Photonic Crystals

From Wikipedia: “Photonic Crystals are periodic optical nanostructures that are designed to affect the motion of photons in a similar way that periodicity of a semiconductor crystal affects the motion of electrons. Photonic crystals occur in nature and in various forms have been studied scientifically for the last 100 years”.

Wikipedia Continued “Photonic crystals are composed of periodic dielectric or metallo-dielectric nanostructures that affect the propagation of electromagnetic waves (EM) in the same way as the periodic potential in a crystal affects the electron motion by defining allowed and forbidden electronic energy bands. Photonic crystals contain regularly repeating internal regions of high and low dielectric constant. Photons (as waves) propagate through this structure - or not - depending on their wavelength. Wavelengths of light that are allowed to travel are known as modes, and groups of allowed modes form bands. Disallowed bands of wavelengths are called photonic band gaps. This gives rise to distinct optical phenomena such as inhibition of spontaneous emission, high-reflecting omni- directional mirrors and low-loss-waveguides, amongst others. Since the basic physical phenomenon is based on diffraction, the periodicity of the photonic crystal structure has to be of the same length-scale as half the wavelength of the EM waves i.e. ~350 nm (blue) to 700 nm (red) for photonic crystals operating in the visible part of the spectrum - the repeating regions of high and low dielectric constants have to be of this dimension. This makes the fabrication of optical photonic crystals cumbersome and complex.

Photonic Crystals: A New Frontier in Modern Optics MARIAN FLORESCU NASA Jet Propulsion Laboratory California Institute of Technology MARIAN FLORESCU NASA Jet Propulsion Laboratory California Institute of Technology

“ If only were possible to make materials in which electromagnetically waves cannot propagate at certain frequencies, all kinds of almost-magical things would happen” Sir John Maddox, Nature (1990)

Two Fundamental Optical Principles Localization of LightLocalization of Light S. John, Phys. Rev. Lett. 58,2486 (1987) Inhibition of Spontaneous EmissionInhibition of Spontaneous Emission E. Yablonovitch, Phys. Rev. Lett (1987) Two Fundamental Optical Principles Localization of LightLocalization of Light S. John, Phys. Rev. Lett. 58,2486 (1987) Inhibition of Spontaneous EmissionInhibition of Spontaneous Emission E. Yablonovitch, Phys. Rev. Lett (1987) Photonic crystals: periodic dielectric structures.  interact resonantly with radiation with wavelengths comparable to the periodicity length of the dielectric lattice.  dispersion relation strongly depends on frequency and propagation direction  may present complete band gaps  Photonic Band Gap (PBG) materials. Photonic crystals: periodic dielectric structures.  interact resonantly with radiation with wavelengths comparable to the periodicity length of the dielectric lattice.  dispersion relation strongly depends on frequency and propagation direction  may present complete band gaps  Photonic Band Gap (PBG) materials. Photonic Crystals  Guide and confine light without losses  Novel environment for quantum mechanical light-matter interaction  A rich variety of micro- and nano-photonics devices  Guide and confine light without losses  Novel environment for quantum mechanical light-matter interaction  A rich variety of micro- and nano-photonics devices

Photonic Crystals History 1987: Prediction of photonic crystals S. John, Phys. Rev. Lett. 58,2486 (1987), “Strong localization of photons in certain dielectric superlattices” E. Yablonovitch, Phys. Rev. Lett (1987), “Inhibited spontaneous emission in solid state physics and electronics” 1990: Computational demonstration of photonic crystal K. M. Ho, C. T Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990) 1991: Experimental demonstration of microwave photonic crystals E. Yablonovitch, T. J. Mitter, K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991) 1995: ”Large” scale 2D photonic crystals in Visible U. Gruning, V. Lehman, C.M. Englehardt, Appl. Phys. Lett. 66 (1995) 1998: ”Small” scale photonic crystals in near Visible; “Large” scale inverted opals 1999: First photonic crystal based optical devices (lasers, waveguides) 1987: Prediction of photonic crystals S. John, Phys. Rev. Lett. 58,2486 (1987), “Strong localization of photons in certain dielectric superlattices” E. Yablonovitch, Phys. Rev. Lett (1987), “Inhibited spontaneous emission in solid state physics and electronics” 1990: Computational demonstration of photonic crystal K. M. Ho, C. T Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990) 1991: Experimental demonstration of microwave photonic crystals E. Yablonovitch, T. J. Mitter, K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991) 1995: ”Large” scale 2D photonic crystals in Visible U. Gruning, V. Lehman, C.M. Englehardt, Appl. Phys. Lett. 66 (1995) 1998: ”Small” scale photonic crystals in near Visible; “Large” scale inverted opals 1999: First photonic crystal based optical devices (lasers, waveguides)

Photonic Crystals- Semiconductors of Light Semiconductors Periodic array of atoms Atomic length scales Natural structures Control electron flow 1950’s electronic revolution Semiconductors Periodic array of atoms Atomic length scales Natural structures Control electron flow 1950’s electronic revolution Photonic Crystals Periodic variation of dielectric constant Length scale ~ Artificial structures Control e.m. wave propagation New frontier in modern optics Photonic Crystals Periodic variation of dielectric constant Length scale ~ Artificial structures Control e.m. wave propagation New frontier in modern optics

Natural opals Natural Photonic Crystals: Structural Colours through Photonic Crystals Natural Photonic Crystals: Structural Colours through Photonic Crystals Periodic structure  striking colour effect even in the absence of pigments

Requirement: overlapping of frequency gaps along different directions  High ratio of dielectric indices  Same average optical path in different media  Dielectric networks should be connected J. Wijnhoven & W. Vos, Science (1998) S. Lin et al., Nature (1998) Woodpile structure Inverted Opals Artificial Photonic Crystals

 Photonic Crystals  complex dielectric environment that controls the flow of radiation  designer vacuum for the emission and absorption of radiation  Photonic Crystals  complex dielectric environment that controls the flow of radiation  designer vacuum for the emission and absorption of radiation Photonic Crystals: Opportunities  Passive devices  dielectric mirrors for antennas  micro-resonators and waveguides  Active devices  low-threshold nonlinear devices  microlasers and amplifiers  efficient thermal sources of light  Integrated optics  controlled miniaturisation  pulse sculpturing  Passive devices  dielectric mirrors for antennas  micro-resonators and waveguides  Active devices  low-threshold nonlinear devices  microlasers and amplifiers  efficient thermal sources of light  Integrated optics  controlled miniaturisation  pulse sculpturing

Defect-Mode Photonic Crystal Microlaser Photonic Crystal Cavity formed by a point defect O. Painter et. al., Science (1999)

3D Complete Photonic Band Gap Suppress blackbody radiation in the infrared and redirect and enhance thermal energy into visible Suppress blackbody radiation in the infrared and redirect and enhance thermal energy into visible 3D Complete Photonic Band Gap Suppress blackbody radiation in the infrared and redirect and enhance thermal energy into visible Suppress blackbody radiation in the infrared and redirect and enhance thermal energy into visible Photonic Crystals Based Light Bulbs S. Y. Lin et al., Appl. Phys. Lett. (2003) C. Cornelius, J. Dowling, PRA 59, 4736 (1999) “ Modification of Planck blackbody radiation by photonic band-gap structures” C. Cornelius, J. Dowling, PRA 59, 4736 (1999) “ Modification of Planck blackbody radiation by photonic band-gap structures”  Light bulb efficiency may raise from 5 percent to 60 percent 3D Tungsten Photonic Crystal Filament Solid Tungsten Filament

Solar Cell Applications –Funneling of thermal radiation of larger wavelength (orange area) to thermal radiation of shorter wavelength (grey area). –Spectral and angular control over the thermal radiation. –Funneling of thermal radiation of larger wavelength (orange area) to thermal radiation of shorter wavelength (grey area). –Spectral and angular control over the thermal radiation.

 Fundamental Limitations  switching time switching intensity = constant  Incoherent character of the switching  dissipated power  Fundamental Limitations  switching time switching intensity = constant  Incoherent character of the switching  dissipated power Foundations of Future CI Cavity all-optical transistor Cavity all-optical transistor I out I in IHIH IHIH H.M. Gibbs et. al, PRL 36, 1135 (1976)  Operating Parameters  Holding power: 5 mW  Switching power: 3 µW  Switching time: ns  Size: 500  m  Operating Parameters  Holding power: 5 mW  Switching power: 3 µW  Switching time: ns  Size: 500  m Photonic crystal all-optical transistor Photonic crystal all-optical transistor Probe Laser Pump Laser  Operating Parameters  Holding power: nW  Switching power: pW  Switching time: < 1 ps  Size: 20  m  Operating Parameters  Holding power: nW  Switching power: pW  Switching time: < 1 ps  Size: 20  m M. Florescu and S. John, PRA 69, (2004).

Single Atom Switching Effect  Photonic Crystals versus Ordinary Vacuum  Positive population inversion  Switching behaviour of the atomic inversion  Photonic Crystals versus Ordinary Vacuum  Positive population inversion  Switching behaviour of the atomic inversion M. Florescu and S. John, PRA 64, (2001)

 Long temporal separation between incident laser photons  Fast frequency variations of the photonic DOS  Band-edge enhancement of the Lamb shift  Vacuum Rabi splitting  Long temporal separation between incident laser photons  Fast frequency variations of the photonic DOS  Band-edge enhancement of the Lamb shift  Vacuum Rabi splitting Quantum Optics in Photonic Crystals T. Yoshie et al., Nature, 2004.

Foundations for Future CI: Single Photon Sources  Enabling Linear Optical Quantum Computing and Quantum Cryptography  fully deterministic pumping mechanism  very fast triggering mechanism  accelerated spontaneous emission  PBG architecture design to achieve prescribed DOS at the ion position prescribed DOS at the ion position  Enabling Linear Optical Quantum Computing and Quantum Cryptography  fully deterministic pumping mechanism  very fast triggering mechanism  accelerated spontaneous emission  PBG architecture design to achieve prescribed DOS at the ion position prescribed DOS at the ion position M. Florescu et al., EPL 69, 945 (2005)

M. Campell et al. Nature, 404, 53 (2000) CI Enabled Photonic Crystal Design (I) Photo-resist layer exposed to multiple laser beam interference that produce a periodic intensity pattern 3D photonic crystals fabricated using holographic lithography 3D photonic crystals fabricated using holographic lithography Four laser beams interfere to form a 3D periodic intensity pattern Four laser beams interfere to form a 3D periodic intensity pattern 10  m  O. Toader, et al., PRL 92, (2004)

O. Toader & S. John, Science (2001) CI Enabled Photonic Crystal Design (II)

S. Kennedy et al., Nano Letters (2002) CI Enabled Photonic Crystal Design (III)

Transport Properties: Photons Electrons Phonons Transport Properties: Photons Electrons Phonons Photonic Crystals Optical Properties Photonic Crystals Optical Properties Rethermalization Processes: Photons Electrons Phonons Rethermalization Processes: Photons Electrons Phonons Metallic (Dielectric) Backbone Electronic Characterization Metallic (Dielectric) Backbone Electronic Characterization Multi-Physics Problem: Photonic Crystal Radiant Energy Transfer

Summary Designer Vacuum: Frequency selective control of spontaneous and thermal emission enables novel active devices Designer Vacuum: Frequency selective control of spontaneous and thermal emission enables novel active devices PBG materials: Integrated optical micro-circuits with complete light localization PBG materials: Integrated optical micro-circuits with complete light localization Photonic Crystals: Photonic analogues of semiconductors that control the flow of light Photonic Crystals: Photonic analogues of semiconductors that control the flow of light Potential to Enable Future CI: Single photon source for LOQC All-optical micro-transistors Potential to Enable Future CI: Single photon source for LOQC All-optical micro-transistors CI Enabled Photonic Crystal Research and Technology: Photonic “materials by design” Multiphysics and multiscale analysis CI Enabled Photonic Crystal Research and Technology: Photonic “materials by design” Multiphysics and multiscale analysis

Wikipedia Continued “Photonic crystals are composed of periodic dielectric or metallo-dielectric nanostructures that affect the propagation of electromagnetic waves (EM) in the same way as the periodic potential in a crystal affects the electron motion by defining allowed and forbidden electronic energy bands. Photonic crystals contain regularly repeating internal regions of high and low dielectric constant. Photons (as waves) propagate through this structure - or not - depending on their wavelength. Wavelengths of light that are allowed to travel are known as modes, and groups of allowed modes form bands. Disallowed bands of wavelengths are called photonic band gaps. This gives rise to distinct optical phenomena such as inhibition of spontaneous emission, high-reflecting omni- directional mirrors and low-loss-waveguides, amongst others. Since the basic physical phenomenon is based on diffraction, the periodicity of the photonic crystal structure has to be of the same length-scale as half the wavelength of the EM waves i.e. ~350 nm (blue) to 700 nm (red) for photonic crystals operating in the visible part of the spectrum - the repeating regions of high and low dielectric constants have to be of this dimension. This makes the fabrication of optical photonic crystals cumbersome and complex.

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

To Begin: A Cartoon in 2d planewave scattering

To Begin: A Cartoon in 2d planewave for most, beam(s) propagate through crystal without scattering (scattering cancels coherently)...but for some (~ 2a), no light can propagate: a photonic band gap a

Photonic Crystals periodic electromagnetic media with photonic band gaps: “optical insulators” (need a more complex topology)

Photonic Crystals periodic electromagnetic media with photonic band gaps: “optical insulators” magical oven mitts for holding and controlling light can trap light in cavitiesand waveguides (“wires”)

Photonic Crystals periodic electromagnetic media But how can we understand such complex systems? Add up the infinite sum of scattering? Ugh!

A mystery from the 19th century e–e– e–e– current: conductivity (measured) mean free path (distance) of electrons conductive material

A mystery from the 19th century e–e– e–e– + current: conductivity (measured) mean free path (distance) of electrons crystalline conductor (e.g. copper) 10’s of periods!

A mystery solved… electrons are waves (quantum mechanics) 1 waves in a periodic medium can propagate without scattering: Bloch’s Theorem (1d: Floquet’s) 2 The foundations do not depend on the specific wave equation.

Time to Analyze the Cartoon planewave for most, beam(s) propagate through crystal without scattering (scattering cancels coherently)...but for some (~ 2a), no light can propagate: a photonic band gap a

Fun with Math 0 dielectric function  (x) = n 2 (x) First task: get rid of this mess eigen-operatoreigen-value eigen-state + constraint

Hermitian Eigenproblems eigen-operatoreigen-value eigen-state + constraint Hermitian for real (lossless)  well-known properties from linear algebra:  are real (lossless) eigen-states are orthogonal eigen-states are complete (give all solutions)

Periodic Hermitian Eigenproblems [ G. Floquet, “Sur les équations différentielles linéaries à coefficients périodiques,” Ann. École Norm. Sup. 12, 47–88 (1883). ] [ F. Bloch, “Über die quantenmechanik der electronen in kristallgittern,” Z. Physik 52, 555–600 (1928). ] if eigen-operator is periodic, then Bloch-Floquet theorem applies: can choose: periodic “envelope” planewave Corollary 1: k is conserved, i.e. no scattering of Bloch wave Corollary 2: given by finite unit cell, so  are discrete  n (k)

Periodic Hermitian Eigenproblems Corollary 2: given by finite unit cell, so  are discrete  n (k)     k band diagram (dispersion relation) map of what states exist & can interact ? range of k?

Periodic Hermitian Eigenproblems in 1d 11 22 11 22 11 22 11 22 11 22 11 22  (x) =  (x+a) a Consider k+2π/a: periodic! satisfies same equation as H k = H k k is periodic: k + 2π/a equivalent to k “quasi-phase-matching”

band gap Periodic Hermitian Eigenproblems in 1d 11 22 11 22 11 22 11 22 11 22 11 22  (x) =  (x+a) a k is periodic: k + 2π/a equivalent to k “quasi-phase-matching” k  0 π/a –π/a irreducible Brillouin zone

Any 1d Periodic System has a Gap 11 k  0 [ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ] Start with a uniform (1d) medium:

Any 1d Periodic System has a Gap 11  (x) =  (x+a) a k  0 π/a –π/a [ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ] Treat it as “artificially” periodic bands are “folded” by 2π/a equivalence

 (x) =  (x+a) a 11 Any 1d Periodic System has a Gap  0 π/a [ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ] x = 0 Treat it as “artificially” periodic

 (x) =  (x+a) a 11 22 11 22 11 22 11 22 11 22 11 22 Any 1d Periodic System has a Gap  0 π/a [ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ] Add a small “real” periodicity  2 =  1 +  x = 0

band gap Any 1d Periodic System has a Gap  0 π/a [ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ] Add a small “real” periodicity  2 =  1 +   (x) =  (x+a) a 11 22 11 22 11 22 11 22 11 22 11 22 x = 0 Splitting of degeneracy: state concentrated in higher index (  2 ) has lower frequency

Some 2d and 3d systems have gaps In general, eigen-frequencies satisfy Variational Theorem: “kinetic” inverse “potential” bands “want” to be in high-  …but are forced out by orthogonality –> band gap (maybe)

algebraic interlude completed… … I hope you were taking notes* algebraic interlude [ *if not, see e.g.: Joannopoulos, Meade, and Winn, Photonic Crystals: Molding the Flow of Light ]

2d periodicity,  =12:1 E H TM a frequency  (2πc/a) = a /  X M  XM  irreducible Brillouin zone gap for n > ~1.75:1

2d periodicity,  =12:1 E H TM  XM  EzEz –+ EzEz (+ 90° rotated version) gap for n > ~1.75:1

2d periodicity,  =12:1 E H E H TMTE a frequency  (2πc/a) = a /  X M  XM  irreducible Brillouin zone

2d photonic crystal: TE gap,  =12:1 TE bands TM bands gap for n > ~1.4:1 E H TE

3d photonic crystal: complete gap,  =12:1 I. II. [ S. G. Johnson et al., Appl. Phys. Lett. 77, 3490 (2000) ] gap for n > ~4:1

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