介质层厚度无序对Bragg微腔模式的影响

Effects of Disorder on the Mode of Photonic Crystal Bragg

Cavity

ZHENG Gai-ge, SHI Lin-xing, WANG Hai-jiao, JIANG Li-yong, LI Xiang-yin Applied Physics Department, Nanjing University of Science and Technology, Nanjing 210094

Email: zgggyh1984@https://www.360docs.net/doc/169193259.html,

Abstract ：Based on eigen-mode expansion method (EME) and perfectly matched layer (PML) absorbing boundary condition, the property of the mode of planar photonic crystal Bragg cavity which composed of and bilayers are investigated. The study of localization behavior

in a randomly layered medium is extended from normal to oblique incidence ，both TE and TM modes are considered. The effects of disorder and incident angle on the mode and the localization length of the cavity are studied in detail. The localization length is found to be very small in gaps and much larger in bands, the transmission decreases in bands while increases in gaps as the disordering degree increases .It is revealed that when light incident upon the system and the disordering degree is small, the localization length of the band is lager than the one in the gap, and their relationship with disordering degree is different. When the disordering degree is bigger, localization length is affected not only by disordering degree and photonic band gap, but also the material. Moreover, the localization length can be changed with period and incident angle, and it’s behaviors are very different for TE and TM modes.

2TiO 2SiO Keywords: Bragg micro-cavity; eigen-mode expansion method; perfectly matched layer absorbing boundary condition; random thickness

1 Introduction

During the past two decades, the characteristic of the photonic crystal micro-cavity has

attracted considerable interests of researchers [1-3]. By introducing a micro-cavity into the PBG

structures, it is possible to obtain highly localized cavity modes inside the photonic stop band, which is analogous to the impurity states inside the semiconductor band gap. Researchers have fabricated planar photonic crystal micro-cavity by using of the methods of PVD, MBE, CVD, MOCVD, or MOVPE [4-6], and so on. However, manufacture of photonic crystals is still a real challenge for the optical domain. In practical cases, disorder usually caused by randomly distributed material defaults or manufacture errors during production process is very common, this may lead to localization phenomenon, like the well-known Anderson localization of electron waves in disordered systems [7]

.

The localization length has been introduced in the study of random medium and is a length representative of the thickness of material necessary to localize light (stop light propagation). Very recently, some studies have dealt with extended defect in 2D and 3D disturbed photonic crystals or in a much simpler 1D model [8-16]. These studies have been presented using different numerical methods. However, depending on the problem tackled, not all methods are equivalent or suitable.

In this paper, we have extended the study from normal incidence to oblique incidence. Both TE and TM waves are considered, the property of the mode of the one-dimensional (1D) photonic crystal micro-cavity which composed of and bilayers is studied by using of the

eigen-mode expansion method (EME) coupled with the perfectly matched layer (PML) boundary 2TiO 2SiO

conditions. Due to the existence of Brewster anomalies for the TM mode, the localization length behaviors are very different for TE and TM waves. These results can provide useful information for the study of multichannel systems in the strong scattering limit [17].

2 Outline of EME method

EME is a powerful method for calculation of electromagnetic propagation which has been well known amongst academic environments and also in microwave fields, representing the electromagnetic fields everywhere in terms of a basis set of local modes. It is at the same time a rigorous solution of Maxwell’s Equations and is able to deal with very long structures [18]. In addition, this algorithm is inherently bi-directional, all reflections are taken into account.

We identify a unit cell in the crystal oriented along a certain propagation direction, and subsequently divide this cell into layers in which the index profile does not change in the propagation direction. In each of these layers, we expand the field in the local eigenmodes of that particular layer [19]. Using mode matching

[20], we can eventually derive reflection and transmission matrices that completely describe the scattering behavior of the unit cell

…

Fig.1 Unit cell in a periodic structure.

1,22,121,22,112B t r F F r t 1B =??????????????????i (1)

121,21,1()i j d t diag e T β+?=i (2)

1,21,1r R += (3)

(4) 12122,11,1()(i j d j d r diag e R diag e β+?=i i )i β?) (5)

122,11,1(i j d t T diag e β+?=F and B are column vectors containing the expansion coefficients of the forward and backward propagating fields, respectively. T relates the transfer matrices of a structure . βis propagating constant , and d is the thickness of certain layer. As we know the scattering matrix of every unit, then we can calculate the matrix of the whole structure

(6) 1,11,111,11,111p p p p p p T R F F R T B B ++++++=???????????????i 1???p p p p +

(7) 11,1,1,1,11,()p p p p p p T t I R r T ?+++=?i i

(8) 11,1,1,1,1,11,1,()p p p p p p p p p p R t I R r R t r ?+++++=?+i i i

(9) 11,1,1,1,1,11,1,()p p p p p p p p R T I r R r T R ?+++=?+i i i i (10) 11,1,1,1,11,()p p p p p p T T I r R t ?++=?i i i Choosing the right boundary conditions is crucially important for the accuracy and efficiency of the EME method. The boundary at 0X imposes a certain relation between forward and

backward-propagating waves [21]:

(11) 0000F r B ?=On right side, we impose a similar boundary condition: (12) 0N N N F r B ?=3 Numerical results and discussions

The system contains N units, each unit is composed by two layers with refractive index and 2a 2.4()n TiO =O 21.45()b n Si =respectively.

The defect is , and it’s thickness is 100nm. To introduce disorder, we choose the width of

the layers to be random variable 2TiO 0(1)n n d d υη=+,where ,n a b =,052a d nm =,, and 058b d nm =(01)υυ≤≤ describes the strength of randomness and η is a random number between (.

0.5,0.5)

?

Fig.2. A schematic diagram of photonic crystal micro-cavity.

When the plane wave propagates in such a system, the transmission and reflection can be tackled in a manner by the TMM method [22,23]. But here we take use of the EME method coupled with the perfectly matched layer (PML) boundary conditions, and compared with the other methods. The localization length (ξ) of wave transmitting through such a finite disorder system can be obtained as

2ln La T ξ=??? (13) Where is the thickness of the the slab of the disordered photonic crystal, represents the logarithmic average of the transmission over different Configurations for the disordered sample. Here, the distribution of T is obtained from the results of 100 configurations.

La ln T ??3.1 The effect of the disordering degree on the defect mode of the cavity.

The unit of the localization length is the same as Λ(a b Λ=+, is the spatial period ) , the relationship between localization length and incident wavelength when light is incident upon the system is shown in Fig.3.

When the thickness of is random, if , the localization length in the super-edge is larger than the sub-edge’s. 2TiO 0.4υ<ξ is decreasing with the disordering degree in the band, but increasing with disordering degree in the gap. In other words, the transmissivity of the band is increasing with disordering degree while the transmissivity of the gap is decreasing. As the thickness of is random, we can get the similar conclusion. If 2SiO 0.4υ>，the localization length in the super-edge is smaller than the sub-edge’s, and it’s value is much smaller than the one corresponding to small υ. But when the thickness of is random, if , the result is

2SiO 0.4υ>

opposite, as seen in Fig.3(d).

3.2 The effect of the incident angle and period on the defect mode of the cavity.

Both TE and TM modes must be considered when oblique incidence is taken into account. The localization length behaviors are very different for TE and TM modes. When 0.3υ=, for TM mode, the localization length in the down band is larger than the one in the upper band obviously as seen in Fig.4(a). The value of ξin the gap doesn’t show any orderliness. For TE mode, the value of ξ is less than the one corresponding to TM mode. We also find that the smallest value is coming towards the short wavelength as the incident angle is increasing. This is the result of contributions of Bragg reflection and photon localization.

When both of the layers’ widths are random, the condition is similar as the conclusion we get as above. We also investigate the relationship between the periods and ξ as seen in Fig.5.

Fig.3 The relation between the localization length and wavelength as normal incident . (a),

(b) ,(c) ,(d) corresponding to the disordering degree of is small, the disordering degree of is small,

the disordering degree of is big, the disordering degree of is big respectively .

For the same υ, the bigger the periods, the larger the localization length is. This is because if the periods are more, L becomes larger . When the number of the periods is 20, the localization length corresponding to0.591m Λ.

μis 1220

Fig.4 The effect of incident angle on localization length when 0.3υ=. (a)、(b) is TM and TE mode respectively.

Fig.5.The relationship between periods and localization length.

3 conclusion

In summary , we have analytically extend the effects of random thickness of the layers on the transmission and localization length,which have been confirmed by numerical simulations. We find out that localization length is very small in gaps and much larger in bands, the transmission decreases in bands while increases in gaps as the disordering degree increases. the localization length can be changed with period of the structure and the incident angle, and it’s behaviors are very different for TE and TM modes. The results we get by using of the EME method, is the same as the conclusions in some literature [24,25], and is expected to be useful in understanding the general intrinsic nature of randomly perturbed PCs.

EME method not only is suitable for the structure that composed of multi-layers, but also can be used to solve the two and three dimensional problems, associated with other theories, it is expected to be a powerful and exact solution. In the future, some works should be made in order to analyze the sensitivity to disorder of the very peculiar properties of photonic crystals such as the superprism phenomenon, the negative dielectric constant and the negative magnetic permeability. References

[1] TOMOYUKI Y, AXEL S , CHEN Hao, et al . Optical characterization of two-dimensional photonic crystal

cavities with indium arsenide quantum dot emitters[J].Appl. Phys. Lett.,2001,79 (1) :114-116.

[2] LAURENT S, VAROUTSIS S, GRATIET L, et al Indistinguishable single photons from a single-quantum dot

in a two-dimensional photonic crystal cavity[J]. Appl. Phys. Lett.,2005,87(31):163107(1)-163107(3).

[3] POSANI K T,TRIPATHI V,ANNAMALA S, et al . Nanoscale quantum dot infrared sensors with photonic crystal cavity[J].Appl. Phys. Lett .,2006, 88 (15):151104(1)-151104(3).

[4] RUSSELL P S J,TREDWELL S, ROBERTS P J. Full photonic bandgaps and spontaneous emission control in 1D multilayer dielectric structures[J]https://www.360docs.net/doc/169193259.html,mun.,1999,160(1): 66-71.

[5] CHEN Su, SONG Zhi-tang, WANG Yang, et al. Preparation of one-dimensional photonic crystal with variable

period by using ultra-high vacuum electron beam evaporation[J]. Microelectronics Journal 2207,38

(2):282–284.

[6] BROWEN E R, PARKER C D, YABNOLOVITCH E [J].Radiation properties of a planar antenna on a

photonic-crystal substrate[J]. J. Opt. Soc. Am. B ,1993,10(2):404-407.

[7] ANDERSON P W. Absence of diffusion in certain random lattices[J].Phys. Rev. 1958,109 (5):1492-1505.

[8] SIGALAS M M, SOUKOULIS C M.Elastic wave propagation through disordered and/or absorptive layered

systems[J]. Phys. Rev. B,1995,51(5):2780-2789.

[9] LIN Lan-lan, LI Zhi-yuan, HO https://www.360docs.net/doc/169193259.html,ttice symmetry applied in transfer-matrix methods for photonic

crystals[J].J.Appl.Phys.,2003,15(2): 811-821.

[10] LI Zhi-yuan LIN Lan-lan. Photonic band structures solved by a plane-wave based transfer-matrix method

[J].Phys. Rev.E, 2003,67(4):046607(1)-046607(11).

[11] TANAKA Y, TAMURA S. Two-dimensional phononic crystals: surface acoustic waves[J]. Physica

B ,1999,77(1):263–264.

[12] T.T. Wu, Z.G. Huang, S. Lin. Surface and bulk acoustic waves in two-dimensional phononic crystal

consisting of materials with general anisotropy[J].Phys.Rev.B,2004,69(9):094301(1)-094301(10).

[13] VASSEYR J O, DEYMIER P A,ZEMMOURI J ,et al. Phononic crystal with low filling fraction and absolute

acoustic band gap in the audible frequency range: A theoretical and experimental study[J]. Phys.Rev.

E.2002, 65 (5) 056608(1)-056608(6).

[14] F.G. Wu, Z.Y. Liu, Y.Y. Liu. Splitting and tuning characteristics of the point defect modes in

two-dimensional photonic crystals[J]. Phys. Rev. E ,2004,69(6: 066609(1)-066609(4).

[15] KHELIF A,DJAFARI-ROUHANI B,VASSERU J O,et al.Transmission and dispersion relations of perfect

and defect-containing waveguide structures in phononic band gap materials[J]. Phys. Rev. B,2003, 68(2):024302(1)-024302(8).

[16] PSAROBAS I E, SIGALAS M . Elastic band gaps in a fcc lattice of mercury spheres in aluminum [J]. Phys.

Rev. B, 2002,6 (5):052302(1)-052302(3).

[17] DENG Wen-ji, ZHANG Zhao-qing.Amplification and localization behaviors of obliquely incident light in

randomly layered media[J].Phys.Rev. B ,1997,55(21):14230-14235.

[18] HUANG K C, BIENSTMAN P, JOANNOPOULOS J D, et al. Field expulsion and reconfiguration in

polaritonic photonic crystals[J]. P hys. Rev. Lett.,2003, 90(19):196402(1)- 196402(4).

[19] HUANG K C, BIENSTMAN P, JOANNOPOULOS J D, et al. Phonon-polariton excitations in photonic

crystals[J].Phys. Rev. B,2003, 68(7):075209(1)- 075209(12).

[20] BIENSTMAN P, BAETS R. Optical modelling of photonic crystals and VCSELs using eigenmode expansion

and perfectly matched layers[J]. Opt. Quantum Electron. 2001,33 (4-5):327-341.

[21]BIENSTMAN P, BAETS R. Rigorous and efficient optical VCSEL model based on vectorial eigenmode

expansion and perfectly matchedlayers[J].IEE.Proc.-Optoelectron.2002,149(4):161-165.

[22] MCGUM A R, CHRISTENSEN K T. Anderson localization in one-dimensional randomly disordered optical

system that are periodic on average[J].Phys.Rev.B,1993,47(13):13120-1312.

[23] DEYCH L ,LISYANSKY A A, ALTSHULER B L. Single Parameter Scaling in One-Dimensional

Localization Revisited[J]. Phys. Rev. Lett. ,2000,84(12):2678-2681

[24]ASATRYAN A A, ROBINSON P A, BOTTEN L C, et al. Effects of disorder on wave propagation in

two-dimensional photonic crystals[J]. Phys. Rev. E ,1999,60(5):6118-6127.

[25] Z. H. Zhu, W. M. Ye ,J. R. Ji, et al. Influence of random errors on the characteristics of typical 2D photonic

crystal microcavity[J]. Appl. Phys. B,2007,88(2):231-236.