Photonic Crystals: Periodic Surprises in Electromagnetism. Steven G. Johnson MIT

Transcription

1 Photonic Cystals: Peiodic Supises in Electomagnetism Steven G. Johnson MIT

2 To Begin: A Catoon in 2d k scatteing planewave E, H ~ e i( k x -wt ) k = w / c = 2p l

3 To Begin: A Catoon in 2d k a planewave E, H ~ e i( k x -wt ) k = w / c = 2p l fo most l, beam(s) popagate though cystal without scatteing (scatteing cancels coheently)...but fo some l (~ 2a), no light can popagate: a photonic band gap

4 Photonic Cystals peiodic electomagnetic media D 2-D 3-D peiodic in one diection peiodic in two diections peiodic in thee diections with photonic band gaps: optical insulatos (need a moe complex topology)

5 Photonic Cystals peiodic electomagnetic media can tap light in cavities 3D Photonic C ysta l with Defects and waveguides ( wies ) magical oven mitts fo holding and contolling light with photonic band gaps: optical insulatos

6 Photonic Cystals peiodic electomagnetic media High index of efa ction Low index of efa ction 3D Photonic C ysta l But how can we undestand such complex systems? Add up the infinite sum of scatteing? Ugh!

7 A mystey fom the 19th centuy e conductive mateial E e + cuent: J = se + conductivity (measued) mean fee path (distance) of electons

8 A mystey fom the 19th centuy e E e cystalline conducto (e.g. coppe) s of peiods! cuent: J = se conductivity (measued) mean fee path (distance) of electons

9 A mystey solved 1 electons ae waves (quantum mechanics) 2 waves in a peiodic medium can popagate without scatteing: Bloch s Theoem (1d: Floquet s) The foundations do not depend on the specific wave equation.

10 Time to Analyze the Catoon k a planewave E, H ~ e i( k x -wt ) k = w / c = 2p l fo most l, beam(s) popagate though cystal without scatteing (scatteing cancels coheently)...but fo some l (~ 2a), no light can popagate: a photonic band gap

11 Fun with Math E = - 1 c H = e 1 c t t H = i w c E + H 0 J = i w c e E Fist task: get id of this mess dielectic function e(x) = n 2 (x) 1 e Ê H = Á w Ë c ˆ 2 H + constaint H = 0 eigen-opeato eigen-value eigen-state

12 Hemitian Eigenpoblems 1 e Ê H = Á w Ë c ˆ 2 H + constaint H = 0 eigen-opeato eigen-value eigen-state Hemitian fo eal (lossless) e well-known popeties fom linea algeba: w ae eal (lossless) eigen-states ae othogonal eigen-states ae complete (give all solutions)

13 Peiodic Hemitian Eigenpoblems [ G. Floquet, Su les équations difféentielles linéaies à coefficients péiodiques, Ann. École Nom. Sup. 12, (1883). ] [ F. Bloch, Übe die quantenmechanik de electonen in kistallgitten, Z. Physik 52, (1928). ] if eigen-opeato is peiodic, then Bloch-Floquet theoem applies: can choose: H ( x,t) = e i ( k x -wt ) H k ( x ) planewave peiodic envelope Coollay 1: k is conseved, i.e. no scatteing of Bloch wave Coollay 2: H given by finite unit cell, k so w ae discete w n (k)

14 Peiodic Hemitian Eigenpoblems Coollay 2: H given by finite unit cell, k so w ae discete w n (k) band diagam (dispesion elation) w w 3 w 1 w 2 map of what states exist & can inteact k? ange of k?

15 Peiodic Hemitian Eigenpoblems in 1d H(x) = e ikx H k (x) e 1 e 2 e 1 e 2 e 1 e 2 e 1 e 2 e 1 e 2 e 1 e 2 Conside k+2π/a: a e(x) = e(x+a) 2p i(k+ a e ) x H 2p (x) = e ikx k + a È e i 2p Í a x H 2p (x) Î Í k + a k is peiodic: k + 2π/a equivalent to k quasi-phase-matching peiodic! satisfies same equation as H k = H k

16 Peiodic Hemitian Eigenpoblems in 1d k is peiodic: k + 2π/a equivalent to k quasi-phase-matching e 1 e 2 e 1 e 2 e 1 e 2 e 1 e 2 e 1 e 2 e 1 e 2 a e(x) = e(x+a) w band gap π/a 0 π/a ieducible Billouin zone k

17 Any 1d Peiodic System has a Gap [ Lod Rayleigh, On the maintenance of vibations by foces of double fequency, and on the popagation of waves though a medium endowed with a peiodic stuctue, Philosophical Magazine 24, (1887). ] Stat with a unifom (1d) medium: e 1 w w = k e 1 0 k

18 Any 1d Peiodic System has a Gap [ Lod Rayleigh, On the maintenance of vibations by foces of double fequency, and on the popagation of waves though a medium endowed with a peiodic stuctue, Philosophical Magazine 24, (1887). ] Teat it as atificially peiodic e 1 bands ae folded by 2π/a equivalence w a e(x) = e(x+a) π/a 0 π/a e + p a x, e - p a x Ê Æ cosá p Ë a x ˆ, sin Ê Á p Ë a x ˆ k

19 Any 1d Peiodic System has a Gap [ Lod Rayleigh, On the maintenance of vibations by foces of double fequency, and on the popagation of waves though a medium endowed with a peiodic stuctue, Philosophical Magazine 24, (1887). ] Teat it as atificially peiodic w 0 π/a Ê siná p Ë a x ˆ Ê cosá p Ë a x ˆ e 1 a x = 0 e(x) = e(x+a)

20 Any 1d Peiodic System has a Gap [ Lod Rayleigh, On the maintenance of vibations by foces of double fequency, and on the popagation of waves though a medium endowed with a peiodic stuctue, Philosophical Magazine 24, (1887). ] w Add a small eal peiodicity e 2 = e 1 + De Ê siná p Ë a x ˆ Ê cosá p Ë a x ˆ a e(x) = e(x+a) e 1 e 2 e 1 e 2 e 1 e 2 e 1 e 2 e 1 e 2 e 1 e 2 0 π/a x = 0

21 Any 1d Peiodic System has a Gap [ Lod Rayleigh, On the maintenance of vibations by foces of double fequency, and on the popagation of waves though a medium endowed with a peiodic stuctue, Philosophical Magazine 24, (1887). ] w Add a small eal peiodicity e 2 = e 1 + De band gap Ê siná p Ë a x ˆ Ê cosá p Ë a x ˆ Splitting of degeneacy: state concentated in highe index (e 2 ) has lowe fequency a e(x) = e(x+a) e 1 e 2 e 1 e 2 e 1 e 2 e 1 e 2 e 1 e 2 e 1 e 2 0 π/a x = 0

22 Some 2d and 3d systems have gaps In geneal, eigen-fequencies satisfy Vaiational Theoem: w 1 ( k ) 2 = min w 2 ( k ) 2 = E 1 ee 1 = 0 min E 2 e E 2 = 0 Ú Ú ee * 1 E 2 = 0 ( ) + ik Ú "L" e E 1 2 E 1 2 c 2 kinetic invese potential bands want to be in high-e but ae foced out by othogonality > band gap (maybe)

23 algebaic intelude algebaic intelude completed I hope you wee taking notes* [ *if not, see e.g.: Joannopoulos, Meade, and Winn, Photonic Cystals: Molding the Flow of Light ]

24 a 2d peiodicity, e=12:1 1 fequency w (2πc/a) = a / l Photonic Band Gap TM bands ieducible Billouin zone M k G X 0 G X M G TM E H gap fo n > ~1.75:1

25 2d peiodicity, e=12: E z (+ 90 otated vesion) Photonic Band Gap E z 0.2 TM bands G X M G + TM E H gap fo n > ~1.75:1

26 a 2d peiodicity, e=12:1 fequency w (2πc/a) = a / l Photonic Band Gap TM bands 0.1 TE bands ieducible Billouin zone M k G X 0 G X M G TM E H TE E H

27 2d photonic cystal: TE gap, e=12:1 TM bands TE bands TE H E gap fo n > ~1.4:1

28 3d photonic cystal: complete gap, e=12:1 I. II % gap z 0.3 U' G X K' U'' U W W' K L I: od laye II: hole laye L' 0 U L G X W K gap fo n > ~4:1 [ S. G. Johnson et al., Appl. Phys. Lett. 77, 3490 (2000) ]

29 You, too, can compute photonic eigenmodes! MIT Photonic-Bands (MPB) package: on Athena: add mpb