Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 1923, 2012 1121

Investigation of Defect Modes on One-dimensional TernaryMetallic-dielectric Nano Photonic Crystal with Metallic Defect

Layer

Abdolrasoul Gharaati and Hadis AzarshabDepartment of Physics, Payame Noor University, Tehran, Iran

Abstract We investigate the characterization of defect mode in one-dimensional ternarymetallic-dielectric photonic crystal structures. The defect layer is made of metallic material. Weconsider defect mode for both symmetric and asymmetric geometries. Additionally, we demon-strate reflectance in terms of wavelength and its dependence on different angles of incidence forboth transverse electric (TE) and transverse magnetic (TM) waves. There is just one defectmode when we use metallic defect layer. All of our calculations have done with transfer matrixmethod (TMM) and the Drude model of metals.

1. INTRODUCTION

In recent years, advances in photonic technology have caused so much attention. The simplest pos-sible photonic crystal consists of alternating layers of material with different dielectric constants [1].A one dimensional ternary metallic-dielectric photonic crystal (1DTMDPC) is a periodic structurecontaining dielectric and metallic elements with different index of refraction. 1DPCs because of itssimple fabrication has many applications such as multilayers coatings [2], narrow band filters, andso on. A range of frequency in which light cannot propagate through the PC is called photonicband gaps (PBG). We have used TMM to calculate the transmitted and reflected coefficients ofincident electromagnetic waves.

In this paper, using TMM, we calculate the defect mode wavelength. We apply this method to1DPC consisting of periodically dielectric-metal-dielectric with metallic defect layer for both TEand TM waves.

2. THEORETICAL ANALYSIS

A 1DTMDPC with a structure of A/(n1n2n3)ND(n3n2n1)N/A, for symmetric geometry and A/(n1n2n3)ND(n1n2n3)N/A, for asymmetric geometry, where A and N mean the usual air and thenumber of periods, and n1, n2 and n3 are the refractive indices of dielectric, metal and dielectriclayers, respectively. 1DTMDPC is made of N -period cells immersed in air (nA = 1) in which themetallic layer 2 is sandwiched by two dielectric layers 1 and 3 in each cell. The Drude model [3] isinvoked to characterize the wavelength dependence of metallic layer. We assume that the temporalpart of the field to be exp(it). So metal permittivity in Drude model is

2() = 12p

2 + i(1)

where p and are the plasma frequency and damping coefficient, respectively. Then metal refrac-tive index is given by n2 =

2. A one layer in Fig. 2 along with the components of the electric

and magnetic fields of the incident, reflected, and transmitted wave [7].Using Maxwell equations we have the relations k0 =

00, where it is the wave vector in

the air, B = n

00E, p` = n` cos `, ` = k0n`d` cos `, where ` = 1, 2, and 3. So, the electricfields become: Ei2 = Et1 exp(i`) and Ei1 = Er2 exp(i`) and when included in the boundaryconditions, Et1 and Ei1 can be solved in terms of Ea and Bb:

Ei1 =(p1Eb + Bb)

2p1exp(i`), Et1 = (p1Eb + Bb)2p1 exp(i`). (2)

And finally, substituting the above expressions in the initial fields components and written in matrixform, we have

(EaBa

)=

(cos` ip` sin`ip` sin` cos`

)(E1B1

)= M1

(E1B1

)(3)

1122 PIERS Proceedings, Moscow, Russia, August 1923, 2012

N-cells

1 2 3 .. 1 2 3 d 3 2 1 .. 3 2 1

N-cells

x

1 2 3 .. 1 2 3 d 1 2 3 .. 1 2 3

z

(a)

(b)

Figure 1: (a) The structure of symmetric 1DTMDPC. (b) The structure of asymmetric 1DTMDPCs.

(a) (b)

Figure 2: A schematic of one layer of 1DTMDPC.

Each layer of PC has its own transfer matrix and the overall transfer matrix of the system is theproduct of individual transfer matrices, so the characteristics matrix M() for a single period isexpressed as [4]

M() =[

M11 M12M21 M22

]=

3

`=1

[cos` 1ip` sin`ip` sin` cos`

](4)

On the basis of Bloch theorem, the half trace is used to compute the Bloch wave vector [3, 4].

a =(M11 + M22)

2= cos1 cos2 cos3 12

(p1p2

+p2p1

)sin1 sin2 cos3

12

(p2p3

+p3p2

)cos1 sin2 sin3 12

(p1p3

+p3p1

)sin1 cos2 sin3 (5)

Then, the total characteristic matrix of the total PC is given by [4]

MT (N) =[

m11 m12m21 m22

]= M()N =

[M11 M12M21 M22

]N, (6)

The reflection coefficient r is given by [7]

r =(m11 + m12p0)p0 (m21 + m22p0)(m11 + m12p0)p0 + (m21 + m22p0)

(7)

where p0 = n0 cos 0. We can calculate the reflectance R = |r|2. The above calculations can beused for TM wave by substituting p` = cos `/n` where ` = 0, 1, 2, and 3, respectively. So thecharacteristic Matrix of Left and Right unit cells with the number of periods equal N in symmetric

Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 1923, 2012 1123

and asymmetric geometry is given by [5]

McellL = (M3 M2 M1)N , McellR = (M1 M2 M3)N (8)McellL = McellR = (M3 M2 M1)N (9)

And therefore defect matrix (Mdef ) is given by Eq. (4).Then the characteristics matrix of entire system is expressed as

Mtot = McellR Mdef McellL (10)3. NUMERICAL RESULTS AND DISCUSSION

In this paper, the layers 1 and 3 are ZnSe and Na3AlF6 which refractive indices and thicknessesare n1 = 2.6, d1 = 90nm, n3 = 1.34, d3 = 90 nm, respectively [6]. The metallic layer is taken to besilver (Ag) with the plasma frequency p = 22.175101515 rad/s [7], d2 = 10nm. The thicknessof defect layer is ddef = 10 nm and its index of refraction is calculated with nd ddef = q0/4, witha design wavelength of 0 = 1550 nm in infrared region. The number of periods for right and leftcells is equal. The substrate is assumed to be air with nA = 1.

3.1. Defect Modes in 1DSTMDPC with Metallic Defect LayerIn Fig. 3, we plot the reflectance response for symmetric 1DTMDPC (1DSTMDPC) as shown inFig. 1, at different angles of incidence at ddef = 10 nm, and N = 5 for TE wave. We see that defectmode move to the shorter wavelengths with increasing angle of incidence.

We now in Fig. 4 show wavelength-dependant reflectance of 1DSTMDPC with the structure of(A/(n1n2n3)5d(n3n2n1)5/A) for TM-wave. For normal incidence, that is at 0, there is no differencebetween TE and TM waves. We see that resonant peak move to the left (shorter wavelengths) asthe angle of incidence increases. We give the wavelength of the defect modes for different angles ofincidence in TE and TM wave in Table 1.

As we see in Table 1 defect mode move to the shorter wavelengths for TE wave as comparedwith TM wave in 1DSTMDPC.

3.2. Defect Modes in 1DATMDPC with Metallic Defect LayerIn Fig. 5, we plot the reflectance response for asymmetric 1DTMDPC (1DATMDPC). We see defectmode move to the shorter wavelengths with increasing angles of incidence.

We show in Fig. 6 wavelength-dependant reflectance of 1DATMDPC with the structure of(A/(n1n2n3)5d(n1n2n3)5/A) for TM-wave. For normal incidence, that is at 0, there is no differencebetween TE and TM wave. We see that resonant peak move to the left (shorter wavelengths) as

Figure 3: The calculated wavelength-dependant reflectance for the 1DSTMDPC for TE-wave.

Figure 4: The calculated wavelength-dependant reflectance in 1DSTMDPC for TM-wave.

1124 PIERS Proceedings, Moscow, Russia, August 1923, 2012

Figure 5: The calculated wavelength-dependant reflectance for the 1DATMDPC for TE-wave.

Figure 6: The calculated wavelength-dependant reflectance for the 1DATMDPC for TM-wave.

Table 1: The location of defect mode in 1DSTMDPC by changing angle of incidence.

Angels (degree) Wavelength in TE wave ( nm) Wavelength in TM wave ( nm)0 529 52915 521 52630 501 51945 473 50960 446 49975 427 490

Table 2: The location of defect mode in 1DATMDPC by changing angles of incidence.

Angels (degree) Wavelength in TE wave ( nm) Wavelength in TM wave ( nm)0 795 79515 790 79030 778 77445 759 74860 740 71675 726 690

the angle of incidence increases. We give the wavelength of the defect modes for different angles ofincidence in TE and TM wave in Table 2.

As we see in Table 2 defect mode in 1DATMDPC move to the shorter wavelengths for TM waveas compared with TE wave.

4. CONCLUSION

In this paper, we have studied properties of defect modes in 1DTMDPC. Defect layer is made ofmetallic element with structure A/(n1n2n3)5d(n3n2n1)5/A for symmetric geometry and A/(n1n2n3)5d(n1n2n3)5/A for asymmetric geometry. As we have shown, there is just one defect mode formetallic defect layer. Moreover defect modes move to the shorter wavelengths for TE and TMwaves, but, defect mode move to the shorter wavelengths for TE wave in 1DSTMDPC as comparedwith TM wave. Also, defect mode move to the shorter wavelengths for TM wave in 1DATMDPCas compared with TE wave in this structure.

Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 1923, 2012 1125

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