Prism coupling method for determining

refractive index and thickness

Soft-matter Optical Spectroscopy Laboratory

발표자 : 이재진

Introduction

problem

Prism coupling

P

S

c

Unknown parameter : reflective index, thickness glass

Necessary Non Destructive method

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

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

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

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

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

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

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

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

Prism coupling

)2

()2

(' pp

A App

'

P

S

Effective refractive index

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

'

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