Wave Incidenceat Oblique angles

Sandra Cruz-PolECE Dept. UPRM

Quiz VIERNES

Uno de los problemas asignados en el prontuario

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Wave Incidence

I. Normal incidenceWave arrives at 90o from the surface

Standing waves

II. Oblique incidence (lossless) Wave arrives at an angle

Snell’s Law and Critical angle Two types: Parallel or Perpendicular Brewster angle

Oblique incidenceFirst, we need to define:

Expression for uniform plane wave traveling in any direction

The Normal , an

Plane of incidence Angle of incidence

zy

z=0

Medium 1 : 1 , 1

Medium 2 : 2, 2

r

i

t

x

Oblique incidence Uniform plane wave in general form

the radius or position vector

the wave number or propagation vector

For lossless unbounded media, k =

22222

)(

ˆˆˆ

ˆˆˆ

]Re[)cos(),(

zyx

zzyyxx

zyx

trkjoo

kkkk

akakakk

azayaxr

eEtrkEtrE

Oblique Incidencex

zy

z=0

Medium 1 : 1 , 1 Medium 2 : 2, 2

r

i

t

kr

ki

kt

Look at the direction of travelx

zy

z=0

Medium 1 : 1 , 1 Medium 2 : 2, 2

r

i

t

kr

ki

kt

kix

kiz

Expression for Waves

)cos(

)cos(

)cos(

tzkxkEE

tzkxkEE

tzkxkEE

tztxtot

rzrxror

izixioi

kikix

kiz

i

iiz

iix

izixi

k

k

kkkwhere

cos

sin

1

1

11122

Tangential E must be Continuous

From this we know that frequency is a property of the wave. So is color.

So 700nm is not always red!!

Give example of bathing suit.

)0()0()0( zEzEzE tri

ri

rxix kk

sinsin 11

Snell Law Equating, we get

or

where, the index of refraction of a medium, ni , is defined as the ratio of the phase velocity in free space (c) to the phase velocity in the medium.

11 rn

Critical angle, c

…All is reflected

When t =90o, the refracted wave flows along the surface and no energy is transmitted into medium 2.

The value of the angle of incidence corresponding to this is called critical angle, c.

If i > c, the incident wave is totally reflected.

)(for

]90[ sinsin

21

1

2

1

2

o

1

2

r

r

ttc

n

n

n

n

Fiber optics

Light can be guided with total reflections through thin dielectric rods made of glass or transparent plastic, known as optical fibers.

The only power lost is due to reflections at the input and output ends and absorption by the fiber material (not perfect dielectric).

Optical fibers have cylindrical fiber core with index of refraction nf, surrounded by another cylinder of lower, nc < nf , called a cladding.

For total reflection:

1

23 sinsin

n

nc o9032

0

22 )(sin

n

nn cf

a

Acceptance angle

[Figure from Ulaby, 1999]

Exercise: optical fiber An optical fiber (in air)

is made of fiber core with index of refraction of 1.52 and a cladding with an index of refraction of 1.49.

Find the acceptance angle:

Answer: 17.5 degrees

Parallel polarization

It’s defined as E is || to incidence plane

yeE

H

ezxEE

yeE

H

ezxEE

yeE

H

ezxEE

tt

tt

rr

rr

ii

ii

zxjtots

zxjtttots

zxjrors

zxjrrrors

zxjiois

zxjiiiois

ˆ

)ˆsinˆ(cos

ˆ

)ˆsinˆ(cos

ˆ

)ˆsinˆ(cos

cossin

2

cossin

cossin

1

cossin

cossin

1

cossin

2

2

1

1

1

1

y

z=0

Medium 1 : 1 , 1

Medium 2 :

2, 2

r

i

t

kr

ki

kt

Er

Ei

Et

x

z

Equating for continuity, the tangent fields

Which components are tangent to the interface between two surfaces?

y and xAt z = 0 (interface):

ttiiii

ttrrii

zxjtozxjrozxjio

zxjtto

zxjrro

zxjiio

eE

eE

eE

y

eEeEeEx

cossin

2

cossin

1

cossin

1

cossincossincossin

211

211

:ˆ

)(cos)(cos)(cos:ˆ

211

:ˆ

coscoscos:ˆ

toroio

ttorroiio

EEEy

EEEx

Reflection and Transmission Coefficients: Parallel Incidence

Reflection

it

i

io

to

it

it

io

ro

E

E

E

E

coscos

cos2

coscos

coscos

12

2||

12

12||

t

i

cos

cos)1( where ||||

Reflection and Transmission Coefficients: Perpendicular Incidence

ti

i

io

to

ti

ti

io

ro

E

E

E

E

coscos

cos2

coscos

coscos

12

2

12

12

1

Java Animation

http://www.amanogawa.com/archive/Oblique/Oblique-2.html

ExerciseA plane wave in air with

Is incident upon planar surface of nonmagnetic dielectric material with r=4, on z>0, Find

The polarization of the incident wave The angle of incidence The time-domain expressions for the reflected

electric and magnetic fields. The average power density carried by the wave

in the dielectric medium.

]/[10ˆ )43( mVeyE zxji

ExerciseA plane wave in free space with

Is incident upon planar surface of nonmagnetic lossless dielectric material with r=4, on z>0, Find

The polarization of the incident wave The angle of incidence and transmission The reflection and transmission coefficients. The E field in reflected and in dielectric Brewster angle

]/)[42cos()ˆ5ˆ10( mVzytzyEi

o

oo

zytzy

zytzy

4.63),7.82cos()ˆ2.3ˆ5.6(

),42cos()ˆ47.1ˆ946.2(

,65.0,295.0,9.12,6.26

:Answers

Brewster angle, B

Is defined as the incidence angle at which the reflection coefficient is 0 (all transmission).

The Brewster angle does not exist for perpendicular polarization for nonmagnetic materials.

221

1221||

12

12

12||

)/(1

)/(1sin

0coscos

0coscos

coscos

B

Bt

Bt

Bt

* BB is

known as the polarizing angle

Reflection vs. Incidence angle.

Reflection vs. incidence angle for different types of soil and parallel or perpendicular polarization.

Exercise (Brewster angle)A wave in air is incident upon the flat boundary of a

nonmagnetic soil medium with r=4, at i=50o. Find reflection and transmission coefficients for both

incident polarizations, and the Brewster angle. Answers:

oB 4.63

52.0,48.0

58.0,16.0

||

||||

EMAG

SummaryProperty Normal

IncidencePerpendicular Parallel

Reflection coefficient

Transmission coefficient

Relation

Power Reflectivity

Power Transmissivity

Snell’s Law:

it

it

coscos

coscos

12

12||

ti

i

coscos

cos2

12

2

ti

ti

coscos

coscos

12

12

t

i

12

22

it

i

coscos

cos2

12

2||

12

12

1 1 t

i

cos

cos1 ||||

2R2

R 2

|||| R

RT 1 RT 1 |||| 1 RT

2222

1 wheresinsin rrit nn

n