Appendix A Light Absorption, Dispersion and Polarization

Transcription

1 73 Aendix A Light Absortion, Disersion nd Polriztion A. Electromgnetic Sectrum The electromgnetic sectrum (Figure A.) is divided into seven min domins rnged ccording to their wvelength λ. We hve λ ct c=ν where c is the seed of light, T the eriod nd ν the frequency. Another imortnt eqution is tht relting hoton energy E nd frequency ν: E hν where h is Plnck s constnt. A. Light Absortion The light bsortion of mterils is relted to the electronic nd tomic (or ionic) olriztions under the electromgnetic field (E nd B, see lso Chter ). Absortion cn be modelled using simle roch considering the movement of the rticles constituting the mtter (e.g. electrons, ions). Their movement cn be obtined from clssicl mechnics solving m r f _r k ŕ qe v ^B qe where r is the rticle osition, m its mss, q its chrge nd f _r nd k r the friction nd ttrction forces resectively. The electromgnetic force reduces in good roximtion to its electricl comonent (vb E). Let us tke the conventionl cdemic nottions for olriztion vector nd wve vector k: qr Glss: Mechnics nd Technology, Second Edition. Eric Le Bourhis. 04 Wiley-VCH Verlg GmbH & Co. KGA. Published 04 by Wiley-VCH Verlg GmbH & Co. KGA.

2 74 Aendix A Light Absortion, Disersion nd Polriztion Wvelength λ γ-rys Ultrviolet rys Infrred rys X-rys Visible light Rdr Rdio UV IR 0.0 nm 0 nm 0.4 µm 0.7 µm 00 µm cm 00 cm Figure A. Electromgnetic sectrum. E E 0 ex iωt ikr k ω c n then we get (Feynmnn, Leighton, nd Snds, 979; Bertin, Froux, nd Renult, 984; Pérez, Crles, nd Fleckinger, 997) q ε 0 m ω ε 0 ω 0 E: where ω 0 k =m nd γ = f/m. Considering tht the concentrtion of rticles is N, then olriztion P is P N ε 0 χe ε 0 n E where ε 0, χ nd n re the vcuum ermittivity, electric suscetibility nd refrctive index resectively. Refrctive index n is determined to be n N q ε 0 m ω 0 ω This reltion cn be generlized tking into ccount ll tyes of olriztion: n X i N i α i where subscrit i refers to olriztion tye, for exmle electronic, tomic, ionic, orienttion. Note tht for dense mterils (fluids, glsses nd crystls), the locl field E l is to be considered insted of the mcroscoic field E. Then we get X n n i N i α i 3 This lst reltion is clled the Lorenz Lorentz reltion. A metl is chrcterized by free electrons so tht ω 0 is zero nd n is comlex. In fct metls re not trnsrent but reflect light, nd this is used in mirrors nd low-e glzing: n N q ε 0 m ω

3 A.4 Light Polriztion 75 Likewise glss bsorbs UV rdition becuse of electronic olriztion. In such cse ω ω 0, the friction term γ is not negligible close to resonnce nd refrctive index becomes comlex number since n N q ε 0 m nd the glss is not trnsrent. A.3 Light Disersion In contrst, wy from the bsortion bnds (ω 6 ω 0 ) we hve n N q ε 0 m ω 0 ω The refrctive index increses with the frequency (decreses with the wvelength). Disersion eqution cn be roximted in the μm region (ω ω 0 )by n N q ε 0 m ω 0 ω =ω 0 N q ω =ω 0 ε 0 m ω 0 q N ε 0 mω λ 0 =λ 0 n A B=λ which is known s the emiricl Cuchy eqution. Disersion is defined s n F n C where n C nd n F re the refrctive indices t λ C = nm nd λ F = nm (H α nd H β hydrogen lines). Recirocl disersion ν is ν n D n F n C where n D is the refrctive index t λ D = nm (sodium D line). A.4 Light Polriztion Polrizers select light with olriztion long given direction (in fct within smll ngle erture round the olrizer xis). In rctice, olymeric mterils

4 76 Aendix A Light Absortion, Disersion nd Polriztion Figure A. Polriztion chnge from liner to elliticl fter rogtion nd hse shift β. re used (see lso Aendix G). Light olriztion chnges when refrctive indices vry with the direction. Consider the three rincil directions, nmely, nd 3, nd suose light rogtes long direction 3. Phse shift is written (see lso Chter ) β π λ δ π λ dn n where n, n re the rincil indices, λ the wvelength nd d the distnce rogted by the bem. Suose initilly, light is olrized linerly nd inclined by 45 to nd rincil xis (Figure A.). We write the olriztion vector s cos ωt cos ωt cos ωt which trnsforms fter rogting through the secimen into cos ωt cos ωt β For β = π/ circulrly olrized light emerges from the secimen. In the generl cse, light is elliticlly olrized (Figure A.). Next consider /4 wve lte (s used for Senrmont comenstor; Chter ) inclined by 45 from rincil directions nd of secimen, sy, rllel to initil liner olriztion (Figure A.3). The olriztion vector

5 A.4 Light Polriztion 77 λ/4 wve lte xis β/ Figure A.3 Polriztion chnge from elliticl to liner fter rogtion through λ/4 lte. is written in the /4 wve lte frme s ffiffiffi cos ωt cos ωt β cos cos ωt β ffiffiffi cos ωt cos ωt β sin sin ωt β which trnsforms fter crossing the λ/4 wve lte into cos cos ωt β cos cos ωt β sin sin ωt β π sin cos ωt β ffiffiffi cos β β cos ωt sin β Tht is, liner olrized light with olriztion vector forming n ngle of β/ with initil olriztion vector. Hence, n nlyser llows determintion of β/ fter rotting to drkness s done when using Senrmont comenstor.