AIR FORC RSARCH LABORATORY The xtinctin Theem as an xample f Reseach Vistas in Mathematical Optics Mach Richad A. Albanese Infmatin Opeatins and Applied Mathematics Human ffectiveness Diectate Bks City-Base TX
Outline Classical xtinctin They An Updated View f the xtinctin Theem Vistas
Classical xtinctin Theem A hamnic wave mving with the speed f light utside a dielectic medium is eplaced within the medium by a hamnic wave f the same fequency but mving at a slwe velcity given as n/c whee n is the efactive index. 3
4 Classical xtinctin Theem R dv R c R t culcul t N P R R c R t P culcul inc j i jl jl jl l l inc j l jl inc j ' ' / ', '(, ' / ( ' ' ' can be eithe within utside the vlume
5 Classical xtinctin Theem R dv R c R t culcul t inc ' ' / ', '(, ' Using the abve integal equatin, and the thee dimensinal Geen functin, the extinctin theem is fund. ik G ds n G n G k S inc exp(, ( (, (, ( ( 4
6 Classical xtinctin Theem Studying the abve integal equatin, it appeas that the incident fields is annihilated within the medium by fields n the suface f the medium. Jacksn: The mathematical analysis shws that f hamnic time dependence the cancellatin f the incident vacuum fields inside the medium is cause by diples n the bunday f the medium. ik G ds n G n G k S inc exp(, ( (, (, ( ( 4
An Updated View f the xtinctin Theem ttal inc diples unifm density f diples ikr diples ( z dx ' dy ' dz ' ttal ( z ' e... 4 ik 3 ˆ ˆ ˆ ˆ k x n x n 3 R R R dx ' dy ' dz ' inc ttal ttal 4 ik 3 ˆ ˆ ˆ ˆ k x n x n 3 R R R ( z' e ik R... 7
An Updated View f the xtinctin Theem dx ' dy ' dz ' ( z ' e ik R... inc ttal ttal 4 ik 3 ˆ ˆ ˆ ˆ k x n x n 3 R R R This integal equatin suggests that the extinctin f the incident field in the dielectic is a functin f all diples in the mateial. 8
An Updated View f the xtinctin Theem What abut liquids? What abut mateials that ae hetegeneus? What abut mateials with iegula sufaces? What abut quantum effects? Sme f these questins have been patially answeed. The liteatue is substantive but nt vewhelming. Thee is n time-dmain extinctin theem. 9
Numeical Findings. time dmain dispesive espnse at distance.4 time dmain n-dispesin espnse at distance.5.3...5. -.5 -. -. -. -.5 -.3 -..5.5.5 3 3.5 x -8 -.4.5.5.5 3 3.5 x -8 Tansient in mateial medium fm squae wave pulse Field in mateial medium neglecting mlecula espnses = pue sinusid & ze integals each peid
Numeical Findings-Dispesive time dmain dispesive espnse at distance time dmain dispesive espnse at distance time dmain dispesive espnse at distance time dmain dispesive espnse at distance.8.8.8.8.6.6.6.6.4.4.4.4.... -. -. -. -. -.4 -.4 -.4 -.4 -.6 -.6 -.6 -.6 -.8 -.8 -.8 -.8 -..4.6.8..4.6.8 x -8 -.5.5.5 x -8 -.5.5.5 x -8 -.5.5.5 x -8 z =.5 metes z =.5 metes z =. metes z =.5 metes
Dispesive vs. Nn-Dispesive.5 diffeence between dispesive and n-dispesin effect.3 diffeence between dispesive and n-dispesin effect.3 diffeence between dispesive and n-dispesin effect.5 diffeence between dispesive and n-dispesin effect..5.......5.5 -.5 -. -. -.5 -. -. -. -. -.5 -. -.3 -.3 -.5 -.5 -.4.5.5.5 3 3.5.5.5.5 3 3.5 x -8 x -8 -.4 -..5.5.5 3 3.5 x -8.5.5.5 3 3.5 x -8 z =.5 metes z =.5 metes z =. metes z =.3 metes
Maxwell s quatins B t D H J J D P B P ap b t D P Faaday's law Ampee s law Gauss s law Nnexistence f mnples Cnstitutive elatin ( t, x is the electic field B( t, x is the magnetic inducatin field D( t, x is the electic displacement field H ( t, x is the magnetic intensity magnetic field P( t, x is the chage density
Specializing t One Dimensin x H y z t H y x Px t t dt P ap b t Cnstitutive elatin
Fequency Dmain Vesin b ( i a x z ( i a x Helmhltz Fequency dmain equatin used f cmputatin Nea suface Laplace and Fuie integals failed thus fa in u hands
Fequency Dmain Cmputatins b b i a i= iw a iw a Such that Is the Phase fact Is the attenuatin fact
Fequency Dmain Cmputatins ab ab ab a a a ab ab ab a a a Ntice: n attenuatin when n -P cupling; als wave speed deceased by -P cupling.
Wk in the Time Dmain 3 3 a ( a b x x 3 x x tz tz t t By P. Petpuls x x x b x a t z cm t z cm a t Fast wave Slw wave assciated with P z z x t t Refeences: Kevkian and Cle (996 Bende and Oszag(999 Sulem and Sulem (999
Recalling the multiple scales Theefe we have Multiple Scales t z cm t t z zz cm t tt b a z zz cm a t tt and 3 3 3 b a tz cm tt zz cm a tt b a t z cm t z cm a t
Multiple Scales 3 3 3 b a tz cm tt zz cm a tt z t z t e e kz i f ( t z / v (, (, cmplex cnjugate such that n n t z f f k b 3 v v v cm a n k k i f 8 af af n 4 ka i 4 k v v (
Multiple Scales nd Ode quatin b a t z cm t z cm a t Geneal Wave Slutin Advectin-Diffusin equatin g( t z/ v / b v v m z c a t a a z / / c v c v m m b b cm cm t z a z a a D t z z
Slutin f Advectin Diffusin quatin f Single Pint Suce Advectin-Diffusin quatin in tems f such that D t z z ( z, t M A Dt e t z 4Dt
vaiable vaiable vaiable Slutin f Advectin Diffusin quatin f Single Pint Suce 5 5 5 45 45 45 4 4 4 signal 35 3 5 35 3 5 35 3 5 5 5 5 5 5 5.5..5..5.3.35.4.45.5 time.5..5..5.3.35.4.45.5 time.5..5..5.3.35.4.45.5 time integal..5..5..5.5..5..5.3.35.4.45.5.5..5..5.3.35.4.45.5.5..5..5.3.35.4.45.5 z =.5 metes z =.5 metes z =. metes
Summay Can study f tansients be a step twad btaining a time dmain extinctin theem? Much wk Refining hamnic case Diffeent shapes and mateials Quantum Time dmain Multiple scales and electmagnetic/ptic they 4
View f Math Algebaic tplgy Algebaic gemety Diffeential gemety Diffeential tplgy Diffeential equatins Gemety Algeba Tplgy Analysis Lgic Set they Applied mathematics usually elates t sme subject matte such as electmagnetic technlgy, quantum technlgy, fluid dynamics, aedynamics, elasticity and the like. In my mind the best applied mathematics extends mathematical cultue while als cntibuting t a specific chemical, bilgical physics cultue. Gd applied mathematics emplys existing mathematics and uses it iginally in the chemical, bilgical physics pblem. 5