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Applications of the 3D electromagnetic model to
somechallenging optical
problemsSeptember 24, 2004
Xiuhong wei, Paul Urbach, Arther Wachters
Supported by the Dutch Ministry of Economic Affairs under project TS01044
• Configurations– 2D or 3D– Non-periodic structure (Isolated pit in
multilayer)– Periodic in one direction (row of pits)– Periodic in two directions (bi-gratings)– Periodic in three directions (3D crystals)
• Source– Unrestricted incident field (plane wave, focused
spot) – Imposed current density
• Materials
– Linear.– In general anisotropic, (absorbing) dielectrics
and/or conductors:– Magnetic anisotropic materials (for
completeness):– Materials could be inhomogeneous:
• Mathematical Model– Given field: incident field
imposed current – Total field:– Maxwell equations’ are equivalent to Vector Helmholtz
Equation:
– Scattered field:
– The scattered field satisfies the Sommerfeld radiation condition.
,)(
,)(ti
tii
erJ
erE
,)]()([)()( 01
002 JirErrΕr rr
)()()( rErErE is
titi erHerE )( ,)(
• Variational formulation– E=E0+Es
• Calculate E0 in Multilayer– S-polarization, i
– P-polarization, j
– is the source term
– Tangential field h(z), e(z) in basis (i,l)
k iz
i
j
l
HE and
– Up and down recursion
– Amplitude for planewave
– Where are the tangential source term.)( and )( zhze
lzClzhzZlze
izCizezYizh
lzClzhzZlze
izCizezYizh
)()()()(
)()()()(
)()()()(
)()()()(
• Numerical calculation– Construction of Matrix
– Matrix property• Complex symmetric
• indefinite
rd
'
321~1
002
rrA
WEA
on Efield zero with terms termSource 0 W
• Iterative solver– RCM(reversing Cuthill-Mckee) reordering– Precondition
• ILUTP(incomplete LU threshold pivoting)– to solve a problem with 300,000 unknows, a fill-in is
needed of more than 600, which takes about 25hours on a Hewlett Packard machine (CPU = 107 FLOPS/sec)..
• Compare with MRILU(Matries reordering ILU)– More suitable for Finite Difference Method– Complex problems give an extra complication
– Krylov subspace method: BICGSTAB (bi-conjugate gradient stabilized algorithm )
• Propagation outside of computational domain– The field of Electric Dipole in free space
– However we need the field of electric dipole in Multilayer
• Calculated by Fourier transformation plane wave expansion• Using recursion as for calculating E0
–
r
e
r
ik
rrrrrkr
rr
e
ikrkr
r
ikrEe
ikrHe
02
2
4
13 )0,(
4
11 )0,(
prp
rrp
rpG
pr
pG
• Stratton-Chu formula
)( ),()()(
)( ),()()(
)(
0
0
,
rSprrGrErEn
rSprrGrHrHn
prE
s
s
s
d
d
i
He
Ee
Js
Observation point
• Results: Near Field Optical Recording• Background
• Geometry
Cross section
In the SIL:
kx nSIL kx
kx nSIL kx
Hence, Saptially frequences of the spot are increased , which means the spot became smaller
/2 nSIL
= 405nm
• NAeffective= 1.9
• Spotsize
/2NAeff=106nm
• Grooves(track)
Track pitch=226nm
Top view
Top view
Energy density, wall angle 55, E // groove
Energy density , wall angle 55, E groove
Top view
Energy density, wall angle 85, E //groove
Energy density , wall angle 85, E groove
Cross section xz-plane
Energy density, wall angle 55, E//groove
Energy density , wall angle 55, E groove
Cross section yz-plane
Energy density, wall angle 55, E // groove
Energy density , wall angle 55, E groove
– Lithography• Background
• Geometry
Incoherent Light source
Condenser
Mask
Aperture stop
Photoresist wafer
Projection lens
• Material: Crome = 193nm
• High NA lithography
• nCr=0.86 + 1.65 I
• Perpendicular incident
planewave
100nm
720nm
260nm
340nm
Top view
Serif mask, ESquare mask, E
Square mask, E Square mask, E
Top view
Square mask,
finite conduct, E
Square mask,
Perfect conduct, E
Top view
Cross section yz-plane
Square mask,
finite conduct, E
Square mask,
Perfect conduct, E
Far field
Square mask, E Square mask, E
acknowledge
• Our cluster in Philips, Paul Urbach, Arthur wachters, Jan Veerman
• Delft mathematical department, Kees Vuik, Kees Oosterlee, Yogi Erlangga, Mari Berglund
• Shell staffs, Ren´e-Edouard Plessix, Wim Mulder
