Phys 322 Chapter 9 Interference Lecture 25 Reminder: Exam 2 and Review quiz, more details on the course website Interferometers Wavefront-splitting interferometers Amplitude-splitting interferometers Mirrored interferometers The Michelson interferometer Albert Abraham Michelson Compensator plate 1852 - 1931 1881 +/2 - reflection What if the path differences 2(d1-d2)=m? - minium, dark spot What if the path differences 2(d1-d2)=m+/2? - maximum, bright spot +/2 - reflection The Michelson interferometer Path difference: 2dcos Phase shift: (internal/external reflection) Minima: 2d cos m What if light is white? (©WIU OptoLab) The Michelson interferometer: speed of light Michelson-Morley experiment Speed of light is constant in all reference systems The Michelson interferometer: application Accurate length measurements: Displacement of M2 by /2 fringe will move to the position occupied by an adjacent fringe Count fringes, N: d = N(0/2) The Michelson Interferometer: Input advanced treatment beam The Michelson Interferometer splits a L beam into two and then recombines them 2 Output Mirror beam at the same beam splitter. Beam- L splitter 1 Suppose the input beam is a plane wave: Delay Mirror * Iout I12 I cRe E 0 exp i ( t kz 2 kL 1 ) E 0 exp i ( t kz 2 kL 2 ) 2 II2 I Re exp 2 ikLL (21 ) since II 12 I ( c 0 / 2) E 0 2IkL 1 cos( ) “Dark fringe” “Bright fringe” Iout where: L = 2(L2 – L1) Fringes (in delay): L = 2(L2 –L1) Input The Michelson beam Interferometer L2 Output Mirror Another application of the beam Michelson Interferometer is to Beam- L splitter 1 measure the wavelength of monochromatic light. Delay Mirror IIout 21cos() kLI 21cos(2 L /) Iout L = 2(L2 –L1) Huge Michelson Interferometers may someday detect gravity waves. Gravity waves (emitted by all massive objects) ever so slightly warp space-time. 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 L2 one direction and shrink it in Mirror another. They should cause Beam- L1 one arm of a Michelson splitter L and L = 4 km! interferometer to stretch and the 1 2 other to shrink. Mirror -16 Unfortunately, the relative distance (L1-L2 ~ 10 cm) is less than the width of a nucleus! So such measurements are very very difficult! The LIGO project The building containing an arm Laser Interferometer Gravitational- CalTech LIGO Wave Observatory A small fraction of one arm of the CalTech 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???? 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. Constructive interference for Delay = 0: all times (coherent) Adding a non- “Bright fringe” monochromatic wave to a delayed Destructive replica of itself Delay = interference for ½ period all times (coherent) (<< c): “Dark fringe”) Incoherent Delay > c: addition No fringes. The Michelson Interferometer is a Fourier Transform Spectrometer L2 Suppose the input beam is not monochromatic Beam- (but is perfectly spatially coherent): splitter L1 Iout = 2I + c Re{E(t+2L1 /c) E*(t+2L2 /c)} Delay Mirror Now, Iout will vary rapidly in time, and most detectors will simply integrate over a relatively long time, T : TT/2 /2 U ItdtU() 2 ITc Re EtLcEtLcdt ( 2 / ) *( 2 / ) out 12 TT/2 /2 Changing variables: t' = t + 2L1 /c and letting = 2(L2 -L1)/c and T UITc2Re(')*('' EtEtdt The Field Autocorrelation! Fourier Transform of the Field Autocorrelation is the spectrum!! Fourier Transform Spectrometer Interferogram A Fourier Transform Spectrometer's detected light energy vs. delay is called an interferogram. Michelson interferometer Spectrum integrated irradiance 2/ 1/ Intensity Integrated irradiance 0 Delay Frequency The Michelson interferometer output—the interferogram—Fourier transforms to the spectrum. Fourier Transform Spectrometer Data Actual interferogram from a Fourier Transform Spectrometer Interferogram This interferogram is very narrow, so the spectrum is very broad. 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: 1 to 18 m Accuracy: 10-3 /cm to 10-4 /cm Dynamic range: 19 bits (5 x 105) Fourier-transform A compact commercial spectrometers are now FT spectrometer from available for wave- Nicolet lengths even in the UV! Strangely, they’re still called FTIR’s. The Unbalanced Misalign mirrors, so beams cross at an angle. Michelson Interferometer Input beam x Now, suppose an object is placed in one arm. In addition Mirror z to the usual spatial factor, Beam- one beam will have a spatially splitter varying phase, exp[2i(x,y)]. Place an Mirror object in this path Now the cross term becomes: exp[i(x,y)] Re{ exp[2i(x,y)] exp[-2ikx sin] } Iout(x) Distorted fringes (in position) x The Unbalanced Michelson Interferometer can sensitively measure phase vs. position. Placing an object in one arm of a Spatial fringes distorted misaligned Michelson by a soldering iron tip in one path interferometer will distort the spatial fringes. Input beam Mirror Beam- splitter Mirror Phase variations of a small fraction of a wavelength can be measured. Michelson Interferometer: summary of simple treatment 1) Compensation plate: Negates dispersion from the beam splitter 2) Extended source: Fringes of equal inclination 3) Unbalanced configuration: Fringes of equal thickness S S1 S2 P M1 M2 Optical path length difference: 2d cos 4 Phase difference: d cos Dark fringes: 2d cosm m The Mach-Zehnder Interferometer Beam- splitter Mirror Output beam Object Input beam Mirror Beam- splitter The Mach-Zehnder interferometer is usually operated misaligned and with something of interest in one arm. Mach-Zehnder Interferogram Nothing in either path Plasma in one path The Sagnac Interferometer The two beams take the same path around the interferometer and the output light can either exit or return to the source. Mirror Beam- Mirror splitter Input beam It turns out that no light exits! It all returns to the source! Why all the light returns to the source in a Sagnac interferometer Reflection off a Mirror front-surface For the exit beam: mirror yields a phase shift of Clockwise path has phase (180 degrees). shifts of , , , and 0. Counterclockwise path Beam- has phase shifts of Input splitter beam 0, , , and 0. Return Perfect beam Mirror cancellation!! Reflection off a back- Reflective Exit surface mirror yields surface beam no phase shift. For the return beam: Clockwise path has phase shifts of , , , and 0. Counterclockwise path has phase shifts of 0, , , and . Constructive interference! Rotating Sagnac Interferometer Interferometer rotates with angular velocity (classical treatment) R 2 Travel AB: tAB c v 2 2R t AB 2c R 2R Travel AD: t AD 2c R 8R2 4A Time difference ( R<<c): t c2 c2 Rotating Sagnac Interferometer: example Michelson and Gale, 1925 Rotation of earth: = 2/24 hours = 7.27×10-5 s-1 side=500 m 2 4A 4500 m 7.27105s-1 t 2 c 8 2 310 m/s t 8.11016 s One period of light wave: /c = (500 nm)/(3×108 m/s)=1.7×10-15 s Sagnac Interferometer: gyroscope.