1 Introduction

Optical devices based on photonic crystals (PCs) have inspired great deal of interest due to their photonic band gaps (PBGs) in which propagation of electromagnetic waves whose frequencies lie within the PBG region is completely prohibited, similar to electronic band gap in semiconductors. PCs have fascinating ability to control and manipulate light wave propagation and thus find their novel scientific, technical and technological applications in photonics and optoelectronics [1–5]. Nowadays, a lot of attention has been paid to design tunable optical filters consisting of PCs due to their vast potential for many applications in modern optical communication [6]. The most significant feature of such tunable filters is the control and monitoring of their transmission properties externally [7–13]. For instance, Kong et al. [13] have focused on how the transmission properties of the defect mode for the transverse electric (TE) wave in 1D PC doped with magnetized plasma are influenced by the external magnetic field, etc. Suthar et al. [10] have investigated the transmission properties of the 1D photonic quantum well (PQW) structure with single defect by transfer matrix method (TMM) and pointed out that defect mode inside the PBG of PQW structure can be utilized as a high-quality single channel filter only by changing the temperature of the structure externally.

All the aforementioned studies have focused on the defect mode properties of the 1D PCs. Though the temperature-dependent tuning of band edge resonant modes in 1D PCs has been studied by Yeong et al. [14] for designing optical thermal sensing devices, none of the studies have investigated the temperature-dependent tuning of PBG for wavelength-selective switching (WSS) applications without using defect in conventional 1D PCs. In the present communication, we have proposed thermally tunable optical filter system (TTOFS) for WSS application. The proposed filter is composed of two 1D silicon–air PCs (PC1 and PC2) and two highly reflecting mirrors M1 and M2. The transmission characteristic of the structure is investigated through TMM [15]. The operating temperature of both the PCs is same and has a marked influence on the transmission spectra as well as on the range of the PBGs of both PCs. The proposed filter may also be used to enhance the capacity and flexibility of wavelength-division multiplexing (WDM) networks by overcoming the fundamental limitations of semiconductor optical amplifier (SOA)-based WSS systems, such as power consumption, signal-to-noise ratio, speed, polarization and size. [16].

The paper is organized as follows. The theoretical formulation is introduced in Sect. 2. Numerical results are presented and discussed in Sect. 3. Finally, conclusions are given in Sect. 4.

2 Theoretical formulation

The structural configuration of the proposed TTOFS for WSS is depicted in Fig. 1. Both PCs are put close to each other in such a way that radiation after coming out of PC1 is allowed to reach normally on PC2 via two highly reflecting parallel mirrors M1 and M2 through air. The two mirrors are inclined at an angle 45° with respect to the direction of periodicity (z axis). Both PCs are denoted by [(AB)N] where A and B represent layers of high and low index materials of PC1 and PC2 with equal number of periods N. The thicknesses of layers A and B of PC1 are denoted by d 1 and d 2 while those of PC2 by d 3 and d 4, respectively. The entire TTOF is assumed to be surrounded by air with refractive index 1.

Fig. 1
figure 1

A schematic drawing of TTOF system consisting of two 1D PCs (PC1 and PC2) along with two highly reflecting mirrors M1 and M2

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Let a plane wave be incident normally from left onto the 1D PC1 along z axis as shown in Fig. 1. The characteristic matrix which connects electric and magnetic fields in the two interfaces of ith layer can be obtained via TMM as

$$ M_{i} = \left( {\begin{array}{*{20}l} {\cos \;\gamma_{i} } \hfill & { - \frac{i}{{p_{i} }}\sin \;\gamma_{i} } \hfill \\ { - ip_{i} \sin \;\gamma_{i} } \hfill & {\cos \;\gamma_{i} } \hfill \\ \end{array} } \right), $$
(1)

where \( \gamma_{i} = \frac{2\pi }{{\lambda_{0} }}n_{i} d_{i} , \) n i d i represents the optical thickness of the ith layer. \( p_{i} = n_{i} , \) n i is the refractive index of ith layer of thickness d i , and λ 0 is the free space wavelength.

The characteristic matrix of the entire structure that connects the electric and magnetic fields of first and last layer can be shown to be

$$ \left( {\begin{array}{*{20}c} {E_{1} } \\ {H_{1} } \\ \end{array} } \right) = M_{1} M_{2} \cdots M_{l} \cdots M_{N} \left( {\begin{array}{*{20}c} {E_{N + 1} } \\ {H_{N + 1} } \\ \end{array} } \right), $$
(2)
$$ \begin{aligned} & = M_{A} M_{B} M_{A} \cdots M_{A} M_{B} \left( {\begin{array}{*{20}c} {E_{n + 1} } \\ {H_{n + 1} } \\ \end{array} } \right) \\ & = M\left( {\begin{array}{*{20}c} {E_{N + 1} } \\ {H_{N + 1} } \\ \end{array} } \right) \\ & = \left( {\begin{array}{*{20}c} {\chi_{11} } & {\chi_{12} } \\ {\chi_{21} } & {\chi_{22} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {E_{N + 1} } \\ {H_{N + 1} } \\ \end{array} } \right). \\ \end{aligned} $$
(3)

where χ 11, χ 12, χ 21 and χ 22 are the matrix elements of transfer matrix representing the whole structure. The transmission coefficient of the electromagnetic wave passing through the structure with N identical periods is

$$ t = \frac{{2\eta_{0} }}{{\chi_{11} \eta_{0} + \chi_{12} \eta_{0} \eta_{N + 1} + \chi_{21} + \chi_{22} \eta_{0} }} $$
(4)

where η 0 and η N+1 represent the coefficients of medium on either side of the structure. Since entire structure is kept in air, η 0 = n 0 and η N+1 = n 0. Here n 0 is the refractive index of air.

Then, transmittance T of 1D PC

$$ T = \left( t \right)^{2} $$
(5)

The wavelength- and temperature-dependent refractive index of silicon (Si) in the ranges 1.2–14 μm and 293–1600 K, respectively, is defined as [17]

$$ n(\lambda ,T) = \left\{ {\varepsilon (T) + \frac{L(T)}{{\lambda^{2} }}(A_{0} + A_{1} T + A_{2} (T))} \right\}^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} $$
(6)

where \( \varepsilon (T) = 11.4445 + 2.7739 \times 10^{ - 4} T + 1.7050 \times 10^{ - 6} T^{2} - 8.1347 \times 10^{ - 10} T^{3} ,L(T) = e^{{{{ - 3\Delta L(T)} \mathord{\left/ {\vphantom {{ - 3\Delta L(T)} {L_{293} }}} \right. \kern-0pt} {L_{293} }}}} \), λ = wavelength in units of µm, T = temperature in units of K, \( A_{0} = 0.8948,A_{1} = 4.3977 \times 10^{ - 4} ,A_{2} = 7.3835 \times 10^{ - 8} , \) and \( \frac{\Delta L(T)}{{L_{293} }} = - 0.071 + 1.887 \times 10^{ - 6} T + 1.934 \times 10^{ - 9} T^{2} - 4.544 \times 10^{ - 13} T^{3} . \) The temperature-dependent thickness of Si layer is given by

$$ d\left( T \right) = d_{293} (1 + \alpha \Delta T) = d_{293} \left( {1 + \frac{\Delta L\left( T \right)}{{L_{293} }}} \right) $$
(7)

where d 293 and d(T) represent the thicknesses of Si layer at temperatures 293 K and T K, respectively, α is the coefficient of thermal expansion of Si with numerical value \( 2.5 \times 10^{ - 6} /{\text{K}} \), and ΔT is the change in temperature of Si layer. Due to the thermal expansion in Si layer, the thicknesses d 1, d 2 and d 3, d 4 of layers A and B of PC1 and PC2 at temperature 293 K will be modified in accordance with Eq. (7) as [7, 10, 11]

$$ d_{1} \left( T \right) = d_{1} +\Delta d_{1} ,d_{2} \left( T \right) = d_{2} { - \Delta }d_{1} \;{\text{and}}\;d_{3} \left( T \right) = d_{3} +\Delta d_{3} ,d_{4} \left( T \right) = d_{4} -\Delta d_{3} $$
(8)

Here \( \Delta d_{i} = d\left( T \right) - d_{i} \,\left( {293} \right) \) is the change in the thickness of Si layer due to corresponding change in \( \Delta T \). The subscripts i = 1 and 3 are used to represent Si layer of PC1 and PC2, respectively.

3 Results and discussion

In our calculations, we have taken (AB)N as 1D Si–air PC (PC1) with refractive index of layer A (Si), dependent on temperature and wavelength both in accordance with Eq. (6). The refractive index of layer B (air) is taken to be 1 [10]. For the proposed 12 period structure of PC1, we have chosen the thicknesses of layers A and B as d 1 = 419 nm and d 2 = 105 nm in order to get wider PBG in the near-infrared spectral region, ranging from 1372 to 1650 nm at a temperature of 293 K because this is a low-loss region in silica fibres [18]. Now, any change in temperature of PC1 and PC2 changes the thicknesses of layers A and B of both PCs according to Eq. (8) which in turn influences the range of PBGs of PC1 and PC2. Using TMM as discussed above, the transmission spectrum of PC1 at normal incidence is plotted as a function of wavelength in Fig. 2. It shows PBG of PC1 at temperature 293 K extending from λ l  = 1372 nm to λ r  = 1650 nm which lies within the spectral band (1260–1675 nm) used for optical fibre communications. The range of PBG of PC1 is dependent on the temperature of Si besides other parameters such as lattice constant and refractive indices of materials.

Fig. 2
figure 2

Transmittance spectra at normal incidence of PC1 by blue colour (d 1 = 419 nm, d 2 = 105 nm, N = 12 and T = 293 K)

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Now consider an input signal consisting of 11 independently modulated light sources, each having a unique wavelength ranging from 1391 to 1591 nm as listed in Table 1, incident normally on PC1 through input port PI (Fig. 1). Since all these wavelengths lie within the PBG of PC1 (Fig. 3), they are reflected back by PC1 and do not reach at output port Po through the combination of mirrors M1 and M2 as shown in Fig. 1. The reason behind the selection of this combination of mirrors M1 and M2 is to keep the phase difference between the signals reflected by mirrors M1 and M2 unaltered. Now we have chosen another 1D PC (PC2) composed of same materials as that of PC1. The thickness of the layers A and B of PC2 is selected as d 3 = 86.5 nm and d 4 = 148 nm, respectively, in order to get a narrow PBG extending from 1250 to 1344 nm in the near-infrared region at the same temperature 293 K (Fig. 4). Total number of periods of PC2 is also 12. Both PCs are put close to each other in such a way that radiation after coming out of PC1 falls normally on PC2 via the two highly reflecting parallel mirrors M1 and M2 through air, to avoid Fabry–Perot resonance between PC1 and PC2 in the event of PC1 and PC2 being put in cascade. Mirrors M1 and M2 are inclined at an angle 45° with respect to direction of periodicity of PC1 and PC2, respectively. This arrangement also ensures the condition of normal incidence for the signals coming out of PC1 and reaching to PC2. Since the refractive index of Si is dependent on the temperature as well as on the wavelength, any change in the temperature of Si layer changes its refractive index, thus altering the PBG of both PC1 and PC2. As we increase the operating temperature of Si from 625 to 1585 K, PBG of PC1 starts to shrink in the region of investigation and shifts towards the higher side of the wavelengths, thus allowing different unique wavelength channels of input signal to pass through PC1 and reach normally at PC2 (Table 2). But due to this temperature variation, the PBG of PC2 starts to expand and moves towards the higher side of the wavelengths in such a way as to select and switch only the desired wavelength channel from PC2 to reach Po. This temperature-dependent tunability of PBG of PC2 reflects all the wavelength channels of input radiation coming out from PC1 and transmits only one single wavelength channel at a time depending upon the temperature as shown in Fig. 5(a) and 5(b). Table 2 shows how temperature-dependent tunability of PBGs of both PCs (PC1 and PC2) of the proposed system can be used to separate and switch specific wavelength channels. The crosstalk between adjacent channels has been avoided by selecting the channel spacing to be 20 times the signal spectral width though the standard value is only 10 times [19]. If we reduce the number of periods of both PCs from the proposed 12, their PBGs contract and also their PBG edges are modified significantly. Thus, the period selection of both PCs is dependent on the number of equally spaced wavelength channels to be switched from the proposed design.

Table 1 List of 11 wavelength channels with specific wavelength, separation and line width of input signal
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Fig. 3
figure 3

Transmittance spectra (blue colour) of PC1 (d 1 = 419 nm, d 2 = 105 nm, N = 12 and T = 293 K) at normal incidence. Also shown are 11 wavelength channels (red colour) injected from input port PI. These channels with separation 20 nm and width 1 nm fall within the PBG of PC1. (Color figure online)

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Fig. 4
figure 4

Transmittance spectra at normal incidence of PC2 by green colour (d 3 = 86.5 nm, d 4 = 148 nm, N = 12 and T = 293 K). (Color figure online)

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Table 2 Control table of TTOFS for selecting different wavelength channels
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Fig. 5
figure 5

Working principle of TTOFS. (a) Transmittance spectra (blue colour) of PC1 (d 1 = 419 nm, d 2 = 105 nm, N = 12 and T = 625, 1181 and 1585 K) at normal incidence. The 11 wavelength channels (red colour) injected from input port PI. (b) Transmittance spectra (green colour) of PC2 (d 3 = 86.5 nm, d 4 = 148 nm and N = 12 and T = 625, 1181 and 1585 K) at normal incidence. The selected wavelengths λ 1, λ 6 and λ 11 by PC2 corresponding to temperatures 625, 1181 and 1585 K. (Color figure online)

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The working of the proposed TTOFS does not suffer from drawbacks of temperature-dependent tunability of defect mode inside PBG of defect PCs (DPCs) as reported in [8–12]. In order to filter number of closed and equispaced wavelength channels (Table 1) one by one at a time from PCs, temperature-dependent feature of defect mode inside PBG plays crucial role. The effect of temperature variation causes significant impact on the intensity as well as full width at half maximum (FWHM) of the defect mode, which limits the application of such structures as wavelength-selective switches for switching number of wavelength channels. Contrary to this, our structure works on the principle of temperature-dependent tunability of PBG which affects neither the intensity nor FWHM of individual and closely spaced wavelength channels to be filtered. In the proposed structure, the left and right band edges of PBGs of PC1 and PC2 can be simultaneously changed to allow only the desired wavelength channel to reach at Po by controlling the temperature of both PCs in accordance with Table 2. Besides this, for achieving WSS from DPCs to switch large number of closely spaced wavelength channels, crosstalk between adjacent channels must also be taken into account. Its value should be as minimum as possible to improve performance of the filter. In the proposed design, crosstalk does not contribute to the performance of the system, because PBGs of both PCs are tuned in such a way as to allow only one wavelength channel at Po without causing any interference between adjacent channels.

Thus, it is seen that for proper switching action of the proposed TTOFS, the only requirement is precise control of temperature. Presently available high-temperature technologies [20–22] can be used for this purpose. Hence, the proposed design is a better alternative to serve the objective over the conventional technologies being used for this purpose. It would be seen that, though, in order to switch the number of wavelength channels one by one from the proposed structure, some amount of signal power is lost due to two main reasons. Firstly, because of two high-quality reflective mirrors incorporated in the design and secondly due to temperature-dependent left and right band edge resonant modes of PC1 and PC2, respectively, as per the findings of Yeong et al. [14]. However, this power loss is much above the threshold value of detector to detect the power significantly.

4 Conclusions

A new and simple design of TTOFS composed of two 1D PCs (PC1 and PC2), and two highly reflecting mirrors (M1 and M2) have been proposed. This design is capable of separating and switching unique wavelength channels without interfering with each other, amongst 11 equally spaced wavelength channels each having FWHM of 1 nm as per the WDM standards developed by International Telecommunication Union (ITU) specifying channel spacing in terms of wavelength for WSS applications. The proposed TTOFS may play an important role in the design of some new kind of optical components in WDM environment which can ensure that optical signal power from one channel does not drift into the spectral territory occupied by adjacent channels.