What is the characteristic timescale for decay of a nonequilibrium charge distribution in a conductor?

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1 Charge ecay in a conuctor What is the characteristic timescale for ecay of a nonequilibrium charge istribution in a conuctor? Continuity: Gauss law: J = σe = ε E = ρ t ρ Combining, σ ρ = ρ ε t σ τ ε c Typical numbers for copper give τ c ~ 0-9 s (!). EM waves an conuctors What happens to EM waves insie conuctors? Electric fiel wave equations incluing currents becomes: E E µ J µε = 0 t t We know efinition of conuctivity: E E E µσ µε = 0 t t Plugging in our trial plane wave solution gives: ( k + iωµσ + µεω ) E 0k e i( k r ωt ) = 0 Solve for the real an imaginary parts of k = a + ib, taking positive roots an assuming that all 3 components of k have the same ratio of real to imaginary parts,

2 EM waves an conuctors a = k 0 b = k ( ωτ ) / c + ( ωτ ) / c k 0 ω εµ If ωτ c <<, the material is a goo conuctor; If ωτ c >>, the material is a poor conuctor; The poor conuctor limit is boring; small corrections to what we alreay have seen, with very slow exponential ecay of wave with propagation (b ~ 0). EM waves an conuctors In the goo conuctor limit, a b k 0 ωτ c This means the EM wave ecays in the conuctor on a length scale given by: δ = ωµσ This is the skin epth. In copper at optical frequencies, this is something like 3 nm. Nanoscale conuctors can be on the same scale as the skin epth, an so can strongly interact with an electromagnetic wave. So, in a goo conuctor the real an imaginary parts of k are of the same size.

3 Waves an conuctor interfaces Consier TE case. Apply cont. of tang. E at x = 0; E0 i + E0r = E0t Also, + i k tx δ Using the tang. H conition gives: E E 0r 0t n cosθ i i = ωτ c E n n cosθ i i = ωτ c E0i n 0i E ε ε 0i E 0i k i +x n i H0i E k t 0t H 0t E 0r H k r 0r t r Can o energy flow an Poynting vector an so forth. Note that a goo conuctor reflects very well. Diffraction We want to solve the general problem of raiation impinging on a screen with holes of some size an shape in it, an ask what the resulting intensity pattern is at some istance away: Source r z = 0 Can treat this as something like a bounary value problem. Remember Huygen s principle? Can apply that mathematically. We ll o scalar iffraction theory (no info on polarization, fiels). 3

4 Scalar iffraction theory Do theory for generic scalar variable ψ, an assume that s not a ba escription for, say, the E-fiel amplitue of an EM wave. Generic time-inepenent wave equation = Helmholtz eq. ψ + k ψ = 0 Green s fn for this: ik e r r' φ( r) = satisfies: ( + k ) φ( r) = 4πδ ( r r' ) r r' Want to solve for region to right of screen, with bounary conitions at slits an at infinity. Solution is calle the Kirchoff iffraction integral: ψ ( r) = π ψ ( r) = π R r r' S S e a' ψ ( r' ) n' i R a' e R ikr n' ' ikr ( ψ ( r') ) i ik eˆ R R r z = 0 R r Scalar iffraction theory ψ ( r) = π ψ ( r) = π R r r' S S e a' ψ ( r' ) n' i R a' e R ikr n' ' ikr ( ψ ( r') ) i ik eˆ R Typically assume either ψ or its graient in the normal irection is zero on screen an unisturbe in the holes. R r z = 0 R r General case = Fresnel iffraction. Fraunhofer iff. assumes incient plane wave, an very large R. R r eˆ r r' Defining k keˆ r leas to ik e ψ ( r) 4π r ikr (cosθ + cosθ ) i S a' ψ ( r' ) i e -ik r ' 4

5 Fraunhofer limit ikr ik e -ik r ' ψ ( r) (cosθ i + cosθ ) ' ( r') 4 a ψ i e π r S For a normally incient planewave, this calculation looks like a Fourier transform of a function that is in the holes an 0 outsie the holes. A tilte incient wave gives the same result, but offset. Classic example: single slit iffraction, for rectangular slit of with a an height b. For normal incience, ψ ( z = 0) = ψ a b 0 x' y' a b ikr ik e i( k xx' + k y y' ) ψ ( r) ( + cosθ ) ψ e 4π r 4 = k k x y i sin k asin k b x y 0 z = 0 Single slit iffraction a b 0 x' y' a b ikr ik e i( k xx' + k y y' ) ψ ( r) ( + cosθ ) ψ e 4π r In sensible spherical coorinates, 4 = k k x y sin k asin k b k x = k sinθ cosφ k y = k sinθ sinφ Diffracte intensity proportional to square of wave: x y I = I 0 k ( + cosθ ) π r sin k asin k b Narrower the slit, the wier the iffraction pattern. x kx k y y

6 Circular aperture For a circular aperture of raius a, we can rewrite the integral: ik e ψ ( r) 4π r ikr ( + cosθ ) ψ a 0 0 ρ ρ π 0 φ e ikxρ cosφ Turns out this can be one using special functions to give: I = I 0 k a 4r 4 J( ka sinθ ) ka sinθ assuming θ is small. J is the first orer Bessel fn of the first kin, an has its first zero when its argument = The result of this iffraction is an Airy pattern, with a central spot surroune by rings. Playing with numbers, that first zero hits at λ θ =. a from Iniana Univ. Diffraction Scalar iffraction theory is just an approximation. To keep track of polarization, bounary conitions at a conucting screen, etc. requires vector iffraction theory. Because of form of Frauenhofer integral, iffracte bright spots in this (far-fiel) limit always have a minimum size on the orer of the wavelength of the incient light. We will revisit iffraction shortly, to consier the Fresnel case an get a feel for what near-fiel optics is about. 6

7 Antireflection coatings Basic iea is straightforwar. Some of incient wave reflects off first interface, an picks up a π phase shift. The transmitte light racks up aitional phase while propagating through the high inex meium. Some of that light is then reflecte off the secon interface (another π phase shift), racks up more phase, an (some) reenters the original meium. If the optical path lengths are chosen correctly, () an () can be chosen to interfere estructively. n = n a =.4 n b =.7 () () n=.5 (lens) One homework problem will be on this. Dielectric mirrors The basic iea here is the same, only this time the interfaces are chosen to give constructive interference, enhancing reflecte intensity. We ll o an example of this here, assuming normal incience an TE wave. n = Going from n to n a, the reflection coefficient r is: n a =.7 n b =.4 E r E 0r 0i () () n n = n + n Note that r < 0 for n a > n, so there s a π phase shift for this reflection. Choose thickness t a to be the right thickness that light reflecte from the a-b interface comes back to the surface with exactly a π phase shift with respect to the incient light: ω ka = n = π a a c k t π c λ Want t a a = = 0 πν na 4 na a a 7

8 Dielectric mirrors Now suppose you continue the stack, an wante constructive interference between the reflecte waves from each succeeing interface. Assume n a > n b. Going from n a to n b, the reflection coefficient r is: na nb r = n + n a b n a =.7 n b =.4 n a =.7 n b =.4 substrate Since n b < n a, another π phase shift at this reflection. Choose thickness t b to be the right thickness; eventually nee a π phase shift wrt the incient light. If we pick π c λ0 tb = =, πν n 4 n π π π π total phase shift is then + + π + + = π + π That s why these structures are sometimes calle quarter wave stacks. b cavity b Dielectric mirrors Each bilayer calle a perio. For normal incience an a limite wavelength range, ielectric mirrors are often superior to metal mirrors higher reflectance, lower loss. Why? We know the reflection coefficient for each interface; we can therefore sum the series of reflection coefficients for a stack of m perios, an fin that the total reflection coefficient for the chosen wavelength is: m α nsn c nb r m = + α na n α a Here n s an n c correspon to the substrate an cavity inices, respectively. Can get higher reflectivity (smaller α) by having larger inex contrast (n b /n a << ) or more perios. 8

9 Dielectric mirrors Can solve for number of perios neee to reach a given reflectance: na ( m 0.5ln n n ( + s c R) R) / ln( n b / n ) a What happens a little away from the esign-optimize wavelength? The half-maximum reflectance happens at: n a nb ± ( λ = 0 / λ) sin π na + nb For problem set work regaring these mirrors, Application: Distribute Bragg reflectors This type of ielectric mirror is also known as a istribute Bragg reflector, an one particular application is in soli state lasers: image from UCSB 9

10 Application: Distribute Bragg reflectors It s clear that by MBE growth of semiconuctor layers, it s possible to achieve very large reflectances, an thus very high quality optical cavities. These cavities can be use for VCSELs, an also for manipulations of optically active nanoparticles, cavity QED, etc. Materials limitations are tricky: how o you make a soli state DBR that works at very high frequencies? Application: fiber gratings Another version of the DBR appears in fiber optic communications: the fiber grating: By varying the inex along the length of an optical fiber, can result in very high reflectance of particular wavelengths; makes possible several switching technologies with less pain (fewer fiber splices / transitions to electrical components). Even better: Chirpe gratings to compensate for ispersion. Different wavelengths propagate ifferent istances before being reflecte back. Can make up for fiber ispersion. 0

11 Complications We ve talke a lot about normal incience, largely because the algebra is less messy. Things get complicate at nonnormal incience. As you might imagine, polarization effects (which come in at every interface) are severe in structures with ozens or hunres of interfaces. On the other han, these effects can be turne to our avantage - see the paper you re suppose to rea for problem 3. Next time: Photonic bangap systems

12 Photonic bangap materials Sania What is a photonic bangap? Why are these metamaterials important an such a hot topic of research? How are photonic bangap systems mae? MIT photonics group NEC Iea of the photonic ban gap We ve seen that by carefully choosing layer thicknesses an inices of refraction, it s possible to create extremely high reflectance ielectric mirrors aroun a specific wavelength. This works for large numbers of perios, even if the ielectric contrast between layers is relatively small. It turns out that the physics here (essentially no available propagating optical moes compatible with bounary conitions impose by the perioic material) is exactly analogous to the formation of electronic ban gaps in solis. To see this in, start from the wave equation: E( x, t) E( x, t) = µ 0ε( x) Assume the usual time epenence: x t κ ( x) E( x, t) E( x, t) = E( x) exp( iωt) = c t

13 Iea of the photonic ban gap Rearranging, x κ ( x) ω E( x) ω E( x) = c c E( x) Remember, here κ(x) is a perioic function in space: κ ( x + ) = κ ( x) As a result, this equation for the electric fiel looks remarkably like the Schroeinger equation for a particle in a perioic potential. This means solutions are in the form of Bloch waves! E( x + ) = exp( iγ ) E( x) Now γ takes the role of the Bloch wavevector (or crystal momentum). Just as in the electronic structure case, We can label allowe EM moes by a Bloch wavevector an a ban inex. For values of γ that coincie with a reciprocal lattice vector of the perioic ielectric array, the Bragg conition is satisfie, an the EM wave is strongly iffracte off the meium: no propagation. Example in We can o a case in, analogous to the Kronig-Penney moel from the Schroeinger equation version. I II III Region I: Region II: Region III: E I E II E III a ( x) = Aexp( ikx) + B exp( ikx) ( x) = f exp( iqx) + g exp( iqx) ( x) = C exp( ikx) + D exp( ikx) k = ω / c, q = κω / c Usual bounary conitions: E an its erivative must be continuous at the interfaces between the regions.

14 Example in As in Kronig-Penney case, can write as a matrix equation: where k q sin( qa) exp( ika) cos( qa) + i + q k = M q k sin( qa) i k q C A = M D B q k sin( qa) i k q k q sin( qa) exp( ika) cos( qa) i + q k C exp( ik) A From E( x + ) = exp( iγ) E( x) we know = exp( iγ) D exp( ik) B exp( ik) 0 Defining: T 0 exp( ik) A We can write: ( TM exp( iγ ) I) = 0 B Example in A ( TM exp( iγ ) I) = 0 B For this to have a solution requires that the eterminant of the left han sie be equal to zero: k q cos( γ) = cos( k( a))cos( qa) + sin( k( a))sin( qa) q k Just like the K-P moel: solutions can only exist if the right han sie is between an ; if ω is chosen so that the rhs is outsie this omain, there are no allowe solutions, an we are in the photonic ban gap. What happens when there are efects? As you might imagine, efects (an surfaces) can introuce states in the gap. Those states are localize (!), an correspon to local resonant moes or staning waves. 3

15 What are the requirements to o this in higher imensions? Nothing too special; we just nee a spatially perioic array of ielectric contrast in all three imensions. Remember the electronic structure case, an the spaghetti iagrams : the size of the photonic ban gap epens in a nontrivial way on the irection of γ in reciprocal space. One other restriction: clearly spatial moulations on a scale vastly shorter than the wavelength of light we care about are not useful (basically our trial plane wave solutions for the perioic potential problem break own). Optimal size again correspons to typical features ~ λ/4. Polarization is also a complication that only really matters in higher imensions (nonnormal incience). Yablanovitch, JOSA B 0 83 (993). Typical ifficulty in early experimental work Yablanovitch, JOSA B 0 83 (993). 4

16 Possible applications of photonic bangap materials Extremely small ( microcavity ) lasers Superefficient semiconuctor optoelectronic evices Infinitely single-moe optical fibers Unconventional optical waveguies an routing Controlle optical switching Microcavity lasers MIT Usually many allowe moes in an optical cavity that is pumpe, an only one is the lasing moe of interest. Result: much pumping energy is waste, + other moes lea to noise in laser. PBG solution: controlle placement of efect in PBG material leas simultaneously to a single well efine moe for lasing (lower threshol) an a minimum size cavity. 5

17 Superefficient optoelectronic evices How oes this work, a little more rigorously? Typically a laser requires an optical cavity (the photons nee to stick aroun long enough to cause stimulate emission). This cavity has some Q. The material oing the raiating also has some effective Q -that is, there is a linewith to the raiation. usual esire Krause et al., Prog. Quant. Elect. 3, 5 (999). Superefficient optoelectronic evices How to get a better match between material an cavity properties? Sharpen up material emission by changing the available ensity of final states for the raiation: a cavity enhancement effect. ω Free space ensity of states for photons: g( ω ) = 3 πc Q Case for cavity of quality Q: g c ( ω) = π ω V Can get an aitional factor of three enhancement if cavity is right shape to couple to emitting system well. Total enhancement is therefore: 3 gc( ω ) 3 Qλ f P = g( ω) 4π V Purcell factor. So, a higher Q an a smaller size for the cavity can greatly improve efficiency, essentially by restricting the available phase space for photon emission. 6

18 Superefficient optoelectronic evices Consier a typical GaAs/AlGaAs light emitting ioe. Quantum efficiency (# of photons prouce per e-h pair) can be exceeingly high (~ 99%). Photons are typically emitte isotropically. However, because of high inex of refraction, only those intersecting the wafer surface within 6 egrees of normal are able to be transmitte at all; the rest experience total internal reflection. Only 3-5% of the light generate is therefore useful. With a correctly esigne PBG LED, though, one can stack the eck so that the LED active region only couples to a single propagating moe! Krause et al., Prog. Quant. Elect. 3, 5 (999). Photonic crystal fibers Nakazawa group, Tohoku Univ. Apply same basic iea of restricte ensity of states available for loss mechanisms to optical fibers. Can cut own ramatically on couplings between moes, an possibly open up new switching technologies (ex.: flui in vois that can be move to allow positioning of lossy segment of fiber). 7

19 Waveguies an optical routing University of Jena, Germany We ve alreay sai that an isolate efect in a PBG material can act as a resonant cavity. Extene series of efects can be use as waveguies, an it s possible (because of the very special properties of the surrouning PBG meium) to steer light in unusual ways - aroun sharp corners, for example. Controlle optical switching Busch, University of Toronto Infiltrate a PBG material with a liqui crystal, which you alreay know is birefringent an tunable. Reorienting LC molecules can open an close the PBG reversibly! 8

20 How are these metamaterials mae? Early implementations Lithography + etching: holes Lithography + etching: posts Lithography + etching: complicate 3 shapes Self-assembly: opals Self-assembly: inverte opals Early implementations Much early work (late 980s) was one using microwaves rather than IR or visible light. Vastly easier to assemble materials - can be one by han with simple tools! D. Smith, UCSD 9

21 Holes an posts For an systems, basic lithography an etching are typically the way to go: Krause et al., Prog. Quant. Elect. 3, 5 (999). Complicate 3 shapes Clearly executing 3 perioic architectures many wavelengths eep by oing purely surface processing (like previous slie) is increibly challenging. Best approach lithographically is to use either multilayere starting substrates (as we ll see in micromachining), or interference lithography an photopolymerization. Sania Big avantage of lithography: controlle fabrication of efects (cavities, waveguies, etc.) 0

22 Opals Natural opal looks opalescent because of intrinsic photonic ban gap properties ue to its structure. Opals consist of hyrate silica spheres ~ hunres of nm in iameter stacke in regular patterns. Through work with collois, it is possible to create artificial opal structures by self assembly: University of Sheffiel, UK Inverte opals Another exciting variant on this is to use self-assembly to create an opal structure, infiltrate a high inex material (Si? Se? TiO?), an remove the original opal spheres, to get the highest possible ielectric contrast. At right, an inverte opal photonic crystal mae by infiltration growth of amorphous Si by low pressure chemical vapor eposition. The original spheres were then etche away. NEC Big avantage of self-assembly: large volume 3 perioic structures are naturally prouce!

23 Conclusions: 3 structuring of ielectics on the nano scale makes possible an enormous variety of optically interesting structures. Tremenous technological potential here, with top-own an bottom-up assembly techniques both likely to play a role. Eventual goal: all-optical processing of information. Moral: possible hien opportunities out there even when the essential physics is well unerstoo.