Advances in Condensed Matter Physics

Volume 2015 (2015), Article ID 959546, 5 pages

http://dx.doi.org/10.1155/2015/959546

## The Quantum Well of One-Dimensional Photonic Crystals

^{1}Institute of Physics, Jilin Normal University, Siping 136000, China^{2}Institute of Physics, Jilin University, Changchun 130012, China^{3}School of Physics and Electronic Engineering, Anqing Normal University, Anqing 246133, China

Received 6 January 2015; Accepted 8 April 2015

Academic Editor: Ashok Chatterjee

Copyright © 2015 Xiao-Jing Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We have studied the transmissivity of one-dimensional photonic crystals quantum well (QW) with quantum theory approach. By calculation, we find that there are photon bound states in the QW structure , and the numbers of the bound states are equal to . We have found that there are some new features in the QW, which can be used to design optic amplifier, attenuator, and optic filter of multiple channel.

#### 1. Introduction

Photonic crystals (PCs) are artificial structures with a periodic dielectric constant in one, two, or three dimensions [1, 2]. They are characterized by photonic band structures owing to the multiple Bragg scatterings [3, 4]. Between photonic bands there may exist a photonic band gap (PBG), in which the propagation of electromagnetic waves or photons is strongly inhibited [5]. This facilitates the manipulation and control of the flow of electromagnetic waves or photons as well as the design of high-performance optoelectric devices [6, 7].

The concept of super lattice and quantum well (QW) stemmed from the pioneering work of Shimuzu and Ishihara [8]. It is well known that there are many interesting and new phenomena for electrons in semiconductor QW structures [9]. The QW structures and super lattices can be used to tailor the electronic band structures of semiconductors [9–13]. Similar to the idea of semiconductor QW structures, one can use different PCs to construct photonic QW structures, provided that the PBG of the constituent PCs are aligned properly. The constituents can be one-dimensional (1D), two-dimensional (2D), or three-dimensional (3D) PCs. It has been shown by the authors [14] that the transmission properties of the 1D and 2D PCs can be tailored by using QW structures. The nontransmission frequency range can be enlarged as desired by using QW. The use of QW exciton embedded in high-finesse semiconductor microcavities of the Fabry-Perot type has allowed observing a modification of spontaneous emission (weak coupling regime) [15–21] as well as the occurrence of a vacuum Rabi splitting (strong coupling regime) [22–25]. The latter effect arises when the radiation-matter coupling energy overcomes the damping rates of QW exciton and microcavities photons.

In [26, 27], we have studied the quantum transmission characteristic of 1D PCs with quantum theory approach and given the quantum transform matrix, quantum transmissivity, and reflectivity. In this paper, we use the quantum method to research the QW transmissivity of 1D PCs. It is found that there are some new features in the QW structure , which can be used to design optic amplifier, attenuator, and optic filter of multiple channel.

#### 2. Quantum Transform Matrix and Transmissivity of QW

The QW structures consisted of two different 1D PCs. The first and second 1D PCs structures are and , respectively, where are the numbers of the second PCs layers. The two 1D PCs can consist of 1D PCs of QW structure, which is .

In quantum theory approach [26, 27], we consider the photon travels along with the -axis, and the QW structure and quantum wave functions distribution are shown in Figure 1. The thicknesses and refractive indexes of layers and are , , , and , respectively. The and are the photon wave functions of the first period media and . The photon wave functions of incident, reflection, and transmission are [26, 27] where , , and are the wave vector of photon in vacuum, mediums and . The constants , , and are the wave function amplitudes of incident, reflection, and transmission wave. By calculation, similarly as [26, 27], we can directly give the wave functions of photon in arbitrary layers and . For medium () of the th (th) layer, the photon wave function can be written as before the th layer medium , there are layers medium and layers medium , and before the th layer medium , there are layers medium and layers medium . The constants , , , and are the wave function amplitudes. By the condition of wave function and its derivative continuation at the interface of two mediums, we can obtain the quantum transfer matrix of th medium layer; it iswhere () is the wave vector of photon in the th (th) layer medium and is the thickness of th layer medium. For the QW structure , its total quantum transfer matrix is and the quantum transmissivity is [26, 27]