Implementation of the Matrix Method

Transcription

1 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Computatonal Photoncs Semnar 0 Implementaton of the Matr Method calculaton of the transfer matr calculaton of reflecton and transmsson characterstcs of stratfed meda calculaton of felds nsde layers 1

2 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Optcs n stratfed meda y Bragg mrror mrror wth chrp for compensatng dsperson nterferometer

3 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Optcs n stratfed meda y Plane of ncdence = --plane no y-dependency

4 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena A stratfed (layered) medum refractve nde n

5 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena A stratfed (layered) medum each layer wth nde s charactered by ts thckness d and t delectrc constant ε (ω) ε an arbtrary contnuous varaton of the refractve nde can be dscreted wth a suffcent large number of layers mportant for so called 'GRIN' graded nde wavegudes

6 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena EM felds n the stratfed meda Requrements: statonarty nfnte etenson of the layers n the -y-plane llumnatng feld ncdent n the --plane ncdent feld feld to be calculated Ansat: Ereal (,, t) Re E( )epk t H t H k t real (,, ) Re ( )ep Separaton nto TE und TM polaraton TE: E 0 H E, H 0 0 H TE y TE TM: H 0 E H, E 0 0 E TM y TM

7 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Boundary condtons Felds: E t and H t contnuous TE: E = E y and H TM: H = H y and E Performng all computatons wth the tangental components, (f necessary the normal components can be derved) transversal wavevector s constant n the stack and s determned by the angle of ncdence: k = ω ε c s sn θ = π ε λ s sn θ 0 substrate ε s k normal component vares n the stack: k depends on the permttvty of each layer dsperson relaton k c k k k k k

8 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Computng the felds by contnuous components (TE) Helmholt-equaton: k Ey( ) 0 c k Soluton: E ( ) cos sn y C k C k 1 0 H ( ) Ey( ) 0H ( ) Ey( ) k C1 sn k cos C k Determnaton of C 1, C : Ey (0) C 1 E k C y 0

9 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Soluton TE: 1 E ( ) cos k E (0) sn k E k y y y 0 E k sn k E (0) cos k E y y 0 TM: 1 H ( ) cos k H (0) sn k H k y y y 1 k 1 H sn k H (0) cos k H y y y 0 0 TE/TM: 1 F( ) cos k (0) sn F k (0) G k G( ) k sn k F(0) cos k G(0) TE: F Ey, G 0H Ey, 1 TM: F H, G 0E H, 1/ y y

10 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Summary: Matr method Need to know: F(0), G(0), k,, d We want to calculate the felds F(D), G(D) N F F ˆ ( ) ˆ F d G G G D m M mˆ d cos k k sn k d k d k d snk d cos TE: F Ey, G Ey, 1 TM: F H, G H, 1/ y y wth, k k k c

11 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Reflecton and transmsson coeffcents of the felds transmsson coeffcent T reflecton coeffcent R F T F n F R feld at substrate sde (nde s): F(, ) ep( k) Fn ep( k ) F s R ep( k ) s G(, ) sk ep( k ) s Fn ep( k ) F s R ep( k ) s feld at claddng sde (nde c): F(, ) ep( k) FT ep k ( D) c G(, ) ck ep( k ) c FTep k ( D) c c c T 1 s s n R F connecton of felds at = 0 and = D by transfer matr FT M11( D) M1( D) Fn FR k F M ( D) M ( D) k F F n substrate, s F n F R claddng, c F T

12 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Reflecton and transmsson coeffcents of the felds transmsson coeffcent T reflecton coeffcent R R T k M k M M k k M s c 11 1 s c 1 s N k s c s c s N k M k M M k k M N s c 11 1 s c 1 s c s c F T F n F R F n substrate, s F n F R claddng, c F T wth TE: TM: F E y, 1 F H y, 1/ k k s/c s/c 0 0

13 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Energy flu defned va the normal component of the Poyntng vector S Substrat, s s s T n s s R n s n s n s R s R R c s Re Re k k c s T s T Claddng, c s T wth TE: 1 TM: 1/ k k s/c 0 s/c 0

14 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Feld dstrbuton Goal: Computaton of F() nsde the entre structure, (the absolute values can be scaled) Intal pont: Take the known entres of the transmtted ampltude F n F R F T F F 1 F FT FT 1 G k now: D c c c D F n F T F R Approach: 1. Reverse the structure (ncdent vector gets (1, c k c ). Calculate the feld vector up to the net nterface 3. From there, calculate the feld to the net -pont of nterest 4. Save the frst value of the vector for ths -pont 5. Iterate untl all -values are calculated and reverse the structure and the feld F n F R F T

15 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena The real feld The observable (real) feld Er H TE: E F r (,, t) Re E( )ep k t (,, t) Re H( )ep k t What you have actually calculated s the comple value of a certan component: TM: H F e e y y

16 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Task I : Transfer matr Goal : calculaton of M Input varables: epslon, thckness, polarsaton, lambda, k functon M = transfermatr(epslon, thckness, polarsaton, lambda, k) % M = transfermatr(epslon, thckness, polarsaton, lambda, k) % Computes the transfer matr for a gven stratfed meda. % All dmensons are n µm. % Arguments: % epslon : Delectrc permttvty of the layers (vector) % thckness : Thcknesses of the layers (vector) % polaraton : Polarsaton of the feld to be computed (strng: 'TE' or 'TM') % lambda : Wavelength of the lght (scalar) % k : Transverse wavevector [1/µm] (scalar) % Returns: % M : Transfer matr (matr) (potentally) useful functons: eye(n): creates the N-dmensonal unty matr error('message'): prnts 'message' on the screen and nterrupts the program strcmp(varable,'strng'): compares varable aganst 'strng

17 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Task II: Reflecton and transmsson coeffcents Goal: computaton of R, T, ρ, τ as a functon of the wavelength Input varables: epslon; thckness; polarsaton, angle_nc = 0; n_s = 1; n_c = 1.5; lambda_vector functon [T, R, tau, rho] = spectrum(epslon,thckness,polarsaton,... lambda_vector,angle_nc,n_s,n_c) % [T, R, tau, rho] = spectrum(epslon,thckness,polarsaton,... % lambda_vector,angle_nc,n_s,n_c) % Computes the reflecton and transmsson of a stratfed medum dependng on the % wavelength. All dmensons are n µm. % Arguments: % epslon : Delectrc permttvty of the layers (vector) % thckness : Thcknesses of the layers (vector) % polarsaton : Polarsaton of the feld to be computed (Strng: 'TE' or 'TM') % lambda_vector : Wavelength of the lght (vector) % angle_nc : Angle of ncdendence n degree (scalar) % n_s, n_c : Refractve ndces of the substrate and claddng (scalars) % Returns: % T : Transmtted ampltude (comple vector) % R : Reflected ampltude (comple vector) % tau : Transmtted energy (real vector) % rho : Reflected energy (real vector) Usng the functon transfermatr

18 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Task III*: Feld dstrbuton voluntary task Goal: Computaton of the comple feld f at predefned values of Input varables: k = 0; n_s = 1; n_c = 1.5; lambda = 0.6; N = 500; functon [f, nde, ] = feld(epslon,thckness,polarsaton,... lambda,k,n_s,n_c,n,l_s,l_c); % functon [f, nde, ] = feld(epslon,thckness,polarsaton,... % lambda,k,n_s,n_c,n,l_s,l_c) % Computes the feld n a stratfed meda. All dmensons are n µm. % The stratfed meda starts at = 0 at the entrance sde % (stratfed meda for > 0). The transmtted feld has a magntude of unty % Arguments: % epslon : Delectrc permttvty of the layers (vector) % thckness : Thcknesses of the layers (vector) % polarsaton : Polarsaton of the feld to be computed (Strng: 'TE' or 'TM') % lambda : Wavelength (scalar) % k : Transverse wavevector [1/µm] (scalar) % n_s, n_c : Refractve nde of the substrate/claddng (scalars) % N : Number of ponts where the feld shall be computed (nteger) % l_s, l_c : Addtonal thckness of the substrate and claddng where the % feld should be computed (scalars) % Returns: % f : Feld structure (comple vector) % nde : Refractve nde dstrbuton (comple vector) % : Spatal coordnate (real vector) Usng the functons transfermatr, flplr

19 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Task IV*: Tme anmaton of the feld voluntary task Goal: Vsualaton of the temporal evoluton of the feld Input varables: steps = 00; perods = 10; functon tmeanmaton(,f,nde,steps,perods) % tmeanmaton(,f,nde,steps,perods) % Anmaton of a quas-statonary feld. % Arguments: % : Spatal coordnates (real vector) % f : Feld (comple vector) % nde : Refractve nde (comple vector, real part plotted wth the feld) % steps : Total number of tme ponts (nteger) % perods : Number of the oscllaton perods (nteger) Usng the functons ma, as, fgure(gcf), pause

20 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena 0 Eample parameters Parameter of a Bragg mrror wth e.g. hgh reflectvty at =600nm: >> eps1 =.5; >> eps = 4; >> d1 = 0.1; %[µm] >> d = 0.075; %[µm] >> N = 50; Generatng the necessary parameters for further computatons: >> epslon(1::*n) = eps1; >> thckness(1::*n)= d1; Note that only every second element of the vectors has a value. In between, the vectors consst of eros. The length of the vectors s *N-1=99. Now we have to defne the remanng values. >> epslon(::*n) = eps; >> thckness(::*n) = d; >> lambda= 0.6; %[µm] >> k = 0.0; The polarsaton s defned to be a strng: >> polarsaton='te'; substrate n s =1 ncdent (TE) R 1 d 1 d 100 layers claddng n c =1.5 T

21 Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Voluntary Homework 1 (due 1 Aprl 017, 3 am) All tasks are voluntary etra tasks but you should be able to solve at least I and II If you want feedback on ths tranng task, you send us a short report about your soluton wth one or two fgures of some calculated eamples. You can submt the matlab m-fles of your program together wth the report electroncally to teachng-nanooptcs@un-jena.de by 1 Aprl 017, 3 am. (Please put everythng together n one sngle emal whch contans your name [FAMILY NAME, Gven Name], your matrculaton number, and the name of ths course, Computatonal Photoncs ) Do not submt later than the deadlne snce n the mornng of Aprl 1th, we wll make the soluton avalable onlne at the lectures homepage >>> Computatonal Photoncs. You are epected to solve the task yourself and a declaraton of ndependent work (for the tasks followng ths tranng sesson) must be sgned by every student at the end of the semester.