prism coupling method for determining refractive index and
TRANSCRIPT
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Prism coupling method for determining
refractive index and thickness
Soft-matter Optical Spectroscopy Laboratory
발표자 : 이재진
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Introduction
problem
Prism coupling
P
S
c
Unknown parameter : reflective index, thickness glass
Necessary Non Destructive method
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Fresnel equation
𝑛1
𝑛2
𝜃𝑖 𝜃𝑟
𝜃𝑡
𝜃𝑖= 𝜃𝑟
𝒔𝒏𝒆𝒍𝒍′𝒔 𝒍𝒂𝒘 𝑛1𝑠𝑖𝑛𝜃𝑖=𝑛2𝑠𝑖𝑛𝜃𝑡
𝒍𝒂𝒘 𝒐𝒇 𝒓𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏
𝜃𝑖 𝜃𝑟
𝜃𝑡
𝑟𝑠=𝐸′
𝐸 𝑇𝐸 𝑟𝑠=
𝐸′
𝐸 𝑇𝑀
𝑡𝑝=𝐸′′
𝐸 𝑇𝐸 𝑡𝑝=
𝐸′′
𝐸 𝑇𝑀
Reflection coefficients are defined as amplitude ratios
z
x
y
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Fresnel equation
𝑟𝑠 =𝑛1𝑐𝑜𝑠𝜃𝑖 − 𝑛2𝑐𝑜𝑠𝜃𝑡
𝑛1𝑐𝑜𝑠𝜃𝑖 + 𝑛2𝑐𝑜𝑠𝜃𝑡=
𝑛1𝑐𝑜𝑠𝜃𝑖 − 𝑛2 1 −𝑛1𝑛2
𝑠𝑖𝑛𝜃𝑖
2
𝑛1𝑐𝑜𝑠𝜃𝑖 + 𝑛2 1 −𝑛1𝑛2
𝑠𝑖𝑛𝜃𝑖
2
𝑟𝑝 =𝑛1𝑐𝑜𝑠𝜃𝑡 − 𝑛2𝑐𝑜𝑠𝜃𝑖
𝑛1𝑐𝑜𝑠𝜃𝑡 + 𝑛2𝑐𝑜𝑠𝜃𝑖=
𝑛1 1 −𝑛1𝑛2
𝑠𝑖𝑛𝜃𝑖
2
− 𝑛2𝑐𝑜𝑠𝜃𝑖
𝑛1 1 −𝑛1𝑛2
𝑠𝑖𝑛𝜃𝑖
2
+ 𝑛2𝑐𝑜𝑠𝜃𝑖
𝑅𝑠= 𝑟𝑠2 𝑅𝑝= 𝑟𝑝
2
Reflectance
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Total reflection
Snell’s law 𝑛1𝑠𝑖𝑛𝜃𝑖=𝑛2𝑠𝑖𝑛𝜃𝑡
Critical angle 𝑛1𝑠𝑖𝑛𝜃𝑐=𝑛2𝑠𝑖𝑛90°
𝑠𝑖𝑛𝜃𝑐=𝑛2
𝑛1 𝜃𝑐=sin−1 𝑛2
𝑛1
𝒏𝒑
𝒏𝒄
𝑟𝑠 =
𝑛1𝑐𝑜𝑠𝜃𝑖 − 𝑖 𝑛2𝑛1𝑛2
𝑠𝑖𝑛𝜃𝑖
2
− 1
𝑛1𝑐𝑜𝑠𝜃𝑖 + 𝑖 𝑛2𝑛1𝑛2
𝑠𝑖𝑛𝜃𝑖
2
− 1
𝒏𝒑
𝑟𝑝 =
𝑖 𝑛1𝑛1𝑛2
𝑠𝑖𝑛𝜃𝑖
2
− 1 − 𝑛2𝑐𝑜𝑠𝜃𝑖
𝑖 𝑛1𝑛1𝑛2
𝑠𝑖𝑛𝜃𝑖
2
− 1 + 𝑛2𝑐𝑜𝑠𝜃𝑖
𝒏𝒑 > 𝒏𝒂 𝜽𝒑 > 𝒔𝒊𝒏−𝟏𝒏𝒂
𝒏𝒑
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Evanescent wave
evanescent waves are formed when waves traveling in a medium undergo
total internal reflection at its boundary
'''' ( )i k r t
transE E e
'' '' ''sin cost tk r k x k z
2 2
'' ''
2
sinsin 1
p t
t
c
nk x ik z
n
1( )'' z i k r t
transE E e e
''
1
sinp p
c
k nk
n
2 2
''
2
sin1
p p
c
nk
n
𝒏𝒑
𝒏𝒄
𝜃𝑡
𝜃𝑝 𝜃𝑟
Introduction to modern optics 2ed, grant R. Fowles, 43p
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Prism coupling
Phase difference = path difference + phase change term = 2m𝝅 (m = 0,1,2,….)
constructive interference = Propagate!
Propagation of electromagnetic wave (in film)
2 cos 2fs fcf fdk m (m = 0, 1, 2, 3, ….)
fsfcFilm-substrate phase change Film-airgap phase change
sinf f fn k
sinp p pn k pn
cn
fn
sn
d
p
f
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Prism coupling
Calculate phase difference
pn
cn
fn
sn
d
p
f
A
l
B B
( 2 ) ( ) 2f fs fc pk B k A m
A
l
2p
sin p
A
l
sin pA l
B B
2
l
f
B
d
2d
2f
2
l
2tan( )
2f
d
l
sin2 tan 2
cos
f
f
f
l d d
2sin f
lB
fs
fc
Film-substrate phase change
Film-airgap phase change
sin sinfp p fn n
2 cos 2f f fs fcdk m
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Prism coupling
phase change
fs
fc
d
S-wave s
ii
s i
aer e
ae
2 22
1
cos sin ( )i nae i
n
2s
2 2sintan tan
2 cos
s n
2si i
sr e e
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Prism coupling
22
2 2
1:1:
,: :
f
s c
s c f
s waves wave
f f nnp wave p wave
n n
sineff f fN n
4cos 2
fs fcf f
dn m
2 2
2 22 2
2 2 2 2
42arctan 2arctan 2
eff
eff
eff cs
f s c
f eff f eff
N nN ndn N f f m
n N n N
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Prism coupling
)2
()2
(' pp
A App
'
P
S
Effective refractive index
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Prism coupling
0
2
0
2 sincossinsin AnANpeff
Effective refractive index
sineff f fN n
ffnNeff sin
ppn sin
)'sin( Anp
p
AnAnppppsin'coscos'sin
0sinsin p
pn
'p
App
'
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Prism coupling
Calculate 𝒏𝒇 , d
)()(
)()()()(22
2222
pFqF
pFqNqFpNn effeff
f
)(2
)(
22 pNn
pFd
efff
(m = p , q)
2 24( ) ( ) ( )
efff
dn m N m F m
2 22 2
2 2 2 2( ) arctan arctan
eff eff cs
s c
f eff f eff
N nN nF m f f m
n N n N