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　　折射率变化型多层光存储的矢量衍射理论

　　作者姓名：郭汉明

　　论文题目：折射率变化型多层光存储的矢量衍射理论

　　作者简介：郭汉明，男， 1977年12月出生，2003年09月师从于上海理工大学庄松林教授，于2007年08月获博士学位。

　　中文摘要

　　多层光存储是解决信息社会对高密度大容量数据存储要求的重要技术之一，而建立相应的电磁理论模型对促进多层光存储技术的迅速发展具有重要意义。折射率变化型多层光存储记录的信息坑是因为其折射率不同于其周围的介质，利用信息坑对光的散射而读出信息，因此可以利用脉冲激光在各种透明材料中记录信息，可以不需要光敏材料，从而不会像光敏聚合物那样存在因收缩或流动而导致的畸变现象和因紫外光产生的异构化问题，也不会像光折变材料那样存在记录数据的固定难题。与全息体存储、双光子多层光存储和荧光多层光存储相比较，折射率变化型多层光存储在下一代高密度光存储中占有显著的优势。根据折射率变化型多层光存储的读入光入射方式，折射率变化型多层光存储可以分成采用传统读取方式(读入光是通过物镜沿垂直于光盘的表面方向聚焦到待读层，进行信息的读取，例如DVD光盘)的多层光存储(简称传统多层光存储)和波导多层光存储(读入光被物镜从光盘的侧面耦合到待读平面波导单元的芯层并激发出导模，该导模遇到信息坑时将散射出光波导，从而实现信息的读取)。本论文的目的就是根据传统多层光存储和波导多层光存储的工作原理，建立折射率变化型多层光存储严格的矢量衍射模型，以便能够精确地描述信息坑的散射光在探测器平面上的像场分布、光盘结构参数对串扰和调制度等光存储性能的影响机理，为折射率变化型多层光存储的实验研究及优化设计提供理论指导，促进多层光存储技术的实用化发展。

　　针对这一目的，本论文首先根据建立折射率变化型多层光存储的工作原理，将其读取系统分解为读系统、多层光盘(即存储介质)和探测系统，然后提出利用光学成像的矢量场理论来解决读系统和探测系统的像场计算，利用李普曼—薛维格尔方程和平面分层均匀介质的并矢格林函数来解决读入光与信息坑相互作用后的散射场计算，从而实现用严格的电磁理论描述折射率变化型多层光存储工作的完整过程。在该思路的指导下，本论文在第二章详细推导出平面分层均匀介质的并矢格林函数后，建立了一种有效的平面分层均匀介质并矢格林函数的数值计算方法。为了解决平面分层均匀介质并矢格林函数中由于被积函数具有高振荡和奇点特性而导致的计算难题，我们深入分析了各种已知的数值算法，利用平面波导的矩阵转移理论和图解法推导出分层均匀介质并矢格林函数极点以及计算这些极点的留数的递推公式;利用复平面变换积分路径方法和正弦或正切变量替换成功地建立了一种有效的分层均匀介质并矢格林函数的数值计算方法。该数值计算方法由于考虑了极点处留数的贡献，即提取了表面波，从而能够用于远场区域的计算，这在多层光存储的模型计算中是必须考虑的。

　　由于高密度光存储的信息坑尺寸接近甚至小于波长，而且光学头的数值孔径会超过0.9(如蓝光存储)，因此光存储中的读系统和探测系统的像场计算必须基于矢量成像理论。然而目前国内外光学系统成像的矢量场理论尚未完善，有较大的应用局限性及不足，不能满足光存储中的理论需求。为此，本论文在第三章系统地建立了消球差光学系统成像的矢量场理论。在这一部分，本论文①利用横向电磁场的模式展开式建立了均匀介质中任意偏振电磁场的矢量平面波谱公式。该矢量平面波谱公式由TM波和TE波组成，且每个平面波的振幅和偏振态完全分离，与文献

　　(Opt. Commun., 1998,

　　152:108-118)中的矢量平面波谱公式相比(每个平面波的振幅和偏振态没有分离)，具有更加明显的物理意义，在某些场合中应用更加便利。我们利用该矢量平面波谱公式成功地证明了“方位角偏振光经过透镜聚焦耦合到平面波导芯层时，将在波导芯层中仅仅激发出

　　TE模;而径向偏振光则仅仅激发出TM模”。②利用电磁波的矢量平面波谱公式和稳相法建立了相干点源位于光轴上任意位置时，消球差共轴光学系统成像的矢量衍射理论，并且通过数值模拟研究验证了物方孔径角、像方孔径角和光源偏振态等对像场结构、光学系统分辨率等方面的影响，证明了经典的Wolf成像矢量场理论(Proc.

　　R. Soc. London, Ser. A, 1959, 253,

　　358-370)的不足，同时表明Wolf成像矢量场理论只适用于点源位于光轴无穷远情况，无法直接应用于物方为大数值孔径角的情况。③首次建立了一种类似于标量衍射理论中相干传递函数的矢量相干传递函数，推导出基于矢量衍射理论的光学系统成像中联系像场和物场的简单关系式。该关系式表明像场的矢量平面波谱等于对应x偏振和y偏振单位振幅点电场源的矢量相干传递函数和相应的横向物场分量的标量谱的乘积之和。④首次利用矢量相干传递函数和分层均匀介质的并矢格林函数建立了消球差光学系统物方和像方分别为分层均匀介质时矢量点源的成像物理模型，并且证明了文献[J.

　　Mod. Optics, 1998,

　　45(8):1681-1698]认为“物空间入射光线和像空间对应出射光线相对于光轴的夹角随散射体在光轴上的位置变化而变化”的错误观念，同时指出了正确的应用方法。这一系列成果基本完善了消球差光学系统成像的矢量场理论，也解决了折射率变化型多层光存储中成像系统的严格矢量衍射理论问题。

　　根据传统多层光存储与波导多层光存储的工作原理及其读入光入射方式的差异，本论文利用在平面分层均匀介质并矢格林函数数值计算方法和光学成像的矢量场理论方面取得的研究成果，在第四章和第五章首次系统地建立了折射率变化型多层光存储的矢量衍射理论，包括传统多层光存储和波导多层光存储的矢量衍射模型。该矢量衍射模型可以适用于动态的信号读出、计算信号功率、探测器表面上的场分布、调制度和串扰随信息坑尺寸的变化、道间距和层间距对串扰的影响、以及码间影响等参数。编制了一个折射率变化型多层光存储的模拟计算软件，分析了光源偏振态、波长、探测系统数值孔径对信号功率及其像场分布的影响;计算了沿信道方向二进制码的信号功率分布;分析了调制度和串扰随信息坑长度、宽度和深度的变化以及光源偏振态和数据层厚度对调制度的影响。通过模拟计算，得到了一系列重要结论。例如，在波导多层光存储中，读入光的最优偏振态为方位角偏振(azimuthal

　　polarization)，因为方位角偏振光经过透镜聚焦耦合到平面波导芯层时，将在波导芯层中仅仅激发出TE模，而TE模相对于TM模而言，可以使波导多层光存储的读出信号功率更高、串扰更低、调制度更大。又如，在传统多层光存储中，探测器接收的来自多层存储介质分界面的反射光功率越大，读出信号的调制度越大，因此每个数据层的厚度最好设计成半波长或者波长的整数倍，此时调制度可以超过0.9，甚至接近1。

　　本论文的研究成果将不仅促进折射率变化型多层光存储的实用化发展和性能提高，还将促进光学系统成像矢量场理论的发展及完善。

　　关键词：多层光存储　波导多层光存储 传统多层光存储 矢量成像理论　并矢格林函数 李普曼—薛维格尔方程

　　Vector diffraction theory of a multilayered optical memory with bits stored

　　as refractive index change

　　Guo Hanming

　　ABSTRACT

　　Multilayered optical memory is one of important technologies used to meet

　　the demands for high density recording in the information society. It is very

　　significant to construct corresponding electromagnetic theoretic models in order

　　to accelerate the rapid development of the technologies of multilayered optical

　　memories. The operation of multilayered optical memory with bits stored as

　　refractive index change is based on scattering by bits because of the refractive

　　index being different from that of their surrounding medium. An ultrashort pulse

　　laser may be used to produce bits in various transparent materials that might

　　not be photosensitive mediums. Therefore, unlike for a photopolymer gel, there

　　are no problems of distortion due to shrinkage and flow or of isomerization due

　　to ultraviolet light. And, unlike for photorefractive materials, the

　　difficulties of fixing the recorded data are entirely avoided. Compared with a

　　holo- graphic optical storage, two-photon multilayered memories, and fluorescent

　　multilayered memories, multilayered optical memory with bits stored as

　　refractive index change has more predominance in the next generation of high

　　density optical memories. In terms of different methods for coupling the

　　incident light, multilayered optical memory with bits stored as a refractive

　　index change may be decomposed into the multilayered optical memory based on the

　　usual readout method [i.e., the reading light is focused by the object lens on

　　the addressing layer in the direction perpendicular to the surface of the

　　optical disc so as to read data, such as in DVD, which is hereafter called the

　　conventional multilayered optical memory (CMOM) for brevity] and the waveguide

　　multilayered optical memory (WMOM) (i.e., the reading light is coupled by the

　　object lens into the core of the addressing waveguide from the side face of the

　　optical disc and excites a series of guided modes, of which partial powers will

　　be scattered from the core by bits when they meet bits so as to read data). The

　　purposes of this dissertation are to create rigorous vector diffraction models

　　of the CMOM and the WMOM according to their operations so as to describe

　　accurately the distribution of image fields of bits and the effects of the

　　structural parameters of the optical disc on the cross talk and the modulation

　　contrast, give a theoretical guidance on their experimental researches, and

　　promote the development of the technologies of multilayered optical

　　memories.

　　For this purpose, in terms of its operations, the multilayered optical

　　memory with bits stored as a refractive index change is first decomposed into

　　the reading system, the multilayered disc, and the detection system in this

　　dissertation. Then, we bring forward a method that utilizes the vector imaging

　　theory to calculate the image fields of the read-in and the detection system,

　　and the Lippman-Schwinger equation and the dyadic Green’s function (DGF)

　　associated with the planar multilayered media to calculate the scattering fields

　　of bits after they interact with the read-in light. With the above methods, a

　　full and rigorous electromagnetic theory of the multilayered optical memory with

　　bits stored as a refractive index change will can be constructed. Under the

　　guidance of this idea, an effectively numerical method for the DGF for a planar

　　layered media is constructed after the DGF is derived in detail in Chapter 2. In

　　order to deal with the difficult problem of computation because of the

　　singularity and highly oscillating behavior of the integrand in the DGF, after

　　we analyzed carefully the known numerical methods for the DGF, we find the poles

　　of the DGF for a planar layered media utilizing the transformation theorem of

　　matrix and the graphic method and then deduce the recursive formula used to

　　compute the residue of a function at these poles. Finally, an effectively

　　numerical method for the DGF for a planar layered media is constructed

　　successfully by the deform path at complex plane and the sine or tangent

　　transformation of variables. This numerical method can be applied to the

　　calculation of the far fields because of the considerations of the contribution

　　of the residues at these poles, which is necessary for the calculation of the

　　model of the multilayered optical memory.

　　The sizes of bits are close to or shorter than the wavelength in the high

　　density optical storage, moreover the numerical aperture of the optical head

　　might be beyond of 0.9 (e.g., the blue laser optical storage), so the

　　computation for the image fields of the read-in and the detection system in the

　　multilayered memory must be based on the vector imaging theories. However, up to

　　now, the vector imaging theories are not constructed successfully at home and

　　abroad, and there are many limits and scarcities in applications so as to miss

　　the theoretical requirements of the optical memory. Hence, a vector field theory

　　for an aplanatic system is created systematically in Chapter 3. In this part, ①

　　by using the method for modal expansions of the independent transverse fields,

　　we derived a formula of vector plane wave spectrum (VPWS) of an arbitrary

　　polarized electromagnetic wave in a homogenous medium, where this formula is

　　composed of TM- and TE-mode plane wave spectrum and the amplitude and unit

　　polarized direction of every plane wave are separable. Compared with the formula

　　of VPWS in Ref. (Opt. Commun., 1998, 152: 108-118), where the amplitude and the

　　unit polarized direction of every plane wave are not separable, our formula of

　　VPWS has more obviously physical meaning and is more convenient to apply in some

　　cases. Utilizing the formula of VPWS derived in this dissertation, we prove

　　successfully that when an azimuthally polarized beam is coupled into a perfect

　　planar waveguide by a lens, only TE-mode plane waves are excited in the

　　waveguide, and when a radially polarized beam is coupled into a perfect planar

　　waveguide by a lens, only TM-mode plane waves are excited in the waveguide. ②

　　With the VPWS and stationary phase method, a vector field theory for an

　　aplanatic system when the polarized point source is at an arbitrary location on

　　the optical axis is presented. And, the computer simulation is used to discuss

　　in detail the effects of various angular semiapertures on the object and image

　　sides on the structures of the image fields and the resolution, which identifies

　　the shortcomings of the classical Wolf theory (Proc. R. Soc. London, Ser. A,

　　1959, 253, 358-370) and indicates that this Wolf theory can only be applied to

　　the study of imaging properties of an aplanatic system when the point source is

　　located at infinity in the direction of the axis and can not be directly applied

　　to the case of high-numerical-aperture on the object side. ③ The vector coherent

　　transfer function (CTF) of an aplanatic system, similar to the CTF of the scalar

　　diffraction theory, is first derived, and on the basis of the vector diffraction

　　theory, a simple formalism relating image fields to object fields is developed

　　for an aplanatic system, which shows that the VPWS of image fields is equal to

　　the product of the vector CTF due to the x- and y-polarized point electric field

　　source and the scalar spectrum of the corresponding transverse object fields. ④

　　Utilizing a DGF, a rigorous imaging theory of an aplanatic system with a

　　stratified medium on the object or image sides is first developed. In addition,

　　the inaccurate concept “the angles that the incident ray in the object space and

　　the corresponding emergent ray in the image space make with the positive z axis

　　vary with the variation of the position of the scatter” in Ref. [J. Mod. Optics,

　　1998, 45(8): 1681-1698] is identified and the correct application method is also

　　indicated. These fruits make the vector field theory for an aplanatic system

　　perfect and resolve the problem of the rigorous vector diffraction theory of the

　　imaging system in the multilayered optical memory with bits stored as refractive

　　index.

　　On the basis of the differences between the CMOM and the WMOM and our

　　fruits in the aspect of numerical computation of the DGF for a planar layered

　　media and the vector field theory for an optical system, the vector diffraction

　　model of the multi- layered optical memory with bits stored as refractive index

　　is first constructed system- atically in Chapter 4 and 5, including those of the

　　CMOM and the WMOM. These models may be applied to calculate the read-out of

　　dynamic signals, the power of the readout signals, the distribution of fields at

　　the detection plane, the variations of the modulation contrast and the cross

　　talk with the sizes of bits, the effects of the distances between the two tracks

　　and between the two layers on the cross talk, and the effects between the two

　　bits. In this dissertation, a program of the multilayered optical memory with

　　bits stored as refractive index change is presented. With this program, the

　　effects of the polarization and the wavelength of a light source and the

　　numerical aperture of the detection system on the power of the readout signals

　　and the distribution of its image field are analyzed in detail. The distribution

　　of binary codes along the information track is calculated. The variations of the

　　modulation contrast and the cross talk with respect to the length, width and

　　depth of bits and the effects of the polarization of a light source and the

　　thickness of the data layer on the modulation contrast are analyzed. Through the

　　simulations, a series of important conclusions are drawn. For example, for a

　　WMOM, the optimum illumination is the azimuthally polarized incident light

　　because when an azimuthally polarized beam is coupled into a perfect planar

　　waveguide by a lens, only TE-mode plane waves are excited in the waveguide,

　　whereas compared with the TM mode, a higher power of the readout signals, a

　　smaller cross talk, and a higher modulation contrast can be obtained when the TE

　　mode is used. Another example is that for a CMOM, the higher the power of the

　　reflected light resulting from the interfaces in the storage medium is, the

　　higher the modulation contrast is. Therefore, the thickness of the data layer

　　should be designed as integer times the semiwavelength—or better, the

　　wavelength—in the data layer. At this time, the modulation contrast can be

　　greater than 0.9 and even close to 1.

　　The achievements in this dissertation will not only promote the development

　　of the multilayered optical memory with bits stored as refractive index, but

　　also make the vector field theory for an optical system developmental and

　　perfect.

　　Key words: multilayered optical memory, waveguide multilayered optical

　　memory, conventional multilayered optical memory, vector imaging theory, dyadic

　　Green’s function, Lippman-Schwinger equation

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