Wave Incidence at Oblique angles Sandra Cruz-Pol ECE Dept. UPRM.
-
Upload
dayana-noss -
Category
Documents
-
view
221 -
download
3
Transcript of Wave Incidence at Oblique angles Sandra Cruz-Pol ECE Dept. UPRM.
Wave Incidenceat Oblique angles
Sandra Cruz-PolECE Dept. UPRM
Quiz VIERNES
Uno de los problemas asignados en el prontuario
Enviar contestaciones a preguntas en Word por email
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