1. Antenna in Plasma (AIP) Code Timothy W. Chevalier Umran S. Inan Timothy F. Bell March 4, 2008

2. 1 Stanford MURI Tasks Scientific Issues:  The sheath surrounding an electric dipole antenna operating in a plasma has a significant effect on the tuning properties.  Terminal impedance characteristics vary with applied voltage.  Active tuning may be needed.  Stanford has developed a general AIP code to determine sheath effects on radiation process. MURI Tasks:  Validation of our AIP code by laboratory experiments using LAPD.  UCLA will provide time measurements of voltage, current and field patterns for dipole antennas to compare with Stanford model.  Locate sources of error in current model and identify means for improvement.  Perform LAPD experiments on magnetic loop antennas.

3. 2 Outline 1.Introduction 2.Cold Plasma Electromagnetic Model 3.Current Distribution and Impedance Results 4.Warm Plasma Electrostatic Model 5.Plasma Sheath Results

4. 3 Coupling Regions Sheath Region  Near field  Reactive Energy (ES)  Highly nonlinear Warm Plasma Region  Transition zone  Reactive/Radiated Energy (EM & ES)  Nonlinear effects still important Cold Plasma Region  Far field  Radiated Energy (EM)  Linear environment R ES: Electrostatic EM: Electromagnetic ( R ¼ ¸ )( R ¿ ¸ m i n )( R À ¸ )

5. 4 Modeling Methodology (Poisson/Maxwell)-Vlasov Formulation (Poisson) (Maxwell) (Lorentz Force)  Near field antenna characteristics  Electrically short dipole antennas  ES & EM approaches r ¢ ~ E = P ® ½ ® ² o ( r £ ~ H = P N ~ J ® + ² o d ~ E d t r £ ~ E = ¡ ¹ o d ~ H d t @f @ t + ( v ¢ r r ) f + · F m ¢ r v ¸ f = 0 F = q ( ~ E + v £ ~ B )

6. 5 Moments of Vlasov Equation F ( v ) = @f @ t + ( v ¢ r r ) f + · F m ¢ r v ¸ f N th moment M n t h = ZZZ v 8 > > > < > > > : m F ( v ) d v m v F ( v ) d v m [ v ¡ u ][ v ¡ u ] F ( v ¡ u ) d ( v ¡ u ) m [ v ¡ u ][ v ¡ u ][ v ¡ u ] F ( v ¡ u ) d ( v ¡ u ) v ´ p h asespaceve l oc i t y u ´ average ° owve l oc i t y c = [ v ¡ u ] ´ ran d omve l oc i t y d ue t o t h erma l mo t i ons

7. 6 Fluid Representation of Plasma Fluid Moments (3 rd : heat flux)…… (2 nd : pressure)...... (1 st : momentum).. (0 th : mass density) Fluid Variables Additional Variables E ´ e l ec t r i c ¯ e ld vec t or B ´ magne t i c ¯ e ld vec t or m ´ mass q ´ c h arge ­ c ´ gyro f requencyvec t or

8. 7 Outline 1.Introduction 2.Cold Plasma Electromagnetic Model 3.Current Distribution and Impedance Results 4.Warm Plasma Electrostatic Model 5.Plasma Sheath Results

9. 8 Cold Plasma Fluid Approximation Fluid Description: Generalized Ohms Law Closure Assumption: d ~ J ® d t + º ® ~ J ® = q ® m ® ³ q ® n ® ~ E + ~ J ® £ ~ B o ´ P = n k T = 0

10. 9 Finite Difference Time and Frequency Domain Techniques (FDTD/FDFD) Time Domain (FDTD) Computational Mesh: FDTD Method:  Time domain solution of Maxwell’s equations.  Wide spread use in EM community Frequency Domain (FDFD) r £ ~ H = X N ¾ ® ~ E + ² o j ! ~ E r £ ~ E = ¡ ¹ o j ! ~ H ¾ ® = ² o ! 2 p ( j ! I ¡ ­ ) ¡ 1 ­ = 0 @ ¡ º ¡ ! b z ! b y ! b z ¡ º ¡ ! b x ¡ ! b y ! b x ¡ º 1 A Solves: Ax=B r £ ~ H = X N ~ J ® + ² o d ~ E d t r £ ~ E = ¡ ¹ o d ~ H d t d ~ J ® d t + º ® ~ J ® = q ® m ® ³ q ® n ® ~ E + ~ J ® £ ~ B o ´

11. Outline 1.Introduction 2.Cold Plasma Electromagnetic Model 3.Current Distribution and Impedance Results 4.Warm Plasma Electrostatic Model 5.Plasma Sheath Results

12. 11 Cold Plasma Simulation Setup Computational Domain: Antenna Properties  Length: 100 m  Diameter: 20 cm  Orientation: Perpendicular to B o  Position: Equatorial Plane

13. 12 Current Distribution for 100 m Antenna in Freespace Current distribution on linear antennaExcitation frequency: 10 kHz I / I o s i n · 2 ¼ ¸ µ L 2 § z ¶¸ L = ¸ 2 L ¿ ¸

14. 13 Current Distributions for 100 m Antenna at L=2 Excitation frequency: f < f LHR Excitation frequency: f > f LHR

15. 14 L=2L=3 Simulation vs. Theory [Wang and Bell., 1969,1970] [Wang., 1970] [Bell et. al., 2006] Input Impedance FormulaPrevious Analytical Work Z i n = V ( f ) I ( f ) = ( R ~ E ¢ dl ) f ee d ( H ~ H ¢ dl ) f ee d

16. 15 Conclusions Based upon Cold Plasma Approximation  Current distribution is triangular for cases demonstrated.  This result supports triangular assumption made in early analytical work.  Input impedance does not vary significantly as a function of frequency  The same antenna can be used over a broad frequency range; self tuning property.  Early analytical work should provide accurate estimates of radiation pattern of dipole antennas in a magnetoplasma [Wang and Bell., 1972].  What about the Sheath?

17. Outline 1.Introduction 2.Cold Plasma Electromagnetic Model 3.Current Distribution and Impedance Results 4.Warm Plasma Electrostatic Model 5.Plasma Sheath Results

18. 17 Warm Plasma Fluid Approximation Isothermal Approximation (2-moments) Closure Assumption: Adiabatic Approximation (3-moments) Closure Assumption: P = n k T r ¢ Q = 0

19. 18 Sheath region < Electrostatic approach is valid Electrostatic Approximation Poisson’s Equation L ¿ ¸ Triangular current distribution Nonlinear Equations Time domain approach Constant Voltage  Removes EM time-stepping constraint  Avoids problems associated with PML r ¢ ~ E = P ® ½ ® ² o ¸ m i n

20. Outline 1.Introduction 2.Cold Plasma Electromagnetic Model 3.Current Distribution and Impedance Results 4.Warm Plasma Electrostatic Model 5.Plasma Sheath Results

21. 20 Warm Plasma Simulation Setup (2-D) Computational Domain: Fluid closure relations:  Isothermal (2 - moments)  Adiabatic (3 - moments) Antenna Properties  Length: Infinite in z-direction  Diameter: 10 cm  Position: Equatorial Plane P = n k T r ¢ Q = 0 m i m e = 200 Plasma Properties L=2:  N = 2e9 # / m 3  f pe = 400 kHz  f pi = 28 kHz  f ce = 110 kHz  f ci = 550 Hz L=3:  N = 1e9 # / m 3  f pe = 284 kHz  f pi = 20 kHz  f ce = 33 kHz  f ci = 163 Hz Mass ratio:

22. 21 Simulation of Infinite Line Source Plane of symmetry: Simulation Properties  25 kHz sinusoid  f>f pi  No magnetic field

23. 22 Simulation of Infinite Line Source Plane of symmetry: Simulation Properties  25 kHz sinusoid  f>f pi  No magnetic field

24. 23 Simulation Properties  25 kHz sinusoid  f>f pi  No magnetic field Simulation of Infinite Line Source Plane of symmetry:

25. 24 IV Characteristics (Sinusoid) Non-magnetized Magnetized 25 kHz (f > f pi )15 kHz (f < f pi )

26. 25 IV Characteristics (Pulse) 15 kHz (f < f pi )25 kHz (f > f pi ) Non-magnetized Magnetized

27. 26 Warm Plasma Simulation Setup (3-D) Computational Domain: Adiabatic (full pressure tensor) Antenna Properties  Length: 20 m  Gap: 2 m  Diameter: 10 cm  Position: Equatorial Plane  Electron gun (removes charge) m i m e = 200 Plasma Properties L=2:  N = 2e9 # / m 3  f pe = 400 kHz  f pi = 28 kHz  f ce = 110 kHz  f ci = 550 Hz L=3:  N = 1e9 # / m 3  f pe = 284 kHz  f pi = 20 kHz  f ce = 33 kHz  f ci = 163 Hz Mass ratio:

28. 27 Simulation of 20 m Dipole at L=3 Orthographic Projection Gap Current Potential and Density Variation Current-Voltage

29. 28 Simulation of 20 m Dipole at L=3 with 20 cm Gap Orthographic Projection Gap Current Potential and Density Variation Current-Voltage

30. 29 Simulation of 20 m Dipole at L=3 without Electron Gun Orthographic Projection Gap Current Potential and Density Variation Current-Voltage

31. 30 Circuit Diagrams Tuning Circuit Diagram of Sheath Impedance:

32. 31 Conclusions Based upon Sheath Calculations  Sheath structure is periodic with both sinusoid and pulse waveform excitation.  Sheath is a quasi-steady state structure.  Proton densities vary significantly throughout sheath region and contribute to current collection.  Commonly used assumption of immobile protons within sheath region for frequencies above and below proton plasma frequency is not necessarily accurate.  Most notable in case of floating antenna.

33. 32 Validity of Fluid Code for Sheath Region  Ma and Schunk [1992], Thiemann et al. [1992]: Compared PIC and 2-moment fluid codes with diagonal pressure tensors surrounding spherical electrodes stepped to 10,000V.  Noisy PIC simulations agreed with results of fluid code with addition of more particles  Under-sampled distribution functions in PIC code are inherently noisy.  Plasma ringing and double layer formation was captured in both fluid and PIC simulations.  Very good qualitative agreement  Borovsky [1988], Calder and Laframboise[1990], Calder et al. [1993]: PIC simulations of spherical electrodes stepped to very large potentials.  Calder and Laframboise [1990], noted ringing effects could be driven to large amplitude by ion-electron two steam instability which a fluid code can capture.  No presence of electron-electron two-stream instability in any of the PIC simulations  Landau damping is negligible since the phase velocity of waves within the sheath region are generally different than thermal velocities.  No need to capture this effect in fluid code.  Though particle trapping within sheath is possible (mainly slow moving ions), the relatively small number of trapped particles results a minimal deviation of the potential variation within the sheath.  A fluid code can provide an accurate and more computationally efficient method for the determination of sheath characteristics!