Diffraction gratings

1
Diffraction gratings

• By M. Ravi Kiran

2
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.
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 ? n?/d
Diffraction Pattern
Intensity of the beam is governed by I
I0 sin ß / ß 2 Where ß (p / ?) d sin ?
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Two Slit Interference
Slit width b Distance between the slits d
I I0 sin ß / ß 2 cos2 µ Where ß (p/?).b
sin ? µ (p/?).d sin ?
Intensity distribution is similar to single slit
and the spacing between the fringes is
determined by (?/d) and width of the envelop by
?/b.
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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 ß / ß 2 (sin Nµ
  )/ (N sin µ) 2
• Where ß (p/?).b sin?
• µ (p/?).d
  sin?
• Principle fringes occur at µ n p ? n ? d sin?
• Secondary fringes occur at µ 3p/2N, 5p/2N,

7
Physics of diffraction

• Ray Propagation through the grating

Grating normal
Grating normal
Incident light
Incident light
Reflected light


-
-
a
Diffracted light
a gt 0, ß1 gt0
ß0
a
Diffracted light
?-1
ß1
ß0 lt 0, ß-1 lt 0
d
?-1
ß1
ß0

-
Diffracted ray
A Reflection grating
A transmission grating
Light diffracted in the same direction of the
incident ray ? ve angle
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• Wave front propagation through the grating

Classical diffraction
Grating equation m? d(sina sinß)
? Gm? sina sinß
? Gm? 2cosK sinØ
B1
A1
G groove frequency 1/d ? wavelength of
the diffracted light K deviation angle
½(a-ß) Ø scan angle ½(aß)
A4
B4
ß
a
a
ß
Littrow configuration aß
? m? 2dsina

B2
A3
A2
B3
d
Conical diffraction
Gm? cose (sina sinß)
Path difference A2A3 B2B3 d sina d sin ß
e angle between the incident light path and
the plane perpendicular to the grooves.
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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.

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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 ?ß/?? G.m.secß
• (2/?)tanß
  --- Littrow condition
• Linear dispersion is the product of angular
  dispersion D and effective focal length r(ß)
• linear dispersion (l) rD
  r.G.m.secß
• Platefactor is change in wavelength when
  we move along the spectrum and is given by P
  1/l dcosß / rm
• Obliquity factor is the factor that
  governs the platefactor when the incident ray is
  not perpendicular to the grooves and is 1/sinØ

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RESOLVING POWER

• This is the ability to separate adjacent spectral
  lines of average wavelength ?. Mathematically,
• R ?/?? ?? -- 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.(sina sinß)/? ? Rmax
  2n.d/ ?

SPECTRAL RESOLUTION

• ?? is the spectral resolution and is measured
  by convoluting the image
• of the entrance aperture with the exit
  aperture.

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• 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 ? ? 1 /m where ?
  1 is the wavelength of light diffracted in
  the mth order.
• The greater the free spectral ranges the less is
  the filters required.

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• 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.

r/W
f/no. input
Source
A
Incident light
r
f/no. output
r/W
a
W
ß
O
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.

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• 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.

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• 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

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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

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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.

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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.

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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


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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

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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.

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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.

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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.

?
a
ß
?
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Efficiency curve
Graph between absolute efficiency or relative
efficiency with respect to wavelength or
sometimes ?/d.

• Depends on
• m (diffraction order)
• angles of incidence and diffraction
• ?/d
• polarization
• P- Plane gt no anomalies
• S- Plane gt anomalies.

m1lt m2lt m3
m2
m1
?B
P-plane is TE polarized light S-plane is TM
polarized light
is the blaze wavelength where highest efficiency
is recorded
?B
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Efficiency for triangular and sinusoidal grooves

• Triangular grooves ( blaze angle)
• Very low BA ? lt 50
• Low B A 50 lt ? lt 100
• Medium B A 100 lt ? lt 180
• Special low anomaly 180 lt ? lt 220
• High BA 220 lt ? lt 380
• Very high B A ? gt 380

• Sinusoidal grooves (modulation)
• µ groove height/ spacing
• very low µ lt 0.05
• low 0.05 lt µ lt 0.15
• Medium 0.15 lt µ lt 0.25
• High 0.25 lt µ lt 0.4
• Very high µ gt 0.4

• Maximum efficiency is obtained through
  triangular grooves.

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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

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Grating spectrometers

• Czerny-turner spectrograph

Entrance slit
collimator
Grating
Detector
Exit slit
Camera