Chapter 12
The Diffraction Grating
Lecture Notes for Modern Optics based on Pedrotti & Pedrotti & Pedrotti
Instructor: Nayer Eradat
Instructor: Nayer Eradat
Spring 2009
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The Diffraction Grating
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Diffraction grating
Grating equation
Free spectral range of a grating
Free spectral range of a grating
Dispersion of a grating
Resolution of a grating
Types of grating
Types of grating
Blazed gratings
Grating replicas
Grating instruments
Grating instruments
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The Diffraction Grating
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Diffraction from many slits For N slits of width b and pitch a the diffraction pattern
has the following form
2
⎛ sin β ⎞ ⎛ sin Να ⎞
I = I0 ⎜
⎟
β ⎠⎟ ⎜⎝
sin
α ⎠
⎝
Diffraction by a
single slit
2
Secondaryy
maxima
Interference
between N slits
with β = (1/ 2 ) kb sin θ and α = (1/ 2 ) ka sin θ
The pattern consists of series of sharp maxima that we call
principal maxima.
I principal maxima ∝ N 2 cantered at values α = 0, ±π , ±2π , ±3π ,...
Between successive peaks there are N − 2 secondary peaks.
Principal maxima
Limiting diffraction envelope
The full irradiance is p
product of the diffraction ppattern and
interference pattern.
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Fraunhofer Diffraction
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The grating equation
In many-slit problem in chapter 11 the plane of the incident wavefronts was parallel to the plane of
tthee slits.
s ts. Here
e e we assume
assu e plane
p a e of
o the
t e incident
c de t wavefront
wave o t makes
a es an
a angle
a g e θi with
w t the
t e plane
p a e of
o the
t e
grating. The net path difference between the successive slits is:
Δ = Δ1 + Δ 2 = a sin θi + a sin θ m
Sign convention: when θi and θ m are on the same side of the normal to the grating,
grating θ m > 0
when the diffracted rays on the opposite side of the normal to the grating, θ m < 0
The grating equation is then:
a ( sin
i θi + sin
i θ m ) = mλ where
h m = 0,
0 ±1,
1 ±2,...
2
For each value of the m different wavelengts will be enhanced at different angles excpt for m = 0
that all wavelengths will be enhanced at θ m = −θi
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The Diffraction Grating
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Diffraction grating and many‐slit
1) Maxima for different wavelengths occur at different positions
2) Higher number of slits result narrower principal
maxima in diffraction pattern.
A diffraction grating is a periodic multiple-slit device that is designed to take advantage of the
sensitivity of its diffraction pattern to the wavelength of the incident light. invented by Fraunhofer
⎧ transmission
There are two kinds of grating: ⎨
⎩reflection grating
Slits are called lines or rulings
Grating spacing is the distance between rulings
Question:
Where is the first order maxima for red compared to blue on the screen?
a ( sin θi + sin θ m ) = mλ where m = 0, ±1, ±2,...
For a given m and θi the longest wavelength has the smallest deviation angle θ m .
Thi is
This
i opposit
it off the
th prism
i deviation
d i ti that
th t the
th largest
l
t deviation
d i ti occur off the
th largest
l
t λ.
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The Diffraction Grating
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Free spectral range of a grating
Example: Visible spectrum is spread between 400 & 700 nm. Find the angular width of the first order visible
spectrum produced by a plane grating with 600 slits per millimeter when white light falls normally on the
grating. Do the first and second order spectrum overlap? What about second and third order spectra?
Does your answer depend on the grating spacing? No
a ( sin θi + sin θ m ) = mλ where m = 0, ±1, ±2,...
θi = 0, a = 1×10−3 / 600 = 1.67 ×10−6
700 × 10−9
a sin θ1r = λr → sin θ1r =
→ θ1r = 24.770
−6
1.67 ×10
400 × 10−9
a sin
i θ1b = λb → sin
i θ1b =
→ θ1b = 13.88
13 880
−6
1.67 ×10
Δθ1 = 10.57 0
sin θ 2 r = sin θ1r / 2 → θ 2 r = 12.0970
Th 1st
The
1 t andd 2nd
2 d order
d do
d nott overlap,
l the
th 2nd
2 d and
d 33rd
d overlap.
l
Free spectral range ( λfsr ) for a diffraction grating: the
non-overlapping wavelength range in a particular order
is called the free spectral range.
range
λ1 shortest wavelength in the spectrum, and λ2 longest wavelength
λfsr for order m is determined by coinciding the λ2 in order m with λ1 in order m + 1
mλ2 = ( m + 1) λ1 → λ fsr = λ2 − λ1 =
λfsr1 = 400, λfsr2
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λ1
Note λfsr is different for different orders.
orders
m
= 200 yes there will be overlap, λfsr3 = 133, only 133 nm is not overlaping.
The Diffraction Grating
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Dispersion of a grating
Higher diffraction orders become less intense under the envelope of the single slit diffraction.
Within an order the wavelengths
g spead
p
better for higher
g
orders.
The angular dispersion defined as
D≡
dθ m
dλ
is the angular separation per unit wavelength.
The grating equation: a ( sin θi + sin θ m ) = mλ
For normal incidene: a ( sin θ m ) = mλ → cos θ m dθ m =
Angular dispersion ≡ D ≡
m
dλ
a
dθ m
m
=
d λ a cos θ m
Linear dispersion on a screen aatt the focal point of a lens with focal length of f :
Linear dispersion ≡
dθ
dy
= f m = fD
dλ
dλ
Example : λ =500nm,
=500nm normal incidence on a grating 5000 grooves/cm used with a lens of focal
length 0.5m. What is the angular and linear dispersion in first order?
a=2 × 10-4 cm; asinθ =mλ gives the diffraction angle m=1 and θ1 = 14.50 ; then the angular dispersion:
D=
m
= 5165rad / cm = 0.02960 / nm; the linear dispersion fD = 0.258mm / nm
a cos θ m
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The Diffraction Grating
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Resolution of a grating
High resolution with diffraction gratings is not achhieved by high disppersion.
Like the Fabry-Perot cavity we define the resolution fo a grating as: ℜ ≡
λ
( Δλ )min
with this definition each wavelength peack appear narrower for high-resolution gratings.
Δλmini is the minimum wavelength interval of two spectral components that are just resolvable by
Rayleigh criteron. For normal incidence the angle of principal maximum of order m is:
a sin θ m = m ( λ + Δλ )
With Rayleigh's criterion this peak coincides with the first minimum of the neighboring wavelength
W
⎛ a sin θ m ⎞ W W sin θ m
where N=
a sin θ m = ( m + 1/ N ) λ → m ( λ + Δλ ) = ( m + 1/ N ) λ → ℜ = mN = ⎜
=
⎟
λ
a
⎝ λ ⎠a
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The Diffraction Grating
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