非局部边界条件下的光学微腔散射矩阵构建

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本文探讨了光学微腔的散射矩阵构建，作为非局部边界值问题的研究。作者提出了一种数值方法，旨在处理光学微腔中的特殊情形，包括具有帕累托-时间（Parity-Time, PT）和其他非Hermitian对称性的情况。这种方法的核心在于引入了一个非局部边界条件，将入射光视为非均匀项，从而解决了R-matrix方法中常见的正常导数不连续的问题。在传统的R-matrix方法中，计算散射矩阵时可能会遇到边界条件处理上的困难，导致物理结果的精度受限。通过采用非局部边界条件，这种方法能够更准确地模拟微腔边缘的光行为，使得计算出的S矩阵更为精确。这种改进不仅适用于一般光学微腔，而且对于那些具有非Hermitian性质的系统，如PT-symmetric结构，有着显著的优势。作者进一步指出，通过排除非均匀项，非Hermitian哈密顿量在他们的方法中不仅决定了系统的行为，还提供了一种替代途径来导出标准的R-matrix结果。这表明，他们的方法不仅提升了理论分析的精度，还在基础理论上有所贡献，可能推动了对非平衡量子光学、非线性光学等领域更深入的理解。此外，文中可能包含了具体数值实现的细节，例如有限元分析、数值求解算法或者矩阵运算技巧，这些都是为了确保计算效率和结果的可靠性。研究结果可能被应用于设计新型光子晶体、超材料或量子光学器件，其中微腔的散射特性至关重要。本文的工作为理解和设计具有非Hermitian性质的光学微腔提供了重要的数值工具，提升了散射矩阵的计算质量，并揭示了非Hermitian系统中新的物理现象和计算策略。这在当前对非平衡物理现象探索和量子技术发展中具有显著的意义。

Constructing the scattering matrix for optical

microcavities as a nonlocal boundary value

problem

LI GE

1,2

Department of Engineering Science and Physics, College of Staten Island, CUNY, Staten Island, New York 10314, USA

The Graduate Center, CUNY, New York, New York 10016, USA (li.ge@csi.cuny.edu)

Received 18 August 2017; revised 17 September 2017; accepted 17 September 2017; posted 18 September 2017 (Doc. ID 304065);

published 20 October 2017

We develop a numerical scheme to construct the scattering (S ) matrix for optical microcavities, including the

special cases with parity-time and other non-Hermitian symmetries. This scheme incorporates the explicit form of

a nonlocal boundary condition, with the incident light represented by an inhomogeneous term. This approach

resolves the artifact of a discontinuous normal derivative typically found in the R-matrix method. In addition, we

show that, by excluding the aforementioned inhomogeneous term, the non-Hermitian Hamiltonian in our

approach also determines the Periels–Kapur states, and it constitutes an alternative approach to derive the stan-

dard R-matrix result in this basis. Therefore, our scheme provides a convenient framework to explore the benefits

of both approaches. We illustrate this boundary value problem using 1D and 2D scalar Helmholtz equations.

The eigenvalue s and poles of the S matrix calculated using our approach show good agreement with results

obtained by other means.

OCIS codes: (140.3945) Microcavities; (290.5825) Scattering theory; (080.6755) Systems with special symmetry.

https://doi.org/10.1364/PRJ.5.000B20

1. INTRODUCTION

Driven by advances in nanofabrication capabilities and their

applications to integrated optics, understanding resonances

and wave transport in optical microcavities [1,2] has been

one of the most energized subjects in modern optics. These

compact optical structures also offer a unique opportunity to

study non-Hermitian phenomena [3] and wave chaos [4,5]

in a well-controlled manner. To probe these properties of

optical microcavities, one approach resorts to the scattering (S)

matrix formalism [6], which was an essential tool in the under-

standing of resonances in nuclear physics [7,8], particle physics

[9], and quantum field theory [10], which also played a crucial

role in the study of wave transport in various fields, including

condensed matter systems [11], optics [12], and microwave

networks [13].

In essence, the S matrix, denoted by an energy- or

frequency-dependent Sω, connects a set of incoming chan-

nels Ψ

−

to their corresponding outgoing channels Ψ



, both

defined outside the scattering potential. Therefore, it takes

the openness of the system into account, and the conservation

of optical flux in the absence of gain and loss is manifested by

the unitarity of Sω [i.e., SωS

†

ω1]. When Sω is

analytically continued into the complex-ω plane, its poles

(i.e., where its eigenvalues approach infinity) correspond to

the resonances of the system, whose wave functions only con-

nect to the outgoing channels, now also evaluated at complex

frequencies [14].

As already pointed out by Wigner and Eisenbud’s early work

in nuclear physics [8], the calculation of the S matrix can be

understood as a nonlocal boundary value problem (BVP),

which was derived using an orthogonal basis of the system

and explicitly contains the real-valued frequencies of this basis.

More specifically, this orthogonal basis was defined with van-

ishing normal derivatives at the boundary of the system, and, as

a result, the expansion of an arbitrary state Ψ using a finite

number of these basis functions has a discontinuous normal

derivative in general as an artifact [8,15]. Alternatively, the

expansion can be carried out using quasinormal modes [16]

(i.e., the Gamow states [14]) or the Periels–Kapur states

[17,18], both defined with purely outgoing boundary condi-

tions. These approaches, however, do not remove the artifact

in the normal derivative, due to the lack of incoming flux that

is inherent in the scattering process. We note that in literature

the modal expansion approach, regardless of the specific basis,

is referred to as the R-matrix method [15] in general.

Although the consequence of the aforementioned artifact

may be insignificant with the introduction of the Bloch oper-

ator [19] and a large number of basis functions [20], having an

B20

Vol. 5, No. 6 / December 2017 / Photonics Research

Research Article

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