Project supported by the National Natural Science Foundation of China (Grant Nos. 61205205 and 6156508508), the General Program of Yunnan Provincial Applied Basic Research Project, China (Grant No. 2016FB009), and the Foundation for Personnel Training Projects of Yunnan Province, China (Grant No. KKSY201207068).
Project supported by the National Natural Science Foundation of China (Grant Nos. 61205205 and 6156508508), the General Program of Yunnan Provincial Applied Basic Research Project, China (Grant No. 2016FB009), and the Foundation for Personnel Training Projects of Yunnan Province, China (Grant No. KKSY201207068).
† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 61205205 and 6156508508), the General Program of Yunnan Provincial Applied Basic Research Project, China (Grant No. 2016FB009), and the Foundation for Personnel Training Projects of Yunnan Province, China (Grant No. KKSY201207068).
The refraction index of the quantized lossy composite right-left handed transmission line (CRLH-TL) is deduced in the thermal coherence state. The results show that the negative refraction index (herein the left-handedness) can be implemented by the electric circuit dissipative factors(i.e., the resistances R and conductances G) in a higher frequency band (1.446 GHz≤ ω ≤ 15 GHz), and flexibly adjusted by the left-handed circuit components (Cl, Ll) and the right-handed circuit components (Cr, Lr) at a lower frequency (ω = 0.995 GHz). The flexible adjustment for left-handedness in a wider bandwidth will be significant for the microscale circuit design of the CRLH-TL and may make the theoretical preparation for its compact applications.
A well-established route to constructing negative refractive index materials (NRMs)[1,2] is based on Veselago’s theory of left-handed materials (LHM), simultaneous negative permittivity (ϵ) and magnetic permeability (μ) with different types of metamaterials.[3–9] Although very exciting from a physics point of view, the negative ϵ and μ produced by electromagnetic resonance may bring about a very highloss[10,11] and narrow bandwidth consequently. Due to the weaknesses of resonant-type structures, three groups almost simultaneously introduced a transmission line (TL) approach for NRM,[12–15] i.e., the composite right–left handed transmission line (CRLH-TL), which refers to the right-handedness accompanying the positive refraction index at high frequencies and to the left-handedness with the negative refraction index (NRI) at lower frequencies.[16] The CRLH-TL, initially the non-resonant-type one, is perhaps one of the most representative and potential candidates due to its low loss, broad operating frequency band, and planar configuration,[16,17] which is often related to the easy fabrication for NRI applications in a suite of novel guided-wave,[18] radiated-wave,[19] and refracted-wave devices and structures.[20,21] Nowadays, the CRLH-TLs show a tendency to the compact applications influenced by the nanotechnology and microelectronics.[22–24] Recently, a new class of miniaturized nonreciprocal leaky-wave antenna is proposed for miniaturization, nonreciprocal properties and wide-angle scanning at the same time.[25] With four unit cells of CRLH-TL a wide-band loop antenna is proposed in a compact size.[26]
However, when the compact size of the CRLH-TL approaches to the Fermi wavelength, the quantum effects[22–24,27] on the CRLH-TL must be taken into account. In our former work, we firstly deduced the quantum features of NRI of the lossless mesoscopic left-handed transmission line (LH-TL).[28,29] Then we quantized the lossy LH-TL and discussed the quantum influence of dissipation on the NRI[30] in a displaced squeezed Fock state. Some novel quantum effects were revealed and the significance for the miniaturization application of LH-TL was pointed out.
In this paper, the flexible adjustment of negative refraction index is achieved from a wider frequency bandwidth than in the former work[16,28–30] in the quantized lossy CRLH-TL in the thermal coherence state. The rest of this paper is organized as follows. In Section
The equivalent unit-cell circuit model of the proposed lossy CRLH-TL is shown in Fig.
In a previous work,[16] the negative refraction index happened in the lower frequency bandwidth of the microwave wave. In order to investigate the minus refraction index, we should work with the refraction index from Eq. (
The frequency bandwidth for negative refraction index (i.e., the left-handedness) is of interest to the mesoscopic CRLH-TL. Besides the role of the imported resistance R and conductance G representing the loss is also important here. Figure
A notable question is whether it is possible to realize negative refraction index (i.e., the left-handedness) in the lower frequency bands. Figures
The refraction index dependent shunt capacitor Cr is provided in Fig.
At the lower frequency ω = 0.995 GHz, figure
Before concluding this paper, we should point out that how to broaden the frequency band for a negative refraction index is an active field for metamaterials, and the metamaterials within a wider band are always a research object. However, the frequency bands for negative refraction index only exist in the microwave band introduced by the first researchers[10–13] in the CRLH-TL. In our current study, we implement the negative refraction index within a wider frequency band in the quantized CRLH-TL and conclude that the negative refraction index in the higher frequency band (1.446 GHz≤ ω ≤ 15 GHz) tuned by resistance R and conductance G and at a lower frequency (ω = 0.995 GHz) tuned by the parameters of the circuit components. These achievements show the new characteristic for the quantized CRLH-TL, which should receive enough attention in the near future.
In the present paper, the negative refraction index has been obtained in a wider frequency band for the quantized CRLH-TL. The frequency domain for a negative refraction index is 1.446 GHz≤ ω ≤ 15 GHz with the values of negative refraction index being opposite through tuning the resistance R and conductance G, respectively. At a much lower frequency (ω = 0.995 GHz), the negative refraction indices can also be flexibly obtained by the left-handed circuit components (Cl, Ll), and the right-handed circuit components (Cr, Lr), respectively. The adjusting of negative refraction index within a wider bandwidth in this quantized CRLH-TL is significant for the microscale circuit design and compact applications for the CRLH-TL.
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