Novel application of a perturbed photonic crystal: High-quality filterXin-Ya Lei, Hua Li, Feng Ding, Weiyi Zhang, and Nai-Ben Ming Citation: Applied Physics Letters 71, 2889 (1997); doi: 10.1063/1.120207 View online: http://dx.doi.org/10.1063/1.120207 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/71/20?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Multichanneled filter based on a branchy defect in microstrip photonic crystal Appl. Phys. Lett. 88, 081106 (2006); 10.1063/1.2176851 Photonic states deep into the waveguide cutoff frequency of metallic mesh photonic crystal filters J. Appl. Phys. 97, 033102 (2005); 10.1063/1.1850993 Angular Response of Narrow Band 2DPBG Guided Wave Filters AIP Conf. Proc. 709, 439 (2004); 10.1063/1.1764047 Enlargement of the nontransmission frequency range of multiple-channeled filters by the use of heterostructures J. Appl. Phys. 95, 424 (2004); 10.1063/1.1634367 Dual-band infrared metallodielectric photonic crystal filters Appl. Phys. Lett. 77, 2098 (2000); 10.1063/1.1314880

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Novel application of a perturbed photonic crystal: High-quality filterXin-Ya Lei,a) Hua Li, Feng Ding, Weiyi Zhang, and Nai-Ben MingNational Laboratory of Solid State Microstructures and Department of Materials Science and Engineering,Nanjing University, Nanjing 210093, People’s Republic of China

~Received 18 June 1997; accepted for publication 15 September 1997!

The transmission of light waves through a perturbed photonic crystal has been investigated. Theperturbed photonic crystal was constructed by randomly repeated stacking of a number of identicalunit cells. Although most of the light waves are localized by the randomness, there still exist lightwaves with special wavelengths which are in extended states. This produces a high-quality resonanttunneling with a very narrow transmission coefficient peak. ©1997 American Institute of Physics.@S0003-6951~97!01246-1#

The concept of photonic crystal stems from early ideasof Yablonovitch and John.1 The idea is to design materials sothat they can affect the properties of photons, in much thesame way that ordinary semiconductor crystals affect theproperties of electrons. In particular, photonic crystals forbidpropagation of photons having a certain range of energies~known as photonic band gaps!, which could be incorporatedinto the design of novel optoelectronic devices.2 Followingthe demonstration of a material with full photonic band gapat microwave frequencies,3 there has been considerableprogress in the fabrication of three-dimensional photoniccrystals with operational wavelength as short as 1.5mm.4

Although most of the efforts focused on the search of a ma-terial that exhibits a full photonic band gap, it has been rec-ognized that the introduction of defects into the photoniccrystal, either locally or in an extended region, will allow usto generate electromagnetic states with specificproperties.5–12 In this letter, we study light transmission in atype of perturbed photonic crystal. Here we employ one-dimensional photonic crystal13,14 as a simple example. This

type of perturbed photonic crystal is shown in Fig. 1. Themotion of light in such a perturbed photonic crystal is analo-gous to the motion along a one-dimensional disordered sys-tem. According to the scaling theory,15 all the wave func-tions are localized in such a system. However, for electronsin a one-dimensional random double barrier system, Dunlapet al.16 have found a small amount of states which are stillextended under this type of randomness. On the basis ofthese findings we think it may be interesting to investigatethe transmission of light through a perturbed photonic crys-tal. The results show that it is possible to produce a high-quality resonant tunneling with a very narrow transmissionpeak. This is a novel mechanism for an optical filter.

The thickness of thei th spacerdi , satisfies a uniformprobability distribution

P~di !5 H1/~d22d1! for d2>di>d1 ,0, otherwise . ~1!

The transmission of light through one unit cell~A andB!can be represented by the characteristic matrixT∧

17

T∧5S cosdA cosdB2nB

nAsin dA sin dB 2

i

nBcosdA sin dB2

i

nAsin dA cosdB

2 inA sin dA cosdB2 inB cosdA sin dB cosdA cosdB2nA

nBsin dA sin dB

D ~2!

in which the phased is given bydA(B)5nA(B)kdA(B) , wherek is the wave vector in vacuum and wheredA(B) is the thick-ness of the components. The transmission of a lightwavethrough the above mentioned perturbed photonic crystal isrepresented by a matrix string

T5TMTS1TMTS2 ...TMTSi ...TMTSNTM , ~3!

where TM is the transfer matrix forM unit cells, TSi( i

51,2,...,N) is the matrix for thei th spacer. Thus, the trans-mission coefficient for tunneling through such a structure canbe calculated as

t54

uT111T22u21uT121T21u2 , ~4!

whereTi j are the elements of the matrixT.TM can be further decomposed,TM5(T∧)M. From the

theory of matrices, theM th power of the 232 unimodularmatrix T∧ can be written as17

TM5mM21~x!T∧2mM22~x!I , ~5!

where x5 12 Tr(T∧), and mM(x) is the Chebyshev polyno-

mial of the second kind, which obeys the recurrence relation

mM11~x!52xmM~x!2mM21~x!I , M>0 ~6!

with m2150, m051, anda!Electronic mail: xylei@nju.edu.cn

2889Appl. Phys. Lett. 71 (20), 17 November 1997 0003-6951/97/71(20)/2889/3/$10.00 © 1997 American Institute of Physics This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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mM21~x!5sin@M cos21~x!#

sin@cos21~x!#. ~7!

For the spacers

Tsi5S cosd i 2

i

nAsin d i

2 inA sin d i cosd i

D , ~8!

whered i5nikdi , di is the thickness of thei th spacer. If

x5cosS lp

M D[xl , l 51,2,...,M21, for M>2, ~9!

we havemM21(xl)50, andmM22(xl)5(21)l 11. From Eq.~5!, we haveTM5(21)l I , and

Ts5S cosd 2i

nAsin d

2 inA sin d cosdD , ~10!

whered5nAkD, D is the total length of all spacers.Under the condition of Eq.~9!, the transmission coeffi-

cient is equivalent to that of a lightwave through a homog-enous region of low-dielectric medium with thicknessD.This means that the states with wavelengthl satisfying Eq.~9! are completely unscattered. We iterate numerically thematrices of Eq.~3!. Figure 2 shows the calculated result ofthe transmission–wavelength curves. It is striking to notethat most states are localized due to the randomness, the

transmission coefficient for these states vanishes, but therestill exist states with certain wavelengths which have high-quality tunneling with very narrow transmission peaks.

In order to investigate directly the physical reason forthis phenomenon, we study the light amplitude as a functionof position in such a photonic crystal. Most injected light-waves are localized in the photonic crystal, such as light ofl54.5, which was in a passband in an ordinary photoniccrystal withnA /nB53.4 anddA /dB50.286. When the pho-tonic crystal is perturbed, it would be localized as shown inFig. 3~a!. But for lightwaves with certain wavelength, itsamplitude is Bloch wavelike and is clearly delocalized~orextended! @see Fig. 3~b!#. It is fundamental to note that bothlocalization and delocalization phenomena are pure conse-quences of Maxwell’s equations and disorder. In otherwords, the phenomena are observed numerically, but no con-dition of localization or delocalization has been introducedinto the theory.

For the application of this novel phenomenon, anotheraspect is the determination of the locations of the resonanttunneling states. In fact, from Eq.~9! we have

cosdA* cosdB* 2S nA

nB1

nB

nAD sin dA* sin dB* 5cosS lp

M D ,

l 51,2,...,M21, ~11!

wheredA* 5nAkldA , dB* 5nBkldB , andkl is the correspond-ing wave vectors. For lightwave with wave vectorkl , thetransfer matrix string of the perturbed photonic crystal isonly composed of matrixTs and (21)l I . This matrix stringis equivalent to that of the homogeneous medium of thespacers with thickness ofD5Sdi . For this perturbed pho-tonic crystal, the light with wave vectorkl is an extendedstate, which is completely unscattered. The wavelength loca-tions of the extended states also can be predicted from Eq.~11!. Figure 4 depicts the wavelength locations which changewith the dielectric constant contrast of the perturbed photo-nic crystal for fixed structural parameters. It can be foundthat the tunneling wavelength is sensitive tonA /nB .

In conclusion, we have shown that for a kind of per-turbed photonic crystal, although most states are localizeddue to the randomness, there exist extended states with cer-tain wavelengths. This produces a high-quality resonant tun-neling with a transmission peak much narrower than that ofthe tunneling through a perfect photonic crystal. By using

FIG. 1. The perturbed photonic crystal is constructed by repeated stackingof a number of identical unit cells, each of which is an alternating array ofM identical high-dielectric components andM identical low-dielectric com-ponents. These unit cells are separated byN spacers, which are of the samematerial as the low-dielectric component, but the thickness of the spacers isa random variable.

FIG. 2. The transmission coefficient as a function of incident wavelength.The parameters:nA /nB53.4, dA /dB50.286, M52, (d22d1)/dB51, andN550.

FIG. 3. Light transmission coefficient as a function of position in the per-turbed one-dimensional photonic crystal. The parameters are the same asFig. 2. ~a! l54.5, ~b! l52.0.

2890 Appl. Phys. Lett., Vol. 71, No. 20, 17 November 1997 Lei et al. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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this kind of photonic microstructure in an appropriate mate-rial system, the optical filtering could be adapted to the de-localization property. We believe that these results are sig-nificant towards the realization of novel optoelectronicdevices based on the new concept of photonic crystals.

The authors are grateful to Yan Chen for discussions.This work was supported in part by a grant for key research

projects from the State Science and Technology Commis-sion.

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FIG. 4. Wavelength locations of unscattered states for samples withdA /dB50.286, M52, (d22d1)/dB51, andN550. Herea and b curvesrepresent peak positionsa andb in Fig. 2, respectively.

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