## 1 Introduction  
The terahertz (THz) technology has been investigated intensively for decades due to its great potential in fundamental and applied research fields, such as high-bandwidth communication, environmental monitoring and public security [1; 2]. Compact intrinsic-Josephson junction (JJ) THz sources made of the high-\(T_{\rm c}\) superconductor Bi\({}_{2}\)Sr\({}_{2}\)CaCu\({}_{2}\)O\({}_{8+\delta}\) (BSCCO) have attracted much attention for their continuous and tunable terahertz radiation at the sub-THz range [3; 4]. Bi\({}_{2}\)Sr\({}_{2}\)CaCu\({}_{2}\)O\({}_{8+\delta}\) naturally forms serial stacks of IJJs, with about 670 Josephson junctions per micron of the stack thickness [5]. For IJJ-THz emitters, radiation is generated by the ac Josephson effect. The supercurrent across the stack biased at a voltage \(V\) can oscillate synchronously at a frequency \(f=V/N\Phi_{0}\), where \(\Phi_{0}\) is the flux quantum, and \(N\) is the number of junctions in the stack. For a single junction, the emission power is typically on the order of pW to nW. To reach a high emission power, \(N\) should be large, and the IJJs in the stack should be synchronized. In 2007, coherent terahertz emission was first detected in a 1 \(\upmu\)m-thick BSSCO stack, with an emission power of up to 0.5 \(\upmu\)W and emission frequencies between 0.5 and 0.85 THz [6]. Since then, IJJ-THz emitters have been studied intensively, both experimentally and theoretically [6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31]. Single stacks with lateral dimensions on the order of a few 100 \(\upmu\)m and consisting of around 1000 IJJs now reach an emission frequency of up to 2.4 THz [29] and an emission power of about tens of microwatts [13; 22]. The maximum power of about 610 \(\upmu\)W was detected from a three-stack array [16]. Demonstrations of applications in terahertz imaging and detection have been given [10; 15; 20; 22; 26]. Theoretically, the frequencies of emission from a stack can be tuned by the applied voltage from a small value to more than 20 THz [3], limited by the gap energy of the IJJs. This, in principle, makes BSCCO emitters to be ideal terahertz sources for the application of wide-band continuous waves. However, the realized frequency range of BSCCO emitters is far from the theoretical values. Much attention has been paid to extending the emission frequency range of BSCCO emitter towards high frequencies. The maximum emission frequency of typical BSCCO emitters with more than 600 IJJs and lateral sizes over 100 \(\upmu\)m was always limited by self-heating, which leads to the formation of a hot spot in the stack [12], decreasing the voltage across the Josephson junction in the stack. In order to improve the emission frequency, sandwich-structure devices were created for good cooling [24; 25; 29; 30], with the highest emission frequency being improved to values above 2 THz [24; 29]. Recently, emission frequencies between 1 and 11 THz have been observed by fabricating smaller stacks and using an on-chip detector [32]. On the low-frequency side of emission, the Josephson plasma frequency (\(f_{\rm pl}\)) and resonant cavity modes play important roles in limiting the emission frequency. The \(f_{\rm pl}\) depends on temperature \(f_{\rm pl}(T)\propto\sqrt{j_{\rm c}(T)}\) and determines a cutoff emission frequency of about 4\(f_{\rm pl}\)[11]. At low bath temperatures \(T_{\rm b}\), \(f_{\rm pl}\) is saturated, leading to a constant cutoff emission frequency at the low-frequency side. With an increase in \(T_{\rm b}\), \(f_{\rm pl}\) decreases and approaches zero when \(T_{\rm b}\) is close to \(T_{\rm c}\), leading to a reduced low-frequency cutoff [11; 33]. Further, the excitation of collective cavity resonances of the whole stack has been shown to enhance terahertz emission [6; 7; 8; 9; 11; 17; 21; 23; 24; 25; 29; 34], with the emission frequency depending on the resonance mode.