1 Introduction

Optical filters capable of switching and routing optical signals, at a particular wavelength channel, carrying optical data from multichannel wavelength network into an optical fiber with minimum distortions are the key elements in modern computation and communication systems because optical signal can handle large bandwidth and large information flow with very high speed [1, 2].

During the past two and half decades, optical filters based on one-dimensional phototonic crystals (1D PCs) have inspired a lot of attention and investigation due to their important applications in modern photonic engineering including optical communication and optoelectronics. PCs are structures with periodically modulated dielectric constants whose distribution follows periodicity of the order of a fraction of the optical wavelength [3]. On account of the wave nature of light and interference caused by the periodic structure, photonic bandgaps (PBGs) are created. Photons with energies lying in the bandgap cannot propagate through the structure, while those with energies lying outside the bandgap can do so [4]. Thus, they have very useful optical properties of technical interest [5], for instance, narrow and high band pass filters, omnidirectional mirrors, tunable optical switches, multichannel, and single-channel optical filters etc. [6–9]. This active research area has been extended to superconductor-dielectric photonic crystals (SDPCs), due to their special features over conventional all dielectric and metal dielectric PCs [10–13]. One such feature is to tune their PBGs externally by changing the temperature of superconductor without disturbing the experimental arrangement [13, 14].

In the present communication, we have proposed a single-channel thermally tunable photonic filter (TTPF) for wavelength-selective switching (WSS) applications by cascading two 1D SDPCs (PC1 and PC2) side by side through low loss silica fiber. The transmission properties of 1D SDPC-based TTPF are investigated through transfer matrix method (TMM) [8, 15]. For proper working of the proposed filter, very precise control of temperatures of PC1 and PC2 is required. Therefore high-precision temperature control and stabilization technique [16] using cryocooler can be used to achieve the same.

The structure of the paper is organized as follows: The model and corresponding analytical formulae, such as effective refractive index of the superconductor (SC) and transmission coefficients are introduced in Section 2. Numerical results are presented and discussed in Section 3. Finally, conclusions are given in Section 4.

2 Theoretical Formulation

Block diagram of the proposed TTPF for WSS-based optical networks by cascading two 1D SDPC (PC1 and PC2) through low loss silica fiber is shown in Fig. 1. Here, PC1 and PC2 are put close to each other in such a way that radiation after passing through PC1 is allowed to fall normally on PC2 through low loss silica fiber. The lattice constants of PC1 and PC2 are denoted by D 1=d 1+d 2 and D 2=d 3+d 4, respectively. Here, d 1, d 2 and d 3, d 4 denote the thicknesses of low temperature SC (niobium [Nb]), dielectric (barium fluoride [BaF2]) of PC1 and high temperature SC (yttrium barium copper oxide [YBa2Cu3O7]), dielectric (barium fluoride (BaF2]) of PC2. The entire TTPF is assumed to be surrounded by air with a refractive index 1. TMM is used to calculate the transmittance of PC1 and PC2 at normal incidence [8, 15]. The characteristic matrix M (Λ) of one period is given by [15]

$$\begin{array}{@{}rcl@{}} M\left( {\Lambda} \right)&=&\mathop {\Pi}\limits_{j=1}^{2} \left( {{\begin{array}{*{20}c} {cos\delta_{j}} \hfill & {-\frac{i}{q_{j}} sin\delta_{j}} \hfill \\ {-i q_{j}} sin\delta_{j} \hfill & {cos\delta_{j}} \hfill \end{array}} } \right) \\ &=&\left[ {\left( {{\begin{array}{*{20}c} {M_{11}}& {M_{12}}\\ {M_{21}}& {M_{22}} \end{array}} } \right)} \right], \end{array} $$
(1)

where \(\delta _{j} =\frac {2\pi } {\lambda _{0}} n_{j} d_{j}\), with q j =n j , (j= SC, D), and λ 0 is the free space wavelength. Here, n j and d j denote the refractive index and thickness of the jth layer respectively.

Fig. 1
figure 1

A schematic design of TTPF system by cascading two 1D SDPC (PC1 and PC2) side by side through low loss silica fiber

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For a N period structure, the characteristic matrix of the medium is given by

$$\begin{array}{@{}rcl@{}} \left[ {M({\Lambda} )} \right]^{N}&=&M_{SC} M_{D} M_{S C} M_{D} \text{....M}_{SC} M_{D} \\ &=&\left( {{\begin{array}{*{20}c} {m}_{11} & {m}_{12} \\ {m}_{21} & {m}_{22} \end{array}} } \right). \end{array} $$
(2)

where m 11,m 12,m 21, and m 22 are the matrix elements of [M(Λ)]N which represents the total transfer matrix.

The transmission coefficient tof the multilayer periodic structure is given by

$$ t=\frac{\textit{2q}_{1}} {(\textit{m}_{11} +\textit{m}_{12} q_{k}) \text{q}_{1} +(\textit{m}_{21} +\textit{m}_{22} q_{k})} $$
(3)

where q 1and q k represent the coefficients of medium on either side of the structure. Since the entire structure is surrounded by air, q 1=q k =n o . Here, n o is the refractive index of air with value 1. The transmittance of the multilayer [15] is given by

$$ T=\left| t \right|^{2} $$
(4)

In order to describe electromagnetic response of both superconductors, Gorter-Casmir two fluid model [11, 17] in the absence of external magnetic field is adopted. The frequency and temperature-dependent effective relative permittivity of a superconductor is given by [17]

$$ \begin{array}{l} \varepsilon_{i} (\omega ,T)=\varepsilon_{c} -\frac{c^{2}}{\omega^{2}\left[ {\lambda_{L} (0)} \right]^{2}}\left[ {1-\left( {\frac{T}{T_{c}} } \right)^{p}} \right] \\ -\frac{c^{2}}{\omega (\omega +i\gamma)\left[ {\lambda_{L} (0)} \right]^{2}}\left( {\frac{T}{T_{c}}}\right)^{p} \end{array} $$
(5)

where ε c is the dielectric constant of the crystal, and ω and c are the frequency and speed of electromagnetic wave in vacuum, respectively. λ L (0) is the London penetration depth at temperature T= 0 K, and T c is the transition temperature of the superconductor. The value of exponent p is 2 and 4 to represent high and low temperature SC, respectively. Here, γ represents damping term due to normal conducting electrons. The third term on the right-hand side of (5) is omitted by neglecting the contribution of normal conducting electrons because the results obtained by two methods—neglecting and not neglecting this contribution are almost the same as described in [10]. Thereby, the temperature-dependent effective refractive index of lossless superconductor takes the form

$$ n_{i} =\sqrt {\varepsilon_{i} (\omega ,T)} =\sqrt {\varepsilon_{c} -\frac{c^{2}}{\omega^{2}\left[ {\lambda_{L} (0)} \right]^{2}}\left[ {1-\left( {\frac{T}{T_{c}} } \right)^{4}} \right]} $$
(6)

3 Results and Discussion

In the following numerical calculations, we have taken (AB) N as 1D SDPC (PC1) composed of two alternating layers of low temperature SC (A) and conventional material (B). The conventional material is taken as dielectric BaF2 with relative permittivity (ε B ) 2.13 and low temperature SC is niobium (Nb). The transition temperature (\(T_{c}^{A} )\) and London penetration depth at zero temperature (λ L (0)) of Nb are 9.2 K and 40 nm, respectively, as reported in [6]. Though in the region of present investigation, superconductors may be lossy even below the transition temperature, but for simplicity, the losses in superconductors Nb and YBa2Cu3O7 both are neglected. For the proposed 12-period structure PC1, we have chosen the thicknesses of layers A (B) as 40 (248) nm in order to get a PBG in the near-infrared spectral region (1250 to 1650 nm) at a temperature of 7.0 K because it is a low-loss region in silica fibers [18].

We have first calculated (using TMM) the transmission spectra at normal incidence to get a PBG in PC1 in the region of investigation. Then, transmittance is plotted as a function of wavelength (Fig. 2). It is observed that a PBG occurs in the near-infrared region which extends from λ l = 1366 nm to λ r = 1650 nm which lies within the spectral band (1260–1675 nm) used for optical fiber communications.

Fig. 2
figure 2

Transmittance spectra (blue color) of PC1 (d 1= 40 nm, d 2= 248 nm, N= 12, and T 1= 7.0 K) at normal incidence. Also shown are 12 discrete wavelength channels (red color) injected from input port P I into PC1. These channels with separation 20 nm and width 1 nm fall within the PBG of PC1

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We present the working principle of the proposed TTPF composed of two 1D SDPCs (PC1 and PC2) for WSS-based optical networks. Consider an input signal consisting of 12 independently modulated light sources emitting signals each of at a unique wavelength ranging from 1391 to 1611 nm (listed in Table 1) is incident normally on PC1 through input port P I (Fig. 1). Since all these wavelengths lie within the PBG of PC1 (Fig. 2), they will be reflected by PC1 and will not reach low loss silica fiber as shown in Fig. 1.

Table 1 List of 12 WDM channels with specific wavelength, separation, and line width of input signal
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Next, we have chosen another 1D SDPC (PC2) composed of two alternating layers of high-temperature SC (C) and conventional material (B). The conventional material is again taken as dielectric BaF2 with relative permittivity (ε B ) 2.13 and thickness 248 nm as that of PC1. The high-temperature SC is taken as YBa2Cu3O7 with λ L (0) = 145 nm and \(T_{c}^{C}=\) 93 K as reported in [19]. The total number of periods of PC2 is also 12. Here, thickness and temperature of the layer C is selected as 350 nm and 41 K, respectively, in order to get a narrower PBG extending from 1250 to 1385 nm in the near-infrared region.

Further, PC1 and PC2 are put close to each other in such a way that radiation after passing through PC1 falls normally at PC2 through low loss silica fiber (Fig. 1). Since the refractive index of SC layer is dependent on London penetration depth and external temperature both, therefore, by changing the temperature of the SC layer, its refractive index can be changed and hence the extent of the PBG. As we increase the temperature (T 1) of PC1 from 7.28 to 8.05 K below \(T_{c}^{A} \), PBG of PC1 starts to shrink in the region of investigation and shift towards the higher side of the wavelength, thus allowing different unique wavelength channels of input signal to pass through PC1 via low loss silica fiber and reach normally at PC2 (Table 2). In order to switch desired wavelength channels from PC2, we vary the temperature of PC2 from 41 to 90 K (\(T_{2} \textless {T_{c}^{C}} )\), which causes the PBG of PC2 to expand and move towards the higher side of the wavelength. This temperature-dependent PBG reflects all the wavelength channels of input radiation coming out from PC1 and transmits only one wavelength channel depending upon the temperature of PC2 to reach at output port (P o) as shown in Fig. 3. Table 2 shows how PC1 and PC2 can be cascaded side by side through low loss silica fiber to separate and switch-specific wavelength channels without interfering with each other amongst equally spaced 12 wavelength channels each having full width at half maximum (FWHM) of 1 nm as per the WDM standards developed by the International Telecommunication Union (ITU) specifying channel spacing in terms of frequency (wavelength) for wavelength selective switching applications [18].

Table 2 Control table of TTPFS for selecting different wavelength channels
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Fig. 3
figure 3

Working principle of TTPFS for wavelength selective switching application. a–c Transmittance spectra at normal incidence (blue color) for PC1 (d 1= 40 nm, d 2= 248 nm, and N= 12) at T 1= 7.28 K, 7.72 K, and 8.05 K. Also shown are 12 discrete wavelength channels (red color) injected from input port P I into PC1. d–f Transmittance spectra (green color) for PC2 (d 3= 350 nm, d 4= 248 nm, and N= 12). The discrete wavelength channels transmitted by PC1 and incident normally on PC2 via low loss silica fiber are shown in red color. PC2 transmits the wavelength channels λ 1, λ 6, and λ 12 at T 2 = 41 K, 78.6 K, and 90 K, respectively

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The WSS behavior of the proposed photonic filter does not affect the intensity and FWHM of individual WDM channels; it thus eliminates the drawbacks for switching applications associated with 1D defect photonic crystals (DPCs) [14–17]. Moreover, for achieving wavelength selective switching from 1D DPCs to switch large number of closely spaced wavelength channels, crosstalk between adjacent channels should also be taken into account. For better performance of the filter, its value must be small enough. Contrary to this, in the proposed design, crosstalk does not have any significant contribution over the filter performance due to the synchronize tuning between the PBGs of PC1 and PC2 to allow only one wavelength channel to reach at P o.

4 Conclusions

A new and simple design of single-channel TTPF composed of two 1D SDPCs (PC1 and PC2) connected through low loss silica fiber, which is capable of separating and switching unique wavelength channels without interfering with each other amongst 12 equally spaced wavelength channels, is presented. Each channel has FWHM of 1 nm as per the WDM standards developed by the ITU specifying channel spacing in terms of frequency (wavelength). The proposed TTPF that performs channel selection for WSS applications may play an important role for designing some new kind of optical cross-connect switches to achieve high degree of path-configuration modularity, capacity of scaling, and flexibility in adding dropping channels at a client site. This idea may also be used to achieve high-precision thermal sensing in remote sensing applications.