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Physics Letters A 321 (2004) 120–126

www.elsevier.com/locate/pla

Analytical solution for band-gap structures in photonic crystal wsinusoidal period

Jun Zheng, Zhicheng Ye, Xiaodong Wang, Dahe Liu∗

Department of Physics, Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing 100875, PR China

Received 10 November 2003; accepted 17 November 2003

Communicated by V.M. Agranovich

Abstract

In this Letter an approximate analytic method was described. The method was used to analyze the band gaps of hophotonic crystals, having sinusoidal periodic structure. The results obtained were compared with those obtainedanalysis by plane wave expansion method. It shows that the method suggested is suitable for analyzing the photonhaving non-step function periodic structure. 2003 Elsevier B.V. All rights reserved.

PACS: 42.40.Pa; 42.70.Qs; 71.20.Tx

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e-ionlso

ho-lledg-rtedtheo-ion

flec-mbeer,c-

anly

Onadeoniod,ly-

Since the end of 1980’s, great interests weretracted by the works on photonic crystals [1–3]. Seral theoretical methods, such as plane wave exsion method [4,5], multi-scattering theory (KorringKohn–Rostoker method) [6], tight-binding formultion [7], etc., were used to calculate the photonic bastructures, and many important properties of photocrystals were revealed. However, analytical solutfor any kind of photonic crystals has been reporlittle up to now, despite the fact that, it is extremeimportant for predicting the characteristics of the phtonic crystals. In 2003, Khorasani and Adibi [8] dveloped a new analytical approach for computatof band structure. But other further studies are aneeded still.

* Corresponding author.E-mail address: dhliu@bnu.edu.cn (D. Liu).

0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2003.11.041

Recently, attention to the relationship between ptonic crystals and optical holography has been cato. Works in fabricating photonic crystals with holoraphy, i.e., multi beam interference had been repo[9,10]. The authors of this Letter have analyzedvolume hologram by the principles relating to phtonic crystals, especially the plane wave expansmethod, and explained the characteristics of the retion hologram successfully [11,12]. Volume holograhas typical periodic dielectric structures, thus it cantaken as a special kind of photonic crystal. Howevdifferent from super lattice, the distribution of dieletric constant in the holographic photonic crystal isnon-step function rather than a mutant. There are oa few lower order terms in its Fourier expansion.the other hand, some kinds of photonic crystals mwith different materials have similar non-step functiperiodic structures, especially the sinusoidal persuch as the photonic crystals made with photopo

.

J. Zheng et al. / Physics Letters A 321 (2004) 120–126 121

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g-sicamspti-ve--ap-fi-lyz-lo-and

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mers [13,14] or liquid crystal [15]. This kind of photonic crystals with sinusoidal period structure is a scial kind of photonic crystal, and to analyze its bagap structures is of interest.

The fabrication of photonic crystals with holoraphy is much easier and cheaper than the clasmethod in solid state physics. Also, volume holograas well as photonic crystals can all be used in ocal communications as wave guide, interlink or walength division demultiplexing (WDM). So, the holographic photonic crystal may have great potentialplications in the future. Therefore, it may be benecial to suggest an easy and simple method for anaing analytically the band gap structure in volume hograms so that the characteristics of its forbidden bcan be revealed easily.

In 1984, Yariv and Yeh [16] suggested an apprimate analytical method for analyzing the forbiddband in crystals. It is much simpler than other methmentioned above for treating photonic crystals. Hoever, as several factors were neglected, it has notused actually in calculating the band gaps in photocrystals which have step function periodic structuIn this Letter, we made some supplements to Yarmethod, and used it in analyzing the band gaps inume hologram, a typical non-step function periostructure. We found that it can give clear and siple analytic solutions, in which the factors influening band gap structures can be determined. It mahelpful in improving the fabrication technology of thholographic photonic crystals. The following is otreatment.

The dielectric constant in a 1D periodic mediucan be expressed as

(1)ε(�r) = ε(�r + lΛ�z),where,l is an integer,Λ is the periodic length alonz direction. Expandingε(�r) into Fourier series by threciprocal lattice vector�G ( �G = l �g = l 2π

Λ�z), we get

(2)ε(�r) =∑

�Gε �Gei �G·�r =

∑�G

ε �GeiGZ.

For volume hologram, though the distributionthe dielectric constant is in three dimensions, its symetry is in one dimension only (for unslanted hogram). For simplicity, we assumed that the reconstrtion beam is iny–z plane, and has an angleθ with

l

Fig. 1. The geometry of reconstruction beam in volume hologra

respect toz axis (see Fig. 1). Generally, the light wapropagating in a periodic medium follows Bloch’s lait can be expanded as

(3)�E(�r) =∑

�G�C�k− �Ge−i(�k− �G)·�r .

By substituting Eq. (3) into wave equation, ausing Eq. (2), we have

(�k − l′g)2Ck−l′g − ω2µε0Ck−l′g

(4)−∑l =l′

ω2µε(l′−l)gCk−lg = 0

so

(5)

Ck−lg = 1

(�k − l′ �g)2 − ω2µε0

∑l =l′

ω2µε(l′−l)gCk−l �g.

Eq. (5) represents a series of equations with difent l′.

Theoretically, the distribution of the dielectric costant in a volume hologram is sinusoidal [17] (alothez axis), so it is a typical non-step function. For anusoidal function, there is only first order term inFourier expansion. Thus, the problem becomes simgreatly.

According to Fig. 1, the wave vector�k of thereconstruction beam inside the volume holograman angle with respect to the reciprocal lattice vecSince there is only first order term in the seriesequations similar to Eq. (5), when the frequency athe wave vector of the reconstruction beam sat|�k − l �g| ≈ |�k|, and k2 ≈ ω2µε0, C�k and C�k−lg

arethe major terms, the other terms could be neglec

122 J. Zheng et al. / Physics Letters A 321 (2004) 120–126

can

the

ate

nd

of

ted

,

e

that

It means that the plane wave termC�k andC�k−lgare

coupled with each other. Thus, the equation seriesbe simplified into two equations:(�k2 − ω2µε0

)C�k − ω2µε−l �gC�k−l �g = 0,

(6)−ω2µεl �gC�k + [(�k − l �g)2 − ω2µε0

]C�k−l �g = 0.

A non-zero solution can be obtained whenfollowing equation is satisfied

(7)

∣∣∣∣�k2 − ω2µε0 −ω2µε−l �g−ω2µεl �g (�k − l �g)2 − ω2µε0

∣∣∣∣ = 0.

In this way, we get

(8)ω2l± = k2

µ(ε0 ± |εlg|) ,

ω2l± are the top and bottom edges oflth forbidden

band. So, the width of the band gap is

(9)�ωl = ω|εlg|ε0

.

Eq. (8) and (9) and the formulae in the approximcondition|�k − l �g| ≈ |�k|, andk2 ≈ ω2µε0 are the basicrelations determining the structures of forbidden bain a holographic photonic crystal.

Comparing with the equation used in the methodplane wave expansion

(10)

det

{(k2 + l2g2 − 2kgl cosθ

)δl,l′ − ω2

c2 ε(l′−l)g

}= 0

it can be seen that Eq. (10) is very much complicawhile Eq. (7) is only a 2× 2 determinant.

From the approximate conditions|�k − l �g| ≈ |�k| andk2 ≈ ω2µε0, we have|�k| = gl

2 cosθ (see Fig. 2). Thusthe top and bottom edges of thelth forbidden band areat positions

(11)� 2l± = (gl)2

(2 cosθ)2µ(ε0 − |εlg|) .

Thus, the width of the forbidden band is

(12)�ωl = gl|εlg|2ε0 cosθ

√ε0µ

.

It is clear that when the value ofθ increase thevalue ofk andω corresponding to the gap will becomlarger. The dispersion relation ofk–ω is shown inFig. 3.

Fig. 2. The relationship of�k andl �g.

Fig. 3. The dispersion relation ofk–ω obtained from formula (12).

Fig. 4. The relationship between�k, l �g andm�g.

In Eq. (5), replacing�k with k − m�g (m is integralnumber), and repeat the above analysis, we foundin this case|�k − m�g − l �g| ≈ |�k − m�g|, i.e.,k = (m +l2)g/cosθ (see Fig. 4), and(k−m�g)2 ≈ ω2µε0, so wecan obtain

(13)� 2l± = (�k − m�g)2

µ(ε0 ± |εlg|) .

J. Zheng et al. / Physics Letters A 321 (2004) 120–126 123

n relation

n relation

Fig. 5. Band gap structures of volume hologram in the case of oblique incidence. The solid lines in bold face represent the dispersioof k–ω for m = 1.

Fig. 6. Band gap structures of volume hologram in the case of normal incidence. The solid lines in bold face represent the dispersioof k–ω for m = 1.

ol-heamill

ndgap

s

s ofcein

apor-

Also

(�k − m�g)2 =(

l �g2

)2

+[(

m + 1

2

)gtgθ

]2

(14)= g2[l2

4+ tg2θ

(m + 1

2

)].

We can find that for the same value ofm (m = 0),different±l will correspond to different�m,l±. Mean-while, for the same value ofl, differentm will alsocorrespond to different�m,l±. Therefore, if the re-construction beam is not normal incident to the vume hologram, there will be no band gap in all tspace. When the incident angle of reconstruction bechanges, the position and the width of the gap w

vary. If the value ofθ increase, we can get wider bagap, and the value of the central frequency of thewill be larger.

We can also find that, ifθ = 0,m = 0, the minimumof ω2

m is at the positionk = mg cosθ , and its value

will be m2g2 sin2 θµε0

, but not zero. The minimum iproportional to the value ofm and sinθ .

Fig. 5 shows the calculated band gap structurea volume hologram in the case of oblique incidenof reconstruction beam. In the figure, the solid linesbold face represent the dispersion relation ofk–ω form = 1.

As a special example, Fig. 6 gives the band gstructures of a volume hologram made with n

124 J. Zheng et al. / Physics Letters A 321 (2004) 120–126

expansion.

e

Fig. 7. Calculated band gap structures of volume hologram in the case of normal incidence. It is obtained by the method of plane wave

Fig. 8. Calculated band gap structures of volume hologram in the case of oblique incidence (θ = 20◦). It is obtained by the method of planwave expansion.

e-

the

etry,the

odd 8. Itofith5,

mal incidence of reconstruction beam, i.e.,θ = 0.In the figure, the solid lines in bold face reprsent the relation ofk–ω for m = 1. From the ap-proximate condition|�k − m�g − l �g| ≈ |�k − m�g|, i.e.,k = (m + l/2)g, and (�k − m�g)2 ≈ ω2µε0, we have� 2

m,l± = (�k − m�g)2/µ(ε0 ± |εlg|).From the formula �k − m�g = �gl/2, it is clear

that in the case of normal incidence,�m,l± willkeep the same value for different values ofm. Itmeans that in the case of normal incidence,

band gap structures possesses translational symmand there are photonic forbidden band in allspace.

The results of numerical simulations by the methof plane wave expansion are shown in Figs. 7 anwhich have been verified by experiments [10,11]is obvious that the calculations from the methodplane wave expansion are in good consistency wthe analytical discussions shown in Figs. 6 andrespectively.

J. Zheng et al. / Physics Letters A 321 (2004) 120–126 125

uggested in

al method

Table 1Comparison of band gap structures between the calculations using plane wave expansion method and the analytical method sthis Letter when the incident anglei = 0◦ and i = 20◦. Where the modulus of the reciprocal lattice vector isg = 4 × π(5145× 10−10 ×1.05/1.52)m−1

Incident Method 1st gap (m = 0, l = ±1) 2nd gap (m = 0, l = ±2)

anglei Central position (ω0,±1) [Hz] Width [Hz] Central position (ω0,±1) [Hz] Width [Hz]

0◦ Method in this Letter 3.4892× 1015 1.607×1014 6.978× 1015 0Plane wave expansion 3.4916× 1015 1.608×1014 6.983× 1015 0.135×1014

20◦ Method in this Letter 3.573× 1015 1.645×1014 7.14568× 1015 0Plane wave expansion 3.575× 1015 1.650×1014 7.352× 1015 0.16× 1014

Table 2Comparison of the lowest frequency of Brillouin zones between the calculations using plane wave expansion method and the analyticsuggested in this Letter when the incident anglei = 20◦

The lowest frequencyωm m = 0 (�0) [Hz] m = 1 (�1) [Hz] m = 2 (�2) [Hz]

Method in this Letter 0 1.5011× 1015 3.0022× 1015

Plane wave expansion 0 1.5010× 1015 3.0005× 1015

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For comparison, the numerical calculations ofband gaps in a volume hologram were computed wboth the method of plane wave expansion andanalytical method suggested in this Letter, the resare given in Tables 1 and 2. In the calculations,periodic length was multiplied by a factor of 1.0because there is a swelling of the recording med(DCG) after processing.

It is easy to fabricate the photonic crystals wnon-step function period structure especially withsinusoidal structure by holography. During the farication of the photonic crystals, if the exposurecontrolled within the linear range of the recordimedium, a sinusoidal period structure will be otained. The period structure with higher orderFourier expansion terms could be obtained by increing the exposure.

As a conclusion, the method described in this Lecan result in analytical solutions, and the calculatiowith it for the band gaps in the volume hologramin good consistency with that from plane wave expsion method. Since the calculations from plane wexpansion method have been verified experimenttherefore, the analytical method developed in this Lter is suitable for analyzing the band gap structuof photonic crystals which have non-step functionriodic structure, especially for holographic photoncrystals with a sinusoidal period.

Acknowledgements

The authors thank National Natural Science Fodation of China for the financial support of graNos. 19874009 and 60277014. We also thank PShouyong Pei and Ruozhen Wang for helpful discsions.

References

[1] E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059.[2] S. John, Phys. Rev. Lett. 58 (1987) 2486.[3] S. Satpathy, Z. Zhang, M.R. Salehpour, Phys. Rev. Lett.

(1990) 1239.[4] K.M. Ho, C.T. Chan, C.M. Soukoulis, Phys. Rev. Lett. 6

(1990) 3152.[5] K.M. Leung, Y.F. Liu, Phys. Rev. Lett. 65 (1990) 2646.[6] X. Wang, X.G. Zhang, Q. Yu, B.N. Harmon, Phys. Rev. B

(1992) 4161.[7] E. Lidoriks, M.M. Sigalas, E.N. Economou, C.M. Soukoul

Phys. Rev. Lett. 81 (1998) 1405.[8] S. Khorasani, A. Adibi, Opt. Commun. 216 (2003) 439.[9] M. Campbell, D.N. Sharp, M.T. Harrison, R.G. Dennin

A.J. Turberfield, Nature 404 (2000) 53.[10] V. Benger, O. Gauthier-Lafaye, E. Costard, J. Appl. Phys.

(1997) 60.[11] Z. Ye, J. Zheng, D. Liu, S. Pei, Phys. Lett. A 299 (2002) 31[12] X. Wang, F. Wang, L. Cui, D. Liu, Opt. Commun. 221 (200

289.

126 J. Zheng et al. / Physics Letters A 321 (2004) 120–126

in,

n,

,.

k,

[13] P. Tondiglia, L.V. Natarajan, R.L. Sutherland, D. TomlT.J. Bunning, Adv. Mater. 14 (2002) 187.

[14] R. Jakubiak, J. Bunning, R.A. Vaia, L.V. NatarajaV.P. Tondiglia, Adv. Mater. 15 (2003) 241.

[15] P. Mach, P. Wiltzius, M. Megens, D.A. Weiz, K.-H. LinT.C. Lubensky, A.D. Yodh, Phys. Rev. E 65 (2002) 031720

[16] A. Yariv, P. Yeh, Optical Waves in Crystals, Wiley, New Yor1984, Chapter 6.

[17] H. Kogelnik, Bell. Syst. Tech. J. 48 (1969) 2909.