Guoqing (Noah) Chang, October 9, 15 Nonlinear optics: a back-to-basics primer Lecture 1: linear optics 1
Suggested references Robert W. Boyd, Nonlinear optics (8) Geoffrey New, Introduction to nonlinear optics (11) George Stegeman and Robert Stegeman, Nonlinear optics: phenomena, materials, and devices (1) David N. Nikogosyan, Nonlinear optical crystals: a complete survey (5) Rick Trebino s course slides on optics (http://frog.gatech.edu/lectures.html)
If you want to talk to a nonlinear optics person, you need to speak his language; that is, you should understand the jargon in this field. David N. Nikogosyan, Nonlinear optical crystals: a complete survey (5) 3
David N. Nikogosyan, Nonlinear optical crystals: a complete survey (5) 4
David N. Nikogosyan, Nonlinear optical crystals: a complete survey (5) 5
A timeline of classical (linear) optics Willibrord Snell (1591-166), Snell s law Johannes Kepler (1571-163), Total internal reflection Pierre de Fermat (161-1665), Fermat principle Christiaan Huygens (169-1695), Wave theory of optics Isaac Newton (164-177), Particle theory of light Thomas Young (1773-189), Interference of optical waves Augustin Fresnel (1788-187), Fresnel coefficients James Clerk Maxwell (1831-1879), Maxwell s equations 6
Operators used in Maxwell s Equations The Del operator:,, x y z f f f The Gradient of a scalar function: f,, x y z The Divergence of a vector function: G G G x x x x y G + + z The Laplacian operator: x y z = + + 7
Operators used in Maxwell s Equations The Laplacian of a scalar function: f f f f f = (,, ) x y z f f f = + + x y z The Laplacian of a vector function is the same, but for each component: G G G (,, ) x y z x y z x y z G G G y y y G G G G x + x + x + + z + z + z 8
Operators used in Maxwell s Equations The Curl of a vector function: G G z y Gx G G z y Gx G (,, ) y z z x x y The curl can be computed from a matrix determinant: G = det x y z x y z Gx Gy Gz 9
Maxwell s Equations of differential form in a medium without free current and free charge Ampere s law: Faraday s law: D B H = E = t t Gauss s law: Gauss s law of magnetism: D = B = Constitutive relations for a nonmagnetic material: D= ε E+ P Polarization. It takes into light-matter interaction. B = µ H ε Electric permittivity in vacuum µ Magnetic permeability in vacuum
Derivation of wave equation Vector Identity: ( ) = ( ) E E E B ( E) = = ( B) t t = ( ( µ H)) = ( µ H) t t E P = ( µ ( ε + )) t t t E P ( ) µ ( ε ) t t E E = +
Derivation of wave equation E P ( ) µ ( ε ) t t E E = + In the linear optics of isotropic source-free media: D = E = In the nonlinear optics, normally we have: ( E) << E Vacuum speed of light: Simplified wave equation: 1 ( ) E = µ c t t P Wave in vacuum Source term
Interaction between EM waves and materials.5 nm Wavelength of green light is about 5 nm. So the optical wave experiences an effective homogeneous medium, which is characterized by Electric permittivity and Magnetic permeability For a nonmagnetic material χ ε µ is the electric susceptibility. The velocity of light is different from the vacuum speed by a factor called the refractive index P = εχe n = ( εµ )/( ε µ ) = 1+ χ ε = ε (1 + χ) µ = µ 13
Lorentz model of light-atom interaction Important assumptions The atomic core is -- positively charged -- static (heavy, fixed within the crystal lattice) -- with the center of charge at x =. The electrons are -- light weight -- elastically bound by a massless spring with spring constant with equilibrium position at x = -- carrying out a damped movement; that is, after removing the force, the movement decreases and finally ends. -- the electron and atomic core form an oscillator with a resonant frequency w. Juergen Popp et al., Handbook of biophotonics (1). H. A. Lorentz (1853-198) 14
Lorentz model: forced electron harmonic oscillator Dipole moment is defined as the product of magnitude of charges and the distance of separation between the charges. Without an applied field, the centers of the negative and the positive charges coincide. The dipole moment is zero. If a field constant in time is applied, the electrons are displaced relative to their position in the absence of an external field. The centers of the positive and negative charges no longer coincide and a static dipole moment is induced. If a time-dependent electric field interacts with the atom, then the electron starts to oscillate around its equilibrium position with the same frequency of the electric field. Such an oscillating dipole moment will emit a new electromagnetic wave at the same frequency as well. Juergen Popp et al., Handbook of biophotonics (1). 15
Lorentz model of light-atom interaction When light of frequency w excites an atom with resonant frequency w : Electric field at atom Electron On resonance (ω = ω ) Emitted field x () E () t e t E () t The crucial issue is the relative phase of the incident light and this emitted light. For example, if these two waves are ~18 out of phase, the beam will be attenuated. We call this absorption. + = Incident light Emitted light Transmitted light Incident Light excites electron oscillation electron oscillation emits new light at the same frequency incident light interferes with the new light leading to the transmitted light. Adapted from Rick Trebino s course slides 16
Forced oscillator and resonance When we apply a periodic force to a natural oscillator (such as a pendulum, spring, swing, or atom), the result is a forced oscillator. Examples: Child on a swing being pushed Periodically pushed pendulum Bridge in wind or an earthquake Electron in a light wave Nucleus in a light wave Tacoma Narrows Bridge oscillating and collapsing because oscillatory winds blew at its resonance frequency. (collapsed under 64 km/h wind conditions the morning of November 7, 194) The forced oscillator is one of the most important problems in physics. It is the concept of resonance. Adapted from Rick Trebino s course slides 17
One more example: child on a swing If you give the swing a push it will swing back and forward. If you just give it one push it will swing back and forth a few times and then come to rest. (That s because of friction and damping.) To keep the swing moving you have to push again each time the swing reaches the closest point to you. You have to match the frequency of the swing to make it swing high. Course note, MIT 6.7 & http://www.lifeinresonance.com/?page_id=47 18
The forced oscillator The amplitude and relative phase of the oscillator motion with respect to the input force depend on the frequencies. Let the oscillator s resonant frequency be ω, and the forcing frequency be ω. Let the forcing function be a light electric field and the oscillator a (positively charged) nucleus in a molecule. Below resonance ω << ω On resonance ω = ω Above resonance ω >> ω Electric field at nucleus Adapted from Rick Trebino s course slides Nucleus Weak vibration. In phase. Strong vibration. 9 out of phase. Weak vibration. 18 out of phase. 19
The forced oscillator Electric field at electron Electron The amplitude and relative phase of the oscillator motion with respect to the input force depend on the frequencies. The electron charge is negative, so there s a 18 phase shift in all cases (compared to the previous slide s plots). Below resonance ω << ω On resonance ω = ω Above resonance ω >> ω Weak vibration. 18 out of phase. Strong vibration. -9 out of phase. Weak vibration. In phase. Adapted from Rick Trebino s course slides
The forced oscillator Electric field at atom Electron Emitted field The amplitude and relative phase of the oscillator motion with respect to the input force depend on the frequencies. Maxwell s Equations will yield emitted light that s 9 phaseshifted with respect to the atom s motion. Below resonance ω << ω On resonance ω = ω Above resonance ω >> ω Weak emission. 9 out of phase. Strong emission. 18 out of phase. Weak emission. -9 out of phase. Adapted from Rick Trebino s course slides 1
Interference depends on relative phase When two waves add together with the same complex exponentials, we add the complex amplitudes, E + E '. ~ ~ Constructive interference: Destructive interference: Quadrature phase: ±9 interference: 1. 1. 1.. + = -. + = -.i + = 1..8 1-.i time time time Laser Absorption Adapted from Rick Trebino s course slides Slower phase velocity (when accumulated over distance)
Dielectric Permittivity: Lorentz model Pt () = Dipole moment Volume = N pt () =εχet () Density (# of atoms per unit volume) χ = N pt () ε Et () Elementary Dipole Lorentz Model: x(t) p( t) = ex( t) xt () is much smaller than the wavelength of electric field. Therefore we can neglect the spatial variation of the E field during the motion of the charge. 3
Response to a monochromatic field: forced electron harmonic oscillator d x m dt mass damping dx + mγ + mω x = ee( t) dt force electron charge frequency of undamped oscillator Et () = Ee jωt x( t) = xe jωt p( t) = ex( t) = pe jωt e / m ω + p = ω E jωγ Ne /( mε ) χ( ω) = ω ω + jωγ 4
Sellmeier equation to model refractive index If the frequency is far away from the absorption resonance χω ( ) = ω p ω ( ω ) ω Ne /( mε ) ω ω ωγ >> Normally there are multiple resonant frequencies for the electronic oscillators. It means in general the refractive index will have the form i n ( ω) = 1 + χω ( ) = 1+ Ai = 1+ a i i ω ω i λ λi ω p = λ 5
P = εχe D= ε E+ P Susceptibility is a tensor Polarization and electric field are vectors: Therefore in general, the susceptibility is a nd order tensor (i.e., 3 by 3 matrix): Px χxx χxy χ xz Ex P = ε χ χ χ E y yx yy yz y P z χzx χzy χ zz E z D = εεe ε = 1+ χ P Dielectric constant P x = P y P z E E x = E y E z A more convenient notation: P χ i = εχ ije ij j D = εε ( ) E = ε δ + χ E i ij j ij ij j is the linear susceptibility tensor. Repeated indices imply summation. δ ij is the identity matrix. 6
Optical anisotropy: birefringence Dx εxx εxy ε xz Ex D = ε ε ε ε E y yx yy yz y D z εzx εzy ε zz E z We can always select a (x,y,z) axes (i.e., principal dielectric axes) to diagonalize the dielectric matrix to the following form: Dx ε xx Ex n x Ex D y ε εyy E y ε ny E = = y D z ε zz E z n z E z nx = ny = nz nx ny no Isotropic ( the same in all directions ) medium (no birefringence) ne > no Positive uniaxial n < n Negative uniaxial = = nz = ne no Uniaxial medium nx ny nz biaxial medium e o 7
Crystal symmetry A crystal or crystalline solid is a solid material whose constituents, such as atoms, molecules or ions, are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. From Wikipedia Every crystal belongs to one of 3 point symmetry classes, which are categorized into 7 crystal systems. If an object is invariant under point reflection through its center, it is said to possess center symmetry or inversion symmetry. The object is centrosymmeric. Otherwise it is non-centrosymmetric. 8
Maxwell s Equations of differential form in a medium without free current and free charge Ampere s law: Faraday s law: D B H = E = t t Gauss s law: Gauss s law of magnetism: D = B = Constitutive relations for a nonmagnetic material: D E P = ε + Polarization B = µ H ε µ Electric permittivity in vacuum Magnetic permeability in vacuum
Light propagation in anisotropic media Gauss s law: D= D exp[j( ω t k r )] B= B exp[j( ω t k r )] D = k D= k D B Gauss s law of magnetism: B = k B= They form a orthogonal basis. Ampere s law: Faraday s law: B = µ H D H = B t H D E = E B t Constitutive relations for a nonmagnetic material: H B D= εe+ P= εεe D E in an anisotropic medium. s = E k H 3
Light propagation in a uniaxial crystal ε xx n x ε yy = ny ε zz n z nx = ny = no nz = ne x y z + + = Index ellipsoid for n n n uniaxial crystal 1 o o e Take BBO as an example.1878 no ( λ) =.7359 +.1354λ λ.18 D dn o dt 6 9.3 1 / =.14 ne ( λ) =.3753+.1516λ λ.1667 C D 1 dn e dt 6 16.6 1 / = C 31
Light propagation in a uniaxial crystal (1) Index ellipsoid is used to find the two refractive indices and the two corresponding directions of D associated with the two independent plane waves that can propagate along k direction. () The plane through the origin and perpendicular to k intersects with the index ellipsoid and generates an ellipse. (3) D 1 lies in the x-y plane and is perpendicular to the optical axis z. D lies in the plane of z-k. D 1 is called ordinary wave and D extraordinary wave. (4) The two axes of the intersection ellipse are n and n ( θ ) o n ( ) eff (5) o is the refractive index for D 1. neff θ is the refractive index for D. 1 cos ( θ) sin ( θ) eff ( ) o e n θ = n + ne( ) = no Two special cases: n n (9 ) = n D D 1 e e 3
Example: BBO at room temperature 1.7 1.68 Refractive index 1.66 1.64 1.6 1.6 1.58 1.56 n e n eff (6 o ) n eff (4 o ) n eff ( o ) n o 1.54.3.4.5.6.7.8.9 1 1.1 wavelength [µm] BBO is a negative uniaxial nonlinear crystal. 33
More on ordinary wave and extraordinary wave Z (optic axis) D, E k, s θ n (θ ) = n o D Index surfaces for a positive uniaxial medium. E Z (optic axis) θ 1 cos ( θ) sin ( θ) n ( θ ) = n + n eff o e k s ρ s = E H Represents the energy flow. For o-wave, D and E are in parallel, and k and s are in the same direction. For e-wave, D and E are NOT in parallel, and the energy flows at the direction different from k. ρ( θ ) = ± {tan 1 n [( n o e ) tanθ ] θ} 34
Take-home message Constitutive relations describe light-matter interaction. Material polarization can be modeled by harmonic electric oscillation. In an anisotropic medium, linear susceptibility is a nd rank tensor. Ordinary wave (o wave) and extraordinary wave (e wave) experience different refractive indices. The refractive index of e wave depends on the incident angle. E wave and o wave carry energies, which may flow at different direction causing double refraction. 35
Suggested reading Classical harmonic oscillator model -- Mark Fox, optical properties of solids, chapter -- George Stegemann and Robert Stegemann, Nonlinear optics, chapter 1 EM wave propagation in anisotropic media -- Amnon Yariv and Pochi Yeh, optical waves in crystals, chapter 1 and 4 -- Geoffrey New, Introduction to nonlinear optics, chapter 3 36