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are also plotted in Fig. 2c-d (right panel). In correspond- ing to the results in Fig. 1, the results in Fig. 2 show that although the peak in δρ ( q , ω ) (then the intensity of the modulation for LDOS) or the peak in δZ ( q , ω ) (then the intensity of the modulation for the ratio of LDOS at opposite energy) is changing rapidly and intricately with energy, the peaks, whose q -vectors oriented towards the 

antinodal direction, appear at finite | q | at very low en- ergy and then move steadily inwards toward the center, while the peaks with q -vectors along the nodal direction, appear and move steadily to large | q | with increasing en- ergy, in qualitative agreement with the STM experimen- tal results $^{4,5}$. However, although the position of the peak obtained from δρ ( q , ω ) is almost the same as that from δZ ( q , ω ), the peak intensity obtained from δZ ( q , ω ) is much stronger than that appeared in δρ ( q , ω ), and there- fore there is an enhancement of the QSI signatures in δZ ( q , ω ). This expected difference of the peak intensity obtained from δρ ( q , ω ) and δZ ( q , ω ) can be understood from the definition of δZ ( q , ω ) in Eq. (18). Although the modulations in δρ ( r , ω ) and δρ ( r , − ω ) are spatially quite similar, the spatial-phase relation for QSI is not known precisely. However, δZ ( q , ω ) is sensitive to the relation signs between δρ ( q , ω ) and δρ ( q , − ω ), namely, taking the ratio of LDOS reduces the in-plane component $^{13}$. This is why the the QSI extinction at the high energy in cuprate superconductors in the SC state is firstly observed from the experimental measurement of δZ ( q , ω ) $^{12–14}$. 

For a better understanding of the physical proper- ties of the QSI phenomenon in cuprate superconduc- tors, we have further performed a calculation for δZ ( q , ω ) with different doping concentrations, and the results of | δZ ( q , ω ) | as a function of momentum along the (a) [0 , 0] → [ π, π ] (nodal) direction and (b) [0 , 0] → [ π, 0] (antinodal) direction with energy ω = − 0 . 02 J at temper- ature T = 0 . 002 J for the doping concentration δ = 0 . 08, δ = 0 . 11, and δ = 0 . 15 (from bottom to top) in the pres- ence of single point-like potential scatterer of strength V = 0 . 1 J are plotted in Fig. 3. Obviously, the peak is doping dependent. With increasing the doping concen- tration, the weight of the peak increases. Furthermore, the position of the peak along the nodal direction ap- pears at finite | q | in the low doping concentration, and then moves towards to the center of the Brillouin zone with increasing doping concentration. In contrast to the case along the nodal direction, the position of the peak along the antinodal direction is located at finite | q | in the low doping concentration, and then is shifted to large | q | with increasing doping concentration. 

The energy dependence of the peaks in Fig. 2 has been used to extract the dispersion of the peaks, and the re- sults of (a) the positions and (b) amplitudes of the lowest energy peaks in | δZ ( q , ω ) | as a function of energy along the [0 , 0] → [ π, π ] direction for temperature T = 0 . 002 J at the doping concentration δ = 0 . 15 in the presence of single point-like potential scatterer of strength V = 0 . 1 J are plotted in Fig. 4. For comparison, the corresponding STM experimental results 14 of cuprate superconductors are also shown in Fig. 4 (inset). Our results show clearly that the sharp peak persists in a large momentum space region along the nodal direction and are in qualitative agreement with the STM experimental result $^{14}$. How- ever, the amplitude of the peak increases with increasing energy at the low energy, and reaches a maximum at the intermediate energy, then decreases rapidly with fur- ther increasing energy. Apparently, there is a substantial difference between theory and experiment, namely, the amplitude of the peak obtained from theory does not di- 

