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By the end of this section, you will be able to:
  • Describe the locations and intensities of secondary maxima for multiple-slit interference

Analyzing the interference of light passing through two slits lays out the theoretical framework of interference and gives us a historical insight into Thomas Young’s experiments. However, much of the modern-day application of slit interference uses not just two slits but many, approaching infinity for practical purposes. The key optical element is called a diffraction grating, an important tool in optical analysis, which we discuss in detail in Diffraction . Here, we start the analysis of multiple-slit interference by taking the results from our analysis of the double slit ( N = 2 ) and extending it to configurations with three, four, and much larger numbers of slits.

[link] shows the simplest case of multiple-slit interference, with three slits, or N = 3 . The spacing between slits is d , and the path length difference between adjacent slits is d sin θ , same as the case for the double slit. What is new is that the path length difference for the first and the third slits is 2 d sin θ . The condition for constructive interference is the same as for the double slit, that is

d sin θ = m λ .

When this condition is met, 2 d sin θ is automatically a multiple of λ , so all three rays combine constructively, and the bright fringes that occur here are called principal maxima . But what happens when the path length difference between adjacent slits is only λ / 2 ? We can think of the first and second rays as interfering destructively, but the third ray remains unaltered. Instead of obtaining a dark fringe, or a minimum, as we did for the double slit, we see a secondary maximum    with intensity lower than the principal maxima.

Picture shows interference with three slits separated by distance d. Rays 1, 2, and 3 travel through the slits at the angles Theta.
Interference with three slits. Different pairs of emerging rays can combine constructively or destructively at the same time, leading to secondary maxima.

In general, for N slits, these secondary maxima occur whenever an unpaired ray is present that does not go away due to destructive interference. This occurs at ( N 2 ) evenly spaced positions between the principal maxima. The amplitude of the electromagnetic wave is correspondingly diminished to 1 / N of the wave at the principal maxima, and the light intensity, being proportional to the square of the wave amplitude, is diminished to 1 / N 2 of the intensity compared to the principal maxima. As [link] shows, a dark fringe is located between every maximum (principal or secondary). As N grows larger and the number of bright and dark fringes increase, the widths of the maxima become narrower due to the closely located neighboring dark fringes. Because the total amount of light energy remains unaltered, narrower maxima require that each maximum reaches a correspondingly higher intensity.

Picture A shows a graph for the interference fringe patterns for two, three and four slits. As the number of slits increases, more secondary maxima appear, but the principal maxima become narrower. Picture B shows photographs of fringe patterns for two, three and four slits. As the number of slits increases, more secondary maxima appear, but the principal maxima become brighter.
Interference fringe patterns for two, three and four slits. As the number of slits increases, more secondary maxima appear, but the principal maxima become brighter and narrower. (a) Graph and (b) photographs of fringe patterns.

Summary

  • Interference from multiple slits ( N > 2 ) produces principal as well as secondary maxima.
  • As the number of slits is increased, the intensity of the principal maxima increases and the width decreases.

Problems

Ten narrow slits are equally spaced 0.25 mm apart and illuminated with yellow light of wavelength 580 nm. (a) What are the angular positions of the third and fourth principal maxima? (b) What is the separation of these maxima on a screen 2.0 m from the slits?

a. 0.40 ° , 0.53 ° ; b. 4.6 × 10 −3 m

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The width of bright fringes can be calculated as the separation between the two adjacent dark fringes on either side. Find the angular widths of the third- and fourth-order bright fringes from the preceding problem.

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For a three-slit interference pattern, find the ratio of the peak intensities of a secondary maximum to a principal maximum.

1:9

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What is the angular width of the central fringe of the interference pattern of (a) 20 slits separated by d = 2.0 × 10 −3 mm ? (b) 50 slits with the same separation? Assume that λ = 600 nm .

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Questions & Answers

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
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
Nwangwu-ike
hi
Siddiquee
hi
Omar
helo
Mcjoi
what is the value of speed of light
Propessor Reply
1.79×10_¹⁹ km per hour
Swagatika
what r dwarf planet
Sivalakshmi Reply
what is energy
Isiguzo Reply
কাজের একক কী
Jasim
কাজের একক কী
Jasim
Energy is ability so capacity to do work.
kenneth
friction ka direction Kaise pata karte hai
Rahul Reply
friction is always in the opposite of the direction of moving object
Punia
A twin paradox in the special theory of relativity arises due to.....? a) asymmetric of time only b) symmetric of time only c) only time
Varia Reply
b) symmetric of time only
Swagatika
fundamental note of a vibrating string
fasoyin Reply
every matter made up of particles and particles are also subdivided which are themselves subdivided and so on ,and the basic and smallest smallest smallest division is energy which vibrates to become particles and thats why particles have wave nature
Alvin
what are matter waves? Give some examples
mallam Reply
according to de Broglie any matter particles by attaining the higher velocity as compared to light'ill show the wave nature and equation of wave will applicable on it but in practical life people see it is impossible however it is practicaly true and possible while looking at the earth matter at far
Manikant
a centeral part of theory of quantum mechanics example:just like a beam of light or a water wave
Swagatika
Mathematical expression of principle of relativity
Nasir Reply
given that the velocity v of wave depends on the tension f in the spring, it's length 'I' and it's mass 'm'. derive using dimension the equation of the wave
obia Reply
What is the importance of de-broglie's wavelength?
Mukulika Reply
he related wave to matter
Zahid
at subatomic level wave and matter are associated. this refering to mass energy equivalence
Zahid
it is key of quantum
Manikant
Practice Key Terms 2

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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