# 9.2 Diffraction grating

 Page 1 / 1
We derive the interference patter for a diffraction grating.

## Diffraction grating

Consider the case of N slit diffraction, We have ${E}_{1}=\frac{{\epsilon }_{L}a}{R}\frac{{\mathrm{sin}}\beta }{\beta }{e}^{i\left(k{R}_{1}-\omega t\right)}$ ${E}_{2}=\frac{{\epsilon }_{L}a}{R}\frac{{\mathrm{sin}}\beta }{\beta }{e}^{i\left(k{R}_{2}-\omega t\right)}$ $\text{.}$ $\text{.}$ $\text{.}$ ${E}_{N}=\frac{{\epsilon }_{L}a}{R}\frac{{\mathrm{sin}}\beta }{\beta }{e}^{i\left(k{R}_{N}-\omega t\right)}$ So we can just follow the steps of the two slit case and extend them and get (using ${R}_{N}=R-\left(N-1\right)d{\mathrm{sin}}\theta$ ) $\begin{array}{c}E=\sum _{n=1}^{N}{E}_{N}\\ =\sum _{n=1}^{N}\frac{{\epsilon }_{L}a}{R}\frac{{\mathrm{sin}}\beta }{\beta }{e}^{i\left(kR-2\left(n-1\right)\alpha -\omega t\right)}\\ =\frac{{\epsilon }_{L}a}{R}\frac{{\mathrm{sin}}\beta }{\beta }\sum _{n=1}^{N}{e}^{i\left(kR-2\left(n-1\right)\alpha -\omega t\right)}\\ =\frac{{\epsilon }_{L}a}{R}\frac{{\mathrm{sin}}\beta }{\beta }{e}^{i\left(kR-\omega t\right)}\sum _{n=1}^{N}{e}^{-i2\left(n-1\right)\alpha }\\ =\frac{{\epsilon }_{L}a}{R}\frac{{\mathrm{sin}}\beta }{\beta }{e}^{i\left(kR-\omega t\right)}\sum _{j=0}^{N-1}{e}^{-i2j\alpha }\end{array}$ This is the same geometric series we dealt with before $\sum _{n=0}^{N-1}{x}^{n}=\frac{1-{x}^{N}}{1-x}$ so $\begin{array}{c}E=\frac{{\epsilon }_{L}a}{R}\frac{{\mathrm{sin}}\beta }{\beta }{e}^{i\left(kR-\omega t\right)}\sum _{j=0}^{N-1}{e}^{-i2j\alpha }\\ =\frac{{\epsilon }_{L}a}{R}\frac{{\mathrm{sin}}\beta }{\beta }{e}^{i\left(kR-\omega t\right)}\frac{1-{e}^{-i2N\alpha }}{1-{e}^{-i2\alpha }}\\ =\frac{{\epsilon }_{L}a}{R}\frac{{\mathrm{sin}}\beta }{\beta }{e}^{i\left(kR-\omega t\right)}\frac{{e}^{-iN\alpha }}{{e}^{-i\alpha }}\frac{{e}^{iN\alpha }}{{e}^{i\alpha }}\frac{1-{e}^{-i2N\alpha }}{1-{e}^{-i2\alpha }}\\ =\frac{{\epsilon }_{L}a}{R}\frac{{\mathrm{sin}}\beta }{\beta }{e}^{i\left(kR-\omega t\right)}\frac{{e}^{-iN\alpha }}{{e}^{-i\alpha }}\frac{{e}^{iN\alpha }-{e}^{-iN\alpha }}{{e}^{i\alpha }-{e}^{-i\alpha }}\\ =\frac{{\epsilon }_{L}a}{R}\frac{{\mathrm{sin}}\beta }{\beta }{e}^{i\left(kR-\left(N-1\right)\alpha -\omega t\right)}\frac{{\mathrm{sin}}N\alpha }{{\mathrm{sin}}\alpha }\end{array}$

Notice that this just ends up being multisource interference multiplied by single slit diffraction.

Squaring it we see that: $I\left(\theta \right)={I}_{0}\frac{{{\mathrm{sin}}}^{2}\beta }{{\beta }^{2}}\frac{{{\mathrm{sin}}}^{2}N\alpha }{{{\mathrm{sin}}}^{2}\alpha }$

Interference with diffractionfor 6 slits with $d=4a$

Interference with diffractionfor 6 slits with $d=4a$

Interference with diffractionfor10 slits with $d=4a$

Interference with diffractionfor10 slits with $d=4a$

Principal maxima occur when $\frac{{\mathrm{sin}}N\alpha }{{\mathrm{sin}}\alpha }=N$ or since $\alpha =kd\left({\mathrm{sin}}\theta \right)/2$ $kd{\mathrm{sin}}\theta =2n\pi \text{ }n=0,1,2,3$ or $\frac{2\pi }{\lambda }d{\mathrm{sin}}\theta =2n\pi$ or ${\mathrm{sin}}\theta =\frac{n\lambda }{d}$

and just like in multisource interference minima occur at ${\mathrm{sin}}\theta =\frac{n\lambda }{Nd}\text{ }n=1,2,3\dots \text{ }\frac{n}{N}\ne integer$ A diffraction grating is a repetitive array of diffracting elements such as slits or reflectors. Typically with N very large (100's). Notice how all butthe first maximum depend on $\lambda$ . So you can use a grating for spectroscopy.

#### Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Waves and optics. OpenStax CNX. Nov 17, 2005 Download for free at http://cnx.org/content/col10279/1.33
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Waves and optics' conversation and receive update notifications?

 By OpenStax By Rylee Minllic By Edgar Delgado By OpenStax By OpenStax By Brooke Delaney By OpenStax By OpenStax By Jill Zerressen By Rhodes