1. Diffraction gratings
By M. Ravi Kiran

2. Introduction FIELD USE
Diffraction grating can be understood as an optical unit that separates polychromatic light into constant monochromatic composition.
Uses are tabulated below
FIELD
USE
Quantum Mechanics
Verification of Hydrogen spectrum
Astrophysics
Composition and processes in stars and planetary atmospheres
chemistry
Concentration of chemical species in samples
Telecommunications
Increase the capacity of fiber optic networks using WDM
When an Electromagnetic radiation falls on a Diffraction Grating, the electric field
and Phase are modified in a predictable manner.

3. Physicist view of Diffraction grating
A Multi-slit arrangement which uses diffraction to separate light wavelengths with high resolution and high intensity. The resolving power is achieved by interference of light.

4. Basics of diffraction Single slit interference P– 1st maximum
Q– 1st secondary maximum
θ = nλ/d
Diffraction Pattern
Intensity of the beam is governed by
I = I0 { sin β / β }2
Where β = (π / λ) d sin θ

5. Two Slit Interference :
Slit width b
Distance between
the slits d
I = I0 { sin β / β }2 cos2 µ
Where β = (π/λ).b sin θ
µ = (π/λ).d sin θ
STT22.1
Intensity distribution is similar to single slit and the spacing between the
fringes is determined by (λ/d) and width of the envelop by λ/b.

6. Multiple slit interference
A N-slits interference pattern is the diffraction pattern and we develop diffraction gratings based on N-slit interference pattern.
Intensity transmission function is
I = I0 { sin β / β }2 {(sin Nµ )/ (N sin µ) }2
Where β = (π/λ).b sinθ
µ = (π/λ).d sinθ
Principle fringes occur at µ = n π  n λ= d sinθ
Secondary fringes occur at µ = 3π/2N, 5π/2N, ……

7. Physics of diffraction
Ray Propagation through the grating
Grating normal
Grating normal
Incident light
Reflected light
Incident light + + - - α
Diffracted light β0
α > 0, β1 >0
Diffracted light α β1
Β-1
β0 < 0, β-1 < 0 d
Β-1 β1 β0 + -
Diffracted ray
A Reflection grating
A transmission grating
Light diffracted in the same direction of the incident ray  +ve angle

8. Wave front propagation through the grating
Classical diffraction:
Grating equation: mλ= d(sinα + sinβ)
 Gmλ= sinα + sinβ
 Gmλ= 2cosK sinØ B1 A1
G – groove frequency = 1/d
λ – wavelength of the diffracted light
K – deviation angle = ½(α-β)
Ø – scan angle = ½(α+β) A4 B4 β α α β
Littrow configuration : α=β
 mλ= 2dsinα A3 B2 A2 B3 d
Conical diffraction:
Gmλ= cosε (sinα + sinβ)
Path difference = A2A3 ~ B2B3 = d sinα + d sin β
ε – angle between the incident light path and
the plane perpendicular to the grooves.

9. Characteristics of Diffraction Grating
Dispersion:
angular dispersion
linear dispersion
Resolving power
Spectral resolution
Band pass
Focal length and f-number
Anamorphic magnification
Free spectral range
Energy distribution
Scattered and stray light
scattered light
instrumental stray light
Signal to noise ratio.

10. DISPERSION
Angular Dispersion is the measure of the separation between diffracted light of different wavelengths. It gives the spectral range per unit angle.
Mathematically,
D= ∂β/∂λ = G.m.secβ
= (2/λ)tanβ Littrow condition
Linear dispersion is the product of angular dispersion D and effective focal length r’(β)
linear dispersion (l) = r’D = r’.G.m.secβ
Platefactor is change in wavelength when we move along the spectrum and is given by P = 1/l = dcosβ / r’m
Obliquity factor is the factor that governs the platefactor when the incident ray is not perpendicular to the grooves and is = 1/sinØ

11. RESOLVING POWER SPECTRAL RESOLUTION:
This is the ability to separate adjacent spectral lines of average wavelength λ. Mathematically,
R = λ/∆λ ∆λ -- limit of resolution, difference in wavelength of equal intensity
Theoretically, it is the product of diffraction order and the total number of grooves illuminated.
R = N.d.(sinα + sinβ)/λ  Rmax = 2n.d/ λ
SPECTRAL RESOLUTION:
∆λ is the spectral resolution and is measured by convoluting the image
of the entrance aperture with the exit aperture.

12. BANDPASS
This is the wavelength interval that passes through the exit slit.
Also, the difference in wavelengths between the points of half-maximum intensity on either side of the intensity maximum.
Mathematically, its estimate is given by
B = w’. P where w’– exit slit width
P – reciprocal of linear Dispersion.
FREE SPECTRAL RANGE
It is the range of wavelengths in a given spectral order for which light from adjacent orders are not superposed.
Mathematically,
F λ = λ 1 /m where λ 1 is the wavelength of light diffracted in the mth order.
The greater the free spectral ranges the less is the filters required.

13. FOCAL LENGTH AND f/NUMBER
If the beam diffracted from the grating of a given wavelength and order converges to a focus, then the distance between the focus and the grating centre is the focal length and the ratio of the focal length to the width of the grating.
f/no. input
= r/W
Source A
Incident light r α
f/no. output
= r’/W W O β
Grating Normal
r,’ B
r/r’ determines the exit slit width
Diffracted light
Image
The more the f/number the less is the spectral aberrations.
ANAMORPHIC MAGNIFICATION
It is the ratio of the width of the collimated diffracted beam to the collimated
incident beam.

14. SCATTERED AND STRAY LIGHT
ENERGY DISTRIBUTION
The distribution of the incident field power of a given wavelength diffracted by a grating to different spectral orders.
This is also called the grating efficiency
SCATTERED AND STRAY LIGHT
The light apart from the energy that is absorbed by the grating and the energy that is diffracted is scattered light.
Scattered light in front of grating surface --- Diffuse scattered light, in dispersion plane --- In-plane scatter. Ghosts are scattered light due to periodic errors in the groove spacing.
Instrumental stray light is the diffracted light due to the light in the atmosphere but not the incident light.

15. SIGNAL TO NOISE RATIO
Ratio of the diffracted energy to unwanted light energy.
The above mentioned characteristics depend on the following parameters of the grating.
Groove profile
Groove frequencies
Groove pattern
Substrate shapes
Surface irregularities
And these parameters depend on the method of manufacturing :
Ruled Gratings or Holographic Gratings

16. Ruled gratings
Mechanically ruled by burnishing grooves with a diamond tool against a thin coating of evaporated metal using Ruling engines.
Michelson engine
servo controlled laser interferometer
20 grooves/mm to 10,800 grooves/mm
Mann engine
automatic interferometric servo system
no ghosts and theoretical resolving power
MIT ‘B’ Engine
double interferometric control system based on frequency stabilized laser
20 grooves/mm to 1500 grooves/mm

17. The Ruling Process
Substrate material BK-7 , fused silica or special grade ZeroDur polished to one tenth of wavelength with gold o aluminum coatings.
Involves interferometric control  requires a monochromatic source  the source environment must have constant temperature and atmospheric pressure.
Vibrations of the ruling engine has to nullified by passing through the diamonds.
VLS gratings
these gratings work on the principle that the variations in the groove spacing modifies the curvature of the diffracted wavefronts which in turn changes the focus of the spectrum.

18. Holographic gratings
Groves are recorded using interference pattern on a photographic plate, which is a photo resist material ( molecular structure changes with the light exposure).
Selected laser should be of the wavelength that the photo resist is sensitive to.
Steps : 1. exposing to Interference pattern\
2. development…..valleys at bright fringe, ridges at dark.
Classification
 single beam : beam reflected upon itself
 double beam : groove pattern defined by the Intersection of the surface of the substrate and the fringe pattern.

19. Comparison ruled grating Property Interference grating
Surface irregularities
yes no
Ruling errors
Yes
Groove placement errors No
Groove frequency
Better
Good
Groove pattern
Need not be equally spaced
Equally spaced

20. Imaging properties
The properties of the image obtained depends mostly on the aberrations in the wave front.
These aberrations depend on the groove pattern.
With respect to groove patterns we divide gratings into
classical gratings  equally spaced lines on tangent plane
1st generation gratings  unequal spacing and curved
2nd generation gratings  toroidal wavefronts
varied line spacing  grooved lines are varied uniformly

21. General definitions
Plane grating – grating whose surface is plane and requires other optical elements for focusing or imaging.
Concave grating – grating whose surface is concave and focusing is done by the grating itself.
Tangential plane – the plane that contains the incident beam and the diffracted rays. Also called as dispersive plane.
Sagittal plane – the plane perpendicular to tangential plane.
Pole rays – the rays that fall on the grating grooves and diffract.
General rays – the rays that fall outside the groove pattern.

22. Aberrations
Defocus - is the blurring of the image along the tangential plane
Astigmatism is the blurring of the image along the Sagittal plane, this occurs generally when the element is placed off- axis.
Spectral resolution is an important imaging property and is maximum when the incident ray is focused into a line parallel to the grooves called the tangential focus and perpendicular to the grooves called the sagittal focus.
Aberrations are reduced by choosing the exact positions of the entrance slit and the exit slit.

23. Efficiency characteristics
Absolute efficiency is the ratio of the diffracted light to the energy of the incident light.
Relative efficiency is the ratio of the energy of the diffracted light to the energy from the light reflected from a polished surface.
Blazing is the control over the magnitude and variation of diffracted energy with the change in wavelength. This control is generally obtained by getting control over the blazing angle or the groove angle. θ α β θ

24. Efficiency curve
Graph between absolute efficiency or relative efficiency with respect to wavelength or sometimes λ/d.
Depends on
m (diffraction order)
angles of incidence and diffraction
λ/d
polarization
P- Plane => no anomalies
S- Plane => anomalies.
m1< m2< m3 m2 m1 λB
P-plane is TE polarized light
S-plane is TM polarized light λB
is the blaze wavelength where highest efficiency is recorded

25. Efficiency for triangular and sinusoidal grooves
Triangular grooves ( blaze angle)
Very low BA θ < 50
Low B A < θ < 100
Medium B A < θ < 180
Special low anomaly 180 < θ < 220
High BA < θ < 380
Very high B A θ > 380
Sinusoidal grooves (modulation)
µ = groove height/ spacing
very low µ < 0.05
low < µ < 0.15
Medium < µ < 0.25
High < µ < 0.4
Very high µ > 0.4
Maximum efficiency is obtained through triangular grooves.

26. Applications Gratings as Principle used FILTERS
Plane gratings blazed for the wavelength of unwanted shorter wavelength radiation
ELECTRON MICROSCOPE CALIBRATION
Replica gratings made from master gratings so that a space is left between the grooves.
LASER TUNING
Plane reflection grating used in littrow mode
BEAM DIVIDERS
Symmetrically shaped grooves and laminar transmission gratings

27. Grating spectrometers
Czerny-turner spectrograph
Entrance slit
collimator
Grating
Detector
Exit slit
Camera