NLO Lect1

Non-Linear Optics: 

Non-Linear Optics Non-linear optical processes, devices, and their applications ~18 Lectures Intro/Basics of Non-linear Optics Non-linear optics of non-resonant systems Non-linear optics of resonant systems Text: Photonics (Saleh and Teich) Helpful: Non-linear Optics (Boyd), Intro. to Electrodynamics (Griffiths) Class time and location: Aug. 00 Thanks Paul Danehy Note Set 1

Assessment: 

Assessment 3 assignments total = 45% 1 simulation maybe = 15% ‘Lecture’ presentation = 40%

What is ‘Non-linear Optics’?: 

What is ‘Non-linear Optics’? Optical processes whose output depends non-linearly on the light intensity are non-linear. Example: Second Harmonic Generation (SHG) Probably the most-used non-linear process I532 = constant • ( I1064 )2 note: 9,398 cm-1 + 9,398 cm-1 = 18,796 cm-1 Frequency: 1064 nm 1064 nm 532 nm Second-order non-linear crystal

Applications of S H G: 

Applications of S H G Theoretical Physics: SHG (second harmonic generation) can exhibit chaotic behaviour Quantum Optics Laboratory produces squeezed light used to produce green to pump OPO squeezer ALDiR : to convert Nd:YAG light: IR to Green Pumps dye laser, which is then used for spectroscopy

Other Non-linear Optics Applications: 

Other Non-linear Optics Applications Optical Parametric Oscillator (OPO) “Pump” converted into two other frequencies Example: 355 nm creates 500 nm and 1224 nm light Tuneable laser beam over Visible, IR Change angle of the crystal to tune squeezing Degenerate Four-Wave Mixing (DFWM) Optical phase conjugation High-resolution spectroscopy Optical multi stability Optical computing?

Required Background Knowledge: 

Required Background Knowledge Required: Maxwell’s Equations Experience with fields and waves Familiarity with plane-wave propagation, notation Multi-dimensional Calculus (Div, Grad, Curl) Helpful: Lasers: how they work Atomic Spectroscopy: 2-level atom Ordinary Differential Equations Quantum Mechanics

What we’ll be (and not be) studying: 

What we’ll be (and not be) studying Types of media: Free space (no non-linearity) Dielectrics (we’ll mostly study these) Also called ‘insulators’ Mostly bound electrons Mostly transparent Low conductivity Examples: crystals, glasses, liquids, gases Plasmas (c3 processes occur) Semiconductors (c2 and c3 processes occur) Metals (Will those beams penetrate?)

Notation for Fields and Waves: 

Notation for Fields and Waves Electric Field and Magnetic Field E - Electric Field [V/m] D - Displacement (or Electric Flux Density) [C/m2] P - Electric Polarisation Density [C/m2] H - Magnetic Field [A/m] B - Magnetic Flux Density [W/m2] All are vectors (bold), and functions of r and t r is the position vector [m] t is time [s]

Maxwell’s Equations: 

Maxwell’s Equations Linear and non-linear optics: Maxwell’s Eqns Where: B = µ0 H (1.5) (nonmagnetic medium) D = e0 E +P (1.6) µ0 = magnetic permeability of free space e0 = electric permittivity of free space (MKS units Source-free medium)

D, E, P and all that...: 

D, E, P and all that... The polarisation is induced by an E-field: P = f(E) D = e0 E +P (for a good discussion see Griffiths, Ch. 4) D is the electric flux density - caused by: An E-field in the absence of the medium: e0 E, Plus the field created by the response of the medium: P P = 0 in free space P ≠ 0 in a dielectric Capacitor: Same q, A Thus, same D [C/m2] E is counteracted by P no dielectric with dielectric E no E with ,P > 0 E no > E with + - E p What about C=Q/V?

Properties of media: 

Properties of media Homogeneous: Ex. Glass P doesn’t vary with location Inhomogeneous: Ex. Graded-index fibre P = P(r) Linear: Ex. Glass, low laser intensity P = e0 c E (where c is the electric susceptibility) Non-linear: Ex. Glass, high laser intensity P µ E n (where n > 1) Isotropic: Ex. Glass P doesn’t depend on direction of E Anisotropic: Ex. Calcite crystal P = Pi (where i = 1,2,3 for x,y,z) Dispersive (and Absorptive) media: Ex. Na Vapour P has a memory - use CEO model: resonance

Linear Susceptibility: 

Linear Susceptibility Linear: P = e0 c E c = the electric susceptibility (dimensionless) (dependent on the frequency) = 0 for vacuum = 0.00059 for air = 1.28 for crown glass e = e0 (1+c) = permittivity of the material resulting in D = e E (includes P) n = (1+c)1/2 = refractive index = c0/c (not valid near a resonance) E is the total field includes the applied field and the field produced by P ! E induces P which changes E which changes P , and so on... c(3) E P + - E p

Non-linear Susceptibility: 

Non-linear Susceptibility Non-linear effects: Intense electric fields of one frequency can generate a P at another P = e0 (c E + c(2) E 2 + c(3) E 3 + …) c(2) = 2nd order susceptibility Only media lacking a centre of symmetry (ie, no gases, liquids) c(3) = 3rd order susceptibility All dielectric media Generally (c E >> c(2) E 2 >> c(3) E 3 ) We and text will use: P = e0 c E +2dE 2 + 4c(3) E 3 + … (1.7) c(2) c(3) Text has similar diagram that Danehy disagrees with c(3)

Anisotropic Media: 

Anisotropic Media P depends on direction of input field(s) ie. Polarisation of input wave(s) Independent nx, ny, nz Pi = e0Sj cij Ej where i,j = 1,2,3 for x,y,z cij is a 3 x 3 tensor with 6 independent variables (cji = cij ) Particular crystal: look up cxx, cxy, cxz, ... Special case: uniaxial crystals nx = ny = no and nz = ne Known as “birefringence” Calcite Crystal

Dispersive Media: 

Dispersive Media Medium doesn’t respond instantaneously to fields: leads to dispersion (n = n(l)) Some colours have more “lag” than others Dispersion implies absorption Kramers-Kronig relations Classical Electron Oscillator Model Forced mass, spring, damper system 2nd order ODE Solution: Real and imaginary parts are dispersion and absorption, respectively (1.8)

Near-resonant Susceptibility: 

Near-resonant Susceptibility Near a resonance: Far from Resonance (l > l0) Dispersion, n Absorption, a l l l Dispersion, n N2 and O2 resonances are in UV Same with Crystals We’re using usually visible light

Wave Equation: 

Wave Equation Consider Maxwell’s Equations Take curl of Eq. (1.2) and use Eqs. (1.5) and (1.6) Use vector identity: Assume medium is homogeneous and isotropic : Define c0 = 1/(e0m0)1/2 The wave equation: Valid for homogeneous, isotropic, non-linear source-free media Find solutions to this!!! (1.9)

Slide18: 

Monochromatic Waves A ‘wave’ = solution to the wave equation We’ll only consider monochromatic fields Fourier methods used for polychromatic light Electric fields are most important: For monochromatic electric fields: (a is real, d is phase) (Re{A} = {A+A*}/2) w = 2pn = the angular frequency E(r) = Complex Amplitude of the Electric Field Vector E(r) contains amplitude, and: The direction of propagation (denoted by (r)) The phase of the light (complex) The polarisation: ‘direction’ of E (as in linear polarisation) H, P, D, B are similarly defined

Monochromatic Plane Waves: 

Monochromatic Plane Waves Plane waves have straight wave fronts As opposed to spherical waves, etc. Suppose E(r) = E0exp(jk . r) E0 still contains: amplitude, polarisation, phase direction of propagation given by wavevector: k = (kx, ky, kz) where | k| = 2p/l = w/c = wn/c0 (w = 2pn) Can also define E = magnitude of E =(Ex, Ey, Ez) Plane wave propagating in z-direction: k plane wave

Intensity of Light: 

Intensity of Light Poynting vector describes flow of E-M power S = E x H Power flow is directed along this vector usually parallel to k Intensity is equal to the magnitude of the time-averaged Poynting vector: I = <S> A plane wave’s intensity is then: Example: 87 V/cm is equivalent to 10 Watt/cm2 (1.10) (1.11) Derive (1.11) starting from (1.10), using Maxwell’s equations.