Download Diffraction gratings

Diffraction gratings
By M. Ravi Kiran
Introduction
•
•
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.
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.
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
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
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.
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, ……
Physics of diffraction
• Ray Propagation through the grating
Grating normal
Incident light
Grating normal
Incident light
Reflected light
+
+
-
α
-
Diffracted light
α
Diffracted light
β1
α > 0, β1 >0
β0
Β-1
β0 < 0, β-1 < 0
d
β1
+
Β-1
β0
Diffracted ray
A Reflection grating
A transmission grating
Light diffracted in the same direction of the incident ray  +ve angle
• 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
A3
β
B2
Littrow configuration : α=β
 mλ= 2dsinα
A2
B3
d
Path difference = A2A3 ~ B2B3 = d sinα + d sin β
Conical diffraction:
Gmλ= cosε (sinα + sinβ)
ε – angle between the incident light path and
the plane perpendicular to the grooves.
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.
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
RESOLVING POWER
•
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.
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.
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.
Source
A
Incident light
r
W
O
f/no. output = r’/W
α
β
r,’
Diffracted light
f/no. input = r/W
B
Image
Grating Normal
r/r’ determines the exit slit width
• 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.
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.
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.
1.
2.
3.
4.
5.
Groove profile
Groove frequencies
Groove pattern
Substrate shapes
Surface irregularities
And these parameters depend on the method of manufacturing :
Ruled Gratings or Holographic Gratings
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
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.
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.
Comparison
Property
ruled grating
Interference grating
Surface irregularities
yes
no
Ruling errors
Yes
no
Groove placement
errors
Yes
No
Groove frequency
Better
Good
Groove pattern
Need not be equally
spaced
Equally spaced
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
1st generation gratings
 equally spaced lines on tangent
plane
 unequal spacing and curved
2nd generation gratings

varied line spacing
 grooved lines are varied uniformly
classical gratings
toroidal wavefronts
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.
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.
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
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
Efficiency for triangular and sinusoidal grooves
Triangular grooves ( blaze angle)
Sinusoidal grooves (modulation)
µ = groove height/ spacing
Very low BA
θ < 50
Low B A
50 < θ < 100
Medium B A
100 < θ < 180
Special low anomaly 180 < θ < 220
High BA
220 < θ < 380
Very high B A
θ > 380
very low
low
Medium
High
Very high
µ < 0.05
0.05 < µ < 0.15
0.15 < µ < 0.25
0.25 < µ < 0.4
µ > 0.4
• Maximum efficiency is obtained through triangular grooves.
Applications
Gratings as
FILTERS
Principle used
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
Grating spectrometers
• Czerny-turner spectrograph
collimator
Entrance slit
Grating
Detector
Exit slit
Camera