equazioni maxwell semplificate nella materia b ... · equazioni maxwell semplificate nella materia...
TRANSCRIPT
µµ
εε
BMBH
EPEDtDHB
tBED
o
o
=−=
=+=∂
∂=×∇=⋅∇
∂∂−=×∇=⋅∇
0
0
Equazioni Maxwell semplificate nella materia
( )
( )
2
2
2
2
2
cn
EEkk
tEE
=
−=××
∂∂−=×∇×∇
µ ε
µ εω
µ ε
Equazioni onde
Onde piane
jiij
zxzyzx
yzyyyx
xzxyxx
ED εεεεεεεεεεε
=
=
iii
z
y
x
EDED εε
εε
=
=
000000
Dielettrici anisotropi
Assi principali
( )
20
20
222
222
222
2
0
ezoyx
z
y
x
yxzzyzx
zyzxyxy
zxyxzyx
nn
EEE
kkkkkkkkkkkkkkkkkk
EEkk
εεεεε
µ εωµ εω
µ εω
µ εω
===
=
−−−−
−−
−=××
Equazioni Onde mezzi anisotropi
Materiali uniassici
0
0
0
0
00
2222
222
2
2222
2
22
22
22
22
22
222
22
=
−
−
−
−−
=
−
−
−−
zyye
zo
zyo
z
y
x
ye
zy
zyzo
zyo
kkkcnk
cnkk
cn
EEE
kcnkk
kkkcn
kkcn
ωωω
ω
ω
ω
Equazioni Onde mezzi uniassici (kx=0)
),,0(0
)0,0,(0
2222
222
2
22
222
22
zyzyye
zo
xzyo
EEEkkkcnk
cn
EEkkcn
==
−
−
−
==
−−
ωω
ω
Equazioni Onde mezzi uniassici (kx=0)
Onda ordinaria
Onda straordinaria
( )
( ) 2
2
2
2
2
2
222
2
222
sincos1
),,0(
)0,0,(
eoe
zyee
xoo
nnn
EEEcnk
EEcnk
θθθ
θω
ω
+=
==
==
Equazioni Onde mezzi uniassici (kx=0)
Onda ordinaria (TE)
Onda straordinaria (TM)y
z
k
θoE
y
z
k
θeE
y
z
cnoω
cne )(θω
oe nn >
y
z
cnoω
cne )(θω
oe nn <
Ghiaccio1.309 1.310Quarzo 1.544 1.553ZnS 2.354 2.358
Tormalina 1.638 1.618 Calcite 1.658 1.486 KDP 1.507 1.467
Superfici isofrequenza
o
BHED
DHkBk
BEkDk
µε
ω
ω
==
−=×=⋅
=×=⋅
0
0
Equazioni Maxwell in onde piane
( )
( )
)0,0,1()(
sin,cos,0
)sin,cos,0(
cos,sin,0)(
)sin,cos,0()0,0,1(
)0,0,1(
cos,sin,0
DncHDE
DDcnk
DncHDE
DDcnk
eo
zxe
e
ee
oo
xo
o
oo
θεθ
εθ
θθ
θθθω
θθε
θθω
=
−=
−=
=
−==
=
=
Onda ordinaria
Onda straordinariay
z
ok
θ
oo DE,
y
z
ek
θ eE
eD
DHk
ω−=×
EDDk
ε==⋅ 0
oH
eH
( )
( ) ( )22
022
22
0
,
sincossincos
cossinsincos
)0,0,1()(
sin,cos,0
)sin,cos,0(
cos,sin,0)(
eo
oeoe
eo
oe
zxe
eelonge
eo
zxe
e
ee
nnnnnnD
nnnnD
DkEkE
DncHDE
DDcnk
+−=−=
=
−=⋅=
=
−=
−=
=
εθθ
εθθ
εθθ
εθθ
θεθ
εθ
θθ
θθθω
Onda straordinaria: Componente longitudinale
y
z
ek
θ eE
eD
eH
( )
( )
kDncS
DncHDE
cnk
kDncS
DncHDE
cnk
zxe
ee
zxe
ee
ox
oo
xo
oo
//cos,sin,0)(
)0,0,1()(
sin,cos,0
cos,sin,0)(
//)cos,sin,0(
)sin,cos,0()0,0,1(
cos,sin,0
2
2
=
−=
−=
=
=
−==
=
εθ
εθ
θ
θεθ
εθ
θθθω
θθε
θθε
θθωOnda ordinaria
Onda straordinariay
z
ok
θ
oo DE,
y
z
ek
θ eE
eD
oH
eH
Velocità di gruppo
( )
gexze
xze
o
z
e
ykg
e
y
o
z
e
y
o
z
eoe
vDn
DncS
cnnk
nkcv
nk
nkc
nk
nkc
nnck
nck
22
0
2
022
2
2
2
2
2
2
2
2
22
2
2
2
222
2
222
)(1cos,sin,0
)(
cos,sin,0)(,,0
sincos
θεεθ
εθ
θ
εθ
εθεθ
ωω
ω
θθθ
ω
=
=
=
=∇=
+=
+=
+==
gegemge
eeo
zxeeem
ee
zxe
e
eem
vvvUvDn
S
nD
nnD
DDEU
DncHDE
DD
vUS
===
=
+=
=
+=⋅=
=
−=
−=
≡
2
0
20
2
2
2
2
2
0
2
222*
)(1
)(sincos
sincos
)0,0,1()(
sin,cos,0
)sin,cos,0(
θε
θεθθ
ε
εθ
εθ
θεθ
εθ
θθ
Velocità energia
( ) ( )( ) ( )
( ) ( ) ( )BACACBCBA
EEEHkHkE
HHHEkEkH
EEHkHk
HHEkEk
EHkHEk
×⋅=×⋅=×⋅
+⋅−=×+×⋅
+⋅=×+×⋅
−−=×+×
+=×+×
−=×=×
ω ε δδ ω εδδ
ω µ δδ ω µδδ
ω ε δδ ω εδδ
ω µ δδ ω µδδ
ω εω µ
Velocità energia=Velocità gruppo 1
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
( ) ( ) ( )( ) ( )
( ) ( )( ) ( ) ( )EEHHEHkHEk
EEHHHEk
EEHHEEHH
EHkHEkHEk
HHHHEEEE
EEEEEHkHEk
HHHHHEkHEk
εδµδωδδ
εµδ ωδ
εδµδωεµδ ω
δδδ
µδµ δεδε δ
ε δωεδ ωδδ
µ δωµδ ωδδ
⋅+⋅+×⋅+×⋅−
=⋅+⋅−×⋅
⋅+⋅+⋅+⋅
=×⋅−×⋅+×⋅
⋅=⋅⋅=⋅
⋅−⋅−=×⋅+×⋅
⋅+⋅=×⋅+×⋅
2
2
Velocità energia=Velocità gruppo 2
( ) ( )( ) ( ) ( )
( ) ( )( ) ( )
( ) ( )
( ) ∴∇⋅=⋅=
⋅+⋅=×⋅
=−×⋅−+×⋅
=⋅+⋅−×⋅
⋅+⋅+×⋅−×⋅
=⋅+⋅−×⋅
kkUSk
EEHHHEk
HEkHEHkE
EEHHHEk
EEHHEkHHkE
EEHHHEk
kel
ωδδ ωδδ ω
εµδ ωδ
ω µδω εδ
εµδ ωδ
εδµδωδδ
εµδ ωδ
21
0
2
2
Velocità energia=Velocità gruppo 3
Doppia rifrazione
NOTA: effetto anche ad incidenza normale
y
z
cω c
ne )(θω
cnoω
Doppia rifrazione
z
cω
cne )(θω
cnoω
Doppia rifrazione n
k
z
cω
cne )(θω
cnoω
Doppia rifrazione n
Sk
Principio di Huygens
Per vedere immagine in movimento: http://www2.polito.it/ricerca/qdbf/fil/indicegenerale/ottica/ottica_fisica/birifrangenza.htm
Principio di Huygens
y
z
cω
cne )(θω
cnoω
Lamine ritardanti
yo
o
ye
e
ecnk
ecnk
ˆ
ˆ
ω
ω
=
=
Lamine ritardanti
yo
o
ye
e
ecnk
ecnk
ˆ
ˆ
ω
ω
=
=
d
dndcn
dndcn
ooo
eee
λπωϕ
λπωϕ
2
2
==
==
( ) ( )
( ) ( )
( ) ( )tkyjjjzx
tkyjjz
jxout
tkyjzxin
eeeeeE
eeeeeEE
eeeEE
ooe
eo
ωϕϕϕ
ωϕϕ
ω
−−
−
−
+=
=+=
+=
)(0
0
0
ˆˆ
ˆˆ
ˆˆ
zx
y
( )oeoe nnd −=−λπϕϕ 2
Lamine λ/2
dndcn
dndcn
eoo
eee
λπωϕ
λπωϕ
2
2
==
==
( ) ( )
( ) ( )
( ) ( )tkyjjzx
tkyjjjzxout
tkyjzxin
eeeeE
eeeeeEE
eeeEE
o
o
ωϕ
ωϕπ
ω
−
−
−
−=
=+=
+=
ˆˆ
ˆˆ
ˆˆ
0
0
0
zx
yoe nn
md−
+= 12
λλ
)12( +=− moe πϕϕ
Lamine λ/4
( ) ( )
( )
( ) ( )tkyjjzx
tkyjjj
zxout
tkyjzxin
eeejeE
eeeeeEE
eeeEE
o
o
ωϕ
ωϕπ
ω
−
−
−
+=
=
+=
+=
ˆˆ
ˆˆ
ˆˆ
0
20
0
oe nnmd
−
+= 14
λλ
)212( +=− moe πϕϕ
Polarizer prisms
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