PHYS 415: OPTICS Review of Interference and Diffraction F. ÖMER ILDAY Department of Physics, Bilkent University, Ankara, Turkey I used the following resources in the preparation of almost all these lectures: Trebino’s Modern Optics lectures from Gatech (quite heavily used), and various textbooks by Pedrotti & Pedrotti, Hecht, Guenther, Verdeyen, Fowles and Das

Interference vs. Diffraction Interference is when we add up multiple but a finite number of E&M waves Diffraction is when we add up a continuum of E&M waves. Fundamentally, there is no difference.

Interference and Interferometers

Varying the delay on purpose Simply moving a mirror can vary the delay of a beam by many wavelengths. 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. 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! Translation stage Input beam E(t) E(t–  ) Mirror Output beam

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. 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. Translation stage Input beam E(t) E(t–  ) Mirrors Output beam [Edmund Scientific]

The Michelson Interferometer Beam- splitter Input beam Delay Mirror Fringes (in delay):  L = 2(L 2 – L 1 ) 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: I out L1L1 where:  L = 2(L 2 – L 1 ) L2L2 Output beam “Bright fringe” “Dark fringe”

The Michelson Interferometer Beam- splitter Input beam Delay Mirror The most obvious application of the Michelson Interferometer is to measure the wavelength of monochromatic light.  L = 2(L 2 – L 1 ) I out L1L1 L2L2 Output beam

Huge Michelson Interferometers may someday detect gravity waves. Beam- splitter Mirror L1L1 L2L2 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. Unfortunately, the relative distance ( L 1 -L 2 ~ cm) is less than the width of a nucleus! So such measurements are very very difficult! L 1 and L 2 = 4 km!

The LIGO project A small fraction of one arm of the CalTech LIGO interferometer… The building containing an arm The control center CalTech LIGO Hanford LIGO

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

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)} Now, I out will vary rapidly in time, and most detectors will simply integrate over a relatively long time, T : The Michelson Interferometer is a Fourier Transform Spectrometer The Field Autocorrelation! Beam- splitter Delay Mirror L1L1 L2L2 Recall that the Fourier Transform of the Field Autocorrelation is the spectrum!! Changing variables: t' = t + 2L 1 /c and letting  = 2(L 2 - L 1 )/c and T 

Fourier Transform Spectrometer Interferogram The Michelson interferometer output—the interferogram—Fourier transforms to the spectrum. The spectral phase plays no role! (The temporal phase does, however.) Integrated irradiance 0 Delay Michelson interferometer integrated irradiance 2/2/ 1/  Frequency Intensity  Spectrum  A Fourier Transform Spectrometer's detected light energy vs. delay is called an interferogram.

Fourier Transform Spectrometer Data 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. Actual interferogram from a Fourier Transform Spectrometer

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. Mirror Supposed velocity of earth through the aether Parallel and anti-parallel propagation Perpendicular propagation Beam- splitter Mirror

Michelson-Morley Experiment: Results Michelson and Morley's results from A. A. Michelson, Studies in Optics Interference fringes showed no change as the interferometer was rotated. 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. Their apparatus

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

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

The Mach-Zehnder Interferometer The Mach-Zehnder interferometer is usually operated “misaligned” and with something of interest in one arm. Beam- splitter Input beam Mirror Beam- splitter Output beam Object

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. Beam- splitter Input beam Mirror Beam- splitter Input beam Mirror 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 For the exit beam: Clockwise path has phase shifts of p, p, p, and 0. Counterclockwise path has phase shifts of 0, p, p, and 0. Perfect cancellation!! 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! Beam- splitter Input beam Mirror Reflection off a front-surface mirror yields a phase shift of p (180 degrees). Reflective surface Reflection off a back- surface mirror yields no phase shift. Exit beam Return beam

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. As a result, one beam will travel more, and the other less distance. 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! q R q  WT d d = R q Sagnac Interfer- ometer

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. You see the color  l when: 2L = m l L You only see bold colors when m = 1 or 2. Otherwise the variation with  l is too fast for the eye to resolve.

Newton's Rings

Multiple-beam interference: The Fabry-Perot Interferometer or Etalon A Fabry-Perot interferometer is a pair of parallel surfaces that reflect beams back and forth. An etalon is a type of Fabry-Perot interferometer, and is a piece of glass with parallel sides. The transmitted wave is an infinite series of multiply reflected beams. Transmitted wave: Incident wave: E 0 Reflected wave: E 0r d = round-trip phase delay inside medium = 2kL Transmitted wave: E 0t r, t = reflection, transmission coefficients from glass to air nn air = 1 L

The Etalon (cont'd) The transmitted wave field is: The transmittance is: where: Dividing numerator and denominator by

Etalon Transmittance vs. Thickness, Wavelength, or Angle The transmittance varies significantly with thickness or wavelength. We can also vary the incidence angle, which also affects d (via L ). As the reflectance of each surface ( r 2 ) approaches 1 ( F increases), the widths of the high-transmission regions become very narrow. Transmission maxima occur when d  / 2 = m p: 2 p L/ l = m p or: Transmittance

Does this look familiar? Recall that a finite train of identical pulses can be written: where g(t) is a Gaussian envelope over the pulse train. g(t) = exp(-t/ t ) The light field trans- mitted by the etalon! The peaks become Lorentzians.

The Etalon Free Spectral Range l FSR = Free Spectral Range The Free Spectral Range is the wavelength range between transmission maxima. l FSR Transmittance

Etalon Linewidth The Linewidth d LW is a transmittance peak's full-width-half-max (FWHM). Setting d equal to d LW /2 should yield T = 1/2 : Substituting and we have: Or: For d << 1, we can make the small argument approx: l l LW Transmittance The linewidth is the etalon’s wavelength-measurement accuracy.

The Interferometer or Etalon Finesse The Finesse is the number of wavelengths the interferometer can resolve. Taking The Finesse, F, is the ratio of the Free Spectral Range and the Linewidth:

How to use an interferometer to measure wavelength 1. Measure the wavelength to within one Free Spectral Range using a grating or prism spectrometer to avoid the interferometer’s inherent ambiguities. 2. Scan the spacing of the two mirrors and record the spacing when a transmission maximum occurs. 3. If greater accuracy is required, use another (longer) interferometer with a FSR ~ the above accuracy (line-width) and with an even smaller line-width (i.e., better accuracy). Interferometers are the most accurate measures of wavelength available.

Anti-reflection Coating Notice that the center of the round glass plate looks like it’s missing. It’s not! There’s an “anti-reflection coating” there (on both the front and back of the glass).

Anti-reflection Coating Math Consider a beam incident on a piece of glass ( n = n s ) with a layer of material ( n = n l ) if thickness, h, on its surface. It can be shown that the Reflectance is (for such thin media, we need to go back to Maxwell’s equations): Notice that R = 0 if:

Multilayer coatings Typical laser mirrors and camera lenses use many layers. The reflectance and transmittance can be tailored to taste!

Stellar interferometry Taken from von der Luhe, of Kiepenheuer-Institut fur Sonnenphysik, Freiburg, Germany. Stars are too small to resolve using normal telescopes, but interferometry can see them. Stellar interferometers operate in the radio (when the signals are combined electronically) and visible (where the beams are combined optically).

“Photonic crystals” use interference to guide light— sometimes around corners! Interference controls the path of light. Constructive interference occurs along the desired path. Augustin, et al., Opt. Expr., 11, 3284, Yellow indicates peak field regions. Borel, et al., Opt. Expr. 12, 1996 (2004)

Convolution

The Convolution The convolution allows one function to smear or broaden another. changing variables: x  t - x

The convolution can be performed visually. Here, rect(x) * rect(x) =  (x)

Convolution with a delta function Convolution with a delta function simply centers the function on the delta-function. This convolution does not smear out f(t). Since a device’s performance can usually be described as a convolution of the quantity it’s trying to measure and some instrument response, a perfect device has a delta-function instrument response.

The Convolution Theorem The Convolution Theorem turns a convolution into the inverse FT of the product of the Fourier Transforms: Proof:

The Convolution Theorem in action

The symbol III is pronounced shah after the Cyrillic character III, which is said to have been modeled on the Hebrew letter (shin) which, in turn, may derive from the Egyptian a hieroglyph depicting papyrus plants along the Nile. The Shah Function The Shah function, III(t), is an infinitely long train of equally spaced delta-functions. t

The Fourier Transform of the Shah Function If  = 2n , where n is an integer, the sum diverges; otherwise, cancellation occurs. So: t III(t) F {III(t)}

Fraunhofer Diffraction

Diffraction Light does not always travel in a straight line. It tends to bend around objects. This tendency is called diffraction. Any wave will do this, including matter waves and acoustic waves. Shadow of a hand illuminated by a Helium- Neon laser Shadow of a zinc oxide crystal illuminated by a electrons

Why it’s hard to see diffraction Diffraction tends to cause ripples at edges. But poor source temporal or spatial coherence masks them. Example: a large spatially incoherent source (like the sun) casts blurry shadows, masking the diffraction ripples. Untilted rays yield a perfect shadow of the hole, but off-axis rays blur the shadow. Screen with hole A point source is required.

Diffraction of a wave by a slit Whether waves in water or electromagnetic radiation in air, passage through a slit yields a diffraction pattern that will appear more dramatic as the size of the slit approaches the wavelength of the wave.

Diffraction of ocean water waves Ocean waves passing through slits in Tel Aviv, Israel Diffraction occurs for all waves, whatever the phenomenon.

Diffraction Geometry We wish to find the light electric field after a screen with a hole in it. This is a very general problem with far-reaching applications. What is E(x 1,y 1 ) at a distance z from the plane of the aperture? Incident wave This region is assumed to be much smaller than this one. x1x1 x0x0 P1P1 0 A(x0,y0)A(x0,y0) y1y1 y0y0

Diffraction Solution The field in the observation plane, E(x 1,y 1 ), at a distance z from the aperture plane is given by a convolution: A very complicated result! And we cannot approximate r 01 in the exp by z because it gets multiplied by k, which is big, so relatively small changes in r 01 can make a big difference!

Fraunhofer Diffraction: The Far Field Let D be the size of the aperture: D 2 = x y 0 2. When kD 2 /2z << 1, the quadratic terms << 1, so we can neglect them: Recall the Fresnel diffraction result: This condition corresponds to going far away: z >> kD 2 /2 =  D 2 / If D = 100 microns and = 1 micron, then z >> 30 meters, which is far!

Fraunhofer Diffraction Conventions As in Fresnel diffraction, we’ll neglect the phase factors, and we’ll explicitly write the aperture function in the integral: This is just a Fourier Transform! Interestingly, it’s a Fourier Transform from position, x 0, to another position variable, x 1 (in another plane). Usually, the Fourier “conjugate variables” have reciprocal units (e.g., t & , or x & k ). The conjugate variables here are really x 0 and k x = kx 1 /z, which have reciprocal units. So the far-field light field is the Fourier Transform of the apertured field! E(x 0,y 0 ) = constant if a plane wave

The Fraunhofer Diffraction formula where we’ve dropped the subscripts, 0 and 1, E(x,y) = const if a plane wave Aperture function We can write this result in terms of the off-axis k-vector components: k x = kx 1 /z and k y = ky 1 /z kzkz kyky kxkx  x = k x /k = x 1 /z and  y = k y /k = y 1 /z and: or:

Fraunhofer Diffraction from a slit Fraunhofer Diffraction from a slit is simply the Fourier Transform of a rect function, which is a sinc function. The irradiance is then sinc 2.

Fraunhofer Diffraction from a Square Aperture The diffracted field is a sinc function in both x 1 and y 1 because the Fourier transform of a rect function is sinc. Diffracted irradiance Diffracted field

Diffraction from a Circular Aperture A circular aperture yields a diffracted "Airy Pattern," which involves a Bessel function. Diffracted field Diffracted Irradiance

Diffraction from small and large circular apertures Recall the Fourier scaling! Far-field intensity pattern from a small aperture Far-field intensity pattern from a large aperture

Fraunhofer diffraction from two slits A(x 0 ) = rect[(x 0 +a)/w] + rect[(x 0 -a)/w] 0 x0x0 a-a kx 1 /z ww

Diffraction from one- and two-slit screens Fraunhofer diffraction patterns One slit Two slits

Diffraction from multiple slits Slit Diffraction Pattern Infinitely many equally spaced slits yields a far-field pattern that’s the Fourier transform

Young’s Two Slit Experiment and Quantum Mechanics Imagine using a beam so weak that only one photon passes through the screen at a time. In this case, the photon would seem to pass through only one slit at a time, yielding a one-slit pattern. Which pattern occurs? Possible Fraunhofer diffraction patterns Each photon passes through only one slit Each photon passes through both slits

Fresnel Diffraction

Fresnel Diffraction: Approximations But, in the denominator, we can approximate r 01 by z. And, in the numerator, we can write: This yields:

Fresnel Diffraction: Approximations Multiplying out the squares: This is the Fresnel integral. It's complicated! It yields the light wave field at the distance z from the screen. Factoring out the quantities independent of x 0 and y 0 :

Diffraction Conventions We’ll typically assume that a plane wave is incident on the aperture. And we’ll explicitly write the aperture function in the integral: And we’ll usually ignore the various factors in front: It still has an exp[i(   t – k z)], but it’s constant with respect to x 0 and y 0.

Fresnel Diffraction: Example Fresnel Diffraction from a single slit: Far from the slit z Close to the slit Incident plane wave Slit

Fresnel Diffraction from a Slit This irradiance vs. position emerges from a slit illuminated by a laser. x1x1 Irradiance

Diffraction Approximated The approximate intensity vs. position from an edge: Such effects can be modeled by measuring the distance on a “Cornu Spiral” But most useful diffraction effects do not occur in the Fresnel diffraction regime because it’s too complex. For a cool Java applet that computes Fresnel diffraction patterns, try