The document discusses the concept of interference in waves, particularly light, detailing definitions, conditions, and types of interference, including constructive and destructive interference. It covers significant experiments such as Young's double slit experiment, thin film interference, Newton's rings, Fresnel's biprism, and Lloyd's single mirror, highlighting the principles and applications of each. The document serves as an educational resource on optical interference phenomena and their experimental verification.

1. OPTOMETRY – Part III
INTERFERENCE
ER. FARUK BIN POYEN
DEPT. OF AEIE, UIT, BU, BURDWAN, WB, INDIA
FARUK.POYEN@GMAIL.COM

2. Contents:
 Definition
 Conditions for Interference
 Classes of Interference
 Types of Interference
 Young’s Double Slit Experiment
 Thin Film Interference
 Newton’s Ring
 Fresnel’s Bi prism
 Lloyd’s Single Mirror
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3. Definition:
 The process in which two or more light, sound or electromagnetic waves of the same
frequency (coherent) combine to reinforce or cancel each other and the amplitude of the
resulting wave being equal to the sum of the amplitudes of the combining waves is
termed as Interference.
 Coherent Sources: Waves (EM and others) in which the phases of all waves at each
point on a line normal to the direction of the source are identical are called coherent
waves and the sources that generate them are termed as coherent sources.
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4. Conditions for Interference :
 Coherent sources of light.
 Amplitudes and intensities must be nearly equal to produce contrast between maxima
and minima.
 The source must be small enough to be considered a point source of light.
 The interfering sources must be near enough to produce wide fringes.
 The source and the screen must be far enough to produce wave fringes.
 The sources must emit light in the same state of polarization.
 The sources must be monochromatic.
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5. Classes of Interference:
 The are two classes of interferences:
1. Division of Wavefront: Coherent sources are obtained by diving the wavefront
originating from a common source, employing mirrors, biprisms or lenses. Requires
essentially a point source or narrow slit source. Instruments used to obtain this are
Fresnel Mirror, Lloyd’ Mirror, Lasers et cetera.
2. Division of Amplitude: Amplitude of incident beam is divide into two or more parts
by partial reflection or refraction. Beams travel different paths and finally brought
together. Interference can be from two or multiple beams. Examples of two beam
interference are thin film interference, Newton’s rings, Michelson Interferometer.
Example of multi-beam interference is Fabry-Perot Interferometer.
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6. Classes of Interference
Division of Wavefront & Division of Amplitude:
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7. Types of Interference:
 There are two types of Interference:
1. Constructive Interference: It occurs when two crests overlap each other and as the
result the amplitude of the resultant wave is increased.
2. Destructive Interference: It occurs when the crest of one overlaps the trough of another
resulting in the drop of the resultant amplitude.
If 𝐴1 and 𝐴2 are the amplitudes of two interfering waves, λ is the wavelength, δ is the path
difference and 𝐴 is the resulting amplitude after interference.
Then for Constructive Interference 𝐴 = 𝐴1 + 𝐴2 and 𝛿 = 𝑚λ, 𝑚 = 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
for Destructive Interference 𝐴 = 𝐴1 − 𝐴2 and 𝛿 = (𝑚 +1/2)λ, 𝑚 = 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
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8. Types of Interference:
 Constructive interference (a): In areas where the path length difference between the
two rays is equal to an odd multiple of half a wavelength (λ/2) of the light waves, the
reflected waves will be in phase, so the "troughs" and "peaks" of the waves coincide.
Therefore, the waves will reinforce (add) and the resulting reflected light intensity will
be greater. As a result, a bright area will be observed there.
 Destructive interference (b): At other locations, where the path length difference is
equal to an even multiple of a half-wavelength, the reflected waves will be 180° out of
phase, so a "trough" of one wave coincides with a "peak" of the other wave. Therefore,
the waves will cancel (subtract) and the resulting light intensity will be weaker or zero.
As a result, a dark area will be observed there. Because of the 180° phase reversal due
to reflection of the bottom ray, the center where the two pieces touch is dark.
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9. Young’s Double Slit Interference Experiment:
 This experiment can demonstrate that light displays characteristics of both waves and
particle.
 It uses two coherent sources of light placed at a small distance apart, usually few orders
of magnitude greater than the wavelength of light used. A screen or photo detector is
placed at a large distance away from the slits.
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10. Young’s Double Slit Interference Experiment:
 The original Young’s experiment used diffracted light from a single source passed into
two more slits to be used as coherent sources.
 Lasers are commonly used as coherent sources in modern day experiments.
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11. Young’s Double Slit Interference Experiment:
 Each source is acting as a coherent source of light wave.
 The waves travel l1 and l2 distances to create a path difference of Δl at any point on the
screen at a distance of “y” from the centre.
 An angle of θ is subtended at the sources with D (>>d) being the distance between the
source and screen.
∆𝑙 = 𝑑𝑠𝑖𝑛𝜃 ≅ 𝑑𝜃
𝑡𝑎𝑛𝜃 =
𝑦
𝐷
≅ 𝜃
∆𝑙 ≈ 𝑑𝜃 ≈ 𝑑
𝑦
𝐷
 The expression is valid if θ is very small implying that ‘y’ is small and ‘D’ is very large.
 The formula works best for fringes close to the central maxima.
 This experiment is monumental is establishing that light behaves as waves.
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12. Young’s Double Slit Interference Experiment:
 For Constructive Interference:
For a bright fringe to be at ‘y’
𝑛λ = ∆𝑙 =
𝑦𝑑
𝐷
𝑜𝑟 𝑦 𝑛𝑡ℎ = 𝑛
λ𝐷
𝑑
𝑤ℎ𝑒𝑟𝑒 𝑛 = ±0,1,2,3 … …
The 0th fringe represents the central bridge fringe.
 For Destructive Interference:
For a dark fringe
(𝑛 +
1
2
)λ = ∆𝑙 =
𝑦𝑑
𝐷
𝑜𝑟 𝑦 𝑛𝑡ℎ = (𝑛 +
1
2
)
λ𝐷
𝑑
𝑤ℎ𝑒𝑟𝑒 𝑛 = ±0,1,2,3 … …
 There are alternate light and dark bands running parallel to the slits.
 The bands are equally spaced.
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13. Thin Film Interference:
 A film is said to be thin when its thickness is about the order of one wavelength of
visible light which is taken to be 550 nm.
 When light is incident on such a film, a small portion gets reflected from the upper
surface and a major portion is transmitted into the film.
 Again a small part of the transmitted component is reflected back into the film by the
lower surface and the rest of it emerges out of the film.
 These reflected beams reunite to produce interference.
 Also the transmitted beams too interfere.
 This type of interference that takes place in thin films is called interference by division
of amplitude.
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14. Thin Film Interference:
 In the figure the rays r12 and t21 interfere and results in a constructive or destructive
interference depending on their path differences, given as,
2𝜇2 𝑑 cos 𝑟12 = 2𝑚 + 1
λ
2
𝑐𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑟𝑓𝑒𝑟𝑒𝑛𝑐𝑒
2𝜇2 𝑑 cos 𝑟12 = 𝑚λ 𝑑𝑒𝑠𝑡𝑟𝑢𝑐𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑟𝑓𝑒𝑟𝑒𝑛𝑐𝑒
 𝜇2 is the refractive index of medium 2.
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15. Newton’s Ring:
 Newton's rings is a phenomenon in which an interference pattern is created by the
reflection of light between two surfaces—a spherical surface and an adjacent touching
flat surface.
 This simple demonstration shows how the interference of light can be used to determine
the thickness of a thin film.
 Apparatus and material required to conduct this experiment are
 Travelling microscope
1. Plano Convex lens
2. Plane glass plate (optically flat)
3. Reflector
4. Sodium light source
5. A thin film whose thickness is to be measured (may be a strip of paper)
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16. Newton’s Ring:
 Rings are fringes of equal thickness.
 They are observed when light is reflected from a plano-convex lens of a long focal
length placed in contact with a plane glass plate.
 A thin air film is formed between the plate and the lens.
 The thickness of the air film varies from zero at the point of contact to some value t.
 If the lens plate system is illuminated with monochromatic light falling on it normally,
concentric bright and dark interference rings are observed in reflected light.
 These circular fringes were discovered by Newton and are called Newton’s rings.
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17. Newton’s Ring:
 A ray AB incident normally on the system gets partially reflected at the bottom curved
surface of the lens (Ray 1) and part of the transmitted ray is partially reflected (Ray 2)
from the top surface of the plane glass plate.
 The rays 1 and 2 are derived from the same incident ray by division of amplitude and
therefore are coherent.
 Ray 2 undergoes a phase change of p upon reflection since it is reflected from air-to-
glass boundary.
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18. Newton’s Ring:
 The condition for constructive and destructive interferences are given as;
for normal incidence cos 𝑟 = 1 and for air film μ = 1.
2𝑡 = 2𝑚 + 1
λ
2
𝑐𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑟𝑓𝑒𝑟𝑒𝑛𝑐𝑒
2𝑡 = 𝑚λ 𝑑𝑒𝑠𝑡𝑟𝑢𝑐𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑟𝑓𝑒𝑟𝑒𝑛𝑐𝑒
 Central Dark Spot: At the point of contact of the lens with the glass plate the thickness
of the air film is very small compared to the wavelength of light therefore the path
difference introduced between the interfering waves is zero. Consequently, the
interfering waves at the centre are opposite in phase and interfere destructively. Thus a
dark spot is produced.
 Circular Fringes with equal thickness: Each maximum or minimum is a locus of
constant film thickness. Since the locii of points having the same thickness fall on a
circle having its centre at the point of contact, the fringes are circular.
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19. Newton’s Ring:
 Fringes are localized: Though the system is illuminated with a parallel beam of light,
the reflected rays are not parallel. They interfere nearer to the top surface of the air film
and appear to diverge from there when viewed from the top. The fringes are seen near
the upper surface of the film and hence are said to be localized in the film.
 Radii of the 𝑚 𝑡ℎ dark rings
𝑟 𝑚 = 𝑚λ𝑅
 Radii of the 𝑚 𝑡ℎ bright ring
𝑟 𝑚 = (2𝑚 + 1)
λ
2
𝑅
 The radius of the dark ring is proportional to the radius of curvature of the lens
𝑟 𝑚 ∝ 𝑅
 Rings get closer as the order (m) increases since the diameter does not increase in the
same proportion.
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20. Newton’s Ring:
 In transmitted light the ring system is exactly complementary to the reflected ring
system so that the centre spot is bright.
 Under white light coloured fringes are obtained.
 The wavelength of monochromatic light can be determined as,
λ =
𝐷 𝑚+𝑝
2 − 𝐷 𝑚
2
4𝑝𝑅
Where, 𝐷 𝑚+𝑝 is the diameter of the (𝑚 + 𝑝) 𝑡ℎ dark ring
And 𝐷 𝑚 is the diameter of the (𝑚) 𝑡ℎ dark ring.
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21. Fresnel’s Biprism:
 Fresnel’s Biprism along with Fresnel’s Double mirror experiment were hugely
important to establish the wave nature of light.
 A Fresnel Biprism is a thin double prism placed base to base and have very small
refracting angle ( 0.5°). This is equivalent to a single prism with one of its angle nearly
179° and other two of 0.5° each.
 Here a thin biprism is used to derive two coherent sources (capable of interfering with
each other) from a single monochromatic source of light.
 The surface of the Fresnel biprism struck by the light emitted from a light aperture
encompass an angle of almost 180°.
 If a diverging beam of light strikes the edge of the biprism, two diverging coherent light
beams are created which appear to emerge from two virtual slits and interfere on the far
side of the biprism.
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22. Fresnel’s Biprism: Procedure
 The interference is observed by the division of wave front.
 Monochromatic light through a narrow slit S falls on biprism , which divides it into two
components.
 One of these component is refracted from upper portion of biprism and appears to come
from S1 where the other one refracted through lower portion and appears to come from
S2.
 Thus S1 and S2 act as two virtual coherent sources formed from the original source.
 Light waves arising from S1and S2 interfere in the shaded region and interference
fringes are formed which can be observed on the screen .
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23. Fresnel’s Biprism: Procedure
 The fringes are much brighter than those produced by Young's slits, because of the very
much greater amount of light that can pass through the prism compared with that
passing through the double slit arrangement.
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Experimental Set-Up

24. Fresnel’s Biprism:
 Fringe Width is given as
𝛽 =
𝐷λ
𝑑
 Determination of Wavelength of light:
λ =
𝑑𝛽
𝐷
 Determination of ‘d’: 𝑑 = 𝑑1 𝑑2
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where d1 and d2 are the distance
between S1 and S2 for two
position of lens.

25. Fresnel’s Biprism:
 Determination of thickness of a thin film:
 If ‘S' is the shift in position of white fringe and μ be the refractive index of mica sheet, β
is the fringe width and Δx is the displacement of the fringe, ‘t' is the thickness of mica
sheet, then
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26. Lloyd’s Single Mirror:
 This is another method for finding the wavelength of light by the division of wavefront.
 Another arrangement for producing an interference pattern with a single light source.
 Light from a slit 𝑆0 falls on a silvered surface at a very small grazing angle of incidence
as shown in the diagram below.
 A virtual image of 𝑆0 is formed at S1.
 Interference occurs between the direct beam from 𝑆0 to the observer (O) and the
reflected beam
 The zeroth fringe will be black because of the phase change due to reflection at the
surface.
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27. Lloyd’s Single Mirror:
 This arrangement can be thought of as a double-slit source with the
 distance between points 𝑆0 and S1 comparable to length d (distance between 𝑆0 and S1).
 An interference pattern is formed.
 The positions of the dark and bright fringes are reversed relative to the pattern of two
real sources.
 This is because there is a 180° phase change produced by the reflection.
 Lloyd mirror has same fringe pattern as Young's double slits experiment (YDSE) but
just reversed.
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28. References:
1. https://byjus.com/physics/double-slit-experiment/
2. http://farside.ph.utexas.edu/teaching/302l/lectures/node151.html
3. https://www.linsgroup.com/Physics/Newton_Ring.htm
4. http://vlab.amrita.edu/?sub=1&brch=189&sim=335&cnt=1
5. https://sites.google.com/site/puenggphysics/home/Unit-II/fresnel-s-biprism
6. https://www.phywe.com/en/fresnel-s-biprism-experiment.html
7. http://www.schoolphysics.co.uk/age16-
19/Wave%20properties/Interference/text/Fresnel_biprism/index.html
8. http://www.schoolphysics.co.uk/age16-
19/Wave%20properties/Interference/text/Lloyds_mirror/index.html
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