• Interface plane is defined as the boundary between two different media.

EElectromagnetic Fields

ᱥ ᯱ ᰆ ᰆ

8㨰㵜 Plane Wave Oblique Incidence

S

Sodqh#Zdyhv#dw#Lqwhuidfhv#+Reoltxh Lqflghqfh,

• In the phase plane, electric and magnetic fields are the same direction, same magnitude and same phase.

• Interface plane is defined as the boundary between two different media.

• Plane of Incidence contains the propagation vector. It is both perpendicular to the interface plane and phase plane of the wave.

x

y

z

EElectromagnetic Fields

When we have e oblique incidence of plane waves on a dielectric interface, the reflection and transmission characteristics become interface, the reflection and transmission characteris

polarization and angle of incidence dependent

We need to distinguish between the two different polarizations~!!!

-

We need to distinguish between We

- Perpendicular Polarization -

Perpendicular Polariz P

P

-- Parallel Polarization

S

Sodqh#Zdyhv#dw#Lqwhuidfhv#+Reoltxh#Lqflghqfh,

• Perpendicular Polarization

-

Electric field is perpendicular

to the plane of incidence.

- Magnetic field is parallel to the plane of incidence.

- The fields are configured as in the Transverse Electric (TE) mode.

•

The fields are configured a Parallel Polarization

-

Electric field is parallel

to the plane of incidence.

- Magnetic field is perpendicular to the plane of incidence.

- The fields are configured as in the Transverse Magnetic (TM) mode.

x

y

z

EElectromagnetic Fields

? Polarization Perpendicular Polarization Q: Plane of incidence is ?

x y

x E

H

E is parallel to the plane of incidence

E is perpendicular to the plane of incidence

Sodqh#Zdyhv#dw#Lqwhuidfhv#+Reoltxh#Lqflghqfh,

x y

·

E H

z ·

z ·

XY plane

Parallel Polarization

> @

> sin cos ) @

ˆ exp

) cos sin

( ˆ exp

1 1

1

i i

o

i i

o i

y j x

j E

z

y x

j E

z E

T E

T E

T T

E



& propagating in medium 1 

component in –x direction component in –y direction Within exponential term p p p p p p p p p

Outside exponential term p p p p p p p

 ˆ cos ˆ sin  exp >

₁

( sin cos ) @

1

r r

o r

r

r

E j x y

y x

H E T T

T K

T * 



^A

&

x and y components of H

_r

*

A

: Perpendicular Reflection Coefficient

Reoltxh#Lqflghqfh#dw#d#Sodqh#Glhohfwulf#Phglxp

EElectromagnetic Fields

>

( sin cos )

@

ˆ _oexp _{1 i i}

i yE j x z

E 

E T



T

?&

Perfect Conductor

z

H_i E_i

a_ni

Incident

Ti

Perpendicular Incidence ^İ

⁰

^{, ȝ}

⁰

Reflected? ^Hr

Er

anr

Tr

Reflected

y

Ti

E1

Ti

zˆE₁cosTi

xˆE₁sinTi

>

j

E

xsin

T

_i j

E

zcos

T

_i

@

exp ₁  ₁

Ÿ

What is propagation direction of incident wave?

z x

R

Reoltxh#Lqflghqfh#dw#d#Sodqh#Frqgxfwru

>

( sin cos )

@

ˆ _oexp _{1 i i}

i yE j x z

E&  E T  T Perfect Conductor

Hi

z

E_i

a_ni

Incident

Ti

>

( sin cos )

@

ˆ _rexp _{1 r r}

r yE j x z

E&  E T  T

Perpendicular Incidence ^İ

⁰

^{, ȝ}

⁰

Reflected? ^Hr

Er

anr

Tr

Reflected

i ?

H&



ˆcos ˆsin



exp

>

₁( sin cos )

@

1

i i

o i

i E j x z

z

x E T T

T K

T   



H_i y

E_i

a_ni

Ti

Ti

Ti

S  5 . 0 Ti

i

Hi

zˆ sinT



ˆsin _i z T

i

Hi

zHˆHHH sinT

i

Hi

xˆ cosT

xHˆHH_icosT_i





xˆcosT_i

xˆE₁sinTr

zˆE₁cosTr



H_r

Er

a_nr

Tr

r

Hr

xˆ cosT Tr

Tr

S 5 . 0

z x

r

Hr

zˆ sinT



ˆcos ˆsin



exp

>

₁( sin cos )

@

1

r r

r r r

r E j x z

z x

H E T T

T K

T   

&

Tr

E1

What is propagation direction of reflected wave?

EElectromagnetic Fields

R

Reoltxh#Lqflghqfh#dw#d#Sodqh#Frqgxfwru

Perfect Conductor

z

H_i E_i

a_ni

Ti

> @



ˆcos ˆsin



exp

>

( sin cos )

@

) cos sin

( ˆ exp

1 1

1

r r

r r r

r

r r

r r

z x

E j z

x H

z x

j E

y E

T T

K E T T

T T

E











&

&

1

*

i r

E 0 E

 _r

o E

E

Apply the boundary condition @ z=0

; Snell’s law of reflection

İ

₀

, ȝ

₀

Hr

Er

anr

Tr

y

>

( sin cos )

@

ˆ _oexp _{1 i i}

i yE j x z

E&  E T  T

>

^{( sin cos )}

@

^{ˆ exp}

>

^{( sin cos )}

@

ˆ exp _{1 1}

1

r r

r i

i o

r i

z x

j E

y z

x j E

y

E E E

T T

E T

T

E    



 &

&

&

>

^{( sin )}

@

^{ˆ exp}

>

^{( sin )}

@

ˆ exp _{1 1}

1 0 o i r r

z

x j E

y x

j E

y

E&  E T   E T

tŒ‹œ”GX tŒ‹œ”GY l_{›ˆ•ŽŒ•›} k_•–™”ˆ“ i_•–™”ˆ“ o_{›ˆ•ŽŒ•›}

s–šš“Œšš

s•Œˆ™

tŒ‹œ”

wŒ™ŒŠ›

j–•‹œŠ›–™

H_4w@#H_5w Î H_4w @#3

G_4q @#G_5q.㼪_v Î G_4q @#㼪_v

E_4q@#E_5q Î E_4q @#3

K_4w@#K_5w.M_v Î K_4w@#M_v

r

i

T

T It must be satisfied for all x.

>

( sin )

@

exp

>

( sin )

@

0

exp ₁   ₁

?E_y E_o jE x T_i E_r jE x T_i

x

z=0 ș_i

Hi

))&

Ei

)&

•

Hr

))&

Er

)&

x

z Tr

Perfect Conductor

Incident wave Reflected wave

Parallel Incidence

KZ=#Reoltxh#Lqflghqfh#dw#d#Sodqh#Frqgxfwru

y

EElectromagnetic Fields

Perpendicular Incidence

Mathematical expression for fields

Incident wave Reflected wave

Transmitted wave

x

y

z

xˆE₁sinTi



Ti yˆE₁cosTⁱ

H

_i

Ti

i

Hi

xˆ cosT



i

Hi

yˆ sinT

E1

> @

 ^{ˆ cos ˆ sin  exp} > ^{( sin cos )} @

cos sin

ˆ exp

1 1

1 1

i i

o i i

i

i i

o i

y x

E j y

x H

y j x

j E

z E

T T

K E T T

T E

T E









&

&

> @

 ^{ˆ cos ˆ sin  exp} > ^{( sin cos )} @

) cos sin

( ˆ exp

1 1

1

i i

o i i

i

i i

o i

y x

E j y

x H

y x

j E

z E

T T

K E T T

T T

E









&

&

> @

 ˆ cos ˆ sin  exp > ( sin cos ) @

) cos sin

( ˆ exp

1 1

1

r r

r r r

r

r r

r r

y x

E j y

x H

y x

j E

z E

T T

K E T T

T T

E







&

&

> @

 ^{ˆ cos ˆ sin  exp} > ^{( sin cos )} @

) cos sin

( ˆ exp

2 2

2

t t

t t t

t

t t

t t

y x

E j y

x H

y x

j E

z E

T T

K E T T

T T

E









&

&

Reoltxh#Lqflghqfh#dw#d#Sodqh#Glhohfwulf#Phglxp

EElectromagnetic Fields

> @

 ^{ˆ cos ˆ sin  exp} > ^{( sin cos )} @

) cos sin

( ˆ exp

1 1

1

i i

o i i

i

i i

o i

y x

E j y

x H

y x

j E

z E

T T

K E T T

T T

E









&

&

> @

 ˆ cos ˆ sin  exp > ( sin cos ) @

) cos sin

( ˆ exp

1 1

1

r r

o r

r r

r r

o r

y x

E j y

x H

y x

j E

z E

T T

K E T

T

T T

E

* 





*

A A

&

&

> @

 ˆ cos ˆ sin  exp > ( sin cos ) @

) cos sin

( ˆ exp

2 2

2

t t

o t

t t

t t

o t

y x

E j y

x H

y x

j E

z E

T T

K E T W

T

T T

E W









A A

&

&

Reoltxh#Lqflghqfh#dw#d#Sodqh#Glhohfwulf#Phglxp

 ^{* *}

^A

^E

^o

 ^W

^A

^E

^oo

R

Reoltxh#Lqflghqfh#dw#d#Sodqh#Glhohfwulf#Phglxp

tŒ‹œ”GX tŒ‹œ”GY l_{›ˆ•ŽŒ•›} k_•–™”ˆ“ i_•–™”ˆ“ o_{›ˆ•ŽŒ•›}

s–šš“Œšš

s•Œˆ™

tŒ‹œ”

s–šš“Œšš

s•Œˆ™

tŒ‹œ”

H_4w @#H_5w G_4q @#G_5q E_4q @#E_5q K_4w @#K_5w

E

E tangential components ( s (z-component) are continuous at the boundary.

At y=0,

   

 0   0   0 

0

0

₂

1

 E y E y

y E

y E y

E

t r

i

 

_o

>

_{i i}

@

_o



_i



i

y z E j x z E j x

E & 0 ˆ exp E

₁

( sin T  0 ˜ cos T ) ˆ exp E

₁

sin T

 

_r

>

_{r r}

@

_r



_r



r

y z E j x z E j x

E & 0 ˆ exp E

₁

( sin T  0 ˜ cos T ) ˆ exp E

₁

sin T

 

_t

>

_{t t}

@

_t



_t



t

y z E j x z E j x

E & 0 ˆ exp E

₂

( sin T  0 ˜ cos T ) ˆ exp E

₂

sin T

EElectromagnetic Fields



_i



_r



_r



_t



_t



o

j x E j x E j x

E ^exp E

₁

^sin T  ^exp E

₁

^sin T ^exp E

₂

^sin T

?

t r

i

E T E T

T

E

₁

sin

₁

sin

₂

sin Î Phase matching:

Arguments of the exponentials must be equal~!!!

r

i

T

T

sin sin

2 1

i

t T

E T E

Snell el ˅ l˅s Law

t

r

E

E E

₀





^{E Z PH}



H T P Z

H P

Z sin

2 2

1

1 i 



_i



_r



_i



_t



_i



o

j x E j x E j x

E exp E

₁

sin T  exp E

₁

sin T exp E

₁

sin T

?

sin

2 2

1

1 Ti

H P

H P

R

Reoltxh#Lqflghqfh#dw#d#Sodqh#Glhohfwulf#Phglxp

tŒ‹œ”GX tŒ‹œ”GY l_{›ˆ•ŽŒ•›} k_•–™”ˆ“ i_•–™”ˆ“ o_{›ˆ•ŽŒ•›}

s–šš“Œšš

s•Œˆ™

tŒ‹œ”

s–šš“Œšš

s•Œˆ™

tŒ‹œ”

H_4w @#H_5w G_4q @#G_5q E_4q @#E_5q K_4w @#K_5w



^{ˆcos ˆsin}



^exp

>

1^{( sin cos )}

@

1

i i

o i i

i E j x y

y x

H E T T

T K

T  

& 



^{ˆcos ˆsin}



^exp

>

₁^{( sin cos )}

@

1r r r

r r

r E j x y

y x

H E T T

T K

T  

&



^{ˆcos ˆsin}



^exp

>

2^{( sin cos )}

@

2t t t

t t

t E j x y

y x

H E T T

T K

T  

& 

H fields have x and y components.

H

H fields have x

H tangential components ( mpon

s (x-component) are continuous at the boundary.

At y=0,

   

 ^{0 0} 

²

 ^{0 0   0} 

1

 H y H y

y H

y H y

H

tx rx

ix

x x



⁰



^{cos exp(} ₁ ^{sin )}

1o i

i

ix E j x

y

H E T

T K





⁰



^{cos exp(} 1 ^{sin )}

1r r

r

rx E j x

y

H E T

T K



⁰



^{cos exp(} ₂ ^{sin )}

2t t

t

tx E j x

y

H E T

T K



EElectromagnetic Fields

R

Reoltxh#Lqflghqfh#dw#d#Sodqh#Glhohfwulf#Phglxp

   

t



t



t r

r r

i i

o E j x

x E j

x E j

T E

K T T

E K T

T E

K

₁

^cos T ^exp

¹

^{sin }

₁

^{cos exp}

¹

^{sin }

₂

^{cos exp}

²

^sin



t t r

r i

o E E

E

T

T K T K

K

₁

^{cos }

₁

^{cos }

₂

^cos



(_E₁sinT_i E₁sinT_r E₂sinT_t)

We need to solve two simultaneous equations.

  1

t r

o E E

E



 

2 cos cos

cos

2 1

1

t t r

r i

o E E

E T

T K T K

K ^{ }



   

 0   0   0 

0

0

₂

1

 H y H y

y H

y H y

H

tx rx

ix

x x

From,

R

Reoltxh#Lqflghqfh#dw#d#Sodqh#Glhohfwulf#Phglxp  

1

t r

o E E

E 

t t r r i

o E E

E T

T K T K

K₁ ^{cos } ₁ ^{cos } ₂ ^cos



¸¸¹

¨¨ ·

©

 §

¸¸¹

¨¨ ·

©

 §

¸¸¹

¨¨ ·

©

 §

t t

t

t r

r

t i

o E E

E

T T K K T

T K K T T K

K cos cos

cos cos

cos cos ²

2 2

1 2

1

 

³

cos cos cos

cos

1 2 1

2

t r t i o

t

i E  E E

 T

T K K T

T K K

(1)+(3)Î

¹ _cos^{cos 1} _cos^{cos 0}

1 2 1

2 ¸¸¹

·

¨¨©

§ 

¸¸ 

¹

·

¨¨©

§  _r

t i o

t

i E E

T T K K T

T K K

i t

i t

t i t i

o r

E E

T K T K

T K T K T T K

K T

T K K

cos cos

cos cos

cos 1 cos

cos 1 cos

2 1

2 1

1 2 1 2



 







t i

t i

r i

r

E E E E

T K

T K

T K

T K

cos cos

cos cos

1 2

1 2

0



* 

?

_A ; Fresnel Reflection Coefficient

for Perpendicular Incidence

EElectromagnetic Fields A

*

A

 W

1

t r

o E E

E



Next, let’s find a transmission coefficient.

From ,

: Fresnel Transmission Coefficient

for Perpendicular Incidence

t i

i t

i t

E E E E

T K T K

T W K

cos cos

cos 2

1 2

2

0 

? _A

cos cos

cos cos

0 1

2

1 2

t t

i

t i

o E E

E 

 

T K T K

T K T K

0 1

2

1 2

cos cos

cos

cos E

E

t i

t i

r

K T K T

T K T K



 

   

t t

i

t i

t

i E E E









T K T K

T K T K T

K T K

cos cos

cos cos

cos cos

1 2

0 1

2 0 1

2

t t

i

t i

t

i E E









0 1

2

1 2

1 2

cos cos

cos cos

cos cos

T K T K

T K T K T K T K

t t

i

i

i E E





0 1

2

2 2

cos cos

cos cos

T K T K

T K T K

t t

i

i E E

 ₁ ⁰

2 2

cos cos

cos 2

T K T K

T

K

¸¸¹

¨¨ ·

©

* §







* 

i t

i t

i i t

i t

cf

T W T

T K T K

T W K

T K T K

T K T K

cos 1 cos

cos cos

cos 2

cos cos

cos cos

.

||

||

1 2

|| 2

1 2

1

|| 2

R

Reoltxh#Lqflghqfh#dw#d#Sodqh#Glhohfwulf#Phglxp

2 1

sin sin

E E T T

i t

Index of Refraction (n)

: Ratio of the speed of light in free space to that in the medium

v

p

n c

t

i

n

n

₁

sin T

₂

sin T

?

r

i

T

T

Snell’s Law of Reflection:

Snell’s Law of Refraction

2 1

n n

2 1

p p

v v

Z Z

v

p

E Z



2 1

p p

v c v

c

v

p

n c



Î Snell’s Laws are independent of wave polarization

EElectromagnetic Fields

1 2

K K

2 2

1 1 2

1

sin sin

H P Z

H P Z E E T T

i t

1 2

K K

Reoltxh#Lqfllghqfh#dw#d#S Sodqh#Glhohfwulf#Phglxp

2 2 2

1 1 1

H K P

H K P

For or nonmagnetic materials ( s (ȝ (ȝ= ȝ=ȝ ȝ

_o

),

1 0 2 0

1 2

H P H P

K K

2 0 1 0

H P H P

2 1 0

1 2 0

H H P

H H P

Let’s see Snell’s law of refraction.

2 1

sin sin

H H T T

i

t

: Snell’s law of refraction for nonmagnetic materials

R

Reoltxh#Lqflghqfh#dw#d#Sodqh#Glhohfwulf#Phglxp Let’s see the reflection coefficient.

t i

t i

T K T K

T K T K

cos cos

cos cos

1 2

1 2



*_A 

i i

i i

H T T H

H T T H

2 1 2

2 1 2

sin cos

sin cos









t i

t i

K T T K

K T T K

cos cos

cos cos

2 1 2 1





t i

t i

H T T H

H T T H

cos cos

cos cos

1 2 1 2

 K 

K

2 1

K K

2 1

sin cos

sin 1 cos

sin 1 sin

1 cos

sin sin

sin

2 1 2 1

2

2 2 1 1 2 1

2

2 2 2 1

2 1 2

2 1 1

i t

i t

i t

t

i i

t

H T T H H H

H T H H T H H H

H T T H

T

H T T H H P

H T P



?







sin

cos ²

1 2 1

2

i

t T

H T H H

H 

Tt

H cos H

1 2

Tt

H cos H

1 2

Ti

H

H 2

1 2 sin

Ti

H

H 2

1 2 sin

1 2 2

1

H H K

K

EElectromagnetic Fields

If İ

₂

>İ

₁

,

i i

i i

H T T H

H T T H

2 1 2

2 1 2

sin cos

sin cos









*_A

real.

A

is

*

. 1

2

1

!

H If İ

₁

>İ

₂

(the wave in medium 1 is incident on a less dense medium 2), H

then the square root is positive.

¸ ¸

¹

·

¨ ¨

©

§

2 1

sin sin

H H T

T

i

From Snell’s law of refraction ,

t

sin 1 sin !

i t

T T

i

t

T

T !

Ÿ ; Transmission angle is higher than incidence angle.

R

Reoltxh#Lqflghqfh#dw#d#Sodqh#Glhohfwulf#Phglxp

2

T

_t

S

^,

sin From sin

2 1

H H T T

i t

1

sin 2

H T_i H

Total Reflection

2 1

sin sin 2

H H T

S¸

¹

¨ ·

©

§

i

¸¸¹

·

¨¨©

 §



1 1 2 1

1 2 sin

sin n

n

ic H

T H

Critical Angle

¸¸¸¸¹

¸¸

·¸¸

¨¨¨¨©

¨¨

§¨¨





1 1 2 1

1 2 sin

sin n

n

ic H

T H

Critical Angle

0 1 cos

0 cos sin

cos

sin cos

, sin

At

2 1 2

2 1 2

1 2













*_A

i i

i i

i i

i T

T H T

T H H T T H

H T H 2? become )

angle(

ion transmiss

Can the

T

_t

S

When?

EElectromagnetic Fields

R

Reoltxh#Lqflghqfh#dw#d#Sodqh#Glhohfwulf#Phglxp

. sin sin

,

When

T

_i !

T

_ic

T

_i !

T

_ic

. sin sin

, sin sin

From

2 1 2 1

ic t

i t

H T T H

H T T H

!

When the incident angle is s larger than the critical angle,

. 1 sin , sin

Because

1

1 2 !



t

ic T

H T H

1 sin 1

sin

cos ²

2

2  r 1 

r _{t i}

t j j T

H T H

T

t

t T

T 1 sin²

cos 

; imaginary

i

t T

H T H sin sin

2

 t 1 Ti

H T H sin sin

2

 1

sin²T_t sin²

2 1

Ti

H H

R

Reoltxh#Lqflghqfh#dw#d#Sodqh#Glhohfwulf#Phglxp

>

( sin cos )

@

ˆ _oexp _{2 t t}

t z E j x y

E& W_A E T  T

When the incident angle is s larger than the critical angle, what is a transmitted electric field?

»»

¼ º

««

¬

ª r 

A exp sin sin 1

ˆ ²

2 1 2 2

1

2 i i

o j x y

E

z T

H E H H T

E H W

»»

¼ º

««

¬ ª

¸¸

¹

·

¨¨

©

§ r 

A exp sin sin 1

ˆ ²

2 1 2

1

2 i i

o j x jy

E

z T

H T H

H E H

W _{cos sin 1}

sin sin

2 2 1 2 1



r _i

t

i t

j T

H T H

H T T H sinT_t cosT_t)

sin

2 1

Ti

H H

¼ º

¸¸

¸¸¹

¸¸

·¸¸

1 sin²

2 1

Ti

H H

»»

¼ º

««

¬ ª

»»

¼ º

««

¬

ªr 

AEo y i j x i

z T

H E H H T

E H

W exp sin 1 exp sin

ˆ

2 1 2 2

2 1 2

> @

»»

¼ º

««

¬

 ª

AEo y j x i

z T

H E H D

W exp exp sin

ˆ

2 1

2 sin² 1

2 1

2 T_i

H E H

D sin² 1

2 1

2 T_i

H E H D

¼ º

««

««¬««

ª««r y T_i 

H

E H sin² 1

2 1 2

> @

^{ yDyy}

; Total Internal Reflection (TIR)

EElectromagnetic Fields cos 0

cos

cos cos

1 2

1 2



*_A 

t i

t i

T K T K

T K T

K₂cosT_i K₁cosT_t K

: Incident angle of no Reflection (ș

_Bŏ

)

T T

Brewster Angle

t

B K T

T

K₂cos _{A 1}cos

t

B K T

T

K₂cos _{A 1} 1sin²

t

B T

K

T K ²

2

1 1 sin

cos _A 



t



B T

K

T K ²

2

2

2 1 1 sin

cos ¸¸¹ 

¨¨ ·

©

§

A

¸¸

¹

·

¨¨

©

§

¸¸¹

¨¨ ·

©

§

¸¸¹

¨¨ ·

©

 _BA § _BA

n

n T

K

T K ²

2

2 1 2

2

2 1 1 sin

sin 1

i t

i t

n n n n

T T

E E T T

2 2

2 2 1

2 1 2 1

sin sin

sin sin

¸¸¹

¨¨ ·

©

? §



Tt

sin2

¹

·

A

TB

sin2

¨¨¨¨©

¨¨§

¨¨

¸¸¹

¨¨ ·

©

§ 

 _BA T_BA

H P

H P H

P H

T P ²

2 2

1 1 1

2 2

2 1 1 sin

sin 1

1 2

2 1

2 2 1 1

2 1

H P

H P H P H P K K

H P

H P

1 2

2 1

¸¸¸¸¹

¸¸·

¨¨¨¨ ¸¸

©¨¨

§¨¨

©K K ²

2 1

2 2

1 1 2 1 2 1

H P

H P E E n n

H P

H P

2 2

1 1

¸¸¸¸¹

¸¸·

¨¨¨¨ ¸¸

©¨¨

§¨¨

n n ²

2 1

A

A ¸¸¹

¨¨ ·

©

§

 _B T_B

P P H P

H

T P ²

2

2 1 1 2

2

2 1 sin

sin 1

: Incident angle of no Reflection (ș

_Bŏ

) R

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A

A ¸¸¹

¨¨ ·

©

§

 _B T_B

P P H P

H

T P ²

2

2 1 1 2

2

2 1 sin

sin 1

1 2

2 2 1

2

2

1 sin 1

1 PH

H T P

P

P _

°¿

°¾

½

°¯

°®

­

¸¸¹

¨¨ ·

©

§ _BA

2

2 1 1 2

2 1 2

1 1 sin

¸¸¹

¨¨ ·

©

§



? _A

P P H P

H P T_B

For nonmagnetic materials (ȝ=ȝ_o),

 f



A 1 1

1

sin ¹

2

2 H

H T_B

Î Brewster angle does s not exist for perpendicular incidence.

EElectromagnetic Fields

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Parallel Incidence

^x

z=0 ș_i O

Hi

))&

Ei

)&

•

Hr

))&

Er

)&

ș_t

x

z

Mathematical expression for the incident fields

  > @

>

^{( sin cos )}

@

ˆ exp

) cos sin

( exp ˆsin

ˆcos

1 1

1

i i

o i

i i

o i i

i

z x

E j y H

z x

j E

z x

E

T T

K E

T T

E T

T











&

&

Tr

Ti

xˆE₁sinTi

E

_i

Ti zˆE₁cosTⁱ E i

xˆ ₀cosT

ș_i

z x

E i

zˆ ₀sinT



H&t

E&t

•

z x

y E

1

R

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Parallel Incidence

^x

z=0 ș_i O

Hi

))&

Ei

)&

•

Hr

))&

Er

)&

ș_t

x

z Tr

  > @

>

^{( sin cos )}

@

ˆ exp

) cos sin

( exp ˆsin

ˆcos

1 1

1

r r

r r

r r

r r r

r

z x

E j y H

z x

j E

z x

E

T T

K E

T T

E T

T













&

&

Mathematical expression for the reflected fields

Tr

xˆE₁sinTr

E

_r

zˆE₁cosTr



r

Er

xˆ cosT

r

Er

zˆ sinT

Tr

Tr

Tr

H&t

E&t

•

z x

y

E

1