High power，wide bandwidth，high efficiency and high frequency electromagnetic radiation sources are in urgent demand，especially when the operating frequencies increase into the millimeter wave and terahertz ranges^［1-3］. Higher power terahertz devices enable longer transmission distance and higher data-rate capability for wireless communications. For example，satellite communications are projected to use w-band（70~110 GHz）^［4-5］. Vacuum electronic devices have high electronic efficiencies with a flexible manner without scattering phenomenon due to the vacuum environment，which are ideal for millimeter wave generations and amplifications. Therefore，vacuum electronic devices have strong advantages over solid state electronic devices，when the operating frequency is in the millimeter wave and higher terahertz band. Traveling wave tubes（TWT）are one of the most used vacuum electronic devices due to its capabilities of power and bandwidth.

However，high power amplifiers in sub-terahertz and terahertz frequency ranges are still lacking. A slow wave structure（SWS）is a core component of TWT devices and plays a crucial role in their performances，whilst also playing an important role in other large facilities such as linear synchrotron radiation light sources^［6］. As the frequency increases，the commonly used SWSs，such as helix and coupled cavity，encounter the problems of power capacity，heat dissipation and micron-size manufacture. Folded waveguides have shown the characteristics of combined capabilities of high power and wide frequency bandwidth，as well as operability compared to traditional helix and coupled cavity structures. The first TWT based on a folded waveguide（FW）was studied by Waterman in 1979. Since then，a great deal of experimental and theoretical studies on FW-TWT have been published and several modified FWs were proposed for improving the performance of FW-TWTs^［7-15］. However，FW will affect the electromagnetic field in the waveguide because the circular electron beam channel is made in the waveguide wall lending to increase reflection of FW and reduce its power capacity in the light of engineering.

In order to further improve TWT’s performance in power，efficiency and bandwidth in the millimeter-wave and terahertz band，one stage phase velocity taper（OSPVT）was studied and applied to a Folded Groove Waveguide（FGW）TWT for the first time in this paper. The technique could be considered to be a simple version of Dynamic Velocity Technology（DVT）which has been used in the helix TWTs^［16-17］. This idea could be applied to a variety of TWTs，such as those based on FWs or staggered double grating structures. FGW is suitable structure to operate in a sheet beam and the manufacture of the beam channel is unnecessary，which the electromagnetic field distribution affected in FGW is no longer existing. Therefore，FGW possesses higher power capability than FW，from the stand point of practical view. The FGW，as a kind of SWS，has been proposed for its low transmission loss because the ohmic loss on its wall is much lower than that of an FW^［18］. Moreover，the loss coefficient of an FGW decreases with the increase in the frequency^［19-20］，which would be helpful for a terahertz TWT. Various FGWs with the cross-sections in the form of rectangular，V-，double ridge and ridge-loaded shapes have been studied^［21-24］.

The article is composed as follows. Section 1 contains the theoretical analysis of the phase velocity of the wave in FGWs and the principle of DVT. Section 2 presents the manufacture of the FGWs and their measured transmission and dispersion characteristics. The simulated performance of a high power millimeter-wave FGW-TWT exploiting OSPVT is presented in Section 3. Finally a brief summary is given in this paper.

A schematic diagram of the FGW with sheet electron beam is shown in Fig. 1. The waveguide is formed between the two plates（1 and 2），both of which have a folded waveguide（3）.

Suppose the half-period of the FGW is P₀，which was designed to support a wave of an appropriate phase velocity V_p0 to efficiently interact with an electron beam with velocity U₀. After electrons loosing enough energy their reduced velocity would be no longer in synchronism with the wave，and hence，wave growth would stop. However，at that moment if the phase velocity of the wave is adjusted to a degree so that it continues to match with the reduced electron velocity，then interaction could in principle continue to extract energy from the electrons and wave continue to grow，hence，improve the electron efficiency of the TWT. In this paper the phase velocity of the electromagnetic wave is changed once by changing the half-period of the latter part of the FGW once，i.e. OSPVT，in the operating frequency. This extends the synchronous condition between the wave and the electron beam，transferring more energy from electrons to wave，hence improving the output power and electronic efficiency. When taking no account of the corner reflection and coupling between adjacent grooves，the analysis of the dispersion of the FGW is similar to that of an FW. The unchanged FGW has a curved groove length $l_{\mathrm{0}}$and a half period $P_{\mathrm{0}}$. The OSPVT FGW has a curved groove length $l_{\mathrm{1}}$and a half period${\mathrm{P}}_{\mathrm{1}}$. The length of the straight part is set to be s and defining the phase constant of the fundamental mode ${\beta }_{\mathrm{0}}$ and its nth spatial harmonic${\beta }_{n\mathrm{0},}$ the phase velocity of the fundamental mode $V_{p\mathrm{0}}$ and its nth spatial harmonic $V_{pn\mathrm{0}}$，then the following equations could be derived.

In formula（2），$\mathrm{\beta }=\sqrt[]{k^{\mathrm{2}}-k_c^{\mathrm{2}}}$，is the cutoff wave number and k is the wave number.

here c is velocity of light，from Eq.（3）${\mathrm{V}}_{Pn\mathrm{0}}$ is a monotone function of half-period $P_{\mathrm{0}}.$ Hence，the phase velocity of the fundamental mode decreases as $P_{\mathrm{0}}$reduces. Only when the electron beam velocity $U_{\mathrm{0}}$ is slightly faster than phase velocity of the wave $V_{p\mathrm{0}}$，can the electron energy be transferred to waves. Supposing the electron beam velocity become $U_{\mathrm{1}}$ when the beam-wave synchronization is about to lose，then electron energy transfer ratio is defined as η=（$U_{\mathrm{0}}$-$U_{\mathrm{1}}$）/$U_{\mathrm{0}}$. In order to maintain the synchronization condition，the electromagnetic wave phase speed $V_{p\mathrm{0}}$needs to reduce as the electron velocity $U_{\mathrm{0}}$decreases to $U_{\mathrm{1}}.$ This decrease in speed of the electromagnetic wave phase speed $V_{p\mathrm{0}}$ is defined as ∆. Then we have：

here $\mathrm{B}=\frac{U_{\mathrm{0}}-V_{p\mathrm{0}}}{C{V}_{p\mathrm{0}}}$ is pierce velocity parameter^［25］$.$

Hence，by properly adjusting the phase velocity of the wave in the latter part of the FGW，the synchronous condition between the wave and electron velocity could maintain.

As satellite communications plan to use w-band in the near future，a high power high efficiency folded groove waveguide millimeter-wave TWT based on OSPVT was designed（85~110 GHz）. Fig. 2（a）is a picture of the upper and lower half of the FGW machined by a high-speed CNC milling machine. The FGW with its input and output couplers was formed by assembling the two half with the help of eight screws and four location pins. The FGW with integrated input/output couplers includes two sections：a FGW section in the green box has unchanged half-period P₀ and an OSPVT section of 14 half-periods in the blue box with reduced period P₁. The dimensions of the structure are listed in Table 1.

The setup for the cold test of the FGWs with and without OSPVT sections is shown in Fig. 2（b）. The S-parameters were measured by a vector network analyzer（Ceyear：AV3672E）with millimeter wave frequency extending module（Ceyar：3640A）. Simulated and measured s-parameters are displayed in Fig. 3. The simulated and measured reflection S₁₁ were below -13 dB in the frequency range of 85 to 110 GHz. It can be seen from Fig. 3 that the measured curve of the reflection S₁₁ is not perfectly consistent with the simulated one，this is because of the reflection of the windows in the input and output couplers. When an effective conductivity of 3.6*10⁷ was used，the simulated transmission coefficient S₂₁ agreed well with the measured one. Both the simulations and the measurements verified that the proposed FGW has good transmission characteristics and a wide bandwidth.

The electric field in the FGW is similar to that of TE₁₁ mode in the circular waveguide. The measured wrapped phase of the operating mode in three FGWs is shown in Fig. 4. Fig. 4 shows the measured phase change of transmission（S₂₁）as a function of frequency for three structures：“structure a” has totally 98 half periods，84 P₀ and 14 P₁；"structure b" 84 P₀，and "structure c" 58 P₀. All three structures have the same input and output couplers. Therefore，the phase change of the wave passing through the OSPVT section（14*P₁）could be obtained by taking the phase difference between case b and a. Similarly，the phase change of the wave of 26*P₀ could be obtained by taking the difference between case c and b. The phase information was firstly unwrapped，then the differences between the unwrapped phases used to calculate the normalized phase velocity of the operating wave in FGWs with period P₀ and P₁ are shown in Fig. 5. It can be seen from Fig. 5 that the measured normalized phase velocities are in very good agreement with the simulated ones.

In order to evaluate the performance of OSPVT，a high power high efficiency FGW millimeter-wave TWT based on OSPVT technology was designed，and it was simulated by using CST particle studio. According to the measured transmission loss a conductivity of 3.6*10⁷ was used in the simulation. The optimized beam voltage was found to be 19.1 kV for the FGW with half-period P₀，and the beam current was set to be 0.15 A. The cross section of the electron beam is 0.12 mm*0.45 mm. In order to ensure no electron interception on the beam tunnel，an axial uniform magnetic field of 0.5 T was used in the simulation. Fig. 6 shows the simulation model and the bunching electron beam in FGW with 14 half periods OSPVT based on the PIC. It can be seen from Fig. 6（b）the phenomenon of electron beam bunching appears in the end of the FGW with 14 half periods OSPVT. Therefore，it is evident that some electrons are retarded and some electrons are accelerated.

Fig. 7 shows the simulated output power as a function of input power at 95 GHz，when the total length of the FGW is set at 104 half periods under five cases，with each of them having different length of OSPVT section. It can be seen from Fig. 7 that proper selection and application of OSPVT can effectively improve the output power. Furthermore，the designed high power high efficiency FGW millimeter-wave traveling wave tube with 20*P1 OSPVT could output 240 W saturated power when the input driving signal is 0.08 W to 0.15 W. This is approximately 70 W higher than the case of without OSPVT section. When the OSPVT was set to be 14*P₁，the saturated power and the electronic efficiency were simulated and are shown in Fig. 8. For comparison the saturated power and the electronic efficiency with the FGW OSPVT and without OSPVT are both shown in Fig. 8. In the whole frequency range of 85~110 GHz，the bandwidth of the saturated power above 100 W for the FGW with 14 half periods（14*P₁）OSPVT is larger than the case of without OSPVT. Fig. 8 also demonstrates that the application of OSPVT is beneficial to increase the electronic efficiency in the range of 85~110 GHz.

In this article，an FGW millimeter-wave traveling wave tube based on OSPVT was proposed. A combination of three FGWs with and without OSPVT section，each with the same input and output couplers was fabricated. The s-parameter measurements using VNA and simulations of these FGWs OSPVT demonstrated good transmission characteristics and wide bandwidth. Simulations predicted that the saturated power and electron efficiency were significantly improved by the application OSPVT in the operating frequency of 85~110 GHz. It is possible that the electronic efficiency of a TWT could be further improved by more sophisticated phase velocity change techniques such as DVT and multiple stage phase velocity change，which will be studied in the future.