Issue 
EPJ Appl. Metamat.
Volume 6, 2019
Metamaterials Research and Development in Japan



Article Number  2  
Number of page(s)  13  
DOI  https://doi.org/10.1051/epjam/2018007  
Published online  16 January 2019 
https://doi.org/10.1051/epjam/2018007
Research Article
Quasitheoretical investigation of four circularly polarized metaloop antennas
Science and Engineering, Hosei University, Koganei, Tokyo, Japan
^{*} Corresponding author: hymat@hosei.ac.jp
Received:
28
August
2018
Accepted:
2
November
2018
Published online: 16 January 2019
Four metamaterial line loop (metaloop) antennas, RNDMTLPC, RNDMTLPN, SQRMTLPC, and SQRMTLPN are investigated to clarify the antenna characteristics. Each metaloop is made up of either a Ctype metamaterial line or an Ntype metamaterial line, where the dispersion characteristics of the unit cell for the C and Ntype metamaterial lines are designed to be as similar as possible. The RNDMTLPC and SQRMTLPC act as a dualband counter circularly polarized antenna across a fast wave frequency region. It is found that, depending on the deviation factor, the SQRMTLPN behaves as a triband circularly polarized antenna, but the RNDMTLPN does not have the triband characteristic. The radiation pattern, gain, and input characteristic in terms of the VSWR for the four antennas are clarified.
Key words: Metaloop antennas / circularly polarized wave / polarization / dualband operation / triband operation
© H. Nakano et al., published by EDP Sciences, 2019
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
A conventional loop antenna, shown in Figure 1a, generates a linearly polarized (LP) beam in the two directions normal to the antenna plane (i.e. ±z directions) at a frequency that leads to a loop circumference of one wavelength [1]. This bidirectional beam can be changed to a unidirectional beam using a conducting plane (reflector), where the spacing between the loop and the reflector is chosen to be onequarter wavelength, so that the direct radiation from the loop toward the positive zdirection and the radiation reflected by the reflector constructively add in the positive zdirection.
The abovementioned unidirectional LP beam can be changed to a circularly polarized (CP) beam by adding perturbation elements to the loop, as shown in Figure 1b [2]. The rotational sense of the CP radiation in this situation is uniquely determined by the location of the feed point and the perturbation elements. In other words, a conventional loop antenna with a fixed feed point and perturbation elements only radiates either a lefthanded or a righthanded CP beam.
However, with increasing investigation of metamaterialbased antennas [3–6], recent research has revealed that a single antenna with a fixed feed point can radiate a lefthanded CP beam in a specific frequency band and a righthanded CP wave in a different frequency band (dualband counter CP radiation). This fixed feed point antenna is made by spirally winding a metamaterial line (metaline) that is realized using the concept of a composite right and lefthanded transmission line [7–9], as shown in Figure 1c; the antenna is called the metaspiral antenna [10]. The metaspiral antenna consists of an arm composed of numerous turns printed on a thin dielectric substrate backed by a ground plane. The antenna height above the ground plane is very small − on the order of 1/100 wavelength.
A question arises as to whether dualband counter CP radiation can be realized when the number of turns for the spirally wound arm is reduced to a single turn and subsequently the single turn arm is formed into a loop structure. The answer to this question has appeared in reference [11], where the dualband counter CP radiation from a square metamaterial line loop (metaloop) antenna is described.
A counterpart to the square metaloop exists and is called the round metaloop. A further question arises as to whether the behaviour of the CP radiation for the round metaloop (RNDMTLP) is the same as that for the square metaloop (SQRMTLP). As will be found later, the behaviour of the CP radiation depends on the type of metaline that constitutes the RND and SQRMTLP antennas.
This paper presents a theoretical investigation of the behaviour of the CP radiation using two types of metaline: a Ctype metaline and an Ntype metaline. The investigation is performed for four metaloops: RNDMTLPC, RNDMTLPN, SQRMTLPC and SQRMTLPN, where the last letters (C and N) denote the type of metaline that is used for the metaloop, the Ctype or the Ntype.
Five sections constitute this paper. Section 2 presents C and Ntype metalines and their radiation characteristics. Section 3 discusses the RNDMTLPC and RNDMTLPN antennas, which approximately realize the radiation models round modelC and round modelN, respectively. Similarly, Section 4 discusses the SQRMTLPC and SQRMTLPN antennas, which approximately realize the radiation models square modelC and square modelN, respectively. Section 5 summarizes the obtained results.
Fig. 1 (a) Conventional loop antenna that forms linearly polarized bidirectional radiation. (b) Loop antenna with perturbation elements that form circularly polarized radiation. (c) Metaspiral antenna that forms circularly polarized radiation. 
2 Metamaterial lines (metalines)
Figure 2a shows a metamaterial line, referred to as a Ctype metaline, based on the concept of a conventional composite right and lefthanded (CRLH) transmission line [7–9], while Figure 2b shows a novel metaline, referred to as an Ntype metaline. Both are printed on a dielectric substrate of relative permittivity ε _{r} and thickness B. The substrate is backed by a conducting ground plane (GP). Point F is the feed point. The metaline end point, T, is terminated through resistive load R _{B}.
The Ctype metaline is composed of periodically arrayed subwavelength segments of width w and length p _{0} with separation Δg. The region across length 2(p _{0} + Δg) ≡ p is called the unit cell. The central segment of the unit cell is shorted to the ground plane through inductance L_{Y} using a conducting vertical probe (via) of radius r _{via}. Neighbouring subwavelength segments are connected through a capacitive element, 2C _{Z}, which is inserted into the gap between segments.
The Ntype metaline in Figure 2b differs from the Ctype metaline in that the unit cell's central segment does not have a conducting vertical probe; instead, it has a horizontal strip of width W _{STP} and length L _{STP} that extends from the central subwavelength segment in the negative ydirection, with the end of the strip being shorted to the ground plane. As seen later, the Ntype metaline acts as a circularly polarized radiation element, as opposed to the Ctype metaline, which acts as a linearly polarized radiation element.
Figure 3 shows the dispersion diagram of the unit cell for Ctype and Ntype metalines of infinite length. The parameters used are summarized in Table 1. These two metalines are designed to have dispersion characteristics that are as similar as possible. The notation is as follows: k (=2π/λ with free space wavelength λ) and β (=2π/λ _{g} with guided wavelength λ _{g}) are the propagation phase constants in free space and along the metaline, respectively; f _{T} is the transition frequency; and f _{L} and f _{U} are the lower and upper band edge frequencies for a fast wave, respectively. Note that the dispersion in terms of guided wavelength, λ _{g}, is shown in Figure 4.
The radiation from Ctype and Ntype metalines of 10 cells (N _{cell} = 10) in length is shown in Figure 5, where the parameters in Table 1 are used; the end of the metaline is terminated with a resistive load of R _{B} = 60 ohms. Propagation phase constant β at frequencies below transition frequency f _{T} (=3 GHz) is negative and hence both metalines form radiation beams in the backward direction, while β at frequencies above f _{T} is positive and hence both metalines form radiation beams in the forward direction.
The current along the periodically arrayed subwavelength segments for the Ctype metaline flows in the longitudinal direction, i.e. the xdirection, and the radiation is linearly polarized (LP), as shown in Figure 5a, where the radiation field is decomposed into the θdirected electric field component, E_{θ} , and the φdirected electric field component, E_{φ} . This result is expected from conventional CRLH transmission line theory [7–9].
In contrast, the radiation from the Ntype metaline is circularly polarized (CP), as shown in Figure 5b, where the radiation field is decomposed into the lefthanded CP electric field component (LHCP wave), E _{L}, and the righthanded CP electric field component (RHCP wave), E _{R}. Note that E _{L} is dominant, i.e. E _{L} is the copolarized component and E _{R} is the crosspolarized component. This is attributed to the fact that the unit cell for the Ntype metaline has currents in the xdirection (longitudinal direction) and negative ydirection (the strip direction), where the negative ydirected current is delayed by 90° with respect to the xdirected current, resulting in LHCP radiation.
Fig. 2 Metalines. (a) Perspective view (upper), exploded view (middle) and side view (bottom) of a Ctype metaline. (b) Perspective view (upper), exploded view (middle) and side view (bottom) of an Ntype metaline. 
Fig. 3 Dispersion diagram for the unit cell. (a) Ctype metaline. (b) Ntype metaline. 
Parameters.
Fig. 4 Guided wavelength λ _{g} for the unit cell as a function of frequency. (a) Ctype metaline. (b) Ntype metaline. 
Fig. 5 Radiation from a 10cell metaline (N _{cell} = 10). The para meters in Table 1 are used, with R _{B} = 60Ω. (a) Ctype metaline. (b) Ntype metaline. 
3 Round metaloop antenna
3.1 Round modelC
The principle for forming CP radiation in the broadside direction (positive zdirection) is to be considered here. Figure 6 shows an antenna model, called round modelC, where the loop is made of a curved narrow width line and has radius r _{LP}.
We express the current at distance s ^{′}, measured from point F in the counterclockwise direction, as , where I _{0} is the amplitude; is the unit vector tangential to the loop. Note that the time dependence, e ^{ jωt }, is omitted.
For deriving the radiation field for the loop, the following assumptions are made: (1) current I (s ^{′}) of phase constant β (β = 2π/λ _{g} > 0 or β = −2π/λ _{g} < 0) is a travelling current without attenuation, i.e. I _{0} is constant; (2) there is no reflection current flowing towards feed point F, i.e. unidirectional current; (3) there is no mutual coupling between current elements at different positions.
Using unit vectors and with rectangular coordinates, unit vector is transformed into(1)where Φ(s ^{′}) ≡ s ^{′}/r _{LP} is the azimuth angle for current I (s ^{′}).
The electric field in the spherical coordinate system, E (R, θ, φ), is calculated using the following equation [12]:(2)where ω (= 2πf ) is the angular frequency at frequency f ; μ is the permeability in free apace; k (= 2π/λ) is the propagation phase constant in free space, as already defined; L = 2πr _{LP} ≡ nλ _{g} is the loop circumference, where n is a positive real number; is the unit vector in the Rdirection of the spherical coordinate system; r ^{′} (s ^{′}) is the position vector from the coordinate origin to a current source point of s ^{′};  r ^{′} (s ^{′})  = r _{LP}.
Substituting the assumed current, I (s ^{′}), into equation (2), we obtain the field in the broadside direction, E _{Brd}^{ n } (R, θ = 0, φ), as(3)where(4)with(5) (6) A _{s}^{ n } in equation (4) is referred to as the current polarization vector, which is composed of an RHCP com ponent and an LHCP component, . It is found that a CP component (either an RHCP component or an LHCP component) is obtained when n is an integer. If n is not an integer, then A _{s}^{ n } becomes elliptically polarized. Our purpose is to obtain broadside CP radiation, and hence n is chosen to be an integer.
For n = 1, equations (5) and (6) become(7) (8)
Consequently, equation (4) yields At the frequency defined by (β > 0, n = 1), called the first Hion frequency and denoted as f _{H1}, the broadside radiation field, E _{Brd}^{ n=1} in equation (3) is RHCP because of the relationship in equation (9a). In contrast, at the frequency defined by (β < 0, n = 1), called the first Nion frequency, f _{N1}, E _{Brd}^{ n=1} is LHCP because of the relationship in equation (9b). The polarization for n = 1 can also be explained qualitatively, taking the sign of β into con sideration: when β is positive, phase −βs ^{′} along the loop from feed point F is negative and regressive as with a conventional natural loop, i.e. the current flows counterclockwise, resulting in RHCP radiation. On the other hand, when β is negative, phase −βs ^{′} is positive and progressive. Hence, the current behaves as if it flows clockwise, resulting in LHCP radiation.
When n = 2 and 3, equations (5) and (6) become(10) (11) and hence equation (4) yields (12) To summarize, the current polarization vector, , is zero at the second Hion frequency, f _{H2}, defined by (β > 0, n = 2) and the second Nion frequency, f _{N2}, defined by (β < 0, n = 2), resulting in no broadside radiation: E _{Brd}^{ n=2,3} = 0. Zero broadside radiation also occurs at the third Hion frequency, f _{H3}, and the third Nion frequency defined by (β > 0, n = 3) and (β < 0, n = 3), respectively. The above discussion leads to shown in Figure 7.
Fig. 6 Round modelC. 
Fig. 7 for round modelC. 
3.2 Round metaloopC antenna (RNDMTLPC): approximation for round modelC
RHCP radiation and LHCP radiation in the broadside direction are realized here by using a single antenna under the condition that the location of feed point F is fixed. For this, the principle in Section 3.1 is implemented by curving the Ctype metaline in Figure 2a, where the radius is r _{LP}, as shown in Figure 8. The antenna is referred to as the round metaloopC antenna or RNDMTLPC. Note that loop end T is terminated through resistive load R _{B} (=60Ω) to the ground plane. The parameter values (w, p _{0}, Δg, ε _{r}, B, 2C _{Z}, L _{Y} , r _{via}) for this metaloop are the same as those shown in Table 1.
The following simulation/analysis is performed using a fullwave analysis tool (HFSS^{1}), whose results include effects of a finitelength nonstraight metaline structure on the current: phase constant (and hence the guided wavelength) and attenuation constant along the loop, and reflection from the loop end. These effects, which are not included in the round modelC, change the intensity of the broadside radiation E presented by the round modelC; however, the discussed behaviour of polarization is still valid. This holds true for RNDMTLPN in Section 3.4, SQRMTLPC in Section 4.2 and SQRMTLPN in Section 4.4, against round modelN, square modelC and square modelN, respectively.
The normalized loop circumference, 2πr _{LP}/λ _{g}, as a function of frequency is shown in Figure 9, where metaloop radius r _{LP} is chosen to be 31.8 mm and λ _{g} in Figure 4a is used. The shaded frequency region of f _{L} to f _{U} shows a fast wave region, where the radiation occurs.
Figure 10a shows the frequency response for the gain, where G _{L} denotes the gain for an LHCP wave, referred to as the LHCP gain, and G _{R} denotes the gain for an RHCP wave, referred to as the RHCP gain. As expected from the results in Figure 7, the gain is maximal at frequency f _{GLmax} near the first Nion frequency, f _{N1} = 2.60 GHz (f _{GLmax} = 2.70 GHz≈f _{N1})_{,} and f _{GRmax} near the first Hion frequency, f _{H1 }= 3.50 GHz (f _{GRmax }= 3.60 GHz≈f _{H1}). Note that the input characteristic in terms of the VSWR illustrated in Figure 10b is desirably small within the G _{L} band (G_{L}BW) and G _{R} band (G _{R}BW), as defined by a 3dB gaindrop criterion.
Figure 11 shows the radiation patterns when gains G _{L} and G _{R} are maximal, where the radiation field is decomposed into the LHCP wave component, E _{L}, and the RHCP wave component, E _{R}. It is found that the halfpower beamwidth (HPBW) for E _{L} is wider than that for E _{R}. This is attributed to the fact that the antenna size relative to the operating freespace wavelength (i.e. the electrical antenna size) at f _{GLmax} is smaller than that at f _{GRmax}.
Fig. 8 Round metaloopC antenna (RNDMTLPC), made by curving the Ctype metaline shown in Figure 2a, where N _{cell} = 19. 
Fig. 9 Normalized loop circumference 2πr _{LP}/λ _{g} as a function of frequency, where the radius of the metaloop is r _{LP} = 31.8 mm. 
Fig. 10 Frequency response for the round metaloopC antenna (RNDMTLPC). (a) LHCP and RHCP gains. (b) VSWR. 
Fig. 11 Normalized radiation patterns for the round metaloopC antenna when gains G _{L} and G _{R} are maximal. (a) At f _{GLmax} = 2.70 GHz. (b) At f _{GRmax} = 3.60 GHz. 
3.3 Round modelN
The model shown in Figure 12 is referred to as round modelN, where the current on the loop made of a curved wide width line is expressed as ; q is called the deviation factor and is a function of frequency f : q = q(f). The assumptions for this current are the same as those for round modelC in Section 3.1.
Vector is the unit vector perpendicular to and is given as(13)Note that the directed current has a 90° phase delay relative to the directed current and travels along the loop with phase constant β.
Substituting current I (s ^{′}) into equation (2), the electric field in the broadside direction, E _{Brd} (R, θ = 0, φ), is formulated as(14)where polarization vector generated by  and directed currents is(15)Note that K ^{+}^{ n } and K ^{−}^{ n } are already defined by equations (5) and (6), respectively.
The current for a deviation factor of q = 0 is and polarization vector is the same as A _{s}^{ n }, i.e. . This allows formation of LHCP radiation at first Nion frequency f _{N1} and RHCP radiation at first Hion frequency f _{H1}, as with round modelC when n =1.
and when n = 2 and 3 are zero, as shown by equations (10) and (11), respectively, and hence equation (15) becomes(16)resulting in no broadside radiation: , which holds true for integers n ≥ 4. The broadside radiation, , is summarized in Figure 13a, which is the same as Figure 7.
When q = 1, equation (15) yields(17)
Hence, equation (17) for n = 1 becomes(18)This means that round modelN forms an LHCP wave in the broadside direction at first Nion frequency f _{N1} (the frequency satisfies the conditions β < 0 and n = 1); however, round modelN does not form an RHCP wave in the broadside direction at first Hion frequency f _{H1} (the frequency satisfies conditions β > 0 and n = 1) as compared to round modelC.
Note that equation (17) for integers n = 2 and 3 becomes zero,(19)and hence no broadside radiation is formed, as with round modelC. This also happens at the nth Hion frequency, f _{Hn}, and the nth Nion frequency, f _{Nn}, for integers n ≥ 4.
The broadside radiation, , is illustrated in Figure 13b.
Next, a situation where deviation factor q is neither 0 nor 1 is considered. As one example, a value of q = 0.5 is arbitrarily chosen. Then, equation (15) becomes(20)At first Nion frequency f _{N1} (where β < 0, n =1), K ^{+}^{ n=1} = 0 from equation (7) and = j2π from equation (8), and hence equation (20) yields(21) leading to LHCP broadside radiation for . At first Hion frequency f _{H1} (where β > 0, n =1), = j2π from equation (7) and K ^{−}^{ n=1} = 0 from equation (8), and hence equation (20) yields(22)leading to RHCP radiation for ; this is not found with round modelC.
At second Hion frequency f _{H2} (where β > 0, n =2) and second Nion frequency (where β < 0, n =2), K ^{+}^{ n=2} = 0 and K ^{−}^{ n=2} = 0 are obtained from equations (5) and (6), respectively, and hence, the polarization vector of equation (20) yields zero, (23)resulting in no broadside radiation. This also happens at the nth Hion frequency, f _{Hn}, and the nth Nion frequency, f _{Nn}, for integers n ≥ 4.
Based on the abovementioned results, broadside radiation for round modelN is depicted in Figure 13c. It is worth emphasizing that round modelN forms an LHCP wave at f _{N1} and an RHCP wave at f _{H1} when q is not zero. However, this RHCP wave at f _{H1} disappears when q = 1, although the LHCP wave at f _{N1} is present, as shown in Figure 13b.
Fig. 12 Round modelN. 
Fig. 13 for round modelN. (a) q = 0. (b) q = 1. (c) q = 0.5. 
Fig. 14 Round metaloopN antenna (RNDMTLPN), made by curving the Ntype metaline shown in Figure 2b, where N _{cell} = 19. 
3.4 Round metaloopN antenna (RNDMTLPN): approximation of round modelN
Round modelN in Section 3.3 is approximated by curving the Ntype metaline in Figure 2b, as shown in Figure 14. This antenna is referred to as the round metaloopN antenna (RNDMTLPN). The radiation occurs roughly between f _{L} and f _{U} shown in Figure 9, because the Ntype metaline is designed such that its dispersion characteristics are as similar as possible to those of the Ctype metaline.
Figures 15a and 15b show the frequency response of the gain and the VSWR, respectively. It is observed that the RNDMTLPN has a small gain at a frequency near f _{H1}. This means that the RNDMTLPN no longer has a deviation factor of q = 1 (discussed in Sect. 3.3), due to the effects of the curved structure. The VSWR across G _{L} and G _{R} gain bandwidths for a 3dB gaindrop criterion is reasonably small, leading to good 50Ω impedance matching. Figures 16a and 16b show the radiation patterns when the gain is maximal at frequencies near f _{N1} and f _{H1}, respectively. The radiated waves below and above transition frequency f _{T} = 3 GHz are LHCP and RHCP, respectively; this behaviour is similar to that observed in Figure 13c.
Fig. 15 Frequency response for the round metaloopN antenna. f _{N1} = 2.55 GHz and f _{H1} = 3.425 GHz. (a) Gain. (b) VSWR. 
Fig. 16 Normalized radiation pattern for the round metaloopN antenna. (a) At 2.575 GHz near f _{N1}= 2.55 GHz. (b) At 3.5 GHz near f _{H1} _{ }= 3.425 GHz. 
4 Square metaloop antenna
4.1 Square modelC
Figure 17 shows an antenna model made of a bent narrow width line, referred to as square modelC, where a traveling current, , flows from point F along the loop, with the same assumptions as those for the current in round modelC: unidirectional current with a constant amplitude, I _{0}, and phase constant β that takes on either a positive or negative value.
The unit vector along the loop, , is for loop section 1, for loop section 2, for loop section 3, for loop section 4 and for loop section 5. Broadside radiation E _{BrdSQ}^{ n } formed by the directed current along the loop is formulated to be(24)where B _{s}^{ n } is referred to as the polarization vector for the directed current:(25)
with n being a real number. Equation (25) becomes(26) (27) (28) Equation (26) means that broadside radiation E _{BrdSQ}^{ n=1} is LHCP at first Nion frequency f _{N1} and RHCP at first Hion frequency f _{H1}, while equation (27) reveals no broadside radiation at second Nion frequency f _{N2} and second Hion frequency f _{H2}. As seen from equation (28), the broadside radiation is RHCP at third Nion frequency f _{N3} and LHCP at third Hion frequency f _{H3}. The broadside radiation, , is illustrated in Figure 18. It should be emphasized that the generation of CP broadside radiation for square modelC differs from that of its counterpart, round modelC, at frequencies below f _{N1} and above f _{H1}.
Fig. 17 Square modelC. 
4.2 Square metaloopC antenna (SQRMTLPC): approximation of square modelC
Square modelC is approximately realized by making square bends in the Ctype metaline in Figure 2a. The antenna shown in Figure 19 is referred to as the square metaloopC antenna or SQRMTLPC, whose parameters are given in Table 1.
Figure 20a and 20b show the frequency response of the gain and VSWR, respectively. The gain is maximal at frequencies near f _{N1} and f _{H1}, both within the fast wave frequency region (from f _{L} to f _{U}, where the radiation occurs), as expected from square modelC. Figure 20b reveals that the VSWR within the G _{L} and G _{R} bandwidths for a 3dB gaindrop criterion is less than 2, as with the RNDMTLPC. Thus, the SQRMTLPC acts as a dualband counter CP radiation element within the fast wave frequency region, as with the RNDMTLPC, although square modelC and round modelC behave differently at frequencies below f _{N1} and above f _{H1}, as shown in Figures 18 and 7.
The radiation patterns when the gain becomes maximal at frequency f _{GLmax} near first Nion frequency f _{N1} and frequency f _{GRmax} near first Hion frequency f _{H1} are shown in Figures 21a and 21b, respectively. The halfpower beam width (HPBW) for the LHCP radiation is wider than that for the RHCP radiation. This difference leads to the same characteristic for the gain described in Section 3.1, i.e. the LHCP gain is smaller than the RHCP gain, as shown in Figure 20a.
Fig. 18 Broadside radiation for square modelC. 
Fig. 19 Square metaloopC antenna (SQRMTLPC), where N _{cell} = 19. 
Fig. 20 Frequency response for the square metaloopC antenna. (a) Gain. (b) VSWR. 
Fig. 21 Normalized radiation pattern for the square metaloopC antenna. (a) At frequency f _{GLmax} = 2.58 GHz near f _{N1} _{ }= 2.60 GHz. (b) At frequency f _{GRmax} _{ }= 3.51 GHz near f _{H1} = 3.50 GHz. 
4.3 Square modelN
The radiation model made of a bent wide width line, shown in Figure 22, is referred to as square modelN, corresponding to round modelN in Figure 12. The current on the loop is expressed as with deviation factor q. Vector is the unit vector perpendicular to , i.e. for loop section 1, for loop section 2, for loop section 3, for loop section 4 and for loop section 5. The assumptions for the current are the same as those for round modelC in Section 3.1.
The radiation in the broadside direction, E _{BrdSQ}^{ n } _{ q } , is formulated to be(29)where the polarization vector due to  and directed currents is(30)with(31) (32)When q = 0, the current is and equation (30) yields equation (25): . Hence, equations (26)–(28) are obtained, resulting in broadside radiation , shown in Figure 23a.
Next, we consider when q = 1 in equation (29). Equation (31) becomes zero: . Hence, equation (29) is written as(33)where for n = 1, 2, and 3 is(34) (35) (36) The broadside radiation for these results, , is shown as Figure 23b.
As performed for round modelN, an arbitrary case where deviation factor q is neither 0 nor 1 is considered, choosing q = 0.5 as one example. When n = 1, equations (31) and (32) become(37) (38)
Hence, equation (30) for n = 1 and q = 0.5 becomes(39) (40)When n = 2,(41) (42)
In addition, when n = 3, (43) (44)
The sum of equations (43) and (44) leads to(45) (46) Using the above results, broadside radiation is summarized in Figure 23c.
Fig. 22 Square modelN. 
Fig. 23 for square modelN. (a) q = 0. (b) q = 1. (c) q = 0.5. 
4.4 Square metaloopN antenna (SQRMTLPN): approximation of square modelN
Figure 24 shows an antenna that approximates square modelN discussed in Section 4.3. This antenna, referred to as the square metaloopN antenna or SQRMTLPN, is formed by bending the Ntype metaline in Figure 2b. The formed metaloop is composed of five arm sections having strips that support the flow of directed current. The configuration parameters for the SQRMTLPN are summarized in Table 1.
It is difficult to realize q = 0 and 1 for the squarebent metaline due to mutual coupling between the arm filaments, as with the RNDMTLPN. Therefore, it is inferred from Figure 23 c that, as the frequency is increased, the broadside radiation will change from LHCP to RHCP and then to LHCP. This is confirmed by the frequency response of the triband gain shown in Figure 25a. It should be emphasized that the triband gain characteristic obtained from the SQRMTLPN cannot be realized by its counterpart, the RNDMTLPN (see Fig. 15a). Also, it should be emphasized that the VSWR within the three bands is desirably small, as shown in Figure 25b. The change in the principal component of the radiation pattern with increase in frequency (from E _{L}, through E _{R}, to E _{L}), shown in Figure 26, is consistent with the change in the components of the triband gain (from G _{L}, through G _{R}, to G _{L}) [13].
Note that the upper bandedge frequency for the fast wave, f _{U}, and the third Hion frequency f _{H3} in Figure 25a are the values (original values) obtained when the metaline is straight. The position of f _{U} and f _{H3} is affected when the metaline is bent, due to mutual coupling; in other words, guided wavelength λ _{g} is changed. This adds an error component to the position of f _{U} and f _{H3}. A true upper bandedge frequency and a true third Hion frequency become larger than the original values, because the maximum LHCP gain appearing on the right side in Figure 25a is the gain due to the radiation at the true third Hion frequency f _{H3}.
Fig. 24 Square metaloopN antenna or SQRMTLPN, where N _{cell} = 19. 
Fig. 25 Frequency response for the square metaloopN antenna (SQRMTLPN). (a) Gain. (b) VSWR. 
Fig. 26 Radiation pattern for the square metaloopN antenna (SQRMTLPN). (a) At 2.55 GHz = f _{N1}. (b) At 3.55 GHz near f _{H1} _{ }=_{ }3.475 GHz. (c) At 4.8 GHz. 
5 Conclusions
The linearly polarized Ctype metaline and circularly polarized Ntype metaline have been designed such that their dispersion diagrams are as similar as possible. The polarization for round modelC has been analysed and the RNDMTLPC antenna, which is an approximation of modelC and implemented using a Ctype metaline, has been investigated. As expected from the analysis results of round modelC, the RNDMTLPC antenna acts as a dualband counter CP radiation element across a fast wave frequency region. In addition, the polarization for round modelN has been analysed. Subsequently, the RNDMTLPN antenna, which is an approximation of modelN and implemented using an Ntype metaline, has been investigated across the fast wave frequency region. It is found that the RNDMTLPN antenna has principal LHCP radiation at frequencies near f _{N1} and additional lowlevel RHCP radiation at frequencies near f _{H1}.
Similar investigation has been performed for square metaloop antennas SQRMTLPC and SQRMTLPN after analysing square modelC and square modelN, respectively. Consistent with the model analysis, the SQRMTLPC acts as a dualband counter CP radiation element across a fast wave frequency region. An interesting point is that the SQRMTLPN acts as a triband CP radiation element when deviation factor q is neither zero nor one. Such behaviour cannot be found in the counterpart antenna, i.e. the RNDMTLPN antenna.
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Cite this article as: Hisamatsu Nakano, Tomoki Abe, Junji Yamauchi, Quasitheoretical investigation of four circularly polarized metaloop antennas, EPJ Appl. Metamat. 6, 2, (2019)
All Tables
All Figures
Fig. 1 (a) Conventional loop antenna that forms linearly polarized bidirectional radiation. (b) Loop antenna with perturbation elements that form circularly polarized radiation. (c) Metaspiral antenna that forms circularly polarized radiation. 

In the text 
Fig. 2 Metalines. (a) Perspective view (upper), exploded view (middle) and side view (bottom) of a Ctype metaline. (b) Perspective view (upper), exploded view (middle) and side view (bottom) of an Ntype metaline. 

In the text 
Fig. 3 Dispersion diagram for the unit cell. (a) Ctype metaline. (b) Ntype metaline. 

In the text 
Fig. 4 Guided wavelength λ _{g} for the unit cell as a function of frequency. (a) Ctype metaline. (b) Ntype metaline. 

In the text 
Fig. 5 Radiation from a 10cell metaline (N _{cell} = 10). The para meters in Table 1 are used, with R _{B} = 60Ω. (a) Ctype metaline. (b) Ntype metaline. 

In the text 
Fig. 6 Round modelC. 

In the text 
Fig. 7 for round modelC. 

In the text 
Fig. 8 Round metaloopC antenna (RNDMTLPC), made by curving the Ctype metaline shown in Figure 2a, where N _{cell} = 19. 

In the text 
Fig. 9 Normalized loop circumference 2πr _{LP}/λ _{g} as a function of frequency, where the radius of the metaloop is r _{LP} = 31.8 mm. 

In the text 
Fig. 10 Frequency response for the round metaloopC antenna (RNDMTLPC). (a) LHCP and RHCP gains. (b) VSWR. 

In the text 
Fig. 11 Normalized radiation patterns for the round metaloopC antenna when gains G _{L} and G _{R} are maximal. (a) At f _{GLmax} = 2.70 GHz. (b) At f _{GRmax} = 3.60 GHz. 

In the text 
Fig. 12 Round modelN. 

In the text 
Fig. 13 for round modelN. (a) q = 0. (b) q = 1. (c) q = 0.5. 

In the text 
Fig. 14 Round metaloopN antenna (RNDMTLPN), made by curving the Ntype metaline shown in Figure 2b, where N _{cell} = 19. 

In the text 
Fig. 15 Frequency response for the round metaloopN antenna. f _{N1} = 2.55 GHz and f _{H1} = 3.425 GHz. (a) Gain. (b) VSWR. 

In the text 
Fig. 16 Normalized radiation pattern for the round metaloopN antenna. (a) At 2.575 GHz near f _{N1}= 2.55 GHz. (b) At 3.5 GHz near f _{H1} _{ }= 3.425 GHz. 

In the text 
Fig. 17 Square modelC. 

In the text 
Fig. 18 Broadside radiation for square modelC. 

In the text 
Fig. 19 Square metaloopC antenna (SQRMTLPC), where N _{cell} = 19. 

In the text 
Fig. 20 Frequency response for the square metaloopC antenna. (a) Gain. (b) VSWR. 

In the text 
Fig. 21 Normalized radiation pattern for the square metaloopC antenna. (a) At frequency f _{GLmax} = 2.58 GHz near f _{N1} _{ }= 2.60 GHz. (b) At frequency f _{GRmax} _{ }= 3.51 GHz near f _{H1} = 3.50 GHz. 

In the text 
Fig. 22 Square modelN. 

In the text 
Fig. 23 for square modelN. (a) q = 0. (b) q = 1. (c) q = 0.5. 

In the text 
Fig. 24 Square metaloopN antenna or SQRMTLPN, where N _{cell} = 19. 

In the text 
Fig. 25 Frequency response for the square metaloopN antenna (SQRMTLPN). (a) Gain. (b) VSWR. 

In the text 
Fig. 26 Radiation pattern for the square metaloopN antenna (SQRMTLPN). (a) At 2.55 GHz = f _{N1}. (b) At 3.55 GHz near f _{H1} _{ }=_{ }3.475 GHz. (c) At 4.8 GHz. 

In the text 
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