AIR FORCE RESEARCH LABORATORY

Transcription

1 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

2 Outline Classical xtinctin They An Updated View f the xtinctin Theem Vistas

3 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 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 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 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

7 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

8 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

9 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

10 Numeical Findings. time dmain dispesive espnse at distance.4 time dmain n-dispesin espnse at distance x x -8 Tansient in mateial medium fm squae wave pulse Field in mateial medium neglecting mlecula espnses = pue sinusid & ze integals each peid

11 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 x x x x -8 z =.5 metes z =.5 metes z =. metes z =.5 metes

12 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 x -8 x x x -8 z =.5 metes z =.5 metes z =. metes z =.3 metes

13 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

14 Specializing t One Dimensin x H y z t H y x Px t t dt P ap b t Cnstitutive elatin

15 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

16 Fequency Dmain Cmputatins b b i a i= iw a iw a Such that Is the Phase fact Is the attenuatin fact

17 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.

18 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

19 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 b a tz cm tt zz cm a tt b a t z cm t z cm a t

20 Multiple Scales 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 (

21 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

22 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

23 vaiable vaiable vaiable Slutin f Advectin Diffusin quatin f Single Pint Suce signal time time time integal z =.5 metes z =.5 metes z =. metes

24 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

25 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