Nonlinear Optics

Basic concept of optical nonlinearity

The polarization density in the presence of an electric field: P ∝ E

Here we assume that the medium is isotropic. When the field is strong, the displacement of electrons from their equilibrium point may deviate from linearity:

2 (3) 3 P = ε 0 χE + 2dE + 4χ E (1)

The coefficients describe the first order, second order and third order nonlinearity.

If the medium has an inversion symmetry, the second order nonlinearity, characterized by d, is zero. Why?

Physical pictures: When the potential as a function of displacement deviates from a perfect parabola, the electrons driven by an electric field may create a polarization. For example: a DC polarization can be created by an electric in an asymmetric potential. An asymmetric shape in the polarization-electric field relation results in a distorted polarization which, in turn, generates a wave which contains the harmonic waves whose frequencies are two times the fundamental.

The polarization created by the radiation then acts as a source for the generation of new waves.

2 (3) 3 P = ε 0 χE + 2dE + 4χ E

Second order nonlinearity

(2) 2 jω1t jω2t 2 P = 2dE = 2d(E1e + E2e + cc)

•Second harmonic generation ω1=ω2 •Optical rectification • •Electrooptic effect ω1=0

ω1≠ω2 •Frequency conversion •Optical parametric process

2 (3) 3 P = ε 0 χE + 2dE + 4χ E

Third order nonlinearity

(3) jω1t jω2t jω3t 3 P = 4χ(E1e + E2e + E3e + cc)

ω1=ω2 =ω3 •Third harmonic generation •Degenerate four-wave mixing •Optical phase conjugation

ω1=ω2 •Optical Kerr effect •Self-phase modulation •Self- focusing

ω1≠ω2 ≠ω3 •Raman gain •Four-wave mixing

I In general, the polarization density and electric field is related by a tensor. For example, (3) Pi = ε 0 χ ij E j + 2dijk E j Ek + 4χ ijkl E j Ek El + ...... (1.1) where each of the i,j,k,l are can be x, y, or z. The coordinate axis does not need to be one that diagonalizes the dielectric tensor.

See Yariv’s.

Nonlinear coefficients- the anharmonic oscillator model

Consider the bound electron model with an electron in a anharmonic potential well driven by electromagnetic wave of frequency ω. The displacement of electron from equilibrium follows the following relations:

.. . eE x+ γ x+ ω 2 + Dx 2 = (e jωt + cc) (1.2) 0 2m Assuming that the solution of this equation has the following form: 1 X = (q e jωt + q e j 2ωt + q e j3ωt + cc) 2 1 2 3

Solve for the q’s eE 1 q1 = ( ) 2 2 (1.3) m (ω0 − ω ) + jωγ 2 2 − De E0 q2 = (1.4) 2 2 2 2 2 2 2m [(ω0 − ω ) + jωγ ] (ω0 − ω + 2 jωγ )

The nonlinear coefficients are larger in the vicinity of resonance lines..

Nonlinear wave equation

From Lecture 2 Eq. (7), the Helmholtz equation with a source is given by

2 2 2 1 ∂ E ∂ P ∇ E − 2 2 = µ0 2 (2) c0 ∂t ∂t The polarization has a linear and nonlinear components. Four-wave mixing : beam deflection by an oscillating grating

P = ε 0 χE + PNL (3)

k3 The linear part can be combined with the other time derivative term to give rise to a speed of light c =c0/n. Eq. (2) can be rewritten as k4 2 k2 2 1 ∂ E k1 ∇ E − 2 2 = −S c ∂t The directions and frequencies of the k’ (4) ∂ 2 P S = −µ NL 0 ∂t 2

Second order nonlinearity

Consider the case of two sinusoidal electric fields in a nonlinear medium

2 P (2) = 2d( E e j 2ω1t + E e jω2t + cc) 2 ω1 ω2 (5)

In general, the two electric fields can be in different direction and the polarization is related to the individual fields through a tensor. The mixing of the field results in four nonlinear polarization densities oscillating at 2ω1, 2ω2, ω1+ω2 , and ω1-ω2 , representing the sources for second harmonic generation of ω1 and ω2 , sum and difference frequency generation. Note the for second harmonic generation, only one wave is needed.

Pockels effect

A special case is when one of the waves is a DC electric field with ω2=0, the polarization density is given by P (2) = 2dE 2 = 2 d E E e jω1t (6) DC ω1 This term contribute to a change in the refractive index

2d ∆n = EDC (7) nε 0 The change of refractive index by applied voltage is the Pockels effect, which has wide applications in optics as optical shutter or modulators.

Frequency conversion 2 2 P (2) = 2d ( E e j 2ω1t + E e j2ω2t + 2E E * e j(ω1 −ω2 )t + 2E E e j(ω1 +ω2 )t ω1 ω2 ω1 ω2 ω1 ω2 (8) Thus the nonlinear medium can be used to mix two optical waves of difference frequencies and generate a third wave at the sum and difference frequencies. Although the waves produce polarization densities at various frequencies, not all of them are generated in a meaningful way unless the following conditions are simultaneously satisfied.

Frequency matching : ω3 = ω1 + ω2 (9) C C C

Phase matching : k3 = k1 + k2 (10)

Physical meaning: Using propagation delay of SHG in one dimension as an example. The waves must have the same speed (phase velocity) to ensure constructive interference along the path.

Coupled-wave theory with second-order nonlinearity

1 ∂ 2 E ∇ 2 E − = −S c 2 ∂t 2 (11) ∂ 2 P ∂ 2 S = −µ NL = −2dµ E 2 0 ∂t 2 0 ∂t 2 From (4) The field is a superposition of three waves 1 1 E = ƒ (Eq exp( jωqt) + cc) = ƒ Eq exp( jωqt) (12) q=1,2,3 2 q=±1,±2,±3 2

* where E-q is defined to be Eq By substitution the electric filed into (11), and carrying out the time derivative

1 2 S = µ0 d ƒ (ωq + ωr ) Eq Er exp[( jωq + ωr )t] 2 q,r=±1,±2,±3

The Hemlholtz equations with sources are

2 2 (∇ + k s )Es = −S s s = 1,2,3 (13) where Ss is the component of S with a frequency, and ks=nsω/c. Among the frequencies in the three equation, the interaction takes place only when

ω3 = ω1 + ω2 (14)

2 2 2 * (∇ + k1 )E1 = −2µ0ω1 dE3 E2 (15-1) 2 2 2 * (∇ + k2 )E2 = −2µ0ω2 dE3 E1 (15-2) 2 2 2 (∇ + k3 )E3 = −2µ0ω3 dE1 E2 (15-3) With this constraint,

This is the Helmholtz equations for three-wave mixing.

Special case: Collinear plane waves

In the normalized form > 1/ 2 Eq = (2π ωq ) aq exp(− jkq z) q = 1,2,3 (16)

The coupled mode equations are, from (15) and (16) and using slow varying amplitude approximation.

da1 * = − jga3a2 exp(− j∆kz) (17) dz da 2 * = − jga3a1 exp(− j∆kz) (18) dz da 3 = − jga a* exp( j∆kz) (19) dz 1 2

2 > 3 2 1/ 2 where g = 2 ω1ω2ω3η d and η = (µ0 / ε 0 ) , and ∆k = k3 − k2 − k1 is the phase mismatch of the three waves.

For an input of two frequencies, the wave of the third frequency cannot build up unless the phase mismatch is zero. Second harmonic generation Angle tuning

The electric displacement vector and the electric field are related through a dielectric tensor

Dk = ε kl El (1) where k and l refer to the Cartesian coordinates (x,y,z). Using the principal axes as the coordinate system, the dielectric tensor is diagonal. The index ellipsoid is described by 2 2 3 x1 x2 x3 2 + 2 + 3 = 1 (2) n1 n2 n3 where 1/ n1 ,1/ n2 ,and 1/ n3 are the principal values. The index ellipsoid of an isotropic medium is a sphere. A wave propagating in the z-direction and linearly polarized in the x-direction travels with phase velocity c0/n1.

What about a wave traveling in the z-direction and an electric field on the z=0 plane and making a 45-degree angle with the x-axis?

In an uniaxial crystal, 1 cos 2 (θ) sin 2 θ 2 = 2 + 2 n (θ ) n0 ne

The ray with index n0 is the ordinary wave and n(θ) the extraordinary wave.

Walk-off

o and e

OA no>ne o

e ne

ne n(θ)