Nonlinear Notes 1 Optical Effect we experience typically in nature are linear Absorption A = cl Raman scattering Ic Linearly proportional: Light interacts with matter in proportion to its concentration 1961 Franken at Univ of Michican Ruby Laser focused on a crystal 1st nonlinear observation Transitted light consisted of both wavelenths both 649.3 and 347.15 nm 2nd Harmonic optical effect Today Since advent of powerful lasers many nonlinear effects used. Sum and frequency difference mixing Kerr electroptical effect third Harmoic generation Four wave mixing Stimulate Raman Scattering Two photon absorption Nonlinear effects normally only appear in the presence of large electric or magnetic fields. Why? Light interacts with matter inducing oscillating dipole moments in the medium. These can be modulated by the vibrations and the oscillating dipoles may emit at frequencies that are the same or are modulated by the vibrations of the molecules Thes depns on the polarizability of molecules at least to a first approximation, since to generate the oscillating dipoles the electron could needed to slosh back and forth. This is generally a harmonic type oscillation which leads to an oscillating field with similar characteristics and the incoming EM. However at high levels of the incoming irradiance, the dipoles no longer responds linearly with the field and the dipoles are anharmonic That is if the response is linear the dipole follow Hookes Law and the potential energy curve is parabolic But at high enough driving forces the dipole restoring force no longer increases in a linear way with the displacement and Hookes law is violated the potential energy curve strays from a perfect parabola! How it strays depends on the molecular structure of the material itself. Generally speaking there are two broad categories for molecular structures Those with a center of symmetry isotropic Those without a center of symmetry anisotropic For the centrosymmetric case, the dipole restoring potential energy curve may look like For the noncentrosymmetric case, the dipole resoring potential energy curve may look like In a concentrated or condensed medium especially each dipole is SUSCEPTIBLE to the local fields of its neighbors as well as the external filed of the light wave Remember in Raman scattering, the polarizability of the molecule was important and one wrote that the Polarization = P = E However now this is no longer the case when the dipole of the neighbors and nearest neighbors influence each other especially with an intense driving force. Intense EM so here then more terms are needed in the Polarization expression. P = 1 E + 2 E:E + 3 E:E:E + ……… where n symbolizes the susceptibility tensor 1 = the polarizability 2 = the hyperpolarizability in microscopic systems, the 3 is referred to as the 3rd order susceptibility. If the field intensity is low, then the electric field is purely linear and the Polarization induced is linear and the material exhibits all of the ordinary optical properties. If the field strength is higher than the induced polarization becomes nonlinear then the Polarization is distorted from the original EM field see figs Fourier analysis of the Polarizaton waves provide us a way to see what optical effects are associated wth the optical properties of the materials Consider the nonlinear response of the noncentrosymmetric media to a high powered laser light source at 1 Fourier analysis of the Polarization reveals that it is three superimposed waveforms 1) sinusoidal wave with twice the frequency of the electric field 2) a negative dc component 3) a sinusoidal wave with the same frequency as the electric field These are 2nd harmonic generation, optical rectification, and normal refraction If two monochromatic waves with frequencies 1 and 2 are now incident on a noncentrosymmetric crystal The resulting polarization wave induced in the material has now the following components 1) component at the fundamental 1 2) component at the fundamental 2 3) 2nd harmonic at 21 4) 2nd harmonic at 22 5) the combination sum 1 +2 6) the combination difference 1 - 2 7) a dc component Generally speaking, the second order nonlinear effect involves wave-mixing processes in which waves of differing frequencies merge to produce others. How does this happen? Light waves are not supposed to exchange energy with each other and they don't in vacuum, in the medium The nonlinear polarization wave itself couples the energy from one wave to another. This effect is called optical parametric interaction. How large is this effect? Depends on the relative phase velocities of the polarization wave as the light waves move through the medium. Most of what we will be looking at are plane waves which is a wave whose phase is the same at a given instant at all points in each plane perpendicular to some direction of propagation E(z,t) = E0 exp(-i(wt ± kz) k is the wave vector = /c kz is the phase factor and is constant for a given t and z no matter what the values of x and y are. Since the electric field initiates a dipole moment which then reradiates The Polarization wave of oscillating dipoles creates the secondary wave, but the secondary induced wave is produced proportionally to the acceleration of the elctrons the lags it by 90o . The sum of this with the original wave produces a refracted wave with a slightly different phase factor that slightly lags the original EM. Also the 2nd Harmonic wave has a frequency that is twice the original EM. Overlap of these beams can add intensity to the beams but in general the index of refraction is different at different frequencies and so the wave propagation speed is different. To allow more overlap of the waves, one needs the medium to have the same index of refraction at the 2nd Harmonic and the original frequency and to get intense 2nd harmonic generation one needs the material to be birefrigent Birefringent the index of refraction depends on the direction and polarization of the propagating light. Thus if a polarized light passes through at just the correct angle, the phase velocites of the induced polarization and the 2nd harmonic wave can be made equal. Other considerations for SHG Materials Nonlinear optical effects can take place in all phases of matter. Historically inorganic crystals have proven to be a very convenient medium. Crystals are grouped into seven general types: Triclinic, Monoclinic, Orthorhombic, Trigonal, Tetragonal, Hexagonal, Cubic Cubic is the most symmetric, triclinic is the least. Additionally, the fit into one of the 32 point groups Based on Symmetry Considerations it is IMPOSSIBLE for centrosymmetric (isotropic) materials to have any nonlinear phenomenon associated with the even order susceptibilities. Only the odd order susceptibilities will contribute. So second harmonic generation is not possible for in a centrosymmetric medium or nonlinear phenomenon associated with the hyperpolarizability are not possible in a centrosymmetric medium. For noncentrosymmetric media both even and odd order susceptibility terms may contribute. A consideration of the points groups shows that 11 of them have controsymmetric symmetry, so only 21 of them could show the effect . Additional symmetry considerations reduce this to 18. Classic crystals KDP - potassium dihydrogen phosphate KD* P - deuterated KDP KTP KTiOPO4 pretty high 2 used to double cw light to get green laser pointers LiNbO3 has low threshold damage BANANNA BaNa2Nb5O15 BBO beta barium borate LBO Lithium triborate Organic NLOs Poly diacetylenes with high 2 Often we might be interested in nonlinear effects in gases and liquids. These are isotropic media since the molecules do not have a fixed position relative to the incoming light. Second order effects are impossible here so we must go to the 3rd order effects. How many susceptibility terms are there that could possibly contribute to the induced polarization? Turns out 1st order there are 9 terms 2nd order there are 27 terms 3rd order there are 81 terms How do these come about? Lets go back and consider the Polarization and rewrite it carefully Pa = b1 ab Eb + bg2 abgbgd 3 abgd EbEgEd + higher order The sums are over the coordinates in the system such as x, y, z. Writing out a few of these terms Writing the Polarization in the x direction from the contribution from the 1st order susceptibility gives Px = xxEx + xyEy + xzEz Py = yxEx + yyEy + yzEz Px = zxEx + zyEy + zzEz From this one can see how the 9 susceptibility terms contribute. It depends on the polarization direction of the induced Polarization and also the polarization of the incident electromagnetic field. Can be thought of as a 3 by 3 matrix of the susceptibility terms. What about the 2nd order susceptibility? Write out the polarization in the x direction. Px = 2 xxxExEx + 2 xxyExEy + 2 xxz ExEz + 2 xyxEyEx + 2 xyyEyEy + 2 xyz EyEz + 2 xzxEzEx + 2 xzyEzEy + 2 xzz EzEz The Px and Py terms would be written similarly. One can see that this would give a total of 27 terms 9 2nd order susceptibility terms for each direction of the Polarization. This then can be thought of as a 3 by 3 by 3 matix Or three of the 3 by 3 matrices of the first order susceptibility Continuing with this then the 3rd order susceptibility can be thought of as three 3 by 3 by 3 matrices for a total of 27 x 3 - 81 terms. Now when writing out the Polarization for the CARS it is written as P3 CARS(3, z) = D 3 abgd(-3, 11, -2) 12 E2 exp [i(2k1-k2)z] One sees that the subscripts on the 3rd order susceptibility correspond to the polarization directions of the induced Polarization wave at 3, the polarization of the incoming wave at electric field E1, the polarization of the 2nd incoming electric field at 1 and the polarization of the 2nd incoming electric field at E2 Note also that the sum of the Energies corresponding to -3, 1, 1 -2 where the negative on the means the energy is negative, must sum to be equal to zero. For the isotropic medium only a few of the third order susceptibility terms are nonzero. In particular these are 1111 = xxxx = yyyy = zzzz 1122 = xxyy = etc. 1212 = xyxy = etc. 1221 = xyyx = etc. 1111 = 1122 + 1212 + 1221 Thus different Combinations of the the 3rd order susceptibility can be accessed using different experimental geometries in CARS D3 CARS (polarization directions) Pcars E1 E1 E2 3rd order susceptibility term Now this is just the induced Polarization We really want the Electromagnetic field produced by the induced polarization Need to solve Maxwell's Eqns. to get the new electric field emitted at 3 due to the induced Polarization from the fields at 1E1E2 Come up with the nonlinear Raman gain eqn. d2 E/dz2 + 2ik3 dE/dz = -432 /c2 CARS E12 E2 exp (i kz) usually neglect the 2nd order term and assuming z = L and E(3) = 0 at z =0 This can be solved to give 2 (w3)/dz2 + 2ik3 dE(w3)/dz = - 432 / c2 cars Ew12 Ew22 exp ikz k = 2k1 - k2 - k3 k3 = n3 3 And the time averaged intensity of the antistokes wave at z can be determined from this Neglecting the 2nd order term and using I(w3) = n3c/(8) E(w3)2 one can get: I(w3) = 256 4 32 / (n12 n2 n3 c4 ) cars 2 I12 I2 L2 [(sin (kL/2)) /kL/2)]2 L is the coherence length or the interaction length