Wider frequency domain for negative refraction index in a quantized composite right-left handed transmission line

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 (1.446GHz$\leq\omega\leq$ 15GHz), and flexibly adjusted by the left-handed circuit components ($C_l$, $L_l$) and the right-handed circuit components ($C_r$, $L_r$) at a lower frequency ($\omega$=0.995GHz) . 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.