Quasi Modes and Density of States (DOS) of 1D Photonics Crystal
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QUASI MODES AND DENSITY OF STATES (DOS) OF1D PHOTONICS CRYSTALShaolin Liao , ∗ and Lu Ou (2008-2010) Physics Department, Queens College, City Universityof New York, 65-30 Kissena Blvd, Flushing NY 11367; College of Computer Science and Electronic Engineering, HunanUniversity, Changsha, Hunan, China 410082. ∗ Corresponding author ([email protected]): S. Liao is now at theDepartment of Electrical and Computer Engineering, Illinois Instituteof Technology, Chicago, IL, USA 60616.
Abstract —1-Dimensional (1D) photonics crystals with and withoutdefects have been numerically studied using efficient Transfer MatrixMethod (TMM). Detailed numerical recipe of the TMM has beenlaid out. Dispersion relation is verified for the periodic PhotonicsBand Gap (PBG) structure. When there are defects, the transmissionspectrum can be decomposed into one or more quasi modes withexcellent agreement. The Density of States (DOS) is obtained fromthe phase derivative of the transmission spectrum. Green’s functionis also obtained showing much sharper mode characteristics when theexcitation source is localized at the peaks of the quasi modes.
1. INTRODUCTION
Photonics crystal has been of great interest due to its potentialapplications in optics filter and low-loss reflection mirror [1], [2]. Thetransmission spectrum and its phase information of different kindsof photonics crystal has been widely reported using Transfer MatrixMethod (TMM) [2]. Experiment on single-mode 1-dimensional (1D)waveguide system has been recently carried out in our group [3].What’s more, the Green’s function is important in description ofrandom laser [4] and periodic Photonics Band Gap (PBG) structurewith defects [3] since the radiation can be understood as convolutionof the Green’s function with the source. Also, Local Density of States(LDOS) is closely related to the imaginary part of the Green’s function[5]. The Density of States (DOS), which is the volume integral of the a r X i v : . [ phy s i c s . c l a ss - ph ] J u l Liao
LDOS, is shown to be the phase derivative of the transmission spectrumwith respect to frequency for 1D system without loss [6].In this article, we numerically study the transmission spectrum,its phase information and the Green’s function, by means of TMM [2].The obtained 1D transmission spectrum is further decomposed intoquasi modes [7]. The DOS is obtained from the phase derivative of thetransmission spectrum.
2. TRANSFER MATRIX METHOD
The electromagnetic wave is governed by the Maxwell’s equations andhas many applications [8]-[70]. Here we are dealing with the 1Dproblem in the optics regime.
The Green’s function is described by ∇ G ( x, x (cid:48) ) + k ( x ) G ( x, x (cid:48) ) = − δ ( x − x (cid:48) ) (1)where the delta source is located at x (cid:48) .The boundary conditions are G ( x + b , x (cid:48) ) = G ( x − b , x (cid:48) ) (2) dG ( x + b , x (cid:48) ) dx − dG ( x − b , x (cid:48) ) dx = − δ ( x b − x (cid:48) )where x ± b denote the right and left sides of the boundary. The Green’s function G ( x, x (cid:48) ) in each layer can be expressed assuperposition of forward- and back-ward propagating waves, G ( x, x (cid:48) ) = t exp − jk ( n r − jn i ) x + r exp jk ( n r − jn i ) x (3)where n r − jn i is the complex index of refractive. Substituting Eq. (3) into Eq. (2) and setting x b = 0, we have, v ± = T ± v ∓ + S ± u (4) uasi Modes and Density of States (DOS) of 1D Photonics Crystal 3 where the superscript ”+” denotes the right side of the boundary and”-” denotes the left side of the boundary. The transfer matrixes T ± and source matrixes S ± are given below, T ± = 12 n ∓ r − jn ∓ i n ± r − jn ± i − n ∓ r − jn ∓ i n ± r − jn ± i − n ∓ r − jn ∓ i n ± r − jn ± i n ∓ r − jn ∓ i n ± r − jn ± i S ± = ± n ± r − jn ± i − n ± r − jn ± i The wave propagations inside each layer can be described by apropagators P ± , v ± m = P ± m v ∓ m (5)where ± denote the left and right ends of layer m respectively, and P ± m = (cid:34) exp ∓ jk ( n mr − jn mi ) d m
00 exp ± jk ( n mr − jn mi ) d m (cid:35) where d m is the thickness of layer m . From Eq. (4), we know that, away from the source location x (cid:48) , v ± = T ± v ∓ (6)At the excitation source location, we have v + = v − + (cid:34) − j k n r − jn i j k n r − jn i (cid:35) (7)where n r − jn i is complex index of refractive at the source location x (cid:48) . The numerical recipe to obtain the 1D Green’s function is as follows,1) individually, calculate the transmission and reflection spectra forthe segments to the left and to the right sides of the source location x (cid:48) ,through Eq. (5) and Eq. (6); 2) connect both segments at the sourcelocation, through Eq. (7). Liao
Figure 1.
Dispersion relation of a typical PBG structure: a = 2 d =2 d = 8 mm; n = 1 and n = 1 . For each segment, we have to cascade the propagator P − in Eq. (5)and the transfer matrix T − in Eq. (6), i.e., the cascading procedurehas to been done from the end of each segment, where there is onlyforward propagating wave, v + (left/right) = (cid:34) N (cid:89) m =1 P − m T − m (cid:35) (cid:20) (cid:21) (8) We now connect both segments at the source location through Eq. (7), t (right) = t (left) − j k n r − jn i (9) r (right) = r (left) + j k n r − jn i uasi Modes and Density of States (DOS) of 1D Photonics Crystal 5
3. PERIODIC PBG STRUCTURE
It is well-known that the dispersion relation of the periodic binarydielectric layers is given by [71],cos( Ka ) = cos ( k d ) cos ( k d ) − k + k k k sin ( k d ) sin ( k d ) (10)where K is the wave vector of the Bloch wave; k , = ω √ µ(cid:15) , are thewave vector of the binary pair; a = d + d is the periodicity of thestructure, with d , being the thickness of each layer. Fig. 1 showsa typical dispersion relation for a = 2 d = 2 d = 8 mm; n = 1and n = 1 .
7; the frequency ranges from 1 GHz to 110 GHz. Thesimulated transmission spectrum for such periodic PBG structure hasbeen carried out for 25 pairs of binary dielectric layers: incident wavefrom left and transmitted wave on the right. The result is shown inFig. 2, which agrees well with Fig. 1.
Figure 2.
Transmission intensity T , reflection intensity R and theirsum T + R for 25 pairs of dielectric layers.
4. DEFECTS AND QUASI MODES4.1. Transmission and reflection spectra
Now let’s look at what happens when one or more defects areintroduced deep inside the PBG structure (25 pairs). First, let’s look
Liao
Figure 3.
Single defect inside PBG structure is simulated: 3 pairs inthe middle are replaced with n . Gray: n = 1; Orange: n = 1 . n (see Fig. 3). The result is shown in Fig. 4 for the first band gap in Fig.2. One quasi modes appears at around f c = 14 .
85 GHz. A closer lookat the intensity and phase of the transmission and reflection spectrumare shown in Fig. 5 and Fig. 6, together with the quasi mode fittingand DOS. Quasi modes will be explained in Section 4.2 and DOS isdiscussed in Section 4.3.Now let’s look at double potential wells (defects) by replacing 3pairs with n (see Fig. 7) at two locations separated by a potentialbarrier n . The intensity and phase are shown in Fig. 8 and Fig.9 respectively. Also see Section 4.2 for quasi mode explanation andSection 4.3 for DOS. Quasi modes can be considered as localized modes around the defectsites and could couple with each other if more than one modes arepresent. Mathematically, the frequency part of the quasi mode can beexpressed as the Lorentzian function,Ψ( f ) = Γ( f − f c ) + j Γ (11) uasi Modes and Density of States (DOS) of 1D Photonics Crystal 7
Figure 4.
First band gap of PBG structure with single defect:transmission intensity T , reflection intensity R and their sum T + R . In Fig. 5 and Fig. 6, we look closer into the quasi mode shown inFig. 4. We also plot the Lorentzian quasi mode as circles in Fig. 5and Fig. 6, which agrees with the simulation very well. We found that f c = 14 .
85 GHz and Γ = 0 .
131 MHz for this quasi mode.
For double defects inside PBG structure., the wave function can beexpressed in sum of two quasi modes,Ψ( f ) = a Γ ( f − f ) + j Γ + a exp jπ Γ ( f − f ) + j Γ (12)with f = 14 .
810 GHz, Γ = 4 . a = 0 .
972 and f = 14 . = 5 . a = 0 . π . Liao
Figure 5.
Single defect (a closer look of Fig. 4): simulated intensityand that of the theoretical Lorentzian quasi mode, with f c = 14 . .
131 MHz.
Figure 6.
Single defect: simulated phase and and that of theLorentzian quasi mode, with f c = 14 .
85 GHz and Γ = 0 .
131 MHz. uasi Modes and Density of States (DOS) of 1D Photonics Crystal 9
Figure 7.
Double defects inside PBG structure is simulated: 3 pairsat two locations are replaced with n , separating by a potential barrier n . Gray: n = 1; Orange: n = 1 . Figure 8.
Double defects: simulated transmission and reflectionintensities and those of the sum of two Lorentzian quasi modes, with f = 14 .
810 GHz, Γ = 4 . f = 14 .
898 GHz, Γ = 5 . For 1D system, DOS is given by [6],DOS = 1 π dφdω (13)
Figure 9.
Double defects: simulated phase and and that of the sumof two Lorentzian quasi modes, with f = 14 .
810 GHz, Γ = 4 . f = 14 .
898 GHz, Γ = 5 . Figure 10.
Schematics of Green’s function for double defects. Gray: n = 1; Orange: n = 1 . uasi Modes and Density of States (DOS) of 1D Photonics Crystal 11 Figure 11.
Simulated transmission spectrum of Green’s function forboth left and right transmissions.
Figure 12.
Simulated phase of the transmission spectrum of Green’sfunction for both left and right transmissions.
We also obtained the Green’s function following the procedure stated inSection 2.3. Here we show the result for double defects: the schematicswith excitation source inside the first defect is shown in Fig. 10. Thetransmission spectra and phases for both left and right sides are shownin Fig. 11 and Fig. 12 respectively. Compared to the transmission andreflection spectra in Fig. 8 and Fig. 9, we can see that the Green’sfunction shows much sharper quasi mode peaks.
5. CONCLUSION
We have simulated the transmission spectrum and the Green’s functionfor 1D photonics crystal with and without defects. It has been shownthat quasi mode decomposition gives excellent agreement with thesimulated result. DOS is also obtained through the phase derivativeof the transmission spectrum. The method can find applications indesign of 1D photonics filter and reflection mirror, defect simulation ofphotonics crystal and its DOS.
ACKNOWLEDGMENT
Shaolin Liao wants to thank Prof. Azriel Genack and his group fortheir help when he worked there as a Postdoc Fellow from 2008-2010.
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