Vol. 34, No. 11 Journal of Semiconductors November 2013

Influence of incident angle on the defect mode of locally doped photonic crystal�

Wang Jin(王瑾)1, Wen Tingdun(温廷敦)1; �, Xu Liping(许丽萍)1, and Liu Zufan(刘祖凡)2

1Department of Physics, North University of China, Taiyuan 030051, China2China Aviation Lithium Battery Co., Ltd, Luoyang 471000, China

Abstract: Bymeans of a transfer matrix method, this paper deduces the transmittance calculation equation of lighttravelling in locally doped (including one defect layer) mirror heterostructure (ABCCBA)PD(ABCCBA)Q photoniccrystals. In the cases of defect layers being either introduced or not introduced, an ORIGIN simulation showsthe influence of incident angle change on the number of photon band gap, bandwidth and defect mode numbers.Studies indicate that when such photonic crystals have 8 mirror cycles and the thickness of defect layer D meetsnDdD D �0=2 or nDdD D 4�0, the photonic crystal defect mode transmission peak changes significantly. Also,with the change of incident angle, the number of defect mode transmission peaks changes. By altering incidentangle and defect layer thickness, we can get photon band gaps and defect mode transmission peaks at differentfrequency domains and different relative angular frequencies. This provides theoretical reference for achievinglight wave multi-channel filtering and tunable filtering.

Key words: photonic crystal; defect mode; incident angle; locally doped; transfer matrix methodDOI: 10.1088/1674-4926/34/11/112003 PACC: 4230D; 4270Q; 7820P

1. Introduction

The concept of “photonic crystal” was advanced in 1987by both YablonovitchŒ1� at the Baer Lab, USA, and JohnŒ2� atPrinceton University, USA, respectively. The former studiedit from the perspective of inhibiting the spontaneous radiationof the light while the latter from the angle of the photon local-ization in the superlattice. Photonic crystal has important char-acteristics such as photonic band gap, Anderson photon local-area effect, and inhibition of spontaneous radiation. In 1990,researchers Ho et al. from Ames Lab, Iowa State University,USA, verified the existence of the photonic band gap. Sincethe photonic band gap excludes the existence of photons in itsfrequency range, it can be used to make high efficiency lowloss reflectors, broadband band-stop filters, very narrow bandfrequency selecting filters, etc. Moreover, the photonic crys-tal’s local area feature obtained by the introduction of a defectlayer can be utilized in photonic crystal waveguides, photoniccrystal micro cavities, photonic crystal quantum cascade lasers,etcŒ3�6�.

Recent years have witnessed considerable investigationefforts both at home and abroad on one-dimensional, two-dimensional and three-dimensional photonic crystal. However,the majority of these studies are concerned with heterogeneousdual-periodical photonic crystal such as AmBnAmBnAmBn,with defect layers introducedŒ7�9�. Chen et al.Œ10� adopted thetransfer matrix method and studied the feasibility of usingSi/C60 multilayer films as one-dimensional photonic band gapcrystal by theoretical calculations. Xiao and YangŒ11� presentedboth theoretical and experimental investigations on the filter-ing characteristics for metal/dielectric photonic crystals withhexagonal round hole arrays in optically thin gold/silicon diox-ide films by varying the array periodicity from 6 to 8 �m ev-

ery 1 �m while the radio of hole radius to array periodicity iskept constant (1/4). Based on the 1D doped photonic crystal(AB)NC(BA)N model, Reference [12] studied the change ofdefect mode frequency with different incident angle, thicknessof impurities and refractive index of impurities. Recently, Ke-dia, Kumar and SinghŒ13� adopted the self-assembly techniqueand produced three-dimensional polymeric photonic crystalthat can yield multiple optical vortices (OVs). However, lit-tle attention has been paid to the locally doped mirror triplyperiodic photonic crystal heterostructures (MTPPCH). In or-der to gain clear understanding of the optical characteristics ofsuch crystals so that they can be better applied, this paper pri-marily studies the influence of incident angle change on thephoton band gap and defect mode when photonic plane wavesapproach the surface of MTPPCH, and research is carried outin the presence or absence of defect layers. In addition, we alsoutilize ORIGIN to simulate the above-mentioned research. Thesimulations and analysis reveal that the photonic crystals have8 mirror cycles and the thickness of defect layer D satisfiesnDdD D �0=2 or nDdD D 4�0, via altering the incident angle,the notable changes of defect mode could be discovered. Thisinvestigation will provide an important reference for the designof multi-channel and optical tunable filters.

2. Modeling

Figure 1 shows the model of a photonic crystal consist-ing of three materials with the refraction index of nA D 1.45(SiO2/, nB D 2.35 (ZnS), nC D 2.60 (TiO2/ respectively. It islocated between two uniform media with a refraction index ofn0 and nG and composed of N mirror cycles. Each mirror cy-cle has six thin layers: A, B, C, C, B, A and N cycles growalternatively. In the middle of the photonic crystal, D repre-

* Project supported by the National Natural Science Foundation of China (Nos. 60776062, 50730009).† Corresponding author. Email: tdwen@nuc.edu.cnReceived 3 April 2013, revised manuscript received 4 June 2013 © 2013 Chinese Institute of Electronics

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Fig. 1. Locally doped photonic crystal model diagram (ABC-CBA)PD(ABCCBA)Q.

sents the defect layer, with refraction index nD D 1.80. Thus,the structural arrangement of the photonic crystal can be ex-pressed as (ABCCBA)PD(ABCCBA)Q, where P and Q rep-resent the number of cycles at both sides of the defect layer,and P C Q D N .

3. Methodology—the transfer matrix method

Assume that the incident monochromatic plane wave is alinearly polarized light, and it can be divided into a TE wavewhose electric vector is perpendicular to the incident surfaceand a TM wave whose electric vector is parallel to the incidentplane. Due to the fact that the boundary conditions of the verti-cal component and the parallel component of E at the mediummutation are independent, the two waves are therefore inde-pendent to each other. Taking TE waves for instance, assumethat the periodic dielectric film is uniform and non-magnetic,and there will be an electromagnetic field distribution in thefilm when the TE wave approaches the surface of the film Afrom the air with an incident angle of � . With Maxwell equa-tions, we can get the wave equationŒ14�16� of light in any layerof medium:

@2Ey

@z2C kzEy D 0: (1)

The general solution of the equation can be expressed as:

Ey.z/ D Ei1ejkj .z�zj/ C Er1e�jkj.z�zj/: (2)

Now consider the example of interface 1 between the airand medium A, and interface 2 between mediums and B. Ac-cording to the electromagnetic field theory, the tangential com-ponent of electric polarization E and magnetic polarization H

is continuous at the interface between two media. With theseboundary consistency conditions, we can get the linear rela-tion between the two fields at the two interfaces, expressed inmatrix form as follows:

�E1

H1

�D

264 cos ıA �jsin ıA

�A

�j�A sin ıA cos ıA

375 �E2

H2

�: (3)

Let

MA D

264 cos ıA �jsin ıA

�A

�j�A sin ıA cos ıA

375, then�

E1

H1

�D MA

�E2

H2

�(4)

where �A D

q"0

�0

p"A cos �A, ıA D �

!c

nAdA cos �A. ıA indi-cates the phase difference of the plane wave of wave vector k

vertically across the two interfaces in the medium.For each period of the photonic crystal consisting of six

layers of dielectric films ABCCBA, each layer has a specificn and d . Thus, the fields of the first and the last interface arerelated by Eq. (4), that is�

E1

H1

�D MAMBMCMCMBMA

�E7

H7

�;

M D MAMBMCMCMBMA: (5)

By means of recursion, we can get the transfer matrixof photonic crystals (ABCCBA)PD(ABCCBA)Q between thetwo homogeneous media whose refractive index is n0 and nG,respectively:�

E1

H1

�D M P MDM Q

�EtGHtG

�D M 0

�EtGHtG

�; (6)

where M 0 D M P MDM Q D

�M 0

11 M 012

M 021 M 0

22

�: It can be calcu-

lated:Transmission coefficient:

t DEtG

E1

D2�0

M11�0 C M12�0�G C M21 C M22�G: (7)

Transmittance:

T D t � t�D jt j2 : (8)

4. Research and analysis of the model

Ignoring the dispersion and polarization of the material,the center wavelength in this article is taken as nAdA D

nBdB D nCdC D �0=6 nm. Discussions about the influenceof the incident angle � on the photonic band gap and defectmode are presented as follows, in the cases of defect layer be-ing introduced and not introduced respectively.

4.1. Influence of incident angle on the photonic band gapof the photonic crystal (ABCCBA)P (ABCCBA)Q

Through ORIGIN software simulation, we find that theunique characteristics of the photonic crystal with such struc-ture are most prominent when it has 8 mirror cycles and its de-fect layer is located in the center of the periodic photonic crys-tal. For the purpose of comparison, Figures 2(a)–2(d) show thechange of photon band gap with varied incident angle whena defect layer is not introduced into the 8-period mirror pho-tonic crystals and the incident angle is 0, � /6, � /4 and � /3.The film thickness of A, B and C are related in the equationnAdA D nBdB D nCdC D �0=6. It can be found that thefrequency domain of the photonic band gap widens with theincrease of incident angle, which means the photonic band gapnarrows and moves towards high frequency. Besides, a newphotonic band gap can be found between a relative angular fre-quency of 1.5–2.0 and 3.0–3.5. The above phenomenon can beexplained by noting that a portion of the light will be com-pletely reflected with the incident angle � changing.

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Fig. 2. Transmission spectrum of (ABCCBA)P (ABCCBA)Q with thechanges of the incident angle. (a) � D 0, (b) � D � /6, (c) � D � /4, (d)� D � /3.

4.2. Influence of incident angle on the defectmode of locally doped photonic crystal (ABC-CBA)PD(ABCCBA)Q

When an nDdD D �0=2 thick defect layer located in thecenter of the photonic crystal is introduced into the above 8 pe-riod photonic crystal, Figure 3 simulates the change of photonband gap with varied incident angle of 0, � /6, � /4 and � /3. Asshown in Fig. 3(a), in the case of normal incidence, a transmis-sion peak with the transmittance of 95.9% can be observed si-multaneously at the relative angular frequency of 0.494, 2.506and 3.494. It is shown in Figs. 3(b)–3(d) that, with the increaseof incident angle, the photon band gap and transmission peakmove towards high frequency, the frequency domain of pho-ton band gap widens and the transmittance of the transmissionpeak tends to decrease. At the same time, along with the gen-eration of new photonic band gap between 1.0–2.0 and 3.0–3.5, some original non-noteworthy photonic band gaps disap-pear by degrees. At the relative angular frequency of 0.512,0.532 and 0.555, the transmittance of the transmission peakare respectively 95.8%, 99.7% and 99.8%. At the relative an-gular frequency of 2.602, 2.714 and 2.844, however, the ratiochanged to 95.8%, 85.3% and 97.0%, respectively. But, over-all, the transmittance of transmission peaks heightens gradu-ally. It can be found that with the increase of incident angle,the original insignificant photon band gap at the relative an-gular frequency of 1.5–2.0 and 2.5–3.5 gradually disappearsand a new photon band gap emerges. This may be due to thestrengthening or weakening of light at a certain relative angularfrequency with the increase of incident angle, which makes theupper band edge of the original insignificant photon band gapcouple with the lower band edge of its adjacent photon bandgap, thus producing a significant photon band gap.

Figure 4 illustrates the change of transmission peak andphoton band gap with varied incident angle when the thickness

Fig. 3. Transmission spectrum of photonic crystals of 8 cycles (ABC-CBA)PD(ABCCBA)Q and the thickness is nDdD D �0=2 with thechanges of the incident angle: (a) � D 0, (b) � D � /6, (c) � D � /4,(d) � D � /3.

of layer A, B, C and defect layer D meets nAdA D nBdB D

nCdC D �0=6 and nDdD D 4�0. In the case of normal in-cidence, two symmetrical transmission peaks with their ratioclose to 1 emerge around the center of relative angular fre-quency 0.5, 2.5 and 3.5. Meanwhile, around the relative angu-lar frequency 1.0 and 2.0, two insignificant transmission peakswith the ration of 1 also emerge. With the increase of incidentangle, these transmission peaks and photon band gap move to-wards high frequency. The transmittance of the transmissionpeak at the lower frequency gradually decreases and the photonband gap frequency domain widens. In the position betweenrelative angular frequency 1.5–2.0 and 3.0–3.5 where there wasno photon band gap, photon band gaps can be observed tomovetowards high frequency. It can also be seen that the transmit-tance of the transmission peak at high frequency changes with-out a common rule. In Figs. 4(a)–4(d), the ratio of the trans-mission peaks at the relative angular frequency 2.544, 2.646,2.763 and 2.896 are respectively 99.8%, 90.6%, 86.2% and 1.Also, in Fig. 4(d), a significant transmission peak with the ra-tio of 1 gradually emerges at the relative angular frequency1.174; while at the relative angular frequency 2.691 and 3.356where there was no transmission peak, new peaks with the ra-tio of 94.7% and 98.1% have emerged. Since the root causeof the resonant transmission is the defect layer, whose opticalthickness is consistent with its defect mode frequency, it canbe concluded that the resonant transmissionŒ17� is caused bythe interference of the reflected wave and the travelling wavewhen the optical thickness of the defect layer equals four op-tical wavelengths. Two transmission peaks with high transmit-tance emerged at certain relative angular frequency in the caseof normal incidence. When the incident angle increases to acertain degree, the lower band edge of the stop band overlapsthe upper band edge of its adjacent stop band, thus producing a

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Fig. 4. Transmission spectrum of photonic crystals of 8 cycles (ABC-CBA)PD(ABCCBA)Q and the thickness is nDdD D 4�0 with thechanges of the incident angle:(a) � D 0, (b) � D � /6, (c) � D � /4,(d) � D � /3.

new defectmode. This can be explained by the tight binding ap-proximation theory in solid state physics. When TEMwaves ofdifferent energy are incident on the interface, multiple reflec-tions occur on the interface which leads to the localization ofelectromagnetic field energy in a special interface. The strengthof the transmission peak is caused by the strength of the fieldintensity of local areaŒ18� .

5. Conclusions

By means of the transfer matrix method and ORIGIN soft-ware, this paper studies the influence of incident angle changeson the photon band gap and defect mode transmission peak, inthe cases of defect layers being introduced or not introducedinto the MTPPCH. Results show that, in the case of normal in-cidence when the defect layer thickness meets nDdD D �0=2

or nDdD D 4�0, a significant transmission peak with ratioof 95.9% emerged at the relative angular frequency of 0.494,2.506 and 3.494, while two symmetrical significant transmis-sion peaks with their ratio close to 1 emerged around the cen-ter of relative angular frequency 0.5, 2.5 and 3.5. It is also ob-served that with the increase of incident angle, the photon bandgap and transmission peak move towards high frequency andthe transmittance decreases. In addition, a new stable photonband gap and significant transmission peaks can be observedto come into being.

By altering the angle of the incident light, we can get thedefect mode transmission peaks at different relative angularfrequencies. This can be used to achieve multi-channel filter-ing. Also, the changing rule of the defect mode transmissionpeak with varied defect layer thickness is able to provide a cer-tain theoretical basis uponwhich tomanufacture tunable filters.

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