Multilayered optical memory with bits stored as refractive index change. II. Numerical results of a waveguide multilayered optical memory

$Page 1: Multilayered optical memory with bits stored as refractive index change. II. Numerical results of a waveguide multilayered optical memory$
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Guo et al. Vol. 25, No. 7 /July 2008 /J. Opt. Soc. Am. A 1799

Multilayered optical memory with bits stored asrefractive index change. II. Numerical

results of a waveguide multilayered opticalmemory

Hanming Guo,1,3 Songlin Zhuang,1,* Shuwen Guo,1 Jiabi Chen,1 and Zhongcheng Liang2

1Shanghai Key Laboratory of Contemporary Optics System, College of Optics and Electronic Information Engineering,University of Shanghai for Science and Technology, 516 Jungong Road, 200093 Shanghai, China

2College of Optoelectronic Engineering, Nanjing University of Posts and Telecommunications,66 Xin Mofan Road, 210003 Nanjing, China

3E-mail: [email protected]*Corresponding author: [email protected]

Received February 20, 2008; accepted May 15, 2008;posted May 21, 2008 (Doc. ID 92933); published June 27, 2008

In terms of the electromagnetic theory described in Part I of our current investigations [J. Opt. Soc. Am. A 24,1776 (2007)], the numerical method for and results of numerical computations corresponding to the electro-magnetic theory of a waveguide multilayered optical memory are presented. Here the characteristics of thecross talk and the modulation contrast, the power of readout signals, the variation of the power of the readoutsignals with the scanning position along the track, and the distribution of the light intensity at the detector areinvestigated in detail. Results show that the polarization of the reading light, the feature sizes of bits, and thedistances between the two adjacent tracks and the two adjacent bits on the same track have significant effectson the distribution of the light intensity at the detector, the power of the readout signals, the cross talk, andthe modulation contrast. In addition, the optimal polarization of the reading light is also suggested. © 2008Optical Society of America

OCIS codes: 210.4680, 260.2110.

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. INTRODUCTIONnovel waveguide multilayered optical memory (WMOM)

1] was first reported in 2002, and the primary experi-ent [2] presented in 2004 demonstrated the feasibility ofWMOM. Recently, Yang et al. proposed a simple model

escribing the side scattering in the WMOM and deducedn expression for the relation between the attenuation co-fficient and the light propagation distance of a wave-uide with a distributed attenuation coefficient [3]. In therst part of the current study [4], which is hereafteralled Part I, we developed an electromagnetic theory of aMOM for the static case with the Lippman–Schwinger

quation, dyadic Green’s functions, and the vector coher-nt transfer function. This theory [4] is rigorous in itselfnd able to be used for the optimum design of a WMOM,ut its effective applications require highly powerful nu-erical methods, which mainly include numerical calcu-

ations of the Lippman–Schwinger equation and the elec-ric dyadic Green’s function (EDGF).

In this paper, we shall focus our attention on the inves-igation of the improvement of the performance of a

MOM. In order to accomplish this aim, first, we outlinehe general numerical method for the electromagneticheory of a WMOM [4], and, second, we introduce briefly aort of numerical method for the EDGF associated withhe planar multilayered media that is a major concern inhe applications of the theory of a WMOM [4] and an im-ortant problem in the electromagnetic field computation

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egion [5–10]. Finally, we investigate in detail the rela-ionships between the modulation contrast and the fea-ure sizes of bits and the modulation contrast and thehickness of the core, the relationships between the crossalk and the feature sizes of bits and the cross talk andhe distance between the two adjacent tracks or the twodjacent bits on the same track, the relationship betweenhe power of the readout signals and the length of bits,he variation of the power of the readout signals with thecanning position along the track, and the distribution ofhe light intensity at the detector. In the following analy-es, the definitions of some variables or concepts have noteen given for simplification. Interested readers are re-erred to Part I [4].

. INTRODUCTION TO THE NUMERICALETHOD

. Numerical Method for the Electromagnetic Theory ofWMOMhe schematic diagram of a WMOM is composed of theeading system, the waveguide multilayered disc (WMD)i.e., storage medium), and the detection system consist-ng of a confocal microscope (Fig. 1). The structure of the

MD is shown in Fig. 2, where the refractive indices ofhe cladding and substrate are identical. In terms of theperation principle [4] of a WMOM, a complete electro-agnetic theory of a WMOM should find the electric field

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1800 J. Opt. Soc. Am. A/Vol. 25, No. 7 /July 2008 Guo et al.

f the reading light, the scattered electric field of bits, andhe electric field at the pinhole plane of the detection sys-em. In Part I, we have introduced in detail how to deter-ine the above three kinds of electric fields. In essence,

ll of the works focus the Lippman–Schwinger equation

E�r� = Er�r� +�D

dr�GJ�r;r�� · V�r��E�r��, �1�

nd the vector imaging theory of an aplanactic system11]. In Eq. (1), Er�r� denotes the electric field of the read-ng light that can be viewed with those of guided modesonfined in the core of the reading layer waveguide ele-ent of the WMD, GJ�r ;r�� is the EDGF associated with

he planar multilayered media, the integration domain Ds a scattering region composed of all of bits, and the func-ion V�r�=−k0

2��p�r� describes the perturbations.Obviously, if the electric field Er�r� of the reading light

nd the EDGF GJ�r ;r�� associated with the planar multi-ayered media have been given, the electric field Ep�r� ofll points located inside the perturbations can be deter-ined from Eq. (1) by numerical methods. After Ep�r� has

een computed, the scattered electric field at the pinholelane can be obtained utilizing the vector imaging theoryimilar to that of [11], which has been introduced in detailn Part I. Therefore, the numerical computations of thelectromagnetic theory of a WMOM [4] include mainlyour parts except for setting initial values for various pa-ameters of a WMOM.

The first part is to calculate the electric field Er�r� ofhe reading light. As already indicated in Part I, once thetructures of the waveguide element are known, the elec-ric field of each guided mode may be found in any funda-ental book concerning theories of a planar waveguide.s for how many sizes of the amplitude of each guided

ig. 1. Schematic diagram of a waveguide multilayered opticalemory.

ig. 2. Structures of a waveguide multilayered disc with theame cladding and substrate.

ode there are, it is still necessary to consider couplingethods and the polarization of the incident light. In this

aper, we do not consider coupling methods but assumehat the power of all guided modes are identical. In termsf this assumption, the electric field Er�r� of the readingight can be generated explicitly.

The second part is to determine the electric field Ep�r�f all points located inside the perturbations from Eq. (1)y numerical methods. One method for the numerical cal-ulation of Eq. (1) is an iterative scheme based on the par-llel use of Lippman–Schwinger and Dyson’s equations,hich was introduced in detail by Martin et al. [12]. Thisumerical method is used in this paper, but it is not anfficient enough numerical method because when theumber of bits computed and that of discretized meshes ofach bit increase, the time and memory requirements in-rease rapidly, which limits the applications of the elec-romagnetic theory of a WMOM [4].

The third part is to calculate the EDGF GJ�r ;r�� associ-ted with the planar multilayered media, which is neces-ary for the numerical calculation of Eq. (1). Because ofhe slowly decaying and highly oscillating behavior of theDGF associated with the planar multilayered media, ex-

ensive research [5–10] has been done in order to acceler-te its numerical calculation. However, all of these ap-roaches [5–10] have individual limits and applied areas.iming at the WMOM, the method used in this paper for

he numerical calculation of the EDGF GJ�r ;r�� associatedith the planar multilayered media will be introduced inubsection 2.B.The fourth part is to calculate the scattered electric

eld at the pinhole plane from Eqs. (28)–(31) in Part I,here the integral equations (29) are evaluated by thedaptive Gauss–Kronrod quadrature [5]. After completinghe numerical calculation of the above four parts, the nu-erical computations of the electromagnetic theory of aMOM [4] will be realized.

. Numerical Method for the EDGF Associated with thelanar Multilayered Mediahe EDGF GJ�r ;r�� associated with the planar multilay-red media is given by the expressions (7)–(11) of [4]. It isoted that the forms of GJ�r ;r�� in [4] are different fromhose in [5], but they are identical in essence. In [5], Pau-us et al. proposed a technique for the accurate computa-ion of GJ�r ;r��. In this subsection, we shall first introducehe main idea of this technique and then give a detailedllustration for some problems, such as the computationsf residues and guided modes that were not indicated ex-licitly in [5].As indicated by Paulus et al. [5], the double integral of

�r ;r��, i.e., Eq. (7) of [4], can be further simplified as aet of one-dimensional semi-infinite integrals, so-calledommerfeld integrals, by introducing a cylindrical coordi-ate system in the spatial and frequency domains,amely, kx=k� cos �, ky=k� sin �, x−x�=� cos �, y−y�� sin � with 0��, ��2�. An integral of this kind cannote performed analytically but has to be evaluated numeri-ally. However, a straightforward implementation wouldail because of the mathematically awkward behavior of

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he integrand. In order to avoid these difficulties, Paulust al. [5] used Cauchy’s integral theorem and deformedhe integration path in the complex k� plane. It is notedhat there are two differences in our practical implemen-ation compared with [5]. One is that the usual time de-endence is exp�j�t� in Part I, whereas it is exp�−j�t� in5], which leads to our integration path running inside therst quadrant (see Fig. 3) in the complex k� plane insteadf the fourth quadrant of [5]. The other is that, in order toccelerate the computation speed of the numerical inte-ral, we adopt the integration path shown in Fig. 3, where0 is a wavenumber in free space and C is the integralath in the complex k�. Because branch points appearnly in the two outermost regions [5], k0 also representshe branch point when the structure of the WMD ishown in Fig. 2.

In addition, Paulus et al. [5] ordered the integrands ofhe nine components of GJ�r ;r�� in TE and TM modes (thelectric fields, respectively, being normal to and in thelane of incidence) with first- and second-order Besselunctions. Thus, seven independent terms can be defined,nd these terms have a similar behavior in the k� plane,hich makes possible their simultaneous integration. Inur practical implementation, we use the zero- and first-rder Bessel functions, and seven similar independenterms are also defined, namely,

U1 =�0

Nm,n� �k��k�J0�k��dk�, �2a�

U2 = 2�−1�0

Nm,n� �k��J1�k��dk� − U1, �2b�

V1 =�0

Nm,n�1 �k��k�J0�k��dk�, �2c�

V2 = 2�−1�0

Nm,n�1 �k��J1�k��dk� − V1, �2d�

V3 =�0

Nm,n�2 �k��k�2J1�k��dk�, �2e�

V4 =�0

Nm,n�3 �k��k�2J1�k��dk�, �2f�

V5 =�0

Nm,n�4 �k��k�J0�k��dk�, �2g�

ith for zz�

Fig. 3. Integration path in the complex k plane.
�
Nm,n� �k�� =1

�n��m,n� exp�− j�m�z − zm��exp�j�n�z� − zn��

�1 + Rm� exp�j2�m�z − zm��

�1 + Rn�� exp�− j2�n�z� − zn−1��, �3a�

Nm,n�1 �k�� =�n

kn2 ��m,n� exp�− j�m�z − zm��exp�j�n�z� − zn��

�1 + Rm� exp�j2�m�z − zm��

�1 + Rn�� exp�− j2�n�z� − zn−1��, �3b�

Nm,n�2 �k�� =1

kn2 ��m,n� exp�− j�m�z − zm��exp�j�n�z� − zn��

�1 + Rm� exp�j2�m�z − zm��

�1 − Rn�� exp�− j2�n�z� − zn−1��, �3c�

Nm,n�3 �k�� =�n

kn2�m

��m,n� exp�− j�m�z − zm��exp�j�n�z� − zn��

�1 − Rm� exp�j2�m�z − zm��

�1 + Rn�� exp�− j2�n�z� − zn−1��, �3d�

Nm,n�4 �k�� =kt

2

kn2�m

��m,n� exp�− j�m�z − zm��exp�j�n�z� − zn��

�1 − Rm� exp�j2�m�z − zm��

�1 − Rn�� exp�− j2�n�z� − zn−1��, �3e�

nd for z�z�

Nm,n� �k�� =1

�n��m,n�� exp�j�m�z − zm−1��exp�− j�n�z� − zn−1��

�1 + Rm� exp�− j2�m�z − zm−1��

�1 + Rn� exp�j2�n�z� − zn��, �4a�

Nm,n�1 �k�� =�n

kn2 ��m,n�� exp�j�m�z − zm−1��exp�− j�n�z� − zn−1��

�1 + Rm�� exp�− j2�m�z − zm−1��

�1 + Rn� exp�j2�n�z� − zn��, �4b�

Nm,n�2 �k�� =1

kn2 ��m,n�� exp�j�m�z − zm−1��exp�− j�n�z� − zn−1��

�1 + Rm�� exp�− j2�m�z − zm−1��

�1 − Rn� exp�j2�n�z� − zn��, �4c�

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Nm,n�3 �k�� =�n

kn2�m

��m,n�� exp�j�m�z − zm−1��exp�− j�n

�z� − zn−1��1 − Rm�� exp�− j2�m�z − zm−1��

�1 + Rn� exp�j2�n�z� − zn��, �4d�

Nm,n�4 �k�� =kt

2

kn2�m

��m,n�� exp�j�m�z − zm−1��exp�− j�n

�z� − zn−1��1 − Rm�� exp�− j2�m�z − zm−1��

�1 − Rn� exp�j2�n�z� − zn��, �4e�

here for the definition of all of the variables, the readers referred to Part I. Therefore, the nine components ofhe EDGF GJ�r ;r�� associated with the planar multilay-red media, i.e., Eq. (7) of Part I, can be expressed as fol-ows for zz�:

Gm,n�xx = − �j/8��U1 + U2 cos 2��, �5a�

Gm,n�xy = − �j/8��U2 sin 2�, �5b�

Gm,n�yx = Gm,n�xy , �5c�

Gm,n�yy = − �j/8��U1 − U2 cos 2��, �5d�

Gm,n�xx = − �j/8��V1 − V2 cos 2��, �6a�

Gm,n�xy = Gm,n�yx = �j/8��V2 sin 2�, �6b�

Gm,n�xz = �1/4��V3 cos �, �6c�

Gm,n�yy = − �j/8��V1 + V2 cos 2��, �6d�

Gm,n�yz = �1/4��V3 sin �, �6e�

Gm,n�zx = �1/4��V4 cos �, �6f�

Gm,n�zy = �1/4��V4 sin �, �6g�

Gm,n�zz = − �j/4��V5. �6h�

or z�z�, the nine components of GJ�r ;r�� can be also de-oted by Eqs. (6) except that the signs of the components

m,n�xz , Gm,n�yz , Gm,n�zx , and Gm,n�zy are opposite to those in thease of zz�.

The singularities of GJ�r ;r�� can be classified into twoypes: branch point singularities and pole singularities5]. For the EDGF GJ�r ;r�� associated with the planarultilayered media, the poles are determined by the poles

f the function

� = �1 − Rn�zn�Rn

��zn−1�exp�− j2�nln��−1, �7�

.e., Eq. (11a) of Part I. These poles correspond physicallyo modes guided by the layered structure [5,13]. For thelanar multilayered media, the guided modes may be ob-ained by the method of the transfer matrix [14]. As we

now, when Cauchy’s integral theorem and the deformedntegration path are applied to an integral, this integral

ay be denoted as

�k0

ka

g�k��dk� =�C

g�k��dk� + j��i=1

n

Res�k�i�, �8�

here k�i represents the ith pole of the integrand g�k�� inhe integration interval �ka ,kb�, C is the integration pathn the complex plane, and n is the number of poles. There-ore, when the numerical method for GJ�r ;r�� of [5] issed, the contribution of the residues must be considered.n what follows, we shall give a detailed illustration of theomputation of the residues.

In Eqs. (2), the integrals may be expressed in the fol-owing general forms:

W�r;r�� =�0

f�k�;r;r��

m�k��dk�, �9�

m�k�� = 1 − Rn�zn�Rn

��zn−1�exp�− j2�nln�, �10�

here W�r ;r�� represents U1, U2, V1, V2, V3, V4, or V5. Wessume that the pole k�i is the first-order pole of function7). Then the residue of the integrand in Eq. (9) may beritten as

Res�k�i;r;r�� =f�k�i;r;r��

m��k�i�, �11�

here m��k�i� denote the value of the first-order differen-ial of the function m�k�� at the pole k�i, namely,

dm�k��

dk�

= j2lnRnRn

�d�n

dk�

−dRn

dk�

Rn� − Rn

dRn

�

dk�

exp�− j2�nln�, �12�

here Rm =Rm

�zm� and Rm� =Rm

��zm−1� are recurrence for-ulas. Having a differential in Eqs. (9c) and (10c) of Partabout the variable k�, we can obtain

dRi

dk�

= �1 + �iRi+1

e−j2�i+1li+1�−2��1 − �Ri+1 e−j2�i+1li+1�2�

d�i

dk�

+ �1 − �i�i

�e−j2�i+1li+1dRi+1

dk�

− j2li+1Ri+1

d�i+1

dk�� ,

�13a�

dRi�

dk�

= �1 + �i�Ri−1

� e−j2�i−1li−1�−2��1 − �Ri−1� e−j2�i−1li−1�2�

d�i�

dk�

+ �1 − �i��i

��e−j2�i−1li−1dRi−1�

dk�

− j2li−1Ri−1�

d�i−1

dk�� .

�13b�

imilarly, in terms of Eqs. (9d), (10d), and (11d) of Part I,he following differentials may be found for the TE mode:

d�i

dk�

=2

��i + �i+1�2�i+1

d�i

dk�

− �i

d�i+1

dk� , �14a�

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d�i�

dk�

=2

��i + �i−1�2�i−1

d�i

dk�

− �i

d�i−1

dk� . �14b�

or the TM mode,

d�i

dk�

=2�i+1�i

��i�i+1 + �i+1�i�2�i

d�i+1

dk�

− �i+1

d�i

dk� , �15a�

d�i�

dk�

=2�i−1�i

��i�i−1 + �i−1�i�2�i

d�i−1

dk�

− �i−1

d�i

dk� . �15b�

s �i= �ki2−k�

2�1/2 for 0�k��ki and �i=−j�k�2−ki

2�1/2 for k�

ki, there are

d�i

dk�

= �− k��ki2 − k�

2�−1/2 0 � k� � ki

− jk��k�2 − ki

2�−1/2 k� ki. �16�

n addition, when i=N,

dRN

dk�

=d�N

dk�

. �17a�

hen i=1,

dR1�

dk�

=d�1

�

dk�

. �17b�

herefore, we shall be able to derive the residues of thentegrand in Eq. (9) from Eqs. (11)–(17).

Up to now, we have introduced a numerical method forhe electromagnetic theory of a WMOM, including thosef the Lippman–Schwinger equation and the EDGF. Theumerical method for the EDGF is not efficient enoughompared with other methods [6–10], but it is accuratend is the simplest compared with the others [6–10]. Nexte shall investigate in detail the performances of aMOM based on the above numerical methods.

. NUMERICAL RESULTSn the design of optical storage, one of the most importantasks is to decrease the cross talk and enhance the modu-ation contrast. For a WMOM, the question of how to in-rease the power of the readout signals is also of vital im-ortance. In what follows, we shall focus on thenvestigation of these problems. Before proceeding fur-her, the definitions of cross talk and modulation contrastill be first given.In Part I, the cross talk of a WMOM is defined as

Cross talk = − 20 log��Pd − PA�/PA�, �17�

here Pd is the power of the readout signals, namely, theower of all light transmitting the pinhole collected by theetector [see Eq. (32) of Part I], and PA is the power of thecattered light, transmitted by the pinhole, of the de-ected bit A [see Eq. (34) of Part I]. At this time, the unitf the cross talk is decibels. As the total electric fields athe detector are the coherent superposition of the scat-ered electric fields radiated by all bits, including the de-ected bit A, the power Pd might be bigger or smaller thanhe power PA due to the coherent extension or the coher-nt subtraction. When P is smaller than P , the defini-
d A
ion of the cross talk given by Eq. (17) will be meaning-ess. In order to treat this case, the cross talk of a WMOMs redefined as

Cross talk = �Pd − PA�/PA. �18�

In this paper, we discuss only the modulation contrastesulting from bits with the smallest size. Shown in Fig., L, W, T, and B represent the length and width of bitsith the smallest size, the distance between the two ad-

acent tracks, and the distance between the two adjacentits on the same track, respectively. The stacking factor athe same track is 0.5, where �xo ,yo� denotes the coordi-ates at the WMD of the reading position and P1 and P0re the powers of the readout signals when the readingosition �xo ,yo� is located at the center of the bit and athe midpoint of the two adjacent bits on the same track,espectively (see Fig. 4). Similar to the cross talk, becausef the coherent extension or the coherent subtraction ofcattered electric fields, the modulation contrast is de-ned as

Modulation Contrast = ��P1 − P0�/P1 for P1 � P0

�P1 − P0�/P0 for P1 � P0.

�19�

t this time, there is −1�modulation contrast�1.In all calculations, if it is not indicated especially, the

efault values of all parameters are as follow: The thick-esses of the core and the clad are, respectively, 1.5 �mnd 4 �m. The refractive indices of the core, the clad, andhe bit are 1.58, 1.406, and 1.406, respectively. A standardit representing the binary code “1” is a cube of size.6 �m, and the stacking factor is 0.5 on the same track.he number of the three-layer planar waveguide element

s 1. Other parameters are the wavelength �=0.65 �m,he numerical aperture NA=0.65 of L2, the diameter R4 mm of L2, the nominal magnification M=40 of the de-

ection system L2L3, the width of the incident light con-isting of guided modes confined in the core being 0.1 mm,nd the output power of laser P=10 mW. The discretizedumbers of a standard bit is 2 2 2. The diameter andiscretized numbers of the pinhole are, respectively,4 �m and 20 20. The sampling number is 80 along thecanning position. The relativity precision of the numeri-al integral is 10−6. In Figs. 7–9, 11–16, curves a, b, and cepresent the type of guided mode confined in the core be-ng mixed mode (i.e., including simultaneously the TEnd TM modes), TE mode, and TM mode, respectively.

ig. 4. Distribution of bits and the position of the reading lightpot.

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. Variation of the Power of the Readout Signals withhe Scanning Position along the Trackn optical storage, the readout of the information recordedn the WMD is realized by the variation of the power ofhe readout signals with the scanning position along therack. As an example, the bits compose the binary code0110111010,” and the type of guided mode confined in theore is the mixed mode. Namely, the number of the bits onhe x axis is 1, that on the y axis (namely, the track) is 6,nd the bits are arrayed as the binary code “0110111010.”bviously, Fig. 5 shows that the detected power varies ac-

urately with the sequence of the binary code0110111010,” which demonstrates the validity of thelectromagnetic theory of a WMOM derived in Part I.

. Effects of the Polarization of the Reading Light on theower of the Readout Signals and the Distributionf the Light Intensity at the Detectorigure 6 describes the distribution of the light intensitycattered by a single bit at the detector plane for variousolarizations of the reading light. Here the size and theiscretized numbers of the detector plane are 76 �m and0 60, respectively. The bit is a cube of size 0.6 �m, andts discretized numbers are 5 5 5.

In terms of the principle of a WMOM [4], the readingight propagates along the x axis. Shown in Fig. 6, whenhe nominal magnification is M=40, the distribution ofhe light intensity scattered by a single bit at the detectorlane is not a square of size 24 �m as we expect, but it isisshapen. For the mixed and TE modes, it is extended

long the x axis, its length is approximately 24 �m on theaxis, and its width on the x axis is about twice as long as

ts length on the y axis [see Figs. 6(a) and 6(b)], whereasor the TM mode, it is extended along the y axis and itsidth is approximately 24 �m on the x axis [see Fig. 6(c)].In addition, the powers of the readout signals are, re-

pectively, 18.9 nW, 36.2 nW, and 1.5 nW for the mixed,E, and TM modes [note the values of the color bar (gray-cale in print) of Fig. 6], which shows that a higher powerf the readout signals may be obtained if the TE mode issed. Moreover, the power of the readout signals for theE mode is about double that for the mixed mode and 24imes that for the TM mode. As indicated in Part I, theypes and sizes of the guided modes confined in the core ofhe reading layer waveguide element of the WMD are re-ated to the structures of the waveguide, the coupling

ethods, and the polarization of the incident light. Whenhe coupling method used is a parallel incident light thats focused into the core of the reading layer waveguide el-

ig. 5. Variation of the power of the readout signals with thecanning position along the track. Here the bits compose the bi-ary code “0110111010.”

ment, the mixed modes always exit for the general polar-zed incident light, such as linearly and circularly polar-zed incident light. In order to further enhance the powerf the readout signals, some specially polarized incidentight may be used. Because a radially or azimuthally po-arized beam can only excite, respectively, TM- or TE-

ode plane waves in the core when they are coupled intoperfect planar waveguide by a lens [15], a higher power

f the readout signals may be obtained when the azimuth-lly polarized incident light is used for a WMOM.As the output power of laser is P=10 mW in this paper,

hough the azimuthally polarized incident light is used,he power �36.2 nW� of the readout signals is still as lows 4 10–6 of the output power of the laser. Moreover, it ishe calculational result under the case of neglecting theosses of powers when the incident light is coupled intohe core, which means that the actual power of the read-ut signals is far smaller than 36.2 nW. Hence, it is verymportant for a WMOM to increase the power of the read-ut signals.

In conclusion, in the design of a WMOM, we should usehe azimuthally polarized incident light so as to obtainE modes in the waveguide. In addition, for the TE mode,ecause of the extension along the x axis of the distribu-ion of the light intensity at the detector plane, the dis-ance between the two adjacent tracks should be 200%onger than the width of the bit in order to meet the re-uirement of the cross talk. Moreover, because of the ab-ence of extension along the y axis, the distance betweenhe two adjacent bits on the same track may be equal tohe length of the bits, namely, the stacking factor being.5.

. Relationship between the Modulation Contrast andhe Feature Sizes of Bitss we know, it is vital for a WMOM to have an optimumesign on the feature sizes of bits so as to improve theodulation contrast. Next we will investigate this prob-

em in detail. In this simulation, the number of the bits onhe x axis is 1, that on the y axis (namely, the track) is 3,nd the bits are arrayed as the binary code “10101.” Theetected bit A is the second “1.”Figure 7 shows a plot of the modulation contrast as a

unction of the depth of bits in the range [0.2�, �] for theixed, TE, and TM modes. It is easy to see that the modu-

ation contrast is an oscillating function of the depth ofits in the case of a certain width and length of bits. Forhe mixed and TE modes, the modulation contrast will in-rease with increasing depth of bits in the range [0.2�,.45�] and will reach its maximum value (being, respec-ively, about 0.7 and 0.75) when the depth of bits is ap-roximate up to 0.45�, whereas for the TM mode, it willrrive at a maximum value of 0.74 when the depth of bitss 0.6�. When the depth of bits is continuously increased,he modulation contrast will be decreased. Moreover,hen the TE mode is used, the modulation contrast is al-ays bigger than 0.6 for the depth of bits in the range

0.2�, �] for the default parameters used in this paper (seehe paragraph before Subsection 3.A). In Fig. 7 it is alsopparent that the TE mode is the best illumination to im-rove the modulation contrast, the mixed mode takes sec-nd place, and the TM mode is the worst.

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Figure 8 describes the relationship between the modu-ation contrast and the width of bits in the case of a cer-ain length and depth of bits. For the mixed and TEodes, the modulation contrast increases with increasingidth of bits within the range �0.2� /NA,0.45� /NA�.hen the width of bits is more than 0.45� /NA, the modu-

ation contrast will drop. For the TM mode, the modula-ion contrast is almost smaller than 0.4 at the total range0.2� /NA,� /NA�, at which it is difficult to detect the in-ormation recorded in the WMD. It is apparent from Fig.that for the TE mode, the modulation contrast may also

e greater than 0.6 despite that the width of bits is only.2� /NA. Hence, when the TE mode is used, we can makese of the smaller width of bits so as to increase the den-ity of the tracks and the recording density of the opticalemories.Figure 9 describes the relationship between the modu-

ation contrast and the length of bits in the case of a cer-ain width and depth of bits. In this simulation, when theength of bits is changed, the distance between the twodjacent bits on the same track is always equal to theength of bits. Figure 9 shows that when the length of bitss within the range �0.2� /NA,0.45� /NA�, the modulationontrast is near zero or slightly smaller than zero, whicheans that at this time, the length of bits is too small to

ecord information. For the mixed and TE modes, whenhe length of bits is greater than 0.45� /NA, the modula-ion contrast will increase with increasing length of bits

ig. 6. Distribution of the light intensity scattered by a single biixed mode, (b) TE mode, (c) TM mode.

ig. 7. Relationship between the modulation contrast and theepth of bits in a case of a certain length and width of bits.urves a, b, and c represent the mixed, TE, and TM modes,espectively.

p to 0.65� /NA, which shows that at this time, the ratiof the effective signals in the readout signals is larger,hich will be helpful for decoding the readout signals.hen the length of bits is longer than 0.65� /NA, theodulation contrast will tend to the limit. The above be-aviors indicate that, for the short length of bits, thehange in length has an obvious effect on the modulationontrast and that the longer the length of bits is, the big-er the modulation contrast is. For the long length of bits,he change in length does not have a significant effect on

e detector plane for various polarizations of the reading light: (a)

ig. 8. Relationship between the modulation contrast and theidth of bits in the case of a certain length and depth of bits.urves a, b, and c represent the mixed, TE, and TM modes,espectively.

ig. 9. Relationship between the modulation contrast and theength of bits in the case of a certain width and depth of bits.urves a, b, and c represent the mixed, TE, and TM modes,espectively.

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he modulation contrast. Finally, it is clear from Fig. 9hat when the length of bits is changed and the width andepth of bits kept fixed, the TE mode is also the best illu-ination to improve the modulation contrast among the

hree kinds of modes.

. Relationship between the Modulation Contrast andhe Thickness of the Coren this subsection, we will discuss the effect of the thick-ess of the core on the modulation contrast. The param-ters used in this simulation are the same as those usedn Subsection 3.C. In terms of the principle of a WMOM4], the types and the sizes of guided modes confined inhe waveguide are very sensitive to the change of thehickness of the core. In other words, the electric fields ofhe reading light are going to vary with the change of thehickness of the core, which will lead to the change of theeadout signals. Figure 10 describes the effect of thishange on the modulation contrast for the mixed mode. Its evident from Fig. 10 that the thickness of the core has aignificant effect on the modulation contrast. The plot ofhe modulation contrast as a function of the thickness ofhe core in the range [1 �m, 2 �m] oscillates drastically,ut its average value is approximately fixed to 0.65 for theefault parameters used in this paper (see the paragraphefore Subsection 3.A).

. Relationship between the Cross Talk and the Featureizes of Bitsn this subsection, the relationship between the cross talknd the feature sizes of bits will be investigated in detail.n order to avoid a vast amount of computation and toake the numerical simulation run successfully, only the

ffects of the bits on the same track with and adjacent tohe detected bit A on the cross talk are considered. At thisime, the number of the bits on the x axis is 1, that on theaxis is 3, and the bits are arrayed as the binary code

10101.” The detected bit A is the second “1.”Figure 11 shows a plot of the cross talk as a function of

he depth of bits in the range [0.2�, �] for the mixed, TE,nd TM modes. It is easy to see that the cross talk is anscillating function of the depth of bits in the case of a cer-ain width and length of bits. For the TE mode, when theepth of bits is within the ranges [0.2135�, 0.4712�] and0.9�, �], the cross talk is bigger than zero, which meanshat the power Pd of the readout signals is strengthened

ig. 10. Relationship between the modulation contrast and thehickness of the core.

ecause of the coherent extension. When the depth of bitss in the range [0.4712�, 0.9�], the cross talk is smallerhan zero, which means that the power Pd of the readoutignals is weakened because of the coherent subtraction.n the design of a WMOM, the basic principles are to ob-ain a higher power of the readout signals, a smaller crossalk, and a higher modulation contrast. Hence, consider-ng the relationship between the modulation contrast andhe depth of bits (see Fig. 7), for the TE mode, an opti-um depth of bits is approximately 0.45�. At this time,

he modulation contrast reaches its maximum value of.75 and the cross talk is about 0.02 (i.e., 34 dB). Figure1 also shows that for the TM mode, the cross talk is al-ays bigger than 0.05 (i.e., always smaller than 26 dB).bviously, the TE mode is the most optimum illuminationmong the mixed, TE, and TM modes for the cross talk.Figure 12 shows a plot of the cross talk as a function of

he width of bits in the range [0.2� /NA, � /NA] for theixed, TE, and TM modes. For the mixed and TE modes,

hese plots are approximate sine functions and the char-cteristic of the cross talk is almost identical when theidth of bits is within the range [0.45� /NA, 0.75� /NA].or the TM mode, the variation of the cross talk is too

arge to meet the requirement of an optical memory. Forhe mixed and TE modes, when the width of bits is withinhe range [0.54� /NA, 0.67� /NA], the cross talk will be inhe range (0, 0.03) (i.e., bigger than 30 dB). Moreover, athis time, if the width of bits is equal to 0.54� /NA, theodulation contrast is also greater than 0.7 in terms ofig. 8.

ig. 11. Relationship between the cross talk and the depth ofits in the case of a certain length and width of bits. Curves a, b,nd c represent the mixed, TE, and TM modes, respectively.

ig. 12. Relationship between the cross talk and the width ofits in the case of a certain length and depth of bits. Curves a, b,nd c represent the mixed, TE, and TM modes, respectively.

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Figure 13 describes the relationship between the crossalk and the length of bits in the case of a certain widthnd depth of bits. In this simulation, when the length ofits is changed, the distance between the two adjacentits on the same track is always equal to the length ofits. It is evident from Fig. 13 that, for the range of theength of bits considered, the cross talk is an approxi-

ately monotonic decreasing function of the length ofits. Shown in Figs. 11–13, the effect of the length of bitsn the cross talk is more obvious than that of the depthnd width of bits. When the length of bits is below.4� /NA, the cross talk is bigger than 0.078 (i.e., smallerhan 22 dB) for the TM mode and bigger than 0.367 (i.e.,maller than 8.7 dB) for the mixed and TE modes. Be-ause the TE mode is more optimum for improving theower of the readout signals and the modulation contrasts indicated before than the mixed and TM modes, nowhe TE mode will be further focused on. Shown in Fig. 13here is a very small range [0.59� /NA, 0.61� /NA] wherehe cross talk will be in the range (0, 0.03) (i.e., biggerhan 30 dB) for the TE mode. When the length of bits isqual to 0.6� /NA, the cross talk is 0.018 (i.e., 35 dB) (seeig. 13) and the modulation contrast is about 0.7 (see Fig.).

. Relationship between the Cross Talk and theistance between the Two Adjacent Tracks or the Twodjacent Bits on the Same Trackext the relationship between the cross talk and the dis-

ance between the two adjacent tracks or the two adjacentits on the same track will be investigated in detail. Forhe former, the number of the bits on the x axis is 3, thatn the y axis is 1, and the bits are arrayed as the binaryode “10101” along the x axis. The detected bit A is theecond “1.” The binary code “0” denotes the distance be-ween the two adjacent tracks, whose variation range is0.5, 4] with the unit of the width of bits. For the latter,he number of the bits on the x axis is 1, that on the y axiss 3, and the bits are arrayed as the binary code “10101”long the y axis. The detected bit A is also the second “1.”he binary code “0” denotes the distance between the twodjacent bits on the same track, whose variation range is0.5, 4) with the unit of the length of bits.

Shown in Figs. 14 and 15, the plots of the cross talk asfunction of the distance between the two adjacent tracks

ig. 13. Relationship between the cross talk and the length ofits in the case of a certain width and depth of bits. Curves a, b,nd c represent the mixed, TE, and TM modes, respectively.

r the two adjacent bits on the same track are an approxi-ately attenuation sine or cosine oscillation function.ne possible reason for the oscillation is the coherent su-erposition of the image fields of many bits. A more de-ailed explanation is the following. Because of the finiteperture of the detection system, the image field of eachit is a diffractive pattern. The interference pattern re-ulting from many diffractive patterns of many bits willary with increasing distance between the two adjacentracks or the two adjacent bits on the same track. Hence,ue to the oscillation characteristic, the distance betweenhe two adjacent tracks and the two adjacent bits on theame track must be selected carefully so as to meet theequirement of the cross talk.

For the mixed and TE modes, the plots are almost iden-ical (see Figs. 14 and 15). Compared with the mixed andE modes, the oscillation amplitude of the plot of the TMode is far bigger. Hence, both the mixed and TE modes

re better than the TM mode for improving the cross talk.n terms of Fig. 14, when the distance between the twodjacent tracks is twice as long as the width of bits, theross talk is 0.03 (i.e., 30 dB) for the mixed and TE modes.nly when this distance is at least four times longer than

he width of bits does the absolution of cross talk start toe consistently smaller than 0.04 (i.e., bigger than 27 dB)or the mixed and TE modes. In Fig. 15, when the dis-ance between the two adjacent bits on the same track isqual to the length of bits, the cross talk is 0.018 (i.e.,5 dB) for the mixed and TE modes. When this distance is.6 times longer than the length of bits, the absolution of

ig. 14. Relationship between the cross talk and the distanceetween the two adjacent tracks. Curves a, b, and c represent theixed, TE, and TM modes, respectively.

ig. 15. Relationship between the cross talk and the distanceetween the two adjacent bits on the same track. Curves a, b,nd c represent the mixed, TE, and TM modes, respectively.

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he cross talk starts to be consistently smaller than 0.03i.e., bigger than 30 dB) for the mixed and TE modes.ased on the above analyses, the distance between the

wo adjacent tracks has a more significant effect on theross talk compared with the distance between the twodjacent bits on the same track. The main reason for thiss that for the mixed and TE modes, the distribution ofhe light intensity scattered by a single bit at the detectorlane is extended along the x axis, whereas it is not ex-ended along the y axis.

. Relationship between the Power of the Readoutignals and the Length of Bitss already indicated, for a WMOM, the question of how to

ncrease the power of the readout signals is of vital impor-ance. By intuition, increasing the feature sizes of bits canncrease the power of the readout signals. However, inerms of the above analyses, if the width and depth of bitsre too long or deep, the modulation contrast will be de-reased, whereas this is not true for the length of bits.herefore, we may obtain a higher power of the readoutignals by increasing the length of bits.

Figure 16 shows a plot of the power of the readout sig-als as a function of the length of bits in the range0.3 �m,1.2 �m� for the mixed, TE, and TM modes. Herenly one bit is considered. Its discretized numbers are 44 4. It is apparent from Fig. 16 that a higher power of

he readout signals may be obtained if the TE mode issed. When the length of bits is shorter than 0.9 �m, theower of readout signals will increase very slowly with in-reasing length of bits. However, when the length of bitss longer than 0.9 �m, for the TE mode, the power of read-ut signals will increase rapidly.

. CONCLUSIONSn this paper, the numerical methods for the electromag-etic theory of a WMOM [4] and the EDGF associatedith the planar multilayered media are described. Basedn these numerical methods, the characteristics of theross talk, the modulation contrast, and the power ofeadout signals are investigated in detail by computerimulations. Some important conclusions are drawn.

First, the polarization of the reading light has signifi-ant effects on the distribution of the light intensity at the

ig. 16. Relationship between the power of the readout signalsnd the length of bits. Curves a, b, and c represent the mixed,E, and TM modes, respectively.

etector, the power of the readout signals, the cross talk,nd the modulation contrast. Among the mixed, TE, andM modes, the optimum illumination is the TE mode,hich can be realized by the azimuthally polarized inci-ent light. When the TE mode is used, we can obtain aigher power of the readout signals, a smaller cross talk,nd a higher modulation contrast. However, at this time,ecause of the extension along the x axis of the distribu-ion of the light intensity at the detector plane, the longeristance between the two adjacent tracks should bedopted in order to meet the requirement of the crossalk.

Second, the feature sizes of bits have significant effectsn the modulation contrast and the cross talk. If theidth and depth of bits are too long or deep, the modula-

ion contrast and the cross talk will be deteriorated. How-ver, if the length of bits become longer, the modulationontrast will tend to the limit and the cross talk will be-ome smaller.

Third, due to the oscillation characteristic of the crossalk as a function of the distance between the two adja-ent tracks or the distance between the two adjacent bitsn the same track, these distances must be selected care-ully so as to meet the requirement of the cross talk.

Finally, in order to enhance the power of the readoutignals, it will be an optimum choice to increase theength of bits and keep the width and depth of bits fixed.

CKNOWLEDGMENTShis work was supported by the National Basic Researchrogram of China (2005CB724304), the Shanghai Lead-

ng Academic Discipline Project (T0501), the Nationalatural Science Foundation of China (60777045 and0377006), the Key Basic Research Program of Sciencend Technology Commission of Shanghai Municipality07JC14056), the Shanghai Educational Developmentoundation (2007CG61), and the Shanghai Foundation

or Cultivation of Outstanding Young Teachers in Univer-ities (slg-07004).

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Documents

Multilayered optical memory with bits stored as refractive index change. II. Numerical results of a waveguide multilayered optical memory