• The Grating Equation and Free Spectral Range

This Lecture

• The Grating Equation and Free Spectral Range

• Grating Dispersion and Resolution

• Types of Gratings

• Grating Instruments Last Lecture

• Fraunhofer versus Fresnel Diffraction

• Diffraction from a Single Slit

• Beam Spreading

• Rectangular and Circular Apertures

• Resolution

Chapter 17. Diffraction Grating

Chapter 17. Diffraction Grating

17-1. Grating equation: normal incidence 17-1. Grating equation: normal incidence

λ θ ^m

a sin =

m=1

m=2 m=0

m=1

The Grating Equation: generalized The Grating Equation: generalized

The grating equation can be easily generalized for the case that the incident light is not at normal incidence,

λ θ

θ a m

a

+

=

= Δ + Δ

=

Δ

sin sin

( ^{sin + sin} ) ^{= m , m = 0 , ± 1 , ± 2 ,...}

a θ _i θ _m λ

m=0 m > 0

m < 0

θ_m> 0

θ_m< 0

Phase matching

( )

( )

( ) ( )

( ) ( )

( θ θ ) λ

θ θ

λ θ

θ

λ θ

θ

θ π λ

θ π λ

π

θ θ

θ θ

m a

m m

m a

m a

m a mG

k k

mG k

k

mG k

k

m i

m m

m i

m i

m i

m i

i m

i x m x

= +

→

−

→

−

−

=

− +

−

=

−

⎟ ⎠

⎜ ⎞

⎝

− ⎛

⎟ =

⎠

⎜ ⎞

⎝

− ⎛

⎟ ⎠

⎜ ⎞

⎝

⎛

−

=

−

+

= +

=

sin sin

, :

sin sin

sin sin

sin 2 sin 2

2

sin sin

sin sin

, ,

17-2. Free Spectral Range of a Grating 17-2. Free Spectral Range of a Grating

( )

1

1

1

2

The free spectral range of the grating can be determined from the condition that the shortest detectable wavelength

in the order m just overlaps with the longest detectable wavelength in the order m

m m

The free spectral rang λ

λ

λ λ

+

+ =

1

2 1

e for order m is then

FSR m

λ λ λ

= − =

FSR ≡ λ

− λ

= λ m

17-3. Dispersion of a Grating 17-3. Dispersion of a Grating

cos

m

m

m

The angular dispersion of the grating is defined by

d m

d a

The linear dispersion is given by d

linear dispersion dy f f

d d

θ

λ θ

θ

λ λ

= =

= = =

D

D

( θ θ ) m λ

a sin

+ sin

=

Angular and linear dispersions of a grating

Angular and linear dispersions of a grating

17-4. Resolution of a Grating 17-4. Resolution of a Grating

( )

sin max

sin 1 min ; sin

The resolution of the grating is found from condition

that for two wavelengths λ and λ+ λ, the maximum for λ+ λ just concides with the first minumum for λ. This gives us

a m

a m Note that N N a

N

θ λ λ

θ λ α π θ

λ

Δ Δ

= + Δ

⎛ ⎞

= ⎜ + ⎟ =

⎝ ⎠

( )

( )

( )

min

min

1 sin 0

sin Nm N

Equating the right hand sides of the equations above we obtain

mN

The resolving power of the grating is defined

R mN

π α

α

λ λ

λ λ

⎛ ⎞ = + ⇒ =

⎜ ⎟

⎝ ⎠

−

Δ =

= =

Δ

( ) λ

λ

≡ Δ

R : Resolving power of a grating

( ) ^mN

R =

≡ Δ

λ

λ

Types of Gratings Types of Gratings

Types of Gratings

• Transmission Amplitude Grating – periodic

transmission in clear sections of glass blank, grooves serve as scattering centers

• Transmission Phase Grating – light is periodically modulated in phase due to refractive index variations

• Reflection Gratings – widely used in practice

• Blazed Gratings – increase intensity in higher orders

Reflection Gratings Reflection Gratings

( ^{sin sin} )

0 0

i m

i m

The grating equation for a reflection grating is

m a

As shown and

λ θ θ

θ θ

= +

> <

m < 0 θ_m< 0

m > 0 θ_m > 0

: Phase matching

Blazed Transmission Gratings

Blazed Transmission Gratings

Blazed Reflection Gratings

Blazed Reflection Gratings

Blazed Reflection Gratings Blazed Reflection Gratings

( )

( )

2

sin sin

sin sin 2

i b m b

i m

b

i m

i b i

To determine the proper blaze angle for the grating, we need to reflect the incident light directly into the desired order m :

But m a

Therefore

m a

θ θ θ θ θ θ θ

λ θ θ

λ θ θ θ

− = +

⇒ = −

= +

= ⎡ ⎣ + − ⎤ ⎦

θ

Grating Instruments

Grating Instruments

Grating Instruments

Grating Instruments

Grating Instruments

Grating Instruments