1. Using controlling chaos technique to suppress self-modulation in a delayed feedback traveling wave tube oscillator Nikita M. Ryskin, Oleg S. Khavroshin Dept. of Nonlinear Physics Saratov State University, Russia

2. Controlling chaos An idea of controlling chaos technique for stabilizing of unstable periodic orbits in dynamical systems was suggested by Ott, Grebogy and Yorke (PRL 64, No. 11. P. 1196-1199 (1990)) A simple and effective method of chaos control by time-delayed feedback named time-delay autosynchronization (TDAS) was introduced by K. Pyragas (Phys. Lett. A 170, No. 6. P. 421-428 (1992)) dynamical system with chaotic dynamics system with time-delayed control with delay time equal to period of motion to be stabilized Usually T is unknown a priori Does not allow to stabilize high-frequency motion Dolov and Kuznetsov (Tech. Phys. 73, No. 8. P. 139-142 (2003)) suppress of self- modulation in a BWO via beam current modulation by external feedback control signal with delay time which depends on self-modulation period. Provides 1.5 times electronic efficiency enhancement and 2 times output power enhancement of the BWO single-frequency generation.

3. TWT delayed feedback oscillator Folded waveguide TWT oscillators for THz radiation sources Self-modulation and chaos is typical for such devices: Bhattacharjee S., Booske J. H., Kory C.L., et al. IEEE Trans. Plasma Sci. 32, 1002-1014 (2004). Han S.-T., So J.-K., Jang K.-H., Shin Y.-M., Kim J.-H., Chang S.-S., Ryskin N.M., Park G.-S. IEEE Trans. Electron Devices 52, 702-708 (2005).

4. Method of chaos control Boundary condition: Idea: To choose delay times and phases so that fundamental waves passing through two feedback legs appear in same phase, while the self- modulational sidebands appear in anti-phase and suppress each other.

5. Method of chaos control Consider propagation of a modulated signal Substituting into the boundary condition one can show that if we adjust the parameters as we obtain same as for the oscillator with single feedback and Sideband waves coming from different feedback legs weaken and for k=1/2 completely suppress each other

6. Basic equations N.M. Ryskin. Study of the Nonlinear Dynamics of a Traveling-Wave-Tube Oscillator with Delayed Feedback, Radiophys. Quantum Electron. 47, 116-128 (2004). N.M. Ryskin et al., Nonstationary Behavior in a Delayed Feedback Traveling Wave Tube Folded Waveguide Oscillator, Phys. Plasmas 11, 1194-1202 (2004). phase of an electron, F normalized field amplitude and normalized coordinate and time L CN, C Pierce gain parameter, N phase length first harmonic of the beam current Boundary conditions

7. Numerical results Parameters: Self modulation appears at L3.17 and Thus we must choose L=3.3, k=0 (no control). Deep self-modulation

8. Numerical results k=0.1. Week control feedback completely suppresses self-modulation

9. Numerical results L=4.1, k=0. Deep self-modulation. Among the sidebands, largest is that with frequency Thus we should change the control parameters:

10. Numerical results With old control parameters we cannot completely suppress self modulation even for k=0.3

11. Numerical results Proper adjustment of control parameters provides complete suppressing of self modulation for k=0.3.

12. However, it is important that in the TWT delayed feedback oscillator maximal efficiency and maximal amplitude of the output signal is attained below the self-modulation threshold (in contrast to BWO and delayed feedback klystron oscillator). The output amplitude decreases with L. It is advantageous to use the control feedback to stabilize single-frequency oscillations not on the fundamental mode but on one of the high-order modes. Higher frequency modes with lower phase velocities have higher interaction efficiency and output power. Dotted line marks steady states that are unstable without the control feedback.

13. Stabilization of a higher-frequency mode L=4.1, k=0.5,. Stabilization of the mode with frequency Output signal amplitude F out =3.1. For fundamental frequency mode and F out =2.39.

14. Stabilization of a higher-frequency mode L=4.1, k=0.5,. Stabilization of the mode with frequency provides even larger output signal amplitude F out =3.4. Mode switching is possible

15. Using control feedback to suppress gain ripples of a TWT amplifier For a TWT amplifier the boundary condition should be modified as follows: F in and are amplitude and frequency of the input signal. In that case it is easy to prove that one should choose Gain ripples arise due to end reflections. Now suppose that the first feedback path is the feedback due to reflections, while the second one is the external control feedback applied to suppress the gain ripples.

16. Using control feedback to suppress gain ripples of a TWT amplifier Gain ripples appear with the increase of (L=5.2, k=0) Additional feedback completely eliminates gain ripples for k=0.5 ( =0.3)