CHAPTER II THEORETICAL BACKGROUND

Transcription

1 3 CHAPTER II THEORETICAL BACKGROUND.1. Lght Propagaton nsde the Photonc Crystal The frst person that studes the one dmenson photonc crystal s Lord Raylegh n He showed that the lght propagaton depend on the forbdden angle for a certan range of frequency. Many optoelectronc devces use one dmenson photonc crystal as a frequency flter or delectrc mrror. When the lght hts the layer, each surface reflects a part of the feld. If we choose the thckness of each layer for a sutable value, the reflected feld wll combne a constructve phase, producng constructve nterference and strong reflectance called Bragg Reflecton. It s shown that the Bragg Reflecton n the perodcal delectrc structure create a photonc band gap (PBG). When the perodcty destroyed by the present of defects n the photonc crystal, the localzaton of the defect mode wll appear nsde the PBG due to of the changes n lght nterference. Fgure.1. Illustraton of the Photonc Crystal Structure

2 4 As dscussed n the prevous chapter, the search of the best way to control the lght propagaton always becomes the man prorty. The man concern of ths research focused on the nteracton between the electromagnetc feld wth soldlke-structure photonc crystal. The Maxwell Equaton s the frst and defntely the most mportant one n ths theory. The frst step s to dervate all the formula n the Maxwell Equaton. The components n the electromagnetc wave, electrc feld and magnetc feld wll move through a medum that s load free and the free wave has been connected through 4 Maxwell Equaton, as follows; E( r, B( r, t H( r, D( r, J( r, t B( r, 0 D( r, ( r, (1) () (3) (4) The standard notaton for electrc feld ( E ), magnetc feld ( H ), electrcty propagaton (D), and magnetc nducton ( B ) have been used n ths equaton. Rememberng agan a certan dentty from the vector arthmetc: ( A) (. A) A (5) and adjust t wth the Maxwell Equaton, where. ( r) 0 and ( r) 1 E 00 E t (6) H 00 H t (7) The equatons are the equaton of standard wave that has many solutons, one of whch s the equaton for an area wth the shape of E ( k. rt ) E0e and

3 5 H H e ( k. rt ) 0, where the wave s vector and frequency are shown by k and ω smultaneously. Furthermore, the equaton for wave produces: 1 [ E( r, ] E( r, () r c (8) 1 [ H ( r, t )] H ( r, t ) () r c (9) The equatons above usually called as master equaton by the researchers n photonc crystal. Although t does not show somethng new from the perspectve of the researchers, the test by usng Schrodnger s Egen-value equaton that s more famlar n the quantum mechancs has opened a new pont of vew. Object wth mass m obey the Schrodnger equaton, and ts energy can be calculated through the correspondng equaton. Therefore, the equaton for electrc feld or magnetc feld has an analogy wth Schrodnger s [4] wave equaton. One of the general solutons for equaton (8) s monochromatc harmonc plan wave that depends on tme E r, t (9), producng EM wave n the frequency doman of t E r e that s renserted nto equaton k E r, 0 (10) In the cartesan coordnate system, equaton (10) reduced nto three scalar equatons for each electrc feld components E z, E x, and E y. Ths equaton can be solved through separaton of varables method. For the TE wave, electrc feld E = (0, E x, 0) s lnearly polarzed n the y drecton and descrbed n the form of scalar functon E y (z, y), so t produces: E z y ky y, E z e (11) By usng the same varable separaton method, a general soluton for harmonc plan wave s gven as follows:

4 6 ( kzzt ) ( kzzt ) ky y E Ae Be e (1).. Modelng and Mathematcal Formulaton Matrx method s the best way to perform an accurate analyss of the EM wave transmsson n a layered medum. Generally, matrx formalsm s used to relate electrc feld and magnetc feld components n each layer [8]. We use the standard transfer matrx method to observe TE and TM wave s transmttance. The advantage of ths transfer matrx method s that t gves the exact numerc soluton from the model made and s relatvely easer modfed f the structure of the model needs to be changed. The feld n the last layer of the photonc crystal for both refractve polarzatons can be calculated from the followng relaton: E / Et 1 TE( TM ) Er / E t 0 (13) where E, E r and E t are electrc felds that came, reflected, and transmtted and matrx: 1 T = P 0 Q 1 P Q P M Q D1 P 1 1 D1 Q 1 P 1 Q 1 P Q P N Q D P 1 1 D Q P Q 1 P Q P L (14) s a transfer matrx where matrx P and Q for TE and TM polarzatons are gven by: P TE 1 1, and k cos k cos Q k d1cos k d1cos e e TE j k d1cos k d1cos k cose k cose P TM cos k cos, and k Q (15) k d1cos k d1cos cose cose TM j k d1cos k d1cos ke ke

5 7 wth k = n ω/c, = 0, 1, for fxed structures, meanwhle k d1 = n d1 ω/c, k d = n d ω/c, are for defect layers, θ shows the comng angle n each layer. The transmttance of the electrc feld s gven below: T E / E t (16).3. The Condton of Quarter-Wave Stack The thckness of each medum layer (n 1 and n ) can be chosen to fulfll the quarter-wave stack condton, that s: d 1 = λ o 4n 1 and d = λ o 4n so both layers have the same optc length (n 1 d 1 = n d ). Moreo v er, λ o s called the operatonal wavelength and s the centre of the frst PBG frequency that s formed (for m=l), equaton: mλ B = n eff L (17) where n eff s the effectve refractve ndex that can be stated as: n eff = n 1d 1 +n d L (18) and L s the crystal s perods, whch s d 1 +d, meanwhle λ o s usually stated n a frequency form ω o = πc = cπ = cπ (19) λ o n 1 d 1 n d and f t s compared wth Bragg, for n 1 d 1 = n d and n eff, and defned n equaton (3), the Bragg frequency can be smplfed nto: cπ cπ ω Bragg = m = m (0) n 1 d 1 n d so from the equaton we can get Bragg = m o (1) wth m = l, 3, 5, and so on, for the quarter- wave stack case.

6 8.4. Felds Dstrbuton nsde the Defect Layer In the photonc crystal wth mperfectly perodcty, there wll be a resonance mode n the PBG range where EM wave frequency that exsts s the same as the frequency of the mperfect mode of the crystal. The wave wth the mperfect mode or frequency wll be reflected smultaneously n harmony (back and forth) around the mperfect mode by DBR (dstrbuted Bragg reflector) n the left and rght sdes of the mperfect layer that functoned as PBG mrror. Ths causes the photons to localze around the mperfects and cause hgh feld enhancement. Feld enhancement n the defect area leads to a full transmttance n the PBG n ts resonance frequency, whch s usually defect mode. Fgure.. Feld Dstrbuton n the Defect nsde the Photonc Crystal The EM feld s profle whch s propagatng nsde the photonc crystal layer can be descrbed by usng the transfer matrx method and by consderng the translaton s symmetry. The soluton for the EM feld that comes n the z drecton whch s vertcal wth the crystal layer and move n layer n 1 and n can be wrtten as: E() z Ae B e k1z k1z 1 1 () E() z A e B e k ( z d1 ) k ( z d1 ) (3).5. Transmttance of Photonc Pass Band n the Defect Cell The effect of defects exstence nsde the photonc crystal leads to a resonance n the defect, causng a very large feld wth hgh transmttance. Ths

7 9 transmttance s known as Photonc Pass Band (PPB) that s located nsde the PBG that should be a forbdden area for the perodc crystal structure. The exstence of defect nsde ths photonc crystal s analog wth the exstence of mpurty nsde the materal structure of a semconductor. Fgure.3. Profle of PBG n the Photonc Crystal Structure (a).pbg for Photonc Crystal wthout Defect. (b) PPB that happens nsde PBG for Photonc Crystal wth Defect.6. One Dmenson Photonc Crystal wth Two Defects One dmenson photonc crystal wth two defects has a more nterestng phenomenon. The structure of one dmenson photonc crystal wth two geometrc asymmetrc defects s llustrated n the Fgure.4. below: Fgure.4. Structures of One Dmenson Photonc Crystal wth Two Geometrc Defects.

8 10 Fgure.5. Curve for One Dmenson Photonc Crystal Transmttance wth Two Geometrc Defects. (a) Changes n the Regulator s Defect Cause the PPB Shftment. (b) Changes n Defect (Receptor) Cause the Maxmum Transmttance of PBB Decreased (H. Alatas, 006) The structural dfferent between the one dmensonal photonc crystal wth two asymmetrc defects and one dmensonal photonc crystal wth one defect s that n the former structure, the refractve ndex n the left corner of the crystal s not the same wth the refractve ndex n the rght corner of the crystal. PPB that used to form n the photonc crystal wth two asymmetrc geometrc defects has the same response toward the changes n defect wdth (d D ). Fgure.6. The Relatonshp between Defect s Refractve Index and the Transmtance of PPB that s Used for Refractve Index Sensor. (H. Alatas, 006)

9 11.7. Photonc Crystal Model for Optcal Bosensor One dmensonal Photonc crystal model that we made conssts of a fxed layer of delectrc layer that crss-crossed along wth two defect layers, they are: n 0 n s (n 1 / n ) M D 1 (n 1 / n ) N D (n 1 / n ) L n s n 0, lke the structure descrbed n Fgure n 1 and n showed the refractve ndex n the fxed layer (n 1 / n ) and ts thckness s s marked by (d 1 / d ). Two defect layers were marked by (D1) (n d1 / n ), and (D) (n d / n ) that were related wth ts thckness (d d1 / d ) and (d d / d ) smultaneously. The refractve ndex of the substrate and background medum are n s and n 0 respectvely. Fgure.7. One Dmenson Photonc Crystal Model wth M = 4, N = 6 and L =. The total number of cell layers n the left sde of D 1, between D 1 and D also after D, s gven by M, L, and N, n orders. In the numercal studes, we assumed that the materals used have low capablty to absorb the TM wave (lowloss meda). The parameter used are gven by n 0 = 1 (ar), n s = 1.5(BK 7 ), n 1 =.1 (OS-5), n = 1.38 (MgF ) and the optcal thckness fulflls the quarter wave stack condton: 55 nm. Frst defect of cell s made by d d1 = λ 0 /, whereas the second one s made empty space to nsert the sugar soluton as sensng materal..8. Sensor Devce In the last year, several applcatons of bosensor were already exst and based on the characterstcs of transmsson spectrum and reflecton n the surface of the object. The Surface Plasmon Resonance (SPR) sensor has been used wdely for screenng the bochemstry nteractons, whle other researcher groups developed an optcal bosensor based on Fabry-Ferot cavtes n the porous

10 1 slcon or guded mode resonance reflectance flters. Other applcaton used the optcal resonance shftment to test the DNA. The unque characterstcs of the PPB s that t s not only used as a flter but also can be developed as an optcal sensor related to the functon of defect, one of whch s as a regulator and the other s as a receptor. A type of an optcal sensor that can be developed s the refractve ndex sensor that can measure the substance concentraton n a soluton, such as sugar soluton sensor or salt concentraton. As an example, the refractve ndex of the sugar soluton for a 30% concentraton s 1.37, meanwhle for a 50% concentraton, the refractve ndex s 1.4. To develop a bosensor, a photonc crystal can be used so that t produces a narrow resonance mode where the wavelength s very senstve toward the modulaton that s nducted by the bochemstry materal deposton n the defect layer. A structure of sensor conssts of a transparent materal that has low refractve ndex wth the perodc surface structure coated wth a thn layer that has hgh ndex. SOLUTION FLOW Fgure.8. Illustraton of the Sensor Devce that s Photonc Crystal-Based to Detect a Soluton s Concentraton