34. Even more Interference Effects
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1 34. Even more Interference Effects The Fabry-Perot interferometer Thin-film interference Anti-reflection coatings Single- and multi-layer Advanced topic: Photonic crystals Natural and artificial periodic structures
2 Interference and Interferometers Same polarizations Different polarizations Same wavelength Different wavelengths I I I 1 c Re E1 E * I I I 1 I I1I I I1 I Interference only occurs when waves have the same wavelength and polarization. Interferometers (like the Michelson) produce fringes but only if L is less than the coherence length: the interferogram mirror beamsplitter input beam output mirror delay
3 Multiple-reflection interference: The Fabry-Perot Interferometer A Fabry-Perot interferometer is a pair of parallel surfaces that reflect beams back and forth. An etalon is one particular type of Fabry-Perot: a thin piece of glass with parallel sides. Incident wave: E 0 Reflected wave: E 0r n = 1 = round-trip phase between the two surfaces Transmitted t E 0 wave: E 0t n n = 1 The transmitted and reflected waves are both an infinite series of multiply reflected beams. d Maurice Paul Auguste Charles Fabry ( ) j t r e E0 j t ( r e ) E0 j 3 t ( r e ) E0 e.g., for normal incidence, = nkd Jean-Baptiste Alfred Perot ( )
4 The Fabry-Perot Interferometer The field of the transmitted wave is an infinite sum: E t E t re E t ( re ) E t ( r e ) E... j j j 3 0t E t E r e r e 1 ( j ) ( j )... 0t 0 Here we can use the identity: 1 xx x... 1 x 3 1 E t 0t j 0 1 re E
5 The Fabry-Perot interferometer So, if the transmitted E-field is: E t 0t j 0 1 re E then the transmittance is: T 4 E 0t t t j j j E0 1 r e (1 r e )(1 r e ) (1 r ) r r r sin ( /)] Dividing numerator and denominator by T (1 r ) 1 r 1 where: F F sin / 1 r F is known as the coefficient of finesse.
6 Etalon Transmittance vs. Thickness, Wavelength, or Angle F r 1 r Transmission maxima occur when sin(/) = 0, or when / = m But = kl = L/ (L = round trip length between the two reflecting surfaces, which depends on propagation angle) So max = L / m transmittance F = 0. F = 1 F = 0 F = 00 = L/ The transmittance varies significantly with thickness or wavelength. We can also vary the incidence angle, which also affects. As the reflectance of each surface (r ) approaches 1, F approaches infinity, and the widths of the transmission peaks become very narrow.
7 The Etalon Free Spectral Range The Free Spectral Range is the wavelength range between transmission maxima. FSR = Free Spectral Range transmittance FSR F = 00 FSR = L/ L L FSR FSR L But usually L >>, so FSR L
8 Etalon Linewidth and Finesse Which means: setting equal to LW / should yield T = 1/. 1 T T 1 F 1 sin / The Linewidth LW is the value of at which a transmittance peak has dropped to half of its largest value: the full-width at half-max (FWHM). 1 Fsin / / or sin LW LW /4 1/ For << 1, we can make the small argument approximation: LW /4 1/ F 4/ The Finesse,, is the ratio of the phase corresponding to one Free Spectral Range to the phase corresponding to one linewidth: LW F 4/ F F F = corresponds to one FSR Substituting F r 1 r we have: r r /[1 ]
9 Etalon Linewidth and Finesse The Finesse tells us the resolving power of the interferometer. Building an etalon with a finesse of 100 requires mirror reflectivities greater than 98%. A finesse of 100 is quite high, but can be achieved. transmittance Finesse = 100 = L/ A high-finesse etalon placed inside of a multimode laser cavity can force the laser to operate on a single mode only.
10 Other uses of Fabry-Perot interferometers and etalons To frequency filter a beam (e.g., for multi-frequency telecommunications systems such as wavelength division multiplexing) To measure the wavelength or spectrum of a beam (but you must know it in advance to within a Free Spectral Range, and you must scan the length of the interferometer and watch for the transmission vs. length). Money is now coated with interferometric inks to help foil counterfeiters. Notice the shade of the 100, which is shown from two different angles.
11 transmittance Thin film interference Finesse = 100 FSR = L/ This effect causes the colors in bubbles and oil films on puddles. FSR L For very thin films (where L approaches ), the free spectral range becomes large. If it is large enough, then only one visible color can be transmitted through the film at a given angle.
12 A single layer on a surface Consider a beam incident on a piece of glass (n = n s ) with a layer of material (n = n c ) of thickness, h, on its surface. It can be shown that the reflectance from the coated surface, for a beam impinging at normal incidence, is: n ( n n ) cos ( kh) ( n n n ) sin ( kh) R n ( n n ) cos ( kh ) ( n n n ) sin ( kh ) c 0 s 0 s c c 0 s 0 s c air, index n 0 coating, index n c & thickness h glass, index n s If h is chosen so that kh = / (i.e., h = / 4), then this becomes: R Notice that R = 0 if: ( nn 0 s n ) ( nn n ) 0 s n c c c n n 0 s If the index of the coating is the geometric mean of n 0 and n s, then the reflectance is zero! - an anti-reflection coating
13 Anti-reflection Coating An anti-reflection coating reduces the reflection from this piece of glass to nearly zero. no coating coating A coating like this, which has a broadband anti-reflection, requires multiple layers. Coatings with hundreds of layers are not unusual.
14 Deposition of multiple layers There are numerous techniques for forming multi-layer structures like these. Common ones include chemical vapor deposition (CVD), physical vapor deposition (PVD), molecular beam epitaxy (MBE), and pulsed laser deposition (PLD). A plasma CVD chamber These techniques can be used to deposit thin films of almost any material. For optical components, commonly used dielectrics include titanium dioxide, magnesium fluoride, zinc sulfide, and silicon dioxide.
15 The opposite of Anti-reflection Multi-layer coatings can also be engineered to provide high reflection, rather than anti-reflection. In this case, we have a mirror which contains no metal! This is known as a Bragg mirror, after William H. Bragg and William L. Bragg, who discovered x-ray crystallography. more layers = better reflectivity
16 Multilayer coatings Typical laser mirrors and camera lenses use many coating layers. The reflectance and transmittance can be tailored to taste! In the X-ray region of the spectrum, multilayer coatings are the only optical components that work. transmission electron micrograph of a crosssection of a silicon-molybdenum multilayer mirror for x-ray optics
17 Multilayer coatings - the best mirrors Multiple layers can be used to make dielectric mirrors that are much better than the best polished metal surface. But they usually work over a smaller range of wavelengths. metal mirror reflectance: dielectric mirror reflectance: 1 inch mirror: $50 1 inch mirror: $175
18 Interference filters An interference filter is any multi-layer structure which has a narrow band pass, and which therefore can be used to filter the spectrum of a light wave. An etalon is one example of an interference filter. But often multiple dielectric layers are used to narrow or tailor the filter function. More layers are added to narrow the band pass. This filter has a 10 nm band pass.
19 1 Many many layers: a 1D crystal! exact dielectric profile dielectric constant dielectric constant 1 Approximate the dielectric of this structure using a periodic dielectric function, of the form: cos 0 jgz 0 e e Gz where G = /(period) jgz The field inside the medium may generally contain an incident and a reflected wave: E Ce C e jkz 1 Plug these into the one-dimensional wave equation: 0 our approximation distance z j kg z d E d E dz c dt
20 Many many layers: a 1D crystal! Wave equation: jkz jkgz kce 1 k G Ce jkz jkz jkz 0 e e Ce 1 Ce c jkgz Match up terms with the same exponents: k 0 C1 C 0 c c C 1kG 0 C 0 c c Two equations in two unknowns: this only has a solution if the determinant vanishes, which gives a condition on k: k 4 G G 0 G c c c Note: it is no longer true that k = n/c - a new dispersion relation for k()!
21 Dispersion of a 1D photonic crystal k 4 G G 0 G c c c G G 0 4 c where G c c Note that is always positive. What if it is large? When G 4 c 0 0, then the k vector has an imaginary component.
22 One-dimensional photonic band gap Frequency cg in units of Re k 0 k G 0 c Photonic band gap For these values of, k is complex. At frequencies in the gap, the propagating wave contains a term of the form exp[im{k} z], which attenuates the wave. In this one-dimensional example, the device is equivalent to a Bragg mirror, with high reflection in a narrow band.
23 Photonic band gaps in D, 3D This can be achieved in two or even three dimensional structures! A cartoon of a 3D photonic crystal A D photonic crystal with an integrated waveguide the end face of a photonic crystal optical fiber What is required for a photonic band gap? A periodic variation of the refractive index, with a period close to the wavelength. A large enough difference between the maximum and minimum values of the index (i.e., the value of ) Eli Yablonovich
24 Natural photonic crystals Butterfly wings are iridescent and brightly colored, not because of a pigment, but because of diffraction and interference! Butterfly wings are natural photonic crystals. the blue morpho butterfly increasingly close-up views of a butterfly wing
25 Other photonic crystals Opals are photonic crystals. At certain angles, the reflected light at a particular wavelength experiences constructive interference, giving rise to colored reflections. Scanning electron microscope (SEM) image of a natural opal Natural opal SEM image of an artificial opal Artificial opal thin film m
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