# 4.4 Diffraction gratings  (Page 2/4)

 Page 2 / 4 (a) This Australian opal and (b) butterfly wings have rows of reflectors that act like reflection gratings, reflecting different colors at different angles. (credit b: modification of work by “whologwhy”/Flickr)

## Applications of diffraction gratings

Where are diffraction gratings used in applications? Diffraction gratings are commonly used for spectroscopic dispersion and analysis of light. What makes them particularly useful is the fact that they form a sharper pattern than double slits do. That is, their bright fringes are narrower and brighter while their dark regions are darker. Diffraction gratings are key components of monochromators used, for example, in optical imaging of particular wavelengths from biological or medical samples. A diffraction grating can be chosen to specifically analyze a wavelength emitted by molecules in diseased cells in a biopsy sample or to help excite strategic molecules in the sample with a selected wavelength of light. Another vital use is in optical fiber technologies where fibers are designed to provide optimum performance at specific wavelengths. A range of diffraction gratings are available for selecting wavelengths for such use.

## Calculating typical diffraction grating effects

Diffraction gratings with 10,000 lines per centimeter are readily available. Suppose you have one, and you send a beam of white light through it to a screen 2.00 m away. (a) Find the angles for the first-order diffraction of the shortest and longest wavelengths of visible light (380 and 760 nm, respectively). (b) What is the distance between the ends of the rainbow of visible light produced on the screen for first-order interference? (See [link] .) (a) The diffraction grating considered in this example produces a rainbow of colors on a screen a distance x = 2.00 m from the grating. The distances along the screen are measured perpendicular to the x -direction. In other words, the rainbow pattern extends out of the page. (b) In a bird’s-eye view, the rainbow pattern can be seen on a table where the equipment is placed.

## Strategy

Once a value for the diffraction grating’s slit spacing d has been determined, the angles for the sharp lines can be found using the equation

$d\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta =m\lambda \phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}m=0,\phantom{\rule{0.2em}{0ex}}±1,\phantom{\rule{0.2em}{0ex}}±2,\phantom{\rule{0.2em}{0ex}}...\phantom{\rule{0.2em}{0ex}}.$

Since there are 10,000 lines per centimeter, each line is separated by 1/10,000 of a centimeter. Once we know the angles, we an find the distances along the screen by using simple trigonometry.

## Solution

1. The distance between slits is $d=\left(1\phantom{\rule{0.2em}{0ex}}\text{cm}\right)\text{/}10,000=1.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}\phantom{\rule{0.2em}{0ex}}\text{cm or}\phantom{\rule{0.2em}{0ex}}1.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\phantom{\rule{0.2em}{0ex}}\text{m}.$ Let us call the two angles ${\theta }_{\text{V}}$ for violet (380 nm) and ${\theta }_{\text{R}}$ for red (760 nm). Solving the equation $d\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}{\theta }_{\text{V}}=m\lambda \phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}{\theta }_{\text{V}},$
$\text{sin}\phantom{\rule{0.2em}{0ex}}{\theta }_{\text{V}}=\frac{m{\lambda }_{\text{V}}}{d},$

where $m=1$ for the first-order and ${\lambda }_{\text{V}}=380\phantom{\rule{0.2em}{0ex}}\text{nm}=3.80\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-7}\phantom{\rule{0.2em}{0ex}}\text{m}.$ Substituting these values gives
$\text{sin}\phantom{\rule{0.2em}{0ex}}{\theta }_{\text{V}}=\frac{3.80\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-7}\phantom{\rule{0.2em}{0ex}}\text{m}}{1.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\phantom{\rule{0.2em}{0ex}}\text{m}}=0.380.$

Thus the angle ${\theta }_{\text{V}}$ is
${\theta }_{\text{V}}={\text{sin}}^{-1}\phantom{\rule{0.2em}{0ex}}0.380=22.33\text{°}.$

Similarly,
$\text{sin}\phantom{\rule{0.2em}{0ex}}{\theta }_{\text{R}}=\frac{7.60\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-7}\phantom{\rule{0.2em}{0ex}}\text{m}}{1.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\phantom{\rule{0.2em}{0ex}}\text{m}}=0.760.$

Thus the angle ${\theta }_{\text{R}}$ is
${\theta }_{\text{R}}={\text{sin}}^{-1}\phantom{\rule{0.2em}{0ex}}0.760=49.46\text{°}.$

Notice that in both equations, we reported the results of these intermediate calculations to four significant figures to use with the calculation in part (b).
2. The distances on the secreen are labeled ${y}_{\text{V}}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{y}_{\text{R}}$ in [link] . Notice that $\text{tan}\phantom{\rule{0.2em}{0ex}}\theta =y\text{/}x.$ We can solve for ${y}_{\text{V}}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{y}_{\text{R}}.$ That is,
${y}_{\text{V}}=x\phantom{\rule{0.2em}{0ex}}\text{tan}\phantom{\rule{0.2em}{0ex}}{\theta }_{V}=\left(2.00\phantom{\rule{0.2em}{0ex}}\text{m}\right)\left(\text{tan}\phantom{\rule{0.2em}{0ex}}22.33\text{°}\right)=0.815\phantom{\rule{0.2em}{0ex}}\text{m}$

and
${y}_{\text{R}}=x\phantom{\rule{0.2em}{0ex}}\text{tan}\phantom{\rule{0.2em}{0ex}}{\theta }_{R}=\left(2.00\phantom{\rule{0.2em}{0ex}}\text{m}\right)\left(\text{tan}\phantom{\rule{0.2em}{0ex}}49.46\text{°}\right)=2.338\phantom{\rule{0.2em}{0ex}}\text{m}.$

The distance between them is therefore
${y}_{\text{R}}-{y}_{\text{V}}=1.523\phantom{\rule{0.2em}{0ex}}\text{m}.$

#### Questions & Answers

what is an atom
Aroyameh Reply
All matter is composed of two sets of three dimensions. The first set (1,2,3) decay with a positive charge. The second set (4,5,6) decay with a negative charge. As they decay, they create space (7 8,9) dimensions.
John
Two sets of (1,2,3,4,5,6) dimensions create a proton, a neutron, and an electron. This is the primordial atom.
John
A 10kg mass lift to a height of 24m and release. what is the total energy of the system
ADEPOJU Reply
mechanics is that branch of physical and mathatics that
ADEPOJU
E=Mgh=10*10*24=2400J
Adamu
what is the difference between a molecule and atom
Natanim Reply
Atoms are single neutral particles. Molecules are neutral particles made of two or more atoms bonded together.
Manfred
what I'd dynamic propulsion
Elias Reply
A body quadruples its momentum when its speed doubles.What was the initial speed in units of c.i.e..what was u/c ?
Lekshmi Reply
what is enthalpy?
prabir Reply
a thermodynamic quantity equivalent to the total heat content of a system
RAMLA
proparty of tharmo dainamic
bloch
What is the meaning of Nuclear Fission?
Benita Reply
what do you mean by dynamics single particles
Peacekamei Reply
عند قذف جسم إلى أعلى بسرعة إبتدائية فإنه سيصل إلى ارتفاع معين (أقصى ارتفاع) ثم يعود هابطاً نحو سطح الأرض .   إذا قُذِفَ جسم إلى أعلى ووجد أن سرعته 18 م / ث عندما قطع 1/4 المسافة التي تمثل أقصى ارتفاع سيصله فالمطلوب إيجاد السرعة التي قُذِف بها بالمتر / ث . إن هذه السرعة هي واحدة من الإجابات التالية
Aml Reply
what is light
Ayebanifesunday Reply
light is a kind of radiation That stimulates sight brightness a source of illumination.
kenneth
Electromagnet radiation creates space 7th, 8th, and 9th dimensions at the rate of c.
John
That is the reason that the speed of light is constant.
John
This creation of new space is "Dark Energy".
John
The first two sets of three dimensions, 1 through 6, are "Dark Matter".
John
As matter decays into luminous matter, a proton, a neutron, and an electron creat deuterium.
John
There are three sets of three protons, 9.
John
There are three sets of three neutrons, 9.
John
A free neutron decays into a proton, an electron, and a neutrino.
John
There are three sets of five neutrinoes, 15.
John
Neutrinoes are two dimensional.
John
A positron is composed of the first three dimensions.
John
An electron is composed of the second three dimensions.
John
What is photoelectric
Hsssan Reply
light energy (photons) through semiconduction of N-P junction into electrical via excitation of silicon purified and cristalized into wafers with partially contaminated silicon to allow this N-P function to operate.
Michael
i.e. Solar pannel.
Michael
Photoelectric emission is the emission of electrons on a metal surface due to incident rays reflected on it
Benita
If you lie on a beach looking at the water with your head tipped slightly sideways, your polarized sunglasses do not work very well.Why not?
Rakhi Reply
it has everything to do with the angle the UV sunlight strikes your sunglasses.
Jallal
this is known as optical physics. it describes how visible light, ultraviolet light and infrared light interact when they come into contact with physical matter. usually the photons or light upon interaction result in either reflection refraction diffraction or interference of the light.
Jallal
I hope I'm clear if I'm not please tell me to clarify further or rephrase
Jallal
what is bohrs model for hydrogen atom
Swagatika Reply
hi
Tr
Hello
Youte
Hi
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hi
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helo
Mcjoi
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Propessor Reply
1.79×10_¹⁹ km per hour
Swagatika
3×10^8
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