Physics 221 Lecture 41 Nonlinear Absorption and Refraction

Transcription

1 Physics 221 Lectue 41 Nonlinea Absoption and Refaction Refeences Meye-Aendt, pp Boyd, Nonlinea Optics, 1.4 Yaiv, Optical Waves in Cystals, p. 22 (Table of cystal symmeties) 1. Intoductoy Remaks. Duing the last lectue, we eviewed the basic featues of the second-ode non-linea susceptibility of a centosymmetic medium. We found that the effects of a quadatic nonlinea tem in the foce on an anhamonic oscillato was to intoduce sum- and diffeence-fequencies into the oscillations thus poducing such phenomena as secondhamonic geneation, sum- and diffeence-fequency mixing, and optical ectification. In this lectue, we extend this idea to a thid-ode nonlineaity a ich field which includes thidhamonic geneation, phase conjugation, and nonlinea absoption and efaction. These ae the ingedients of all-optical switching schemes fo communication and computing. A. Ou pesciption fo finding the nonlinea susceptibilities in a medium with a non-zeo second-ode susceptibility was to model it as an anhamonic oscillato with a quadatic tem in the potential. This gave ise to oscillatoy steady-state solutions of the equation of motion involving the fundamental, second hamonic and dc tems (optical ectification). By elating the polaization pe unit volume to the displacement of the oscillato, we wee able to find the susceptibilities: P (1) (ω ) = χ (1) (ω) E(ω ) = Nex (1) (ω) χ (1) (ω) = Nex(1) (ω) E(ω ) = N(e 2 m) D(ω ) The second-ode tems wee found analogously, by solving the second-ode analog to the definition of the fist-ode polaization. The susceptibility tuned out to be. ( P 2 ) ( 2ω ) = Nex (2) (2ω) χ (2) (2ω) = a Ne m 2 1 D(2ω ) D 2 (ω ) B. Oigin and size of the nonlinea polaization esponse. The chief Ansatz on which we shall ely in descibing nonlinea optics is the powe-seies expansion in the electic field. If we wite the polaization the dipole moment pe unit volume as an expansion in the electic field, and conside only the fist thee tems. P (t) = χ (1) E (t) +χ (2) E (t) [ ] 2 + χ ( ) E [ (t)] +... Notice that the fom of the thid-ode tem is popotional to the cube of the electic field. As we shall see, this intoduces an intensity-dependent tem into the absoption coefficient and into the index of efaction. These polaizations can, in tun, be elated to a simila expansion in the dipole moment pe unit volume: P (t) = P (1) (t) + P (2) (t) + P () (t) +... = en x (1) (t) en x (2) (t) en x () (t)... C. The technique we used to aive at the values fo the nonlinea susceptibility in a noncentosymmetic medium was simila to the odinay Rayleigh-Schödinge petubation

2 expansion. We wote the solution as a powe seies in a paamete λ which measues the stength of the inteaction: x (t) = λx (1) (t) + λ 2 x (2) (t) +λ x () (t) If we now substitute this expession fo x(t) into ou equation of motion, and collect tems in like powes of the expansion paamete λ. [Question: Why do we not have a tem in λ 0 x (0) in ou petubation expansion?] This led us a model-dependent expession fo the k-th ode susceptibilities which have the fom: [ ] k χ (k) = Nex (k) (t) E (t) 2. Nonlinea susceptibility of a classical anhamonic oscillato. In ode to bette undestand the application of ou definitions, we now apply them to the paticula case of the classical anhamonic oscillato. The values of the nonlinea susceptibilities which we calculate ae at least qualitatively ealistic, but it would be necessay to eplace them by detailed quantum-mechanical calculations if we wanted them to eplicate detailed measuements. A. We take as ou next example a thee-dimensional medium which is not centosymmetic, and in which the estoing foce on an electon subjected to a magnetic field is 2 F estoing = mω o + mb ( ) with b the paamete descibing the stength of the (thid-ode) nonlinea inteactions of the oscillato with the applied field. 1. Note and discuss the significance of the signs. As befoe, we also add a fictional (damping) foce with the damping facto 2γ chosen to make the FWHM of the atomic absoption pofile exactly equal to 2γ. 2. The estoing potential may be found by integation of the foce. In one dimension, it would then include a quatic tem in the potential: U (x) = Fest dx = [ ω 2 o x + bx ] dx = ω o 2 x bx 4 4. The applied (diving) field is chosen to have thee components, since that is the most geneal possibility in the case of thid-ode nonlinea inteactions. E (t) = n= E(ω n )e iω nt The equation of motion is found fom Newton's 2nd Law of Motion, in the following way: F = m = k 2mγ + mb( ) ee (t) Afte dividing though by m and eaanging tems, we find the following equation of motion: 2

3 +ω 2 o 2γ b( ) = ee (t) B. As befoe, we conside a petubation expansion in powes of a coupling constant λ which measues the stength of the diving field and which, it tuns out, dops out (o equivalently, is set equal to 1) at the end of the computation. λ (1) (t) + λ 2 (2) (t) +λ () (t) ω 2 o (t) = λ (1) (t) + λ 2 (2) (t) +λ () (t) +... ( λ (1) (t) + λ 2 (2) (t) + λ () (t)) [ ( )] +b ( λ (1) (t) + λ 2 (2) (t) + λ () (t) ) λ (1) (t) +λ 2 (2) (t) + λ () (t) ( ) = λ e m ( λ (1) (t) + λ 2 (2) (t) + λ () (t))+2γ λ (1) (t) + λ 2 (2) (t) + λ () (t) Collecting tems in like powes of λ, we get seveal equations of motion just as in the noncentosymmetic case: λ (1) (t) ω 2 o ( λ (1) (t) ) 2λ λ 2 (2) (t) ω 2 o λ () (t) ω 2 o ( λ 2 (2) (t) ) + 2γ λ 2 ( λ () (t) ) + b λ E (t) fist ode in λ. (1) (t) = λ e m ( (2) (t)) = 0 second ode in λ. [( (1) (t)) ( λ (1) (t) )] λ (1) (t) thid ode in λ. ( ) + 2γ λ E (t) ( () (t)) = 0 The fist-ode equation has pecisely the same stuctue as the fist-ode equation fo the non-centosymmetic case. It must theefoe have the same solution: (1) (t) = n (1) (ω n )e iω nt If we now substitute this solution into the fist of ou thee equations, we find that it can wok only if the tem satisfies (1) (ω n ) = ee(ω n) md(ω n ), D(ω n ) ω o 2 ω n 2 2iγω n C. We go about finding the fist-ode susceptibility in a manne analogous to what we used to get solutions in the pevious case, although things ae a bit moe complicated this time because thee is a susceptibility coesponding to each of the possible input fequencies. P (1) (ω ) = χ (1) (ω) E(ω ) = Nex (1) (ω) χ (1) (ω) = Nex(1) (ω) E(ω ) = N(e 2 m) D(ω ) The next-highe-ode equation is an inteesting case. It has no steady-state oscillatoy solution; it is, in fact, the diffeential equation fo the damped, fee-unning oscillato. It is the homogeneous solution of the fist-ode equation.

4 The thid-ode solutions ae the next "level" fo a centosymmetic medium. Since thee ae thee waves which may be "mixed" by the thid-ode susceptibility we guess that the solutions must have the fom () (t)= () (ω q ) e iω qt, ω q =ω m +ω n + ωp q Diffeentiating as indicated on the left-hand-side of the thid-ode equation we obtain the equation: ω 2 2 [ q + ω o 2iγω q ] () (ω q ) D(ω q ) ( ) (ω q ) = b m e E(ω m ) m D(ω m ) Recalling the definition of the thid-ode polaization and its elation to the thid-ode nonlinea susceptibility, we find fo the solutions to the polaization equation: ijkl (ω q) = Ne e m χ () bδjkδil D(ω q )D(ω m )D(ω n )D(ω p ) whee the Konecke deltas ae defined by δ ij = 1, i = j, δ ij = 0, i j.. This solution is not unique, but it can be made self-consistent. Next time we shall have to deal with the question of the symmeties of this object. Fo now, we content ouselves with estimating the size of the thid-ode susceptibility and pointing out a featue which is distinctive compaed to the phenomena aising fom the second-ode susceptibility. III. Applications: Thee ae many diffeent phenomena which aise because of the thid-ode susceptibility. These include fou-wave mixing; thid-hamonic geneation, nonlinea efaction and nonlinea absoption; and phase conjugation. A. Appoximate size of the thid-ode susceptibility. Assume that the linea and nonlinea contibutions to the estoing foce ae compaable when the size of the displacement vecto is compaable to a lattice spacing d in a solid, o the atomic o molecula adius in a gas. This leads to the appoximate equality mω o 2 d mbd b ω o 2 d 2 Let us futhe assume that N = 1 d. If we then substitute easonable values fo these numbes [ into ou expession fo the thid-ode susceptibility, we find that χ () = esu. ω o ad s -1 e esu m g d 10-8 cm 4

5 B. Degeneate fou-wave mixing. One of the most widely used thid-ode lase spectoscopy techniques is DFWM. [It is called that not because of the moals of its pactitiones but because it mixes thee waves all at the same fequency; the wave obseved is also at the same fequency.] The expeimental geomety in a typical situation is shown in the diagam below: Inteaction Zone Nonlinea Medium Thus the thid-ode susceptibility in question is:χ ()(ω = ω ω + ω). This is a qualitatively diffeent phenomenon than we could obseve in second-ode non-lineaities; thee, the only way to obseve a wave at the fequency of the input wave was via the fistode susceptibility. In a centosymmetic medium, on the othe hand, it is possible to have both fist- and thid-ode contibutions to the wave aising at the input fequency. Thought question: How can you sepaate out these two contibutions fom diffeent physical mechanisms? 5