SME
University Physics III: Waves, Optics & Modern · Problem type
The diffraction grating
Use this for “A grating has 300 lines per mm. At what angle is the second-order line of 656 nm light, and how many orders can be seen?”
Diagram
d
=
1
N
=
1
600
lines/mm
=
1.667
μm
Order
2:
sin
θ
=
mλ
d
=
2
×
500
nm
1,667
nm
=
0.6,
θ
=
36.87°
Highest
order:
mₘₐₓ
=
⌈
d
λ
⌉
−
1
=
⌈3.333⌉
−
1
=
3;
m
=
4
would
need
sin
θ
=
1.2,
past
1.
This is an example. Type your own values to start a new problem.
d (μm)
sin(
θ
°)
=
m
×
λ (nm)
N
Lines per millimetre
example
mₘₐₓ
Highest order
worked out (not typed)
Formulas
d
=
1,000
N
1.6667
=
1,000
600
d
×
1,000
×
sin(
θ
)
=
m
×
λ
1.6667 × 1,000 × sin(36.8699°) = 2 × 500
mₘₐₓ
=
⌈
d
×
1,000
λ
⌉
−
1
3
=
⌈1.6667
×
1,000
500
⌉
−
1
Assumptions
•
Light falls square on the grating, and every line sends out light in step with the others.
•
Bright orders where d sin θ = mλ, with m = 0, ±1, ±2, …; they are sharp because thousands of lines must agree.
•
An order needs sin θ below 1, so the highest is mₘₐₓ = ⌈d ÷ λ⌉ − 1, the largest whole number below d ÷ λ.
•
N is in lines per mm, so d = 1000 ÷ N μm; d in nm is d × 1000.
Step-by-step
WE KNOW
N = 600, λ = 500 nm, m = 2
FIND
line spacing (d), angle of the order (θ), highest order (mₘₐₓ)
Step 1 · Find line spacing (d)
Use d = 1,000 ÷ N
One
millimetre
is
1000
μm,
shared
among
the
N
lines.
d
=
1,000
600
d
=
1.6667
μm
Step 2 · Find angle of the order (θ)
Use d × 1,000 × sin(θ) = m × λ
sin
θ
=
mλ
d
,
with
d
turned
to
nm
(1000
nm
per
μm).
θ
=
sin⁻¹(
m
×
λ
d
× 1,000
)
θ
=
sin⁻¹(2
×
500
1.6667 × 1,000
)
θ
=
36.8699°
Step 3 · Find highest order (mₘₐₓ)
Use mₘₐₓ = ⌈d × 1,000 ÷ λ⌉ − 1
The
largest
whole
number
below
d
λ
(
d
in
nm):
one
order
more
would
need
sin
θ
of
1
or
more.
mₘₐₓ
=
⌈1.6667
×
1,000
500
⌉
−
1
mₘₐₓ = 3
Check
Put the numbers back in. Both sides should match.
1.6667
=
1,000
600
✓
1.6667 × 1,000 × sin(36.8699°) = 2 × 500 ✓
3
=
⌈1.6667
×
1,000
500
⌉
−
1
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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