1. Peng Zhang 1, Brad Hoff 2, Y. Y. Lau 1, David French 2, John Luginsland 3, R. M. Gilgenbach 1 1 Department of Nuclear Engineering and Radiological Sciences University of Michigan,Ann Arbor, MI 2 Air Force Research Laboratory, Kirtland AFB, NM 3 Air Force Office of Scientific Research, Arlington, VA Fifteenth Annual Directed Energy Symposium Albuquerque, New Mexico November 26-30, 2012 Excitation of Slow Wave Structure Work supported by AFOSR Grant no. FA9550-09-1-0662, AFRL, L-3 Communications Electron Devices, and Northrop Grumman Corporation.

2. Nonlinear Transmission Line ~ I(z, t) V(z,t) + - V s (0, t) + - 2 The nonlinearity in C and L will generate output pulse of various shapes from a voltage input pulse V s (0,t)

3. 3 Slow wave structure may be used as a radiator for the NLTL-generated pulses A NLTL-based radiation source will not require an electron beam Key question: Conversion of the NLTL output of a general temporal pulse shape into radiation

4. Motivation 4 A slow wave structure (SWS) readily converts an input voltage pulse into radiation when a circuit mode is excited [e.g. traveling wave tube (TWT)] Since the NLTL output voltage pulse may consist of a waveform of a rather general temporal shape, the Green’s function for the SWS is of fundamental interest So we calculate the Green’s function for textbook example

5. Excitation of a Slow Wave Structure Green’s Function L (x 0,y 0 ) y x d h (a,a’) 5 Response (R) to will give the green’s function response by superposition

6. Excitation of a Slow Wave Structure 6 Ideal current source: (a heavy sheet beam) Ideal current source (a heavy sheet beam)

7. Excitation of a Slow Wave Structure m-th eigenfunction H zm satisfies Eigenfrequency  m is root of 7 Represent the frequency domain solution in terms of the vacuum eigenmode solution of the slow wave structure

8. Excitation of a Slow Wave Structure Green’s Function Constructed Note: a). Too much information b). Difficult to compare with simulations 8 The above equation can be calculated numerically

9. Excitation of a Slow Wave Structure 9

10. 10 Q is adjusted to match the peak value with the numerical simulation Finite response at the resonant frequency by including a finite quality factor, Q

11. ICEPIC particle-in-cell simulation (Taper and finite) Analytical model (Uniform and infinite) 11

12. Analytical vs. ICEPIC results 12 The peak locations in the ICEPIC simulations were found to occur within 1.5% of the analytically predicted values in spite of the difference in geometries m = 1

13. 13 Analytical vs. ICEPIC results The peak locations in the ICEPIC agree with analytical values within 1.5% despite the differences in the geometries m = 2 m = 3

14. Conclusions * 14 Green’s function for a slow wave structure is developed to include all spatial harmonics and all radial modes Analytic solution was compared to particle-in-cell simulation. Resonant peak locations from ICEPIC were found to agree with analytically predicted values to within 1.5% in spite of the difference in geometries Analytical prediction not only furnishes a vital tool to guide experimental efforts, but also provides capabilities to benchmark computational algorithm development * P. Zhang, B. Hoff, Y. Y. Lau, D. M. French, J. W. Luginsland, Phys. Plasmas 19, 123104 (2012).