Terahertz multichanneled filter in a superconducting photonic crystal

Wei-Hsiao Lin; Chien-Jang Wu; Tzong-Jer Yang; Shoou-Jinn Chang

doi:10.1364/OE.18.027155

1. Introduction

It is known that a narrowband transmission filter can be achieved by adding a defect layer in a one-dimensional dielectric-dielectric photonic crystal (DDPC). The transmission peak (also called the defect mode) in such a filter is generally designed to be located near the center of the photonic band gap (PBG). A typical and simple narrowband transmission filter has a structure of (HL)^ND(HL)^N, in which D is the defect layer, N is the stack number of the two periodic bilayer structure, and H, L are the high- and low-index layers. Here, both H and L are equal to the quarter wavelength layers, i.e., n_Hd_H = n_Ld_L = λ_c/4, where n_H, n_L and d_H, d_L are their refractive indices and physical thicknesses, respectively, and λ_c is the design wavelength which is chosen in the vicinity of the center of the PBG. The shape of the transmission resonant peak can become very sharp like a comb when N is large enough. This is a single channel filter because there is only one transmission peak within the PBG. The position and height of the transmission peak are dependent on the angle and polarization of the incident wave [1]. A detailed discussion on the filter design is available in an excellent book by Orfanidis [2].

In order to improve the spectral efficiency in utilizing PBG, it is of technical interest to design a multichanneled transmission filter (MTF), that is, more than one transmission peaks within the PBG are expected. In fact, such an MTF has first been suggested and realized by replacing the single defect layer D by a photonic quantum well (PQW) [3]. The insertion of PQW into a PC gives rise to a series of discrete resonant transmission peaks which in turn leads to the function of multiple channeled filtering which is of particular use in the signal processing. With this idea, many related defective PCs containing the PQW for the purpose of achieving the MTFs have been feasible and reported recently [4–8]. Moreover, the MTFs based on the PQW containing double-negative or single-negative metamaterials are also of interest to the community recently [9–12].

Narrowband filters based on the PCs at terahertz (THz) have also attracted much attention over the past decade [13]. The THz waves are known to lie in the frequency range of from 0.1 THz to 10 THz. In this paper, we propose a novel design for the THz MTF based on a high-temperature superconductor-dielectric photonic crystal (SDPC). The design principle of this SDPC MTF is to make use of the photonic passband instead of utilizing the PBG to design an MTF in the usual dielectric PCs. This MTF has three features. First, it will be seen that there exist discrete comb-like transmission resonant peaks. Second, the number of peaks can be easily controlled by the number of spatial periods of the structure. Third, unlike the use of the PQW as a defect in PC, it is not necessary to introduce defect layer in such an SDPC in order to make such a filter. On the other hand, the use of superconducting material in the PCs has two main advantages compared with the usual metal-dielectric photonic crystals (MDPCs). First, with the damping of electromagnetic waves in metals, several potentially useful properties of MDPCs will be suppressed. The metallic loss issue can be remedied by using a superconductor in place of the metal. In fact, the metallic loss can be greatly reduced and be negligibly small when metal becomes a superconductor [14]. Second, the wave properties of an SDPC can be tunable because the response to an electromagnetic wave is mainly dependent on the London penetration depth which is a function of the temperature and the external magnetic field as well. This tunable feature is of technical use in the superconducting electronics and photonics. There have been many reports on the photonic band structures for the one- and two-dimensional SDPCs thus far [15–21].

The format of this paper is as follows: Section 2 first describes the electrodynamics of a superconductor from which the refractive index of a superconducting material is obtained. Next, the transfer matrix method (TMM) in a stratified media is used to calculate the reflectance, the transmittance, and the absorptance for a finite SDPC [22]. The Bloch-Floquet equation is then used to calculate the band structure for an infinite SDPC [23]. The calculated multichannel filtering properties will be given and discussed in Section 3. A summary will be given in Section 4.

2. Theory

In the beginning, let us introduce the superconductor electrodynamics from which the expression for the complex refractive index can be obtained. If the temporal part for all fields is take to be exp(-iωt), the refractive index of a superconducting material is complex-valued and given by

n_{1} = {n^{'}}_{1} + i {n^{″}}_{1} = \sqrt{ε_{1}},

where the imaginary part of the refractive index,

{n^{″}}_{1}

, is called the extinction coefficient, and the complex relative permittivity is related to the complex conductivity by

ε_{1} (ω, T) = {ε^{'}}_{1} (ω, T) + i {ε^{″}}_{1} (ω, T) = i \frac{σ_{1} (ω, T)}{ω ε_{0}},

where ε ₀ is the permittivity of free space. The conductivity in Eq. (2) can be expressed within the framework of the two-fluid model together with the local London electrodynamics [24]. In two-fluid model, the total conducting electrons in a superconductor consist of two kinds of electrons, the normal electrons and the superelectrons. In the circuit representation, the total current density in the superconducting state is contributed by two separate channels. The channel contributed by the normal electrons is made of the resistive and inductive elements in parallel. The other channel arising from the superelectrons is purely inductive. The total conductivity, which is a function of the frequency and the temperature, is the sum of two channel conductivities, namely [25]

\begin{matrix} σ_{1} (ω, T) = {σ^{'}}_{1} (ω, T) + i {σ^{″}}_{1} (ω, T) = \frac{ε_{0} ω_{p}^{2} τ}{1 - i ω τ} x_{n} + i \frac{1}{ω μ_{0}^{} λ_{L}^{2}} x_{s} \\ = \frac{ε_{0} ω_{p}^{2} τ}{1 + ω^{2} τ^{2}} x_{n} + i (\frac{ε_{0} ω_{p}^{2} τ^{2} ω}{1 + ω^{2} τ^{2}} x_{n} + \frac{1}{ω μ_{0}^{} λ_{L}^{2}} x_{s}), \end{matrix}

where

ω_{p} = {(n_{t} e^{2} / m ε_{0})}^{1 / 2}

is the plasma frequency corresponding to the total electrons,

τ = 1 / γ

is the momentum relaxation time of normal electrons with γ being the phenomenological damping factor. In Eq. (3),

x_{s} = N_{s} / N_{t}

and

x_{n} = N_{n} / N_{t} = 1 - x_{s}

are the temperature-dependent fractions of the superelectrons and the normal electrons, where N_s, N_n, and N_t = N_s + N_n, are the concentrations of the superelectrons, the normal electrons, and the total electrons, respectively. At zero or weak magnetic field,

x_{s}

is expressible as

x_{s} = 1 - {(T / T_{c})}^{4},

where T_c is the critical temperature of the superconducting material. In addition, the penetration length in Eq. (3) is dependent on

x_{s}

given by

λ_{L} (T) = \frac{λ_{0}}{\sqrt{1 - {(T / T_{c})}^{4}}},

where λ ₀ is the penetration depth at T = 0 K. It can be seen from Eq. (3) that the real part of complex conductivity, which indicates the loss in the superconductor, is mainly from the normal electrons. At temperatures well below T_c,

x_{s} >> x_{n}

, the loss contributed by

x_{n}

can be negligibly small.

The two-fluid conductivity in Eq. (3) is widely employed to describe the electromagnetic response of a superconductor. However, it should be noted that it is valid for frequencies below the superconducting gap frequency ω_g, i.e., $ω < ω_{g} = 2 Δ / ℏ$ , where $2 Δ$ is known as the superconducting gap and ℏ is the Planck constant. For most superconductors, ω_g is on the other of THz. At frequency less than ω_g, and substituting Eq. (3) into Eq. (2), it can be seen that the real part of the permittivity is negative. Thus a superconductor can be referred to as an epsilon-negative (ENG) medium which is a class of single-negative (SNG) material. Therefore, a wave cannot propagate inside a superconductor since it is evanescent. However, it does propagate through a layered structure such as an SDPC. At frequency above ω_g, the superconducting state is destroyed and the material becomes a normal metal. In this case, with $x_{s} \to 0$ , the conductivity in Eq. (3) goes back to the Drude conductivity for the normal metal.

The one-dimensional SDPC modeled as a periodic superconductor-dielectric superlattice of (S/D)^N immersed in free space is depicted in Fig. 1 , where S is the superconductor layer, D is the dielectric layer, N is the number of periods, and the spatial periodicity is defined by a = d₁ + d₂ with d₁ and d₂ being the thicknesses of S and D, respectively. We also denote the refractive indices for layers S and D as n₁ (given in Eq. (1)) and $n_{2} = \sqrt{ε_{2}}$ (ε₂ is the relative permittivity), respectively. Here, the normal incidence will be considered in this work. The terahertz spectroscopic properties will be analyzed from the reflectance R and the transmittance Γ calculated based on the matrix formulation. The transmittance $Γ = {| t |}^{2}$ and the reflectance $R = {| r |}^{2}$ are directly determined by the transmission coefficient t and the reflection coefficient r, respectively. According to TMM, t and r are expressed as [22]

t = \frac{1}{M_{11}}, r = \frac{M_{21}}{M_{11}},

where

M_{11}

and

M_{21}

are the matrix elements of the total transfer matrix M_sys for the entire system given by

M_{s y s} = (\begin{matrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix}) = D_{0}^{- 1} M_{a}^{N} D_{0},

where

M_{a}

is the transfer matrix of a single period, namely

M_{a} = D_{1} P_{1} D_{1}^{- 1} D_{2} P_{2} D_{2}^{- 1} .

Here, the propagation matrix in S and D is expressed as

P_{ℓ} = (\begin{matrix} e^{- i ϕ_{ℓ}} & 0 \\ 0 & e^{i ϕ_{ℓ}} \end{matrix}),

where

ϕ_{ℓ} = k_{ℓ x} d_{ℓ} = \frac{2 π d_{ℓ}}{λ} n_{ℓ}, ℓ = 1, 2,

and the dynamical matrix in each region is defined by

D_{ℓ} = (\begin{matrix} 1 & 1 \\ n_{ℓ} & - n_{ℓ} \end{matrix}),

where

ℓ = 0

, 1 and 2, and

ℓ = 0

represents the free space. With the calculated R and Γ, the absorptance A is calculated by

A = 1 - Γ - R,

according to the power balance equation.

Fig. 1 The structure of a one-dimensional superconducting photonic crystal denoted by (S/D)^N immersed in free space. The wave is normally incident at the plane boundary, x = 0. The spatial periodicity is a = d ₁ + d ₂, where d ₁ and d ₂ are the thicknesses of S and D, respectively.

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If the number of periods is sufficiently large, i.e., N >> 1, the structure in Fig. 1 can be regarded as an infinite SDPC. In this case, the photonic band structure, ω versus K, can be computed and plotted by making use of the Bloch-Floquet equation [23], namely

\cos (K a) = \cos (k_{1} d_{1}) \cos (k_{2} d_{2}) - \frac{1}{2} (\frac{k_{1}}{k_{2}} + \frac{k_{2}}{k_{1}}) \sin (k_{1} d_{1}) \sin (k_{2} d_{2}),

where K is the Bloch wave vector, and

k_{1} = ω n_{1} / c

,

k_{2} = ω n_{2} / c

are the wave vectors in S and D, respectively. The photonic band structure is usually plotted in terms of the normalized quantities, i.e.,

K a / 2 π

vs.

ω a / 2 π c

. Here,

c = 1 / \sqrt{μ_{0} ε_{0}}

is the speed of light in free space.

3. Numerical results and discussion

Let us now present the calculated results for the transmittance, the reflectance, and the absorptance for the SDPC in Fig. 1. The material parameters of superconductor layer S are taken to be on the order of the typical high-temperature superconducting system, YaBa₂Cu₃O_7-x, with T_c = 92 K, λ ₀ = 200 nm, ω_p = 1.7 x 10¹⁵ rad/s, γ = 1.3 x 10¹³ rad/s, and the superconducting gap frequency $2 Δ / ℏ = 8$ THz [26]. The relative permittivity ε₂ of the dielectric layer D is taken to be 15. The spatial periodicity a = 150 μm is taken in order to make the normalized frequency $ω a / 2 π c$ below 0.5 at terahertz frequency and thus the first few allowed passbands can lie below the gap frequency. The operating frequency ω is varied around 4.8 THz.

In Fig. 2 , we plot the frequency-dependent Γ (top), R (middle), and A (bottom) at a low temperature of 4.2 K. The temperature 4.2 K, which is the boiling point of liquid helium, is frequently used in the community of superconductivity. Here, N = 8 and d ₁ = 0.1 μm are used. It is of interest to see that there are multiple resonant peaks in Γ and dips in R. The corresponding absorptance A is very small because the fraction of normal electrons is relatively smaller than that of the superelectrons at low temperature. The presence of comb-like peaks in Γ can be ascribed to the interactions between the evanescent wave in S and the normal wave in D. With these sharp peaks, the SDPC can thus function as an MTF. In addition, the number of resonant peaks or dips can be concluded from Fig. 3 , in which the reflectance spectra are plotted for different periods of N = 1, 2, 5, and 10. For a single period of N = 1, the structure reduces to a simple bilayer. In this case, it is seen that there are no resonant transmission peaks and the structure, in effect, behaviors like a reflector because of the high reflectance, $R \approx 1$ . To obtain the resonant transmission, the required number of periods must be larger than one. The more number of periods, the more multiple wave interactions between the interfaces will have, leading to the occurrence of the more resonant transmission peaks. Conclusively, Figs. 2 and 3 indicate that the number of resonant peaks (or dips) is equal to N-1 for a given N. Moreover, these peaks are all located within a photonic passband which can be calculated from Eq. (13) and illustrated in the solid curve of Fig. 4 , where the lower and higher band edges of the passband are $ω_{L} = 0.3832 (2 π c / a)$ and $ω_{H} = 0.3873 (2 π c / a)$ , respectively. Here, the Bloch wave vector in Eq. (3) is denoted as $K = K^{'} + i K^{″}$ , from which the passband is represented by the real part $K^{'}$ whereas the imaginary solution $K^{″}$ indicates the forbidden band (the dash curve in Fig. 4). The point P₁ marks the resonant point for the case of N = 2, as will be discussed next.

Fig. 2 The calculated transmittance Γ (top), reflectance R (middle), and absorptance A (bottom) at temperature T = 4.2 K. The comb-like distributions of the resonant peaks in Γ and R are shown.

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Fig. 3 The resonant dips in R in the normalized frequency domain at different numbers of periods, N = 1, 2, 5, and 10, respectively. The number of resonant dips is equal to N-1 for a given N.

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Fig. 4 The calculated band diagram at T = 4.2 K. The solid curve corresponding to the real part of the Bloch wave vector indicates the passband, whereas the photonic band stop bands have the imaginary parts of the Bloch wave vector which are shown in the dash curve. P ₁ marks the resonant point for N = 2.

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The number of resonant peaks in Γ (or dips in R) can be analytically obtained as follows: The explicit expression for the reflectance in an N-period structure in Fig. 1 is given by [22]

R = \frac{{| C |}^{2}}{{| C |}^{2} + {[\sin (K a) / \sin (K N a)]}^{2}},

where K is given by Eq. (13), and C is one of the matrix elements of the transfer matrix in a single period. In the photonic passband, K is real, i.e.,

K = K^{'}

. With the fact that K is a function of the frequency, Eq. (14) then gives the condition for R = 0, namely

\sin (K^{'} N a) = 0 \to K^{'} (ω_{m}) N a = m π, m = 1, 2, ... N - 1,

where ω_m is the frequency at which the m-th resonant dip in R occurs. Therefore, there are N-1 dips in R for a given N, as illustrated in Fig. 3. The correlation between Figs. 3 and 4 can be further seen by taking a look at the simple case of N = 2. In this case, we have

K^{'} (ω_{1}) a / 2 π = 0.25

indicated by point P ₁ in Fig. 4, which in turn graphically corresponds to

ω a / 2 π c = 0.38533

, consistent with the resonant position at N = 2 in Fig. 3. Similar conclusions can be drawn for N >2. In fact, Eq. (15) can be equivalently regarded as the Fabry-Perot resonant transmission condition for a single slab of thickness Na. Moreover, the appearance of N-1 comb-like peaks in Γ (or dips in R) in an N-period structure can be explained by the coupled-resonance interference mechanism [27,28]. Since layer S is an ENG medium, waves within layers S are all evanescent. Thus, each layer D may be regarded as a resonator cavity. When the coupling of evanescent waves between two consecutive layers D is established, a high transmittance can be achieved in the structure. For N = 1, there is no resonant peak since, in this case, no coupling can be obtained in a single layer D. For N = 2, there is one coupling established in two layers D and thus one peak is seen. Inductively, there will be N-1 peaks because there are N-1 couplings in an N-period structure. If the system is lossless, then all resonant peak height will be equal to unity. However, a closer look at Fig. 2 reveals that they do not have the same height. This fact is directly reflected in the absorptance A, the bottom panel of Fig. 2, in which the absorption strongly decreases as the resonant frequency increases. The absorptance A arises from the existence the real part of superconducting conductivity contributed by the normal-fluid. It can be seen from Eq. (3) that the real part of conductivity is inversely proportional to 1 + ω ² τ ² which, in turn, elucidates the trend of the absorptance spectrum in Fig. 2.

The similar terahertz oscillating behavior in spectroscopy can also be seen in a two-dimensional SDPC with superconducting rods in a square lattice [26]. However, the comb-like shape in the transmission spectrum cannot be obtained in the two-dimensional case, in which there exists a nonzero and broad dip between two adjacent peaks. In view of filtering function, the one-dimensional SDPC having the comb-like filtering feature is superior to the two-dimensional one. The above results suggest that a superconducting MTF at THz can be achieved without inserting any defect into the periodic structure. This design is fundamentally different from the usual design of an MTF, in which a PQW must be employed as a defect in a PC in order to produce the multiple resonant peaks located within the PBG [3].

The above spectroscopic properties are calculated at T = 4.2 K well below T_c, indicating that the loss arising from the normal-fluid is negligibly small. To examine the loss effect, we fix at N = 8 and increase the temperature to 42 K, which is larger than 4.2 K by one order of magnitude only for the purpose of comparison. The corresponding Γ, R, and A are plotted in Fig. 5 . It can be seen that the resonant positions remain unchanged, the same as those in Fig. 2. However, the comb-like resonant shape in Fig. 2 has been significantly changed. The peak values in Γ also drop greatly from unity to below 0.02. Although the drop in the peak value of transmittance due to the inclusion of loss is similar to that of in a usual single-slab Fabry-Perot transmission filter, the dropping behavior among these peaks is not uniform. The first and second peaks at lower frequencies are nearly smeared out. The salient drop in the first peak is also seen in a two-dimensional SDPC [26]. The seventh peak at the highest frequency remains a comb-like shape. The severe drop in the peak of Γ is due to the large value in the imaginary part of the superconducting permittivity, ${ε^{″}}_{1}$ , at 42 K compared to that of 4.2 K. In fact, the difference is almost of four orders of magnitude, that is, ${ε^{″}}_{1} \approx$ 1800 at 42 K, whereas ${ε^{″}}_{1} \approx$ 0.18 at 4.2 K.

Fig. 5 The calculated transmittance Γ (top), reflectance R (middle), and absorptance A (bottom) at temperature T = 42 K. The comb-like shapes in Γ, R, and A shown in Fig. 2 are now suppressed and broadened.

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Figure 6 shows the transmittance for three different thicknesses of the superconductor layer, d ₁ = 0.3, 0.5, and 0.7 μm, at the same conditions as in Fig. 2, i.e., a = 150 μm, N = 8, and T = 4.2 K. It is worth noting that distribution of the resonant peaks in Γ is very sensitive to d ₁. The peaks are closely piled together and moved to the higher frequencies as d ₁ increases. At d ₁ = 0.7 μm, the peak height has been dropped to a value less than 0.01, indicating the filtering property is no longer available. If we define the filling factor as ρ = d ₁ / a, then the condition of retaining the filtering feature must be less than 1/300. The thickness dependence of resonant transmission can be qualitatively understood as follows: As mentioned previously, a superconductor is an ENG material at THz. If the superconductor film thickness increases, then the interaction between the evanescent waves in superconductor and the normal wave in dielectric will become very weak, causing the resonant transmission to disappear, as illustrated at d ₁ = 0.7 μm.

Fig. 6 The calculated transmittance at different thicknesses of the superconducting layer, d ₁ = 0.3 (top), 0.5 (middle), and 0.7 μm (bottom), respectively. The resonant transmission peaks are negligibly small at d ₁ = 0.7 μm.

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Finally, to highlight the importance of the superconducting layer for achieving the comb-like response, it is worth calculating the optical properties in the non-superconducting state above T_c, that is, the normal-state response. At temperature above T_c, the superconducting state reduces to the normal state. In this case, the permittivity can be described by the conventional Drude equation [26],

ε_{1} (ω) = 1 - \frac{ω_{p}^{2}}{ω^{2} + i γ ω} .

With the same material parameters in Fig. 2, the calculated normal-state THz spectroscopy is plotted in Fig. 7 , in which the transmittance Γ (top), reflectance R (middle), and absorptance A (bottom) are given. It can be seen that there is no discrete comb-like response as in Fig. 2. The spectra are continuous as a function of the frequency. In fact, in the normal state, the entire structure behaves like a good mirror because of its high reflectance > 0.95 and negligibly small value in transmittance in the frequency of interest. The results illustrate the role played by the superconducting layers in obtaining the comb-like response at THz.

Fig. 7 The calculated normal-state transmittance Γ (top), reflectance R (middle), and absorptance A (bottom). The discrete comb-like behaviors in Γ, R, and A are no longer present.

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We have so far investigated the THz response for an SDPC in the superconducting state and normal state. However, it is known that most high-temperature cuprates are type-II superconductors and they can be in the mixed state when the externally applied static magnetic field is larger than the critical field. In the mixed state, vortices are produced inside the superconductor. Thus, to study the electromagnetic response of a superconductor in the presence of vortices, a relevant vortex dynamical model should be employed. The vortex model is more complicated and totally different from the theory in Section 2. In fact, the study of microwave transmission and reflection in a one-dimensional SDPC in the mixed state has been reported [29]. However, to the best of our knowledge, the THz response for a one-dimensional SDPC in the mixed state remains unseen. It is thus of interest to extend our current work to be in the mixed state and explore whether the anomalous comb-like transmission can be found or not. If it is existing, then how it will be affected by the static magnetic field. This will be discussed and presented in a separate paper.

4. Summary

We have investigated the THz spectroscopy for a one-dimensional high-temperature SDPC. The results suggest that a multichanneled narrowband transmission filter can be achieved without adding any defect layer in an SDPC, which is structurally different form the usual design of multichanneled filter based on the dielectric-dielectric photonic crystal containing a defect of PQW. In addition, this filter utilizes the photonic passband instead of using the photonic band gap in a usual filter. At low temperature, this filter does have a comb-like distribution in the transmission peaks with their number being directly determined by the number of periods. The resonant peaks can be successfully explained by the Fabry-Perot interference condition together with the band diagram calculated by the Bloch-Floquet equation. The dependences of the temperature, the filling factor of the superconductor layer are also illustrated. The proposed filter with comb-like transmission peaks is of technical use to the superconducting electronics at terahertz for it can work as a frequency-selective device or as a sampler for the purpose of signal processing.

Acknowledgments

C.-J. Wu wishes to acknowledge the financial support from the National Science Council of the Republic of China (Taiwan) under grant No. NSC-97-2112-M-003-013-MY3.

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Terahertz multichanneled filter in a superconducting photonic crystal

Abstract

1. Introduction

2. Theory

3. Numerical results and discussion

4. Summary

Acknowledgments

References and links

Cited By

Figures (7)

Equations (16)

Optics Express