The Quantum Well of One-Dimensional Photonic Crystals

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 (BA)6(BBABB)n(AB)6, and the numbers of the bound states are equal to n + 1. 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.


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][10][11][12][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][16][17][18][19][20][21] as well as the occurrence of a vacuum Rabi splitting (strong coupling regime) [22][23][24][25].The latter effect arises when the radiationmatter 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 () 6 ()  () 6 , which can be used to design optic amplifier, attenuator, and optic filter of multiple channel.

Quantum Transform Matrix and Transmissivity of QW
The QW structures consisted of two different 1D PCs.The first and second 1D PCs structures are () 6 () 6 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 () 6 ()  () 6 .
The QW structure of 1D PCs.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  1  and  1  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 (  + 1)th ((  + 1)th) layer, the photon wave function can be written as (2) before the (  + 1)th layer medium , there are   layers medium  and   layers medium , and before the (  + 1)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 is where   ( −1 ) is the wave vector of photon in the th (( − 1)th) layer medium and   is the thickness of th layer medium.For the QW structure () 6 ()  () 6 , its total quantum transfer matrix is and the quantum transmissivity  is [26,27]

Numerical Result
In this section, we report our numerical results of the QW quantum transmissivity.The refractive indexes and the thicknesses of medium  and medium  are as follows:   = 1.35,   = 2.45 and  = 890 nm,  = 469 nm.The quantum transmissivity of () 12 and () 6 () 6 is shown in Figures 2(a) and 2(b).In Figures 3 and 4, we only change the refractive index of medium  in relation to Figure 2; they are   = 2.45 − 0.0001 (active medium) and   = 2.45 + 0.0001 (absorbing medium).From the 1D PCs () 6 () 6 ; that is, the quantum transmission peaks are gained and attenuated when the medium  is active medium (  = 2.45 − 0.0001) and absorbing medium (  = 2.45 + 0.0001).
(2) The forbidden band of 1D PCs () 6 () 6 is in the range of / 0 = 0.8 ∼ 1.2, and the conduction band of 1D PCs () 12 is inside the forbidden band.So, the PCs () 6 () 6 play a role similar to a barrier to PCs () 12 , and the PCs () 12  act as a well in the forbidden band.We put the 1D PCs ()  and () 6 () 6 together to constitute the 1D PCs of QW structure () 6 ()  () 6 , which are shown in Figure 1.In Figures 5, 6, and 7, we should study the quantum transmissivity of the QW structure () 6 ()  () 6 .As mentioned above, the conduction band of PCs ()  is inside the forbidden band of PCs () 6 () 6 ; that is, the PCs () 6 () 6 prohibit the propagation of photon in its forbidden band; then the photon will be confined in the PCs ()  .Because of the quantum effect of photon in QW of 1D PCs, the photon should form the bound state in the QW, which is analogous to the bound state of electron in semiconductor QW.The photon can pass the QW by the resonance perforation way and form the very sharp peaks of quantum transmissivity within the forbidden band of the PCs () 6 ()     6 ()  () 6 .In Figures 5, 6, and 7 refractive indexes of medium  are real number,   = 2.45 (convention medium), and complex numbers   = 2.45−0.0001(active medium) and   = 2.45+ 0.0001 (absorbing medium), respectively.From Figures 5 to  7, we can obtain some results.(1) The numbers of the sharp peaks (bound states) are equal to  + 1; that is, when  = 1,  = 2, and  = 3, the numbers of the sharp peaks are 2, 3, and 4. (2) In Figure 5, the quantum transmissivity of the sharp peaks  = 1 for  = 1,  = 2, and  = 3, which can be designed optic filter of multiple channel.(3) In Figure 6, the quantum transmissivity of the sharp peaks  > 1 for  = 1,  = 2, and  = 3.When  increase, the sharp peaks value  increase, which can be used to design optic amplifier and optic filter of multiple channel.(4) In Figure 7, the quantum transmissivity of the sharp peaks  < 1 for  = 1,  = 2, and  = 3; when  increase, the sharp peaks value  decrease, which can be used to design optic attenuator.

Conclusion
In summary, we have studied the quantum transmissivity of the QW of 1D PCs with quantum theory approach.By calculation, we find that there are photon bound states in QW structure () 6 ()  () 6 , and the numbers of the bound states are equal to  + 1, which are formed by the quantum effect of photon in QW.We also find that the QW () 6 ()  () 6 can be used to design optic amplifier, attenuator, and optic filter of multiple channel.
Chinese Ministry of Education (no.213009A), and Scientific and Technological Development Foundation of Jilin Province (no.20130101031JC).