ECSE 527 Optical Engineering: Assignments 4 Solutions
Question 1
i) For propagation over a distance
z
the electric field evolves according to
E(
z
)=E
0
exp(-j
k
z
)
Since
for
x
-directed
polarization
k
x
=2
π
n
0
/
λ
and
for
y
-directed
polarization
k
y
=2
π
n
e
/
λ,
and the propagation distance is
d
we can write:
( )
(
)
( )
(
)
( )
(
)
(
)
λ
π
λ
π
λ
π
/
2
exp
/
2
exp
/
2
exp
0
0
0
0
d
n
n
j
d
E
d
n
j
E
d
E
d
n
j
E
d
E
e
x
e
y
x
−
−
=
−
=
−
=
So we can write in matrix-vector form:
( )
( )
(
)
(
)
( )
( )
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
−
−
=
⎥
⎦
⎤
⎢
⎣
⎡
0
0
/
2
exp
0
0
1
0
y
x
e
y
x
E
E
d
n
n
j
d
E
d
E
λ
π
and where we have omitted the phase term that is common to both. So the Jones matrix is
given as above.
ii) For an
m
th
half-wave plate designed for wavelength
λ
0
we have thickness:
d
|(n
0
-n
e
)|=(2
m
+1)
λ
0
/2
(
)
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
+
−
λ
λ
π
0
1
2
exp
0
0
1
m
j
So we just have to substitute this value of
d
into the matrix to obtain:
iii) A linear polarizer at -45° has a Jones matrix:
⎥
⎦
⎤
⎢
⎣
⎡
−
−
1
1
1
1
2
1
So the combination of the HWP and linear polarizer is
(
)
(
)
(
)
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
+
−
−
⎟
⎠
⎞
⎜
⎝
⎛
+
−
−
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
+
−
⎥
⎦
⎤
⎢
⎣
⎡
−
−
=
λ
λ
π
λ
λ
π
λ
λ
π
0
0
0
1
2
exp
1
1
2
exp
1
2
1
1
2
exp
0
0
1
1
1
1
1
2
1
m
j
m
j
m
j
M
iv) We have +45° polarized incident light, so the output electric field is given by

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