum!! And this result is true for continuous light also!!
Fourier transform spectrometer interferogram A Fourier transform spectrometer's detected light energy vs. delay is called an interferogram. Michelson interferometer integrated irradiance Spectrum Integrated irradiance 1/n 0 0 Intensity 1/Dn Delay Dw = 2 p. Dn w 0 Frequency The Michelson interferometer output—the interferogram—Fourier transforms to the spectrum. The spectral phase plays no role!
Fourier Transform Spectrometer Data Actual interferogram from a Fourier Transform Spectrometer Interferogram This interferogram is very narrow, so the spectrum is very broad. 4000 3200 2400 1600 800 Fourier Transform Spectrometers are most commonly used in the infrared where the fringes in delay are most easily generated. As a result, they are often called FTIR's.
Fourier Transform Spectrometers Maximum path difference: 1 m Minimum resolution: 0. 005 /cm Spectral range: 2. 2 to 18 mm Accuracy: 10 -3 /cm to 10 -4 /cm Dynamic range: 19 bits (5 x 105) A compact commercial FT spectrometer from Nicolet Fourier-transform spectrometers are now available for wavelengths even in the UV! Strangely, they’re still called FTIR’s.
Technical point about Michelson interferometers: Input the compensator beam Beamsplitter plate Output beam Mirror So a compensator plate (identical to the beam splitter) is usually added to equalize the path length through glass. Mirror If reflection occurs off the front surface of beam splitter, the transmitted beam passes through beam splitter three times; the reflected beam passes through only once.
Irradiance vs. position for crossed beams Fringes occur where the beams overlap in space and time. L = l/(2 sinq)
x The Michelson interferometer and spatial fringes Input beam z Suppose we misalign the mirrors so the beams cross at an angle when they recombine at the beam splitter. And we won't scan the delay. If the input beam is a plane wave, the cross term is (as before): Crossing beams maps delay onto position. And moving a mirror by d. L shifts the fringes by a phase k d. L: Fringes (in position) I L x
Michelson-Morley experiment 19 th-century physicists thought that light was a vibration of a medium, like sound. So they postulated the existence of a medium whose vibrations were light: aether. Michelson and Morley realized that the earth could not always be stationary with respect to the aether. And light would have a different path length and phase shift depending on whether it propagated parallel and anti-parallel or perpendicular to the aether. Parallel and anti-parallel propagation Mirror Perpendicular propagation Beamsplitter Mirror Supposed velocity of earth through the aether
Michelson-Morley Experiment: Details 1 If light requires a medium, then its velocity depends on the velocity of the medium. Velocity vectors add. Parallel velocities Anti-parallel velocities
Michelson-Morley Experiment: Details 2 In the other arm of the interferometer, the total velocity must be perpendicular, so light must propagate at an angle. Perpendicular velocity to mirror Perpendicular velocity after mirror
Michelson-Morley Experiment: Details 3 Let c be the speed of light, and v be the velocity of the aether. Parallel and anti-parallel propagation Perpendicular propagation The delays for the two arms depend differently on the velocity of the aether! If v is the earth’s velocity around the sun, 3 x 104 m/s, and L = 1 m, then:
Michelson-Morley Experiment: Results The Michelson interferometer was (and may still be) the most sensitive measure of distance (or time) ever invented and should’ve revealed a fringe shift as it was rotated with respect to the aether velocity. Interference fringes showed no change as the interferometer was rotated. Their apparatus Michelson and Morley's results from A. A. Michelson, Studies in Optics
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