Analysis of optical properties in cylindrical dielectric photonic crystal

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Analysis of optical properties in cylindrical dielectric photonic crystal

Chung-An Hu

a

, Chien-Jang Wu

b,n

, Tzong-Jer Yang

c

, Su-Lin Yang

a

Department of Electrophysics, National Chiao Tung University, Hsinchu 300, Taiwan

b

Institute of Electro-Optical Science and Technology, National Taiwan Normal University, Taipei 116, Taiwan

c_{Department of Electrical Engineering, Chung Hua University, Hsinchu 300, Taiwan}

a r t i c l e

i n f o

Article history:

Received 6 October 2012 Received in revised form 19 November 2012 Accepted 20 November 2012 Available online 5 December 2012 Keywords:

Cylindrical photonic crystal Transfer matrix method Wave propagation

a b s t r a c t

In this work, theoretical formulas for the H-polarization electromagnetic propagation in a cylindrical multilayer structure (CMS) are given. The relationships between two modes, H- and E-polarization are pointed out. With the derived formulae, we present the numerical results for three model structures such as the single cylindrical interface, the single cylindrical slab, and the cylindrical photonic crystal (CPC). In the single cylindrical interface, it is found that there exists a Brewster starting radius at which a minimum reflectance is attained in H-polarization. In the single cylindrical slab, the result illustrates that the reflectance response in the wavelength domain contains the oscillating and nonoscillating regions. As for the CPC, we find the PBG structure at zero azimuthal mode number is very similar to that of planar photonic crystal. The PBG, however, can be strongly influenced by increasing the azimuthal mode number in a CPC.

1. Introduction

The propagation of electromagnetic waves in dielectric strati-ﬁed structures has been studied for a long time [1,2]. Such an issue has been of much interest to the optical and electromagnetic communities again in 1987 since the concept of the photonic crystals (PCs) was introduced by two pioneering works of Yablonovitch and John[3,4]. Since then a ﬂood of research topics in PCs were triggered in the past two and half decades. At present, research in PCs, which also are called the photonic band gap (PBG) materials, continue to be hot in the communities of photonics, electromagnetics, and material physics.

A simple periodic dielectric multilayer structure known as a one-dimensional (1D) PC is easier to fabricate compared to the two- and three-dimensional PCs. In addition, 1D PCs can be used to explore many fundamental and interesting optical properties, such as the existence of PBGs as well as the feature of omnidirec-tional mirror[5,6]. In 1D PCs the wave propagation properties can be analytically investigated by the familiar transfer matrix method (TMM) in Cartesian coordinates [1,2,7]. The TMM described in Ref.[7]is generally referred to as the Abeles theory. In addition to the usual planar 1D PC, wave propagation in a cylindrical multilayer structure (CMS) has also received much attention in recent years[8–17]. A PC with a periodic CMS is called a cylindrical photonic crystal (CPC) or cylindrical Bragg reﬂector (CBR). The reﬂection or transmission response of a CPC can be analytically investigated based on another version of TMM. In fact, such cylindrical wave TMM has been developed by Kaliteevski et al.

[18]. They developed an elegant TMM in cylindrical coordinates which, in fact, is an analogous version of Abeles theory in Cartesian coordinates. With this cylindrical TMM, the reﬂection response for the CPC can be studied and then a comparison with planar 1D PC can be made[19]. Moreover, based on the use of such TMM, studies of photonic band structures in metallic and superconducting CPCs have also been available[20,21].

In a CMS, it is known that there will be two possible propaga-tion modes called the E-polarizapropaga-tion and the H-polarizapropaga-tion as well [18]. Previous studies [18–21] were all focused in the E-polarization only, which is partially because the formulae of the H-polarization for the reflection and transmission in a CMS still remain unavailable. The purpose of this paper is thus to give a detailed theoretical description on the wave propagation in a CMS under the condition of H-polarization. We shall derive the formulae of reflection and transmission for a CMS. With these formulae, we next give some numerical studies on three model structures, including the single cylindrical interface, the single cylindrical slab, and the CPC. The first structure is similar to Fresnel’s formulae in a planar interface between two different media. The study of second structure is reminiscent of the Airy slab problem in optics. The third one, the CPC, is of particular interest in this work because of the current interest. For the purpose of comparative study, all the numerical results of these three structures are given for both E- and H-polarization.

2. Transfer matrix method in cylindrical system

In this section, we ﬁrst derive the transfer matrix method (TMM) for the electromagnetic propagation in the CMS as shown Contents lists available atSciVerse ScienceDirect

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Optics Communications

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Corresponding author. Tel.: þ886 2 77346724; fax: þ886 2 86631954. E-mail address: [email protected] (C.-J. Wu).

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in Fig. 1.We assume that the temporal part of all the ﬁelds is expðj

o

tÞ. In what follows, the SI-unit will be used in this work. For a given layer with permeability

m

¼

m

r

m

0and permittivity

e

¼

e

r

e

0,

Maxwell’s curl equations are written as,

r

E ¼ j

om

H, ð1Þ

r

H ¼ j

oe

E ð2Þ

In cylindrical coordinate, (

r

,

f

, z), Eq.(1)can be expanded as 1

r

@Ez @

f

@Ef @z ¼ j

om

Hr, ð3aÞ @Er @z @Ez @

r

¼ j

om

Hf, ð3bÞ 1

r

@

r

Ef @

r

@Er @

f

¼ j

om

Hz, ð3cÞ

and expansion of Eq.(2)gives 1

r

@Hz @

f

@Hf @z ¼j

oe

Er, ð4aÞ @Hr @z @Hz @

r

¼j

oe

Ef, ð4bÞ 1

r

@ð

r

HfÞ @

r

@Hr @

f

¼j

oe

Ez ð4cÞ

Let us ﬁrst consider the propagation of cylindrical wave diverging from or converging to the axis of symmetry

r

¼0 (z axis). In this case, the derivatives of the ﬁelds with respect to z can be omitted and hence Eq.(3)can be reduced to

1

r

@Ez @

f

¼ j

om

Hr, ð5aÞ @Ez @

r

¼j

om

Hf, ð5bÞ 1

r

@

r

Ef @

r

@Er @

f

¼ j

om

Hz ð5cÞ

Likewise, Eq.(4)can be reduced to 1

r

@Hz @

f

¼j

oe

Er, ð6aÞ @Hz @

r

¼ j

oe

Ef, ð6bÞ 1

r

@ð

r

HfÞ @

r

@Hr @

f

¼j

oe

Ez ð6cÞ

Solutions for Eqs. (5) and (6) can be classiﬁed as two modes. One is called the E-polarization which has three non-zero com-ponents, Ez, Hf, and Hr. The other is H-polarization having

non-zero components Hz, Ef, and Er. The solutions for E-polarization

have been available[18]. Thus, in this work, we shall limit to the H-polarization. In this case, with Eqs. (6a) and (6b), Eq. (5c)

becomes the governing equation for Hz, namely

@ @

r

1

e

r

@Hz @

r

þ1

e

1

r

@ @

f

@Hz @

f

þ

o

2

_mr

_H z¼0, ð7Þ

which can be further expressed as

r

@ @

r

@Hz @

r

21

e

@

e

@

r

@Hz @

r

þ @ @

f

@Hz @

f

þ

o

2

_mer

2_H z¼0 ð8Þ

For E-polarization, the above differential equation is read as[18]

r

@ @

r

@Ez @

r

21

m

@

m

@

r

@Ez @

r

þ @ @

f

@Ez @

f

þ

o

2

_mer

2_E z¼0 ð9Þ

It is seen that the governing differential equations for E-polarization can be obtained from H-polarization by using H-E and

e

-

m

, and vice versa.

To solve Eq. (8), we let Hz¼Vð

r

Þ

F

ð

f

Þ. Then the angular part

takes the form,

d2

F

d

f

2þm 2

_F

¼0, ð10Þ

which has a solution

F

ejmf_, _ð11Þ

where m can be zero, or a positive or negative integer. The radial part of Eq.(8)is,

r

d d

r

dV d

r

21

e

d

e

d

r

dV d

r

þ

o

2

_mer

2_Vm2_{V ¼ 0} _ð12Þ

If the permittivity is homogeneous, @

e

=@

r

¼0, Eq. (12)

reduces to

r

d d

r

dV d

r

þk2

r

2_m2 _{V ¼ 0,} _ð13Þ

which is a standard Bessel’s equation with a solution expressible as,

Vð

r

Þ ¼AJmðk

r

Þ þBYmðk

r

Þ, ð14Þ

where Jmis a Bessel function, Ymis a Neumann function and A, B

are constants. Here,

k ¼

o

cn ð15Þ

is the wave number in medium where c is the speed of light in free space and n is the refractive index of medium. Based on Eqs.(11) and (14), the magnetic ﬁeld can be written as

Hz

r

,

f

¼Vð

r

Þejmf_{¼ ½AJ}

mðk

r

Þ þBYmðk

r

Þejmf ð16Þ

For the non-zero electric ﬁelds, Eq.(6a)leads to

Er¼ m

oe

Vð

r

Þ

r

e jmf_, _ð17aÞ

and Eq.(6b)becomes

Ef¼ 1 j

oe

@V @

r

e jmf_U

_r

_eimf_, _ð17bÞ Fig. 1. A portion of CMS, in which the m-layer system/1/2/y/m/ is bounded by

the media of refractive indices, n0and nf. The subscript ‘‘0’’ is known as the

starting medium, whereas the ﬁnal medium is indexed by the subscript ‘‘f’’. The dimension in z direction is assumed to be much larger than the dimensions in x and y directions.

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where Uð

r

Þand Vð

r

Þare related by dV

d

r

¼ j

oe

U

r

ð18Þ

With Eq.(14), the function U can be found to be Uð

r

Þ ¼jp AJ0 mðk

r

þBY0 mðk

r

ÞÞ,, ð19Þ where p ¼ ffiffiffiffi

m

e

r , ð20Þ

is known as the intrinsic impedance of medium (p¼ 377

O

for free space). We see that V in Eq.(14)and U in Eq.(19)can be used to respectively determine the non-zero electric ﬁeld components Er

and Ef according to Eqs.(17a) and (17b).

In the transfer matrix formalism, it is convenient to set the column vector Vð

r

,Uð

r

ÞÞTat a radial position

r

from the above solutions and relate it to the corresponding vector at some other point, say

r

0(assuming

r

0o

r

), by the matrix multiplication[18]:

Vð

r

Þ Uð

r

Þ ! ¼M V

r

0 U

r

0 ! ¼ M11 M12 M21 M22 ! V

r

0 U

r

0 ! , ð21Þ

This matrix equation relates the two non-zero electric ﬁelds at two distinct radial positions

r

0and

r

. The elements of transfer

matrix M can be found with the help of Eqs.(14) and (19)when the vector V

r

0 ,U

r

0 Þ

has been set at a special value of (1,0) or (0,1) [18]. These two choices are similar to those in the Abeles theory in dealing with the planar multilayer structure[7]. With

V

r

0 ,U

r

0 Þ ¼ð1,0Þ , Eq.(21)leads to M11¼Vð

r

Þ, M21¼Uð

r

Þ ð22Þ

The explicit expressions for M11 and M21 can be further

obtained as follows. Eqs.(14) and (19)under this special values can be written by AJmk

r

0 þBYm k

r

0 ¼1, ð23aÞ AJ0 mk

r

0 þBY0 m k

r

0 ¼0 ð23bÞ It is easy to have A ¼ Y 0 m k

r

0 2=

p

k

r

0 , B ¼ J 0 mk

r

0 2=

p

k

r

0 : ð24Þ

Therefore, M11and M21, can be expressed as

M11¼V

r

¼

p

2k

r

0 Y 0 m k

r

0 Jm k

r

J0 m k

r

0 Ymk

r

, ð25Þ M21¼U

r

¼j

p

2k

r

0p Y0m k

r

0 J0 m k

r

J0 mk

r

0 Y0 mk

r

: ð26Þ Similarly, with V

r

0 ,U

r

0 Þ ¼ð0,1Þ , we have M12¼Vð

r

Þ, M22¼Uð

r

Þ ð27Þ

Again, from Eqs. (14) and (19), we can solve for A and B, namely A ¼ j p Ym k

r

0 2=

p

k

r

0 , B ¼j p Jm k

r

0 2=

p

k

r

0 ð28Þ

The other two matrix elements are thus given by

M22¼U

r

¼

p

2k

r

0 Jmk

r

0 Y0 m k

r

Ym k

r

0 J0 m k

r

, ð29Þ M12¼V

r

¼ j

p

2 k p

r

0 Jm k

r

0 Ymk

r

Ymk

r

0 Jm k

r

, ð30Þ

In the case of E-polarization, the matrix elements are of the same form as, (25), (26), (29), and (30), but with a replacement ofp ¼pffiffiffiffiffiffiffiffi

e

=

m

[18].

Finally, it is worth knowing the determinant of the transfer matrix M. With Eqs.(25), (26), (29), and (30), it is direct to get the determinant, namely

detM ¼ M11M22M12M21¼

r

0

r

ð31Þ

In obtaining the above result, the following identity has been used, Jm k

r

0 Y0 m k

r

0 J0 m k

r

0 Ym k

r

0 ¼ 2

p

k

r

ð32Þ

It is noted that the determinant is simply equal to the ratio of the initial and ﬁnal radii. Based on Eq. (21), we arrive at the conclusion that the transfer matrix for a region containing two or more different layers is simply the product of the transfer matrices that correspond to all layers. Furthermore, Eq. (31)

reveals that the determinant of the total transfer matrix for a CMS is given by the ratio of its internal and external radii.

3. Propagation waves as basic functions

As shown in previous section, we have known that the ﬁeld solution for the cylindrical wave can be made of the radial and angular parts. The radial part is described by the Bessel function Jmas well as the Neumann function Ym. However, for the problem

of wave propagation, it is convenient to express the ﬁeld solution as the sum of two contrary propagating waves, i.e., a super-position of ingoing (converging) and outgoing (diverging) waves. These two waves are generally represented by two Hankel functions. For H-polarization, the magnetic and electric ﬁelds of outgoing cylindrical wave take the form

Hzþ¼AHð2Þmðk

r

Þexpðjm

f

Þ, ð33Þ

Efþ¼jpAHð2Þ0m ðk

r

Þexpðjm

f

Þ, ð34Þ

where Hð2Þm is the Hankel function of the second kind. On the other

hand, the ingoing wave is the Hankel function of the ﬁrst kind, namely

Hz¼BHð1Þmðk

r

Þexpðjm

f

Þ ð35Þ

Ef¼jpBHð1Þ0m ðk

r

Þexpðjm

f

Þ, ð36Þ

For a field with azimuthal variation specified by m, the total field of both Hzand Efcan be written as

Hz¼HzþþHz, ð37Þ and Ef¼EfþþEf¼jpCmð2ÞHzþþjpCð1ÞmHz, ð38Þ where Cð1,2Þ m ¼H 1,2 ð Þ0 m ðk

r

Þ=H 1,2 ð Þ m ðk

r

Þ ð39Þ

In order to refer to the layer where the ﬁeld exists, we introduce the layer label as a second subscript on the coefﬁcient Cð1,2Þ

m , and as a subscript of the wavevector k, namely

Cð1,2Þ ml ¼H 1,2 ð Þ0 m kl

r

=Hð1,2Þ m kl

r

, ð40Þ

For a layer, say l.

Next, we would like to relate the magnetic ﬁelds at the two boundaries for a single layer. The relationship can be described by a matrix P which plays the similar role as the propagation matrix in a standard TMM in usual planar geometry[2]. This matrix can be constructed as follows. From Eq.(33), the outgoing magnetic

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ﬁeld is rewritten by Hzþ

r

¼ H ð2Þ mðk

r

Þ Hð2Þm k

r

0 AHð2Þm k

r

0 exp jmð

f

Þ ¼ H ð1Þ mðk

r

Þ Hð1Þm k

r

0 Hzþ

r

0 , ð41Þ And the ingoing magnetic ﬁeld is expressed as

Hz

r

¼ Hð1Þ mðk

r

Þ Hð1Þm k

r

0 BHð1Þm k

r

0 exp jmð

f

Þ ¼ H ð1Þ mðk

r

Þ Hð1Þm k

r

0 Hz

r

0 ð42Þ

Eqs. (41) and (42) can be combined as a matrix equation, namely Hzþð

r

Þ Hzð

r

Þ ! ¼P Hz þ

_r

0 Hz

r

0 ! , ð43Þ

where the propagation matrix is diagonal and given by

P ¼ Hð2Þ mðkrÞ Hð2Þmðkr0Þ 0 0 Hð1ÞmðkrÞ Hð1Þ mðkr0Þ 0 B B @ 1 C C A ð44Þ

This matrix converts the magnetic ﬁeld at inner boundary

r

¼

r

0to a point of

r

¼

r

inside the layer. It should be mentioned

that this matrix P is exactly the same as that for the E-polarization. In the usual planar TMM, there is another important matrix called the dynamical matrix D that relates the boundary condi-tions between two media [2]. Similarly, there also exists a corresponding matrix in the CMS. To derive this, let us consider the interface between two layers. Based on the continuity of the tangential components of the electric and magnetic ﬁelds ðHz,EfÞ,

we can express the interface conditions in terms of H zþ,Hz. To

this end, we introduce matrix D that can convert the basis Hzþ,Hz to ðHz,EfÞ, namely Hz Ef ! ¼D Hz þ Hz ! , ð45Þ where D is written by D ¼ 1 1 jpCð2Þm jpC ð1Þ m ! ð46Þ

Eq.(46) can be easily seen from Eqs.(37) and (38). For the E-polarization, the D-matrix in Eq.(46)still holds with a replace-ment of j-j. In addition, for the nonmagnetic medium,

m

r¼1,

the determinant of D is found to be

detD ¼ 1

e

r ffiffiffiffiffiffi

m

0

e

0 r ₄

p

K

r

1 Hð2Þmðk

r

ÞH ð1Þ mðk

r

Þ , ð47Þ

where K ¼

o

/c is the wave number of free space and, in arriving at Eq.(47), we have used the following Wronskian

W Hh ð1Þ_n ðzÞ,Hð2Þ_n ðzÞi¼Hð1Þ_n ð ÞHz ð2Þ0_n ð ÞHz ð1Þ0_n ð ÞHz ð2Þ_n ð Þ ¼ z 4j

p

z ð48Þ

Also, the inverse of D can be calculated to be

D1¼

e

_e

0

m

0 r

_p

4K

r

H ð2Þ m k

r

Hð1Þm k

r

jpCð1Þm 1 jpCð2Þ m 1 ! ð49Þ

Next, from Eq.(45)together with the condition of continuity of the tangential ﬁeld components at the interface of two layers labeled 1 and 2, we have

D1 Hz1þ Hz1 ! ¼D2 Hz2þ Hz2 ! ð50Þ

Eq.(50)can be rewritten by Hz2þ Hz2 ! ¼D21D1 Hz1þ Hz1 ! ¼D21 Hz1þ Hz1 ! ¼ d11 d12 d21 d22 ! Hz1þ Hz1 ! ð51Þ The matrix D21¼D21D1 is regarded as transmission matrix

that links the amplitudes of the waves on the two sides of the interface. With Eqs.(46) and (49), the matrix elements of D21can

be obtained, with the results

d11¼ j

e

_e

0

m

0 r

_p

4K

r

H ð2Þ mk2

r

Hð1Þmk2

r

p2Cð1Þmp1Cð2Þm h i , ð52aÞ d21¼ j

e

_e

0

m

0 r

_p

4K

r

H ð2Þ m k2

r

Hð1Þm k2

r

p1C ð2Þ mp2C ð2Þ m h i ,, ð52bÞ d12¼ j

e

_e

0

m

0 r

_p

4K

r

H ð2Þ mk2

r

Hð1Þmk2

r

p2Cð1Þmp1Cð1Þm h i , ð52bÞ d22¼ j

e

_e

0

m

0 r

_p

4K

r

H ð2Þ m k2

r

Hð1Þm k2

r

p1C ð1Þ mp2C ð2Þ m h i ð52dÞ

4. Reﬂection and transmission in a single cylindrical interface

We are in the position to determine the wave reflection and transmission in a cylindrical interface. These formulae of wave reflection and transmission between two media are analogous to the usual Fresnel’s equations in the planar geometry. Let us consider a diverging wave incident from medium 1 on the inter-face between layers 1 and 2. Based on Eq.(51), it is direct to have the reflection coefficient rdand the transmission coefficient that

should satisfy the following relation, td 0 ¼D21 1 rd ! ð53Þ

It can be seen from Eq. (53)that rd and td are respectively

expressed as rd¼ d21 d22 ¼p2C ð2Þ m2p1Cð2Þm1 p1C ð1Þ m1p2C ð2Þ m2 , ð54Þ and td¼d11þd12rd¼d11d12 d21 d22 ¼detD21 d22 ð55Þ

Here, the determinant of D21can be calculated from Eqs.(51)

and (47), namely detD21¼detD21detD1

¼

e

_e

0

m

0 r

_p

4K

r

H ð2Þ mk2

r

Hð1Þmk2

r

h i2 jp₂Cð1Þ_m2 1 jp2Cð2Þm2 1 U4

p

K

r

1 Hð2Þmk1

r

Hð1Þmk1

r

ð56Þ

To simplify Eq.(56), we ﬁrst consider the fact that

I ¼ D2D21¼

e

_e

0

m

0 r

_p

4K

r

H ð2Þ mk2

r

Hð1Þmk2

r

jp2C ð1Þ m2þjp2C ð2Þ m2 0 0 jp2Cð1Þm2þjp2C ð2Þ m2 0 @ 1 A, which indicates that

e

_e

0

m

0 r

_p

4K

r

H ð2Þ mk2

r

Hmð1Þk2

r

jp2C ð1Þ m2þjp2C ð2Þ m2 h i ¼1 ð57Þ

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By making use of Eq.(57), Eq.(56)becomes detD21¼ Hð2Þ mk2

r

Hð1Þmk2

r

Hð2Þ mk1

r

Hð1Þmk1

r

ð58Þ

Therefore, the transmission coefﬁcient in Eq.(55)can be cast as td¼ j4

e

1 r ffiffiffiffiffiffiffiffiffiffiffiffiffi

m

0=

e

0 p

p

K

r

Hð2Þmk1

r

Hð1Þmk1

r

p2C ð2Þ m2p1C ð1Þ m1 h i ð59Þ

Eqs.(54) and (59)are the reflection coefficient and transmis-sion coefficient for the diverging wave. Similarly, if we are interested in the converging wave, expressions for reflection coefficient and transmission coefficient can be obtained, with the results rc¼ p2C ð1Þ m2p1C ð1Þ m2 p1C ð1Þ m1p2C ð2Þ m2 , ð60Þ tc¼ j4

e

1 r ffiffiffiffiffiffiffiffiffiffiffiffiffi

m

0=

e

0 p

p

K

r

Hð2Þm k2

r

Hð1Þm k2

r

p2C ð2Þ m2p1C ð1Þ m1 h i ð61Þ

The reflection coefficient and transmission coefficient described in Eqs.(54), (59), and (60), (61)are the Fresnel-like equations in the cylindrical interface between the media 1 and 2. In addition, these four equations are of the same forms as in the E-polarization with different definitions in parameter p.

5. Reﬂection and transmission in a cylindrical slab

Let us now focus on the wave reﬂection and transmission in a single cylindrical slab. This issue is reminiscent of the case of Airy formula for a single slab in the usual planar geometry. Fig. 2

depicts a single layer of index n1and of thickness

r

1–

r

0. Here, for

the purpose of illustration, we have intentionally plotted the straight line to represent the cylindrical interface. Reflection coefficient r and transmission coefficient t can be derived as follows. Let us denote the transmission and reflection coefficients at the second interface

r

¼

r

1 as t1d and r1d. In addition, the

coefﬁcients of the ﬁrst interface

r

¼

r

0as t0d, t0c, r0d, and r0c, the

subscripts d and c mean the outgoing and incoming waves, respectively. The explicitly expression for r and t are given by

r ¼r0dþðt0ct0dr0cr1dÞr1d

Y

1r0cr1d

Y

, ð62Þ t ¼ t0dt1d 1r0cr1d

Y

Hð2Þm k1

r

1 Hð2Þm k1

r

0 , ð63Þ where

Y

¼H ð1Þ m k1

r

1 Hð2Þ m k1

r

1 H ð2Þ m k1

r

1 Hð1Þ m k1

r

1 , ð64Þ

and k1is the wave vector in layer 1.

The derivation of Eqs.(62)–(64)can be described as follows. Referring toFig. 2, at the ﬁrst boundary

r

¼

r

0, we have

CHð2Þ m k1

r

0 ¼1Ut0dþr0cUDHð1Þm k1

r

0 , ð65Þ 1Ur ¼ 1Ur0dþt0cUDHð1Þ_m k1

r

0 , ð66Þ

Similarly, at the second boundary

r

¼

r

1, we have

tU1 ¼ t1dUCHð2Þm k1

r

1 , ð67Þ DHð1Þm k1

r

1 ¼r1dUCHð2Þ_mk₁

r

₁ ð68Þ Eliminating D from Eqs.(65) and (68)leads to

C Hð2Þm k1

r

0 r1d Hð2Þm k1

r

1 Hð1Þm k1

r

1 r0cHð1Þm k1

r

0 " # ¼t0d ð69Þ

Again, eliminating D from Eqs.(66) and (68)gives

rr0d¼Cr1d Hð2Þ m k1

r

1 Hð1Þ m k1

r

1 t0cHð1Þm k1

r

0 ð70Þ

From Eqs. (69) and (70), it is direct to have the reflection coefficient given in Eq.(62). Similarly, with Eqs.(67) and (69), we can arrive at the result of transmission coefficient given in Eq.(63). In addition, Eqs. (62)–(64) are of the same from as those in the E-polarization.

6. Reﬂection and transmission in cylindrical multilayer structure

We now turn attention to the main issue of this work, that is, the wave propagation in the CMS as depicted inFig. 1. We would like to derive the analytical expressions for reflection and trans-mission coefficients. Using these formulae, we shall examine the wave properties in a periodic cylindrical multilayer structure, which is also known as a circular photonic crystal (CPC) or a circular Bragg reflector (CBR).

Referring to Fig. 1, an outgoing wave is incident on the interface,

r

¼

r

0, between 0 and 1, and then propagates into the

ﬁnal medium f, which is assumed to extend from

r

¼

r

fto

r

¼ 1.

The amplitudes of the magnetic ﬁeld and electric ﬁelds at

r

0and

r

f can be written in terms of the amplitude reﬂection and

transmission coefﬁcients rd and td together with the transfer

matrix M deﬁned in Eq.(21), with the results 1 þrd jp0C ð2Þ m0þjp0C ð1Þ m0rd ! ¼M1 td jpfC ð2Þ mftd ! ð71Þ

The inverse matrix in the right hand side of Eq. (71) is denoted by M1 ¼ M11 M12 M21 M22 !1 ¼ 1 detM M22 M12 M21 M11 ! M 0 11 M 0 12 M0 21 M 0 22 ! ð72Þ Eqs.(71) and (12)gives expressions for reﬂection coefﬁcient rd

and transmission coefﬁcient tdcan be determined, namely

rd¼ M0 21jp0C ð2Þ m0M 0 11 þjpfC ð2Þ mf M 0 22jp0C ð2Þ m0M 0 12 jp0C ð1Þ m0M 0 11M 0 21 þjpfC ð2Þ mf jp0C ð1Þ m0M 0 12M 0 22 , ð73Þ

Fig. 2. A circular single layer of index n1and of thicknessr1–r0bounded by media

0 and 2. Here, the curved interfaces are intentionally plotted as straight lines for convenience of illustration.

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td¼ 4e1 r ffiffiffiffiffiffiffiffiffiffiffiffiffi m0=e0 p pKr0Hð2Þm k0r0 Hð1Þ m k0r0 jp0C ð1Þ m0M 0 11M021 þjpfC ð2Þ mf jp0C ð1Þ m0M 0 12M022 h i ð74Þ Eqs. (73) and (74) can be used to analyze the photonic band gap structure in a periodic CMS. Finally, it is worth mentioning that Eqs.(73) and (74)can be reduced to those for the E-polarization with a replacement of j-j.

7. Numerical results and discussion

In the analysis that follows, we would like to study the reflectance responses for three model structures. The first is a simple geometry of single cylindrical interface. This problem is similar to a single planar interface, from which the reflectance can be calculated by the familiar Fresnel reflection formula. The second geometry is a single cylindrical slab. This issue is remi-niscent to the problem of Airy slab in the planar geometry. The third structure we are interested in is the circular photonic crystal (CPC). The photonic band gap structure will be investigated.

a. Reﬂection properties in a single cylindrical interface

To present the numerical results, we ﬁrst investigate the reﬂection properties in a single cylindrical interface that separates two regions of refractive indices n0 (internal

med-ium) and n1 (external medium). Unlike in the usual planar

interface, the starting radius

r

0inFig. 1may play an important

role in the study of wave properties in a cylindrical system. In the problem of a single cylindrical interface, the starting radius is simply the radius of interface.Fig. 3depicts the reﬂectance, R ¼ 9r92, versus the starting radius

r

0 at different azimuthal

mode numbers m¼0, 1, 2, and 4, respectively. Here, n0¼1 and

n1¼3 are used, and the results of E- and H-polarization

are respectively given in the upper and lower panels. For E-polarization, the reflectance at m¼0 is small. The reflectance first increases with

r

0near the region of

r

0o0.2

l

, and then

remains nearly a constant. However, the reﬂectance is decreas-ing function of

r

0for m 40. A quick drop in reﬂectance can be

seen at m¼1. At m41, the reﬂectance ﬁrst is unity and then decreases with the increase in

r

0. At sufﬁciently large value of

r

0, the reﬂectance at m 40 converges to that of m ¼0.

The reﬂectance in the H-polarization is quite different from E-polarization. For m¼0, it is a decreasing function of

r

0.

However, at m 40, the reflectance exhibits a dip, as marked by the arrow. This dip with a minimum in reflectance can be regarded as an effective Brewster effect. The dip radius is thus defined as an effective Brewster radius, which is shifted to a larger value as m increases. In addition, at large radius,

r

0*

l

,

both polarizations have the same behavior in reflectance, that is, they remain nearly a constant and approach the limit of m¼0. Conclusively, the reflectance for both polarization modes can be significantly changed as a function of the starting radius when

r

0o

l

.

Fig. 4shows the E-polarization reflectance at distinct refractive index of the medium 1. The left panel is for m¼0, whereas the right one is for m¼4. It can be from the figures that the overall reflectance will be increased as n1increases. Such an increase

in reﬂectance can be ascribed to the impedance mismatch as in the case of planar boundary between two media. The same behavior is also seen in H-polarization, as illustrated inFig. 5. The Brewster radius at m¼4 has been slightly shifted to the right when n1is decreased. It means that the Brewster effect

can be controlled by n1at a ﬁxed azimuthal mode number.

InFig. 6, we ﬁx the index ratio n1/n0with different

combina-tions of (n0,n1). We see that the overall reﬂectance has been

shifted to the left as the index difference, n1n0, increases. The

Brewster dip in the H-polarization is narrowed down and moves toward the shorter radius of

r

0.

b. Reﬂection properties in a single cylindrical slab

Referring to Fig. 1, we now consider a single cylindrical slab occupying the space of

r

0o

r

o

r

1. The slab, which is assumed

to be immersed in free space (n0¼1), has a refractive index of

n1and a quarter-wavelength thickness of d ¼

r

1

r

0 ¼

l

0/4n1,

where

l

0 is called the design wavelength. By taking n1 ¼3,

r

0¼100 nm and

l

0¼500 nm, the wavelength-dependent

reflectance is depicted in Fig. 7, where different azimuthal mode numbers, m¼ 0, 2, and 4 are taken, and the left and right panels are for E- and H-polarization, respectively. It can be seen that the reflectance in the wavelength domain is divided into two regions. The first region located at

l

/

r

0o2 shows an

oscillating behavior. The other nonoscillating region occurs at

l

/

r

042. It means that the starting radius

r

0 has a strong

influence in the reflectance even the thickness of slab is fixed. For both polarizations, at higher m-number, say m ¼4, the wave is completely reflected because of the unity reflectance. If we keep the thickness unchanged and increase the starting radius to a large value of

r

0¼1000 nm, the reﬂectance is

shown inFig. 8. In this case, the oscillating region has been squeezed to a small region of

l

/

r

0o0.4, i.e.,

l

o400 nm. In the Fig. 3. The calculated reﬂectance as a function of the radius of the interfacer0.

Here,r0is normalized to the wavelength of the incident wave. The left panel is for

the E-polarization wave and the H-polarization is shown in the right panel. The refractive indices of inner and outer media are taken to be n0 ¼1 and n1¼3,

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Fig. 4. The calculated E-polarization reﬂectance as a function of the radius of the interfacer0at n0¼1 for different values in n1¼2, 3, 4, and 5, respectively. The left panel is

at m¼ 0, whereas the right one is at m¼ 4.

Fig. 5. The calculated H-polarization reﬂectance as a function of the radius of the interfacer0at n0¼1 for different values in n1. The left panel is at m¼ 0, whereas the right

one is at m¼ 4.

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nonoscillating region, the reﬂectance is inﬂuenced substan-tially at m¼0, and 2 compared toFig. 7.

We now investigate the thickness-dependent reﬂectance. The thickness d can be changed by varying the design wavelength

l

0when the starting radius

r

0 is ﬁxed. The reﬂectance as a

function of d is plotted inFig. 9, where the staring radius is

ﬁxed at

r

0¼100 nm and the operating wavelength is at

l

¼500 nm. Here, the left and right panels are for E- and H-polarization, respectively. It can be seen from the figure that the reflectance will oscillate as a function of slab thickness at m¼ 0, and 2. The reflectance is enhanced at a larger m-value. At m¼ 4, the magnitude in reflectance has been highly raised

Fig. 7. The calculated E- and H-polarization reﬂectance as a function of the wavelength (normalized to the starting radiusr0) at m¼ 0, 2, and 4, respectively. Here,

l0¼500 nm andr0¼100 nm.

Fig. 8. The calculated E- and H-polarization reﬂectance as a function of the wavelength (normalized to the starting radiusr0) at m¼ 0, 2, and 4, respectively. Here,

(9)

up to unity. The results shown inFig. 9are quite different from those of the planar slab, where the reﬂectance is a decreasing function of slab thickness[2].

c. Reﬂection properties in a cylindrical photonic crystal

For the third case study, we present the reﬂectance response for a periodic cylindrical multilayer structure, the cylindrical photonic crystal (CPC). The structure is air/(H/L)N_{/air, where}

H and L are the high- and low-index layers, respectively, and N is number of periods. The refractive indices and the thick-nesses of H and L are denoted by nH, nL, and dH, dL, respectively.

In the following results, we shall take H as TiO2with nH¼2.40

and L as MgF2with nL¼1.38[22]. In addition, both H and L are

taken to be quarter-wavelength layers, i.e., nHdH¼nLdL ¼

l

0/4.

The wavelength-dependent reﬂectance for E-polarization (left) and H-polarization (right) at m¼0 is plotted inFig. 10. Here,

l

0¼ 550 nm and

r

0¼400 nm are taken. It is seen that, in the

visible region, there is a high-reﬂectance region (HRR) or photonic band gap (PBG) at a number of periods of N ¼8. For N ¼ 2 and 4, the PBG cannot be clearly opened up. At m ¼0, there is no obvious difference between the E- and H-polariza-tion. In addition, the reﬂectance spectra shown here for the CPC are very similar as those in a planar 1D PC[22]. In other words, at m¼0, the study of PBG structure in a CPC can be effectively replaced by the simple planar 1D PC. At m¼0, The PBG structure does not related to the geometric curvature in the interfaces.

However, the PBG structure in a CPC can be affected by the nonzero azimuthal mode number. In Figs. 11 and 12, we respectively plot the reﬂectance spectra of E- and H-polariza-tion at m40 and N¼ 8. The PBG structure at m¼0 inFig. 10is now apparently inﬂuenced by m40. In E-polarization (Fig. 11), the overall PBG is red-shifted. In addition, the

Fig. 9. The calculated E- and H-polarization reﬂectance as a function of the slab thickness (normalized to the starting radiusr0) at m¼ 0, 2, and 4, respectively. Here,

l¼500 nm andr0¼100 nm.

Fig. 10. The calculated E- and H-polarization reﬂectance as a function ofl/l0at m ¼0 for three different numbers of periods, N ¼ 2, 4, and 8, respectively. Here,l0¼550 nm

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bandwidth of PBG is nearly unchanged as m increases. A pronounced influence in the side lobes (pass bands) can be seen at a large value of m. For example, at m¼8, the right band edge becomes more sharp and the pass band behind the band edge is highly raised up. In Fig. 12, we can see a salient influence in the PBG structure in H-polarization as m increases. The bandwidth of PBG at m¼4 is smaller than that at m ¼2. At m ¼6, a small dip with reflectance of 0.8 is generated and consequently the size of PBG is reduced. At m¼ 8, the right pass band, like in E-polarization, has been lifted up to become a stop band, leading to an enhancement of PBG, except the sharp deep near

l

/

l

0¼1.1.

Before going into the conclusion, it is worth mentioning the potential applications for the CPC. It can be used to design an annular Bragg reﬂector (ABR) useful in laser systems [23]. By surrounding a radial defect layer, annular Bragg lasers (or resonators) have been realized and demonstrated[24]. Recently, CPC-based circular-grating microcavity has been available[25]. Moreover, the CPC is an attractive and useful structure in the recent emerging ﬁeld of transformation optics[26,27].

8. Conclusion

The wave propagation in the cylindrical multilayer structure has been theoretically treated based on the cylindrical wave under H-polarization. The transfer matrix method in a cylindrical coordinates has been described. Formulae such as the reflection and transmission coefficients have been given and they strongly related to the azimuthal mode number because of the cylindrical waves. With the derived formulae, we have investigated the reflection responses for three basic cylindrical structures.

In a single cylindrical interface, the reﬂection responses in both polarizations are strongly dependent on the starting radius as well as the azimuthal mode number. In H-polarization, there exists a Brewster radius at which the reﬂectance attains a minimum. This Brewster radius will increase as the azimuthal mode number increases. On the other hand, it will decrease by increasing the difference in the refractive indices of the two media.

In the single cylindrical slab, the wavelength-dependent reﬂectance can be divided into the oscillating and nonoscillating regions. The ranges of these two regions are strongly dependent on the starting radius. In the thickness domain, it is found the reﬂectance has an oscillating behavior. This oscillation is then smeared out at a high value of azimuthal mode number.

In the cylindrical photonic crystal, the PBG structure at m¼0 is very similar to that of the planar 1D PC. The PBG structure can be signiﬁcantly changed only at m40. In E-polarization, the PBG structure is red-shifted when m is increased. In H-polarization, the PBG can be greatly enhanced at a higher m-value.

Acknowledgments

C.-J. Wu acknowledges the ﬁnancial support from the National Science Council of the Republic of China (Taiwan) under Contract No. NSC-100-2112-M-003-005-MY3 and from the National Taiwan Normal University under NTNU100-D-01.

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