Analysis and Optimization of a New Photonic Crystal Filters in Near Ultraviolet 2012

8/9/2019 Analysis and Optimization of a New Photonic Crystal Filters in Near Ultraviolet 2012

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Optik 123 (2012) 187418 78

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Optik

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Analysis and optimization of a new photonic crystal lters in near ultravioletband

Huihuan Guan a, b , Peide Han a ,b,, Yuping Lia,b , Xue Zhang a,b , Wenting Zhang a,ba College of MaterialsScience and Engineering,Taiyuan University of Technology, No. 79, Yingze Street, Wanbolin District,Taiyuan 030024, Chinab Key Laboratory of Interface Science and Engineering in Advanced Materials, Taiyuan University of Technology, Ministry of Education, Taiyuan 030024, China

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Article history:Received 26 October 2011Accepted 9 March 2012

Keywords:Photonic crystalOmni-directional reectionPhotonic crystal lterUltravioletvisible range

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Transmission characteristics of one-dimensional (1D) photonic crystals (PCs) heterostructures containingdefective are studied using the transfer matrix method. The key is to search the best combination stylefor different 1D PCs to form heterostructures. It is shown that the non-transmission range over nearultraviolet and visible range can be substantially enlarged and the phenomenon of narrow band PC lterin near ultraviolet can be realized by adjusting the repeat cycle counts of various photonic crystals. Thetheoretical results on multiple heterostructures containing TiO2 /SiO2 multilayer lms are presented.With a perfect omni-directional and high peak transmission lters for TE modes, this structure opensa promising way to fabricate ultra-narrow band PC lters with wide non-transmission range in nearultraviolet and visible range.

2012 Elsevier GmbH. All rights reserved.

1. Introduction

Since last two decades, photonic band gap (PBG) materi-

als, which are structured materials with periodically modulateddielectric function and showing electromagnetic band gaps, haveattracted considerable interest because of the many novel prop-erties in respect of fundamental physics and their potentialapplications in devices [1,2] . Recently, much attention has beenpaid to the properties of defect modes that emerge in the PBGbecause of theexistenceof structural defects [3,4] . Defect modecanbe readily achieved by introducing structure defect into a perfectphotonic crystal (PC), and this defect state allows the transmis-sion of light at corresponding frequency. Narrow band PC lteringphenomena of one-dimensional (1D)periodicstructures havebeenstudied extensively [57] . However, the non-transmission rangeof a 1D PC lter constructed by a single 1D dielectric PC is some-times narrow. Thus the application of the narrow band PC lters is

restricted. If several 1D PCs with different layer thickness or dif-ferent lling factor are combined, multiple heterostructures areformed. Since different photonic crystals have different band gapwidths, multiple heterostructures can show many characteristics,for example, some literature revealed that the non-transmissionrange can be enlarged using heterostructure [810] . The periodic

Corresponding author at: College of Materials Science and Engineering, TaiyuanUniversityof Technology, No. 79, YingzeStreet, Wanbolin District, Taiyuan 030024,China.Tel.: +86 351 6018843; fax: +86 351 6010311.

E-mail addresses: [email protected] , [email protected] (P. Han).

structures can, however, be optimized by introducing defects inmultiple heterostructures.

It is well known that, as the incident angle changes from nor-

mal to oblique incidence, the defect modes with heterostructureswill shift into higher frequencies. It leads to the fact that light of unwanted frequencies cannot be cut off at some incident angles,which will restrict the applications in some outdoors detections.Thus, lters withthe ability of large omni-directional lteringband(i.e., omni-directional reection band except the frequency of thedefect mode) [11,12] are in urgent need. Recently, most of the 1DPC lters with multiple heterostructures reported are focused onthe microwave and infrared regions [13,14] . There are few reportsabout1D PC lters through whichboth TE andTM modes canprop-agate at oblique incidence [15] . Because, a PC lters at UVvisregions have more important applications. It is necessary to dofurther work for investigation of large omni-directional lteringband in ultraviolet and visible light region. Considering the above

issue, in this paper, we will try to apply the transfer matrix methodin optimizing the structure of the multiple heterostructures withdefects and nally obtain an efcient PC lter with enlarged non-transmission range. Efforts are focused on optimization design foromni-directional lters in near ultraviolet region. These wouldidentify the potential applications in the future optical devices atvisible and near ultraviolet range.

2. Structural model and computational method

The heterostructure containing defect is schematically shownin Fig. 1 , where (AB) and (EF) are different PCs with different lling

0030-4026/$ seefrontmatter 2012 Elsevier GmbH. All rights reserved.

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H.Guan etal./ Optik 123 (2012) 18741878 1875

Fig. 1. Schematic of thecalculated heterostructures containing defect. (AB) and(EF) aretwo different1D PCswith differentlled factor. (CD) is a defective PC. n1 , n2 aretherefractive indices of TiO 2 and SiO 2 .

factors (CD)is the defective PCs introduced in the heterostructures.In the calculation, TiO 2 and SiO 2 are chosen as high and low refrac-tive index media. The refractive indices of TiO 2 and SiO 2 are n1and n2 shown in Fig. 1 , which obtain from Refs. [1619] . The het-erostructure can be expressed as (EF) s(AB)m (CD)1 (AB)m . A, C andE all represent TiO 2 , B, D and F represent SiO 2 , where m is the

periodic layers of the potential barrier (AB), and s is the periodiclayer of (EF). The thickness of repeat cycle is represented as a,a =82nm, whereas dA (0.2 a ), dB (0.8 a), dC (0.64 a), dD (0.36 a), dE(0.6 a ) and dF (0.9 a) represent the thicknesses of A, B, C, D, E and F,respectively.

The transfer matrix method (TMM) [20] was applied to studythe transmittance characteristics and the defect modes of theheterostructures containing defect. According to the Maxwellsboundary condition, the tangential components of electric andmagnetic elds must be continuous at the interface between arbi-trary two arbitrary layers. A characteristic matrix for k-th layer isobtained [21,22] . Where there are two modes: transverse mag-netic (TM) and electric magnetic (TE). For the TE (TM) mode, theelectric (magnetic) eld is perpendicular to the x y plane dened

by the wave vector together with the periodic direction. Finally,the total reectivity and transmissivity for heterostructure isobtained.

In a 1D PC, the propagation of an optical pulse is governed bythe wave equation:

2

z 2E ( z, t )

1c 2

2

t 2E ( z, t ) =

10 c 2

2

t 2P ( z, t ), (1)

where E ( z , t )istheelectriceldand P ( z , t ) is theelectricpolarization.Suppose that the pulse canbe expressedas an integral of its Fouriercomponents:

E ( z, t ) =

E ( z, )e it d, (2)

and the Fourier component of the polarization P ( z , ) = ( z , )E ( z ,) under linear response approximation. The equation for theFourier component of electric eld can be written as

d2

dz 2E ( z, ) +

2

c 2 ( z, )E ( z, ) = 0, (3)

where ( z , ) = 1 + ( z , ) is the dielectric function.Our method is that we rstly solve Eq. (3) layer by layer, and

then integrate E ( z , ) byusingEq. (2) to obtain the temporal-spatialbehavior of the pulse. The integral is a coherentsuperposition of allFourier components including the forward waves and backwardwaves with complicated relations of phases.

We consider normal propagation in z direction (layers in x y

plane). The electric eld is polarized along x direction. In the jth

layer the electric eld E j( z , ) satises

d2

dz 2E j( z, ) +

2

c 2 n2 j ()E j( z, ) = 0, ( z j 1 < z < z j, j = 1, 2,...,N ),

(4)

where n j()=

j() is the refractive index that is a constant inthe same layer for a given . The general solution of Eq. (4) can beexpressed as

E j( z, ) = E + j()exp ic

n j()( z z j 1 )

+ E j()exp ic

n j()( z z j 1 ) , (5)

From E = (/t )B= iB we can get the magnetic eld

B j( z, ) =n j()

c E + j()exp i

c

n j()( z z j 1 )

E j()exp i

c n j()( z z j 1 ) , (6)

which is polarized along y axis. Dene

1 j( z, ) = E j( z, ), (7)

2 j( z, ) = icB j( z, ), (8)

such that the electric component 1 ( z , ) and the magneticcomponent 2 ( z , ) can be measured by the same unit. The elec-tromagnetic eld then can be expressed by a two component wavefunction vector,

j( z, ) =1 j( z, )

2 j( z, ) (9)

From Eqs. (5) and (6) w e can obtain the transfer matrix relating j( z + z , ) to j( z , ),

j( z + z, ) = M j( z, ) j( z, ), (10)

Where

M j( z, ) =cos

c

n j() z 1

n j() sin

c

n j() z

n j()sinc

n j() z cosc

n j() z

(11)Because 1 ( z , ) and 2 ( z , ) are proportional to the tangential

components of the electric eld and the magnetic eld, respec-

tively, they are continuous function of z across the layer interfaces.


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1876 H. Guanet al. / Optik 123 (2012) 18741878

At any position z , ( z , ) connects with ( z 0 , ) through a propa-gation matrix. For example, in the jthlayer ( z j 1 < z < z j),theeld at z = z j 1 + z is

( z j 1 + z, ) = Q ( z j 1 + z, ) ( z 0 , ), (12)

where the propagation matrix

Q ( z j

1

+ z, ) = M j( z, )

j 1

i= 1

M i(d

i, ) (13)

From Eqs. (12) and (13) we can calculate the electric eldand the magnetic led at any position provided that ( z 0 , ) =

1 ( z 0 , ), 2 ( z 0 , )T

is known. ( z 0 , ) can be determined bymatchingthe boundarycondition.Assume thelight is incidentfromthe region of z < 0. In this region the eld is a superposition of for-ward eld and backward eld,

E ( z, ) = E i( z, )eikz + E r ( z, )e ikz , (14)

where k = /c . At the incident end we have

(0 , ) = E i (0, ) + E r (0, )

i [E i (0, ) E r (0, )]

(15)

Generally,the forward (incident) eld E i(0, ) is given,while thebackward (reection) eld E r (0, ) is tobe solved. Inorder tosolveE r (0, ), another boundary conditionat theexit end z = z N should beutilized. In the region of z > z N , there is only forward (transmitted)eld,

E ( z ) = E t ( z, )eik( z z N ) (16)

Thus,

( z N , ) = E t ( z N , )

inSE t ( z N , ), (17)

where nS is the refractive index of the substrate. Suppose that the

matrix connecting the incident end and the exit end is X N (), wehave

( z N , ) = X N () (0, ) , (18)

where

X N () =N

j= 1

M j(d j, ) = x11 () x12 ()

x21 () x22 () (19)

Eq. (18) may be rewritten in a form of

(0, ) = X 1N () ( z N , ) (20)

Because det M j(d j)= 1, we have det X N ()= 1 and

X 1N () = x22 () x12 ()

x21 () x11 () (21)

Substituting Eqs. (15), (17) and (21) into Eq. (20) yields

E r (0, ) = r () E i (0, ) , (22)

E t ( z N , ) = t ()E i(0 , ), (23)

where r () is the reection coefcient of a monochromatic planewave of frequency that can be expressedin terms of theelementsof the matrix X N (),

r () =[ x22 () nS x11 ()] i [nS x12 () + x21 ()][ x

22() + n

S x

11()] i [n

S x

12() x

21()]

, (24)

and t () is the transmissioncoefcient of themonochromatic planewave that can be written as

t () =2

[ x22 () + nS x11 ()] i [nS x12 () x21 ()] (25)

In terms of r () and t (), the reectivity and transmissivity canbe obtained by [12]

R = r ()2, T = t ()

2(26)

Thus, we can express the electromagnetic eld at z = 0 with theincident eld E i() as follows:

(0 , ) = E i(0 , ) 1 + r ()

i[1 r ()] (27)

With (0, ) in hand, we are able to calculate the temporal-spatial behavior of the pulse. By multiplying a time factor, e i t toeach side of Eq. (12) and integrating all frequency components, wehave

1 ( z, t ) = 1 ( z, ) e it d = E i (0, ) [1 + r ()] Q 11 ( z, )+ i [1 r ()] Q 12 ( z, ) e

it d (28)

and

2 ( z, t ) = 2 ( z, ) e it d = E i (0, ) [1 + r ()] Q 21 ( z, )+ i [1 r ()] Q 22 ( z, ) e it d, (29)

where Q ij( z , ) arethe elements of the propagation matrix (13) . Theelectric displacement D ( z , t ) can be obtained through

D( z, t ) = 0 3 ( z, t ) = 0 () 1 ( z, )e it d, (30)where 0 is the vacuum permittivity. The unit of 3 ( z , t ) is thesame as 1 ( z , t ) and 2 ( z , t ). Obviously, the eld functions givenby Eqs. (28)(30) are the coherent superposition of all frequencycomponents including forward waves and backward waves. Thewaves enter the integral have complicated relations of phases.

The calculated transmission spectrum shown in Fig. 2 is forthe heterostructure inserted with defective PC. The position andnumber of the defect modes are affected when the periodic layersof the potential barrier (AB) is greater than that of the potentialtrap (CD) by adjusting the repeat layers m of the potential barrier(AB).Fig.2 illustratesthe transmittancespectraof (AB) m (CD)1 (AB)m(m = 25) heterostructures, which the repeat cycle count s of thepotential well-width (CD) remain a constant with one, and therepeat cycle count m of potential barrier are from 2 to 5, respec-tively. Thefull widthat half maximum (FWHM)of the defectmodesgradually narrow with the increasing number m , but the trans-

mittance of defect mode decreases gradually. When m reaches5, the transmittance of defect mode become small, which is only37.93%. To obtain defect mode with high transmittance in UVregion, there should be a proper ratio in the number of repeatedcycle counts m of potential barrier (AB). Generally, the idealcondition is obtained when m is equal to 3 in (AB) 3 (CD)1 (AB)3heterostructure (to see Fig. 2(b)). In Fig. 2(b), we have gained thedefect mode with high transmittance and the FWHM (0.48 nm)is small in UV region. However, the non-transmission range(260.0482.9 nm) is narrow, which could not over the wholevisible and near ultraviolet region. Fig. 3(a )(c) show the transmit-tance spectra of (AB) 3 (CD)1 (AB)3 (EF)2 , (EF)2 (AB)3 (CD)1 (AB)3 and(EF)2 (AB)3 (CD)1 (AB)3 (EF)2 different heterostructures. In Fig. 3(a)and (c), the non-transmission ranges for these heterostruc-

tures of (AB) 3 (CD)1 (AB)3 (EF)2 and(EF) 2 (AB)3 (CD)1 (AB)3 (EF)2 could


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H.Guan etal./ Optik 123 (2012) 18741878 1877

300 400 500 600 700 800

0.0

0.5

1.00.0

0.5

1.00.0

0.5

1.00.0

0.5

1.0

m=3

m=5

m=2

m=4

Wavelength (nm)

(d)

(c) T r a n s m i t i v i t y

(b)

(a)

Fig. 2. The relationship between the transmittance characteristic of defect modeand the potential barrier (AB) repeat layers m of (AB) m (CD)1 (AB)m (m =25) het-erostructures.

not over the whole visible and near ultraviolet region. But,(EF)2 (AB)3 (CD)1 (AB)3 heterostructure has a non-transmissionranges of 260900nm, whichoverthe wholevisible andnearultra-violet region [ Fig. 3(b)]. It is obvious that the non-transmissionrange can be substantially enlarged as desired by choosing properheterostructures. There is a PC lter wavelength corresponding tothe peak at 364.7nm appeared in the non-transmission ranges.

300 400 500 600 700 800 900 10000.0

0.5

1.0

0.0

0.5

1.0

0.0

0.5

1.0 (AB)3(CD)

1(AB) 3(EF) 2

(EF)2(AB) 3(CD) 1(AB) 3(EF) 2

(EF)2(AB) 3(CD) 1(AB) 3

Wavelength (nm)

(c)

T r a n s m i t i v i t y

(b)

(a)

Fig. 3. The relationship between the location of the (EF) 2 (right and left) with the

transmittance characteristic of the defect mode.

300 400 500 600 700 800 900 10000.0

0.5

1.0

0.0

0.5

1.0

0.0

0.5

1.0

364.4 364.8 365.20.45

0.50

0.55

s=6,m=3

s=4,m=3

s=2,m=3

Wavelength (nm)

(c)


(b)

[]

(a)

0.47 nm


Wavelength (nm)

Fig. 4. The relationship between the defect modes and the number s of (EF) s (AB)3(CD)1 (AB)3 (s = 2, 4, 6) heterostructures.

Since our model is unknownprior to optimization, thesearchingareasof the heterostructures thicknessand number of periods haveto be determined. In this section, we will analyze the optimizedstructural parameters of multiple heterostructures. That a perfectnon-transmission range over the whole visible and near ultravioletregion, whichhas an ultra-narrow band PC lter in near ultravioletregion, is thedestination. Fig.4 showsthe trasmission spectra of thePC lter of the herterostructures (EF) s (AB)3 (CD)1 (AB)3 , where s = 2,s =4 and s = 6, respectively. We can see from Fig. 4(a), (b) and (c)that the transmission channels appear in the stop gap formed bythe heterostructures. The transmittances of the lter are 99.38%,97.36% and 97.98%, respectively. The FWHM is ultra-narrow, lessthan 0.48nm. From the theoretical result, the non-transmissionwavelength range canbe enlarged from 260to 900 nm. Thus, closeranalysis shows that (EF) 4 (AB)3 (CD)1 (AB)3 heterostructure exhibitstheideal result, inwhich the transmittance of thePC lteris 97.89%,in which the FWHM is 0.47nm [ Fig. 4(b)].

It is known that the optical properties of conventional l-ters are relevant to incidence angle and polarization of theincident light, even those based on the defect modes in omni-directional gaps [15,23] . Based on aforementioned optimal design,

Fig. 5(a) shows the transmission and absorption spectrum of (EF)4 (AB)3 (CD)1 (AB)3 heterostructures in incidence angles 0 forthe TM and TE modes. Fig. 5(b), (c) and (d) are shown the trans-mission properties of (EF) 4 (AB)3 (CD)1 (AB)3 heterostructures, andare respectively corresponding to the incident angles from 0 to90 with an interval of 30 . The solid and dot curves correspondto TE and TM modes, respectively. And the absorption spectrumof the (EF) 4 (AB)3 (CD)1 (AB)3 PC in short wavelength (260400nm)obtained have been shown in the small gure in Fig. 5. We cansee that the lters for the TE mode has no angular effect when theincident angle is less than 90 , a perfect omni-directional and highpeak transmission lters for TE mode can be formed. The lter forthe TM mode has no angular effect when the incident angle is lessthan40 . With theincrease of theincident anglesfrom0 to90 ,the

location of the lters move to a shorter band called blue shift, and


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1878 H. Guanet al. / Optik 123 (2012) 18741878

300 400 500 600 700 800 900 10000.0

0.5

1.0

0.0

0.5

1.0

0.0

0.5

1.0

0.0

0.5

1.0

270 300 330 360 3900.00.2

0.4

0.6

90 Degree

(a)

(b)

Wavelength (nm)

60 Degree(c) T r a n s m i t i v i t y

30 Degree

0 Degree

(d)

TMTE

A b s o r b t i v i t y

avelength (nm)W

Fig. 5. Transmission spectra of (EF) 4 (AB)3 (CD)1 (AB)3 heterostructure containingdefect at different angles. From (a) to (d) are corresponding to the incident anglesfrom 0 to90 , respectively, with an interval 30 . Thesolidand dotcurves representTE and TM modes, respectively.

the variation in the TM mode is more signicant than that in the TEmode ( Fig.5 ). The structure is suitable for ultra-narrow band PC l-ter at visible and near ultraviolet range, the band width of 0.47nmis quite a narrow one.

3. Conclusion

In this paper, an ultra-narrow band PC lter with widenon-transmission range and high transmissivity was obtainedby using the PC heterostructures in UVvis region. When the(EF)4 (AB)3 (CD)1 (AB)3 heterostructures have the optimized struc-ture parameters, a =82nm, s =4 and m =3, the PC lter with hightransmission of 97.89% located in the UV region (364.7nm), whichthe FWHM is 0.47nm. The new PC heterostructure provides a per-fect omni-directional and high peak transmission lters for TEmode. This structure opens a promising way to fabricate ultra-narrow band PC lters with wide non-transmission range in nearultraviolet and visible band.

Acknowledgements

This research was supported by the National Natural Sci-ence Foundation of China (Grant No. 50874079, 51002102),

and the Taiyuan Science and Technology Project (Grant No.100115105).

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Analysis and Optimization of a New Photonic Crystal Filters in Near Ultraviolet 2012