Lecture 11: Introduction to nonlinear optics I. Petr Kužel Formulation of the nonlinear optics: nonlinear polarization Classification of the nonlinear phenomena • Propagation of weak optic signals in strong quasi-static fields (description using renormalized linear parameters) ! Linear electro-optic (Pockels) effect ! Quadratic electro-optic (Kerr) effect ! Linear magneto-optic (Faraday) effect ! Quadratic magneto-optic (Cotton-Mouton) effect • Propagation of strong optic signals (proper nonlinear effects) — next lecture Nonlinear optics

Experimental effects like • Wavelength transformation • Induced birefringence in strong fields • Dependence of the refractive index on the field intensity etc. lead to the concept of the nonlinear optics

The principle of superposition is no more valid The spectral components of the electromagnetic field interact with each other through the nonlinear interaction with the matter Nonlinear polarization Taylor expansion of the polarization in strong fields:

= ε χ + χ(2) + χ(3) + Pi 0 ij E j ijk E j Ek ijkl E j Ek El !

()= ε χ~ − (′ )′ (′ + ) Pi t 0 ∫ ij t t E j t dt + χ(2) −()()()′ − ′′ ′ ′′ ′ ′′ + ∫∫ ijk t t ,t t E j t Ek t dt dt ()()()() + χ(3) − ′ − ′′ − ′′′ ′ ′′ ′′′ ′ ′′ + ∫∫∫ ijkl t t ,t t ,t t E j t Ek t El t dt dt + !

()ω = ε χ ω ()ω + ()ω χ(2) ω ω ω (ω )ω (+ ) ( ) Pi 0 ij E j ∫ d 1 ijk ; 1, 2 E j 1 Ek 2 %"$"""ω"=ω +"#ω """" ()()()()1 2 + ω ω χ(3) ω ω ω ω ω ω ω + ∫∫d 1d 2 ijkl ; 1, 2 , 3 E j 1 Ek 2 El 3 ! %"$""""ω"="ω +ω"#+ω"""""" 1 2 3 Linear electro-optic effect (Pockels effect)

Strong low-frequency field Es (renormalization of optical constants due to second-order susceptibility) Propagation of a weak high-frequency optical field E in such a disturbed linearized medium

= ε χ + χ(2) ( S + S ) Pi 0 ij E j ijk E j Ek Ek E j

New effective permittivity tensor:

ε = ε ()+ χ + χ(2) S = εL + χ(2) S ij 0 1 ij 2 ijk Ek ij 2 ijk Ek

Wave equation (anisotropic media): ()⋅ − 2 + ω2µ ε⋅ = k k E k E 0 E 0 Pockels effect: continued Description using deformations of the indicatrix

We define the reciprocal dielectric tensor: = ε (ε−1 ) Kij 0 ij

Indicatrix in the system of the principal dielectric axes: 2 2 2 x + y + z = 2 2 2 1 nx ny nz Indicatrix in a general system of axes: = Kij xi x j 1

Now the external strong electric field is switched on… Indicatrix in electro-optics Reciprocal dielectric tensor is renormalized due to external field:

′ = + S Kij Kij rijk Ek rijk … Pockels electro-optic tensor

′ = Deformed indicatrix: Kij xi x j 1

′ 2 + ′ 2 + ′ 2 + ′ + ′ + ′ = K11x K22 y K33z 2K12 xy 2K13xz 2K23 yz 1

• Length of principal axes is modified • Orientation of the ellipsoid is modified

The Pockels tensor and the second order susceptibility tensor are related by the following equation: ε ε χ(2) = − ii jj r ijk ε ijk 2 0 Pockels tensor

ε Intrinsic symmetry: related to the fact that ij is a hermitic tensor: = rijk rjik

For symmetric tensors Voigt notation can be introduced: indices (ij) 11 22 33 23 or 32 13 or 31 12 or 21 contraction (l)123456

= A 6×3 matrix rlk is introduced, where l 1…6 is a contracted index, and k =1…3.

 r r r   11 12 13   r21 r22 r23    r r r r =  31 32 33  lk  r r r   41 42 43   r51 r52 r53     r61 r62 r63  Pockels effect: cubic media

In cubic non-centrosymmetric media (GaAs, InP, GaP, CdTe, ZnTe…):

   0 0 0   0 0 0    0 0 0 =   rlk = = r 0 0  r63 r52 r41  41   0 r41 0     0 0 r41 

= For an applied electric field E (Ex, Ey, Ez):

x2 y2 z2 + + + 2r E yz + 2r E xz + 2r E xy =1 n2 n2 n2 41 x 41 y 41 z Example: E//c x2 y2 z2 + + + 2r E xy =1 n2 n2 n2 41 Rotation by 45° in the xy plane will diagonalize the equation: 1 1 x = ()η + ξ η = ()x + y 2 2 1 1 y = ()η − ξ ξ = ()x − y 2 2 One gets:

n 1 3 nξ = ≈ n + n r E − 2 2 41 1 n r41E

n 1 3 nη = ≈ n − n r E + 2 2 41 1 n r41E = nz n Example: E//c, continued

n 1 3 nξ = ≈ n + n r E − 2 2 41 1 n r41E

n 1 3 nη = ≈ n − n r E + 2 2 41 1 n r41E = nz n

Transverse geometry: Longitudinal geometry:

(001) (110)

(110) (110)

n n − ∆n Optical n + ∆n Optical n + ∆n beam beam (110) (001) Example: E // (1,1,0)

x2 y2 z2 + + + 2r E yz + 2r E xz =1 n2 n2 n2 41 41 1) Rotation by 45° in the xy plane (as previously); new variables ξ, η ξ2 η2 z2 + + + 2r E ηz =1 n2 n2 n2 41 2) Rotation by 45° in the ηz plane; new variables η’, z’ (110) 1 3 n ′ ≈ n + n r E z 2 41

1 3 nη′ ≈ n − n r E 2 41 (001) n + ∆n nξ = n n − ∆n Optical beam Electro-optic modulation quarter-wave polarizer plate analyzer π/4 π/4 δ 0 π , 0 /2 δ1 EO modulator

It was shown previously 1 1 1+ δ I = ()1− cos(δ + δ ) = ()1+ sin δ ≈ 1 2 0 1 2 1 2 Electro-optic dephasing:

π π 3 U δ = 2 3 Transverse setup: δ = n r L Longitudinal setup: 1 n r41 U 1 λ 41 d λ Quadratic electro-optic effect (Kerr effect) Kerr effect can be of a great importance in the media where the Pockels effect vanishes (centrosymmetric media): = ε χ + χ(3) ( S S + S S + S S ) Pi 0 ij E j ijkl E j Ek El Ek E j El El E j Ek

ε NL = χ(3) S S ij 3 ijkl Ek El

The same treatment can be used as for the linear Pockels effect; the following differences should be emphasized: • it is a 3rd order nonlinear effect (the electric field should be very strong) • it is a quadratic effect in E, so the induced birefringence is proportional ∆ ∼ S 2 to n (E ) • the effect depends on a 4th rank tensor, so the symmetry properties are different comparing to the linear EO effect Magneto-optic effects (Faraday, Cotton-Mouton)

The polarization, and consequently, the permittivity can depend on the magnetic field: = ε χ + χ(2) + χ(3) + (2) + (3) Pi 0 ij E j ijk E j Ek ijkl E j Ek El hijk Bk E j hijkl Bl Bk E j !

Faraday Cotton-Mouton effect effect

ε ()= (2) + (3) + ij B hijk Bk hijkl Bl Bk ! Intrinsic symmetry properties in the magnetic field (Onsager theorem):

ε ()= ε − (= )ε∗ = ()ε∗ − ( ) ij B ji B ji B ij B

From this relation we find conditions for hijk and hijkl Faraday effect ε ()= (2) ij B hijk Bk ε ()= ε − (= )ε∗ = ()ε∗ − ( ) ij B ji B ji B ij B

These conditions are similar as for the spatial dispersion and the optical activity

The tensor hijk is • purely imaginary • antisymmetric in first two indices Similarly as for the optical activity, we can define the effective permittivity tensor:

 ε0   xx ig12 ig13  ε = − ig ε0 ig   12 yy 23  − − ε0   ig13 ig23 zz  Effect of a longitudinal field

Cubic non-centrosymmetric medium Wave equation:  2 − 2 ∆    n0 n i B 0   Ex   − ∆ 2 − 2  = i B n0 n 0  Ey  0  2     0 0 n0   Ez 

Eigenvalues Eigenvectors (effective refractive indices) (Polarization direction) ∆ B  1  1 n ≈ n ±     I ,II 0 2n − i ,  i  0      0  0

• Cumulative character of the Faraday effect Effect of a transverse field Wave equation:  2 − 2    n0 n 0 0   Ex   0 n2 − n2 i∆ B  E  = 0  0  y  − ∆ 2     0 i B n0   Ez 

Eigenvalues: = nI n0 ∆2 2 ≈ − B nII n0 4 2n0

Cotton-Mouton effect: very small correction to a linear birefringence