Di
ff
raction grating
:
There are di
ff
erent kinds of di
ff
raction grating, with many applications in spectroscopy. We
consider here one type of grating, which is a set of
N
1 slits or rulings, each of width
b
,
at distance
a
from one another, so that the total width of the grating
d
= (
N

1)
a
≈
Na
.
We have already found the result for the case
N
= 2. We can treat the di
ff
raction grating
as a collection of pairs of slits. For each pair we can take the solution we have found, and
only superpose the waves from such pairs. We give each pair an index
j
∈
[1
,
N
2
]:
ψ
P
≈
E
L
r
0
e
i
(
kr
0

ω
t
)
N
2
j
=1

(2
j

1)
a

b
2

(2
j

1)
a
+
b
2
e
iks
sin
θ
ds
+
(2
j

1)
a
+
b
2
(2
j

1)
a

b
2
e
iks
sin
θ
ds
≡
E
L
r
0
e
i
(
kr
0

ω
t
)
N
2
j
=1
F
j
82
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F
j
=
1
ik
sin
θ
e
iks
sin
θ

(2
j

1)
a

b
2

(2
j

1)
a
+
b
2
+
e
iks
sin
θ
(2
j

1)
a
+
b
2
(2
j

1)
a

b
2
=
1
ik
sin
θ
e

ik
sin
θ
(2
j

1)
a

b
2

e

ik
sin
θ
(2
j

1)
a
+
b
2
+
e
ik
sin
θ
(2
j

1)
a

b
2

e
ik
sin
θ
(2
j

1)
a
+
b
2
=
b
2
i
β
e

i
(2
j

1)
α
(
e
i
β

e

i
β
)
+
e
i
(2
j

1)
α
(
e
i
β

e

i
β
))
=
b
2
i
β
(
e
i
β

e

i
β
) (
e
i
(2
j

1)
α
+
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 Spring '09
 LIORBURKO
 Physics, Calculus, Diffraction, Sin, Mathematics in medieval Islam, sinc function

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