Full-wave method of calculation of electromagnetic fields...
Transcript of Full-wave method of calculation of electromagnetic fields...
STANFORDE L E C T R I C A LE N G I N E E R I N G
Full-wave method of calculation of electromagneticfields in stratified media
Nikolai G. Lehtinen, Timothy F. Bell, Umran S. Inan
STAR Laboratory, Stanford University, Stanford, CA, U.S.A.ICEAA ’11, Torino, Italy
September 15, 2011
STANFORDE L E C T R I C A LE N G I N E E R I N G
N. Lehtinen (Stanford) Full-wave method September 15, 2011 1
Stanford Full-Wave Method (SFWM) code STANFORDE L E C T R I C A LE N G I N E E R I N G
Outline
1 Stanford Full-Wave Method (SFWM) codeCapabilitiesDescription
2 Applications of SFWMWave propagation through ionosphereModulated electrojet VLF radiationVLF transmitter radiationRadiation from lightningEarth-ionosphere waveguide modesScattering on ionospheric disturbances
3 Conclusions
4 Extra slides
N. Lehtinen (Stanford) Full-wave method September 15, 2011 2
Stanford Full-Wave Method (SFWM) code Capabilities STANFORDE L E C T R I C A LE N G I N E E R I N G
Outline
1 Stanford Full-Wave Method (SFWM) codeCapabilitiesDescription
2 Applications of SFWMWave propagation through ionosphereModulated electrojet VLF radiationVLF transmitter radiationRadiation from lightningEarth-ionosphere waveguide modesScattering on ionospheric disturbances
3 Conclusions
4 Extra slides
N. Lehtinen (Stanford) Full-wave method September 15, 2011 3
Stanford Full-Wave Method (SFWM) code Capabilities STANFORDE L E C T R I C A LE N G I N E E R I N G
Capabilities:Arbitrary plane stratified medium, e.g., a horizontally-stratifiedmagnetized plasma with an arbitrary direction of geomagnetic field(such as ionosphere)Arbitrary configuration of harmonically varying currentsProvides full wave 3D solution of both whistler waves launchedinto ionosphere and VLF waves launched into Earth-ionospherewaveguideStable against the “swamping” instability by evanescent wavesEfficient use of the computer resources, easily parallelized
Applications:Trans-ionospheric propagationEarth-ionosphere waveguide propagationScattering on D-region disturbances
N. Lehtinen (Stanford) Full-wave method September 15, 2011 4
Stanford Full-Wave Method (SFWM) code Description STANFORDE L E C T R I C A LE N G I N E E R I N G
Outline
1 Stanford Full-Wave Method (SFWM) codeCapabilitiesDescription
2 Applications of SFWMWave propagation through ionosphereModulated electrojet VLF radiationVLF transmitter radiationRadiation from lightningEarth-ionosphere waveguide modesScattering on ionospheric disturbances
3 Conclusions
4 Extra slides
N. Lehtinen (Stanford) Full-wave method September 15, 2011 5
Stanford Full-Wave Method (SFWM) code Description STANFORDE L E C T R I C A LE N G I N E E R I N G
z
xy
Ne
B
z2
zk
zk+1
zM
z1=0
εk
We work in Fourier (horizontal wave vector k⊥) domain:1 For each k⊥ = const (Snell’s law) =⇒ find kz , E and H in each
layer for each of 4 plane wave modes: 2 up (u) and 2 down (d)2 Use continuity of E⊥ and H⊥ between layers to find reflection
coefficients Ru,d and mode amplitudes u, dRecursion order Ru
k+1 → Ruk and uk → uk+1 provides stability
against “swamping” of solution by evanescent wavesRepresent source currents as boundary conditions on E⊥ and H⊥between layers
3 Inverse Fourier transform from k⊥ to r⊥N. Lehtinen (Stanford) Full-wave method September 15, 2011 6
Stanford Full-Wave Method (SFWM) code Description STANFORDE L E C T R I C A LE N G I N E E R I N G
Full-wave method backgroundGeneral description of waves in stratified media and reviews:
Altman, C., and K. Suchy (1991), Reciprocity, Spatial Mapping and Time Reversal in Electromagnetics, Kluwer AcademicPublishers, Boston.
REVIEW: Budden, K. G. (1985), The Propagation of Radio Waves: The Theory of Radio Waves of Low Power in theIonosphere and Magnetosphere, Cambridge Univ. Press, Cambridge, Chapter 18.
Clemmow, P. C., and J. Heading (1954), Coupled forms of the differential equations governing radio propagation in theionosphere, Math. Proc. of the Cambridge Phil. Soc., 50(2), 319–333, doi:10.1017/S030500410002939X.
Wait, J. R. (1970), Electromagnetic Waves in Stratified Media, 2nd ed., Pergamon, New York.Methods using subtraction of growing evanescent mode proposed by Pitteway [1965], The numerical calculation of wave-fields,reflexion coefficients and polarizations for long radio waves in the lower ionosphere I, Phil. Trans. R. Soc. London, Ser. A,257(1079), 219–241:
Nagano, I., K. Miyamura, S. Yagitani, I. Kimura, T. Okada, K. Hashimoto, and A. Y. Wong (1994), Full wave calculationmethod of VLF wave radiated from a dipole antenna in the ionosphere — analysis of joint experiment by HIPAS andAkebono satellite, Electr. Commun. Jpn. Commun., 77(11), 59–71, doi:10.1002/ecja.4410771106.
Scarabucci, R. R. (1969), Analytical and numerical treatment of wave-propagation in the lower ionosphere, Tech. ReportNo. 3412-11, Stanford University, Stanford, CA.
Yagitani, S., I. Nagano, K. Miyamura, and I. Kimura (1994), Full wave calculation of ELF/VLF propagation from a dipolesource located in the lower ionosphere, Radio Sci., 29(1), 39–54, doi:10.1029/93RS01728.
Methods using the same recursion order as the one described here:
Nygren, T. (1982), A method of full wave analysis with improved stability, Planet. Space Sci., 30(4), 427–430,doi:10.1016/0032-0633(82)90048-4.
Wang, T.-I. (1971), Intermode coupling at ion whistler frequencies in a stratified collisionless ionosphere, J. Geophys.Res., 76(4), 947–959, doi:10.1029/JA076i004p00947.
Other methods:Using “matrizants”, e.g.: Bossy, L. (1979), Wave propagation in stratified anisotropic media, J. Geophys., 46, 1–14.
N. Lehtinen (Stanford) Full-wave method September 15, 2011 7
Applications of SFWM STANFORDE L E C T R I C A LE N G I N E E R I N G
Outline
1 Stanford Full-Wave Method (SFWM) codeCapabilitiesDescription
2 Applications of SFWMWave propagation through ionosphereModulated electrojet VLF radiationVLF transmitter radiationRadiation from lightningEarth-ionosphere waveguide modesScattering on ionospheric disturbances
3 Conclusions
4 Extra slides
N. Lehtinen (Stanford) Full-wave method September 15, 2011 8
Applications of SFWM Wave propagation through ionosphere STANFORDE L E C T R I C A LE N G I N E E R I N G
Outline
1 Stanford Full-Wave Method (SFWM) codeCapabilitiesDescription
2 Applications of SFWMWave propagation through ionosphereModulated electrojet VLF radiationVLF transmitter radiationRadiation from lightningEarth-ionosphere waveguide modesScattering on ionospheric disturbances
3 Conclusions
4 Extra slides
N. Lehtinen (Stanford) Full-wave method September 15, 2011 9
Applications of SFWM Wave propagation through ionosphere STANFORDE L E C T R I C A LE N G I N E E R I N G
TE and TM modes
θi
θB
B0
H
ETE mode
transmitted wave
reflected wave
θi
θB
B0
E
HTM mode
transmitted wave
reflected wave
N. Lehtinen (Stanford) Full-wave method September 15, 2011 10
Applications of SFWM Wave propagation through ionosphere STANFORDE L E C T R I C A LE N G I N E E R I N G
Highly-collisional plasma (low altitude)
−5 0 5−5
−4
−3
−2
−1
0
1
2
3
4
5
nx
n y
h=73 kmX=ω
p2/ω2=11.201, Y=ω
H/ω=139.96, Z=ν/ω=68.509
UP: n
1
UP: n2
DOWN: n3
DOWN: n4
Geomagnetic field Solid = Re(nz)
Dashed = Im(nz)
Z � XThe modes at thesame frequency inthe lower D-regionare (almost) like invacuum, with |n| = 1
N. Lehtinen (Stanford) Full-wave method September 15, 2011 11
Applications of SFWM Wave propagation through ionosphere STANFORDE L E C T R I C A LE N G I N E E R I N G
Waves in magnetized plasma (high altitude)
−500 0 500−500
−400
−300
−200
−100
0
100
200
300
400
500
nx
n y
h=300 kmX=ω
p2/ω2=3.6738e+05, Y=ω
H/ω=139.96, Z=ν/ω=6.2761e−05
UP: n
1 (evanescent)
UP: n2 (whistler)
DOWN: n3 (whistler)
DOWN: n4 (evanescent)
Geomagnetic field
Solid = Re(nz)
Dashed = Im(nz)
In VLF/ELFfrequency range, the4 solutions inionosphere andmagnetosphere areupward anddownward whistlerand evanescentwaves
N. Lehtinen (Stanford) Full-wave method September 15, 2011 12
Applications of SFWM Wave propagation through ionosphere STANFORDE L E C T R I C A LE N G I N E E R I N G
Comparison with absorption from Helliwell [1965]Total losses (including absorption and reflection) for TE (circles) and TM (crosses)incident at nighttime ionosphere at θ = 45◦ with vertical in the plane of B0.
10 20 30 40 50 60 70 80 9010
0
101
102
Geomagnetic latitude, deg
Tota
l lo
ss, dB
Nighttime loss, θ=45o; φ=0
o
Helliwell: 2 kHz
Helliwell: 20 kHz
FWM: 2 kHz
FWM: 20 kHz TE and TM modesbehave differently
Reflection losses∼4 dB (not taken intoaccount by Helliwell[1965])
The transmittedenergy is maximized‖ B0
Helliwell, R. A. (1965), Whistlers and related ionospheric phenomena, Stanford University Press, Stanford, CA.
N. Lehtinen (Stanford) Full-wave method September 15, 2011 13
Applications of SFWM Modulated electrojet VLF radiation STANFORDE L E C T R I C A LE N G I N E E R I N G
Outline
1 Stanford Full-Wave Method (SFWM) codeCapabilitiesDescription
2 Applications of SFWMWave propagation through ionosphereModulated electrojet VLF radiationVLF transmitter radiationRadiation from lightningEarth-ionosphere waveguide modesScattering on ionospheric disturbances
3 Conclusions
4 Extra slides
N. Lehtinen (Stanford) Full-wave method September 15, 2011 14
Applications of SFWM Modulated electrojet VLF radiation STANFORDE L E C T R I C A LE N G I N E E R I N G
SchematicsThe HF radiation (∼3–10 MHz) modulates the polar electrojet at ELF/VLF frequencies(fmod ∼ 300 Hz–3 kHz)
N. Lehtinen (Stanford) Full-wave method September 15, 2011 15
Applications of SFWM Modulated electrojet VLF radiation STANFORDE L E C T R I C A LE N G I N E E R I N G
Vertical slice of fields [Lehtinen and Inan, 2008]HAARP HF beam at 3.2 MHz, ERP = 24 MW, fmod = 1875 Hz.
Upper: Bx , formation of upward whistler “column” (∼3 W)Lower: Ez , radiation into the Earth-ionosphere waveguide (∼1 W)
y, km
z,
km
Bx at x=0 km, pT
−400 −300 −200 −100 0 100 200 300 4000
50
100
150
200
250
−20
−10
0
10
20
y, km
z,
km
Ez at x=0 km, mV/m
−400 −300 −200 −100 0 100 200 300 4000
50
100
150
200
250
−0.6
−0.4
−0.2
0
0.2
0.4
(a)
(b)
N. Lehtinen (Stanford) Full-wave method September 15, 2011 16
Applications of SFWM VLF transmitter radiation STANFORDE L E C T R I C A LE N G I N E E R I N G
Outline
1 Stanford Full-Wave Method (SFWM) codeCapabilitiesDescription
2 Applications of SFWMWave propagation through ionosphereModulated electrojet VLF radiationVLF transmitter radiationRadiation from lightningEarth-ionosphere waveguide modesScattering on ionospheric disturbances
3 Conclusions
4 Extra slides
N. Lehtinen (Stanford) Full-wave method September 15, 2011 17
Applications of SFWM VLF transmitter radiation STANFORDE L E C T R I C A LE N G I N E E R I N G
SchematicsWaves are launched both into ionosphere and into the Earth-ionosphere waveguide.
N. Lehtinen (Stanford) Full-wave method September 15, 2011 18
Applications of SFWM VLF transmitter radiation STANFORDE L E C T R I C A LE N G I N E E R I N G
SFWM resultsFeatures of VLF radiation from NPM (f = 21.4 kHz, P = 424 kW, B = 34 µT, d = 38.4◦):
Mode interference (both on the ground and in space)
Higher attenuation westward on the ground
Radiation higher along B into space
West-East asymmetry for radiation into space
x, km (← geomag. W-E→)
y,km
(←geom
ag.S-N→)
NPM: iri 00h 22−Jun−2010Field on the ground
−600 −400 −200 0 200 400 600−600
−400
−200
0
200
400
600
log10
B⊥ , T
−11
−10.5
−10
−9.5
−9
−8.5
x, km (← geomag. W-E→)
y,km
(←geom
ag.S-N→)
NPM: iri 00h 22−Jun−2010Upward power flux at h=137.5 km
−600 −400 −200 0 200 400 600−600
−400
−200
0
200
400
600
log10
Sz, W/m2
−10.5
−10
−9.5
−9
−8.5
−8
−7.5
−7
−6.5
N. Lehtinen (Stanford) Full-wave method September 15, 2011 19
Applications of SFWM VLF transmitter radiation STANFORDE L E C T R I C A LE N G I N E E R I N G
Comparison with satellite dataDEMETER pass over NWC transmitter on October 24, 2006, starting at 14:50:40 UT:(a) Ne; (b) VLF energy flux at 700 km; (c) data and SFWM results for two profiles ofelectron-neutral collision rate νe [Lehtinen and Inan, 2009].
Ew
spectrogram
f, k
Hz (c)
19.2
19.4
19.6
19.8
102
103
104
105
Ew
, µ
V/m
Ew
data
FWM, νen
[V02]
FWM, νen
[H65]
Bw
spectrogram
f, k
Hz
19.2
19.4
19.6
19.8
−24 −22 −20 −18 −16 −14 −12 −10 −810
0
101
102
103
Latitude, deg
Bw
, p
T
Bw
data
FWM, νen
[V02]
FWM, νen
[H65]
100
101
102
103
104
80
85
90
95
100
105
110
Ne, cm
−3
h,
km
(a)
(b)
Longitude, deg
La
titu
de
, d
eg
DEMETER passes over NWC
110 112 114 116 118
−24
−23
−22
−21
−20
−19
−18
−17
dB
re
W/m
2
−95
−90
−85
−80
−75
−70
−65
−60
The deficit of measured VLF energy points to scattering on irregularities in the ionosphere
N. Lehtinen (Stanford) Full-wave method September 15, 2011 20
Applications of SFWM Radiation from lightning STANFORDE L E C T R I C A LE N G I N E E R I N G
Outline
1 Stanford Full-Wave Method (SFWM) codeCapabilitiesDescription
2 Applications of SFWMWave propagation through ionosphereModulated electrojet VLF radiationVLF transmitter radiationRadiation from lightningEarth-ionosphere waveguide modesScattering on ionospheric disturbances
3 Conclusions
4 Extra slides
N. Lehtinen (Stanford) Full-wave method September 15, 2011 21
Applications of SFWM Radiation from lightning STANFORDE L E C T R I C A LE N G I N E E R I N G
V- and U-shaped streaks in satellite spectrogramsDEMETER observations [Parrot et al., 2008]
⇐On path: V-shaped
Off-path: U-shaped⇒
N. Lehtinen (Stanford) Full-wave method September 15, 2011 22
Applications of SFWM Radiation from lightning STANFORDE L E C T R I C A LE N G I N E E R I N G
Calculated VLF “streaks” on and off-path of a satellite
x, km
f, kH
z
log10(Sz) at 120 km, W/m2/Hz
P0=1 W/Hz, ∆y=0 km
(a)
−1000 −500 0 500 1000
10
20
30
40
−16
−15
−14
−13
−12
−11
x, km
f, kH
z
log10(Sz) at 120 km, W/m2/Hz
P0=1 W/Hz, ∆y=300 km
(b)
−500 0 500
10
20
30
40
−15
−14
−13
−12
N. Lehtinen (Stanford) Full-wave method September 15, 2011 23
Applications of SFWM Radiation from lightning STANFORDE L E C T R I C A LE N G I N E E R I N G
Spectrogram of E field on the ground
x, km
f, kH
zR1/2E
z at 0 km, dB re (V/m−m1/2)2/Hz, for I2=1 (A−m)2/Hz
200 400 600 800 1000
5
10
15
20
25
30
35
40
−140
−135
−130
−125
−120
−115
−110
−105
−100
−95
−90
N. Lehtinen (Stanford) Full-wave method September 15, 2011 24
Applications of SFWM Radiation from lightning STANFORDE L E C T R I C A LE N G I N E E R I N G
Time-domain sferic waveform calculationApply an inverse Fourier transform: ω→ t to the field calculated for a set ofωValidated using the sferic database [Said, personal communication, 2010]
0 0.5 1 1.5−50
0
50
100
150150 km
(B/I),
pT
/kA
theory
experiment
0 0.5 1 1.5−50
0
50
100300 km
0 0.5 1 1.5−60
−40
−20
0
20
40535 km
t, ms
(B/I),
pT
/kA
0 0.5 1 1.5−60
−40
−20
0
20
40940 km
t, ms
N. Lehtinen (Stanford) Full-wave method September 15, 2011 25
Applications of SFWM Earth-ionosphere waveguide modes STANFORDE L E C T R I C A LE N G I N E E R I N G
Outline
1 Stanford Full-Wave Method (SFWM) codeCapabilitiesDescription
2 Applications of SFWMWave propagation through ionosphereModulated electrojet VLF radiationVLF transmitter radiationRadiation from lightningEarth-ionosphere waveguide modesScattering on ionospheric disturbances
3 Conclusions
4 Extra slides
N. Lehtinen (Stanford) Full-wave method September 15, 2011 26
Applications of SFWM Earth-ionosphere waveguide modes STANFORDE L E C T R I C A LE N G I N E E R I N G
Modal (dispersion) relation: det(1 − RdRu) = 0Attenuation is due to both absorption and radiation into ionosphere
Strongest modes at R0 = 2000 km
10−5
10−4
10−3
10−2
0
20
40
60
80
100
120
E, V/m, at 2000 km from transmitter (P=250 kW)
h,
km
QTM1
QTM2
QTM3
QTM4
N. Lehtinen (Stanford) Full-wave method September 15, 2011 27
Applications of SFWM Earth-ionosphere waveguide modes STANFORDE L E C T R I C A LE N G I N E E R I N G
Waveguide radiation leaking into ionosphere
−200 0 200 400 600 800 1000−114
−112
−110
−108
−106
−104
−102
−100
−98
x=(R−R0), km
Sz, d
B r
e W
/m2
Waveguide leakage into ionosphere, at R0=2000 km
QTM1QTM2QTM3QTM4
N. Lehtinen (Stanford) Full-wave method September 15, 2011 28
Applications of SFWM Earth-ionosphere waveguide modes STANFORDE L E C T R I C A LE N G I N E E R I N G
Role of whistler radiation in the total attenuation
10−2
10−1
100
101
QTM1
QTM2
QTM3
QTM4
dB/Mm
Attenuation
TotalDue to whistler radiation
N. Lehtinen (Stanford) Full-wave method September 15, 2011 29
Applications of SFWM Scattering on ionospheric disturbances STANFORDE L E C T R I C A LE N G I N E E R I N G
Outline
1 Stanford Full-Wave Method (SFWM) codeCapabilitiesDescription
2 Applications of SFWMWave propagation through ionosphereModulated electrojet VLF radiationVLF transmitter radiationRadiation from lightningEarth-ionosphere waveguide modesScattering on ionospheric disturbances
3 Conclusions
4 Extra slides
N. Lehtinen (Stanford) Full-wave method September 15, 2011 30
Applications of SFWM Scattering on ionospheric disturbances STANFORDE L E C T R I C A LE N G I N E E R I N G
SchematicsELVES are transient luminous events caused by lightning EMP (electromagnetic pulse)
N. Lehtinen (Stanford) Full-wave method September 15, 2011 31
Applications of SFWM Scattering on ionospheric disturbances STANFORDE L E C T R I C A LE N G I N E E R I N G
Scattering in Born approximation
Born approximation: neglect the scattered field Es compared tothe incident field E0 inside the scattering regionE0 acting together with the perturbation ∆σ creates currents whichradiate Es
N. Lehtinen (Stanford) Full-wave method September 15, 2011 32
Applications of SFWM Scattering on ionospheric disturbances STANFORDE L E C T R I C A LE N G I N E E R I N G
Scattered VLF wave (∆A) on the ground[Lehtinen et al., 2010]
VLF wave propagates from left to right (x = R − R0)scattering region is indicated by a circle at x = 0, y = 0
∫∆Ne dz, m−2
x, km
y, km
Vert 20V/m ×1
(a)
−200 0 200
−200
−100
0
100
200
0
2
4
6
x 109
x, km
Horiz 5V/m ×60
(b)
−200 0 200
−200
−100
0
100
200
−4
−3
−2
−1
0x 10
10
x, km
y, km
Vert 20V/m ×1
(a)
−200 0 200
−200
−100
0
100
200
0
2
4
6
x 109
x, km
Horiz 5V/m ×60
(b)
−200 0 200
−200
−100
0
100
200
−4
−3
−2
−1
0x 10
10
Amplitude change
y,
km
∆ A in dB, for mode QTM1, φB=270
°
Vert 20V/m ×1
−200
−100
0
100
200
−0.01
0
0.01
x, km
y,
km
Horiz 5V/m ×60
−200 0 200 400 600 800 1000−200
−100
0
100
200
−0.05
0
0.05
N. Lehtinen (Stanford) Full-wave method September 15, 2011 33
Conclusions STANFORDE L E C T R I C A LE N G I N E E R I N G
Outline
1 Stanford Full-Wave Method (SFWM) codeCapabilitiesDescription
2 Applications of SFWMWave propagation through ionosphereModulated electrojet VLF radiationVLF transmitter radiationRadiation from lightningEarth-ionosphere waveguide modesScattering on ionospheric disturbances
3 Conclusions
4 Extra slides
N. Lehtinen (Stanford) Full-wave method September 15, 2011 34
Conclusions STANFORDE L E C T R I C A LE N G I N E E R I N G
Future work
Method of moments instead of Born approximation (moreaccurate)Effects of the Earth’s curvatureLong-distance propagation by using segmented path of a VLFsignal
N. Lehtinen (Stanford) Full-wave method September 15, 2011 35
Conclusions STANFORDE L E C T R I C A LE N G I N E E R I N G
Summary
We have developed a full-wave method (Stanford FMW) which isstable against “swamping” and easily parallelized. It has been appliedto calculate
Plane wave transport through ionosphereModulated electrojet current radiationRadiation from ground-based transmittersRadiation from lightningEarth-iononsphere waveguide modesScattering on ionospheric disturbances
N. Lehtinen (Stanford) Full-wave method September 15, 2011 36
Conclusions STANFORDE L E C T R I C A LE N G I N E E R I N G
Acknowledgments
This work was supported byONR grant N0014-09-1-0034NSF grant ATM-0836326DARPA grant HR-0011-10-1-0058DoAF grant FA9453-11-C0011DTRA grant HDTRA-10-1-0115
to Stanford University.
N. Lehtinen (Stanford) Full-wave method September 15, 2011 37
Conclusions STANFORDE L E C T R I C A LE N G I N E E R I N G
SFWM Bibliography
Lehtinen, N. G., and U. S. Inan (2008), Radiation of ELF/VLFwaves by harmonically varying currents into a stratified ionospherewith application to radiation by a modulated electrojet, J. Geophys.Res., 113, A06301, doi:10.1029/2007JA012911.
Lehtinen, N. G., and U. S. Inan (2009), Full-wave modeling oftransionospheric propagation of VLF waves, Geophys. Res. Lett.,36, L03104, doi:10.1029/2008GL036535.
Lehtinen, N. G., R. A. Marshall and U. S. Inan (2010), Full-wavemodeling of “early” VLF perturbations caused by lightningelectromagnetic pulses, J. Geophys. Res., 115, A00E40,doi:10.1029/2009JA014776.
N. Lehtinen (Stanford) Full-wave method September 15, 2011 38
Extra slides STANFORDE L E C T R I C A LE N G I N E E R I N G
Outline
1 Stanford Full-Wave Method (SFWM) codeCapabilitiesDescription
2 Applications of SFWMWave propagation through ionosphereModulated electrojet VLF radiationVLF transmitter radiationRadiation from lightningEarth-ionosphere waveguide modesScattering on ionospheric disturbances
3 Conclusions
4 Extra slides
N. Lehtinen (Stanford) Full-wave method September 15, 2011 39
Extra slides STANFORDE L E C T R I C A LE N G I N E E R I N G
Amplitudes u, d and reflection coefficients Ru, Rd
Separation into u and d:
Im kz > 0⇒ upward mode, Im kz 6 0⇒ downward mode
Total electromagnetic field ∝ ei(k⊥·r⊥) is a linear combination(E(z)H(z)
)= F
(u(z)d(z)
), F is a 6× 4 matrix
Propagation up or down within a uniform layer:
u(z > 0) = Pu(z)u(0), d(z < 0) = Pd (z)d(0)
Pu(z) =(
eikuz1z 00 eiku
z2z
), Pd (z) =
(eikd
z1z 00 eikd
z2z
)||Pu || 6 1, ||Pd || 6 1⇒ numerical stabilityReflection coefficients “from above” Ru and “from below” Rd (2× 2 matrices):
d = Ruu u = Rd d
Transporting the reflection coefficients through a layer of thickness h:
Ru(z < 0) = Pd (z)Ru(0)Pu(z), Rd (z > 0) = Pu(z)Rd (0)Pd (z)
u, d are transported “forward”; Ru , Rd are transported “backward.”N. Lehtinen (Stanford) Full-wave method September 15, 2011 40
Extra slides STANFORDE L E C T R I C A LE N G I N E E R I N G
At the boundaries between layers
∆E⊥ = 0 and ∆H⊥ = 0 at each layer boundaryWe find Ru′ in terms of Ru and Rd′ in terms of Rd
We also find u ′ = Uu and d ′ = Dd
The source currents are assumed to be flowing in thin layers⇒ theygive ∆E⊥ 6= 0 and ∆H⊥ 6= 0.
N. Lehtinen (Stanford) Full-wave method September 15, 2011 41
Extra slides STANFORDE L E C T R I C A LE N G I N E E R I N G
Upward flux in k⊥-space
Earth-ionosphere waveguide modes manifest as maxima in k⊥-space
k⊥ = {kx , ky }
r⊥ = {x , y }k⊥ ↔ r⊥
by Fourier transform.
Pup =
∫∫Sz(k⊥)
d2k⊥(2π)2
– gives a moreaccurate result than∫∫
Sz(r⊥) d2r⊥:
Pup
P= 13%
nx=k
x/k
0 (k
0=w/c)
n y=k y/k
0
NPM: iri 00h 22−Jun−2010Upward power flux at h=137.5 km in k⊥ −space
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
log10
Sz, W
7
8
9
10
11
12
13
14
N. Lehtinen (Stanford) Full-wave method September 15, 2011 42
Extra slides STANFORDE L E C T R I C A LE N G I N E E R I N G
Scattering in Born approximation
We must solve the wave equation:
∇× (∇× E) − k20 ε̂E = 0 (k0 = ω/c)
where E = E0 + Es, ε̂ = ε̂0 + ∆ε̂,E0 — incoming wave (e.g., a waveguide mode in stratified ε̂0);Es — scattered wave;∆ε̂ — inhomogeneous change in the dielectric permittivity tensor.
In a stratified waveguide ∇× (∇× E0) − k20 ε̂0E0 = 0 =⇒
∇× (∇× Es) − k20 ε̂0Es = k2
0∆ε̂(E0 + Es)
Born approximation: neglect Es compared to E0 inside thescattering region (rhs).rhs gives the source currents for scattered waves:
∇× (∇× Es) − k20 ε̂0Es ≈ k2
0∆ε̂E0 = ik0Z0∆Jwhere Z0 is the impedance of free space.
N. Lehtinen (Stanford) Full-wave method September 15, 2011 43