8. Propagation in Nonlinear Media

Transcription

1 8. Propagation in Nonlinear Media

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

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

4 8... 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 evb

5 Simple case where electric field polarized along x, magnetic field lies on y E E x,, H, H y, f em e EvB ˆ ˆ x y yˆ e E zb x yexb z d x t dx t e x t Ex z () t By() t dt dt m dy t y t dt dt d y t d z t dz t e z t t x () t By () t dt dt m 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: ee e m m e z i i xby. m r x i i z By

6 x z D b ee x D b m ee D b m x. D i e b i B m y. Equations obtained from method in 5. Ignore magnetic field, we get linear response Express magnetic field in terms of electric field Order of magnitude relation between x and z displacements keib z eex eex E x me mc x By ic Solve for when electric field and associated irradiance where x and z are comparable mc Ex I E x xz e V, W m m

7 z x E Using e E b x D mc e mc x D b e mc E x x D. D D eex m D b ee m we have: it it x x z E e E e Electron oscillates at on z axis, DC term is called optical rectification Ex cos t.

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

9 8.. Second Order Susceptibility d x t dx t E t x t ax t dt dt m Want solution of form where is the nonlinear perturbation to linear solution. x x x d x t dx t ee t x t d x t dx t t dt dt m x dt dt a x t a x t x t a x t. Simplify by neglecting and x t x t x t a x t D x a x vx a m D D e E E v,

10 Illuminating the nonlinear crystal with two monochromatic fields of different frequencies x * * E E E E E e E E a v a e E ' E ' m D D ' D ' m DD D d '. av b ab

11 E E All possible frequencies: E ve

12 e e x t a g t cc.. g t cc.. x a f f f f m D m it it E e E EEe E E g t DD D D D DD Fourier f EE DD Transform * it EE e * EE cc.. * D DD E E it it E e E EEe g t f EE DD DD D D D DD * * it EE EE e cc.. * D D D Equation 8.6 indicates that, as the result of the second-order nonlinear interaction, the resulting field has components that oscillate at frequencies (second harmonic of ), (second harmonic of ), + (sum frequency), - (difference frequency) and (optical rectification terms).

13 Nonlinear susceptibility from induced polarization * * i j i j; i, j i j i j i j,,, P E E P Nex i j Ne x ;, i j i j i i ane m g t g t cc.. Importantly, vanishes in centrosymmetric media r r P r r E r E r r E r E r r. P P r Ner Ne P r r. Fulfilled simultaneously only if P r So second order nonlinear processes require noncentrosymmetric media

14 8... Second Harmonic Generation (SHG) ane m D D ;, ane ;,. m D D Nonlinear susceptibility a s function of linear chi: Linear: a) SHG: pumping the chi() material at omega yields both the fundamental frequency (omega) and its second harmonic (omega). b) Description in terms of virtual energy levels. Ne. m D Nonlinear: a m ;,. Ne

15 8... Optical Rectification (OR) ;, m D D D ane a m. Ne OR: a DC polarization is created in a chi() material.

16 8... Sum Frequency Generation (SFG) ;, m D D D ane a m Ne.

17 8..4. Difference Frequency Generation (DFG) ;, m D D D ane a m Ne.

18 8..5. Optical Parametric Generation (OPG) The time reverse process of SFG

19 8.. Third Order Susceptibility Anharmonic oscillator ee t x t x t x t a x t m. Solve using perturbation theory, solution of form ee t x t x t x t m. x t x t x t a x t x t First term vanishes, approximate a x x a x x t x t x t a x t,

20 x t x e E md, * * E E * E E E E E. So, int e Ee n x t cc.. m D n n For scalar fields perturbation displacement is: e EEEe m n p x t x t x t a m D D D i m n p mn,, p m n p

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

22 General expression for * ane ;,,, ijkl q m n p D a a d m DiqDjmDk ndlp i Express in terms of am ijkl q; m, n, p 4 i q j m k n l p, dn e a Ne m D a

23 8... Third Harmonic Generation (THG) signal contained by terms that oscillate at ω The THG susceptibility for one component of the electric field (scalar case) is 4 ;,, d m D D ane am 4, Ne D i

24 8... Two Photon Absorption (TPA) and Intensity Dependent Refractive Index If,,, 4 ;,, d m D D ane am 4 Ne i, R I am 4 R I ir I Ne

25 Define effective refractive index n n nn n I E E, Find expression for n n I ( ) n 4n n c R ii 4n c in ' ''.

26 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).

27 8... Four Wave Mixing ;,, ane d m D D D D k kk k k vector conservation, phase matching condition

28 a) Generic four wave mixing process. b) Momentum conservation. PNL * i 6 AA A e. k k k r

29 8..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 E, i.e. E4=E*.

30 8..5. Stimulated Raman Scattering (SRS) Spontaneous Raman Scattering Population of the excited vibrational levels obeys the Maxwell Boltzmann distribution P E e z E kt B, Spontaneous Raman scattering: a) Stokes shift b) anti Stokes shift

31 P E P E e h kt B Thus, the Stokes component is typically orders of magnitude stronger!

32 p E. the molecular optical polarizability, Pt t x t. x v x v v N P N xv t E t x v xv. v * x t Ae A e i t ivt v v v Harmonic vibration occurs spontaneously due to Brownian motion, L NL ilt P t P t P t PL t N ALe cc.. P t N A A e cc.. A A e cc... * i L v t i L v t NL v L v L x v

33 Stimulated Raman Scattering To derive expression for SRS susceptibility, solve equation of motion for vibrational mode of resonant frequency and damping, d xv t dxv t F t v xv t, dt dt m dw Fdx v. W t p t E t E d dx v t t xv E t t F t dw dx v d dx v t d F E E dx v v E t. ilt is t L S.. * * E t Ae Ae cc E A A A A L L S S L L S S

34 x v F m d mdx i v v v A A * L S L S v i P N xv E xv NEN xv E x P L P NL. v Induced polarization for SRS v * N d AL AS L S NL L L L L mdx v v i * P A A P NL t R * * d AL AS L S AL L AL L * is t * i LS t AL AS ALe AL e N d m dx v v L S i L S nonlinear contribution

35 S t N dx m dx v v L S i L S L v S S t N dx i m dx v L S 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.

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

37 8..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 A ;,, CSRS S CARS is a powerful method currently used in microscopy we will hear about it during student presentations.

38 8.4. Solving the Nonlinear Wave Equation Nonlinear Helmholtz Equation Br, t Er, t t Dr, t Hr, t jr, t t Dr, t Br, t. follow the standard procedure of eliminating B and H from the equations j., t, t Dr Dr Er,, t Er, t Pr, t t Er, t P r, t P r, t r L NL Er, tp r, t NL, t Er E E E.

39 , t, t Er P r,, c dt t n NL Ert Nonlinear wave equation after approximation, D E term negligible Er, Er, P r,, NL

40 Propagation of the Sum Frequency Field SFG nonlinear polarization has the form P r; ;, ;, E r, t E r, t NL r r r r r r E t A e cc i tz,.. E t A e cc i tz,..,.. i t z E t A e cc n c Er, tn E r, t E r, t E r, t c E x E y d Er, t A e dt z i t z da dz dz d A i A e z i t z.

41 i d A t da z dt i AA e. dz Simplifying approximation, A and A, do not change with z (do not deplete). d A t da t i dt B AA k, dt Be ikz z amplitude is slowly varying d A z da t. dz dz i ikz da t e dz

42 ib ikz A L e dz ikl ib e ik ikl sin kl ib e k ikl sin kl ibl e kl L ib e ikl sinc kl. The intensity of SFG field is Iz A z BL 8 sinc kl, n Net output power can vanish at kl,,... 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>

43 Parametric Processes: Phase Matching k n n n c c c n, n n. Normal dispersion Abnormal dispersion.

44 Normal dispersion curves for a positive uniaxial crystal. Negative n e o n Positive ne no Type I n n n n n n e o o o e o Type II n n n n n n e e o Table 8-. o o e

45 cos sin phase matching can be achieved by angle tuning, that is, selecting the angle that ensures n n n k e Type I o,, n n n Type II o, n n n o Type II phase matching by angle tuning in a positive uniaxial crystal: o ordinary wave, e extraordinary wave, c optical axis.

46 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() is the static field. These sketches should be interpreted as D representations.

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

48 Pockles effect P ;, r E E. r i ii jj ijk j k ijk r ijk r jik ijk ii jj n n ijk i j Change in the rank of the tensor, from to r r r k k r k k r r k k r r r k k 6k r r r k k 5k. k k 4 k r r r Electro-optic tensor can be represented by a x6 matrix. This tensor contraction, allowed by the permutation symmetry, reduces the number of independent elements from 7 to 6 8.

49 Electro Optic Effect in Uniaxial Crystals Use KDP (KH PO 4, or potassium dihydrogen phosphate) as a specific example of uniaxial crystal r r4 r4 r 6 The KDP refractive index tensor is (in the normal coordinate system of interest) no n no n e assume that the voltage is applied only along z, such that only r 6 is relevant.

50 D n EP D n E n n r E E 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). D n E n r E E 4 x o x o 6 y z D n E n r E E D 4 y o y o 6 x z n E z e z Electro optic effect in a KDP crystal. For E() parallel to z, the normal axes rotate by 45 degrees around the z axis. 4 o o 6 z 4 no r6ez no n n r E. n e

51 express ij in a reference system rotated about the z-axis by an arbitrary angle, say R R ' cos sin n cos sin o D sin cos D n sin cos o n Dsin Dcos, 4 Dcos n Dsin D n r E o o o 6 z n D o ' no D. n e

52 Electro Optic Modulators (EOMs) defining the normal refractive indices n ' ' ij ij n' x n' n' y n' n' n D x o D no no n n r E n n n r E n' n. o o 6 z ' y o o 6 z z e z E z phase retardation n' xn' y L c no r6ez L, V. d half-wave voltage V d n o r6 L V V. V.

53 a) Longitudinal modulators (electrodes are made of transparent material), d=l. b) Transverse modulator, d<l.

54 a) Electro optic (EO) crystal operating as phase modulator: incident polarization parallel to the new normal axis. b) Amplitude modulation: input polarization parallel to original (i.e. when V=) normal axis; P is a polarizer with its axis parallel to x. KDP longitudinal modulator reads inokl E' V Ee e Ee in k L o e kn o r6v i V.

55 (x,y ) is rotated by 45 with respect to (x,y) Ex ' V R45W VR45 Ey ' Ex ' isin i e Ix' Ex' W V i e V sin, R 45. voltage is modulated sinusoidally, at frequencies,, much lower than the optical frequency, t V V t V sin t V sin t.

56 ' phase and intensity modulations x t V V sint I t V t x ' sin sin. V phase modulators also modulate the frequency of the light ' t V t dx ' dt V t. V I x ' t cos sin t V V sin sin t V V sin t. V Liquid crystal spatial light modulator: a) phase mode; b) amplitude mode.

57 Acousto Optic Effect The acousto-optic effect is the change in optical properties of a material due to the presence of an acoustic wave Elasto Optic Tensor the medium acts as a (phase) grating, which is capable of diffracting the light Light wave (wavevector k, frequency omega, speed c) interacts with a travelling grating induced by a sound wave (wavevector script k, frequency Omega, speed v). k, v

58 induced polarization in the normal coordinate system P n n P S E i i j ijkl kl j jkl,, strain tensor S kl uk xl u x l k. Kerr electro-optic effect, induced polarization P n n S E E E, i i j ijkl k l j ijkl P S S E E. ijkl kl ijkl k l. electro-optic tensor, due to symmetry, the elasto-optic tensor can also be used using the following (Voigt) contraction P P, i, j, k, l,,; m, n,,...,6 ijkl mn ij, kl ; m, n ij, kl ; m, n ij, kl ; m, n ij, kl ; m 6, n 6 ij, kl ; m 5, n 5 ij, kl ; m 4, n 4.

59 Photoelastic effect: the sound wave induces strain, which in turn modulates the refractive index. a) Longitudinal sound wave; b) Transverse (shear) sound wave. Lambda is the sound wavelength, with Omega the frequency and v the propagation speed.

60 Acousto Optic Effect in Isotropic Media Let us investigate in more detail the acousto-optic effect in isotropic media elasto-optic tensor for isotropic media p mn p ' p ' p ' p ' p ' p ' p ' p ' p ' p ' p ' p ' p ' p ' p '

61 S S S kl expressions for the polarization 4 P n S p E p E p E x x 6 y 5 z 4 P n S p E p E p E y 6 x y 4 z 4 P n S p E p E p E z 5 x 4 y z. Dielectric displacement n E E, D P p4 p5 p6 P 4 n SpEx 4 n S p E x P P y 4 n S p E z z. y '. n p ' n p ' ns n p dielectric tensor medium becomes uniaxial, i.e. xx yy zz

62 S u z z ikae i t kz plane acoustic wave of amplitude A t kz. i uz Ae a) Brogg regime; b) Raman Nath regime.

63 Bragg Diffraction Regime wave equation for the total field (incident plus diffracted) reads itkr r r, r, r,, n E r, t P r, t Er, t c t t P, t BAe E r, t E t E t E t Fourier transform r, r, r, r, ikr BAe E n E F E F r,. r r r, r,. E, n E, i t e E t E,, E F e d iqr q r r V q k k A A k k. Bragg condition,

64 . k k The Brogg condition: a) k, k and script k are, respectively, the incident, diffracted and acoustic wavevectors. b) Triangle that illustrates geometrically k=k+script k. sin sin k n

65 Raman Nath Diffraction Regime a) Bragg diffraction (see solution for k). b) Raman Nath diffraction when sound wave is curved (multiple solutions for k). scattering potential in the (x,z) coordinates,, i x i z z F x z BAe e

66 diffracted field is the Fourier transform of F q x Eqx, qz, qz. q is the scattering wavevector, q k k. k k z z k k x x. if the sound beam curvature satisfies k k x x m, Where m=,,,, the diffracted beam has multiple solutions k k m. z z

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