1. Reminder: E-Dynamics in homogenous media and at interfaces

Transcription

1 0. Introduction 1. Reminder: E-Dynamics in homogenous media and at interfaces 2. Photonic Crystals 2.1 Introduction 2.2 1D Photonic Crystals 2.3 2D and 3D Photonic Crystals 2.4 Numerical Methods 2.5 Fabrication 2.6 Non-linear optics and Photonic Crystals 2.7 Quantumoptics 2.8 Chiral Photonic Crystals 2.9 Quasicrystals 2.10 Photonic Crystal Fibers Holey Fibers 3. Metamaterials and Plasmonics 3.1 Introduction 3.2 Background 3.2 Fabrication 3.3 Experiments

2 Bibliography Optik, E. Hecht, Addison-Wesley (just as a reminder) Nanophotonics, P.N. Prasad, John Wiley & Sons (2004) (recent comprehensive overview, nothing in depth, good for finding further references and original work) Photonic Crystals, J.D. Joannopoulos, R.D. Meade, J.N. Winn, Princeton University Press (nice textbook introduction into the theory, mostly 2D) Photonic Crystals, K. Busch et al., eds., Wiley-VCH (2004) (collection of recent review papers, incl. experimental ones) Optical Properties of Photonic Crystals, K. Sakoda, Springer (2001) (advanced theory, mostly 2D, good introduction into symmetry properties)

3 0. Introduction 1. Reminder: E-Dynamics in homogenous media and at interfaces 2. Photonic Crystals 2.1 Introduction 2.2 1D Photonic Crystals 2.3 2D and 3D Photonic Crystals 2.4 Numerical Methods 2.5 Fabrication 2.6 Non-linear optics and Photonic Crystals 2.7 Quantumoptics 2.8 Chiral Photonic Crystals 2.9 Quasicrystals 2.10 Photonic Crystal Fibers Holey Fibers 3. Metamaterials and Plasmonics 3.1 Introduction 3.2 Background 3.2 Fabrication 3.3 Experiments

4 Optical properties of periodic structures Lattice constant a Wavelength λ

5 Optical properties of periodic structures Lattice constant a Wavelength λ

6 Optical properties of periodic structures Lattice constant a Wavelength λ

7 Theoretical framework Relevant parameter: wavelength λ / lattice constant a

8 Geometrical optics is only valid in the limit λ / a << 1. neglects wave effects (diffraction, interference). treats light propagation in terms of rays. is employed, e.g., in raytracing programs.

9 Normal crystals... have lattice constants much smaller than the wavelength of light (λ / a >> 1). can be treated as homogeneous media (Q.M. ε,µ,n,z).... are common optical materials. have a refractive index n > 0.

10 Photonic crystals... have lattice constants comparable to the wavelength of light (λ / a 1). are (in most cases) artificial materials. exhibit a photonic band structure (Maxwell).... can have a complete photonic bandgap.

11 Metamaterials... have lattice constants smaller than the wavelength of light (λ / a > 1). are artificial materials. can be treated as homogeneous media (Maxwell ε,µ,n,z). Structure made at UCSD by David Smith... can have a negative index of refraction n < 0.

12 Photonics Photonics is the science and technology of generating and controlling photons, particularly in the visible and near infra-red light spectrum. The science of photonics includes the emission, transmission, amplification, detection, modulation, and switching of light. Photonic devices include optoelectronic devices such as lasers and photodetectors, as well as optical fibers, photonic crystals, planar waveguides and other passive optical elements.

13 Example I DWDM (Dense Wavelength Division Multiplexing) Can we fabricate these devices on a micron scale? see Photonic Crystals

14 Example II Conventional optical fibers guide the light inside a glass core, thus showing dispersion. After a certain travel distance, information sent in the form of short laser pulses, smears out. Therefore, repeaters and amplifiers are needed. Can we transmit light without dispersion? see Photonic Crystal fibres

15 Example III With the further downscaling of conventional electronic components, quantum effects become important. What can we expect from photonic structures on a wavelength or even smaller scales? see Photonic Crystals, Quantumoptics

16 Example IV All known natural materials exhibit a positive index of refraction. Can we design and fabricate artificial materials with a negative index of refraction? see Metamaterials

17 0. Introduction 1. Reminder: E-Dynamics in homogenous media and at interfaces 2. Photonic Crystals 2.1 Introduction 2.2 1D Photonic Crystals 2.3 2D and 3D Photonic Crystals 2.4 Numerical Methods 2.5 Fabrication 2.6 Non-linear optics and Photonic Crystals 2.7 Quantumoptics 2.8 Chiral Photonic Crystals 2.9 Quasicrystals 2.10 Photonic Crystal Fibers Holey Fibers 3. Metamaterials and Plasmonics 3.1 Introduction 3.2 Background 3.2 Fabrication 3.3 Experiments

18 All of macroscopic electromagnetism can be described within the framework of the macroscopic Maxwell equations: D= ρ B E= t B= 0 D = ε 0E + B = µ 0H + with D H = j+ t E D P ρ : electric field : dielectric displacement : polarization : free charge density B H M j P M : magnetic induction : magnetic field : magnetization : free current density

19 The material properties enter via the constitutive relations. For low light intensities, one usually finds a linear relationship between the polarization and the electric field as well as between the magnetization and magnetic field: P(t, r ) = ε 0 M (t, r ) = µ 0 χ e (t, t, r, r ) E (t, r ) dt dr χ m (t, t, r, r ) H (t, r ) dt dr Tensors!

20 Here, we consider only isotropic materials with a local response: χ e (t, t, r, r ) = χ e (t, t ) δ (r r ) χ m (t, t, r, r ) = χ m (t, t ) δ (r r ) The response functions must be causal and do not explicitly depend on time (homogeneity in time): χ e (t, t ) = χ e (t t ) Θ (t t ) χ m (t, t ) = χ m (t t ) Θ (t t ) t P(t ) = ε 0 χ e (t t ) E (t ) dt t M (t ) = µ 0 χ m (t t ) H (t ) dt

21 In the frequency domain, we get: t F.T. P(t ) = ε 0 χ e (t t ) E (t ) dt P(ω ) = ε 0 χ e (ω ) E (ω ) t F.T. M (t ) = µ 0 χ m (t t ) H (t ) dt M (ω ) = ε 0 χ m (ω ) H (ω ) Electric permittivity This finally leads to D(ω ) = ε 0 (1 + χ e (ω ) ) E (ω ) = ε 0 ε (ω ) E (ω ) B(ω ) = µ 0 (1 + χ m (ω ) ) H (ω ) = µ 0 µ (ω ) H (ω ) Magnetic permeability

22 Electromagnetic waves in homogenous media without dispersion, free charges and free currents (ε = const, µ = const, ρ = 0, j = 0 ): B E= t B = µ 0µ H D H= t D = ε 0ε E t 2 E = µ 0µ 2 H t t H = ε 0ε E t ( ) With = and B = 0 we obtain the wave equation: 2 H (t, r ) ε 0 µ 0ε µ 2 H (t, r ) = 0 t see Physik II and THEORIE D

23 Electromagnetic waves in homogenous media without dispersion, free charges and free currents (ε = const, µ = const, ρ = 0, j = 0 ): B E= t B = µ 0µ H D H= t D = ε 0ε E t E = µ 0µ H t 2 H = ε 0ε 2 E t t ( ) With = and E = 0 we obtain the wave equation: 2 E (t, r ) ε 0 µ 0ε µ 2 E (t, r ) = 0 t see Physik II and THEORIE D

24 With the complex ansatz for E and H The physical electric and magnetic fields are obtained by taking the real parts of the complex quantities! [( )] exp[i ( k r ω t )] E(t, r ) = E0 exp i k r ω t H(t, r ) = H 0 we obtain from the wave equations: Case 1: Plane waves k 2 = ε 0 µ 0ε µ ω 2 ε µ > 0 k i ℜ, i { x, y, z} Case 2: Evanescent modes ε µ < 0 k i ℑ, i { x, y, z }

25 Plane waves E(t, r ) = B(t, r ) = [( [( E0 exp i k r ω t B0 exp i k r ω t )] )] are transversal: ik y E z ik z E y! E = ik x E z ik z E x = ik E = iω B ik E ik E y x x y! B = i ( k x B x + k y B y + k z B z ) = ik B = 0 Moreover, E and B are in phase.

26 The planes of constant phase propagate with the phase velocity c : c c ω 1 c2 = 2 = = 0 = 02 k ε 0ε µ 0 µ ε µ n The material properties enter via the refractive index n2= ε µ n = ± ε µ Dielectric materials: (ε, µ > 0) n= εµ and the impedance Z= µ 0µ ε 0ε Z0 = µ Ω ε0 see Metamaterials

27 The energy density w of an electromagnetic wave in a nondispersiv medium is given by ( 1 w = ℜ E0 D0 + H 0 B0 4 ) This formula is valid only for nondispersive media! The corresponding time averaged energy flux density is given by the Poynting vector ( ) 1 S = ℜ E0 H 0 µ > 0 S k 2 The magnitude of S denotes the intensity of the electromagnetic field: 1 2 I = S = ε 0ε c E0 2 These quantities are spatially constant for plane waves.

28 Plane waves Wavelength k r = const E and B are in phase! λ

29 Evanescent modes E(t, r ) = B(t, r ) = [ [ E0 exp B0 exp ] ] k ek r exp[ iω t ] k ek r exp[ iω t ] exhibit exponentially decaying field strengths (E and B).

30 Evanescent modes do not transport energy since the time-averaged normal component of the Poynting vector vanishes : ( [ 1 S ek = ℜ ek E0 H 0 2 ]) ( [ 1 = ℜ ek E0 (k E0 ) 2ω µ 0 µ ])= 0 Purely imaginary! Therefore, evanescent modes do only have a noticeable field strength at interfaces.

31 Classification of electromagnetic modes n2 = ε µ < 0 n2 = ε µ > 0 evanescent waves propagating waves n2 = ε µ > 0 n2 = ε µ < 0 propagating waves evanescent waves

32 Electromagnetic fields at interfaces df n Use 3rd Maxwell equation x df Use Gauss-Theorem 0= V dr B= 3 ( df B x 0 df n B2 B1 S ( V ) Normal component of B must be continous )

33 Electromagnetic fields at interfaces n l2 Use 4th Maxwell equation Use Stokes-Theorem df H = F F df rot H = F = Ft Fdf jf + t Fdf D x 0 l ( jf t ) x t ( ds H x 0 l ( t n ) H 2 H1 ) l1 ( ( t n ) H 2 H1 = jf t ) F (no surface current) Tangential component of H must be continous

34 Electromagnetic fields at interfaces With a similar derivation for E and D follows: D normal E tangential B normal H tangential have to be continous across charge- and current-free interfaces. We obtain for the other field components: D2t = ε2 D1t ε1 E2 n = ε1 E1n ε2 B2t = µ2 B1t µ1 H 2n = µ1 H1n µ2

35 Refraction at an interface Fresnel formulas p-polarization Θi= Θ r ni sin ( Θ i )= nt sin ( Θ t ) Er ( n / µ ) cos( Θ i ) ( ni / µ i ) cos( Θ t ) = t t rp = Ei p ( ni / µ i ) cos( Θ t ) + ( nt / µ t ) cos( Θ i ) Et 2( ni / µ i ) cos( Θ i ) t p = = Ei p ( ni / µ i ) cos( Θ t ) + ( nt / µ t ) cos( Θ i )

36 Refraction at an interface Fresnel formulas s-polarization Θi= Θ r ni sin ( Θ Er ( ni / µ i ) cos( Θ rs = = Ei s ( ni / µ i ) cos( Θ i )= nt sin ( Θ t ) ) ( nt / µ t ) cos( Θ t ) i ) + ( nt / µ t ) cos( Θ t ) i Et 2( ni / µ i ) cos( Θ i ) t s = = Ei s ( ni / µ i ) cos( Θ i ) + ( nt / µ t ) cos( Θ t )

37 Refraction at an interface Fresnel formulas Brewster s angle Parameters: ε1=1.0, µ1=1.0, ε2=2.25, µ2=1.0

38 A slab of matter: Fabry-Perot modes E0 tt r 7 e i 4 δ nslab E 0 tt r e 2 E 0 tt r e 2 6 i 7δ E0tt r 5ei 3δ 4 i 5δ E0 tt r 3 e i 2 δ E 0 tt r 2 e i E0 tt r e iδ E0 r E 0 tr e i E0t ϑt E0 δ E 0 tt e i δ 3δ Phase difference due to propagation (single round trip): δ = k0 2nslab d cos(ϑ t ) d

39 A slab of matter: Fabry-Perot modes The total transmitted electric field is given by the superposition of all partially transmitted electric fields: Et = E0tt e iδ 2 + E 0 tt r e 2 i 3δ 2 + E0 tt r e 4 i 5δ 2 + E0 t t r e 6 i7δ After a little bit of algebra we obtain the intensity of the total transmitted electromagnetic field (for lossless media) It = I F sin 2 (δ / 2) 2r with the finesse factor F = 2 1 r 2 See, e.g., Optik, E. Hecht, Addison-Wesley

40 A slab of matter: Fabry-Perot modes F=0.1 F=1 F=10 F=100 The transmittance is maximal for δ = 2mπ, m {0,1, 2,...} since all partial waves are in phase: Et = E0tt e i 2 (1 + r 2 + r 4 + r ) δ

41 A slab of matter: Fabry-Perot modes The total reflected electric field is given by the superposition of all partially reflected electric fields: Er = E0 r + E0tt r e iδ + E0tt r 3ei 2δ + E0tt r 5e i 3δ +... After a little bit of algebra we obtain the intensity of the total reflected electromagnetic field (for lossless media) F sin 2 (δ / 2) Ir = I0 1 + F sin 2 (δ / 2) 2r with the finesse factor F = 2 1 r 2 See, e.g., Optik, E. Hecht, Addison-Wesley

42 A slab of matter: Fabry-Perot modes F=0.1 F=1 F=10 F=100 The reflectance is maximal for δ = (2m + 1)π, m {0,1, 2,...}

43 Classification of optical materials??

44 Classification of optical materials Optical frequencies: µ=1??

45 Lorentz Oscillator modell Equation of motion: d 2x dx m 2 + mγ 0 + mω 02 x = qe0 e iω t dt dt Lorentz dielectric function: Nq 2 1 ε (ω ) = 1 + mε 0 ω 02 ω 2 iω γ 0

46 Lorentz Oscillator modell without losses

47 Lorentz Oscillator modell with losses

48 0. Introduction 1. Reminder: E-Dynamics in homogenous media and at interfaces 2. Photonic Crystals 2.1 Introduction 2.2 1D Photonic Crystals 2.3 2D and 3D Photonic Crystals 2.4 Numerical Methods 2.5 Fabrication 2.6 Non-linear optics and Photonic Crystals 2.7 Quantumoptics 2.8 Chiral Photonic Crystals 2.9 Quasicrystals 2.10 Photonic Crystal Fibers Holey Fibers 3. Metamaterials and Plasmonics 3.1 Introduction 3.2 Background 3.2 Fabrication 3.3 Experiments

49 Silicon, a semiconductor crystal Is there such a thing as a semiconductor for light? S. John, Phys. Rev. Lett. 58, 2586 (1987) E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987)

50 Semiconductors Periodic potential for electrons Band structure for electrons

51 Photonic Crystals Dielectric or metallic materials with a dielectric function ε ( r, ω ) that is periodically modulated along at least one spatial direction: 1D 2D 3D

52 Photonic Crystals Periodic potential for photons Band structure for photons

53 1D Photonic Crystals in nature Mother-of-pearl Aragonite [CaCO3] / protein layers taken from:

54 2D Photonic Crystals in nature 300 nm 20 cm Sea-mouse Taken from: A.R. Parker et al., Nature 409, 36 (2001)

55 3D Photonic Crystals in nature Pachyrhynchus argus Taken from: A. R. Parker et al., Nature 426, 786 (2003)

56 3D Photonic Crystals in nature 1.2µm Morpho Rhetenor und Parides Sesostris Overview: P. Vukusic and J.R. Sambles, Nature 424, 852 (2003)

57 Opals: 3D Photonic Crystals Taken from: ebay.com

58 A closer look at an Opal 400nm Taken from: J.B. Pendry, Current Science 76, 1311 (1999)

59 Visions for Photonic Crystals Custom designed electromagnetic vacuum Control of spontaneous emission Zero threshold lasers Ultrasmall optical components Ultrafast all-optical switching Integration of components on many layers

60 Visions for photonic crystals Photonic Micropolis J. Joannopoulos Research Group (MIT) Optical Microchip S. John Research Group (Toronto)