Understanding Nonlinear Media Propagation

8. Propagation in Nonlinear Media

8.1. Microscopic Description of Nonlinearity. 8.1. Microscopic Description of Nonlinearity. 8.1.1. Anharmonic Oscillator. Use Lorentz model (electrons on a spring) but with nonlinear response, or anharmonic spring ( ) ( ) 2 ( ) dt ( ) t m f x t m 2 E d x t dt f dx t e ( ) NL + + + = 2 x t 0 ( ) ( ) 2 3 = + + a x t a x t NL m 2 3 b) Harmonic spring; Low intensity c) Anharmonic spring (overstretched); High intensity

At strong incident linearly polarized fields electron trajectories no longer straight lines

8.1.2. Lorentz Force as Source of Nonlinearity At strong incident linearly polarized fields electron trajectories no longer straight lines Molecule oscillates over other degrees of freedom Lorentz force becomes significant e v B

Simple case where electric field polarized along x, magnetic field lies on y ( ) ,0,0 x E = E 0, ,0 y H = H ( ) ( ) 2 ( ) dt dy t dt dz t dt 2 d x t dt d y t dt d z t dt dx t e m ( ) ( ) t + + = + 2 ( ) z t B t ( ) x t E 0 x y ( ( ) = + v B E f e ( ) 2 ( ) 2 em ( ) + + = 2 0 y t ) = + y exB z + 0 0 e E zB x x y y ( ) 2 ( ) 2 e m ( ) z t + + = 2 ( ) x t B t ( ) 0 y Equations of motion from E and H fields, no force acting in y direction. Solve for x and z displacements. Take Fourier transform, obtain following equations: ( ) m eE e m ( ) ( ) ( ) = + r 2 2 x i i z B 0 y e m ( ) ( ) ( ) = 2 2 . z i i x B 0 y

( ) ( ) m D eE ( ) ( ) = x x = 2 2 D i ( ) b ( ) 2 2 D b 0 e m Equations obtained from method in 5.1 Ignore magnetic field, we get linear response ( ) ( ) m ( ) ( ) = eE . b i B ( ) = x . z y ( ) ( ) 2 2 D b Order of magnitude relation between x and z displacements Express magnetic field in terms of electric field ( ) ( ) z x eE me eE ( ) ( ) = k E B i = x x 2 2 mc E ic ( ) = B x y Solve for when electric field and associated irradiance where x and z are comparable 2 2 xx z E e Vm 1 mc = 2 I E = x 2 ( ) ( ) = W 21 10 12 10 , 2 m

e E ( ) ( ) we have: = b D x Using mc 2 e ( ) ( ) . ( ) ( ) 2 = i t i t 2 + z E z E e E e ( ) ( ) x x x 2 2 2 m c D b ( ) = 1 cos2 + 2 2 E t 2 e x ( ) 2 E ( ) x 2 2 m c D Electron oscillates at 2? on z axis, DC term is called optical rectification ( ) D eE m eE m ( ) = x x ( ) 1 ( ) 2 2 D b . x ( ) D

8.1.3. Dropping the Complex Analytic Signal Representation of Real Fields. Real field Complex analytic signal = Ae = cos U A t i t U r 1 2 A ( ) 2 = 2 i t i t Re[ ] A cos(2 ) U t U = + 2 U A e e r r ( ) = 1 cos2 + 2 t Complex analytic signal does not capture DC term. = 2 2 i t . U A e Whenever we deal with fields raised to powers higher than one, we use 1 2 ( ) = + * U U U r

8.2. Second 8.2. Second- -Order Susceptibility Order Susceptibility ( ) ( ) ( ) 0 2 x t a x t dt dt ( ) m 2 2 d x t dx t E t ( ) 2 + + + = 2 2 ( ) 1 ( ) 2 Want solution of form where ?(2) is the nonlinear perturbation to linear solution. x x x = + 2 Simplify by neglecting 2?2?1?2 and ?2?2 ( )( ) 2 dt ( )( ) dt ( ) m 1 1 2 d x t dx t eE t 2 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 2 2 2 1 + + = 2 x t x t x t a x t 1 + + + + 2 x t 0 2 0 ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) 2 dt ( )( ) dt 2 1 1 = 2 2 D x a x x 2 d x t dx t ( )( ) 2 2 + + + + 2 x t ( ) ( ) 0 2 E D E D e m = , a 2 2 2 ( )( ) ( )( ) ( )( ) ( )( ) 1 1 2 2 + + + = 2 0. a x t a x t x t a x t 2 2 2

Illuminating the nonlinear crystal with two monochromatic fields of different frequencies ( ) ( ) ( ( D ( ) ( ) ( ( 2 E D E D 1 e m ( )( ) 2 = x a ( ) 2 D ( ) ( ) ( ) = + + + * E E E ) ) ' ) ) ' 2 ' ' 1 1 1 1 E E D 1 e m = '. a d ( ) ( ) + + + * E E ( ) 2 D 2 2 2 2 ( ) ( ) ( ) + + = a b + a b

( ) ( ) E E 2 1 1 2 ( ) ( ) E E All possible frequencies: 2 2 = 2 2 0 + + 2 2 2 1 2 1 1 1 2 1 2 1

2 e m ( )( ) ( ) ( ) t 2 = + + + 2 . . cc . . cc x t a g t g 1 e m ( )( ) ( ) ( ) ( ) ( ) 2 1 2 2 = + + + x a f f f f ( ) 2 1 1 2 2 D ( ) ) ( D 2 + i t E 2 i t 2 E e E E e + 1 2 1 ( ) = + + + 1 ( ) ( ) + g t 1 1 2 + 2 + + 2 2 E E ( ) ( ) ( ) ( D ) 1 ( ) 0 ( ) 2 2 2 D D D 1 E E 1 1 D D Fourier Transform 1 1 1 2 1 2 1 1 ( ) ( ) = + + f ( ) ( D ) ( ) 1 1 2 1 2 i t * D E E e 1 2 + + . . cc 1 2 1 1 ( ) + * E E ( ) ( D ) ( ) * D D 1 2 1 2 1 2 1 2 ( ) ( ) + 2 ( ) ) ( D 2 + 2 + 2 i t E E E 2 i t 2 E e E E e + 1 2 2 ( ) t 2 E E 2 2 = + + + 2 2 1 2 g 1 ( ) ( ) ( ) ( D ) 2 ( ) 0 ( ) 2 ( ) ( ) 2 2 D D D = + + D D f 2 2 1 2 1 2 ( ) ( D ) 2 2 1 2 1 2 D 2 2 ( ) ( ) i t + * * E E e E E 1 2 ) + + . . cc 1 2 ) ( D 1 2 1 2 ( ( ) * D D 1 2 1 2 Equation 8.26 indicates that, as the result of the second-order nonlinear interaction, the resulting field has components that oscillate at frequencies 2 2 (second harmonic of 2), 2 1 (second harmonic of 1), 1 + 2 (sum frequency), 1 - 2 (difference frequency) and 0 (optical rectification terms).

Nonlinear susceptibility from induced polarization Nex ( )( ) ( )( ) ( )( ( )( ) ) ( )( ) ( ) 2 2 = ; , ( ) 2 2 = * * ; , P E E i j i j i i 0 i j i j i j i j 0 ( )( ) 2 2 = = 3 , , 1,2 a Ne P Nex i j ( ) ( ) t = + + . . cc 2 g t g i j i j 1 2 2 m 0 ( )( ) ( )( ) 2 2 = ( ) 2 r r Importantly, vanishes in centrosymmetric media ( )( ) ( ) ( ) ( )( ) ( ) . r 2 ( )( ) ( )( ) 2 = = 2 = P r r P r r E r E r Ne Fulfilled simultaneously only if 0 ( ( )( ) ) ( )( ) 2 r Ne ) ( r E ) 2 = P r 0 = = r E r 0 ( 2 = P r . P So second-order nonlinear processes require noncentrosymmetric media

8.2.1. Second Harmonic Generation (SHG) 3 1 a Ne ( ) = 2 ; , 2 ( ) ( ) 1 1 1 2 2 2 m D D 0 1 1 3 1 a Ne ( ) = 2 ; , . 2 ( ) ( ) 2 2 2 2 2 2 m D D 0 2 2 Nonlinear susceptibility a s function of linear chi: a) SHG: pumping the chi(2) material at omega yields both the fundamental frequency (omega) and its second harmonic (2omega). b) Description in terms of virtual energy levels. Linear: 2 1 Ne ( )( ) 1 = . ( ) m D 0 Nonlinear: 2 2 0 a N e m 2 ( )( ) ( )( ) ( )( ) 2 1 1 = 2 ; , 2 . 1 1 1 1 1 2 3

8.2.2. Optical Rectification (OR) 3 1 a Ne ( )( ) 2 = 0; , 2 ( ) ( 0 ) ( D ) 1 1 2 m D D 0 1 1 2 2 0 a N e m ( )( ) ( )( ) ( )( ) 1 1 1 = 0 . 1 1 2 3 OR: a DC polarization is created in a chi(2) material.

8.2.3. Sum Frequency Generation (SFG) 3 1 a Ne ( )( ) 2 + = ; , 2 ( ) ( D ) ( D ) 1 2 1 2 + 2 m D 0 1 2 1 2 2 2 0 a N e m ( )( ) ( )( ) ( )( ) 1 1 1 = + . 1 2 1 2 2 3

8.2.4. Difference Frequency Generation (DFG) 3 1 a Ne ( )( ) 2 = ; , 2 ( ) ( D ) ( D ) 1 2 1 2 2 m D 0 1 2 1 2 2 2 0 a N e m ( )( ) ( )( ) ( )( ) 1 1 1 = . 1 2 1 2 2 3

8.2.5. Optical Parametric Generation (OPG) 2 The time reverse process of SFG = + ( ) 2 1 2 3 3

8.3. Third 8.3. Third- -Order Susceptibility Order Susceptibility Anharmonic oscillator ( ) m eE t ( ) ( ) ( ) ( ) + + + = 2 3 . x t x t x t a x t 0 3 ? ? = ?1? + ?3? Solve using perturbation theory, solution of form ( ) m eE t ( )( ) ( )( ) ( )( ) 1 1 1 + + + + 2 x t x t x t 0 3 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 3 3 3 1 3 + + + + + = 2 0. x t x t x t a x t x t 0 3 ( ) ( ) 3 ( )( ) ( )( ) ( )( ) ( )( ) 3 3 ( ) 1 ( ) 3 ( ) 1 3 3 3 1 + + = 2 , x t x t x t a x t + a x x a x First term vanishes, approximate 3 3 0 3

( ) ( ) ( ) ( ) ( ) E = + + + e m D * E E E ( )( ) ( )( ) 1 1 = , x t x So, 1 1 1 1 ( ( ) ) ( ( ) ) + + + + + + + * E E i t 3 E e D e m n ( )( ) 2 2 2 2 1 = + . . cc n x t ( ) * 3. E E = 1 n 3 3 3 n For scalar fields perturbation displacement is: ( ) + + i 3 E E E e D m n p 3 e m ( )( ) ( )( ) ( ) 3 3 m n p + + = 2 x t x t x t a ) ( D ) ( ) ( D 0 3 = , , 3 m n p m n p

Induced polarization both for electromagnetic fields in terms of displacement and susceptibility ( )( ) ) ( )( ) 3 3 = P Nex i q q 3 3 ( )( ( )( ) ( ) ( ) ( ) 3 3 = ; , , P E E E 0 i q ijkl q m n p j m k n l p = = , , 1 j k l , , 3 m n p ( )( ) ( ) ( ) ( ) 3 = ; , , , d E E E 0 ijkl q m n p j m k n l p jkl Where d is the degeneracy factor

( ) 3 General expression for * 1 a Ne d m ( ) ( )( ijkl ) = 2 2 D i 3 = ; , , , 3 ( ) ( ) 0 a a ( ) ( ) q m n p 3 D D D D 0 i q j m k n l p ( ) 1 ( ) 3 Express in terms of 3 a m dN e 2 ( )( ijkl ) ( )( ) ( )( ) 1 Ne ( )( ) ( )( ) ( )( ) 3 1 1 1 1 = 1 = ; , , , 3 0 ( ) a q m n p i q j m k n l p m D 3 4 0 a

8.3.1. Third Harmonic Generation (THG) signal contained by terms that oscillate at 3 The THG susceptibility for one component of the electric field (scalar case) is 4 a Ne d m 1 ( )( ) 3 3 ; , , = 3 ( ) ( ) 3 3 3 D D 0 ( ) = 2 2 D i 0 3 a m 3 ( )( ) ( )( ) 1 1 = 3 , 3 N e 0 3 4

8.3.2. Two-Photon Absorption (TPA) and Intensity-Dependent Refractive Index = = = , , , If 2 1 3 4 1 a Ne d m ( )( ) 3 ; , , = 3 ( ) ( ) 2 3 2 D D 0 3 a m 2 2 ( )( ) ( )( ) 1 1 = 3 N e 0 3 4 3 a m 2 2 2 ( )( ) ( )( ) ( )( ) R ( )( ) I ( )( ) R ( )( ) I 3 1 1 1 1 1 = + 2 3 N e 0 i 3 4 ( )( ) R ( )( ) I 3 3 = + , i

Define effective refractive index ( ) = ( )( ) ( )( ) ( ) ( ) 2 1 3 + 3 E ( )( ) 2 3 = 1 3 + 2 n E 0 = 2 1, n = + n n n I 0 2 Find expression for ?2 ( ) 3 3 n ( ) + = + 2 2 1 ( ) n n I = n 0 2 2 2 4 c 0 0 3 ( )( ) R ( )( ) I 3 3 = + i 2 4 n n c 0 ' 0 ( ) ( ) = + '' . in 2 2

A plane wave undergoes an intensity-dependent loss of factor e^-alpha. b) Energy level diagram for single photon absorption (left) and two-photon absorption (right).

8.3.3. Four Wave Mixing a Ne d m 1 ( )( ) 3 = ; , , 3 ( ) ( ) ( D ) ( D ) 1 2 3 3 D D 0 1 2 3 = + + = + + k k k k 1 2 3 1 2 3 k-vector conservation, phase matching condition

a) Generic four-wave mixing process. b) Momentum conservation. ( ) ( ) 3 + + k k k r i = * 6 . P A A A e 1 2 3 0 1 2 3 NL

8.3.4. Phase Conjugation via Degenerate (all are the same) Four-Wave Mixing Phase conjugation via degenerated four wave mixing: field E4 emerges as the phase conjugate of E3, i.e. E4=E3*.

8.3.5. Stimulated Raman Scattering (SRS). 8.3.5.1. Spontaneous Raman Scattering Population of the excited vibrational levels obeys the Maxwell-Boltzmann distribution E 1 z ( ) P E = k T , e B Spontaneous Raman scattering: -a) Stokes shift -b) anti-Stokes shift

( ) h k T 10 P E = 1 e 1 B ( ) P E 0 Thus, the Stokes component is typically orders of magnitude stronger !

= p E . the molecular optical polarizability, ( ) = + i t i t * , x t A e A e ( ) v v = P t N P v v v ( ) t ( ) = + . x t Harmonic vibration occurs spontaneously due to Brownian motion ( ) ( ) 0 v v x x = + . N x t E t = 0 0 v v x x v = 0 v ( ) ( ) ( ) ( ) t = + P t P t P L NL = + i t . . cc P t N A e L 0 L L ( ) t ( ) ( ) + i t i t = + + + * . . cc . . . cc P N A A e A A e L v L v NL v L v L x v 0

8.3.5.2. Stimulated Raman Scattering To derive expression for SRS susceptibility, solve equation of motion for vibrational mode of resonant frequency and damping ??,? ( ) 2 ( ) ( ) m 2 d x t dt dx t F t ( ) + + = v v dt 2 , x t v v 1 2 1 2 1 2 dW dx ( ) ( ) p t E t ( ) = = W F t t v ( ) t 1 2 d dx = 2 ( ) t E = = 2 . E . dW Fdx t v t v 0 d dx ( ) t 1 2 d dx = + 2 ( ) ( ) ( ) x E = F E E 0 v t v 0 v 0 ( ) ( ) = = + + i t i t . . cc E t A e A e S L L S ( ) ( ) ( ) ( ) + + + + + * * E A A A A L L S S L L S S

( ) m ( ) ( ) ( ) = + Induced polarization for SRS F P N x E 1 ( ) = 0 v x x v v 2 2 0 i v ( ) 2 ( ) ( ) ( ) ( ) E + = + * A A N E N x 1 m dx d 0 v = L S L S x v 2 0 2 i ( ) ( ) v v 0 = + . P P L NL 2 ( ) 2 ( ) + * A A N m dx d ( ) ( ) ( ) = + + L S L S * P A A NL L L L L 2 2 i nonlinear contribution v v 0 ( )( ) ( ) ( ) ( ) R = + + * * d A A A A 0 L S L S L L L L ( ) ) ( ) 2 2 i t + i t * * A A A e A e L S S N m dx d ( ) t L S L L = P ( ) ( NL 2 2 2 i v 0 v L S L S

2 1 N m dx dx ( )( ) S = t ( ) ( ) 2 12 + 2 i 0 v 0 v L S L S 2 1 N m dx dx ( )( ) = + S = t i ( ) L v S 12 0 v L S 0 By resonantly enhancing the vibration mode the Stokes component can be amplified significantly; in practice, this amplification can be many orders of magnitude higher than for the spontaneous Raman.

2 susceptibility associated with the anti-Stokes component 1 N m d dx )( ) ( AS = t i ( ) ( ) 12 = = 0 v L S 0 L AS v L S ( )( ) * S = t Strong attenuation a) Raman susceptibility at Stokes frequency; indicates amplification. b) Raman susceptibility at anti-Stokes frequency; indicates absorption.

8.3.6. Coherent Anti-Stokes Raman Scattering (CARS) and Coherent Stokes Raman Scattering (CSRS) Coherent Anti-Stokes Raman Scattering (CARS) and Coherent Stokes Raman Scattering (CSRS) are also established methods for amplifying Raman scattering. These techniques involve two laser frequencies for excitation ( )( CARS ) 3 = 2 ; , , 1 2 1 1 2 A ( )( CSRS ) 3 = 2 ; , , 2 1 2 2 1 S CARS is a powerful method currently used in microscopy we will hear about it during student presentations.

8.4. Solving the Nonlinear Wave Equation. 8.4.1. Nonlinear Helmholtz Equation follow the standard procedure of eliminating B and H from the equations ( ) B r , t ( ) = E r , t t ( ) D r , t ( ( ( ) ) ) ( ) = + H r j r , , t t = 0 0. t = = D r = , t j B r , 0. t ( ) ( ( E r ) ) , ( ) t ( t ) = = = + + ( ) ( ) D r E r P r , , , t t t 2 D r , t = 2 E r E E , t ( ) + = E r 0 , 0, t 0 2 ( ) r ( ) + E r P r P r , , , t t 2 E . 0 L NL ( ) ( ) + P , t t 0 r NL

( ) ( 2 ) 2 2 E r P r , , t t 2 n c ( ) = NL 2 E r , , t Nonlinear wave equation after approximation, term negligible ( ) = D E 0 2 2 dt t 1 ( ) ( ) ( ) + = 2 2 2 E r E r P r , , , , 0 NL 0

Propagation of the Sum Frequency Field SFG nonlinear polarization has the form ( )( NL P ) ( )( ) ( ) ( ) 2 2 = + = r r r ; ; , ; , , , E t E t 3 1 2 1 2 0 3 1 2 1 2 ( ( ( ) ) ) ( ) ( ) ( ) r ( ) i t z = + r r , . . cc E t A e 1 1 1 1 2 ( ) ( ) ( ) ( ) ( ) ( ) i t z ( ) 2 = + r r , . . cc E t A e + = 2 2 2 r r r r , , , , 2 2 3 2 E t n E t E t E t 2 2 3 3 3 0 3 0 1 2 c ( ) i t z = + r , . . cc E t A e 3 3 3 3 E E = = 0 3 3 x y ( ) = n 1 c 1 1 2 d dt ( ) ( ) ( ) 3 i t z z = 2 r , E t A e 3 3 3 2 2 dA dz d dz ( ) ( ) 3 i t z z = + 2 2 . 3 A i A e 3 3 3 3 3 2

( ) 2 ( ) dz 2 d A t dt dA z ( ) ( ) 2 + i z + = 3 3 2 2 . i A A e 1 2 3 3 0 3 0 1 2 1A and 2 A , do not change with z (do not deplete). Simplifying approximation, ( ) 2 ( ) 2 d A t dt B k = dA t i kz + = 3 3 dt 2 i Be 3 ( ) 2 = 2 A A 3 1 2 + , 1 2 3 amplitude is slowly varying i 2 ( ) 2 ( ) ( ) 2 d A z dz dA t dz i kz = dA t e dz 3 3 . 3 2 3 3

L iB ( ) i kz = A L e dz The intensity of SFG field is 3 2 3 0 1 ( ) z ( ) 2 i kL 1 iB e = I A z = = = 3 3 2 B L 0n i k 2 3 ( ) 2 2 kL kL = 2 sinc , 2sin i kL iBe 2 2 2 8 = 2 3 2 k 3 kL = ,2 ,... Net output power can vanish at 2 kL sin i kL iBLe 2 = 2 kL 2 2 3 ( ) i kL iB kL = sinc . e 2 2 2 3 a) The SFG output field has a phase that depends on the position where the conversion took place. b) The overall SFG intensity oscillates with respect to <eq>

Parametric Processes: Phase Matching = 0 k = + 3 1 2 ( ) n ( ) ( ) = + = + 1 c 1 , 3 2 n n 3 1 2 3 2 c c ( ) ( ) + + n n ( ) = 1 1 2 2 . n 3 1 2 Abnormal dispersion. Normal dispersion

Normal dispersion curves for a positive uniaxial crystal. Negative ( ) Positive n n n n e o e o ( ) ( ) ( ) ( ) ( ) ( ) Type I = + = + n n n n n n 3 3 1 1 2 2 3 3 1 1 2 2 e o o o e o ( ) ( ) ( ) ( ) ( ) ( ) Type II = + = + n n n n n n 3 3 1 1 2 2 3 3 1 1 2 2 e e o o o e Table 8-1.

2 2 1 cos n sin = + = 0 k phase matching can be achieved by angle tuning, that is, selecting the angle that ensures ( ) 2 2 2 n n 0 e Type I ( ) ( ) ( ) = + , , on n n 3 3 1 1 2 2 Type II ( ) ( ) ( ) = + , n n n 3 3 1 1 2 2 o o Type II phase matching by angle tuning in a positive uniaxial crystal: o-ordinary wave, e-extraordinary wave, c-optical axis.

Electro-Optic Effect The electro-optic effect is the charge in optical properties of a material due to an applied electric field that oscillates at much lower frequencies than the optical frequency. Electro-Optic Tensor a) Linear interaction with a birefringent crystal. b) Electro-optic (nonlinear) interaction. p is the induced dipole, E(omega) is the optical field, and E(0) is the static field. These sketches should be interpreted as 3D representations.

The induced polarization for the Pockels effect ( ) 0 ; ,0 i ijk P = ( )( ) ( ) ( ) 0 . 2 ; ,0 E E j k Kerr effect ( ) ( )( ijkl ) ( ) ( ) ( ) 3 ; ,0,0 = ; ,0,0 , P E E E 0 i j k l Due to the electro-optic effect, an optical field can suffer voltage-dependent polarization and phase changes.

Pockles effect 1 ( )( ) ( ) ( ) 0 . 2 ; ,0 = P jj ijk r E E i ii j k 0 2 ijk = = 0 ijk r ijk 2 2 n n ii jj i j = ijk r r jik Change in the rank of the tensor, from 3 to 2 r r r r r r r r r r r r = = Electro-optic tensor can be represented by a 3x6 matrix. This tensor contraction, allowed by the permutation symmetry, reduces the number of independent elements from 27 = to 3 6 18 = . = = = = = 11 1 k k 23 22 2 k k 33 3 k k = = r r 12 21 6 k k k 13 31 5 4. k r k k k 23 32 k k

Electro-Optic Effect in Uniaxial Crystals Use KDP (KH2PO4, or potassium dihydrogen phosphate) as a specific example of uniaxial crystal The KDP refractive index tensor is (in the normal coordinate system of interest) 0 0 0 r 0 0 0 0 r 0 0 0 0 0 0 n 0 0 n n o = 0 0 n o = r 0 e 41 0 0 41 0 63 r assume that the voltage is applied only along z, such that only 63 r is relevant.

( ) ( ) 0 = = = + 2 2 2 2 D E P D n E n n r E E 0n 0 0 i i i i j ijk j k ( ) , E ij j electric displacement can be expressed for each component as (i=x, j=y, k=z) ( ) ( ) ( ) 0 . z e z D n E = ( ) ( ) ( ) ( ) 0 = 2 4 0 D n E n r E E 0 0 63 x o x o y z = 2 4 D n E n r E E 0 0 63 y o y o x z 2 ( ) 0 2 4 0 0 . n n n r E 63 n o o z ( ) 0 = 4 2 n r E 0 63 0 o z o 2 0 e Electro-optic effect in a KDP crystal. For E(0) parallel to z, the normal axes rotate by 45 degrees around the z-axis.