Chapter 12. Diffraction Grating
Last Lecture• Fraunhofer versus Fresnel Diffraction• Diffraction from a Single Slit• Beam Spreading• Rectangular and Circular Apertures
This Lecture
• Resolution
This Lecture• The Grating Equation and Free Spectral Range• Grating Dispersion and Resolution• Grating Dispersion and Resolution• Types of Gratings• Grating InstrumentsGrating Instruments
12-1. Grating equation: normal incidence
m=0
m=1
m=2
gratinggrating
λθ ma =sinm=1
a θ
m=1
The Grating Equation: generalizedm > 0θm > 0
y
Phase matching
, ,
sin siny m y ik k mG
k k mGθ θ
= − +
= +sin sinsin sin2 2 2sin sin
m i
i m
k k mGk k mG
m
θ θθ θ
π π πθ θ
= − ++ =
⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟a
m=0
( )
sin sin
sin sin
i m
i m
ma
a m
θ θλ λ
θ θ λ
+ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⇒ + =
m < 0 θm < 0
,The grating equation can be easily generalized for the case that the incident light is not at normal incidence
λθθ maa mi =+=Δ+Δ=Δ sinsin21
( ) ,...2,1,0 ,sinsin ±±==+ mma mi λθθ * Sign convention
12-2. Free Spectral Range of a Grating
The free spectral range of the grating can be determinedfrom the condition that the shortest detectable wavelength
( )
11
2
in the order m just overlaps with the longest detectablewavelength in the order mλ
λ+
( ) 1 21m m
The free spectral rang
λ λ+ =
e for order m is then
12 1FSR
mλλ λ= − =
mFSR 1
22λλλ =−≡
12-3. Dispersion of Grating
The angular dispersion of the grating is defined by
cosm
m
d md aθλ θ
= =D ( ) λθθ ma mi =+ sinsin
The linear dispersion is given by
mddylinear dispersion f fd d
θλ λ
= = = D dy fdθ=
Angular and linear dispersions of a grating
12-4. Resolution of Grating
( )minλλ
Δ≡R : Resolving power of a grating 2 2
0sin sin
sinPNI I β α
β α⎛ ⎞ ⎛ ⎞== ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
The resolution of the grating is found from conditionthat for two wavelengths λ and λ+ λ,Δ
( )minThe principal maxima occur for
, = sin
The first minimum of the neighboring wavelength's peak
mm aπα π α θλ
⎝ ⎠
=
the maximum for λ+ λ just concides withthe first minum
that for two wavelengths λ and λ λ,
.This gives us
um for λ
ΔΔ
g g g pin the same order occurs at
1( 1) ( )N Nm mN
α π α π= + ⇒ = +
( )sin : max
1sin : min
a m
a m
θ λ λ
θ λ
= + Δ
⎛ ⎞= +⎜ ⎟
2sin
⎟⎟⎠
⎞⎜⎜⎝
⎛ββ
2
sinsin
⎟⎠⎞
⎜⎝⎛
ααN
sin : min a mN
Equating the right hand s
θ λ= +⎜ ⎟
−
⎝ ⎠
ides of the equations above we obtain
N2
( )min mNThe resolving power of the grating is defined
λλ
λ
Δ =
( )min
R mNλλ
= =Δ
( ) mNR =Δ
≡minλ
λ
F-P interferometer and Diffraction grating
A good Fabry-Perot interferometer may have, overall, a resolution power in the range 106 – 107,
whereas the resolving power of a good diffraction grating is in the range of 105 106 an order of magnitude smallerwhereas the resolving power of a good diffraction grating is in the range of 105 – 106, an order of magnitude smaller.
Types of Gratings
Types of Gratings
• Transmission Amplitude Grating – periodic transmission in clear sections of glass blank groovestransmission in clear sections of glass blank, grooves serve as scattering centers
• Transmission Phase Grating – light is periodically• Transmission Phase Grating – light is periodically modulated in phase due to refractive index variations
• Reflection Gratings – widely used in practiceReflection Gratings widely used in practice • Blazed Gratings – increase intensity in higher orders
Reflection Gratings
The path difference between equivalent reflected rays
m < 0θm < 0
with same sign convention,
( )sin sin a 0 0
the grating equation for a reflection grating is
h dθλ θ θ θ> <+( ) ,sin sin a 0 0i i mmm a s shown andθλ θ θ θ>= <+
m > 0θm > 0
12-6. Blazed Transmission Gratings
unblazed
굴절
blazed굴절
Blazed Reflection Gratings
Blazed Reflection GratingsTo determine the properblaze angle for the grating,we need to reflect the incidentwe need to reflect the incidentlight directly into the desired order m :
θ θ θ θi b m b
i m
θ θ θ θ
θ θθ
− = +
−⇒ = θm
22
b
m i b
θ
θ θ θ
⇒ =
⇒ = −
( )sin sin ,
, 2
i m
i mm m b
But m a with sign conventionλ θ θθ θθ θ θ
= +
+⇒ →− =
s2
m aλ = ( )in sin 2i b iθ θ θ+ −⎡ ⎤⎣ ⎦
θb 가 정해져 있을 때, θi 로 입사하는 빛은 모두 특정한 θm으로 회절될 수 있다.
Littrow mounting of a blazed reflection gratings
Littrow mountingim θθ +=
i mθ θθ +ib θθ =
[ ])2sin(sin ibiam θθθλ −+=
2i m
bθ =im θθ +=
ib θθ
-12 sin or sin2b bmm a
aλλ θ θ ⎛ ⎞= = ⎜ ⎟
⎝ ⎠
Normal mounting : 0=iθ
2/mb θθ +=
-11 sinbmλθ ⎛ ⎞= ⎜ ⎟
⎝ ⎠
0=iθ
2b a⎜ ⎟⎝ ⎠
Example 12-3.
In a Littrow mounting
⎞⎛ λ 1for 1.212amsin 1- ==⎟
⎠⎞
⎜⎝⎛= mb
λθ
In a normal mountingIn a normal mounting
03.23a
msin21 1- =⎟
⎠⎞
⎜⎝⎛=
λθb⎠⎝
In a Littrow mounting
Interference gratings
( )θλ i2/d ( )θλ sin2/=d
( ) λθθ =+ mia sinsin
Grating Instruments : spectrometer
Echelle spectrometer
Czerny-Turner spectrometer
Concave gratingConcave gratingPaschen-Runge spectrometer
Wadsworth spectrometer
