More Interference Michelson Interferometer (cont'd) Unbalanced Prof. Rick Trebino Georgia Tech www.physics.gatech.edu/frog/lectures Michelson Interferometer (cont'd) Unbalanced Other interferometers Mach-Zehnder Sagnac Fizeau Wedge Newton's Rings The Fabry-Perot Interferometer Anti-Reflection Coatings Single- and Multi-layer Photonic Crystals

The Unbalanced Michelson Interferometer Misalign mirrors, so beams cross at an angle. The Unbalanced Michelson Interferometer z x Input beam Now, suppose an object is placed in one arm. Now, one beam will have an extra spatially varying phase, exp[2if(x,y)]. The cross term becomes: Re{ exp[2if(x,y)] exp[-2ikx sinq] } Mirror Beam- splitter Fringes exp[if(x,y)] Place an object in this path Mirror Distorted fringes (in position) Iout(x) x

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

The Mach-Zehnder Interferometer Beam- splitter Input beam Mirror Output beam Object 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 Not only do both beams take the same path through the device, they also pass through the same amount of glass in the beam splitter! Beam- splitter Mirror 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 front-surface mirror yields a phase shift of p (180 degrees). Since the paths are the same, all that matters are the phase shifts on the reflections. Mirror For the exit beam: Clockwise path has phase shifts of p, p, p, and 0. Beam- splitter Input beam Return beam Counterclockwise path has phase shifts of 0, p, p, and 0. Perfect cancellation!! Reflection off a back-surface mirror yields no phase shift. Mirror Reflective surface Exit beam For the return beam: Clockwise path has phase shifts of p, p, p, and 0. Counterclockwise path has phase shifts of 0, p, p, and p. Constructive interference!

The Sagnac Interferometer senses rotation. Suppose that the beam splitter moves by a distance, d, in the time, T, it takes light to circumnavigate the Sagnac interferometer. So one beam will travel more (d), and the other less distance (-d). If R = the interferometer radius, and W = its angular velocity: Thus, the Sagnac Interferometer's sensitivity to rotation depends on its area. And it need not be round! Sagnac Interfer- ometer (fiber) q R q = WT d d = Rq

Newton's Rings

You see the color l when constructive interference occurs. Newton's Rings Get constructive interference when an integral number of half wavelengths occurs 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). You see the color l when constructive interference occurs. You only see bold colors when m = 1 (possibly 2). Otherwise the variation with l is too fast for the eye to resolve. L This effect also causes the colors in bubbles and oil films on puddles.