Implementation of the Matrix Method

Transcription

1 Computatonal Photoncs, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch Computatonal Photoncs Semnar 03, 7 May 01 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, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch Optcs n stratfed meda y Bragg mrror mrror wth chrp for compensatng dsperson nterferometer

3 Computatonal Photoncs, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch Optcs n stratfed meda y wavegudes composed of many layers Braggg wavegudes

4 Computatonal Photoncs, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch Optcs n stratfed meda y Plane of ncdence = --plane no y-dependency

5 Computatonal Photoncs, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch A stratfed (layered) meda n

6 Computatonal Photoncs, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch A stratfed (layered) meda each layer wth nde s charactered by ts thckness d and t delectrc constant () 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

7 Computatonal Photoncs, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch EM felds n the stratfed meda Requrements: Statonarty nfnte etenson of the layers nto the y-plane Illumnatng feld ncdents n the --Ebene ncdent feld feld to be calculated Ansat: Ereal(, t, ) Re E( )epkt (, t, ) Re H( )epk t Hreal Separatton n TE und TM TE: E 0 H E, H 0 0 H TE y TE TM: H 0 E H, E 0 0 E TM y TM

8 Computatonal Photoncs, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch Boundary condtons Felds: E t and H t contnuous TE: E=E y und H TM: H=H y und E Performng all computatons wth the tangental components, (f necessary the normal components can be deduced) transversal wavevector s constant n the stack and s determned by the angle of ncdence: denoted by k normal component vares n the stack: k depends on the permttvty of each layer k c k k k k k k

9 Computatonal Photoncs, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch Computng the felds by contnuous components (TE) k E y( ) 0 c k H( ) Ey( ) 0 Soluton: E ( ) cos sn y C k C k 1 0H( ) Ey( ) k C1sn k cos C k Determnaton of C 1, C Ey (0) C 1 E k C y 0

10 Computatonal Photoncs, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch 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 y 0 TM: 1 H ( ) cos k H (0) sn k H k y y y 1 k 1 H sn k H (0) cosk 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 Hy, G E Hy, 1/ 0

11 Computatonal Photoncs, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch 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 cos k k sn kd kd kd snk d cos TE: TM: F Ey, G Ey, 1 F Hy, G Hy, 1/ wth, k k k c

12 Computatonal Photoncs, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch Reflecton and transmsson coeffcents of the felds transmsson coeffcent T reflecton coeffcent R k M k M M k k M R N sks T N N k M k M M k k M s s c c 11 1 s s c c 1 s s c c 11 1 s s c c 1 F T F n F R F n Substrat, 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, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch 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 s T Claddng, c wth TE: 1 TM: 1/ k k s/c s/c 0 0

14 Computatonal Photoncs, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch Feld dstrbuton Goal: Computng of F() nsde the entre structure, (the absolute values can be scaled) Intal pont: Taken the known entres of the transmtted ampltude F F 1 F FT FT 1 G D c ck now: c D F n F n F R F T F T F R Approach: 1. Reversng the structure (ncdent vector gets (1, - c k c ). Calculatng the feld vector tll the net nterface 3. From there calculate the feld to the net -pont of nterest 4. Savng 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, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch The real feld The observable (reel) feld Er H TE: E F TM: r (, t, ) Re E( )ep kt (, t, ) Re H( )ep k t e y What you have actually calculated s the comple value of a certan componnt H F e y

16 Computatonal Photoncs, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch Task I : Transfer matr Am : calculaton of ˆM Input varables:, d, polarsaton, lambda, k functon M=transfermatr(epslon, thckness, polarsaton, lambda, k); % Computes the transfer matr for a gven stratfed meda % All dmensons are n µm % Functon call M=transfermatr(epslon, thckness, polarsaton, lambda, k); % 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: Component of the wavevector n transvers drecton [1/µm] (scalar) % M: Transfer matr (matr) (potentally) usefull functons: eye(n): creates the N-dmensonal unty matr error('message'): prnts 'Message' on the screen and nterrupts the program strcmp(varable,'strng'): verfes whether Strng and Varable are equal

17 Computatonal Photoncs, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch Task II: Reflecton and transmsson coeffcents Goal: computaton of R, T,, as a functon of the wavelength Input varables:, d, polarsaton, angle_nc=0; n_s=1; n_c=1.5; lambda_vector functon [T,R,tau,rho]=spektrum(epslon,thckness,polarsaton,lambda_vector,angle_nc,n_s,n_c); % Computes the reflecton and transmsson of a stratfed meda dependng on the wavelength % All dmensons are n µm % Call [T,R,tau,rho]=spektrum(epslon,thckness,polarsaton,lambda_vector,angle_nc,n_s,n_c); % epslon: Delectrc permttvty of the layers (cevtor) % thckness: Thcknesses of the layers (Vector) % polarsaton: Polarsaton of the feld to be computed (Strng: 'TE' or 'TM') % lambda_vector: Wavelength for whch the computaton shall be conducted (Vector) % angle_nc: Angle of ncdendence n degree % n_s,n_c: Refractve ndces of the substrate and claddng Substrat/Claddng; % T: Transmtted ampltude (komple vector) % R: Reflected ampltude (komple vector) % tau: Transmtted energy (reel vector) % rho: Reflected energy (reel vector) Usng the functon: transfermatr

18 Computatonal Photoncs, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch 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); % 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 strength of unty % Call [f,nde,]=feld(epslon,thckness,polarsaton,lambda,k,n_s,n_c,n,l_s,l_c); % epslon: Delectrc permttvty of the layers (cevtor) % thckness: Thcknesses of the layers (Vector) % polarsaton: Polarsaton of the feld to be computed (Strng: 'TE' or 'TM') % lambda: wavelength of lght for whch the computaton shall be conducted (scalar) % k: Component of the wavevector n transversal drecton [1/µm] (scalar) % n_s,n_c: Refracve nde of the substrat/claddng; % N: number of ponts where the feld shall be computed % l_s/c: addtonal thckness of the substrat and claddng n where the % feld should be computed % f: feld structure (vector) % nde: nde dstrbuton (vector) % : spatal coordnate (vektor) Usng the functon: transfermatr, flplr

19 Computatonal Photoncs, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch Task IV*: Tme anmaton of the feld voluntary task Goal: Vsualaton of the temporal evaoluton of the feld Input varables: steps=00; perods=10; functon tmeanmaton(,f,nde,steps,perods); % Anmaton of a quasstatonary feld % Call: tmeanmaton(,f,nde,steps,perods); % : spatal coordnates (reel vector) % f: feld (comple vector) % steps: Number of the ponts n the dscrete tme to be represented % perods: Number of the oscllaton perods Usng the functons: ma, as, fgure(gcf), pause

20 Computatonal Photoncs, Summer Term 01, Abbe School of Photoncs, FSU Jena, Prof. Thomas Pertsch Homework 1 (7 May 01) Solve tasks I & II. Tasks III & IV are voluntary etra tasks. Prepare a one page report about your soluton wth a fgure of some calculated eample. Submt your matlab m-fles of your program together wth your one page report electroncally to thomas.pertsch@un-jena.de by 17 May 01. (Please put everythng together n one sngle emal whch contans your name (FAMILY NAME, Gven Name) and matrculaton number. Late submssons wll not be accepted! 18 May the solutons of the tasks wll be avalable onlne at the lectures homepage >>> Computatonal Photoncs. You are epected to solve the task yourself and a declaraton of ndependent work must be sgned by every student at the end of the semester.