19. Theory of Multilayer Films 19. Theory of Multilayer Films

19. Theory of Multilayer Films 19. Theory of Multilayer Films

Transfer Matrix

Reflectance at Normal Incidence Anti-reflecting Films

High-Reflectance Films

r

t

Transfer Matrix for a Single Film on a Substrate:

TE Mode

Transfer Matrix for a Single Film on a Substrate:

TE Mode

( ) ( )

( ) ( )

1 2

1 01 1 1

2 02 2 2

:

, cos

, cos

Consider two waves E and E that have the same frequency

E r t E k r t

E r t E k r t

ω

ω ε

ω ε

= − +

= − +

r r

r r r r r

r r r r r

n₀ n₂

E0 E_t2

1

Er

n₁

1

Et E_i2

1

Ei E_r2

mode) -

ˆ (TE

0 E z

Er = _z

mode) -

(TE ˆ yˆ

0 Bxx By

Br = + n

0

∑^∞

=

=

1 1 1

m m r

r E

E

E0 1

1

Er

2

1

Er

m

Er 1

1

1

Et

2

1

Et

1 2

Ei 1

1

Ei

2 2

Ei 1

2

Er 2

1

Ei E^r22

m

Ei₁ E^rm₂

m

Et 1

m

Ei₂

1 2

Et 2

2

Et m

Et₂

B0

n

1

n₂=n_s

∑^∞

=

=

1 2 2

m m t

t E

E

∑^∞

=

=

1 1 1

m m i

i E

E

∑^∞

=

=

1 1 1

m m t

t E

E

∑^∞

=

=

1 2 2

m m i

i E

E

∑^∞

=

=

1 2 2

m m r

r E

E

Er0

∑^∞

=

=

1 2 2

m m t

t E

E

∑^∞

=

=

1 1 1

m m r

r E

E

Transfer Matrix for a Single Film on a Substrate:

TE Mode

Transfer Matrix for a Single Film on a Substrate:

TE Mode

Tangential components of E and B-fields are continuous across the interface.

On (a) :

0 1 1 1

0 0 1 0 1 1 1 1

2 2 2

2 1 2 1 2 2

cos cos cos cos

cos cos cos

a r t i

a r t t i t

b i r t

b i t r t t t

E E E E E

B B B B B

E E E E

B B B B

θ θ θ θ

θ θ θ

= + = +

= − = −

= + =

= − =

On (b) :

(a) (b)

0 0

0 0 0 0 0

1 1 0 0 1

2 0 0 2

Note :

:

cos cos

cos

t

s t

E n

B E n E

v c

Define n n

n

ε μ

γ ε μ θ

γ ε μ θ

γ ε μ θ

= = ⎛ ⎞⎜ ⎟⎝ ⎠ =

=

=

=

( ) ( )

( )

0 0 1 1 1 1

1 2 2 2

a r t i

b i r s t

B E E E E

B E E E

γ γ

γ γ

= − = −

= − =

Transfer Matrix for a Single Film on a Substrate:

TE Mode

Transfer Matrix for a Single Film on a Substrate:

TE Mode

0 0 1 0 1 1 1 1

2 1 2 1 2 2

( ) cos cos cos cos

( ) cos cos cos

a r t t i t

b i t r t t t

a B B B B B

b B B B B

θ θ θ θ

θ θ θ

= − = −

= − =

( ) ( )

( ) ( )

( )

2 2

0 1 1

2 1 1 2

1 2 2 1

0

1

1 1 2

2

exp exp

exp

exp ex

o

p

. c s _t

i t i

b i r t

r

i t

b i r t

From the geometry of the problem we can see that :

E E i and E E i

where

Using these exp

k n

ressions we find

E E E E i E i E

B

:

E E E

π t

δ θ

δ δ

λ

δ δ

γ γ

⎛ ⎞

= Δ = ⎜ ⎟

⎝ ⎠

= + = − + + =

= − =

= − = −

( )−iδ − E_i1 exp( )+iδ = γ_s E_t2

⎡ ⎤

⎣ ⎦

Transfer Matrix for a Single Film on a Substrate:

TE Mode

Transfer Matrix for a Single Film on a Substrate:

TE Mode

t

( )

( )

( ) ( )

1 1

1

1 1

1

1 1

2 exp

2 exp

:

exp exp

2

b b

t

b b

i

a t i

t1

b

i1 b b

We can solve the last two equations for :

E B

E i

E B

E i

we then obtain

i i

At the first bou

E and E in terms of E and

E E E

ndary

B

E

γ δ

γ

γ δ

γ

δ δ

⎛ + ⎞

= ⎜ ⎟ +

⎝ ⎠

⎛ − ⎞

= ⎜ ⎟ −

⎝ ⎠

+ + −

⎡ ⎤

= + = ⎢ ⎥

⎣ ⎦

( ) ( )

( ) ( ) ( ) ( ) ( )

( )

1

1

1 1 1 1 1

1

1

exp exp

2 cos sin

exp exp exp exp

2 2

sin cos

b

b b

a t i b b

b b

i i

B

E B i

i i i i

B E E E B

E i B

δ δ

γ δ δ

γ

δ δ δ δ

γ γ γ

γ

γ δ δ

+ − −

⎡ ⎤

+ ⎢ ⎥

⎣ ⎦

⎛ ⎞

= + ⎜ ⎟

⎝ ⎠

+ − − + + −

⎡ ⎤ ⎡ ⎤

= − = ⎢ ⎥ + ⎢ ⎥

⎣ ⎦ ⎣ ⎦

= +

Transfer Matrix for a Single Film on a Substrate:

TE Mode

Transfer Matrix for a Single Film on a Substrate:

TE Mode

1

1

11 12

21 22

cos sin

sin cos

2 2 :

:

a b

a b

a a

This can be written in matrix form :

E i E

B B

i

The x transfer matrix is given by

m m

m m

Generalizing to a system with N layers

E E

B

δ δ

γ

γ δ δ

⎡ ⎤

⎡ ⎤ = ⎢ ⎥⎡ ⎤

⎢ ⎥ ⎢ ⎥⎢ ⎥

⎣ ⎦ ⎢⎣ ⎥⎦⎣ ⎦

⎡ ⎤

= ⎢ ⎥

⎣ ⎦

⎡ ⎤ =

⎢ ⎥⎣ ⎦ 1 2 3K N

M

M M M M ^{N N}

N N

E

B B

⎡ ⎤ ⎡ ⎤

⎢ ⎥ = ⎢ ⎥

⎣ ⎦ M^N ⎣ ⎦

Transfer Matrix for a Single Film on a Substrate:

TE Mode

Transfer Matrix for a Single Film on a Substrate:

TE Mode

0 0 0 0 0

1 1 0 0 1

2 0 0 2

cos cos

cos

t

s t

n n n

γ ε μ θ

γ ε μ θ

γ ε μ θ

=

=

=

for a multilayer

11 12

1

21 22

1

cos sin

sin cos m m i

m m

i δ δ

γ

γ δ δ

⎡ ⎤

⎡ ⎤ ⎢= ⎥

⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎢⎣ ⎥⎦

( )

1

1

cos sin

sin cos

a b b

a b b

E E B i

B E i B

δ δ

γ

γ δ δ

⎛ ⎞

= = + ⎜ ⎟

⎝ ⎠

= +

( )

( )

0 1 2

0 0 1 2

0 1 2 11 12 2

1

0 0 1 2 21 22 2

1

:

cos sin

sin cos

a r b t

a r b s t

r t t

r s t s t

0 r1 t2 :

E E E E E

B E E B E

In matrix form we obtain

E E i E m m E

E E E m m E

i

The am

In terms of E , E , and E

li p t

γ γ

δ δ

γ γ γ γ

γ δ δ

= + =

= − =

⎡ ⎤

⎡ + ⎤ = ⎢ ⎥⎡ ⎤ = ⎡ ⎤⎡ ⎤

⎢ − ⎥ ⎢ ⎥⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦⎢⎣ ⎥⎦

⎣ ⎦ ⎢⎣ ⎥⎦

( ) ^{11 12}

0 1

1 2

0

2 0

2 2

:

1

:

1

s s r t

reflection and transmission coefficients

E

ude are given by

In terms of r and t we obtai

r m t m t

r E t

E E

r m t t

n

m γ

γ γ

+ = +

− = +

≡ ≡

0

0 11 0 12 21 22

0 11 0 12 21 22

0 11 0 12 21 22

2

. :

s s

s s

s s

t m m m m

m m m m

r m

Solving for r and t we have

These equations allow us to evaluate the properties of multilayer m m

ms m

fil γ

γ γ γ γ

γ γ γ γ

γ γ γ γ

= + + +

+ − −

= + + +

0 0 0 0 0

1 1 0 0 1

2 0 0 2

cos cos

cos

t

s t

n n n

γ ε μ θ

γ ε μ θ

γ ε μ θ

=

=

=

11 12

1

21 22

1

cos sin

sin cos m m i

m m

i δ δ

γ

γ δ δ

⎡ ⎤

⎡ ⎤ ⎢= ⎥

⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎢⎣ ⎥⎦

Reflection coefficient (r) and transmission coefficient (t) Reflection coefficient (r) and transmission coefficient (t)

( ) ( )

( ) ( )

2

1 0 0 1

2

1 0 0 1

11 12

1 0 0

21 1 0 0 22

cos si

: cos sin

sin

n

cos

cos sin

s s

s s

At normal incidence the matrix elements become

m m i

n

m i n m

At normal incidence the re

n n n i n n n

flectance

r n n

coefficient becomes :

The powe

n i n n n

δ δ

ε μ

ε μ δ δ

δ δ

δ δ

− + −

= +

=

+ +

=

= =

( ) ( )

( ) ( )

2 2

2 2 2 2

1 0 0 1

2 2

2 2 2 2

1 0 0 1

cos sin

cos sin

s s

*

s s

r reflectance coefficient is given by :

n n n n n n

R = r r

n n n n n n

δ δ

δ δ

− + −

= + + +

11 12

1

21 22

1

cos sin

sin cos m m i

m m

i δ δ

γ

γ δ δ

⎡ ⎤

⎡ ⎤ ⎢= ⎥

⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎢⎣ ⎥⎦

0 0 0 0 0

1 1 0 0 1

2 0 0 2

cos cos

cos

t

s t

n n n

γ ε μ θ

γ ε μ θ

γ ε μ θ

=

=

=

19-2. Reflectance at Normal Incidence:

Single-Layer Film

19-2. Reflectance at Normal Incidence:

Single-Layer Film

1 1 1

0 0

2 2

cos _t

n t n t

π π

δ θ

λ λ

⎛ ⎞ ⎛ ⎞

= ⎜ ⎟ =⎜ ⎟

⎝ ⎠ ⎝ ⎠

(Reflectance)

/λ nt /λ0

Δ =

0 1 2

1 (air)

1.52 (glass)

s

n

n n n n

=

=

= =

s 1.52 n= n =

0

2 2n λ Δ = =λ

0

4 4n λ Δ = =λ

4.3%

(Reflectance at normal incidence)

Reflectance at Normal Incidence:

Single-Layer Quarter-Wave Film Reflectance at Normal Incidence:

Single-Layer Quarter-Wave Film

0

1 1

0 0 1

11 12

1 0 0 1 0 0 1

21 1 0 0 1 0 1

0

0 22

1

:

2 2

cos =0, sin =1

4 2

cos 0 sin 0

sin co 0

4

s 4

0 For the quarter wave thickness

n t n

n

i i i

m m

n n

m i n i n m i

At normal incidence t

t

he r

n λ

π π π

δ δ δ

λ λ

δ δ

ε μ ε μ γ

γ λ

ε μ δ ε μ δ

− λ

⎛ ⎞

= = ⎜ ⎟ = →

⎝ ⎠

⎛ ⎞

= = = = ⎜ ⎟

⇒ ⎜⎜ ⎟⎟

= = = = ⎝

= =

⎠

( )

( ) ( )

( )

( )

( )

2 2

0 1 0 1

2 2

0 1 0

2 2

0 1

2 2

0 1

1

s s

* s s

s

s

i n n n n n n

r

i n n n n n n

eflectance coefficient becomes :

The power reflectance coeff n

icient is given b n n

R = r

n n

y : r

n

−

+

−

−

= +

= =

+

= 0 R

A single λ/4 film can produce perfect antireflection!

1 0

4

4 n

t ₌ λ ₌ λ

ns

n n₁ = ₀

Reflectance at Normal Incidence:

Two-Layer Quarter-Wave Films Reflectance at Normal Incidence:

Two-Layer Quarter-Wave Films

1 1

2

1

1 2

1

1 2

2

: 0

0 :

0 0 0

0 0 0

two quarter wa

i The transfer matrix for a quarter wave thickness film is given by

i For

i i

i i

The reflectance coefficient ve fi

be lms

γ γ

γ

γ γ γ

γ γ γ

γ

⎡ ⎤

⎢ ⎥

− = ⎢ ⎥

⎢ ⎥

⎣ ⎦

⎡− ⎤

⎡ ⎤ ⎡ ⎤ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

= ⎢⎢⎣ ⎥ ⎢⎥ ⎢⎦ ⎣ ⎥⎥⎦ = ⎢⎢ − ⎥⎥

⎣ ⎦

−

1

1 2

M

M = M M

2 1

0 2 2 2 2

1 2

0 11 0 12 21 22 2 0 1 2 0 1

2 2 2 2

0 11 0 12 21 22 2 1 2 0 1 2 0 1

0

1 2

s

s s s s

s s s s

s

*

comes :

m m m m n n n n

r m m m m n n n n

The power reflectance coefficient is given by : R = r r

γ γ

γ γ

γ γ

γ γ γ γ γ γ γ γ

γ γ γ γ γ γ γ γ γ γ γ γ

γ γ

⎛ ⎞ ⎛ ⎞

− − −

⎜ ⎟ ⎜ ⎟

+ − − ⎝ ⎠ ⎝ ⎠ − −

= = = =

+ + + ⎛ ⎞ ⎛ ⎞ + +

− + −

⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

2 2 2

2 0 1

2 2

2 0 1

2 2

2 0 1

2

1 0

s s

s

s

n n n n n n n n

Zero reflectance occurs when n n

n n

n n n n

⎛ − ⎞

= ⎝

⇒

⎜

=

+ ⎠

−

⎟

Reflectance at Normal Incidence:

Two-Layer Quarter-Wave Films Reflectance at Normal Incidence:

Two-Layer Quarter-Wave Films

0 1

2

n n n

n = _s

Although the reflectance is greater, it remains less values over the broad range by introducing a λ/2 layer!

Reflectance at Normal Incidence:

Two-Layer Films

Reflectance at Normal Incidence:

Two-Layer Films

19-4. Reflectance at Normal Incidence:

Three-Layer Antireflecting Films

19-4. Reflectance at Normal Incidence:

Three-Layer Antireflecting Films

Reflectance at Normal Incidence:

Three-Layer Antireflecting Films Reflectance at Normal Incidence:

Three-Layer Antireflecting Films

1 3

0 2

s

For the quarter - wave film, zero reflectance occurs when : n n n n

n =

CVI Antireflecting Films CVI Antireflecting Films

CVI Broadband Antireflecting Films CVI Broadband Antireflecting Films

19-5. High-Reflectance Layers 19-5. High-Reflectance Layers

by reversing the order of deposition of the low - and high refractiv

A high - reflectance film rather than an anti - reflecting film is made

The transfer matrix for a quarter wave t e

hi index la

ckness fi yers

lm .

is gi

− 0

:

0

, :

0 0 0

0 0 0

L H

H L

HL H L

H

H L

L

i ven by

i For two quarter wave films with the high refractive index layer on top

i i

i i

The power reflectance coefficient fo the two l ye

r a

γ γ

γ

γ γ γ

γ γ γ

γ

⎡ ⎤

⎢ ⎥

= ⎢ ⎥

⎢ ⎥

⎣ ⎦

− −

⎡− ⎤

⎡ ⎤ ⎡ ⎤ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

= ⎢⎢⎣ ⎥ ⎢⎥ ⎢⎦ ⎣ ⎥⎥⎦ = ⎢ − ⎥

−

⎢ ⎥

⎣ ⎦

M

M = M M

2 2 2

0

2 2

0

* L s H

L s H

r film at normal incidence

n n n n

R = is giv

r r n n n

en by :

n

⎛ − ⎞

= ⎜⎝ + ⎟⎠

High-Reflectance stack of N Double-Layers High-Reflectance stack of N Double-Layers

( 1 1)( 2 2) ( ) ( )

0 0

0 0

N N

H L H L HN LN H L HL

N N L L

H H

N

H H

L L

For a high - reflectance film with N pairs of high - and :

The ampl

low refractive index layers

itude and powe γ γ

γ γ

γ γ

γ γ

= = =

⎡⎛ ⎞ ⎤

⎡− ⎤ ⎢⎜− ⎟ ⎥

⎢ ⎥ ⎢⎝ ⎠ ⎥

⎢ ⎥

= ⎢⎢⎣ − ⎥⎥⎦ = ⎢⎢⎢⎣ ⎛⎜⎝− ⎞⎟⎠ ⎥⎥⎥⎦

M M M M M K M M M M M

M

( )( )

2 0

0

2 0

0

2 0

1

1

N

N N

L

L H

s

s H

H L

N N N

L H L

s

N

s L H

*

H L s H

r reflectance coefficients for the multi layer film at normal incidence are given by :

n n

n n

n n

n n

n n

r

n n n n

R = r r

n

n n n

n n

n n n n

−

⎛ ⎞⎛ ⎞

⎛ ⎞ ⎛ ⎞

− −

− − − ⎜ ⎟⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎝ ⎠

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

= =

⎛ ⎞

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

− + − ⎜ ⎟ − +

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

= ( )( )

2

2 0

1

N 1

s L H

n n n n

⎡ − ⎤

⎢ ⎥

⎢ + ⎥

⎣ ⎦

High-Reflectance Layers High-Reflectance Layers

N=2

N=2 +1(high) N=6

Reflectance of high-low λ/4 Layers Reflectance of high-low λ/4 Layers

CVI High-Reflectance Coatings CVI High-Reflectance Coatings

CVI High-Reflectance Coatings CVI High-Reflectance Coatings