Interference. Reminder: Exam 2 and Review quiz, more details on the course website

Transcription

1 Chapter 9 Interference Phys 322 Lecture 25 Reminder: Exam 2 and Review quiz, more details on the course website Interferometers Wavefront-splitting interferometers Amplitude-splitting interferometers ed interferometers

2 The Michelson interferometer Albert Abraham Michelson Compensator plate /2 - reflection What if the path differences 2(d 1 -d 2 )=m? - minium, dark spot What if the path differences 2(d 1 -d 2 )=m+/2? - maximum, bright spot +/2 - reflection

3 The Michelson interferometer Path difference: 2dcos Phase shift: (internal/external reflection) Minima: 2d cos m What if light is white? ( WIU OptoLab)

4 The Michelson interferometer: speed of light Michelson-Morley experiment Speed of light is constant in all reference systems

5 The Michelson interferometer: application Accurate length measurements: Displacement of M 2 by /2 fringe will move to the position occupied by an adjacent fringe Count fringes, N: d = N( 0 /2)

6 The Michelson Interferometer: advanced treatment The Michelson Interferometer splits a beam into two and then recombines them at the same beam splitter. Suppose the input beam is a plane wave: Beamsplitter Input beam Delay * I I I cre E exp i( t kz 2 kl ) E exp i( t kz 2 kl ) out II2IRe exp 2 ik( L L) since II I ( c / 2) E 2I 1cos( kl) Dark fringe L 2 L 1 Output beam 2 Bright fringe where: L = 2(L 2 L 1 ) I out Fringes (in delay): L = 2(L 2 L 1 )

7 The Michelson Interferometer Another application of the Michelson Interferometer is to measure the wavelength of monochromatic light. L 2 Beamsplitter Input beam L 1 Delay Output beam I 2I 1cos( kl) 2I 1cos(2 L/ ) out I out L = 2(L 2 L 1 )

8 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 one direction and shrink it in another. They should cause one arm of a Michelson interferometer to stretch and the other to shrink. L 2 Beamsplitter L 1 L 1 and L 2 = 4 km! Unfortunately, the relative distance (L 1 -L 2 ~ cm) is less than the width of a nucleus! So such measurements are very very difficult!

9 The LIGO project Laser Interferometer Gravitational- Wave Observatory A small fraction of one arm of the CalTech LIGO interferometer CalTech LIGO The building containing an arm Hanford LIGO The control center

10 The LIGO folks think big The longer the interferometer arms, the better the sensitivity. So put one in space, of course.

11 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.

12 Adding a nonmonochromatic wave to a delayed replica of itself Delay = 0: Delay = ½ period (<< c ): Constructive interference for all times (coherent) Bright fringe Destructive interference for all times (coherent) Dark fringe ) Delay > c : Incoherent addition No fringes.

13 The Michelson Interferometer is a Fourier Transform Spectrometer L 2 Suppose the input beam is not monochromatic (but is perfectly spatially coherent): I out = 2I + c Re{E(t+2L 1 /c) E*(t+2L 2 /c)} Beamsplitter Now, I out will vary rapidly in time, and most detectors will simply integrate over a relatively long time, T : L 1 Delay T /2 T /2 out 1 2 T /2 T /2 U I () t dt U 2IT c Re E( t 2 L / c) E*( t 2 L / c) dt Changing variables: t' = t + 2L 1 /c and letting = 2(L 2 -L 1 )/c and T U 2IT c Re E( t') E*( t' dt' The Field Autocorrelation! Fourier Transform of the Field Autocorrelation is the spectrum!!

14 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 2/ 1/ Intensity 0 Delay Frequency The Michelson interferometer output the interferogram Fourier transforms to the spectrum.

15 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.

16 Fourier Transform Spectrometers Maximum path difference: 1 m Minimum resolution: /cm Spectral range: 1 to 18 m Accuracy: 10-3 /cm to 10-4 /cm Dynamic range: 19 bits (5 x 10 5 ) 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.

17 The Unbalanced 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[2i(x,y)]. Now the cross term becomes: Misalign mirrors, so beams cross at an angle. Beamsplitter Input beam x z Place an object in this path exp[i(x,y)] Re{ exp[2i(x,y)] exp[-2ikx sin] } Distorted fringes (in position) I out (x) x

18 The Unbalanced Michelson Interferometer can sensitively measure phase vs. position. Spatial fringes distorted by a soldering iron tip in one path Placing an object in one arm of a misaligned Michelson interferometer will distort the spatial fringes. Input beam Beamsplitter Phase variations of a small fraction of a wavelength can be measured.

19 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 P S S 1 S 2 M 1 M 2 Optical path length difference: 2d cos Phase difference: Dark fringes: 2d cos 4 d cos m m

20 The Mach-Zehnder Interferometer Object Beamsplitter Output beam Input beam Beamsplitter The Mach-Zehnder interferometer is usually operated misaligned and with something of interest in one arm.

21 Mach-Zehnder Interferogram Nothing in either path Plasma in one path

22 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. Beamsplitter Input beam It turns out that no light exits! It all returns to the source!

23 Why all the light returns to the source in a Sagnac interferometer For the exit beam: Clockwise path has phase shifts of,,, and 0. Counterclockwise path has phase shifts of 0,,, and 0. Perfect cancellation!! Return beam Input beam Reflective surface Exit beam Beamsplitter Reflection off a front-surface mirror yields a phase shift of (180 degrees). Reflection off a backsurface mirror yields 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!

24 Rotating Sagnac Interferometer Interferometer rotates with angular velocity (classical treatment) R 2 Travel AB: t AB c v 2 2R t AB 2 c R Travel AD: t AD 2R 2 c R 2 8R 4A Time difference ( R<<c): t 2 2 c c

25 Rotating Sagnac Interferometer: example Michelson and Gale, 1925 Rotation of earth: = 2/24 hours = s -1 side=500 m 2 4A t m s 2 c m/s 2 t s One period of light wave: /c = (500 nm)/( m/s)= s

26 Sagnac Interferometer: gyroscope