Part 4, Chapter 1

Electromagnetic Waves

Dispersion relations in magnetized plasmas

1.1 Introduction

This chapter provides comprehensive coverage of Dispersion relations in magnetized plasmas.

Topics include fundamental theory, mathematical derivations, physical interpretations, and practical applications in fusion energy, space physics, and astrophysics.

1.2 Key Concepts

Essential concepts and equations covered in this chapter with detailed explanations and examples.

MIT: Principles of Plasma Diagnostics

Lectures on refractive index diagnostics from MIT's Principles of Plasma Diagnostics course.

Lecture 5: Refractive Index Diagnostics I

Lecture 6: Refractive Index Diagnostics II

Lecture 7: Refractive Index Diagnostics III

Lecture 8: Refractive Index Diagnostics IV

Lecture 9: Refractive Index Diagnostics V: Abel and Faraday

Lecture 10: Refractive Index Diagnostics VI: Faraday and Reflectometry

Interactive Simulations

EM Wave Propagation in Plasma: Cutoffs and Resonances

Python

script.py54 lines

import numpy as np

# Electromagnetic wave dispersion in unmagnetized plasma:
# omega^2 = omega_pe^2 + k^2*c^2
# Cutoff at omega = omega_pe (n^2 = 0)
# In magnetized plasma: O-mode and X-mode with different cutoffs

eps0 = 8.854e-12
e = 1.602e-19
m_e = 9.109e-31
c = 3e8

print("EM Wave Propagation in Plasma")
print("=" * 60)

n_e = 1e18    # m^-3
omega_pe = np.sqrt(n_e * e**2 / (eps0 * m_e))
f_pe = omega_pe / (2 * np.pi)

print(f"n_e = {n_e:.1e} m^-3")
print(f"Plasma frequency: f_pe = {f_pe/1e9:.3f} GHz")
print(f"Cutoff wavelength: {c/f_pe*100:.1f} cm")
print()

# Dispersion relation: omega vs k
print("Dispersion relation (unmagnetized):")
print(f"{'f/f_pe':>8} {'f (GHz)':>10} {'k (1/m)':>12} {'lambda (cm)':>12} {'n_refr':>8} {'v_g/c':>8}")
print("-" * 60)

f_ratios = np.array([0.5, 0.8, 0.9, 1.0, 1.01, 1.1, 1.5, 2.0, 3.0, 5.0, 10.0])

for fr in f_ratios:
    f = fr * f_pe
    omega = 2 * np.pi * f

if omega > omega_pe:
        k = np.sqrt(omega**2 - omega_pe**2) / c
        n_refr = c * k / omega
        v_group = c * np.sqrt(1 - (omega_pe/omega)**2)
        lam = 2 * np.pi / k
        print(f"{fr:8.2f} {f/1e9:10.3f} {k:12.2f} {lam*100:12.2f} {n_refr:8.4f} {v_group/c:8.4f}")
    else:
        print(f"{fr:8.2f} {f/1e9:10.3f} {'evanescent':>12} {'---':>12} {'---':>8} {'---':>8}")

# Critical density for common frequencies
print(f"\nCritical density (cutoff) for common frequencies:")
print(f"{'Frequency':>15} {'n_critical (m^-3)':>18}")
print("-" * 35)
freqs = [("1 GHz", 1e9), ("10 GHz", 1e10), ("100 GHz", 1e11),
         ("1 THz", 1e12), ("CO2 laser", 2.83e13), ("Nd:YAG", 2.83e14)]
for name, f in freqs:
    n_crit = eps0 * m_e * (2*np.pi*f)**2 / e**2
    print(f"{name:>15} {n_crit:18.3e}")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Cold Plasma Dielectric Tensor Components

Fortran

program.f9065 lines

program cold_plasma_dielectric
  implicit none
  ! Stix parameters for cold magnetized plasma:
  ! S = 1 - sum(omega_ps^2/(omega^2-omega_cs^2))
  ! D = sum(omega_cs*omega_ps^2/(omega*(omega^2-omega_cs^2)))
  ! P = 1 - sum(omega_ps^2/omega^2)
  ! R = S + D, L = S - D
  real(8) :: e_ch, m_e, m_p, eps0, pi
  real(8) :: n_e, B, omega_pe, omega_pi
  real(8) :: omega_ce, omega_ci
  real(8) :: omega, S, D, P_stix, R, L_stix
  real(8) :: n2_O, n2_X, n2_R, n2_L
  integer :: i

e_ch = 1.602d-19
  m_e = 9.109d-31
  m_p = 1.673d-27
  eps0 = 8.854d-12
  pi = 4.0d0 * atan(1.0d0)

n_e = 1.0d18    ! m^-3
  B = 1.0d0       ! Tesla

omega_pe = sqrt(n_e * e_ch**2 / (eps0 * m_e))
  omega_pi = sqrt(n_e * e_ch**2 / (eps0 * m_p))
  omega_ce = e_ch * B / m_e
  omega_ci = e_ch * B / m_p

write(*,'(A)') "Cold Plasma Stix Parameters"
  write(*,'(A)') "==========================="
  write(*,'(A,ES10.3,A)') "omega_pe = ", omega_pe, " rad/s"
  write(*,'(A,ES10.3,A)') "omega_ce = ", omega_ce, " rad/s"
  write(*,'(A,ES10.3,A)') "omega_ci = ", omega_ci, " rad/s"
  write(*,'(A,F8.2)') "omega_pe/omega_ce = ", omega_pe/omega_ce
  write(*,*)

write(*,'(A12,A10,A10,A10,A10,A10)') &
    "omega/w_ce", "S", "D", "P", "R", "L"
  write(*,'(A)') repeat("-", 62)

do i = 1, 20
    omega = dble(i) * 0.2d0 * omega_ce

! Stix parameters (electron + ion contributions)
    S = 1.0d0 - omega_pe**2/(omega**2 - omega_ce**2) &
              - omega_pi**2/(omega**2 - omega_ci**2)

D = omega_ce*omega_pe**2/(omega*(omega**2 - omega_ce**2)) &
      - omega_ci*omega_pi**2/(omega*(omega**2 - omega_ci**2))

P_stix = 1.0d0 - omega_pe**2/omega**2 &
                    - omega_pi**2/omega**2

R = S + D
    L_stix = S - D

write(*,'(F12.2,F10.3,F10.3,F10.3,F10.3,F10.3)') &
      omega/omega_ce, S, D, P_stix, R, L_stix
  end do

write(*,*)
  write(*,'(A)') "Cutoffs: R=0 (R-cutoff), L=0 (L-cutoff), P=0"
  write(*,'(A)') "Resonances: S=0 (hybrid), R->inf, L->inf"
end program cold_plasma_dielectric

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

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