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Experimental investigation of photonic band gap in one-dimensional photonic crystals with metamaterials
Physics Letters A 376 (2012) 1396–1400
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Experimental investigation of photonic band gap in one-dimensional photonic crystals with metamaterials Yihang Chen a,b,∗ , Xinggang Wang b , Zehui Yong a , Yunjuan Zhang a , Zefeng Chen b , Lianxing He a , P.F. Lee a , Helen L.W. Chan a , Chi Wah Leung a , Yu Wang a a b
Department of Applied Physics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China Laboratory of Quantum Information Technology, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou, China
a r t i c l e
i n f o
Article history: Received 18 July 2011 Received in revised form 21 January 2012 Accepted 28 January 2012 Available online 1 February 2012 Communicated by R. Wu Keywords: Electromagnetic optics Photonic crystal Metamaterials
a b s t r a c t Composite right/left-handed transmission lines with lumped element series capacitors and shunt inductors are used to experimentally realize the one-dimensional photonic crystals composed of singlenegative metamaterials. The simulated and experimental results show that a special photonic band gap corresponding to zero-effective-phase (zero-ϕeff ) may appear in the microwave regime. In contrast to the Bragg gap, by changing the length ratio of the two component materials, the width and depth of the zero-ϕeff gap can be conveniently adjusted while keeping the center frequency constant. Furthermore, the zero-ϕeff gap vanishes when both the phase-matching and impedance-matching conditions are satisfied simultaneously. These transmission line structures provide a good way for realizing microwave devices based on the zero-ϕeff gap. © 2012 Elsevier B.V. All rights reserved.
1. Introduction The interaction of electromagnetic (EM) wave with matter is determined by the electric permittivity ε and magnetic permeability μ, which are two intrinsic material parameters that describe the response of charges and currents to an applied EM field. Both ε and μ of all transparent natural materials are positive, resulting in a positive index of refraction. However, if the sign and magnitude of the index could be tuned at will, the flow of EM wave could be controlled in unconventional ways [1,2]. In recent years, metamaterials have attracted considerable attention for their ability to precisely control the dispersion and propagation of EM wave. Metamaterials have artificial EM properties that are defined by their sub-wavelength structure rather than their chemical composition. Through variation of the constituent elements and dimensions, metamaterials allow for adjustability of ε and μ that can span between positive, negative, and nearzero values, leading to many interesting applications such as subdiffraction-limited superlenses [3–5] and cloaking at specific EM wavelength ranges [1,6,7]. Metamaterials that exhibit simultaneously negative ε and μ are called double-negative (DNG) materials. Similarly, the metamaterials with either negative ε or negative
*
Corresponding author at: Department of Applied Physics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China. E-mail addresses: [email protected] (Y. Chen), [email protected] (Y. Wang). 0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2012.01.044
μ are referred to as single-negative (SNG) materials. There are two kinds of SNG metamaterials: ε -negative (ENG) material and μ-negative (MNG) material. Since both ε and μ of metamaterials depend on frequency, the characteristics of DNG or SNG can only be realized within a certain frequency range. Over the last two decades, another kind of artificial materials, photonic crystal, has been the intriguing subject of great attention due to their unique EM properties and potential applications [8,9]. Conventional photonic crystals have periodically modulated dielectric functions and thus possess the photonic band gap as a result of Bragg scattering. As such, the frequency of the Bragg gap is proportional to the lattice constant while randomness generally destroys the band gap [10]. Recently, it was shown that stacking alternating layers of ENG and MNG materials may lead to a special type of photonic band gap, known as the zero-effective-phase (zero-ϕeff ) gap [11]. Comparing to conventional Bragg gap, the zero-ϕeff gap can be insensitive to incident angle and thickness fluctuations [12, 13]. It has been theoretically pointed out that the zero-ϕeff gap possesses many unique properties which may lead to important applications, such as omnidirectional and multi-channel filters and broadband wave plates [14–17]. In this Letter, we experimentally investigate the properties of the zero-ϕeff gap in one-dimensional (1D) photonic crystals comprising SNG materials. The SNG materials were fabricated using composite right/left-handed transmission line by periodically loading lumped-element series capacitors (C ) and shunt inductors ( L ). Experimental results show that, as the length ratio between the ENG and MNG materials varies, the width and depth of the zero-
Y. Chen et al. / Physics Letters A 376 (2012) 1396–1400
1397
ϕeff gap changes significantly while the central frequency remains invariant. The experimental results are in accordance with the numerical simulations. 2. The model of the SNG materials Up to now, there are two main approaches to form SNG or DNG metamaterials have been reported: resonant structures made of arrays of wires and split-ring resonators [18–20] and nonresonant transmission line (TL) structures made of lumped elements [21–24]. The TL approach towards metamaterial with lefthanded and right-handed attributes, known as composite right/ left-handed transmission line (CRLH TL), presents the advantage of lower losses over a broader bandwidth. SNG material can be realized in the frequency range between the right-handed passband and left-handed passband of the CRLH TL. The CRLH TL is fabricated by the repetition of the TL unit cell, which consist of a host transmission line with lumped elements of series capacitors C and shunt inductors L [21,22]. When the average lattice constant li is much smaller than the guided wavelength λ g , the structure exhibits a macroscopic behavior which can be rigorously characterized in terms of the constitutive parameters ε and μ. In practice, li < λ g /4 can be considered as the sufficient condition for the validity of homogeneous approximation [24]. In our experiment, 50Ω TL was used to fabricate CRLH TL which has FR-4 substrates with thickness h = 1.6 mm and relative permittivity εsub = 4.75. The thickness of copper strip on the FR-4 substrate was t = 0.018 mm. The width of the copper strip was w = 2.945 mm corresponding to the characteristic impedance of Z 0 = 50 . For a one-dimensional CRLH TL, the constitutive parameters can be obtained by mapping the telegrapher’s equations to Maxwell equations [21], the effective relative permittivity and permeability are given by the following approximate expressions:
εi ≈ C 0 − 1/ω2 L i li /(ε0 p ),
μi ≈ p L 0 − 1/ω2 C i li /μ0 , (1) √
where p is a structure constant which given by p = μ0 /(ε0 εre )/ Z 0 [24], C 0 and L 0 represent the distributed capacitance and inductance of the host TL, and i represents the type of the CRLH TLs. The effective relative dielectric constant εre of the microstrip line can be obtained as [25]
εre =
εsub + 1 2
+
εsub − 1
2
1+
12h w
−1/2
−
εsub − 1 t /h 4.6
w /h
. (2)
For the microstrip lines considered here, the effective relative dielectric constant εre ≈ 3.556. Then it can be obtained that √ p = 3.99, C 0 = εre ε0 μ0 / Z 0 ≈ 128 pF/m, and L 0 = Z 02 × C 0 ≈ 320 nH/m. Two types of CRLH TLs with 12 TL units were designed and fabricated to realize the SNG materials. TL1 unit possesses a unit length of l1 = 7.2 mm and loaded lumped elements C 1 = 5.1 pF and L 1 = 5.6 nH, and TL2 unit has a unit length of l2 = 8.4 mm and loaded lumped elements C 2 = 2 pF and L 2 = 8 nH, respec√ tively. For the frequency of the guided wave f g < 1/4li ε0 μ0 ≈ 8.9 GHz, both TL1 and TL2 can be regarded as homogeneous media described by effective permittivities and permeabilities. Based on the parameters given above, the effective material parameters can be written as
ε1 = 3.57–6.92 × 1020 /ω2 , μ1 = 1.03–8.75 × 1019 /ω2 , for TL1, and
(3)
Fig. 1. The calculated effective permittivity (εi ) and permeability (μi ) for the CRLH TL1 (a) and TL2 (c). The simulated and measured transmittance of TL1 (b) and TL2 (d) with 12 TL units, respectively.
ε2 = 3.57–4.15 × 1020 /ω2 , μ2 = 1.03–1.91 × 1020 /ω2 ,
(4)
for TL2. The calculated relative permittivity ε1 and permeability μ1 of TL1 according to Eq. (3) are presented in Fig. 1(a). The simulated (by Advanced Design System (ADS)) and measured (by Agilent 8720ES S-parameter Network Analyzer) S 21 parameters for TL1 containing 12 units are shown in Fig. 1(b). It is seen that both ε1 and μ1 are positive in the higher frequency range, and a corresponding right-handed passband exist in Fig. 1(b). On the other hand, both ε1 and μ1 are negative in the lower frequency range and a corresponding left-handed passband can be observed. In the frequency range 1.47–2.21 GHz, ε1 < 0 and μ1 > 0, TL1 is equivalent to an ENG material, a stopband appears since the wave √ number k = 2π ε1 μ1 /λ g is imaginary. As shown in Fig. 1(b), the measured result agrees with the simulations although inevitable losses led to a reduced transmittance. There is a cutoff frequency for the CRLH TL given by [21]
fi =
1
4π
√
Li Ci
.
(5)
It can be calculated from Eq. (5) that f 1 ≈ 0.47 GHz for TL1, and the propagation of EM wave will be prohibited when the frequency is lower than this cutoff frequency. Similarly, ε2 and μ2 as functions of frequency, and the simulated and measured results for TL2 with 12 units are shown in Fig. 1(c) and (d), respectively. It can be seen that, between the two passbands, a stopband corresponding to ε2 > 0 and μ2 < 0 appears in the frequency range 1.72–2.17 GHz. It means that TL2 can be equivalent to a MNG material in the corresponding range of
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Fig. 2. Photograph of the fabricated 1D photonic crystals (a) ( A 1 B 1 )12 , (b) ( A 3 B 1 )6 , and (c) ( A 5 B 1 )4 fabricated based on CRLH TLs, respectively.
the stopband. Consequently, in the frequency range 1.72–2.17 GHz, both TL1 and TL2 can be regarded as homogeneous SNG materials.
η A + ηB = 0
3. Properties of photonic band gaps in 1D PCs containing metamaterials
k Ad A = kB dB
We then consider a 1D photonic crystal with a periodic structure of (AB) N , where A represents a layer of ENG material with thickness d A and B represents a layer of MNG material with thickness d B , and N is the number of periods. Let a plane wave be injected from vacuum into the PC from the + z direction, assuming that the wave in the jth layer has a wave vector k j = k j y yˆ + k jz zˆ , √ √ whose magnitude is ω ε j μ j /c (where c is the speed of light in vacuum, j = A , B). For the case of normal incidence (k j = k jz ), the band structure of the PC can be obtained by the characteristic equation [13]
cos κ (d A + d B ) = cos(k A d A ) cos(k B d B )
η A ηB + sin(k A d A ) sin(k B d B ), − 2 ηB ηA 1
√
√
(6)
where κ is the Bloch wave number and η j = μ j / ε j is the impedance. The real (imaginary) solution for κ corresponds to the propagating (evanescent) state of the electromagnetic fields in the target system. It has been demonstrated that a zero-ϕeff gap may appear in the SNG PC [13]. Such zero-ϕeff gap will vanish when both the impedance-matching condition
(7)
and the phase-matching condition
(8)
are satisfied simultaneously. However, if the two conditions cannot be satisfied simultaneously, the zero-ϕeff gap will open from the impedance-matching frequency. Next, we use the TL1 and TL2 units to construct the SNG photonic crystals. For convenience, we use ( A m B n ) N to denote the photonic crystals fabricated using the CRLH TL. A m and B n are made of the units of TL1 and TL2, where m and n are the number of the TL1 units and TL2 units, respectively. Three kinds of 1D PCs, ( A 1 B 1 )12 , ( A 3 B 1 )6 , and ( A 5 B 1 )4 were fabricated, as shown in Fig. 2. Let us first consider the structure ( A 1 B 1 )12 . Fig. 3 presents the absolute value of the impedance η j and the square of the phase thickness Φ j (= k j d j ) against frequency for the considered structure. It is seen from Fig. 3(a) that the impedance-matching condition in Eq. (7) can be satisfied at 1.93 GHz, where the phasematching condition in Eq. (8) is close to being met. In this case, the zero-ϕeff gap should be close to vanishing. Fig. 4(a) shows the dispersion relationship of infinite periodic structure ( A 1 B 1 ) N obtained from Eq. (6). Transmission spectra calculated by transfer matrix method is shown in Fig. 4(b), while Fig. 4(c) gives the simulated and measured S 21 parameters of the structure with 12 periods, respectively. The band structure and
Y. Chen et al. / Physics Letters A 376 (2012) 1396–1400
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Fig. 4. (a) Dispersion relationship of an infinite periodic structure ( A 1 B 1 ) N ; (b) The transmission spectra of the ( A 1 B 1 )12 calculated by transfer matrix methods; (c) The simulated and measured S 21 parameters of the structure ( A 1 B 1 )12 . Fig. 3. (a) The absolute value of the impedance |η j | and (b) the square of the phase thickness (Φ j )2 against frequency for the photonic structure ( A 1 B 1 ) N .
transmittance clearly show that two gaps emerge. In the frequency range 1.5–2.2 GHz in which TL1 or/and TL2 units can be seen as SNG materials, the Bloch wave number κ is real except for a narrow range (1.88–1.96 GHz), as shown in Fig. 4(a). It can be seen from Fig. 4(b) and (c) that a zero-ϕeff gap with narrow width and shallow depth appears, the simulated and measured results agree well with the theoretical results. Pronounced fringes can be observed in Fig. 4(b). However, the losses existing in the CRLH TL make such fringes become less distinct, as shown in Fig. 4(c). For a single TL1 or TL2, the EM waves corresponding to the SNG frequency range are evanescent, as shown in Fig. 1. However, by combining both TL1 and TL2 units, the periodic SNG structure becomes transparent except for the range of the zero-ϕeff gap. Moreover, another photonic gap with the center frequency of 0.64 GHz can be observed in Fig. 4. Such gap should be a Bragg gap since it locates in the left-handed passband of the CRLH TL. It is well known that the Bragg gap may appear when the Bragg condition is satisfied
Φ A + ΦB = k A d A + k B d B = pπ ,
(9)
where p = ±1, ±2, . . . . As discussed in Section 2, both TL1 and TL2 units are equivalent to DNG materials in frequencies f < 1.4 GHz. In this case, the fabricated photonic crystal based on CRLH TL should be regarded as 1D periodic structure consisting of DNG materials. When the Bragg condition corresponding to p = −1 is satisfied, a Bragg gap will appear inside the left-handed passband. Next, we change the length (d A ) of the ENG layer and investigate the properties of the photonic band gap in photonic crystal ( A 3 B 1 )N and ( A 5 B 1 )N . Fig. 5(a) shows the square of the phase thickness (Φ j )2 as a function of frequency for structure ( A 3 B 1 ) N . It can be seen that the phase thicknesses of A and B will not be equivalent in the SNG frequency range 1.5–2.2 GHz and the phase-matching condition cannot be met. It means that compared to ( A 1 B 1 ) N , an evident zero-ϕeff gap can be observed in ( A 3 B 1 ) N . Moreover, (Φ j )2 as a function of frequency for ( A 5 B 1 ) N is also calculated and shown in Fig. 5(b). Comparing to Fig. 5(a), the difference between the phase thicknesses Φ A and Φ B is larger, suggesting an even wider and deeper zero-ϕeff gap will appear in ( A 5 B 1 )N . Fig. 6 show the band structure and transmittance for structure ( A 3 B 1 )N and ( A 5 B 1 )N , respectively. It is seen from Fig. 6(a) that a zero-ϕeff gap appears and locates in the range 1.69–2.08 GHz for ( A 3 B 1 ) N . The ADS simulated and the experimentally measured
Fig. 5. The square of the phase thickness (Φ j )2 as a function of frequency for structure (a) ( A 3 B 1 ) N and (b) ( A 5 B 1 ) N .
results shown in Fig. 6(b) are in accordance with the band structure. It is also noted that the center frequency of the zero-ϕeff gap remains at the impedance-matching frequency of 1.93 GHz. Moreover, two Bragg gaps corresponding to p = +1 and −1 appear on both sides of the zero-ϕeff gap with center frequencies 3.71 and 0.98 GHz, respectively. For ( A 5 B 1 ) N , a deeper zero-ϕeff gap covers the frequency range 1.63–2.13 GHz, as shown in Fig. 6(c) and (d). The central frequency of the zero-ϕeff gap still remains nearly unchanged. On the other hand, the center frequencies of the Bragg gap corresponding to p = +1 and −1 change visibly to about 3.08 and 1.18 GHz, respectively. 4. Summary An approach based on the composite right/left-handed transmission lines (CRLH TLs) has been introduced to fabricate 1D photonic crystals containing metamaterials. The zero-ϕeff gap was observed in the microwave frequency range where the TL units are equivalent to homogenous SNG materials. We experimentally demonstrate that the width and depth of the zero-ϕeff gap are
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the phase-matching and impedance-matching conditions approach to be met simultaneously. The CRLH TL structures can further be used to realize microwave devices based on the zero-ϕeff gap, such as multichannel filters and broadband wave plates. Acknowledgements This work was funded by ITF (ITP/026/09NP) and the Hong Kong Polytechnic University (J-BB9P). This work is also supported by National Natural Science Foundation of China (Grant No. 10704027), Natural Science Foundation of Guangdong Province (Grant No. 9151063101000040). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
Fig. 6. Dispersion relationship of infinite periodic structure (a) ( A 3 B 1 ) N and (c) ( A 5 B 1 ) N . The simulated and measured S 21 parameters of the structure (b) ( A 3 B 1 )6 and (d) ( A 5 B 1 )4 .
sensitive to the lengths ratio of the ENG and MNG materials while the center frequency of zero-ϕeff gap remains at the impedancematching frequency. The zero-ϕeff gap tends to close when both
[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
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Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Experimental investigation of photonic band gap in one-dimensional photonic crystals with metamaterials Yihang Chen a,b,∗ , Xinggang Wang b , Zehui Yong a , Yunjuan Zhang a , Zefeng Chen b , Lianxing He a , P.F. Lee a , Helen L.W. Chan a , Chi Wah Leung a , Yu Wang a a b
Department of Applied Physics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China Laboratory of Quantum Information Technology, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou, China
a r t i c l e
i n f o
Article history: Received 18 July 2011 Received in revised form 21 January 2012 Accepted 28 January 2012 Available online 1 February 2012 Communicated by R. Wu Keywords: Electromagnetic optics Photonic crystal Metamaterials
a b s t r a c t Composite right/left-handed transmission lines with lumped element series capacitors and shunt inductors are used to experimentally realize the one-dimensional photonic crystals composed of singlenegative metamaterials. The simulated and experimental results show that a special photonic band gap corresponding to zero-effective-phase (zero-ϕeff ) may appear in the microwave regime. In contrast to the Bragg gap, by changing the length ratio of the two component materials, the width and depth of the zero-ϕeff gap can be conveniently adjusted while keeping the center frequency constant. Furthermore, the zero-ϕeff gap vanishes when both the phase-matching and impedance-matching conditions are satisfied simultaneously. These transmission line structures provide a good way for realizing microwave devices based on the zero-ϕeff gap. © 2012 Elsevier B.V. All rights reserved.
1. Introduction The interaction of electromagnetic (EM) wave with matter is determined by the electric permittivity ε and magnetic permeability μ, which are two intrinsic material parameters that describe the response of charges and currents to an applied EM field. Both ε and μ of all transparent natural materials are positive, resulting in a positive index of refraction. However, if the sign and magnitude of the index could be tuned at will, the flow of EM wave could be controlled in unconventional ways [1,2]. In recent years, metamaterials have attracted considerable attention for their ability to precisely control the dispersion and propagation of EM wave. Metamaterials have artificial EM properties that are defined by their sub-wavelength structure rather than their chemical composition. Through variation of the constituent elements and dimensions, metamaterials allow for adjustability of ε and μ that can span between positive, negative, and nearzero values, leading to many interesting applications such as subdiffraction-limited superlenses [3–5] and cloaking at specific EM wavelength ranges [1,6,7]. Metamaterials that exhibit simultaneously negative ε and μ are called double-negative (DNG) materials. Similarly, the metamaterials with either negative ε or negative
*
Corresponding author at: Department of Applied Physics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China. E-mail addresses: [email protected] (Y. Chen), [email protected] (Y. Wang). 0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2012.01.044
μ are referred to as single-negative (SNG) materials. There are two kinds of SNG metamaterials: ε -negative (ENG) material and μ-negative (MNG) material. Since both ε and μ of metamaterials depend on frequency, the characteristics of DNG or SNG can only be realized within a certain frequency range. Over the last two decades, another kind of artificial materials, photonic crystal, has been the intriguing subject of great attention due to their unique EM properties and potential applications [8,9]. Conventional photonic crystals have periodically modulated dielectric functions and thus possess the photonic band gap as a result of Bragg scattering. As such, the frequency of the Bragg gap is proportional to the lattice constant while randomness generally destroys the band gap [10]. Recently, it was shown that stacking alternating layers of ENG and MNG materials may lead to a special type of photonic band gap, known as the zero-effective-phase (zero-ϕeff ) gap [11]. Comparing to conventional Bragg gap, the zero-ϕeff gap can be insensitive to incident angle and thickness fluctuations [12, 13]. It has been theoretically pointed out that the zero-ϕeff gap possesses many unique properties which may lead to important applications, such as omnidirectional and multi-channel filters and broadband wave plates [14–17]. In this Letter, we experimentally investigate the properties of the zero-ϕeff gap in one-dimensional (1D) photonic crystals comprising SNG materials. The SNG materials were fabricated using composite right/left-handed transmission line by periodically loading lumped-element series capacitors (C ) and shunt inductors ( L ). Experimental results show that, as the length ratio between the ENG and MNG materials varies, the width and depth of the zero-
Y. Chen et al. / Physics Letters A 376 (2012) 1396–1400
1397
ϕeff gap changes significantly while the central frequency remains invariant. The experimental results are in accordance with the numerical simulations. 2. The model of the SNG materials Up to now, there are two main approaches to form SNG or DNG metamaterials have been reported: resonant structures made of arrays of wires and split-ring resonators [18–20] and nonresonant transmission line (TL) structures made of lumped elements [21–24]. The TL approach towards metamaterial with lefthanded and right-handed attributes, known as composite right/ left-handed transmission line (CRLH TL), presents the advantage of lower losses over a broader bandwidth. SNG material can be realized in the frequency range between the right-handed passband and left-handed passband of the CRLH TL. The CRLH TL is fabricated by the repetition of the TL unit cell, which consist of a host transmission line with lumped elements of series capacitors C and shunt inductors L [21,22]. When the average lattice constant li is much smaller than the guided wavelength λ g , the structure exhibits a macroscopic behavior which can be rigorously characterized in terms of the constitutive parameters ε and μ. In practice, li < λ g /4 can be considered as the sufficient condition for the validity of homogeneous approximation [24]. In our experiment, 50Ω TL was used to fabricate CRLH TL which has FR-4 substrates with thickness h = 1.6 mm and relative permittivity εsub = 4.75. The thickness of copper strip on the FR-4 substrate was t = 0.018 mm. The width of the copper strip was w = 2.945 mm corresponding to the characteristic impedance of Z 0 = 50 . For a one-dimensional CRLH TL, the constitutive parameters can be obtained by mapping the telegrapher’s equations to Maxwell equations [21], the effective relative permittivity and permeability are given by the following approximate expressions:
εi ≈ C 0 − 1/ω2 L i li /(ε0 p ),
μi ≈ p L 0 − 1/ω2 C i li /μ0 , (1) √
where p is a structure constant which given by p = μ0 /(ε0 εre )/ Z 0 [24], C 0 and L 0 represent the distributed capacitance and inductance of the host TL, and i represents the type of the CRLH TLs. The effective relative dielectric constant εre of the microstrip line can be obtained as [25]
εre =
εsub + 1 2
+
εsub − 1
2
1+
12h w
−1/2
−
εsub − 1 t /h 4.6
w /h
. (2)
For the microstrip lines considered here, the effective relative dielectric constant εre ≈ 3.556. Then it can be obtained that √ p = 3.99, C 0 = εre ε0 μ0 / Z 0 ≈ 128 pF/m, and L 0 = Z 02 × C 0 ≈ 320 nH/m. Two types of CRLH TLs with 12 TL units were designed and fabricated to realize the SNG materials. TL1 unit possesses a unit length of l1 = 7.2 mm and loaded lumped elements C 1 = 5.1 pF and L 1 = 5.6 nH, and TL2 unit has a unit length of l2 = 8.4 mm and loaded lumped elements C 2 = 2 pF and L 2 = 8 nH, respec√ tively. For the frequency of the guided wave f g < 1/4li ε0 μ0 ≈ 8.9 GHz, both TL1 and TL2 can be regarded as homogeneous media described by effective permittivities and permeabilities. Based on the parameters given above, the effective material parameters can be written as
ε1 = 3.57–6.92 × 1020 /ω2 , μ1 = 1.03–8.75 × 1019 /ω2 , for TL1, and
(3)
Fig. 1. The calculated effective permittivity (εi ) and permeability (μi ) for the CRLH TL1 (a) and TL2 (c). The simulated and measured transmittance of TL1 (b) and TL2 (d) with 12 TL units, respectively.
ε2 = 3.57–4.15 × 1020 /ω2 , μ2 = 1.03–1.91 × 1020 /ω2 ,
(4)
for TL2. The calculated relative permittivity ε1 and permeability μ1 of TL1 according to Eq. (3) are presented in Fig. 1(a). The simulated (by Advanced Design System (ADS)) and measured (by Agilent 8720ES S-parameter Network Analyzer) S 21 parameters for TL1 containing 12 units are shown in Fig. 1(b). It is seen that both ε1 and μ1 are positive in the higher frequency range, and a corresponding right-handed passband exist in Fig. 1(b). On the other hand, both ε1 and μ1 are negative in the lower frequency range and a corresponding left-handed passband can be observed. In the frequency range 1.47–2.21 GHz, ε1 < 0 and μ1 > 0, TL1 is equivalent to an ENG material, a stopband appears since the wave √ number k = 2π ε1 μ1 /λ g is imaginary. As shown in Fig. 1(b), the measured result agrees with the simulations although inevitable losses led to a reduced transmittance. There is a cutoff frequency for the CRLH TL given by [21]
fi =
1
4π
√
Li Ci
.
(5)
It can be calculated from Eq. (5) that f 1 ≈ 0.47 GHz for TL1, and the propagation of EM wave will be prohibited when the frequency is lower than this cutoff frequency. Similarly, ε2 and μ2 as functions of frequency, and the simulated and measured results for TL2 with 12 units are shown in Fig. 1(c) and (d), respectively. It can be seen that, between the two passbands, a stopband corresponding to ε2 > 0 and μ2 < 0 appears in the frequency range 1.72–2.17 GHz. It means that TL2 can be equivalent to a MNG material in the corresponding range of
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Fig. 2. Photograph of the fabricated 1D photonic crystals (a) ( A 1 B 1 )12 , (b) ( A 3 B 1 )6 , and (c) ( A 5 B 1 )4 fabricated based on CRLH TLs, respectively.
the stopband. Consequently, in the frequency range 1.72–2.17 GHz, both TL1 and TL2 can be regarded as homogeneous SNG materials.
η A + ηB = 0
3. Properties of photonic band gaps in 1D PCs containing metamaterials
k Ad A = kB dB
We then consider a 1D photonic crystal with a periodic structure of (AB) N , where A represents a layer of ENG material with thickness d A and B represents a layer of MNG material with thickness d B , and N is the number of periods. Let a plane wave be injected from vacuum into the PC from the + z direction, assuming that the wave in the jth layer has a wave vector k j = k j y yˆ + k jz zˆ , √ √ whose magnitude is ω ε j μ j /c (where c is the speed of light in vacuum, j = A , B). For the case of normal incidence (k j = k jz ), the band structure of the PC can be obtained by the characteristic equation [13]
cos κ (d A + d B ) = cos(k A d A ) cos(k B d B )
η A ηB + sin(k A d A ) sin(k B d B ), − 2 ηB ηA 1
√
√
(6)
where κ is the Bloch wave number and η j = μ j / ε j is the impedance. The real (imaginary) solution for κ corresponds to the propagating (evanescent) state of the electromagnetic fields in the target system. It has been demonstrated that a zero-ϕeff gap may appear in the SNG PC [13]. Such zero-ϕeff gap will vanish when both the impedance-matching condition
(7)
and the phase-matching condition
(8)
are satisfied simultaneously. However, if the two conditions cannot be satisfied simultaneously, the zero-ϕeff gap will open from the impedance-matching frequency. Next, we use the TL1 and TL2 units to construct the SNG photonic crystals. For convenience, we use ( A m B n ) N to denote the photonic crystals fabricated using the CRLH TL. A m and B n are made of the units of TL1 and TL2, where m and n are the number of the TL1 units and TL2 units, respectively. Three kinds of 1D PCs, ( A 1 B 1 )12 , ( A 3 B 1 )6 , and ( A 5 B 1 )4 were fabricated, as shown in Fig. 2. Let us first consider the structure ( A 1 B 1 )12 . Fig. 3 presents the absolute value of the impedance η j and the square of the phase thickness Φ j (= k j d j ) against frequency for the considered structure. It is seen from Fig. 3(a) that the impedance-matching condition in Eq. (7) can be satisfied at 1.93 GHz, where the phasematching condition in Eq. (8) is close to being met. In this case, the zero-ϕeff gap should be close to vanishing. Fig. 4(a) shows the dispersion relationship of infinite periodic structure ( A 1 B 1 ) N obtained from Eq. (6). Transmission spectra calculated by transfer matrix method is shown in Fig. 4(b), while Fig. 4(c) gives the simulated and measured S 21 parameters of the structure with 12 periods, respectively. The band structure and
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Fig. 4. (a) Dispersion relationship of an infinite periodic structure ( A 1 B 1 ) N ; (b) The transmission spectra of the ( A 1 B 1 )12 calculated by transfer matrix methods; (c) The simulated and measured S 21 parameters of the structure ( A 1 B 1 )12 . Fig. 3. (a) The absolute value of the impedance |η j | and (b) the square of the phase thickness (Φ j )2 against frequency for the photonic structure ( A 1 B 1 ) N .
transmittance clearly show that two gaps emerge. In the frequency range 1.5–2.2 GHz in which TL1 or/and TL2 units can be seen as SNG materials, the Bloch wave number κ is real except for a narrow range (1.88–1.96 GHz), as shown in Fig. 4(a). It can be seen from Fig. 4(b) and (c) that a zero-ϕeff gap with narrow width and shallow depth appears, the simulated and measured results agree well with the theoretical results. Pronounced fringes can be observed in Fig. 4(b). However, the losses existing in the CRLH TL make such fringes become less distinct, as shown in Fig. 4(c). For a single TL1 or TL2, the EM waves corresponding to the SNG frequency range are evanescent, as shown in Fig. 1. However, by combining both TL1 and TL2 units, the periodic SNG structure becomes transparent except for the range of the zero-ϕeff gap. Moreover, another photonic gap with the center frequency of 0.64 GHz can be observed in Fig. 4. Such gap should be a Bragg gap since it locates in the left-handed passband of the CRLH TL. It is well known that the Bragg gap may appear when the Bragg condition is satisfied
Φ A + ΦB = k A d A + k B d B = pπ ,
(9)
where p = ±1, ±2, . . . . As discussed in Section 2, both TL1 and TL2 units are equivalent to DNG materials in frequencies f < 1.4 GHz. In this case, the fabricated photonic crystal based on CRLH TL should be regarded as 1D periodic structure consisting of DNG materials. When the Bragg condition corresponding to p = −1 is satisfied, a Bragg gap will appear inside the left-handed passband. Next, we change the length (d A ) of the ENG layer and investigate the properties of the photonic band gap in photonic crystal ( A 3 B 1 )N and ( A 5 B 1 )N . Fig. 5(a) shows the square of the phase thickness (Φ j )2 as a function of frequency for structure ( A 3 B 1 ) N . It can be seen that the phase thicknesses of A and B will not be equivalent in the SNG frequency range 1.5–2.2 GHz and the phase-matching condition cannot be met. It means that compared to ( A 1 B 1 ) N , an evident zero-ϕeff gap can be observed in ( A 3 B 1 ) N . Moreover, (Φ j )2 as a function of frequency for ( A 5 B 1 ) N is also calculated and shown in Fig. 5(b). Comparing to Fig. 5(a), the difference between the phase thicknesses Φ A and Φ B is larger, suggesting an even wider and deeper zero-ϕeff gap will appear in ( A 5 B 1 )N . Fig. 6 show the band structure and transmittance for structure ( A 3 B 1 )N and ( A 5 B 1 )N , respectively. It is seen from Fig. 6(a) that a zero-ϕeff gap appears and locates in the range 1.69–2.08 GHz for ( A 3 B 1 ) N . The ADS simulated and the experimentally measured
Fig. 5. The square of the phase thickness (Φ j )2 as a function of frequency for structure (a) ( A 3 B 1 ) N and (b) ( A 5 B 1 ) N .
results shown in Fig. 6(b) are in accordance with the band structure. It is also noted that the center frequency of the zero-ϕeff gap remains at the impedance-matching frequency of 1.93 GHz. Moreover, two Bragg gaps corresponding to p = +1 and −1 appear on both sides of the zero-ϕeff gap with center frequencies 3.71 and 0.98 GHz, respectively. For ( A 5 B 1 ) N , a deeper zero-ϕeff gap covers the frequency range 1.63–2.13 GHz, as shown in Fig. 6(c) and (d). The central frequency of the zero-ϕeff gap still remains nearly unchanged. On the other hand, the center frequencies of the Bragg gap corresponding to p = +1 and −1 change visibly to about 3.08 and 1.18 GHz, respectively. 4. Summary An approach based on the composite right/left-handed transmission lines (CRLH TLs) has been introduced to fabricate 1D photonic crystals containing metamaterials. The zero-ϕeff gap was observed in the microwave frequency range where the TL units are equivalent to homogenous SNG materials. We experimentally demonstrate that the width and depth of the zero-ϕeff gap are
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the phase-matching and impedance-matching conditions approach to be met simultaneously. The CRLH TL structures can further be used to realize microwave devices based on the zero-ϕeff gap, such as multichannel filters and broadband wave plates. Acknowledgements This work was funded by ITF (ITP/026/09NP) and the Hong Kong Polytechnic University (J-BB9P). This work is also supported by National Natural Science Foundation of China (Grant No. 10704027), Natural Science Foundation of Guangdong Province (Grant No. 9151063101000040). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
Fig. 6. Dispersion relationship of infinite periodic structure (a) ( A 3 B 1 ) N and (c) ( A 5 B 1 ) N . The simulated and measured S 21 parameters of the structure (b) ( A 3 B 1 )6 and (d) ( A 5 B 1 )4 .
sensitive to the lengths ratio of the ENG and MNG materials while the center frequency of zero-ϕeff gap remains at the impedancematching frequency. The zero-ϕeff gap tends to close when both
[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
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