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

18
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 Instruments Grating Instruments

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

Page 1: Chapter 12. Diffraction Grating - Hanyangoptics.hanyang.ac.kr/~shsong/12-Diffraction grating.pdf · Chapter 12. Diffraction Grating Last Lecture • Fraunhofer versus Fresnel Diffraction

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

Page 2: Chapter 12. Diffraction Grating - Hanyangoptics.hanyang.ac.kr/~shsong/12-Diffraction grating.pdf · Chapter 12. Diffraction Grating Last Lecture • Fraunhofer versus Fresnel Diffraction

12-1. Grating equation: normal incidence

m=0

m=1

m=2

gratinggrating

λθ ma =sinm=1

a θ

m=1

Page 3: Chapter 12. Diffraction Grating - Hanyangoptics.hanyang.ac.kr/~shsong/12-Diffraction grating.pdf · Chapter 12. Diffraction Grating Last Lecture • Fraunhofer versus Fresnel Diffraction

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

Page 4: Chapter 12. Diffraction Grating - Hanyangoptics.hanyang.ac.kr/~shsong/12-Diffraction grating.pdf · Chapter 12. Diffraction Grating Last Lecture • Fraunhofer versus Fresnel Diffraction

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 =−≡

Page 5: Chapter 12. Diffraction Grating - Hanyangoptics.hanyang.ac.kr/~shsong/12-Diffraction grating.pdf · Chapter 12. Diffraction Grating Last Lecture • Fraunhofer versus Fresnel Diffraction

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=

Page 6: Chapter 12. Diffraction Grating - Hanyangoptics.hanyang.ac.kr/~shsong/12-Diffraction grating.pdf · Chapter 12. Diffraction Grating Last Lecture • Fraunhofer versus Fresnel Diffraction

Angular and linear dispersions of a grating

Page 7: Chapter 12. Diffraction Grating - Hanyangoptics.hanyang.ac.kr/~shsong/12-Diffraction grating.pdf · Chapter 12. Diffraction Grating Last Lecture • Fraunhofer versus Fresnel Diffraction

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

Page 8: Chapter 12. Diffraction Grating - Hanyangoptics.hanyang.ac.kr/~shsong/12-Diffraction grating.pdf · Chapter 12. Diffraction Grating Last Lecture • Fraunhofer versus Fresnel Diffraction

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.

Page 9: Chapter 12. Diffraction Grating - Hanyangoptics.hanyang.ac.kr/~shsong/12-Diffraction grating.pdf · Chapter 12. Diffraction Grating Last Lecture • Fraunhofer versus Fresnel Diffraction

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

Page 10: Chapter 12. Diffraction Grating - Hanyangoptics.hanyang.ac.kr/~shsong/12-Diffraction grating.pdf · Chapter 12. Diffraction Grating Last Lecture • Fraunhofer versus Fresnel Diffraction

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

Page 11: Chapter 12. Diffraction Grating - Hanyangoptics.hanyang.ac.kr/~shsong/12-Diffraction grating.pdf · Chapter 12. Diffraction Grating Last Lecture • Fraunhofer versus Fresnel Diffraction

12-6. Blazed Transmission Gratings

unblazed

굴절

blazed굴절

Page 12: Chapter 12. Diffraction Grating - Hanyangoptics.hanyang.ac.kr/~shsong/12-Diffraction grating.pdf · Chapter 12. Diffraction Grating Last Lecture • Fraunhofer versus Fresnel Diffraction

Blazed Reflection Gratings

Page 13: Chapter 12. Diffraction Grating - Hanyangoptics.hanyang.ac.kr/~shsong/12-Diffraction grating.pdf · Chapter 12. Diffraction Grating Last Lecture • Fraunhofer versus Fresnel Diffraction

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.

Page 14: Chapter 12. Diffraction Grating - Hanyangoptics.hanyang.ac.kr/~shsong/12-Diffraction grating.pdf · Chapter 12. Diffraction Grating Last Lecture • Fraunhofer versus Fresnel Diffraction

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⎜ ⎟⎝ ⎠

Page 15: Chapter 12. Diffraction Grating - Hanyangoptics.hanyang.ac.kr/~shsong/12-Diffraction grating.pdf · Chapter 12. Diffraction Grating Last Lecture • Fraunhofer versus Fresnel Diffraction

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

Page 16: Chapter 12. Diffraction Grating - Hanyangoptics.hanyang.ac.kr/~shsong/12-Diffraction grating.pdf · Chapter 12. Diffraction Grating Last Lecture • Fraunhofer versus Fresnel Diffraction

Interference gratings

( )θλ i2/d ( )θλ sin2/=d

( ) λθθ =+ mia sinsin

Page 17: Chapter 12. Diffraction Grating - Hanyangoptics.hanyang.ac.kr/~shsong/12-Diffraction grating.pdf · Chapter 12. Diffraction Grating Last Lecture • Fraunhofer versus Fresnel Diffraction

Grating Instruments : spectrometer

Echelle spectrometer

Czerny-Turner spectrometer

Page 18: Chapter 12. Diffraction Grating - Hanyangoptics.hanyang.ac.kr/~shsong/12-Diffraction grating.pdf · Chapter 12. Diffraction Grating Last Lecture • Fraunhofer versus Fresnel Diffraction

Concave gratingConcave gratingPaschen-Runge spectrometer

Wadsworth spectrometer