Dispersion Ref Feynman Vol-I, Ch-31

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1 Dispersin Ref Feynman Vl-I, Ch-31 n () = 1 + q N q /m i ( b/m) We have learned that the index f refractin is nt just a simple number, but a quantity that varies with the frequency f the light. (36) ћ Dielectric prperties f silicn between 1 and 5 ev. REF: kzt1rmxvkhijltte2tj3rper57ksex2yveuqegm7gygcibrnhxgyjnvl34_1vysce1dblsz_1ijeskkkk- M4MI6KhIJtET0wqrZeKRWRs0QJbuGBwqEgmKYONUOunhUhG9LfO9_1HhYMiSCexGMbYXDaXEU61ZjmphWkG&tb=u&sa=X&ved=2ahUKEwjt7ZXC8sHaAhWnr1QKHRW1DO4Q9C96BAgAEBs&biw=1091& bih=667&dpr=2.75#imgrc=pejytcsadku6sm: Let s discuss it under different circumstances. (We will use as a reference the graph abve crrespnding t silicn material, but the behavir f the index f refractin is mre r less generally valid): Lw frequency regime << Fr mst rdinary gasses (air, hydrgen, helium), their natural frequency crrespnds t ultravilet light. Visible light frequency are smaller than the ultravilet light. Hence, fr frequencies in the regime f visible light, ne can cnsider in expressin (36). We find then that the index f refractin n is apprximately cnstant.

2 It turns ut this is als valid fr mst transparent materials, like glass. Nrmal dispersin As increases, the denminatr in (36) decreases making the index f refractin t increase. That is, n increases with. This explains the higher index fr blue light than fr red light. (A prism bends mre the blue light than the red light) The fact that the index f refractin depends n frequency is called dispersin. Expressin (36) is called the dispersin equatin. Regime ~ The index f refractin displays a resnance behavir. In this regime, the damping factr b/m becmes imprtant. This regime is under full experimentatin these days, revealing interesting phenmena (slw dwn f light, electrmagnetically induced transparency, etc.). Regime > This ccurs if we shine x-ray radiatin n a piece f glass; the frequency f x-rays is greater than sme f the resnance frequencies f the atms in the glass. Similar situatin wuld happen n a gas f free electrns ( = 0). In the upper atmsphere electrns are liberated frm their atms by uv light cming frm the sun and they sit up there as free electrns. Free electrns d nt exert restring frce, hence (w = 0). Ntice hwever that in these cases expressin (36) (putting aside the cmplex number in the denminatr) gives an index f refractin smaller than 1! The speed f the waves in such media is faster than c! Can this be right? It is crrect. In spite the fact that it is said that yu cannt send signals any faster than the speed f light, it is nevertheless true that the index f refractin f materials at a particular frequency can be either greater r less than 1. This just means that the phase shift which is prduced by the scattered light (the sheet f charges in ur mdel) can be either psitive r negative. It can be shwn, hwever, that the speed at which yu can send a signal is nt determined by the index at ne frequency, but depends n what the index f refractin is at many frequencies. What the index tells us is the speed at which the ndes and crests f the wave travels.

3 The nde f a wave is nt a signal by itself. In a perfect wave (purely steady mdulatin) yu cannt really say when it starts ; s yu cannt use it fr a timing signal In rder t send a signal, yu have t change such a perfect wave smehw (make a ntch in it, make it a bit fatter here r thinner there,. That means (in the language f Furier) yu have t have mre than ne frequency in the signal wave. It can be shwn that the speed at which signals travel is nt dependent upn the index alne, but upn the way the index changes with frequency. (In sme apprximatins, such velcity is the grup velcity). Such calculatin indicates that the signal will nt be faster than the speed f light c, althugh the ndes d travel faster than the speed f light. Further clue n what happens at high frequencies can be btained frm the slutin f ur elementary mdel f the electrn respnse t the applied field, x ( ω) 2 q E 2 /m i ( b/m) At lw frequency the displacement is in phase with the field. At higher frequencies the displacement ges ppsite t the field. Hw des the charge happen t be ging in the ppsite directin? It certainly des nt start ff in the ppsite directin when the field is first turned n. When the mtin first starts there is a transient, which settles dwn after a while, and nly then is the phase f the scillatins f the charge ppsite t the driving field. And it is then that the phase f the transmitted field can appear t be advanced with respect t the surce wave. It is this advance in phase which is meant when we say that the phase velcity r velcity f the ndes is greater than c. The figure belw gives a schematic idea f hw the waves might lk fr a case where the wave is suddenly turned n (t make a signal)

4 (15) Absrptin Here we highlight that the index f refractin is a cmplex number. It therefre has the frm, n= n i n (37) where n and n are real numbers. Frm expressin (35) yu can verify that n turns ut t be a psitive number. Ging back t the expressin fr the field after passing a slab f thickness d and index f refractin n, we btained (see expressin (29) in Sectin 3.2 B1), ( - z/c - E E e iω t ) (38) where d ( n -1) c ( iω iω t - z/c - E e- E e ) ) (39)

5 Using n= n i n E e E e - -iω (-i n" )d/c -iω t - z/c - e E e ) ) -ω ( n" )d/c -iω t - z/c - e E e ) ) ω ( n" ) d/c - iω t - z/c -) ) E e e E e Damping Travelling Decreases with thickness d. (40) n (the cmplex part f the index f refractin) is respnsible fr the absrptin