Chapter 12. Diffraction Grating - Hanyangoptics.hanyang.ac.kr/~shsong/12-Diffraction grating.pdf ·...

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