Evolution Operators and Boundary Conditions for Propagation and Reflection Methods

Transcription

1 voluto Operators ad for Propagato ad Reflecto Methods Davd Yevck Departmet of Physcs Uversty of Waterloo Physcs 5/3/9

2 Collaborators Frak Schmdt ZIB Tlma Frese ZIB Uversty of Waterloo] atem l-refae Nortel Networks Ia Betty Nortel Networks Chul Yu Quee s Uversty Physcs 5/3/9

3 Outle Fudametal quatos No-Local Improvg Accuracy Fast Reflecto Calculatos Physcs 5/3/9 3

4 Part I - Fudametal quatos Physcs 5/3/9 4

5 Physcs 5/3/9 5 Scalar Wave quato Scalar Moochromatc lectrc Feld we have ad Defg Ε y N Y X k r N y k Y k X y r k y referece

6 Forward Soluto Defe X Y N. For forwardtravellg waves e ω t tme-depedece k y We the have wth δ k y e δ y Physcs 5/3/9 6

7 Physcs 5/3/9 7 Modal Aalyss Modal Decomposto Appromate Forward Soluto wth y y y y k y y a y m m m m m m β m m y y e y m β

8 Physcs 5/3/9 8 Fresel Appromato Fresel Appromato Slowly-Varyg velope e y y δ y N Y X δ

9 Physcs 5/3/9 9 Wde-Agle Appromatos Taylor Seres paso Padé [] appromat: Padé [] appromat O / 4 / O

10 Square-Root Operator Recurso Recurso Relato f Thus f we have f / f Physcs 5/3/9

11 Cotued Fracto paso Iteratg the recurso relato yelds Note that we have employed f to termate the fracto yeldg a real epresso.... Physcs 5/3/9

12 Physcs 5/3/9 Padé Represetatos The Padé appromat ca be factored as I a partal fracto represetato s r s r s r cos s π π s r s r s r s cos s π π

13 Physcs 5/3/9 3 Fte Dfferece Method 4 / 4 / 3 δ δ δ δ δ O e e N Y X Applyg a [] Padé appromat yelds the Crak-Ncholso procedure

14 Physcs 5/3/9 4 O a oe-dmesoal trasverse grd Dscrete Represetato { }. represets ad for ay operator where 4 4 O O O D D k k D k k

15 Part II - Nolocal Boudary Codtos Physcs 5/3/9 5

16 Objectve To smulate o a fte dscrete computatoal grd the feld radated from a local source to a homogeeous semfte medum. Physcs 5/3/9 6

17 lectrorefracto Modulator Physcs 5/3/9 7

18 Stadard Physcs 5/3/9 8

19 Improved Physcs 5/3/9 9

20 Boudary Layers The appromate propagato operators troduced above are utary. To remove the outward propagatg electrc feld at the boudary we ca troduce absorbg or mpedace-matched boudary layers. N Physcs 5/3/9

21 Trasparet Boudares N Set ad to be cosstet wth purely outgog waves at the boudary. Local : N are computed from at the last propagato step. Nolocal : N are obtaed from prevous values of. Physcs 5/3/9

22 Impedace-Matched Layer For a o-equdstat grd X b X the goverg equato a homogeeous refractve de layer ear the boudary s k For cotuous Thus f k d y k b d b a k o spurous effects. we have e k a k Physcs 5/3/9

23 Impedace-Matched Layer Atteuato e k a k l l e b sθ l a l b r Z L a b θ / taθ L a Physcs 5/3/9 3

24 Appromate ad act Results Physcs 5/3/9 4

25 Cotuous Nolocal Boudary Physcs 5/3/9 5

26 Cotuous Nolocal Boudary Physcs 5/3/9 6

27 Gaussa Beam - Cotuous N.L. Physcs 5/3/9 7

28 Remag Power - Cotuous Physcs 5/3/9 8

29 act Nolocal Boudary Physcs 5/3/9 9

30 Remag Power - Dscrete Physcs 5/3/9 3

31 Physcs 5/3/9 3 Padé [] [] Padé Appromato Claerbout s quato Boudary Codto quato 4 / / 4 y δ b 4 4 s X s X δ

32 Padé [] [] Padé quato s X X s δ j 8 j Laplace trasform ths equato wth respect to the eteror rego. Requrg that o poles are preset the rght-had plae of the trasform yelds the desred boudary codto. Physcs 5/3/9 3

33 [] Boudary Codto Results Physcs 5/3/9 33

34 Physcs 5/3/9 34 Padé [NN] For the [NN] case where g a a g a a g g a a g a a g k k k k k k k s

35 Geeral Itroducg a vector g g wth j j g j k g k yelds A g wth boudary codtos g B g g B g Physcs 5/3/9 35

36 Geeral 3 After Laplace trasformg ths yelds p A g p A pg g or defg C A Problem: Costruct of I C have ˆ p I C gˆ p pg g C p Rp > j such that all poles Physcs 5/3/9 36

37 [NN] Boudary Codto Results Physcs 5/3/9 37

38 Part III - Improvg Accuracy Fast Reflecto Calculatos Physcs 5/3/9 38

39 Facet Reflecto Coeffcet y y Matchg ad at the boudary gves Ψ y Ψ o e k o ol L l Ψ o e k o ol L l A B k yr [ R] T k L B LA yr yr y oa oa L L A A y ob ob L L or B B Physcs 5/3/9 39

40 Reflecto Coeffcets Ar d co cl Wavegude Geometry Physcs 5/3/9 4

41 Stadard Operator Results Physcs 5/3/9 4

42 Calculated Reflecto rror Sce the Padé appromato for L has poles the evaescet spectral rego ucotrollable errors ca develop. Oe method to resolve ths - Geerate a appromat wth comple coeffcets by selectg a magary termato codto for the cotued fracto represetato of. Physcs 5/3/9 4

43 Comple Padé Reflecto.45.4 Comple Pade Real Pade Reflecto Power Pade Order Physcs 5/3/9 43

44 Rotated Padé Appromats A secod method: Wrte e [ e α / α ] ad perform a Padé epaso the varable y α e Physcs 5/3/9 44

45 Rotated Padé Reflecto.4.38 Rotato Agle Rotato Agle 3 Rotato Agle 6 Rotato Agle 9 Reflecto Power Pade Order Physcs 5/3/9 45

46 Refractve Ide Dscretato Ψ Physcs 5/3/9 46

47 Physcs 5/3/9 47 Trasto Propagato Operator j j o o j j o o j j o o j j o o j L L L L L L L L T j j j j j j j j Ψ j out Ψ j out Ψ j Ψ j L jk L jk m m om o m om o e e P

48 Dstrbuted Feedback Normaled power Wavelegth µ m Reflectvty usg rotated [/] Padé Reflectvty usg rotated [3/3] Padé Reflectvty usg rotated [5/5] Padé Coupled wave theory [4] Total power usg [/] Padé Total power usg [3/3] Padé Total power usg [5/5] Padé Physcs 5/3/9 48

49 Coclusos Procedures ow est for costructg eact olocal boudary codtos for wde-classes of two-dmesoal parabolc partal dfferetal equatos. Modfed Padé operators ca be employed to crease the accuracy of reflecto calculatos at abrupt terfaces. Physcs 5/3/9 49