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

Quantum Mechanics Astrophysics chemistry Telecommunications

USE

Verification of Hydrogen spectrum Composition and processes in stars and planetary atmospheres Concentration of chemical species in samples 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 – 1 st maximum Q – 1 st secondary maximum θ = nλ/d

Diffraction Pattern

Intensity of the beam is governed by I = I 0 { sin β / β } 2 Where β = (π / λ) d sin θ

Two Slit Interference :

Slit width b Distance between the slits d I = I 0 { sin β / β } 2 cos 2 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 = I 0 { 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 Incident light Grating normal Reflected light + Diffracted light Grating normal Incident light + α Diffracted light β 1 α β 0 Β -1 d

α > 0, β 1 >0 β 0 < 0, β -1 < 0

A Reflection grating β 1 + β 0 Β -1 Diffracted ray A transmission grating Light diffracted in the same direction of the incident ray  +ve angle

• Wave front propagation through the grating B1

Classical diffraction: Grating equation:

m λ= d(sinα + sinβ)  Gm λ= sinα + sinβ  Gm λ= 2cosK sinØ A1 A4 G – groove frequency = 1/d λ – wavelength of the diffracted light K – deviation angle = ½(α-β) Ø – scan angle = ½(α+β) B4 α β A3 A2 B2 α β

Littrow configuration :

α=β  m λ= 2dsinα B3 d Path difference = A2A3 ~ B2B3 = d sin α + d sin β

Conical diffraction:

Gm λ= 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 = dcos β / r’m Obliquity factor is the factor that governs the platefactor when the incident ray is not perpendicular to the grooves and is = 1/sin Ø

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.(sin α + sinβ)/λ  R max = 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 m th order.

The greater the free spectral ranges the less is the filters required.

FOCAL LENGTH AND f/NUMBER

•

W

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

O

Source

A

Incident light r α r,’ Diffracted light β

B

Image Grating Normal f/no. output = r’/W 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 -- dispersion plane ---

Diffuse scattered light, In-plane scatter. Ghosts

in 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  source environment must have constant temperature and atmospheric the 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 Surface irregularities Ruling errors Groove placement errors Groove frequency Groove pattern ruled grating yes Yes Yes Better Need not be equally spaced Interference grating no no No Good 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 classical gratings  1 st generation gratings  equally spaced lines on tangent plane unequal spacing and curved 2 nd generation gratings  toroidal wavefronts varied line spacing  grooved lines are varied uniformly

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 m 2 λ B m 1 P-plane is TE polarized light S-plane is TM polarized light λ B is the blaze wavelength where highest efficiency is recorded

Efficiency for triangular and sinusoidal grooves

Triangular grooves ( blaze angle) Very low BA θ < 5 0 Low B A 5 0 < θ < 10 0 Medium B A 10 0 < θ < 18 0 Special low anomaly 18 0 < θ < 22 0 High BA 22 0 < θ < 38 0 Very high B A θ > 38 0 Sinusoidal grooves (modulation) µ = groove height/ spacing very low µ < 0.05

low 0.05 < µ < 0.15

Medium 0.15 < µ < 0.25

High 0.25 < µ < 0.4

Very high µ > 0.4

• Maximum efficiency is obtained through triangular grooves.

Applications Gratings as Principle used

FILTERS Plane gratings blazed for the wavelength of unwanted shorter wavelength radiation ELECTRON MICROSCOPE Replica gratings made from master gratings CALIBRATION 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 Exit slit Detector Camera