2. Background Material

Transcription

1 S. Blair Sptmbr 3, Background Matrial Th rst of this cours dals with th gnration, modulation, propagation, and ction of optical radiation. As such, bic background in lctromagntics and optics nds to b covrd in ordr to proprty dscrib lrs, modulators, imaging systms, and ctors..1. Complx Function Formalism Th complx function formalism is usd oftn in lctrical nginring, and will b brifly rviwd hr. In lctromagntics, all solutions to th wav quation can b dcomposd into a st of fundamntal ignmods that ar sinusoidal. Th radial frquncy ω is dfind ω = πν, whr ν is th ral frquncy. A ral sinusoidal function at) can b writtn in trms of complx functions at) = A cos ωt φ a ) = A R [ jωtφ ) a = A [ jωtφ ) a jωtφ ) a Mor gnral = R [A jωt whr A = A jφ a complx amplitud In ral form, th drivativ of at) is Gnral = 1 A R[ jωtφ a ) c.c. = 1 [Ajωt c.c. dat) = ω A sin ωt φ a ). Now, lt s s what happns to th complx functions Gnral [ [ dat) da = R jωt at)bt) = 1 da jωt da jωt = R [jωa jωt = 1 = R [jω A jωt [jωajωt jωa jωt = A [ jω jωtφ ) a jω jωtφ ) a = R [jω A cos ωt φ a ) = A j sin ωt φ a )) [jω j) sin ωt φ a) = ω A sin ωt φ a ) = ω A sin ωt φ a ) Sam answr! Now what happns whn you multiply two functions at) and bt)? In ral form, at)bt) = cos ωt φ a ) cos ωt φ b ) = [cos ωt φ a φ b ) cos φ a φ b )

2 S. Blair Sptmbr 3, In complx form, at)bt) = R [A jωt R [B jωt works! but at)bt) = R [A jωt B jωt = R [AB jωt = R [ jωtjφ a jφ b = cos ωt φ a φ b ) Dos not work! Moral: Can t us complx form to multiply! Gnral at)bt) = 1 4 [Ajωt A jωt ) B jωt B jωt ) = 1 4 [ABjωt AB A B A B jωt = A B [ 4 iωtφ a φ ) b jφ a φ b ) jφ a φ b ) iωtφ a φ ) b = A B [cos ωt φ a φ b ) cos φ a φ b ) As long you kp th c.c. trm, you can t go wrong. B carful, th book will simply us at) = A jωt shorthand. You must always intrprt this to man at) = R [A jωt. Eithr mthod will work; th choic is ntirly up to you. I will us th book s mthod at all tims xcpt during discussions of nonlinar optics whr products of filds appar in th matrial polarization. Tim avrag of product at)bt) = 1 T T 0 [cos ωt φ a φ b ) cos φ a φ b ) thn T = multipl of oscillation priod π/ω at)bt) = cos φ a φ b ) = 1 R [AB.. Enrgy and powr in lctromagntics curl quations H = I D E = B H = magntic fild, I = currnt dnsity D = lctric displacmnt E = lctric fild B = magntic displacmnt constitutiv rlations [ D = ɛo E P B = µ o H M ) P = lctric polarization M = magntic polarization divrgnc quations W can us ths quations to driv s ) E H n da = [ D = ρf B = 0 V ρ f = fr charg [ ) E I ɛ o E E µo ) H H E P µ M oh This quation tlls us that th total powr flowing into a volum of surfac ara s lft-hand sid) quals th powr xpndd on moving charg within that volum E I) plus th rat

3 S. Blair Sptmbr 3, ) ɛ of incr in th stord lctromagntic nrgy in vacuum of volum V o E E µo H H )) plus th avrag powr pr unit volum xpndd by th fild on th lctric dipols within th volum V E P ) plus th powr xpndd on th magntic dipols µ oh M ). Now, lt s look at th avrag powr pr unit volum xpndd by th fild on th lctric dipols in th mdium: powr volum = E P For a scalar monochromatic plan wav, w writ th lctric fild and matrial polarization Et) = Et) = R [ E 0 jωt Pt) = Pt) = R [ P 0 jωt. Th filds E and P ar rlatd by th lctric suscptibility χ so that P 0 = ɛ o χ E 0. With ths sumptions, powr volum [ P0 = R [E 0 jωt R jωt = R [E 0 jωt R [jωɛ o χ E 0 jωt = ω o ɛ o E 0 R [jχ. Sinc χ is in gnral complx, consisting of rfractiv ral) and ractiv imaginary) componnts, χ = χ jχ, w arriv at our final rsult powr volum = ωɛ oχ E 0. This rsult tlls us that χ mdiats powr xchang btwn th lctromagntic fild and th mdium. Thrfor, χ must b rsponsibl for absorption/gain of th mdium, and must play an ssntial rol in lr thory..3. Wav Propagation in Isotropic Mdia E = µ H H = ɛ E B = µ o H M ) µ H D = ɛ o E P ɛ E To simplify our rsults, lt s sum that th lctric fild is ˆx-polarizd and propagating E x 0, H y 0 along th ẑ-dirction. Ths sumptions giv us th conditions E y = H x = 0 What E z = H z = 0 w hav lft is th following wav quation E x z = µε E x and a similar dual quation for H y ).

4 S. Blair Sptmbr 3, xt) π ω t x z) z π k This quation h plan-wav solutions of th form E ± x = E ± 0 jωt kz), whr E is forward-going, E is backward-going, and k = ω µɛ. For th fild amplitud E x constant, w must hav that ωt kz = constant. From this rlationship, w can valuat th ph vlocity v dz = ω k. Givn that th frquncy ν = ω/π and th wavlngth λ = π/k, th vlocity can b writtn v = ω = 1 k µɛ. In vacuum, c = 1 µo ɛ o = m/s. Th rfractiv indx n = c = ɛ/ɛ v o. Th magntic fild amplitud is givn by H y = E x η, whr η = µ/ɛ. In practic, optical radiation, vn from a lr, is not compltly monochromatic, and consists of a rang of frquncis. Th instantanous intrfrnc of ths diffrnt frquncis crats a wavpackt, which travls at a vlocity diffrnt than th ph vlocity. Th group vlocity is givn by v g = dω dk, and is th vlocity at which modulation, or information, propagats. Sinc th rfractiv indx of matrials varis with frquncy i.. th mdium is disprsiv) such that k = ωnω)/c, w can valuat dk dω = nω) ω c c n ω).

5 S. Blair Sptmbr 3, In gnral, for a monochromatic plan wav solution, th total lctromagntic fild is writtn E x z,t) = E 0 jωt k r) E0 jωt k r) H y z,t) = 1 [ E 0 jωt k r) E0 jωt k r) η Hr, k is th dirction of wav propagation and r is th position vctor. Sinc th optical wavfront is dfind th surfac of constant ph, th wavfronts can b dscribd by th condition that k r=const. Th dot product producs th projction of r along th dirction of k, so that this projction is th sam for all points lying in a plan prpndicular to k. Now w ar in a position to valuat th powr flow. Not that th avrag powr pr unit ara is calld th intnsity, which h units of W/m. Th intnsity can b valuatd I = E H = E x H y for our monochromatic plan-wav xampl) = 1 R [E E 0H0 0 E 0 = η η, whr th minus sign indicats th opposit dirction of powr flow. W can also valuat th stord nrgy pr unit volum lctric nrgy volum = ɛ E E magntic nrgy = µ H volum H. Considring only th forward-going wav, th avrag nrgy pr unit volum is thn avrag nrgy volum = ɛ E E µ H H = ε E x ) µ E 0 E 0 = ε 4 µ 4η E 0 = ε H y ) Th intnsity of th forward-going wav is thn Intnsity = E 0 η = c avrag nrgy. volum Rmmbr that η = µ/ɛ. In fr-spac, η = µ o /ɛ o = 377Ω.