Understanding Dipole Radiation's Impact on Optical Properties of Materials - Prof. David H, Study notes of Chemistry

Download Understanding Dipole Radiation's Impact on Optical Properties of Materials - Prof. David H and more Study notes Chemistry in PDF only on Docsity! Optical Properties of Materials – OSE 5312 Summer 2002 Monday, June 24, 2002 Effect of dipole radiation field from Lorentz oscillators. Another way of looking at the optical properties of materials is to consider the filed re-radiated by the induced dipoles of the classical Lorentz oscillators. This dipole radiation field will interfere with the incident field in such a way as to produce absorption or refraction. Due to time constraints in this semester, we will only have time to look very briefly at this subject. Dipole radiation: We make use of the vector potential, A. We know that 0 B  and since there is a general identity   0 V  , which applies for any arbitrary vector field, V. This implies that there can exist a vector potential, A, such that: AB   . (Actually there are a set of possible A’s. We use the “Coulomb gauge” or “transverse gauge” where 0 A  . The wave equation for A in the Coulomb gauge is:     J t A A    02 2 0 2  , where J  is the transverse component of the current. This has a general solution:        rr rdtrJ c trA c rr     3 2 0 ),( 4 1 ),(  , which we can use to find the dipole radiation field. We assume a “point” dipole, placed at the origin: ),( trp  r  r̂  00 ˆ)()cos(),( prtptrp           00 00 ˆ)()sin( ˆ)()sin( prtpJ prtp t P J       Now  0p̂ is the component of the unit vector that is perpendicular to r: 0p̂ r  r̂ )ˆˆ(ˆ 0prr  Error: Reference source not found    00 ˆˆˆˆ prrp  , the magnitude of which is cos(). A polar plot of cos  so shown below: Error: Reference source not found  0̂p r  Hence, the effect of the dipole is to impress a small “shadow” on the transmitted field behind it. If we integrate the outward-flowing component of <Sabs>t over all angles, the result is the total absorbed power, Pabs, i.e.   drSP tabsabs ˆ  , where the integral is over all solid angle . This yields,    IIabs ErEpP )(Imˆˆ2 * 0    Where,       )( )(ˆˆ)(ˆˆ )( 0 22 0 0      D EEp m e i EEp m e r I I      , Hence       2 20 * 20* )( ˆˆ )( 1 Im ˆˆ )(Im     D E m Epe D E m Epe Er I II          . Now we may define a molecular absorption cross section, abs, that relates the absorbed power to the incident irradiance, by: tIabsabs SP  . Hence    IIIabs ErEpS )(Imˆˆ2 * 0     . Since 202 1 II EncS  , we obtain:   2 2 0 2 )( ˆˆ 2     Dm Ep Iabs   Where the averaging is now over all angles between the dipole coordinates and the applied field. For a medium containing randomly oriented molecules, (i.e. isotropic), we have already seen that   3 1ˆˆ 2 0   IEp . Since Pabs is an absorbed power per molecule, we can find the net absorbed power per unit volume as N Pabs = NabsSI. But the power absorbed per unit volume for a plane wave is just –dSI/dz. Hence: Nz I z II IIabs I eSeSzS SSN dz dS      )0()0()( Hence, the absorption coefficient is given by: Nabs )()(   . Scattered Power: Recall that SSscatt BES   0 1  , where,       00 0 2 ˆˆˆ..exp 4 prrcctrkip r k ES       and      00 0 0 2 ˆˆ..exp 4 prcctrkip cr pk BS        . This simplifies down to:     .ˆcos 42 ˆcos 42 2 2 2 0 0 32 4 2 2 0 2 2 0 4 r r p c r r pck Sscatt           The scattered power is then given by:   2 03 0 4 0 2 2 0 12 ˆsin p c SrdrdP tscattscatt          - Hence scattering power is proportional to 4 or -4 – Rayleigh scattering. We can similarly define a scattering cross section: tIscattscatt SP    2 2 0 242 0 44 )( ˆˆ 6    D Ep mc e scatt    Refractive Index: Unlike absorption, we cannot calculate the refractive index from the field of a single dipole. We must look at the field due to an ensemble of induced dipoles. To calculate the refractive index (real and imaginary), we look at the field produced by a sheet of dipoles all induced by the same incident plane wave.