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Surface wave modes in single-negative metamaterials fibers
Optics Communications 291 (2013) 232–237
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Surface wave modes in single-negative metamaterials fibers Jianfeng Dong n, Jinjing Liu, Xiaoyang Luo Institute of Optical Fiber Communication and Network Technology, Ningbo University, Ningbo 315211, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 September 2012 Received in revised form 28 October 2012 Accepted 12 November 2012 Available online 3 December 2012
The features of surface wave modes in the fiber composed of single-negative (SNG) metamaterials have been investigated. The geometric and frequency dispersion of surface wave modes in SNG fibers are considered. The dispersion curves, energy flux and power of lower-order (m¼ 0,1) surface wave modes are presented. Some interesting properties, such as totally backward or forward wave, zero power, signvarying energy flux in the core or cladding region, single mode operation, are found. & 2012 Elsevier B.V. All rights reserved.
Keywords: Surface wave mode Dispersion curve Energy flux Single-negative metamaterial Fiber
1. Introduction The waveguides made from novel electromagnetic materials (metamaterials) have attracted much attention and lots of papers on this topic have been published recently. Metamaterials are generally referred to as materials or structures that exhibit unique electromagnetic properties not usually found in nature. Double-negative (DNG) materials, also called left-handed materials, negative refractive index materials, in which the permittivity and permeability are negative simultaneously, are a class of metamaterials [1]. DNG materials have many unique properties such as negative refraction, inverse Doppler effect, etc. [2,3], and they have fascinating potential applications including super resolution, cloaking, and slow light [4–6]. Other interesting class of metamaterials are single-negative (SNG) materials in which only one of the material parameters, permittivity (epsilon) or permeability (mu), is negative. SNG materials can be constructed more easily than DNG materials because SNG materials need only one material parameter negative. These so called single-negative (SNG) materials, i.e., epsilon-negative (ENG) or mu-negative (MNG) materials also possess interesting properties and have many potential applications [7,8]. For example, remarkable features such as resonance, transparency, anomalous tunneling, and zero reflection through layers consisting of SNG materials have been proved theoretically [8–10]. Guided modes in the parallel-plate waveguide filled with a pair of ENG–MNG layers have been investigated theoretically, and monomodal propagation can be achieved in parallel-plate
n
Corresponding author. Fax: þ86 574 87600940. E-mail addresses: [email protected], [email protected] (J. Dong).
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waveguides with arbitrary thickness [11]. Dispersion properties of surface wave modes in a pair of juxtaposed ENG and MNG slabs have been investigated, interesting properties such as simultaneous TE and TM surface wave modes with opposite total energy flow directions are found [12]. The special type of waveguide composed of impedance-matched SNG materials has also been studied, and it supports only a single mode having TE and TM polarization degeneracy but power flow directions of degenerate TE and TM modes are always opposite to each other [13]. Wave propagation in a planar dielectric waveguide covered by the MNG material layers has been considered, and slow light can be achieved at the critical waveguide thickness [14]. Slow and even stopped electromagnetic wave can be supported by an active dielectric waveguide cladded by SNG materials [15]. The cylindrical symmetry is a simple realistic geometry. Due to their geometrical simplicity and important applications, the metamaterial circular waveguides (fibers) containing DNG materials in the core or cladding have been extensively studied [16–24]. The peculiar mode properties of a fiber with DNG core have been found, such as the perfect phase matching of the TE and TM slow modes, sign-varying energy flux, and the existence of TEM modes [16]. Slow-light can be supported in the practical circular waveguide with metamaterial cladding [22]. Guided optical waves in fibers with negative dielectric constant in the core or cladding have also been investigated in the literature [25–29], and the unusual result is the existence of modes with negative group velocity. Moreover, fiber metamaterials with negative permittivity or negative magnetic permeability in the THz frequency band have been fabricated recently [30–32]. However, the surface wave modes in the fiber composed of a pair of SNG materials have not been examined to our knowledge. The surface wave mode has an evanescent behavior both in the
J. Dong et al. / Optics Communications 291 (2013) 232–237
core and cladding and can propagate along the interface between ENG and MNG materials. Because the field maximum is at the interface, the surface wave mode is a very sensitive and convenient tool for studying of the physical properties of the surfaces. Thus the investigations of surface wave modes in SNG fibers are very important. In this paper, the surface wave modes that exist near the interface of the fiber composed of ENG–MNG materials have been investigated. The results of the geometric and frequency dispersion of lower-order surface wave modes in SNG fibers are presented, showing some interesting properties such as totally backward or forward wave, zero power, sign-varying energy flux in the core or cladding region, single mode operation.
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imposing the conditions of continuity of the tangent electromagnetic fields at r ¼a, the characteristic equation of surface wave modes can be obtained as [18]: 2 hm ihe m e2 i b 1 1 2 1^ 1^ Im 2 K^ m Im K^ m ¼ m2 2þ 2 ð3Þ u v u v k0 u v pffiffiffiffiffiffiffiffiffiffi where k ¼ o m e , u ¼k a,v ¼k a, ^I ¼ I’m ðuÞ, K^ ¼ K’m ðvÞ. I ’ ðUÞ, 0
0 0
t1
m
t2
Im ðuÞ
m
K m ðvÞ
m
Km ’ ðUÞstand for differentiation with respect to argument. When m¼ 0, using relations I0’ ðxÞ ¼ I1 ðxÞ and K 0’ ðxÞ ¼ K 1 ðxÞ, the characteristic Eq. (3) reduces to:
m1 I1 ðuÞ
¼
u I0 ðuÞ
m2 K 1 ðvÞ
ð4Þ
v K 0 ðvÞ
which corresponds to the TE mode, and 2. Formulations
e1 I1 ðuÞ
A single-negative metamaterial fiber (SNG fiber) consists of SNG materials in the core and cladding. The relative dielectric permittivity and magnetic permeability of SNG material are e1,m1 in the core and e2,m2 in the cladding; the core radius is a (see Fig. 1). The cladding is supposed to extend infinitely. Assume electromagnetic guided waves propagate along z direction. Two pairs of SNG materials are considered: the ENG–MNG fiber with e1 o0,m1 40; e2 40,m2 o0 (ENG core and MNG cladding) and the MNG–ENG fiber with e1 40,m1 o0; e2 o0,m2 40 (MNG core and ENG cladding). Because e1m1 o0, e2m2 o0, only surface wave modes, whose electromagnetic fields exponentially decay with distance from interface both in the core and cladding, can exist. This is distinct from DNG fibers where the ordinary guided modes with an oscillatory behavior within the core can also exist [16]. Following the standard procedure, in the cylindrical coordinate system (r, j, z), the longitudinal electromagnetic field components of surface wave modes in SNG fibers can be derived as following: In the core (0rr ra): " # Ez1 A ¼ ð1Þ Im ðkt1 r Þeimj eibz Hz1 B In the cladding (r Za): " # Ez2 C ¼ K m ðkt2 r Þeimj eibz Hz2 D
ð2Þ
where A, B, C, D are unknown constants, Im(U) Km(U) are the first and second modified Bessel functions, m is an integer specifying the azimuthal field dependence, b is the longitudinal propagation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 constant. kt1 ¼ b o2 m0 e0 m1 e1 , kt2 ¼ b o2 m0 e0 m2 e2 are the transverse attenuation factors in the core and cladding, respectively. The transverse components of the electromagnetic fields can be derived from the relationships between longitudinal and transverse components of the electromagnetic fields. Then,
u I0 ðuÞ
¼
e2 K 1 ðvÞ
ð5Þ
v K 0 ðvÞ
which corresponds to the TM mode.It is shown from Eqs. (4) and (5) that the cutoff frequencies of TE and TM surface wave modes (when b ¼0) satisfy following equations respectively: sffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi m1 I1 e1 m1 k0 a m2 K 1 e2 m2 k0 a pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð6Þ e I e K e1 m1 k0 a e2 m2 k0 a 1 0 2 0 sffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi e1 I1 e1 m1 k0 a e K e2 m2 k0 a pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi m I m K e m k a e m k a 1
0
1
1 0
0
2
2
ð7Þ
2 0
when the fiber core radius is large enough, i.e., k0a-N, applying properties of modified Bessel functions, I1(x)¼I0(x), K1(x)¼K0(x) for large x in Eqs. (4) and (5), we can obtain normalized propagation constants for the planar interface between ENG and MNG materials: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m21 e2 m2 m22 e1 m1 b ð8Þ ¼ k0 m21 m22 for the TE mode, and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e21 m2 e2 e22 m1 e1 b ¼ k0 e21 e22
ð9Þ
for the TM mode. It is noted that similar formulas have been found in the single-negative or double-negative metamaterial waveguides [24,33,34]. The energy flux of the mode along the z-axis in the fiber can be derived by: 1 1
ð10Þ Sz ¼ Re E Hn z^ ¼ Re Er Hnj Ej Hnr 2 2 The normalized power of the mode is defined as [16] Pz ¼
P z1 þP z2 9P z1 9 þ9Pz2 9
ð11Þ
where Pz1 and Pz2 are the power of the mode in the core and cladding respectively: Z a Z 2p Z a Pz1 ¼ rSz1 drdj ¼ 2p rSz1 dr ð12aÞ 0
Pz2 ¼
0
Z 2p Z 0
0 1
rSz2 drdj ¼ 2p a
Z
1
rSz2 dr
ð12bÞ
a
3. Numerical results and discussion
Fig. 1. Geometry and material parameters of a single-negative metamaterial fiber.
The longitudinal propagation constant b of the surface wave mode is a function of the fiber core radius (geometric dispersion) and of the frequency through the SNG material’s dispersion relation (frequency dispersion). The characteristic Eqs. (3)–(5)
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are nonlinear equations of b, and can be solved by Newton– Raphson method. After the b has been obtained, it is easy to calculate the energy flux and power of the surface wave mode. In the characteristic equations, the single unknown variable is b2.Thus, the sign of b can be positive or negative. Here we choose b positive. Note that if we change the sign of b, the energy flux and power also change signs. In this section, we will give numerical results for the geometric dispersion and frequency dispersion of lower-order (m¼ 0,1) surface wave modes. 3.2. Geometric dispersion Here we consider properties of surface wave modes varying with the fiber core radius for a fixed frequency. In the conventional fibers, guided modes can exist only for condition e1m1 4 e2m2, however, in the SNG fibers, there is no restriction for e1m1 and e2m2. We will discuss three cases: (i) 9e1m1949e2m29, (ii) 9e1m19o9e2m29 and (iii) 9e1m19¼9e2m29. Only results of the ENG–MNG fiber are presented, the results of the MNG–ENG fiber are similar. 9e1 m1 9 4 9e2 m2 9
ðiÞ
when keeping e2 ¼1,m2 ¼ 1 and e1m1 ¼ 4 unchanged, the normalized propagation constants b/k0 and the normalized power Pz of lower-order (m¼0,1) surface wave modes versus the normalized fiber core radius k0a for different relative permittivity and permeability in the core (e1,m1)¼( 1.6,2.5), ( 2,2), ( 2.5,1.6), (4,1), ( 8,0.5) are presented in Fig. 2. Solid and dashed lines correspond to m¼0 (TM) and m¼1 modes (TE modes are not shown in Fig. 2). It is noted that TM modes exist only for 9e1/e2941. It can be seen from Fig. 2 that for TM modes, as k0a increases, b/k0 decrease and Pz increase monotonously. The bigger 9e19, the bigger Pz. For (e1,m1)¼(1.6,2.5), ( 2,2), TM modes have no cutoff normalized fiber core radii. For (e1,m1)¼( 2.5,1.6), (14,1), ( 8,0.5), the cutoff
normalized fiber core radii are k0a¼3.187, 1.034, 0.502, which satisfy Eq. (7) and decrease as 9e19 increases. This means that for larger 9e19, only below cutoff fiber core radius, the TM mode can propagate. It is obvious from Eqs. (4) and (5) that when e1m1 and e2m2are unchanged, for the TE mode, b/k0 is the same as that for the TM mode if m1/m2 ¼ e1/e2. However, Pz is exactly opposite, i.e., for the TE mode, Pz is negative and the absolute value is the same as that for the TM mode. Pz is negative means that the power flow is opposite to wave vector. When e1 ¼ 2, p m1ffiffiffiffiffiffiffiffiffiffiffiffi ¼2, ffiimpedances pffiffiffiffiffiffiffiffiffiffiffiffiffi of ENG and MNG materials are matched, i.e. m1 =e1 ¼ m2 =e2 , TM and TE modes are degenerate, but their power flow are opposite to each other. This property has also been found in the SNG slab waveguide, and has potential application in the polarization beam splitter [13]. When e1 ¼ 2.5,m1 ¼1.6, for m¼1 surface wave mode, the dispersion curve splits into two branches from the critical point C (k0a ¼1.459). The upper branch has no high cutoff normalized fiber core radius, while the lower branch has high cutoff normalized fiber core radius k0a¼3.431. In the range of k0a ¼1.459– 3.431, b/k0 has double values. Two branches of the dispersion curve have different normalized power Pz. Pz is positive for lower branch of the dispersion curve and negative for upper one (Fig. 2(b)). At the critical point C, Pz is zero. Zero power has been found in the slab [15] or circular waveguides [16,22] containing metamaterials. This implies that for appropriate electromagnetic parameters and size of the fiber, the SNG fiber can also be used to store light [15]. It can be shown from Eq. (3) that when e1m1 and e2m2 are unchanged, dashed line in Fig. 2(a) also corresponds to m¼1 surface wave mode with e1 ¼ 1.6,m1 ¼2.5. However, the sign of Pz is also exactly opposite to e1 ¼ 2.5,m1 ¼1.6 case. Fig. 3(a and b) show the normalized energy flux Sz distribution of the TM mode and TE mode for k0a¼1.5, b/k0 ¼1.360, where
Fig. 2. The normalized propagation constant b/k0 (a), and the normalized power Pz and (b) dependence of the normalized fiber core radius k0a for different (e1,m1) and fixed e2 ¼1,m2 ¼ 1.
Fig. 3. The normalized energy flux Sz for k0a ¼1.5, b/k0 ¼1.360, where e1 ¼ 2,m1 ¼2, e2 ¼ 1,m2 ¼ 1. (a) TM mode and (b) TE mode.
J. Dong et al. / Optics Communications 291 (2013) 232–237
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Fig. 4. The normalized energy flux Sz for m ¼1 surface wave mode, where e1 ¼ 2.5,m1 ¼1.6, e2 ¼ 1,m2 ¼ 1. (a) k0a ¼1.5, b/k0 ¼ 1.572, (b) k0a¼1.48, b/k0 ¼1.352, (c) k0a¼ 1.61, b/k0 ¼ 1.124 and (d) k0a¼ 1.5, b/k0 ¼ 1.302.
Fig. 5. The normalized propagation constant b/k0 (a) and the normalized power Pz (b) dependence of the normalized fiber core radius k0a for different (e2,m2) and fixed e1 ¼ 1,m1 ¼1.
e1 ¼ 2,m1 ¼2, e2 ¼1,m2 ¼ 1. The sign of Sz is opposite for the TM mode and TE mode. Sz have absolute maxima in the cladding at the interface. When e1 ¼ 2.5,m1 ¼1.6, for m¼1 surface wave mode, Sz is always positive in the core and negative in the cladding for upper branch of the dispersion curve. Fig. 4(a) shows Sz distribution for k0a¼ 1.5, b/k0 ¼1.572. However, for lower branch of the dispersion curve, Sz distribution is quite distinct. For k0a¼1.48, b/k0 ¼1.352, Sz is positive in the core, and positive near the interface and negative away from the interface in the cladding (Fig. 4(b)). Sz has a maximum in the core at the interface. For k0a¼ 1.61, b/k0 ¼1.124, Sz is positive in the cladding, and negative near the interface and positive away from the interface in the core (Fig. 4(c)). Sz has a maximum in the cladding at the interface. It is a novel feature that Sz changes sign (sign-varying energy flux) in the core or cladding region. This phenomenon has also been observed in the DNG circular waveguide [18]. For k0a¼ 1.5, b/k0 ¼1.302, Sz is positive both in the core and cladding
(Fig. 4(d)). Thus under this condition, m¼ 1 surface wave mode is the totally forward wave. This fact can also be seen from Fig. 2(b) where Pz ¼1 at k0a ¼1.5. When e1 ¼ 1.6,m1 ¼ 2.5, it can be found that Sz is negative both in the core and cladding for k0a¼1.5, b/k0 ¼1.302. Thus under this condition, m ¼1 surface wave mode is the totally backward wave. 9e1 m1 9 o 9e2 m2 9
ðiiÞ
when keeping e1 ¼ 1,m1 ¼1 and e2m2 ¼ 4 unchanged, the normalized propagation constants b/k0 and the normalized power Pz of surface wave modes versus the normalized fiber core radius k0a for different relative permittivity and permeability in the cladding (e2,m2) ¼(1.6, 2.5),(1, 4),(0.5, 8) (m¼0) or (0.8, 5) (m¼1) are presented in Fig. 5. It can be seen from Fig. 5(a) that for TM modes, the dispersion curves are quite distinct for different e2,m2. TM modes have cutoff normalized fiber core radii k0a ¼3.726, 1.440, 0.698 for (e2,m2)¼ (1.6, 2.5), (1, 4), (0.5, 8), respectively. There also exist zero power for m ¼1 surface wave mode when
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J. Dong et al. / Optics Communications 291 (2013) 232–237
e2 ¼0.8, m2 ¼ 5 (C point in Fig. 5(b)). 9e1 m1 9 ¼ 9e2 m2 9
ðiiiÞ
when keeping e1 ¼ 4, m1 ¼ 1 and e2m2 ¼ 4 unchanged, the normalized propagation constants b/k0 of surface wave modes versus the normalized fiber core radius k0a for different relative permittivity and permeability in the cladding (e2,m2)¼(2.5, 1.6), (2, 2), (1.6, 2.5), (1, 4), (0.5, 8) are presented in Fig. 6. For this case, characteristic Eq. (3) has no solution for m ¼1, modes exist only for m¼0. b/k0 decrease with k0a increases for TM modes. The cutoff normalized fiber core radii are k0a ¼ 1.175, 0.834, 0.641, 0.409, 0.236 for (e2,m2)¼(2.5, 1.6),(2, 2), (1.6, 2.5),(1, 4),(0.5, 8). The normalized power is positive and almost the same for different fiber core radii. 3.3. Frequency dispersion The single-negative metamaterials are innately frequency dispersive. Here we present the numerical results for the MNG–ENG fiber where the core and cladding are MNG and ENG materials, respectively. The results for the ENG–MNG fiber are similar except that the TE and TM modes are exchanged. We adopt the Drude models for the dielectric permittivity and magnetic permeability with frequency dependence [12]:
e2 ðoÞ ¼ 1
2 ep 2 i
o
o
oGe
, m1 ðoÞ ¼ 1
2 mp 2 i G m
o
o
o
ð13Þ
where oep,omp are the electric and magnetic plasma frequencies respectively, Ge,Gm are their damping rates. In the following calculation, we assume Ge ¼0,Gm ¼ 0, that means we neglect loss
for simplicity; e1 ¼2, m2 ¼1, the electric and magnetic plasma frequencies are fep ¼ oep/2p ¼10 GHz and fmp ¼ omp/2p ¼8 GHz, respectively. In this case, the frequency range in which both m1 and e2 are negative extends up to 8 GHz. Fig. 7 shows the normalized propagation constants b/k0 and the normalized power Pz of lower-order (m ¼0,1) surface wave modes as a function of frequency f in the MNG–ENG fiber where fiber core radius a ¼5 mm. Solid and dashed lines correspond to m¼0 (TM, TE) and m¼1 modes. For the TM mode (right-sided line), b/k0 decreases with f increases. The slope of the curve for the TM mode is negative with a high cutoff frequency (8.820 GHz). Pz of the TM mode is negative, thus the TM mode is a backward wave. It is noted that the TM mode can exist even if f 48 GHz (in this frequency range, m1 4 0 in the core, only cladding is ENG material). This result is consistent with reference [26]. For the TE mode (left-sided line), b/k0 increases with f increases. The slope of the curve for the TE mode is positive with a low cutoff frequency (3.829 GHz). Pz of the TE mode is positive, thus the TE mode is a forward wave. The energy flux Sz of the TM mode is positive in the core and negative in the cladding. Sz of the TE mode is negative in the core and positive in the cladding. For m¼1 surface wave mode, there exist two curves. b/k0 has double values in some frequency ranges (5.132–5.208 GHz, and 6.069–6.394 GHz). Pz is negative for the negative slope curve and positive for the positive slope curve. At the critical points (C1, 5.132 GHz, C2, 6.394 GHz), Pz are zeros, i.e., the electromagnetic wave can also be stored in the dispersive SNG fiber. Sz of m ¼1 mode is positive in the core and negative in the cladding for the right-sided curve. Sz of m¼1 mode is negative in the core and positive in the cladding for the left-sided curve. It can be seen from Fig. 7 that the TE and TM modes are separated in the MNG–ENG fiber, single mode operation can be achieved in the lower frequency range 3.829–5.132 GHz (TE mode) and higher frequency range 6.394–8.820 GHz (TM mode). It is very interesting that in the around frequency range 5.0–6.3 GHz, the curves of m¼0 (TE, TM) and m¼1 surface wave modes are very steep, and b/k0 are very large, Pz close to zero. These features may have potential applications such as high-sensitivity sensor and slowing or even storing electromagnetic waves. It is noted that if the loss is considered in the SNG materials, i.e., when Ge a0, Gm a0, there will no critical point of the dispersion curve and no zero power. However, the loss may be compensated by introducing gain in the SNG materials [15].
4. Conclusion Fig. 6. The normalized propagation constant b/k0 of surface wave modes (m¼0, TM) dependence of the normalized fiber core radius k0a for different (e2,m2) and fixed e1 ¼ 4,m1 ¼1.
The properties of surface wave modes in two types of SNG fibers consisting of (I) ENG core and MNG cladding, (II) MNG core
Fig. 7. Frequency dispersion curves (a) and the normalized power Pz (b) of lower-order (m ¼0,1) surface wave modes in the MNG–ENG fiber where fiber core radius a¼ 5 mm.
J. Dong et al. / Optics Communications 291 (2013) 232–237
and ENG cladding have been investigated. The characteristic equation of the surface wave mode and equations of cutoff frequencies for TE and TM surface wave modes are given. When the fiber core radius becomes large enough, the formulas of the propagation constants of the TE and TM surface wave modes in the planar interface between ENG and MNG materials are obtained. The geometric and frequency dispersion of the propagation constants of the lower-order (m¼ 0,1) surface wave modes are examined. Three kinds of electromagnetic parameters of the geometric dispersion are considered for the ENG–MNG fiber. The dispersion curves, energy flux and power of surface wave modes are presented. Some interesting properties of m¼1 surface wave mode such as totally backward wave, totally forward wave, zero power, sign-varying energy flux in the core region or cladding region, are found. Using the Drude models, the frequency dispersion has been studied for the MNG–ENG fiber. It is shown that the TM mode is a backward wave while the TE mode is a forward wave. The TE and TM single mode operation can be achieved. m ¼1 surface wave mode also has zero power points. The results presented here will be helpful for potential applications in novel waveguide devices.
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Surface wave modes in single-negative metamaterials fibers Jianfeng Dong n, Jinjing Liu, Xiaoyang Luo Institute of Optical Fiber Communication and Network Technology, Ningbo University, Ningbo 315211, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 September 2012 Received in revised form 28 October 2012 Accepted 12 November 2012 Available online 3 December 2012
The features of surface wave modes in the fiber composed of single-negative (SNG) metamaterials have been investigated. The geometric and frequency dispersion of surface wave modes in SNG fibers are considered. The dispersion curves, energy flux and power of lower-order (m¼ 0,1) surface wave modes are presented. Some interesting properties, such as totally backward or forward wave, zero power, signvarying energy flux in the core or cladding region, single mode operation, are found. & 2012 Elsevier B.V. All rights reserved.
Keywords: Surface wave mode Dispersion curve Energy flux Single-negative metamaterial Fiber
1. Introduction The waveguides made from novel electromagnetic materials (metamaterials) have attracted much attention and lots of papers on this topic have been published recently. Metamaterials are generally referred to as materials or structures that exhibit unique electromagnetic properties not usually found in nature. Double-negative (DNG) materials, also called left-handed materials, negative refractive index materials, in which the permittivity and permeability are negative simultaneously, are a class of metamaterials [1]. DNG materials have many unique properties such as negative refraction, inverse Doppler effect, etc. [2,3], and they have fascinating potential applications including super resolution, cloaking, and slow light [4–6]. Other interesting class of metamaterials are single-negative (SNG) materials in which only one of the material parameters, permittivity (epsilon) or permeability (mu), is negative. SNG materials can be constructed more easily than DNG materials because SNG materials need only one material parameter negative. These so called single-negative (SNG) materials, i.e., epsilon-negative (ENG) or mu-negative (MNG) materials also possess interesting properties and have many potential applications [7,8]. For example, remarkable features such as resonance, transparency, anomalous tunneling, and zero reflection through layers consisting of SNG materials have been proved theoretically [8–10]. Guided modes in the parallel-plate waveguide filled with a pair of ENG–MNG layers have been investigated theoretically, and monomodal propagation can be achieved in parallel-plate
n
Corresponding author. Fax: þ86 574 87600940. E-mail addresses: [email protected], [email protected] (J. Dong).
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waveguides with arbitrary thickness [11]. Dispersion properties of surface wave modes in a pair of juxtaposed ENG and MNG slabs have been investigated, interesting properties such as simultaneous TE and TM surface wave modes with opposite total energy flow directions are found [12]. The special type of waveguide composed of impedance-matched SNG materials has also been studied, and it supports only a single mode having TE and TM polarization degeneracy but power flow directions of degenerate TE and TM modes are always opposite to each other [13]. Wave propagation in a planar dielectric waveguide covered by the MNG material layers has been considered, and slow light can be achieved at the critical waveguide thickness [14]. Slow and even stopped electromagnetic wave can be supported by an active dielectric waveguide cladded by SNG materials [15]. The cylindrical symmetry is a simple realistic geometry. Due to their geometrical simplicity and important applications, the metamaterial circular waveguides (fibers) containing DNG materials in the core or cladding have been extensively studied [16–24]. The peculiar mode properties of a fiber with DNG core have been found, such as the perfect phase matching of the TE and TM slow modes, sign-varying energy flux, and the existence of TEM modes [16]. Slow-light can be supported in the practical circular waveguide with metamaterial cladding [22]. Guided optical waves in fibers with negative dielectric constant in the core or cladding have also been investigated in the literature [25–29], and the unusual result is the existence of modes with negative group velocity. Moreover, fiber metamaterials with negative permittivity or negative magnetic permeability in the THz frequency band have been fabricated recently [30–32]. However, the surface wave modes in the fiber composed of a pair of SNG materials have not been examined to our knowledge. The surface wave mode has an evanescent behavior both in the
J. Dong et al. / Optics Communications 291 (2013) 232–237
core and cladding and can propagate along the interface between ENG and MNG materials. Because the field maximum is at the interface, the surface wave mode is a very sensitive and convenient tool for studying of the physical properties of the surfaces. Thus the investigations of surface wave modes in SNG fibers are very important. In this paper, the surface wave modes that exist near the interface of the fiber composed of ENG–MNG materials have been investigated. The results of the geometric and frequency dispersion of lower-order surface wave modes in SNG fibers are presented, showing some interesting properties such as totally backward or forward wave, zero power, sign-varying energy flux in the core or cladding region, single mode operation.
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imposing the conditions of continuity of the tangent electromagnetic fields at r ¼a, the characteristic equation of surface wave modes can be obtained as [18]: 2 hm ihe m e2 i b 1 1 2 1^ 1^ Im 2 K^ m Im K^ m ¼ m2 2þ 2 ð3Þ u v u v k0 u v pffiffiffiffiffiffiffiffiffiffi where k ¼ o m e , u ¼k a,v ¼k a, ^I ¼ I’m ðuÞ, K^ ¼ K’m ðvÞ. I ’ ðUÞ, 0
0 0
t1
m
t2
Im ðuÞ
m
K m ðvÞ
m
Km ’ ðUÞstand for differentiation with respect to argument. When m¼ 0, using relations I0’ ðxÞ ¼ I1 ðxÞ and K 0’ ðxÞ ¼ K 1 ðxÞ, the characteristic Eq. (3) reduces to:
m1 I1 ðuÞ
¼
u I0 ðuÞ
m2 K 1 ðvÞ
ð4Þ
v K 0 ðvÞ
which corresponds to the TE mode, and 2. Formulations
e1 I1 ðuÞ
A single-negative metamaterial fiber (SNG fiber) consists of SNG materials in the core and cladding. The relative dielectric permittivity and magnetic permeability of SNG material are e1,m1 in the core and e2,m2 in the cladding; the core radius is a (see Fig. 1). The cladding is supposed to extend infinitely. Assume electromagnetic guided waves propagate along z direction. Two pairs of SNG materials are considered: the ENG–MNG fiber with e1 o0,m1 40; e2 40,m2 o0 (ENG core and MNG cladding) and the MNG–ENG fiber with e1 40,m1 o0; e2 o0,m2 40 (MNG core and ENG cladding). Because e1m1 o0, e2m2 o0, only surface wave modes, whose electromagnetic fields exponentially decay with distance from interface both in the core and cladding, can exist. This is distinct from DNG fibers where the ordinary guided modes with an oscillatory behavior within the core can also exist [16]. Following the standard procedure, in the cylindrical coordinate system (r, j, z), the longitudinal electromagnetic field components of surface wave modes in SNG fibers can be derived as following: In the core (0rr ra): " # Ez1 A ¼ ð1Þ Im ðkt1 r Þeimj eibz Hz1 B In the cladding (r Za): " # Ez2 C ¼ K m ðkt2 r Þeimj eibz Hz2 D
ð2Þ
where A, B, C, D are unknown constants, Im(U) Km(U) are the first and second modified Bessel functions, m is an integer specifying the azimuthal field dependence, b is the longitudinal propagation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 constant. kt1 ¼ b o2 m0 e0 m1 e1 , kt2 ¼ b o2 m0 e0 m2 e2 are the transverse attenuation factors in the core and cladding, respectively. The transverse components of the electromagnetic fields can be derived from the relationships between longitudinal and transverse components of the electromagnetic fields. Then,
u I0 ðuÞ
¼
e2 K 1 ðvÞ
ð5Þ
v K 0 ðvÞ
which corresponds to the TM mode.It is shown from Eqs. (4) and (5) that the cutoff frequencies of TE and TM surface wave modes (when b ¼0) satisfy following equations respectively: sffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi m1 I1 e1 m1 k0 a m2 K 1 e2 m2 k0 a pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð6Þ e I e K e1 m1 k0 a e2 m2 k0 a 1 0 2 0 sffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi e1 I1 e1 m1 k0 a e K e2 m2 k0 a pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi m I m K e m k a e m k a 1
0
1
1 0
0
2
2
ð7Þ
2 0
when the fiber core radius is large enough, i.e., k0a-N, applying properties of modified Bessel functions, I1(x)¼I0(x), K1(x)¼K0(x) for large x in Eqs. (4) and (5), we can obtain normalized propagation constants for the planar interface between ENG and MNG materials: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m21 e2 m2 m22 e1 m1 b ð8Þ ¼ k0 m21 m22 for the TE mode, and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e21 m2 e2 e22 m1 e1 b ¼ k0 e21 e22
ð9Þ
for the TM mode. It is noted that similar formulas have been found in the single-negative or double-negative metamaterial waveguides [24,33,34]. The energy flux of the mode along the z-axis in the fiber can be derived by: 1 1
ð10Þ Sz ¼ Re E Hn z^ ¼ Re Er Hnj Ej Hnr 2 2 The normalized power of the mode is defined as [16] Pz ¼
P z1 þP z2 9P z1 9 þ9Pz2 9
ð11Þ
where Pz1 and Pz2 are the power of the mode in the core and cladding respectively: Z a Z 2p Z a Pz1 ¼ rSz1 drdj ¼ 2p rSz1 dr ð12aÞ 0
Pz2 ¼
0
Z 2p Z 0
0 1
rSz2 drdj ¼ 2p a
Z
1
rSz2 dr
ð12bÞ
a
3. Numerical results and discussion
Fig. 1. Geometry and material parameters of a single-negative metamaterial fiber.
The longitudinal propagation constant b of the surface wave mode is a function of the fiber core radius (geometric dispersion) and of the frequency through the SNG material’s dispersion relation (frequency dispersion). The characteristic Eqs. (3)–(5)
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are nonlinear equations of b, and can be solved by Newton– Raphson method. After the b has been obtained, it is easy to calculate the energy flux and power of the surface wave mode. In the characteristic equations, the single unknown variable is b2.Thus, the sign of b can be positive or negative. Here we choose b positive. Note that if we change the sign of b, the energy flux and power also change signs. In this section, we will give numerical results for the geometric dispersion and frequency dispersion of lower-order (m¼ 0,1) surface wave modes. 3.2. Geometric dispersion Here we consider properties of surface wave modes varying with the fiber core radius for a fixed frequency. In the conventional fibers, guided modes can exist only for condition e1m1 4 e2m2, however, in the SNG fibers, there is no restriction for e1m1 and e2m2. We will discuss three cases: (i) 9e1m1949e2m29, (ii) 9e1m19o9e2m29 and (iii) 9e1m19¼9e2m29. Only results of the ENG–MNG fiber are presented, the results of the MNG–ENG fiber are similar. 9e1 m1 9 4 9e2 m2 9
ðiÞ
when keeping e2 ¼1,m2 ¼ 1 and e1m1 ¼ 4 unchanged, the normalized propagation constants b/k0 and the normalized power Pz of lower-order (m¼0,1) surface wave modes versus the normalized fiber core radius k0a for different relative permittivity and permeability in the core (e1,m1)¼( 1.6,2.5), ( 2,2), ( 2.5,1.6), (4,1), ( 8,0.5) are presented in Fig. 2. Solid and dashed lines correspond to m¼0 (TM) and m¼1 modes (TE modes are not shown in Fig. 2). It is noted that TM modes exist only for 9e1/e2941. It can be seen from Fig. 2 that for TM modes, as k0a increases, b/k0 decrease and Pz increase monotonously. The bigger 9e19, the bigger Pz. For (e1,m1)¼(1.6,2.5), ( 2,2), TM modes have no cutoff normalized fiber core radii. For (e1,m1)¼( 2.5,1.6), (14,1), ( 8,0.5), the cutoff
normalized fiber core radii are k0a¼3.187, 1.034, 0.502, which satisfy Eq. (7) and decrease as 9e19 increases. This means that for larger 9e19, only below cutoff fiber core radius, the TM mode can propagate. It is obvious from Eqs. (4) and (5) that when e1m1 and e2m2are unchanged, for the TE mode, b/k0 is the same as that for the TM mode if m1/m2 ¼ e1/e2. However, Pz is exactly opposite, i.e., for the TE mode, Pz is negative and the absolute value is the same as that for the TM mode. Pz is negative means that the power flow is opposite to wave vector. When e1 ¼ 2, p m1ffiffiffiffiffiffiffiffiffiffiffiffi ¼2, ffiimpedances pffiffiffiffiffiffiffiffiffiffiffiffiffi of ENG and MNG materials are matched, i.e. m1 =e1 ¼ m2 =e2 , TM and TE modes are degenerate, but their power flow are opposite to each other. This property has also been found in the SNG slab waveguide, and has potential application in the polarization beam splitter [13]. When e1 ¼ 2.5,m1 ¼1.6, for m¼1 surface wave mode, the dispersion curve splits into two branches from the critical point C (k0a ¼1.459). The upper branch has no high cutoff normalized fiber core radius, while the lower branch has high cutoff normalized fiber core radius k0a¼3.431. In the range of k0a ¼1.459– 3.431, b/k0 has double values. Two branches of the dispersion curve have different normalized power Pz. Pz is positive for lower branch of the dispersion curve and negative for upper one (Fig. 2(b)). At the critical point C, Pz is zero. Zero power has been found in the slab [15] or circular waveguides [16,22] containing metamaterials. This implies that for appropriate electromagnetic parameters and size of the fiber, the SNG fiber can also be used to store light [15]. It can be shown from Eq. (3) that when e1m1 and e2m2 are unchanged, dashed line in Fig. 2(a) also corresponds to m¼1 surface wave mode with e1 ¼ 1.6,m1 ¼2.5. However, the sign of Pz is also exactly opposite to e1 ¼ 2.5,m1 ¼1.6 case. Fig. 3(a and b) show the normalized energy flux Sz distribution of the TM mode and TE mode for k0a¼1.5, b/k0 ¼1.360, where
Fig. 2. The normalized propagation constant b/k0 (a), and the normalized power Pz and (b) dependence of the normalized fiber core radius k0a for different (e1,m1) and fixed e2 ¼1,m2 ¼ 1.
Fig. 3. The normalized energy flux Sz for k0a ¼1.5, b/k0 ¼1.360, where e1 ¼ 2,m1 ¼2, e2 ¼ 1,m2 ¼ 1. (a) TM mode and (b) TE mode.
J. Dong et al. / Optics Communications 291 (2013) 232–237
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Fig. 4. The normalized energy flux Sz for m ¼1 surface wave mode, where e1 ¼ 2.5,m1 ¼1.6, e2 ¼ 1,m2 ¼ 1. (a) k0a ¼1.5, b/k0 ¼ 1.572, (b) k0a¼1.48, b/k0 ¼1.352, (c) k0a¼ 1.61, b/k0 ¼ 1.124 and (d) k0a¼ 1.5, b/k0 ¼ 1.302.
Fig. 5. The normalized propagation constant b/k0 (a) and the normalized power Pz (b) dependence of the normalized fiber core radius k0a for different (e2,m2) and fixed e1 ¼ 1,m1 ¼1.
e1 ¼ 2,m1 ¼2, e2 ¼1,m2 ¼ 1. The sign of Sz is opposite for the TM mode and TE mode. Sz have absolute maxima in the cladding at the interface. When e1 ¼ 2.5,m1 ¼1.6, for m¼1 surface wave mode, Sz is always positive in the core and negative in the cladding for upper branch of the dispersion curve. Fig. 4(a) shows Sz distribution for k0a¼ 1.5, b/k0 ¼1.572. However, for lower branch of the dispersion curve, Sz distribution is quite distinct. For k0a¼1.48, b/k0 ¼1.352, Sz is positive in the core, and positive near the interface and negative away from the interface in the cladding (Fig. 4(b)). Sz has a maximum in the core at the interface. For k0a¼ 1.61, b/k0 ¼1.124, Sz is positive in the cladding, and negative near the interface and positive away from the interface in the core (Fig. 4(c)). Sz has a maximum in the cladding at the interface. It is a novel feature that Sz changes sign (sign-varying energy flux) in the core or cladding region. This phenomenon has also been observed in the DNG circular waveguide [18]. For k0a¼ 1.5, b/k0 ¼1.302, Sz is positive both in the core and cladding
(Fig. 4(d)). Thus under this condition, m¼ 1 surface wave mode is the totally forward wave. This fact can also be seen from Fig. 2(b) where Pz ¼1 at k0a ¼1.5. When e1 ¼ 1.6,m1 ¼ 2.5, it can be found that Sz is negative both in the core and cladding for k0a¼1.5, b/k0 ¼1.302. Thus under this condition, m ¼1 surface wave mode is the totally backward wave. 9e1 m1 9 o 9e2 m2 9
ðiiÞ
when keeping e1 ¼ 1,m1 ¼1 and e2m2 ¼ 4 unchanged, the normalized propagation constants b/k0 and the normalized power Pz of surface wave modes versus the normalized fiber core radius k0a for different relative permittivity and permeability in the cladding (e2,m2) ¼(1.6, 2.5),(1, 4),(0.5, 8) (m¼0) or (0.8, 5) (m¼1) are presented in Fig. 5. It can be seen from Fig. 5(a) that for TM modes, the dispersion curves are quite distinct for different e2,m2. TM modes have cutoff normalized fiber core radii k0a ¼3.726, 1.440, 0.698 for (e2,m2)¼ (1.6, 2.5), (1, 4), (0.5, 8), respectively. There also exist zero power for m ¼1 surface wave mode when
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J. Dong et al. / Optics Communications 291 (2013) 232–237
e2 ¼0.8, m2 ¼ 5 (C point in Fig. 5(b)). 9e1 m1 9 ¼ 9e2 m2 9
ðiiiÞ
when keeping e1 ¼ 4, m1 ¼ 1 and e2m2 ¼ 4 unchanged, the normalized propagation constants b/k0 of surface wave modes versus the normalized fiber core radius k0a for different relative permittivity and permeability in the cladding (e2,m2)¼(2.5, 1.6), (2, 2), (1.6, 2.5), (1, 4), (0.5, 8) are presented in Fig. 6. For this case, characteristic Eq. (3) has no solution for m ¼1, modes exist only for m¼0. b/k0 decrease with k0a increases for TM modes. The cutoff normalized fiber core radii are k0a ¼ 1.175, 0.834, 0.641, 0.409, 0.236 for (e2,m2)¼(2.5, 1.6),(2, 2), (1.6, 2.5),(1, 4),(0.5, 8). The normalized power is positive and almost the same for different fiber core radii. 3.3. Frequency dispersion The single-negative metamaterials are innately frequency dispersive. Here we present the numerical results for the MNG–ENG fiber where the core and cladding are MNG and ENG materials, respectively. The results for the ENG–MNG fiber are similar except that the TE and TM modes are exchanged. We adopt the Drude models for the dielectric permittivity and magnetic permeability with frequency dependence [12]:
e2 ðoÞ ¼ 1
2 ep 2 i
o
o
oGe
, m1 ðoÞ ¼ 1
2 mp 2 i G m
o
o
o
ð13Þ
where oep,omp are the electric and magnetic plasma frequencies respectively, Ge,Gm are their damping rates. In the following calculation, we assume Ge ¼0,Gm ¼ 0, that means we neglect loss
for simplicity; e1 ¼2, m2 ¼1, the electric and magnetic plasma frequencies are fep ¼ oep/2p ¼10 GHz and fmp ¼ omp/2p ¼8 GHz, respectively. In this case, the frequency range in which both m1 and e2 are negative extends up to 8 GHz. Fig. 7 shows the normalized propagation constants b/k0 and the normalized power Pz of lower-order (m ¼0,1) surface wave modes as a function of frequency f in the MNG–ENG fiber where fiber core radius a ¼5 mm. Solid and dashed lines correspond to m¼0 (TM, TE) and m¼1 modes. For the TM mode (right-sided line), b/k0 decreases with f increases. The slope of the curve for the TM mode is negative with a high cutoff frequency (8.820 GHz). Pz of the TM mode is negative, thus the TM mode is a backward wave. It is noted that the TM mode can exist even if f 48 GHz (in this frequency range, m1 4 0 in the core, only cladding is ENG material). This result is consistent with reference [26]. For the TE mode (left-sided line), b/k0 increases with f increases. The slope of the curve for the TE mode is positive with a low cutoff frequency (3.829 GHz). Pz of the TE mode is positive, thus the TE mode is a forward wave. The energy flux Sz of the TM mode is positive in the core and negative in the cladding. Sz of the TE mode is negative in the core and positive in the cladding. For m¼1 surface wave mode, there exist two curves. b/k0 has double values in some frequency ranges (5.132–5.208 GHz, and 6.069–6.394 GHz). Pz is negative for the negative slope curve and positive for the positive slope curve. At the critical points (C1, 5.132 GHz, C2, 6.394 GHz), Pz are zeros, i.e., the electromagnetic wave can also be stored in the dispersive SNG fiber. Sz of m ¼1 mode is positive in the core and negative in the cladding for the right-sided curve. Sz of m¼1 mode is negative in the core and positive in the cladding for the left-sided curve. It can be seen from Fig. 7 that the TE and TM modes are separated in the MNG–ENG fiber, single mode operation can be achieved in the lower frequency range 3.829–5.132 GHz (TE mode) and higher frequency range 6.394–8.820 GHz (TM mode). It is very interesting that in the around frequency range 5.0–6.3 GHz, the curves of m¼0 (TE, TM) and m¼1 surface wave modes are very steep, and b/k0 are very large, Pz close to zero. These features may have potential applications such as high-sensitivity sensor and slowing or even storing electromagnetic waves. It is noted that if the loss is considered in the SNG materials, i.e., when Ge a0, Gm a0, there will no critical point of the dispersion curve and no zero power. However, the loss may be compensated by introducing gain in the SNG materials [15].
4. Conclusion Fig. 6. The normalized propagation constant b/k0 of surface wave modes (m¼0, TM) dependence of the normalized fiber core radius k0a for different (e2,m2) and fixed e1 ¼ 4,m1 ¼1.
The properties of surface wave modes in two types of SNG fibers consisting of (I) ENG core and MNG cladding, (II) MNG core
Fig. 7. Frequency dispersion curves (a) and the normalized power Pz (b) of lower-order (m ¼0,1) surface wave modes in the MNG–ENG fiber where fiber core radius a¼ 5 mm.
J. Dong et al. / Optics Communications 291 (2013) 232–237
and ENG cladding have been investigated. The characteristic equation of the surface wave mode and equations of cutoff frequencies for TE and TM surface wave modes are given. When the fiber core radius becomes large enough, the formulas of the propagation constants of the TE and TM surface wave modes in the planar interface between ENG and MNG materials are obtained. The geometric and frequency dispersion of the propagation constants of the lower-order (m¼ 0,1) surface wave modes are examined. Three kinds of electromagnetic parameters of the geometric dispersion are considered for the ENG–MNG fiber. The dispersion curves, energy flux and power of surface wave modes are presented. Some interesting properties of m¼1 surface wave mode such as totally backward wave, totally forward wave, zero power, sign-varying energy flux in the core region or cladding region, are found. Using the Drude models, the frequency dispersion has been studied for the MNG–ENG fiber. It is shown that the TM mode is a backward wave while the TE mode is a forward wave. The TE and TM single mode operation can be achieved. m ¼1 surface wave mode also has zero power points. The results presented here will be helpful for potential applications in novel waveguide devices.
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