1
An Extended Equivalent Circuit Based FDTD Scheme for the Efficient Simulation of Composite Right/Left-Handed Metamaterials A. Rennings1, S. Otto2, A. Lauer2, C. Caloz3, P. Waldow2 1
Duisburg-Essen University, Bismarkstr. 81, D-47048 Duisburg, Germany Phone +49-203-379-3183,
[email protected] 2 IMST GmbH, Carl-Friedrich-Gauß Str. 2, D-47475 Kamp-Lintfort, Germany 3 École Polytechnique de Montréal, 2900 Édouard-Montpetit, Montréal (Québec) H3T 1J4, Canada
Abstract— A novel formulation of the well-known finitedifference time-domain (FDTD) scheme is proposed. This formulation consists in an extension of the equivalent circuit (EC) description of the Yee cell including the left-handed (LH) series capacitance CL and shunt inductance LL in addition to the right-handed (RH) series inductance LR and shunt capacitance CR, so as to model general composite right/left-handed (CRLH) transmission line (TL) metamaterials (MTMs). The resulting extended EC (EEC) FDTD tool is demonstrated to successfully analyze structures fully or partially loaded with 3D CRLH TL MTMs in two specific examples. First, the EEC FDTD is applied to the 1D case of a partially CRLH MTM loaded parallel-plate waveguide (PPWG) structure, where recently discovered phenomena such as the dual existence of a lowerfrequency LH band and higher-frequency RH band, antiparallel phase and group velocities and transition frequency infinite wavelength mode propagation are verified and illustrated. For this 1D case, the effective material parameters of a homogeneous TL are extracted and compared with analytical formulas. Next, the EEC FDTD is applied to a patch antenna fully loaded by a CRLH MTM substrate, where so-called dual LH-negative and RHpositive resonances are observed and improvements in terms of miniaturization are predicted. Index Terms— Extended equivalent circuit (EEC), FDTD, metamaterial (MTM), composite right/left-handed (CRLH), transmission line (TL) modeling.
I. INTRODUCTION Originally the fruit of a theoretical speculation by Veselago [1], left-handed (LH) metamaterials (MTMs), as well as other related exotic effectively homogeneous structures, were firstly implemented under the form of resonant electromagnetic particles [2]-[4] and later also under the form of nonresonant transmission line (TL) MTMs exhibiting broader bandwidth and lower insertion loss [5]-[7]. Both of these approaches have quickly matured and a number of novel microwave components, antennas and structures were subsequently developed [8][10]. In 2006, just five years after their real start with the first experimental demonstration by Smith et al. [4], MTMs are widely recognized as representing a new paradigm in engineering and physics, and may lead to revolutionary microwave and optical applications in the future. Several frequency-domain and time-domain numerical
techniques have been used to investigate LH MTMs. Time-domain approaches provide particularly useful insight into the formation of steady waveforms and unusual LH phenomena such as backward-wave propagation and negative refractive index. However, they require inclusion of frequency dispersion, which is an inherent property of these media [1] arising as a consequence of the fundamental entropy law [11]. In typical FDTD schemes, phenomenological Drude/Lorentz-model expressions of the permittivity and permeability are introduced ab initio in Maxwell’s equations before the finite-differences are formed [12, 13]. Alternatively, Kokkinos et al. modeled 2D dispersive MTMs of infinite extent by application of periodic boundary conditions at the edges of a material unit cell constituted by LH lumped elements embedded in a RH host medium discretized with a sufficient number of Yee cells [13]. In the same publication the authors applied their technique also to structures with finite extent where the computational domain was truncated by resistors to absorb the incoming wave. In this contribution, we propose an alternative extended equivalent circuit (EEC) FDTD formulation, where LH series capacitance CL and shunt inductance LL have been added to the conventional RH series inductance LR and shunt capacitance CR. This formulation models thus a composite right/left-handed (CRLH) transmission line (TL) MTM [9], which may be regarded as the most general and most fundamental transmission material possible. In contrast to what is done in the formulation of [13], here the unit cell of the MTM is modeled by only one Yee cell. Drude-model equivalent expressions of the constitutive parameters are then obtained as a natural result of the introduction of the CRLH passive circuit elements in the homogeneous limit (where the electrical size of the cells become infinitely small). This EEC FDTD approach, although very different in its mathematical form, is essentially equivalent to the expanded 3D TLM node recently pioneered and applied to LH media [14] and CRLH MTMs [9] by Hoefer et al. The most fundamental differences between these two FDTD and TLM extended schemes are the same as those existing in the conventional case: the former updates the electric and
2
magnetic field components on a staggered grid while the latter expresses these total field components in terms of incident and reflected voltage impulses; in addition, the latter, due to the necessity to define a characteristic impedance, requires twice the number of unknowns of the former. The paper is organized as follows: Sec. II establishes, from an integral expression of Maxwell’s equation, an initial EC FDTD scheme including only RH components. Sec. III extends this scheme to the CRLH-EEC FDTD scheme, which includes in addition the LH components; first this scheme is established for the 1D case in a parallel-plate waveguide (PPWG) structure and subsequent simulation results show several CRLH MTM phenomena; next, the dispersive constitutive parameters of the CRLH are derived and shown to follow a Drude model in the homogeneous limit; finally, the scheme is generalized to the 3D case. The specific example of a patch antenna on a CRLH substrate is presented in Sec. IV as an illustration of the proposed CRLH-EEC FDTD scheme; in addition, this section verifies the existence of the LH-band and RH-band paired resonances and describes their radiation mechanisms. Finally, conclusions are given in Sec. V. II. EQUIVALENT CIRCUIT (EC) FDTD SCHEME The proposed EC based FDTD scheme is derived from an approximation of Maxwell’s equations in integral form including electric (σe ) and magnetic losses (σm ) and a current source (soft excitation)
G e2
Δ0
Δ1
G e1 G e0
Δ2
H0 E2
H1 H2
E1 E0
Δ2
Δ1
Δ0
Fig. 1. Unit cell of Yee’s scheme at a discrete position described by a index vector X = (X0, X1, X2)T
Fig. 1 depicts the Yee unit cell of the well-known FDTD scheme, with the field components, cell dimensions and constitutive parameters located by the index vector X = (X0, X1, X2)T ∈ N . The relative permittivity εr and the electric conductivity σe are approximated by blocks with constant material parameters within the main grid subcells, while the relative permeability μr and the magnetic conductivity σm (needed for absorbing boundaries) are approximated by constant material blocks within the dual grid subcells, as depicted in Fig. 2. All of the parameters (material constitutive parameters, cell dimensions and later on also lumped elements) corresponding to the dual mesh are overlined. 3
G G G G G G G src G d v∂∫A H ⋅ ds = + dt ∫∫A εE ⋅ dA + ∫∫A σe E ⋅ dA − ∫∫A J ⋅ dA, G G G G G G d (2) v∫ E ⋅ ds = − ∫∫ μH ⋅ dA − ∫∫ σ m H ⋅ dA. dt ∂A A A (1)
G e2
Δ0
Δ1
G e1 G e0
εr
By defining the normalized fields
Δ2
σe
G G G G E′ = ε 0 E and H′ = μ 0 H,
G G G G G G 1 d ε r E′ ⋅ dA + Z0 ∫∫ σe E′ ⋅ dA − μ 0 ∫∫ J src ⋅ dA, ∫∫ dt 0 ∂A A A A G 1 G G G G G 1 d (4) v∫ E′ ⋅ ds = − μ r H′ ⋅ dA − ∫∫ σ m H′ ⋅ dA, c0 dt ∫∫ Z0 A ∂A A (3)
G
μr
Δ2
these relations change to G
v∫ H′ ⋅ ds = + c
where c0 = 1/ ε0 μ 0
and Z0 = μ 0 / ε0 .
Normalization of the field components in (3) and (4) yields the same unit for both primed fields, resulting in a better comparability of the fields in terms of the amount of electric and magnetic energy in the computational domain. Moreover, normalization benefits to numerical processing since the updated primed fields are represented by values in the same range.
σm
Δ0
Δ1
Fig. 2. Magnetic and electric subcell form a complete unit cell at X = (X0, X1, X2)T – material parameters are approximated by constant values within these blocks
For the sake of simpler and compacter notation, component and direction indices are numbered in a modulo-three sense (for the 3D case), e.g. E0 ≡ Ex , E1 ≡ Ey , E2 ≡ Ez and E3 ≡ Ex , E-1 ≡ Ez [16]. For further compactness, we introduce the spatial shift operator S and its inverse S -1 defined by the properties G G G (5) Sdm ξ = Sdm ξ X = ξ X + med , d, ν ∈ {0, 1, 2}, m, n ∈ ], G G G G (6) Sdm D Sνn ξ = Sdm ⎡⎣Sνn [ ξ ]⎤⎦ X = ξ X + neν + med
( ) (
{
}
) ( ) (
)
3
G (7) 1 + Sdm [ ξ Δ d Δ ν ] = 1 + Sdm [ ξ Δ d Δ ν ] X = G G G ξ X Δ d ( X d ) Δ ν ( X ν ) + ξ X − med Δ d ( X d − m ) Δ ν ( X ν )
{ ( )
}
{ (
( )
}
)
where ξ represents a scalar function depending on the index vector X (the discretization Δμ in Eq. (7) is only a function of one component Xμ of the index vector X) and where we have also used the three unit index vectors G G T G T T (8) e0 = (1, 0, 0 ) , e1 = ( 0,1, 0 ) and e2 = ( 0, 0,1) . In particular, Eq. (7) shows the compactness attained in the formulas when several shift operations are involved.
laws is used to derive the EC depicted in Fig. 3. Since in Sec. III the EC FDTD will be generalized by adding LH elements to the Yee equations, we have introduced by anticipation the superscript R to designate the RH nature of the reactive lumped elements in Eqs. (12) and (13). For even greater notational compactness, we substitute the inductance and resistance with an impedance and the capacitance and the conductance with an admittance, which yields, for the three directions of space: Yd −1 = G d −1 + jωCdR−1 and Zd −1 = R d −1 + jωLdR−1
G ed −1
The Yee scheme depicted in Fig. 1 has been used for the spatial discretization of (1) and (2). This results in the time continuous Yee scheme equations: (10)
{ } 1 {1 + S + S 4 −1 d
+ε0
{
}
−1 d +1
}
+ Sd−1 D Sd−1+1 [ ε r Δ d Δ d +1 ]
1 d E d −1 Δ d −1 Δ d −1 dt
+
1 1 1 + Sd−1 + Sd−1+1 + Sd−1 D Sd−1+1 [ σe Δ d Δ d +1 ] E d −1 Δ d −1 4 Δ d −1
−
1 1 + Sd−1 + Sd−1+1 + Sd−1 D Sd−1+1 [ Δ d Δ d +1 ] J src d −1 4
{
}
H d +1
Sd vd −1 at
(11) −μ0 −
{Sd − 1} E d +1 Δ d +1 − {Sd +1 − 1} E d Δ d = Δ Δ d 1 {1 + Sd + Sd +1 + Sd D Sd +1} μ r d d +1 H d −1 Δ d −1 4 Δ d −1 dt
1 Δ Δ {1 + Sd + Sd +1 + Sd D Sd +1} σm d d +1 H d −1 Δ d −1 4 Δ d −1
By defining local voltages and currents vd = E d Δ d and i d = H d Δ d (one may call them also electric and magnetic voltages), the following equations are obtained d v d −1 + G d −1 v d −1 − isrc d −1 , dt d (13) {Sd − 1} v d +1 − {Sd +1 − 1} vd = − Ld −1 id −1 − R d −1 i d −1 , dt (12)
Sd Yd −1
{1 − S } i −1 d
d +1
{
}
− 1 − Sd−1+1 i d = + Cd −1
with the lumped elements 1 1 1 + Sd−1 + Sd−1+1 + Sd−1 D Sd−1+1 [ε r Δ d Δ d +1 ] , 4 Δ d −1
{
Cd −1 = ε0 G d −1 =
1 1 1 + Sd−1 + Sd−1+1 + Sd−1 D Sd−1+1 [ σe Δ d Δ d +1 ] , 4 Δ d −1
{
Ld −1 = μ 0 R d −1 =
}
}
Δ Δ 1 {1 + Sd + Sd +1 + Sd D Sd+1} μ r d d +1 and 4 Δ d −1
1 Δ Δ {1 + Sd + Sd+1 + Sd D Sd +1} σm d d +1 . 4 Δ d −1
Eqs. (12) and (13) describe a passive circuit. This set of equations together with Kirchhoff’s voltage and current
G ed
Sd −1Yd +1
id Sd +1isrc d −1 Hd
vd −1
}
{
id +1
Sd −1Yd
d ∈ {0,1, 2}
Sd −1vd +1 at
id
Sd −1vd at
1 − Sd−1 H d +1 Δ d +1 − 1 − Sd−1+1 H d Δ d =
with
Zd +1 4
Zd R d LR = + jω d 4 4 4
Yd +1
H d −1 Zd −1 4
G ed +1
Sd +1vd at Sd +1Yd Sd +1 ⎡⎣G d + jωCdR ⎤⎦
Sd vd +1 at Sd Yd +1
Fig. 3. Equivalent circuit (EC) derived from (12) and (13). The shaded planes correspond to faces of the electric sub cell at the discrete position X = (X0, X1, X2)T. The three magnetic field components Hd are located in the center of these planes.
Fig. 3 shows the EC derived from (12) and (13). The three shaded planes in the figure correspond to three faces of the electric subcell at the discrete position located by the index vector X = (X0, X1, X2)T. The voltages at the edges of the subcell (along the admittances) can be interpreted as the electric field components in the conventional FDTD scheme. The ring currents id that are flowing through the impedances Zd correspond to the magnetic field components Hd in the conventional approach. Without the four ideal (1:1)transformers in each plane the four currents that are flowing from the edges of the plane to its center are not equal in general. The transformers force the four currents to be equal. Thus we have just one ring current in each plane (instead of four single currents) corresponding to the orthogonal magnetic field component located in the center of these planes. We also included one current source modeling a soft excitation at an arbitrary position (here at the right edge of the face in the back of Fig. 3).
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III. CRLH EXTENDED EQUIVALENT CIRCUIT (EEC) FDTD SCHEME A. Theory for the One-Dimensional Case We will now use the TEM fields propagating in a parallel-plate waveguide (PPWG) to derive the EEC based FDTD scheme. The two existing field components E0 and H1 only depend on the waveguide direction X2, which is also the case for the discretization step Δ2 and for the four material constitutive parameters. Therefore shift operators exist only in that direction. The cross section of the PPWG with height h and width w is discretized by two halved electric cells. The top and bottom walls are perfect electric conductors (PECs) while the sidewalls are perfect magnetic conductors (PMCs). The reduced EC appropriate for the PPWG is shown in Fig. 4. R1 j ⎛ R 1 ⎞ + ⎜ ωL1 − ⎟ 4 4⎝ ωC1L ⎠ Z = 1 4
G e0
i1
i0L
v0
v1L 4
Z1 4
Z1 4
H1
Y0 = ⎛ 1 ⎞ G 0 + j ⎜ ωC0R − L ⎟ ω L0 ⎠ ⎝
{
}
}
{
{
}
Δ2 h d ⎛ 1 ⎞ ⎜ μ 0 2 {1 + S2 } μ r w dt H1 w ⎟ ⎟= {S2 − 1} E 0 h = − ⎜ ⎜ + 1 {1 + S } σ Δ 2 h H w ⎟ ⎜ ⎟ 2 m 1 2 w ⎝ ⎠ d (15) {S2 − 1} v0 = − L1 i1 − R1 i1 dt The CRLH extension to the ordinary EC FDTD is straightforward in the 1D case since the CRLH transmission line model can be directly mapped onto the FDTD scheme. At the edge of the electric cell the admittance Y0 is extended by a LH inductance LL in parallel yielding the new admittance R L Y0 = G 0 + j ωC0 − 1/ ωL0 while on the face of the
)
(
G e2
{
Since the voltages v2 and S0v2 are short circuited by the PEC walls, the upper and lower transformer can be removed. That is also the case for the transformer on the left and right of the electric cell because the incoming current on the left is equal to the outgoing current on the right. Consequently, no more transformers exist in the EC, as shown in the simplified model depicted in Fig. 5.
S−21 i1
Z1 2
h S−21 i1
S2−1Δ 2
i1
v0
S2 v0
Y0
S2 Y0
i1
Z1 2
Δ2
)
extended by a LH capacitor C . In the time domain (TD), this extension yields a time integral in each equation
}
(16) − 1 − S2−1 i1 = + C0R
S−21Z1
}
L
Fig. 4. Equivalent circuit (EC) for the PPWG: the upper and lower admittances are short circuited by the PEC
G e2
}
electric cell the impedance (which is equally distributed along the ring structure) Z1 = R1 + j ωL1R − 1/ ωC1L is
G e1
G e0 G e1
{
{
(
S2 Y0 Z1 4
Z1 4
The EC based time continuous Yee scheme for this simplified model is w d ⎛ 1 ⎞ −1 ⎜ ε0 2 1 + S2 [ ε r Δ 2 ] h dt E 0 h ⎟ ⎜ ⎟ 1 w −1 −1 ⎜ − 1 − S2 H1 w = + + 1 + S2 [ σe Δ 2 ] E 0 h ⎟ = ⎜ ⎟ 2 h ⎜ ⎟ 1 ⎜⎜ ⎟⎟ − 1 + S2−1 [ Δ 2 ] wJ src 0 2 ⎝ ⎠ d (14) − 1 − S2−1 i1 = + C0 v0 + G 0 v0 − isrc 0 dt
S2 Δ 2
Fig. 5. Simplified equivalent circuit for the PPWG
t
d 1 v0 + G 0 v0 − isrc v0 dt ′ 0 + dt LL0 ∫0
t
(17) i 0L =
(18)
1 d v dt ′ ⇒ v0 = LL0 i 0L L ∫ 0 L0 0 dt
{S2 − 1} v0 = −L1R
t
d 1 i1 − R1 i1 − L ∫ i1dt ′ dt C1 0
t
(19) v1L =
1 d i dt ′ ⇒ i1 = C1L v1L L ∫ 1 C1 0 dt
Kokkinos et al. [14] investigated a TL material loaded with LH elements. In contrast to our scheme, their approach locates the additional LH elements at the edge of the electric cell and uses a discrete time integration for the LH inductive load; on the other hand, for the capacitive LH contributions no special treatment was necessary in this approach, since the LH capacitor is just added to the RH intrinsic CR (normally the intrinsic one can be neglected compared to the LH lumped capacitor). In contrast to [13] here a two-step approach is used for the two LH elements, as pointed out in (17) and (19). The continuous time derivatives are now replaced by finitedifferences
5
{
}
n
i 0L
n − 0.5 1
(20) − 1 − S−2 1 i
(21) v0 = LL0
(22)
(23)
{S2 − 1} v0
i1
n + 0.5
n + 0.5
n n −1 ⎛ R v0 n − v0 n −1 v + v0 ⎞ + G0 0 ⎜ C0 ⎟ 2 Δt = +⎜ ⎟ ⎜ ⎟ src n − 0.5 L n − 0.5 −i 0 + i0 ⎝ ⎠
− i 0L Δt
n − 0.5
⎛ R i1 ⎜ L1 = −⎜ ⎜ ⎝
n
v1L
= C1L
n +1
− v1L Δt
n + 0.5
− i1 Δt
n − 0.5
+ R1 + v1L
i1
n + 0.5
n
n − 0.5 1
+i 2
⎞ ⎟ ⎟ ⎟ ⎠
here. This update system has been implemented in our FDTD simulator for d ∈ {0, 1, 2} G0 Δt 2 ε0 n n −1 (24) v′0 = v′0 G R 0 Δt C′0 + 2 ε0 C′0R −
(25) i′0L
n + 0.5
n + 0.5 1
(26) i′
= i′0L
n − 0.5
+
(28) L1R = μ 0
1 Δ h {1 + S2 } μ r 2 2 w
(30) C0R = ε 0
1 w 1 1 + S2−1 [ ε r Δ 2 ] ⇒ (31) L0L = 2 2 h 2πf 0sh C0R
{
}
n − 0.5 ⎛ ⎞ 1 − S2−1 i1′ ⎜ ⎟ ⎜ − μ isrc n −0.5 + i′L n −0.5 ⎟ 0 0 0 ⎝ ⎠
c0 Δ t n v′0 L′0L
R1 Δ t 2 μ0 n −0.5 c0 Δ t = i1′ − R Δ R Δ L1′R + 1 t L1′R + 1 t 2 μ0 2 μ0 L1′R −
= v1′L
n −1
+
⎛ {S2 − 1} v′0 n ⎞ ⎜ ⎟ ⎜ + v′ L n ⎟ ⎝ ⎠ 1
c0 Δ t n + 0.5 i1′ C1′L
CdR−1 ε0
(
L1R
)
RH Material CRLH Material
0.5
where
C′dR−1 =
( 2π f )
As a first verification of the proposed approach, the fields within the PPWG loaded with a lossless MTM are analyzed. The first half of the waveguide is filled with air while the second half is loaded with a CRLH material. The left and right ends of the waveguide are terminated by perfectly matched layers (PML) with twelve cells to absorb the incoming waves. A source (soft excitation) is placed at the interface between the PML and the air-filled region. The PPWG is excited at a single frequency fsrc. The resonances fse and fsh are chosen to be 5 GHz = f0 (transition frequency). In the 1D case, the subscripts at the resonance frequencies in Eqs. (29) and (30), indicating the direction, are unnecessary since an anisotropy is not possible. Three different frequencies are used for the excitation: 4.5 GHz, 5.0 GHz and 5.5 GHz. For the first case, the expected LH is observed in Fig. 6 in the transient phase and in Fig. 7 in the steady state. The voltage v0(X2) is plotted for three different time steps to show propagation. The straight line corresponds to the last step, the dashed one to the intermediate time and the chain line to the first step, respectively.
and L′dR−1 =
LdR−1 . μ0
If a CRLH area is to be simulated, Eqs. (25) and (27) are additionally used within the update leap-frog scheme. For a RH region, they are left out and iL together with uL remain zero. The simulation of single-negative (SNG) materials is also possible, in this case either Eq. (25) for positive-ε/negative-μ (MNG) materials or Eq, (27) for negative-ε/positive-μ (ENG) materials are skipped. The
v0 in V
n +1
}
1 se 2 1
B. One-Dimensional Simulation Example: TEM Mode in a PPWG structure
1.0
(27) v1′L
{
⇒ (29) C1L =
For the 2D or 3D case it is possible to define different resonant frequencies depending on the direction. Anisotropic CRLH materials can thus be modeled easily.
n
to obtain the update equations. The final step consists in isolating the latest values in each of the four equations. The already mentioned benefits of the normalization v′d = ε 0 v d and i′d = μ 0 i d in (24) – (27) as also hold
c0 Δ t − G Δ C′0R + 0 t 2 ε0
RH EC elements can be calculated as a function of geometry, material parameter and cell dimensions via Eqs. (28) and (30). The LH EC contributions on the other hand are determined with Eqs. (29) and (31) by defining corresponding series and shunt resonance frequencies.
0.0 -0.5
-1.0
0
100
200
300
X 2 = x2 / Δ 2
400
Fig. 6. Transient state for fsrc = 4.5 GHz < f0 = 5 GHz.
500
6
RH Material CRLH Material
1.0
0.5
v0 in V
v0 in V
0.5 0.0 -0.5
-1.0
0.0 -0.5
0
100
200
300
X 2 = x2 / Δ 2
400
-1.0
500
Fig. 7. Steady state for fsrc = 4.5 GHz < f0 = 5 GHz (LH prop.).
0.5
100
200
300
X 2 = x2 / Δ 2
400
500
RH Material CRLH Material
1.0 0.5
v0 in V
RH Material CRLH Material
1.0
0
Fig. 10. Transient state for fsrc = 5.5 GHz > f0 = 5 GHz.
In the second case, so-called infinite-wavelength (λg=2π /|β|, β=0) propagation is achieved at the transition frequency f0 of the MTM. Fig. 8 and Fig. 9 show the time evolution for the transient and steady states, respectively.
v0 in V
RH Material CRLH Material
1.0
0.0 -0.5
0.0
-1.0
0
100
-0.5
200
300
X 2 = x2 / Δ 2
400
500
Fig. 11. Steady state for fsrc = 5.5 GHz > f0 = 5 GHz (RH prop).
-1.0
0
100
200
300
X 2 = x2 / Δ 2
400
500
Fig. 8. Transient state for fsrc = f0 = 5 GHz (β = 0 prop.).
RH Material CRLH Material
1.0
C. Extraction of Dispersion Characteristic and Effective Material Parameters of Homogeneously Loaded PPWG
v0 in V
0.5 0.0 -0.5
-1.0
0
100
200
300
X 2 = x2 / Δ 2
400
In other numerical investigations on wave propagation in MTMs usually the steady state solution is shown but here we depicted both the interesting transient behavior at the beginning of the simulation (in Figs 6, 8 and 10) and also the steady state solution (in Figs 7, 9 and 11).
500
Fig. 9. Steady state for fsrc = f0 = 5 GHz (β = 0 prop.).
In the last case, expected RH propagation is observed in Fig. 10 for the transient phase and in Fig. 11 for the steady state.
A PPWG filled with a homogeneous lossless MTM and discretized with a homogenous mesh (Δ2 = const) is considered now. The waveguide is terminated by PMLs with 12 layers. The width of this waveguide is 15 mm while the height is chosen to be 2 mm in order to achieve a 50 Ω line impedance. The line impedance Znum(f) and the dispersion βnum(f) are calculated using the Fouriertransformed voltage U0(f) and current I1(f) on the line and the phase of the transmission s21(f) of the line with length L, respectively: 1 {S2 + 1} U0 (f ) (32) Znum ( f ) = 2 I1 (f ) (33) βnum ( f ) = −
unwrapped phase ⎡⎣s 21 ( f ) ⎤⎦ L
+ξ
7
60
Znum
(36)
Line Impedance Znum in Ω
50
2
2
⎛ 1 ⎞ w = Y0′ = j ⎜ ωC′0R − = jωε0 ε rh (ω) L ⎟ ′ L h ω 0 ⎠ ⎝
⎛ Z 1 ⎞ h (37) lim 1 = Z1′ = j ⎜ ωL1′R − = jωμ 0 μ hr (ω) L ⎟ ′ C w ω Δ 2 →0 Δ 2 1 ⎠ ⎝ R ⎛ ⎡ f sh ⎤ 2 ⎞ ⎛ ⎞ h ′ C h 1 ⎜1 − ⎢ 0 ⎥ ⎟ ⎟ (38) ε hr ( f ) = 0 ⎜1 − L R ( ) = ε ∞ r ⎜ ⎟ f ε0 w ⎜ L′0 C′0 ( 2πf )2 ⎟ ⎝ ⎠ ⎝ ⎣ ⎦ ⎠ ⎛ ⎡ f se ⎤ 2 ⎞ ⎞ L′R w ⎛ 1 h ⎜ ⎟ ⎜1 − ⎟ (39) μ hr ( f ) = 1 = μ ( ∞ ) 1− 1 r ⎜ ⎢⎣ f ⎥⎦ ⎟ μ 0 h ⎜ L1′R C1′L ( 2πf )2 ⎟ ⎝ ⎠ ⎝ ⎠
40
30
20
⎛ ⎡ f sh ⎤ 2 ⎞ ⎛ ⎡ f se ⎤ 2 ⎞ βh = 2πf ε rh ( f ) μ rh ( f ) = 2πf ε rh (∞) ⎜1 − ⎢ 0 ⎥ ⎟ μ rh (∞) ⎜ 1 − ⎢ 1 ⎥ ⎟ ⎜ ⎣ f ⎦ ⎟ ⎜ ⎣ f ⎦ ⎟ ⎝ ⎠ ⎝ ⎠
10
0
Y0
lim Δ →0 Δ
4.0
4.2
4.4
4.6
4.8 5.2 5.4 5.0 Frequency f in GHz
5.6
5.8
6.0
Fig. 12. Numerically extracted line impedance of the homogeneously MTM filled PPWG.
The shift ξ is chosen to ensure the zero phase origin at f0 = 5 GHz. The numerically extracted line impedance and the dispersion are plotted in Fig. 12 and Fig. 13, respectively. The numerically extracted line impedance Znum can be interpreted as a Bloch impedance defined for periodic structures [9].
2.0
2π L′0L C′0R
and f1se =
1 2π L1′R C1′L
εnum r μnum r εhr μhr
1.0
0.0
and
βh
εr
5.8
βnum
1
The comparison between numerically extracted and analytically derived effective material parameters is shown in Fig. 14.
μr
6.0
with f 0sh =
5.6
-1.0
Frequency f in GHz
5.4 5.2
RH propagation
-2.0
5.0 4.8
4.0
LH propagation
4.6
4.4
4.6
4.8 5.0 5.2 5.4 Frequency f in GHz
5.6
5.8
6.0
Fig. 14. Comparison between extracted and theoretically/analytically derived effective εr and μr for the homogeneously MTM filled PPWG [εr(∞) = μr(∞) = 5.0].
4.4 4.2 4.0
4.2
-200
-150
-100
-50
0
50
100
150
200
β in 1/m
Fig. 13. Comparison between numerically extracted and theoretically/analytically derived dispersion relation for the ideal homogeneously MTM filled PPWG.
The effective material parameters of the substrate material can be extracted by using the before calculated values Znum and βnum:
h β c (34) ε num ( f ) = w num 0 r Znum 2πf Z0
(35) μ num (f ) = r
Znum βnum c0 h 2πf Z0 w
The numerically extracted values in Eqs. (34) and (35) are compared to their ideal homogeneous CRLH TL [9] counterparts given in Eqs. (38) and (39). The primed lumped elements are defined either in a per-unit-length (RH ones) or in a times-unit-length (LH ones) sense.
Interestingly, the proposed CRLH-EEC FDTD approach leads to Drude-type constitutive parameters in the homogeneous limit (Δ2 → 0), as confirmed in Fig. 14. At the edges of the considered frequency band there is a small deviation between the numerically extracted and the ideal effective constitutive parameters. In another publication [17] that is mainly about the stability of the EEC FDTD scheme we also showed that the EEC Yee cell restores rigorously the dispersion/attenuation relation of CRLH TL-based MTMs [9], including certain stopbands. Since the Drude model can not model these effects, we get the above mentioned derivation between the perfect homogeneous case (Drude) and the periodic structure case (EEC FDTD). D. Three-Dimensional Case The general 3D update system can be established in a straightforward manner by adding the LH elements to Eqs. (12) and (13) and isolating the latest normalized values. The only difference to the update system of the 1D case given in Eqs. (24) to (27) is an additional term in
8
Eqs. (40) and (42) for the approx. of the circulation integrals given in Eqs (3) and (4). G Δ C′dR−1 − d −1 t 2 ε0 n n −1 (40) v′d −1 = v′ G d −1 Δ t d −1 R C′d −1 + 2 ε0
c0 Δ t − G Δ R C′d −1 + d −1 t 2 ε0 L n − 0.5 d −1
}
{
= i′
(42) i′d −1
R d −1 Δ t 2 μ0 n − 0.5 i′ = R d −1 Δ t d −1 R L′d −1 + 2 μ0 ⎛ {Sd − 1} v′d +1n − {Sd +1 − 1} v′d n ⎞ ⎜ ⎟ ⎜ + v′L n ⎟ ⎝ d −1 ⎠
c0 Δ t n + 0.5 i′d −1 L C′d −1 This update system was implemented for all six components (d ∈ {0, 1, 2}). n +1
n −1
= v′dL−1
+
IV. ANALYSIS OF A PATCH ANTENNA LOADED WITH CRLH SUBSTRATE MATERIAL In this section, the CRLH-EEC FDTD scheme established in the previous section is used to analyze a resonant patch antenna fully loaded with a CRLH MTM substrate above an infinite ground plane. The patch is 50 mm long and 20 mm wide. However, the latter dimension is increased by using PMC walls in the y-direction, as depicted in Fig. 15. The thickness of the substrate is 1 mm. The MTM substrate is discretized by 4 CRLH-EEC unit cells. For this area below the substrate the update system consisting of the four equations (40) to (43) is used. In the surrounding air-filled region the ordinary EC FDTD based on Eqs. (12) and (13) is applied.
f+2
0
-5
L′dR−1 −
c0 Δ t − R Δ L′dR−1 + d −1 t 2 μ0 (43) v′dL−1
f+1
c Δ n + 0 L t v′d −1 L′d −1
(41) i′
n + 0.5
}
S11 in dB
L n + 0.5 d −1
{
⎛ 1 − Sd−1+1 i′d n −0.5 − 1 − Sd−1 i′d +1n −0.5 ⎞ ⎜ ⎟ ⎜ + μ isrc n −0.5 + i′L n −0.5 ⎟ 0 d −1 d −1 ⎝ ⎠
To compare the case of an antenna on a MTMsubstrate with that of conventional antenna, let us first consider a patch on air. The simulated return loss characteristic is plotted in Fig. 16, where the λ/2 and λ resonances at f+1=2.8 GHz and f+2=5.7 GHz can be identified. The position of the feeding probe has been tuned for matching of the λ/2-mode, because this resonance is usually used in a patch. This optimized position has been found to be 6 mm off the center of the resonator.
-10
-15
-20
-25
0
1
2
3
4
5
6
7
Frequency f in GHz
Fig. 16. Return loss of a patch antenna above air. The λ/2 and λ resonances at f+1=2.8 GHz and f+2=5.7 GHz can be identified.
Our design goal in the case of a MTM substrate is to keep the resonance frequency f+1 and the matching at that frequency unchanged. Additional LH resonances in the lower frequency band yielding a potential for miniaturization and zeroth order resonance are expected from the CRLH theory [9]. This objective can be achieved by proper design of the MTM. By using Eqs. (38) and (39) and choosing εr(∞) = μr(∞) = 2.0, the frequencies fsh and fse can be calculated approximately by
(44) f sh = f +1 1 − (45) f se = f +1 1 −
Perfectly Matched Layer with 10 Cells
ε hr ( f +1 ) ε (∞) h r
μ hr ( f +1 ) μ hr ( ∞ )
=
f +1 ≈ 2 GHz, 2
=
f +1 ≈ 2 GHz. 2
Once these frequencies have been determined, the optimized frequency-dependant εr(f) = μr(f) are described by
z y
⎛ ⎡ f ⎤2 ⎞ ⎛ ⎡ f ⎤2 ⎞ (46) ε hr ( f ) = ε hr ( ∞ ) ⎜1 − ⎢ sh ⎥ ⎟ = 2.0 ⎜1 − ⎢ +1 ⎥ ⎟ , ⎜ ⎣f ⎦ ⎟ ⎜ ⎣ 2f ⎦ ⎟ ⎝ ⎠ ⎝ ⎠
Probe Feed
x Perfect Magnetic Conductor Walls
Fig. 15. Simulation setup with patch antenna above a substrate.
⎛ ⎡ f ⎤2 ⎞ ⎛ ⎡ f ⎤2 ⎞ se (47) μ ( f ) = μ ( ∞ ) ⎜1 − ⎢ ⎥ ⎟ = 2.0 ⎜1 − ⎢ +1 ⎥ ⎟ . ⎜ ⎣f ⎦ ⎟ ⎜ ⎣ 2f ⎦ ⎟ ⎝ ⎠ ⎝ ⎠ h r
h r
The return loss for the loaded patch is plotted in Fig. 17. As expected from the above considerations, the resonance frequency f+1 is unchanged with respect to the
9
conventional case. That is also valid for the matching at that frequency, but not for the bandwidth. Since the field distribution below the patch is the same for the RH resonance f+1 and for the LH mode f-1 the matching for both modes is approximately the same. This is also the case for the radiation pattern, as shown in Figs. 18 and 19. Matching for the other resonances is not optimized here, since the feeding position is not changed compared to the patch above air. f-3
f-1
f+1
f 0 = 1.95 GHz
f -1 = 1.36 GHz
f -2 = 990 MHz
f -3 = 750 MHz
f+3
0
S11 in dB
-5
-10
f-2
f+0
f+2
-15
-20
-25
f-4 0
1
2
3
4
5
6
7
Frequency f in GHz
Fig. 17. Return loss of patch antenna above MTM substrate. The resonances in the LH and RH regions can be identified.
For the additional LH and the zeroth order resonances the field component Ez below the patch and the qualitatively 3D radiation pattern are visualized. As a reference the mentioned values are plotted firstly for the RH resonance f+1 in Fig. 18.
Fig. 19. Ez (x, y = 10 mm, z = 0.5 mm) distributions below patch together with the 3D radiation patterns for different resonance frequencies: For f-1 (upper right) and f-3 (lower right) there is a out-of-phase Ez at the edges yielding a broadside radiation, whereas for f0 (upper left) and f-2 (lower left) there is a in-phase Ez at the edges yielding a end-fire radiation.
For the odd modes the field at the radiating edges has a 180° phase-shift yielding a broadside radiation. For the even modes including the zeroth order one this is not the case since the field is now in-phase. Therefore end-fire radiation is predicted by simulation. If the PMCs are replaced by PMLs in Fig. 15, the patch radiates like an electric dipole at the zeroth order resonance f0 as depicted in Fig. 20.
Ez (x, 10 mm, 0.5 mm)
z y
Fig. 20. Electric field E (x, y = 10 mm, z) around the patch together with the 3D dipole-like radiation pattern at the zeroth order resonance. The PMCs in y-direction are replaced by PMLs.
x
f +1 = 2.80 GHz
Fig. 18. Ez (x, y = 10 mm, z = 0.5 mm) below the patch (bottom) together with radiation pattern (top) at f +1 = 2.8 GHz.
Fig. 19 shows the normal field component Ez below the patch together with qualitatively 3D radiation pattern for the additional resonances at f0, f-1, f-2 and f-3.
In a future work, the patch loaded with a CRLH MTM will be optimized by adjusting the size of the patch to achieve the same resonance frequency for the mentioned modes (f+1 = f0 = f-1 = f-2 = f-3) for fair comparison of gain. The present antenna problem suggests a potential for antenna miniaturization by using the modes in the LH low-frequency band.
10
V. CONCLUSION AND OUTLOOK A novel extended equivalent circuit (EEC) formulation of the FDTD scheme has been proposed. This formulation consists in an extension of the equivalent circuit (EC) description of the Yee cell to the most general model of a CRLH transmission line (TL) metamaterial (MTM), which includes LH (series C, shunt L) elements in addition to the conventional RH (series L, shunt C) elements. The proposed CRLH-EEC FDTD scheme also accounts for losses. It represents the FDTD counterpart of a MTM-expanded transmission line method (TLM) recently introduced by Hoefer et al., with the FDTDintrinsic advantage of requiring half the number of unknowns used in the TLM. In this approach, Drude-type constitutive parameters are obtained as a consequence of the addition of the LH elements to the RH host medium, in the limit where the electrical size of the material-Yee cell tends to zero, where the MTM becomes perfectly homogeneous. This EEC scheme can be extremely useful for the analysis of future MTMs which, as a consequence of improved homogenization (potentially from today’s typical average lattice constant of λ/5-λ/15 to λ/50λ/100), will be constituted by a huge number of cells while still being of finite extent and interfaced with other media [19]. Therefore the proposed CRLH MTM simulation feature will be available in one of the next releases of the commercial FDTD solver Empire™ [20]. The proposed CRLH-EEC FDTD scheme was demonstrated to successfully describe a parallel-plate waveguide (PPWG) structure partially filled with a CRLH medium and a resonant patch antenna sitting on a CRLH substrate. In the former case, the phenomena of LH frequency range backward wave, RH frequency range forward and transition frequency infinite wavelength wave were verified and described in both their transient and steady state regimes. In the latter case, the existence of paired LH-negative, RH-positive and of a transition frequency infinite wavelength resonance was verified and these resonances were characterized in terms of their radiation patterns. REFERENCES [1] V. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Uspekhi, vol. 10, no. 4, pp. 509–514, Jan.–Feb. 1968. [2] J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructure,” Phys. Rev. Lett., vol. 76, no. 25, pp. 4773– 4776, June 1996. [3] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena, IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2075–2084, Nov. 1999. [4] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 84, no. 18, pp. 4184–4187, May 2000. [5] A. K. Iyer and G. V. Eleftheriades, “Negative refractive index metamaterials supporting 2-D waves,” in Proc. IEEE MTT Int. Symp. 2002, vol. 2, pp. 412–415.
[6] C. Caloz and T. Itoh, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH transmission line,” in Proc. IEEE-AP-S USNC/URSI National Radio Science Meeting 2002, vol. 2, pp. 412–415. [7] A. A. Oliner, “A periodic-structure negative-refractiveindex medium without resonant elements,” in URSI Dig., IEEE-AP-S USNC/URSI National Radio Science Meeting 2002, p. 41. [8] G. V. Eleftheriades and K. G. Balmain (Editors), Electromagnetic Metamaterials, Transmission Line Theory and Microwave Applications. Wiley – IEEE Press, 2005. [9] C. Caloz and T. Itoh, Electromagnetic Metamaterials, Transmission Line Theory and Microwave Applications. Wiley – IEEE Press, 2005. [10] N. Engheta and R. W. Ziolkowski (Editors), Electromagnetic Metamaterials: Physics and Engineering Aspects. Wiley – IEEE Press, 2005. [11] E. M. Lifshitz, L. D. Landau, and L.P. Pitaevskii, Electrodynamics of Continuous Media, ButterworthHeinemann, 2nd edition, 1984. [12] R. W. Ziolkowski and E. Heyman. “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E, vol. 64, pp. 056625:1–15, 2001. [13] S.A. Tretyakov, S.I. Maslovski, I.S. Nefedov, M.K. Kärkkäinen, “Evanescent modes stored in cavity resonators with backward-wave slabs,” Microwave and Optical Technology Letters, vol. 38, no. 2, pp. 153-157, 2003. [14] T. Kokkinos, R. Islam, C. D. Sarris, and G. V. Eleftheriades. “Rigorous analysis of negative refractive index metamaterials using FDTD with embedded lumped elements,” in Proc. IEEE MTT Int. Symp. 2004, vol. 3, pp. 1783–1786. [15] P. P. M. So, H. Du, and W. J. R. Hoefer. “Modeling of metamaterials with negative refractive index using 2Dshunt and 3D-SCN TLM networks,” IEEE Trans. Microwave Theory Tech., pp. 1496–1505 vol. 53, no. 4, April. 2005. [16] A. Lauer and I. Wolff, “Stable and efficient ABCs for graded FDTD simulations,” in Proc. IEEE MTT Int. Symp. 1998, vol. 2, pp. 461–464. [17] A. Rennings, S. Otto, C. Caloz, A. Lauer, W. Bilgic, P. Waldow, “Composite right/left-handed extended equivalent circuit (CRLH-EEC) FDTD: stability and dispersion analysis with examples”, International J. Numerical Modelling, special issue “Microwave Numerical Modelling of Metamaterial Properties, Structures and Devices,” accepted, to be published. Taflove and S. Hagnes, Computational [18] A. Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Third Edition, 2005. [19] C. Caloz, A. Lai, and T. Itoh, “The Challenge of homogenization in metamaterials,” New Journal of Physics, vol. 7, no. 167, pp. 1-15, August 2005. [20] IMST GmbH, “User and reference manual for the 3D EM time domain simulator EMPIRE”, www.empire.de, Jan. 2004.
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Andreas Rennings was born in Xanten, Germany on October 29th, 1973. He carried out an apprenticeship as an electrician for three years. Afterwards he studied electrical engineering at Duisburg-Essen University, Germany. From 1999 to 2000, he was a visiting student in the Microwave Electronics Laboratory of University of California at Los Angeles (UCLA). He received the Diploma degree (with distinction) from Duisburg-Essen University in 2000. Currently he is working towards the PhD degree. His research interests are in the area of computational electromagnetics with emphasis on time-domain techniques, including wavelet-based and other multi-grid methods, microwave metamaterials and antennas. He has authored 20 papers and filed two patents in the above mentioned areas. He received the VDE price for his diploma thesis and is the recipient of a Student Paper Award (2nd place) presented at the 2005 IEEE Antennas and Propagation Society (AP-S) International Symposium, Washington, DC. Simon Otto was born in Duisburg, Germany in 1978. He received his Diplom-Ingenieur degree from Duisburg University in 2004. In 2004 he started his research career as a visiting student at UCLA, where his final thesis was carried out under the supervision of Prof. Caloz and Prof. Itoh. So far he has generated more than 6 papers related to MTM, filed one patent and won the 2. place of the Antenna and Propagation Symposium (AP-S) student paper award 2005 in Washington. Now he is with the Antennas and EM modelling department at IMST in Kamp-Lintfort, Germany and working towards his PhD. His research interests include antennas, metamaterials, em-theory and numerical modelling. Andreas Lauer was born in Duisburg, Germany on Dec., 15th 1966. He received his Diploma-Degree at Duisburg University in 1992, his PhD Degree in 1998. Now he is with the Antennas and EM modeling department at IMST in Kamp-Lintfort, Germany. His research interests are electro-magnetic simulations (FDTD, MOM etc. ), microwave propagation, antennas, waveguides, electromagnetic theory etc.
Christophe Caloz was born in Sierre, Switzerland, in 1969. He received the Diplôme d'Ingénieur en électricité and the Ph.D. degree from École Polytechnique Fédérale de Lausanne (EPFL), Switzerland, in 1995 and 2000, respectively. From 2001 to 2004, he was a Postdoctoral Research Engineer in the Microwave Electronics Laboratory of University of California at Los Angeles (UCLA), where he conducted research on PBG structures and the theory and applications of microwave metamaterials (MTMs). In March 2004, he received the UCLA Chancellor’s Award for Postdoctoral Research over 875 postdoctoral fellows of all disciplines. In June 2004, Dr. Caloz joined École Polytechnique of Montréal, where he is now an Associate Professor and a member of the Microwave Research Group PolyGrames. In April 2005, he was awarded a Canada Research Chair (CRC) entitled “Future Intelligent Radiofrequency Metamaterials” (FIRMs), associated with a novel Canadian Foundation for Innovation (CFI) infrastructure. Dr. Caloz has authored and co-authored over 120 technical conference, letter and journal papers, three book chapters, and holds several patents. He has also authored the first textbook on MTMs, which is entitled “Electromagnetic Metamaterials, Transmission Line Theory and Microwave Applications” (Wiley - IEEE Press, 2005). He is currently the Guest Editor of two special issues on MTMs in the International Journal for Numerical Methods (IJNM) and in the IEICE
(Institute of Electronics, Information and Communication Engineers) Transactions on Electronics. Dr. Caloz’ current interests, in addition to electromagnetic MTMs, include nonlinear and active devices, integrated front ends and antennas, ferroelectric and ferromagnetic thin film devices, ultra wideband (UWB) systems and terahertz technology. Peter Waldow received the Dipl.-Ing. and Dr.Ing degree in 1982 and 1986 from the Mercator-University Duisburg, respectively. He is currently a director of IMST - Institute of Mobile and Satellite Communication Techniques. He took the chair for Electromagnetic Engineering and Field Theory at the University of Duisburg-Essen from 1999 to 2004. His fields of interest are electromagnetic field computation, wavelet theory and metamaterials.
