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```Applications of the 3D
electromagnetic model to
some
challenging optical problems
September 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: E i ( r )e it ,
J ( r )e it ,
incident field
imposed current
– Total field: E ( r )eit , H ( r )eit
– Maxwell equations’ are equivalent to Vector Helmholtz
Equation:
 2 0 0  r (r ) Ε (r )    [  r (r ) 1   E ]( r )  i 0 J ,
– Scattered field: E s (r )  E (r )  E i (r )
– The scattered field satisfies the Sommerfeld radiation
condition.
• Variational formulation
– E=E0+Es
iz
k
j
• Calculate E0 in Multilayer
– S-polarization, i
– P-polarization, j
– E  and H  is the source term
– Tangential field h(z), e(z) in basis (i,l)
i
l
– Up and down recursion
h( z )  i  Y  ( z )e( z )  i  C  ( z )  i

e( z )  l   Z  ( z ) h ( z )  l  C  ( z )  l
h ( z )  i  Y  ( z ) e( z )  i  C  ( z )  i

e( z )  l  Z  ( z ) h ( z )  l  C  ( z )  l
– Amplitude for planewave


– Where e ( z ) and h ( z ) are the tangential source term.
• Numerical calculation
– Construction of Matrix
A E '  W






A  ~  2 0 0  r    r1         1 2 3dr

W  Source term  terms with zero field E0 on 
– Matrix property
• Complex symmetric
• indefinite
• 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 (biconjugate gradient stabilized algorithm )
• Propagation outside of computational domain
– The field of Electric Dipole in free space
1  eikr r

G ( r,0) p  k 1 
p

 ikr  4r r
ikr
 2r 
r  r
r
 1 ik   e
E
Ge ( r,0) p  k   p     3  p  p  2   
r  r
r
r   40 r r
 r
 r 
H
e
– However we need the field of electric
dipole in Multilayer
• Calculated by Fourier transformation plane wave expansion
• Using recursion as for calculating E0
• Stratton-Chu formula
 iE s , J ( rs )  p 

 



n  E ( r )  E ( r ) G
n  H ( r )  H 0 ( r )  GeE ( r , rs ) p dS ( r )

0
H
e
( r , rs ) p dS ( r )
Observation point


• Results: Near Field Optical Recording
• Background
• Geometry
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
Cross section
• = 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
Aperture stop
Projection lens
Photoresist
wafer
•
•
•
•
•
Material: Crome
= 193nm
High NA lithography
nCr=0.86 + 1.65 I
100nm
Perpendicular incident
planewave
260nm
720nm
340nm
Top view
Top view
Top view
finite conduct, E
Perfect conduct, E
Cross section yz-plane
finite conduct, E
Perfect conduct, E
Far field