Superlattices and Microstructures 52 (2012) 678–686

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Superlattices and Microstructures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . co m / l o c a t e / s u p e r l a t t i c e s

A detailed derivation of eigenvalue equation in twodimensional and three dimensional photonic crystals

Guanghui Liu, Kangxian Guo ⇑Department of Physics, College of Physics and Electronic Engineering, Guangzhou University, Guangzhou 510006, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 7 June 2012Accepted 12 July 2012Available online 20 July 2012

Keywords:Eigenvalue equationPhotonic crystals

0749-6036/$ - see front matter � 2012 Elsevier Lthttp://dx.doi.org/10.1016/j.spmi.2012.07.011

⇑ Corresponding author.E-mail address: axguo@sohu.com (K. Guo).

A detailed derivation of eigenvalue equation in two dimensionaland three dimensional photonic crystals is given by the plane-wave expansion method. Some mathematical formulas such asthe rotation of vector, the gradient of scalar, the divergence ofthe vector, the vector triple product and the conversion betweenscalar and vector are employed. The eigenvalue equation in pho-tonic crystals has become the important base for obtaining theband structure and the distribution of eigenmode.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Since photonic crystals were first put forward by Yablonovitch and John in 1987 [1,2], much atten-tion has been focused on photonic crystals. Photonic crystals, namely photonic bandgap materials, areartificially arranged periodic electromagnetic structures in optical wavelength scale. Periodic arrange-ment of high and low dielectric or metallo-dielectric nanostructures modulates the propagation ofelectromagnetic waves in photonic crystal. It is well known that the electron modulated by the peri-odic potential in a semiconductor crystal has allowed and forbidden electronic energy bands. Analo-gous to the electron modulated by the periodic potential, it is the periodic modulation that photoniccrystals possess photonic band gaps that inhibit the propagation of electromagnetic waves in certainfrequency ranges. This gives rise to distinct optical phenomena such as inhibition of spontaneousemission [1], laser [3], optical-waveguide [4,5], nanocavity [6,7] and so on. In conventional electronicsemiconductor, electrons as the carriers that transfer and process information are restricted by thequantum effects and the thermal vibration, which has greatly restricted the advances of semiconduc-tor technology towards high speed and high integration density. However, in photonic crystals, pho-tons can make up for the disadvantages over electrons, which makes photonic crystals become the

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G. Liu, K. Guo / Superlattices and Microstructures 52 (2012) 678–686 679

ideal materials instead of electronic semiconductor. Therefore, photonic crystals are often called pho-tonic semiconductor.

Researches on photonic crystals have been done from experiment and theory. For now, severalmethods for fabricating photonic crystals have been proposed. It is easy to fabricate one dimensionalphotonic crystals by using conventional thin films technology [8]. For two dimensional photonic crys-tals, we can use methods borrowed from the silicon microelectronics industry such as laser etchingtechnology, electron beam lithography, reactive ion beam etching and electrochemistry lithography[9]. For three dimensional photonic crystals, although it is difficult to fabricate it, various techniqueshave been used including process methods of micro-fabrication [10], direct laser writing technique[11], laser hologram technique [12], layer by layer method [11] and colloidal crystal templating meth-od [13]. From theory, computing photonic band structure is one of the main tasks. To design photoniccrystal systems, it is essential to engineer the location and size of the bandgap. This is done by com-putational modeling using any of the following methods: plane wave expansion method (PWM) [14],transfer matrix method (TMM) [15], finite difference time domain method (FDTD) [16], Korringa KohnRostoker method (KKR) [17], order-n spectral method [18] and multiple scattering theory (MST) [19].In the methods above, PWM, TMM and FDTD are widely used by many researchers. Here, we give andetailed introduction to PWM that is used in our paper. Plane wave expansion method refers to a com-putational technique in electromagnetics to solve the Maxwell’s equations by formulating an eigen-value problem out of the equation. This method is popular among the photonic crystal communityas a method of solving for the band structure (dispersion relation) of specific photonic crystal geom-etries. PWM is also useful in calculating modal solutions of Maxwell’s equations over an inhomoge-neous or periodic geometry. However, there are some disadvantages for PWM. We cannot obtaintransmission and reflection spectra in finite periodic structures. We also cannot obtain dynamic char-acteristic of photonic crystals. Besides, if dielectric constant are dependent on the frequency, we can-not handle this case by PWM. It is well known that in quantum mechanics, the Schr€odinger equation isan equation that describes how the quantum state of a physical system changes with time. If we solvethe Schr€odinger equation, we can obtain all the information about a quantum system. Just like theimportance of the Schr€odinger equation, based on Maxwell’s equation, eigenvalue equation in twodimensional and three dimensional photonic crystals has become the important base for obtainingthe band structure and the distribution of eigenmode. Therefore, it is necessary to have an overallunderstanding of it. However, the references[20] (pp. 15–17) only gives several simple steps. There-fore, we decides to reveal the detailed process.

In this paper, PWM is used to obtain the eigenvalue equation. In the calculation, a large number ofmathematical formulas are used repeatedly. This paper shows that how eigenvalue equation is ob-tained in two dimensional and three dimensional photonic crystals. This paper is organized as follows.In Section 2, we obtain eigenvalue equation in three and two dimensional photonic crystals by PWMand mathematical formulas. Finally, a brief conclusion is made in Section 2.1.

2. Theory

2.1. Eigenvalue equation in three dimensional photonic crystals

The eigenvalue equation with respect to the magnetic field (H) in three dimensional photonic crys-tals is given by [20] (pp. 17. Eq. (2.32))

XG0

X2

j¼1

MijkðG;G

0ÞhG0 jkn ¼

x2kn

c2 hGikn; ð1Þ

with

MijkðG;G

0Þ ¼ jkþ Gjjkþ G0jjðG � G0Þ �eG2 _eG02 �eG2 � eG01

�eG1 � eG02 eG1 � eG01

� �; ð2Þ

where k and n are, respectively, the wave vector in the first Brillouin zone and the band index. xkn andc are, respectively, the eigenvalue of the electromagnetic field and the speed of light. The references

680 G. Liu, K. Guo / Superlattices and Microstructures 52 (2012) 678–686

[20] (pp. 15–17) only gives several simple steps but does not give specific procedures. Next, we willgive a detailed derivation of Eq. (1). The derivation is divided into two sections. In the first section,both the periodic vectorial functions uknðrÞ and the reciprocal 1

eðrÞ of periodic dielectric constant withrespect to the spatial coordinate r are expanded in Fourier series made of plane waves. In the secondsection, the results obtained in the first section are expressed as the matrix form of Eq. (1).

The eigenvalue equation that the magnetic field satisfies is given by [20] (pp. 15. Eq. (2.21))

r� 1eðrÞr � HðrÞ� �

¼ x2

c2 HðrÞ; ð3Þ

where just like Bloch functions, the magnetic field can be written as

HðrÞ ¼ HknðrÞ ¼ vknðrÞeik�r : ð4Þ

Due to the periodicity of the vectorial functions vknðrÞ, it can be expanded in Fourier series as

uknðrÞ ¼X

G0HknðG0Þ expfiG0 � rg: ð5Þ

Substituting Eq. (3) into Eq. (4) allows us to obtain

HknðrÞ ¼X

G0HknðG0Þ expfiðG0 þ kÞ � rg: ð6Þ

The rotation of the magnetic field H(r) can be obtained as

r� HknðrÞ ¼X

G0r � ½HknðG0Þ expfiðG0 þ kÞ � rg� ¼

XG0r expfiðG0 þ kÞ � rg � HknðG0Þ

¼X

G0iðG0 þ kÞ � HknðG0Þ expfiðG0 þ kÞ � rg: ð7Þ

1eð�rÞ are expanded as

1eðrÞ ¼

XG

jðGÞ expðiG � rÞ: ð8Þ

Therefore, from Eqs. (7) and (8), the rotation of 1eðrÞ r �HðrÞn o

can be obtained as

r� 1eðrÞr � HknðrÞ� �

¼ r�X

G

jðGÞ expðiG � rÞX

G0iðG0 þ kÞ � HknðG0Þ exp½iðG0 þ kÞ � r�

( )

¼XGG0

ijðGÞr � ðG0 þ kÞ � HknðG0Þ exp½iðG0 þ G þ kÞ � r�� �

¼ �XGG0

jðGÞðG0 þ G þ kÞ � ½ðG0 þ kÞ � HknðG0Þ� exp½iðG0 þ G þ kÞ � r�: ð9Þ

Letting us set G þ G0 ¼ G00 and using Eq. (9), Eq. (3) can be reexpressed as

�XG00G0

jðG00 � G0ÞðG00 þ kÞ � ½ðG0 þ kÞ � HknðG0Þ� exp½iðG00 þ kÞ � r�

¼X

G00

x2kn

c2 HknðG00Þ exp½iðG00 þ kÞ � r�: ð10Þ

Comparing both sides of Eq. (10), we can obtain the following equation:

�X

G0jðG00 � G0ÞðG00 þ kÞ � ½ðG0 þ kÞ � HknðG0Þ� ¼

x2kn

c2 HknðG00Þ: ð11Þ

After G00 is replaced by G, Eq. (11) can be changed into the following form:

�X0

G

jðG � G0ÞðG þ kÞ � ½ðG0 þ kÞ � HknðG0Þ� ¼x2

kn

c2 HknðGÞ ð12Þ

G. Liu, K. Guo / Superlattices and Microstructures 52 (2012) 678–686 681

To date, the first section has already been finished. Next, we will continue the second section.According to properties of plane wave, HknðGÞ is perpendicular to kþ G. Therefore, HknðGÞ can be

expressed by a linear combination of two orthogonal normal vectors, eG1 and eG2:

HknðGÞ ¼ hG1kn eG1 þ hG2

kn eG2 ð13Þ

Besides, kþGjkþGj is a unit vector so that eG1; eG2;

kþGjkþGj

n oconstitute a right-hand system. The three unit

vectors above have the following relations:

kþ Gjkþ Gj ¼ eG1 � eG2;

kþ G0

jkþ G0j¼ eG01 � eG02: ð14Þ

Therefore, the term ðkþ GÞ � ½ðkþ G0Þ �HknðG0Þ� in Eq. (12) can be written as

ðkþ GÞ � ½ðkþ G0Þ � HknðG0Þ� ¼ ðeG1 � eG2Þ � ½ðeG01 � eG02 Þ � ðhG01kn eG001 þ hG02

kn eG02Þ�jkþ Gjjkþ G0j: ð15Þ

The term ðeG01 � eG02Þ � eG01 in Eq. (15) can be obtained as

ðeG01 � eG02Þ � eG01 ¼ �eG01 � ðeG01 � eG02Þ ¼ eG02: ð16Þ

The term ðeG01 � eG02Þ � eG02 in Eq. (15) can be obtained as

ðeG01 � eG02Þ � eG02 ¼ �eG02 � ðeG01 � eG02Þ ¼ �eG01: ð17Þ

From Eqs. (16) and (17), Eq. (15) can be changed into the following form:

ðkþ GÞ � ½ðkþ G0Þ � HknðG0Þ� ¼ jkþ Gjjkþ G0jðeG1 � eG2Þ � ðhG01kn eG02 � hG02

kn eG01Þ ð18Þ

The term ðeG1 � eG2Þ � eG02 in Eq. (18) can be obtained as

ðeG1 � eG2Þ � eG02 ¼ �eG02 � ðeG1 � eG2Þ ¼ �½ðeG02 � eG2ÞeG1 � ðeG02 � eG1ÞeG2�: ð19Þ

The term ðeG1 � eG2Þ � eG01 in Eq. (18) can be obtained as

ðeG1 � eG2Þ � eG01 ¼ �eG01 � ðeG1 � eG2Þ ¼ �½ðeG01 � eG2ÞeG1 � ðeG01 � eG1ÞeG2�: ð20Þ

Substitution of both Eq. (19) and Eq. (20) into Eq. (18) leads to the following result:

ðkþ GÞ � ½ðkþ G0Þ � HknðG0Þ� ¼ jkþ Gjjkþ G0jf�½ðeG02 � eG2ÞeG1hG01

kn � ðe02G � eG1ÞeG2hG01

kn �

þ ½ðeG01 � eG2ÞeG1hG02

kn � ðeG01 � eG1ÞeG2hG02

kn �g

¼ jkþ Gjjkþ G0j½�ðeG02 � eG2ÞeG1hG01

kn þ ðeG01 � eG2ÞeG1hG02

kn � þ jk

þ Gjjkþ G0j½ðeG02 � eG1ÞeG2hG01

kn � ðeG01 � eG1ÞeG2hG02

kn �: ð21Þ

From Eqs. (12), (13) and (21), we can obtain the following equations:

�X

G0jkþ Gjjkþ G0jjðG � G0Þ½�ðeG02 � eG2ÞhG01

kn þ ðeG01 � eG2ÞhG02kn � ¼

x2kn

c2 hG1kn ; ð22Þ

�X

G0jkþ Gjjkþ G0jjðG � G0Þ½ðeG02 � eG1ÞhG01

kn � ðeG01 � eG1ÞhG02kn � ¼

x2kn

c2 hG2kn : ð23Þ

The short forms of Eqs. (22) and (23) are as follows:

XG0

X2

j¼1

M1jk ðG;G

0ÞhG0jkn ¼

x2kn

c2 hG1kn ;X

G0

X2

j¼1

M2jk ðG;G

0ÞhG0 jkn ¼

x2knc2hG2

k n:

ð24Þ

682 G. Liu, K. Guo / Superlattices and Microstructures 52 (2012) 678–686

From Eq. (24), we can obtain the following equation:

XG0

X2

j¼1

MijkðG;G

0ÞhG0jk

n ¼ x2kn

c2 hGikn; ði ¼ 1;2Þ ð25Þ

which is in good agreement with Eq. (1). So far, we have completely proved that how Eq. (1) isobtained.

Next, we will give a detailed derivation of eigenvalue equation with respect to the electric field (E)in three dimensional photonic crystals. The equation is given by [20] (pp. 16. Eq. (2.28))

�X

G0jðG � G0ÞðG0 þ kÞ � ½ðG0 þ kÞ � EknðG0Þ� ¼

x2kn

c2 EknðGÞ: ð26Þ

This equation is analogous to Eq. (12) with respect to the magnetic field. Therefore, after Eq. (26) isproved, eigenvalue equation with respect to the electric field analogous to Eq. (1) is easily obtained.The references [20] (pp. 15. Eq. (2.20)) gives the following eigenvalue equation:

1eðrÞr � r� EðrÞf g ¼ x2

c2 EðrÞ: ð27Þ

Similar to Eq. (4), Eq. (3) and Eq. (6), we can obtain the following equation as

EknðrÞ ¼X

G0EknðG0Þ exp iðG0 þ kÞ � r

� �: ð28Þ

Similar to Eq. (7), the rotation of the electric field HðrÞ can be obtained as

r� EknðrÞ ¼X

G0iðG0 þ kÞ � EknðG0Þ expfiðG0 þ kÞ � rg: ð29Þ

the rotation of r� EknðrÞ can be expressed as

r� fr� EknðrÞg ¼ r�X

G0iðG0 þ kÞ � EknðG0Þ exp½iðG0 þ kÞ � r�

( )�X

G0ðG0 þ kÞ � ½ðG0

þ kÞ � EknðG0Þ� exp½iðG0 þ kÞ � r� ð30Þ

Therefore, 1eðrÞr � fr� EknðrÞg can be obtained as

1eðrÞr � fr� EknðrÞg ¼ �

XGG0

jðGÞðG0 þ kÞ � ½ðG0 þ kÞ � EknðG0Þ� exp½iðG0 þ G þ kÞ � r�

¼ �XGG0

jðG� G0ÞðG0 þ kÞ � ½ðG0 þ kÞ � EknðG0Þ� exp½iðGþ kÞ � r�

¼X

G

x2kn

c2 EknðGÞ exp½iðG þ kÞ � r�: ð31Þ

Comparing both sides of Eq. (31), we can obtain the following equation as

�X

G0jðG� G0ÞðG0 þ kÞ � ½ðG0 þ kÞ � EknðG0Þ� ¼

x2kn

c2 EknðGÞ: ð32Þ

The result is in good agreement with Eq. (26). By using method in terms of the magnetic field, wecan obtain such equation as

XG0

X2

j¼1

MijkðG;G

0ÞEG0 jkn ¼

x2kn

c2 EGikn; ð33Þ

with

MijkðG;G

0Þ ¼ jkþ G0j2jðG � G0Þ �1 00 1

� �ð34Þ

G. Liu, K. Guo / Superlattices and Microstructures 52 (2012) 678–686 683

and

EknðGÞ ¼ EG1kn eG1 þ EG2

kn eG2: ð35Þ

2.2. Eigenvalue equation in two dimensional photonic crystals

In this section, we mainly focus on deriving eigenvalue equation with respect to both the magneticfield HðrÞ and the electric field EðrÞ in two dimensional photonic crystals in detail. For simplicity, weassume that the wave vector k is parallel to the 2D plane of two dimensional photonic crystals and thedielectric structure is uniform in the z direction. We also assume that the electromagnetic waves tra-vel in the x–y plane and are uniform in the z direction. Besides, k ¼ k== ¼ ðkx; kyÞ, r ¼ r== ¼ ðx; yÞ andG ¼ G== ¼ ðGx;GyÞ are defined. The eigenvalue equation with respect to the magnetic field (H) intwo dimensional photonic crystals is given by [20] (pp. 21. Eq. (2.62))

XG0

HzðG0ÞjðG � G0Þðkþ GÞ � ðkþ G0Þ ¼ x2kn

c2 Hz;knðGÞ: ð36Þ

This equation corresponds to the H-polarization for which the magnetic field is parallel to the zaxis. Next, we will give a detailed derivation of Eq. (36). By employing Maxwell’s equation, the follow-ing equation is given by [20] (pp. 20. Eq. (2.58))

� @

@x1

eðrÞ@

@x

� �þ @

@y1

eðrÞ@

@y

� �� �HzðrÞ ¼

x2kn

c2 HzðrÞ: ð37Þ

Eq. (37) can be expressed as

� @

@x1

eðrÞ

� �@HzðrÞ@x

þ @

@y1

eðrÞ

� �@HzðrÞ@y

� �þ 1

eðrÞ@2

@x2 þ@2

@y2

( )HzðrÞ

" #¼ x2

kn

c2 HzðrÞ: ð38Þ

The left of Eq. (37) can be divided into two parts. The first part is

� @

@x1

eðrÞ

� �@HzðrÞ@x

þ @

@y1

eðrÞ

� �@HzðrÞ@y

� �; ð39Þ

and the second part is

� 1eðrÞ

@2

@x2 þ@2

@y2

( )HzðrÞ: ð40Þ

According to Bloch’s theorem and plane-wave expansion method, we can obtain the followingequation:

HzðrÞ ¼ Hz;knðrÞ ¼X

G

Hz;knðGÞ expfiðkþ GÞ � rg: ð41Þ

1eðrÞ can be expanded as

1eðrÞ ¼

XG;G0

jðG � G0Þ expfiðG � G0Þ � rg: ð42Þ

Substitution of both Eq. (41) and Eq. (42) into � @@x ð 1

eðrÞÞ@HzðrÞ@x of Eq. (39) leads to the following result:

� @

@x1

eðrÞ

� �@HzðrÞ@x

¼XG;G0

jðG � G0ÞðGx � G0xÞ expfiðG � G0Þ � rgX

G0Hz;knðG0Þðkx

þ G0xÞ exp iðkþ G0Þ � r� �

¼XG;G0

Hz;knðG0ÞjðG � G0ÞðGx � G0xÞðkx þ G0xÞ expfiðkþ GÞ � rg: ð43Þ

684 G. Liu, K. Guo / Superlattices and Microstructures 52 (2012) 678–686

Analogously, we can obtain the following equation as

� @

@y1

eðrÞ

� �@HzðrÞ@y

¼XG;G0

jðG � G0ÞðGy � G0yÞ expfiðG � G0Þ � rgX

G0Hz;knðG0Þðky

þ G0yÞ expfiðkþ G0Þ � rg

¼XG;G0

Hz;knðG0ÞjðG � G0ÞðGy � G0yÞðky þ G0yÞ expfiðkþ GÞ � rg: ð44Þ

Eq. (43) + Eq. (44) leads to the following equation:

� @

@x1

eðrÞ

� �@HzðrÞ@x

þ @

@y1

eðrÞ

� �@HzðrÞ@y

� �¼XG;G0

Hz;knðG0ÞjðG�G0ÞexpfðkþGÞ �rg

�½ðGx�G0xÞðkxþG0xÞþðGy�G0yÞðkyþG0yÞ�

¼XG;G0

Hz;knðG0ÞjðG�G0ÞexpfðkþGÞ �rg

�½ðGxG0xþGxkx�GxG0x�G0xkxÞþðGyG0yþGyky�GyG0y�G0ykyÞ�

¼XG;G0

Hz;knðG0ÞjðG�G0ÞexpfðkþGÞ �rg ½ðGxG0xþGyG0yÞ

þðGxkxþGykyÞ�ðG02x þG02y Þ�ðG0xkxþG0ykyÞ�

¼XG;G0

Hz;knðG0ÞjðG�G0ÞexpfðkþGÞ �rg ½G �G0 þG �k�G02�G0 �k�

¼XG;G0

Hz;knðG0ÞjðG�G0ÞexpfðkþGÞ �rgðG�G0Þ � ðkþG0Þ: ð45Þ

The first part is calculated out. Next we will calculate the second part Eq. (40). We know the fol-lowing expression as

r2 ¼ r � r ¼ @2

@x2 þ@2

@y2 : ð46Þ

Let us consider the following calculation:

r2 expfiðkþ G0Þ � rg ¼ r � r expfiðkþ G0Þ � rg ¼ ir � ðkþ G0Þ expfiðkþ G0Þ � rg

¼ �jkþ Gj2 expfiðkþ G0Þ � rg: ð47Þ

The magnetic field can be written as

HzðrÞ ¼ Hz;knðrÞ ¼X

G0Hz;knðG0Þ expfiðkþ G0Þ � rg: ð48Þ

Therefore, from Eqs. (42), (47) and (48), the second part Eq. (40) can be obtained as

� 1eðrÞ

@2

@x2 þ@2

@y2

( )HzðrÞ ¼

XG;G0

Hz;knðG0ÞjðG � G0Þ expfiðkþ GÞ � rgjkþ G0j2: ð49Þ

Eqs. (45) and Eq.(49) results in the following equation:

XG;G0

Hz;knðG0ÞjðG � G0Þ expfir � ðkþ GÞg½ðG � G0Þ � ðkþ G0Þ þ jkþ G0j2�

¼XG;G0

Hz;knðG0ÞjðG � G0Þ expfir � ðkþ GÞgðkþ G0Þ � ½G � G0 þ kþ G0�

¼XG;G0

Hz;knðG0ÞjðG � G0Þ expfir � ðkþ GÞgðkþ G0Þ � ðkþ GÞ: ð50Þ

G. Liu, K. Guo / Superlattices and Microstructures 52 (2012) 678–686 685

From Eq. (37) and Eq. (50), we have the following result:

XG;G0

Hz;knðG0ÞjðG � G0Þ expfir � ðkþ GÞgðkþ GÞ � ðkþ G0Þ

¼ x2kn

c2

XG

Hz;knðGÞ expfiðkþ GÞ � rg: ð51Þ

Comparing both sides of Eq. (51), we can obtain the following equation as

XG0

Hz;knðG0ÞjðG � G0Þðkþ GÞ � ðkþ G0Þ ¼ x2kn

c2 Hz;knðGÞ: ð52Þ

So far, we have proved the eigenvalue equation with respect to both the magnetic field HðrÞ in twodimensional photonic crystals in detail. The eigenvalue equation with respect to the electric field (E)isgiven by [20](pp. 20. Eq. (2.57))

� 1eðrÞ

@2

@x2 þ@2

@y2

( )EzðrÞ ¼

x2

c2 EzðrÞ ð53Þ

This equation corresponds to the E-polarization for which the electric field is parallel to the z axis.The electric field can be written as

EzðrÞ ¼ Ez;knðrÞ ¼X

G0Ez;knðG0Þ expfiðkþ G0Þ � rg: ð54Þ

Comparing the left of Eq. (49) and the left of Eq. (53) and using Eq. (54), we can easily obtain thefollowing equation as

� 1eðrÞ

@2

@x2 þ@2

@y2

( )EzðrÞ ¼

XG;G0

Ez;knðG0ÞjðG � G0Þ expfiðkþ GÞ � rgjkþ G0j2: ð55Þ

From Eq. (53), Eqs. (54) and (55), we can obtain the following equation as

XG;G0

Ez;knðG0ÞjðG � G0Þ expfiðkþ GÞ � rgjkþ G0j2 ¼ x2kn

c2

XG

Ez;knðGÞ expfiðkþ GÞ � rg: ð56Þ

Comparing two sides of Eq. (56), we finally obtain the following equation with respect to the elec-tric field as

XG0

jðG � G0Þjkþ G0j2Ez;knðG0Þ ¼x2

kn

c2 Ez;knðGÞ: ð57Þ

So far, we have fully showed that how can we obtain four eigenvalue equations with respect to theelectric field and the magnetic field in three and two dimensional photonic crystals.

3. Conclusion

A derivation of eigenvalue equation in two dimensional and three dimensional photonic crystals isgiven in detail in our paper. The method used is the plane-wave expansion method, which is based onthe Fourier expansion of electromagnetic field and the dielectric function. In actual calculation, therotation of vector, the gradient of scalar, the vector triple product, the divergence of the vector andthe conversion between scalar and vector are used. For three dimensional eigenvalue equation, notonly the expressions for vector are derived, but also the expressions for matrix are derived. For twodimensional eigenvalue equation, the expressions for vector are derived. The results obtained are ingood agreement with the references, which shows that our calculations are correct. Finally we hopethat our work can make positive contributions to our researches on photonic crystals, especially tobeginners.

686 G. Liu, K. Guo / Superlattices and Microstructures 52 (2012) 678–686

Acknowledgments

This work is supported by the National Natural Science Foundation of China (under Grant No.61178003), Guangdong Provincial Department of Science and Technology (under Grant No. 2011B010400006), and the Science and Information Technology Bureau of Guangzhou (under Grant No.11C62010688).

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