Physics 218, Spring February 2004

Transcription

1 Physis 8 Spring February 004 Today in Physis 8: dispersion Motion of bound eletrons in matter and the frequeny dependene of the dieletri onstant Dispersion relations Ordinary and anomalous dispersion The world s largest-selling (nearly 30 million opies!) illustration of dispersion A relatively aurate depition at that 8 February 004 Physis 8 Spring 004 model So far we have treating the propagation of light in matter as if the permittivity perability and ondutivity do not depend upon the frequeny of light This is not generally true Most materials that transmit light exhibit a notieable variation of refrative index with wavelength This is alled dispersion after the ation of the best example of the phenonon the glass prism To find out how onsider the effet of a plane wave of light on a simple model of a bound eletron in an atom or moleule 8 February 004 Physis 8 Spring 004 model (ontinued) x Eletron i t E E0e ω q Bond z K -Zq Nuleus γ Its bond holds the eletron in so equilibrium position; we an represent the bond with a restoring fore (like a spring) and a resistane hanism for losing energy and montum to its surroundings (like a shok absorber) The nuleus an be onsidered stationary to good approximation Restoring fore: Kx Resisting fore: γ dx dt 8 February 004 Physis 8 Spring () University of Rohester

2 Physis 8 Spring February 004 model (ontinued) The net fore in the x diretion is dx d x F qe γ Kx a dt dt so the eletron s equation of motion is d x dx q iωt + γ + ω0x E0e dt dt where ω 0 Km e This is the equation of motion of a damped driven harmoni osillator whih you have solved in PHY 35 and (maybe) in PHY 7 8 February 004 Physis 8 Spring model (ontinued) i t With a trial solution x x0 e ω we have d x dx + γ + ω0 x E() t dt dt iωt iωt iωt q iωt ω xe 0 iωγxe 0 + ω0xe 0 Ee 0 or q E 0 x0 ω ω iωγ ( 0 ) If the total dipole mont of the dium ontaining this atom is zero before the field displaes the eletron the dipole mont afterward is: 8 February 004 Physis 8 Spring model (ontinued) q Ee iωt 0 p qx ω ω iωγ ( 0 ) Beause of the omplex denominator p and x are out of phase with the eletri field Now this is not the only eletron in the dium nor is this the only kind of bond around Suppose there are N moleules per unit volu and M different kinds of bonds in all different diretions and onsider for the mont ust the th kind 8 February 004 Physis 8 Spring () University of Rohester

3 Physis 8 Spring February 004 Dispersion relations If there are eletrons per moleule that are in this situation with damping onstant γ and natural frequeny ω then the ontribution of the th kind of bond to the dipole mont per unit volu P is iωt fe 0 e P Nfp m e ω ω iγ ω ( ) where E 0 is the omponent of E in the bond diretion The net dipole mont per unit volu from all of the bonds is the vetor sum M P E() t m e ω ω iγ ω ( ) 8 February 004 Physis 8 Spring Dispersion relations (ontinued) The are ommonly known as osillator strengths beause they indiate the response of a eah kind of bond (damped osillator in this simple model) to the applied eletri field To alulate them is generally a ob for quantum hanis Nevertheless we have one onstraint on the quantum result Sine atoms with Z protons only o with Z eletrons that s what the have to add up to: M Z This is the simplest of a family of equations relating the that are known as sum rules 8 February 004 Physis 8 Spring Dispersion relations (ontinued) The vetor P the total dipole mont per unit volu indued by the applied eletri field is not new to all of us In PHY 7 we alled it the polarization and noted that it was related to the applied eletri field by ( χε n MK ) P χee e 0 E i S In linear dia the eletri suseptibility χ e is a salar (by definition) and is related to the dieletri onstant ε by ( r 0 ( e) 0 in MKS; gs rmks) ε + 4 πχe ε ε ε + χ ε ε ε This gives us an expression for ε in terms of the osillator strengths natural frequenies and damping onstants: 8 February 004 Physis 8 Spring () University of Rohester 3

4 Physis 8 Spring February 004 Dispersion relations (ontinued) M 4π ε + ω ω iγ ω ( ) Dispersion relation The damping term whih represents the hanisms by whih eletrons an rid themselves of energy and montum piked up from the applied field makes the dieletri onstant omplex The imaginary part of the dieletri onstant leads to exponential attenuation of eletromagneti waves propagating in the dium: sine ω k µε it endows an imaginary part to k 8 February 004 Physis 8 Spring Dispersion in a dilute gas Example: dieletri onstant of an ideal monatomi gas For standard temperature and pressure (300 K atm 6-0 dyne m ) P 9-3 N 0 moleules m kt so that reall esu 9 0 q m 8 e gm ( ) 4π se 5 - For visible light λ 05 µ m ω π λ 4 0 rad se 4π ω 00 8 February 004 Physis 8 Spring 004 Dispersion in a dilute gas (ontinued) Thus taking µ M 4 k ω ω π ε + ω ω iγ ω ( ) If ω is far from any of the natural frequenies (resonanes) ω then the seond term under the square root is of order /00 as we have ust seen and we an use the binomial theorem again to approximate + x + x : M f k ω π + or m e ω ) iγ ω 8 February 004 Physis 8 Spring 004 () University of Rohester 4

5 Physis 8 Spring February 004 Dispersion in a dilute gas (ontinued) ( ) ( ) ( ) ω ) + iγ ω ω ) M ω π ω ω + iγ ω k + m e ω ω iγ ω ω ω + iγ ω M ω π + f k+ iκ ; M ω π k + and 8 February 004 Physis 8 Spring Dispersion in a dilute gas (ontinued) M π ω γω κ m e It is ustomary to define from this a omplex refrative index for gases: M π ω ) n k + ω m e M 4π ω γω Absorption α κ m e oeffiient n n+ α ω 8 February 004 Physis 8 Spring Dispersion in a dilute gas (ontinued) As light propagates through the gas the eletri field amplitude is exponentially attenuated as e κ z so the intensity of light will derease exponentially as it propagates aording to I( z) I0 e α z Caveat: all the formulas in this example pertain to the propagation of light through gases Solids and liquids are muh denser so we ould not have used the binomial approximation on them Nor do they apply so well at lower frequenies Nor do they work at all near resonanes 8 February 004 Physis 8 Spring () University of Rohester 5

6 Physis 8 Spring February 004 Ordinary and anomalous dispersion Far from any resonanes the index of refration rises gently as frequeny inreases This is what one might all ordinary dispersion Most ommon glasses exhibit ordinary dispersion at visible wavelengths This inrease however turns out to be due to those distant resonanes As frequeny passes through a resonane the index dereases sharply then resus its gentle inrease The sudden derease is alled anomalous dispersion The absorption oeffiient is small far from resonane but peaks sharply on resonane (See example on next page) 8 February 004 Physis 8 Spring Ordinary and anomalous dispersion (ontinued) The resonanes are frequenies at whih the eletrons an get rid of energy and montum effiiently suh as the frequenies orresponding to transitions between atomi energy levels Espeially near a resonane n α ω0 the index and absorption oeffiient look like they must be related by integration or differentiation Indeed they turn out to be related by a set of integral transforms known as the Krars-Kronig relations 8 February 004 Physis 8 Spring α n ω () University of Rohester 6