1. Introduction

Photonic crystals (PCs) have attracted extensive attention in the past years [1–5]. Due to multiple scattering of the electromagnetic (EM) waves suffered from periodic dielectric structures, PCs can generate series of photonic band gaps (PBGs), in which the propagation of EM waves is inhibited. However, when structural defects are introduced into an ideal PC, the defect states may be produced within the PBGs [4–7] and the EM waves aligned to these defect states can be allowed through the PC. By inserting photonic quantum—wells (PQWs) into an ideal PC, a series of the discrete defect states may be created and they provide the function of multiple channeled filtering. Qiao et al. [8] have reported that the number of the confined states is equal to the number of the PQWs and it can be tuned via altering the number of the PQWs. However, the frequencies of the defect states cannot be changed with freedom. In general, these frequencies are governed by the specific structural configuration of the PQWs. For example, in the short-periodic PQWs, defect states will always be equally spaced, the interval is decided by the thickness and dielectric constant of the PQWs. However, in practice, the favorable design of optical multiple channeled filters need to pass arbitrarily preassigned frequencies. Therefore, an important issue is naturally raised i.e. how can we design the specific PQWs having the preassigned filtering channels? This results in solving of a difficult inverse source problem in optics.

Recently, Huang et al. report how to use stimulated annealing (SA) algorithm to conduct optimum design of optical thin-film filters [9]. They have successfully improved the designs of highpass, lowpass, and DWDM filters with a finite band width. Motivated by these works, we hereby discuss the design of the aperiodic PQW structures sandwiched by two finite-length prototype photonic crystals to achieve discrete multiple channeled filtering functions at arbitrary preassigned frequencies with the use of the SA algorithm, the line width of each filtering channel is quite narrower than that obtained in Ref. 9. We now present a more efficient merit function in the SA algorithm to conduct the designs.

 

Fig. 1. Schematic diagram of the APQW structure. The APQWs are sandwiched by two finite-length prototype PCs.

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2. Basic theory for design

The model structure is sketched in Fig. 1. The aperiodic PQWs (APQWs) are sandwiched by two finite-length ideal PCs (the so-called prototype PCs), which consist of two alternately stacked layers A and B with different dielectric constants of e_A and e_B , respectively. Their thicknesses are denoted by d_A and d_B , respectively, and a=d_A +d_B is the lattice constant of the one-dimensional (1D) PC. The APQWs are composed of two different alternately stacked basic constituent layers with the dielectric constants of ε_C and ε_D . However, the thickness of each individual layer may not be equal and the individual layer thickness is determined by the merits of the desirable filters.

We calculate the transmission spectrum of designed APQW structures by using the transfer-matrix method [8,9]. The transfer-matrix in each individual layer can be obtained by solving Maxwell equations with a combination of boundary conditions. For a normally incident EM plane wave with the TE polarization, the transfer-matrix for the j-th layer is given by

where j ∈{1,2, …,N} for N number of layers. GĜ _j is the transfer matrix. The propagating matrix P̂ _j reads

for air at the most left-hand side of the sample and

for the jth layer in sample. Ĝ _j reads

where d_j denotes the thickness of the j-th layer, $k_j=2\pi \sqrt{{\epsilon }_j}⁄\lambda$, ε_j is the dielectric constant of the j-th layer; λ the wavelength of the incident light wave in vacuum. Thus, the total transfer matrix can be obtained by multiplying all individual transfer matrixes in sequence. The transmission and reflection coefficients of EM waves of the sample can be calculated from

for instance, the transmission probability is given by

For the wave which is not normal incident or TM mode, the similar approach can be used to obtain the transmission probability.

Designing the APQWs to produce specified defect states located at the preassigned frequencies ${\omega }_{\alpha }^{\left(0\right)}$ within a given range of [ω_a ω_b ], we need initially to choose a perfect PC, served as the prototype PC, into which the APQWs are implanted. It requires that the chosen prototype PC should have an appropriate PBG located at this frequency range and with a certain width, not narrower than the range of [ω_a ω_b ]. After determining this prototype PC, we can then proceed with the design of the APQWs sandwiched by two finite-length prototype PCs by using the SA algorithm. It is desirable to define a merit function. Here we present a non-traditional merit function, much appropriate to our designs, complete different from that used in Huang et al. [8]. That is

with

where ${\omega }_{\alpha }^{\mathit{\left(}s\mathit{\right)}}$ denotes the frequencies of the defect states appearing in [ω _α-1,ω _α], which are generated in every routed configuration of the APQWs during the SA routine. ω_α denote a series of the frequencies to partition the whole region of [ω_a,ω_b ] into several subregions. In each subregion, only one of the preassigned confined states appears. For instance, ${\omega }_{\alpha }^{\left(\mathrm{o}\right)}$ is located in the subregion of [ω _α-1,ω_α ]. In the SA routine, more than one defect states may occur in one subregion, thus, s may be larger than 1, therefore, the sum over s should be take into account. The optimal design of the APQWs corresponds to a search for the minimum of O. The sandwiched part in the sample is divided into n unit blocks with the thickness δd; the dielectric constant of each individual block is chosen as one of the binary of ε_C and ε_D , decided by the SA algorithm. At the beginning of the SA routine, the initial temperature T_e and dropping rate ΔT_e are selected, a random constructed APQWs is used to compute the initial merit function O_o . Then a new merit function O ₁ is calculated with changing the dielectric constant of a unit block and the difference ΔO=O_o -O ₁ is obtained. A random number p, 0≤p≤1, is generated by computer. If p satisfies p=exp(-ΔO/T_e ), this change will be accepted and O_o will be replaced by O ₁, else the dielectric constant of this unit block will be retired and O_o does not change. After test all of blocks, this procedure will be repeated with a new temperature T_e =T_e ×ΔT_e until no dielectric constant changes in the APQWs. Thus, the favorable arrangement of the APQWs can be completely determined.

 

Fig. 2. Transmission spectrum of the designed APQW sample for two channeled filtering at the preset frequencies of ${\omega }_1^{\left(o\right)}$=0.420 and ${\omega }_2^{\left(o\right)}$=0.480(2πc/a).

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3. Simulation results

In our first design example, we design the APQWs for achieving two channeled filtering in a given range of [0.352 0.506](2πc/a), c is the speed of light in vacuum. The dielectric constants are selected as ε_A =ε_C =13.0 and ε_B =ε_D =1.0. The thicknesses of the constituent layers A and B in the prototype PC are set to be d_A =d_B =0.5a, thus, the second PBG of the prototype PC just is located at [0.352 0.506](2πc/a). In the following calculations, we employ five AB layers, (AB)₅, on either side of the APQWs. The sandwiched part is divided into n=100 blocks and the thickness of the basic block takes δd=0.02a. Two filtering frequencies are preassigned as ${\omega }_1^{\left(\mathrm{o}\right)}$=0.420(2πc/a) and ${\omega }_2^{\left(\mathrm{o}\right)}$=0.480(2πc/a). The transmission spectrum of the designed sample is displayed in Fig. 2. The frequency increment in the scan is taken δω=1.0×10^-5(2πc/a) to ensure that any unwanted extra stray frequency peak does not occur. Two dashed vertical lines remark the positions of the second PBG of the prototype (AB)₅ PC. It is evident that there exist only two expected defect states in the desired frequency range. The frequencies of the defect states accord exactly with the preset values. Their transmittances are 0.90 and 0.86 for ${\omega }_1^{\left(\mathrm{o}\right)}$=0.420(2πc/a) and ${\omega }_2^{\left(\mathrm{o}\right)}$=0.480(2πc/a), respectively. In our second design example, criteria paribus, we have the two filtering frequencies given as ${\omega }_1^{\left(o\right)}$=0.380(2πc/a) and ${\omega }_2^{\left(o\right)}$=0.410(2πc/a). The transmission spectrum obtained is depicted in Fig. (3). It is apparent that only two desired defect states occur in the range of the [0.352 0.506](2πc/a), their transmittances are 0.96 and 0.99 for ${\omega }_1^{\left(o\right)}$=0.380(2πc/a) and ${\omega }_2^{\left(o\right)}$=0.410(2πc/a), respectively. These results illustrate that the two design examples can achieve the desired specification.

 

Fig. 3. Transmission spectrum of the designed APQW sample for two channeled filtering at the preset frequencies of ${\omega }_1^{\left(o\right)}$=0.380 and ${\omega }_2^{\left(o\right)}$=0.410(2πc/a)

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Next, we show a four channeled filtering design example. In this case, the thickness of the unit block sets δd=0.02a and the number of blocks n is chosen as 200. The four preassigned frequencies are ${\omega }_{\mathrm{\alpha }}^{\left(o\right)}$={0.370,0.410,0.420,0.450} (2πc/a). Fig. 4(a) presents the gray-scaled diagram of the constructed APQWs, the black (or white) strips correspond to the layers with dielectric constant of 13.0 (or 1.0). It is evident that the structure of the obtained PQW exhibits fully aperiodic. Its transmission spectrum is shown in Fig. 4(b). It is found that the frequencies of the defect states are located at {0.370,0.409,0.420,0.450}(2πc/a), and they matches well with the target values. Only one frequency is not exactly the same, this is because that the number of blocks in APQWs is limited. The corresponding transmittances are 0.97, 0.98, 0.99, and 1.00, respectively. Because only five AB layers are employed on either side of APQWs, the transmittances for three design examples are not equal to 1.

 

Fig. 4. Calculated results of the designed APQWs for four channeled filtering: (a) gray scaled diagram of the designed APQWs; (b) transmission spectrum.

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We also employ the conventional merit function in the SA algorithm [9], the obtained results are quite poor, much more stray peaks appear beside the desired filtering frequencies. It should be pointed out that the frequency line width of each filtering channel is narrower than 1.2×10^-3(2πc/a) with high isolation.

4. Conclusion

In summary, we have employed the SA algorithm with the new merit function to design the APQWs for achieving multiple channeled filtering at arbitrary preassigned frequencies. In our design example simulations, we demonstrate the obtained APQWs can achieve the desired specification. The frequency line width of each filtering channel is quite narrow due to the confined state effect. This width can be further reduced by increasing the number of prototype PC’s layer.