2.10. Interference by theDivision of Amplitude
Consider a thin transparent film with upper surface PQ and lower surface RS as
shown in Fig. 2.17. When a light ray AB is incident on the upper surface PQ of the film,
apart of it is reflected and a part is
transmitted. The reflected ray is BC C G
ar REFLECTED
and the transmitted ray is BD. The at?r PART
transmitted ray BD is again incident
on the lower surface RS of the film P B
and sufers partial reflection and at atr2 atr3
atr
partial transmission at this surface.
The reflected ray from the surface RS R
H
S
is DE and the transmitted ray is DF. at2 TRANSMITTED
at2
PART
Thus there is partial reflection and
partial transmission of light on each F
surface of the film, As a result, we get
the interfering light waves BC, EG, Fig. 2.17. Interference by division of amplitude
in the reflected part and the interfering light waves DE, HI, ...in the transmitted part.
Let the amplitude of the incident wave AB be a and for the surfaces PÌ and RSs of
the film, the reflection coefficient be r and the transmission coeficient be t. It is clear from
Fig. 2.17 that in the reflected part, the ratio of amplitudes of the interfering waves is
ar : at'r: ar3:.=1:2:P2: ...(2.26)
, 100 Unified Physics :Second Year (First Paper)
and in the transmitted part, the ratio of amplitudes of the interfering waves is
a: afy: afy:.. =1:2: surface, the . 2.21)
According to Fresnel, for normal incidence on a transparent
coefficient is cxpressed as reflection
For glass, l =1:5 0:5
15-1
-=02
Reflection coefficient for glass r = 1:5+1 2:5
Byconservation of energy,
Transmitted enerov
Incident energy = Reflected energy +
a= (ar) +(at or l=P+
2=1-2= | -(0-2) =0-96
.. Transmission coefficient for glass t = 0:98
Similarly for water (u =4/3), r= 1/7and t= 0-99.
amplitudes
It is thus clear from egns. (2.26) and (2.27) that in the reflected part, the
waves are
of the first two interfering waves are nearly equal and the amplitudes of other
negligible (since t 1), while in the transmitted part, the amplitude of interfering waves
decreases rapidly (since r << 1). Hence in thin films, a distinct interference pattern can be
seen in the reflected part (since the interfering waves are nearly of equal amplitudes), while
the interference pattern will not be distinct in the transmitted part (since the interfering
waves have much different amplitudes). This is why Newton's rings are observed in
reflected part.
It may be mentioned here that the coefficient of reflection r can be increased by
coating or silvering the surfaces ofthe film (i.e., the value of transmission coefficienttcan
be decreased). Then we can obtain the interfering waves of nearly equal amplitudes in the
transmited part and then a distinct interference pattern can be seen in the transmitted part
in place of the reflected part. For example, if on silvering the glass surface, its refiection
coefficientr becomes0-9, then the transmission coefficient - 1-=1-(0-9) -0436
(neglecting the absorption by the surface) and then the ratio of amplitudes of the two
interfering waves in the reflected part will be 1:0·19, while the ratio of amplitudes of the
two interfering waves in the transmitted part will be 1:0-81. This is done in
and Fabry-Perot's interferometer where fringes are Michelson's
observed in the transmitted part.
2.11. Interference in Thin Films
Interference in thin films occurs by the division of amplitude, We have read
when light falls on a thin film, we get the interfering waves in thal
the reflected part as well
as in the transmitted part due to the successive
partial reflection and partial transmission
from the upper and lower surfaces of the film. If the reflection
film is less (e.g., in case of pair of water film in coefficient of the surfaces 01
between the two glass plates), the distin
interference pattern is seen in the refilected part and not in the transmitted part. Butifthe
reflection coeficient of each surtace of film is increased by
interference pattern is seen in the transmitted part and not in thesilvering them, the distinu
reflected part.
(a) Phase difference due to reflection from a denser
mnedium (Stokes law)
According to the Stokes' law, when a light wave coming from ararer mediut
reflected from the surface of a denser medium, an additional phase n is introduced in it.
, Interference of Light 101
In Fig. 2. I8, PQ is a boundary surface separating the rarer and denser medium. Let
rav of light AB passes from the rarer medium to the denser mcdum and is incident on
the boundary surface at the point B. The ray of light suffers partial reflection and partial
transmission at the surface PQ. The reflected ray is BC and the transmitted ray is BD. If the
ampitude of the incident ray AB is a and for the upper surface, reflection coefficient is r.
transmission coefhcient is , then theamplitude of the reflected ray BC will be ar and the
amplitude of the transmitted ray BD will be at [Fig. 2.18(a).
But if these rays travel in opposite direction then in Fig. 2.18(b), for the incident ray
CB. the reflected ray is BA and the transmitted ray is BE. Similarly, in Fig. 2.18 (b), for
the incident ray DB, the reflected ray is BE and the transmitted ray is BA. Ifr' and / be the
A
att
ar. arr ar RARER
RARER
MEDIUM
MEDIUM
P B B/
art at DENSER
DENSER at MEDIUM
MEDIUM atr
D E D
(a) REFRACTION AND REFLECTION (b) PARTIAL REFRACTION AND
FROM THE RARER MEDIUM TO REFLECTION FROM DENSER
DENSER MEDIUM MEDIUM TO RARER MEDIUM
two media
Fig. 2.18. Partial reflection and refraction on the boundary of
the
coefficients of reflection and transmission respectively for the lower surface, then since
the upper
amplitude of the ray CB is ar, hence the amplitude of the ray BA reflected from
amplitude of
surface willbe arr and that of the transmitted ray BE willbe art. Similarly, the will be
from the lower surface
the ray DB is at, hence the amplitude of the ray BE reflected
amplitudes of the ray BA
atr and that of the transmitted ray BA willbe att. But the sum of surface
the upper
transmitted from the lower surface and that of the ray BA reflected from
must be equal to the amplitude of the incident ray AB, i.e.,
att'+ arr = a
But by conservation of energy,
.2.28)
Hence
from the lower surface and
Similarly, the sum of amplitudes of the ray BE reflected
zero, i.e.,
that of the ray BE transmitted from the upper surface must be
art + atr' = 0
..2.29)
or
medium sutters refiection from the
Thus when a ray of light incident from a rarer
occurs in it, but there is no
Surface of a denser medium, an additional phase change of n
phase change in the ray suffering the transmission.
interfering waves
(b)Path Difference between the two coherent
vary much,can be treated
Asmall part of a thinfilm of which the thickness does not RS.
parallel surfaces PO and
as a parallel film. Fig. 2.19 shows a film in between the two u.
The thickness of the film is tand the refractive index of medium is
Consider a thin transparent film with upper surface PQ and lower surface RS as
shown in Fig. 2.17. When a light ray AB is incident on the upper surface PQ of the film,
apart of it is reflected and a part is
transmitted. The reflected ray is BC C G
ar REFLECTED
and the transmitted ray is BD. The at?r PART
transmitted ray BD is again incident
on the lower surface RS of the film P B
and sufers partial reflection and at atr2 atr3
atr
partial transmission at this surface.
The reflected ray from the surface RS R
H
S
is DE and the transmitted ray is DF. at2 TRANSMITTED
at2
PART
Thus there is partial reflection and
partial transmission of light on each F
surface of the film, As a result, we get
the interfering light waves BC, EG, Fig. 2.17. Interference by division of amplitude
in the reflected part and the interfering light waves DE, HI, ...in the transmitted part.
Let the amplitude of the incident wave AB be a and for the surfaces PÌ and RSs of
the film, the reflection coefficient be r and the transmission coeficient be t. It is clear from
Fig. 2.17 that in the reflected part, the ratio of amplitudes of the interfering waves is
ar : at'r: ar3:.=1:2:P2: ...(2.26)
, 100 Unified Physics :Second Year (First Paper)
and in the transmitted part, the ratio of amplitudes of the interfering waves is
a: afy: afy:.. =1:2: surface, the . 2.21)
According to Fresnel, for normal incidence on a transparent
coefficient is cxpressed as reflection
For glass, l =1:5 0:5
15-1
-=02
Reflection coefficient for glass r = 1:5+1 2:5
Byconservation of energy,
Transmitted enerov
Incident energy = Reflected energy +
a= (ar) +(at or l=P+
2=1-2= | -(0-2) =0-96
.. Transmission coefficient for glass t = 0:98
Similarly for water (u =4/3), r= 1/7and t= 0-99.
amplitudes
It is thus clear from egns. (2.26) and (2.27) that in the reflected part, the
waves are
of the first two interfering waves are nearly equal and the amplitudes of other
negligible (since t 1), while in the transmitted part, the amplitude of interfering waves
decreases rapidly (since r << 1). Hence in thin films, a distinct interference pattern can be
seen in the reflected part (since the interfering waves are nearly of equal amplitudes), while
the interference pattern will not be distinct in the transmitted part (since the interfering
waves have much different amplitudes). This is why Newton's rings are observed in
reflected part.
It may be mentioned here that the coefficient of reflection r can be increased by
coating or silvering the surfaces ofthe film (i.e., the value of transmission coefficienttcan
be decreased). Then we can obtain the interfering waves of nearly equal amplitudes in the
transmited part and then a distinct interference pattern can be seen in the transmitted part
in place of the reflected part. For example, if on silvering the glass surface, its refiection
coefficientr becomes0-9, then the transmission coefficient - 1-=1-(0-9) -0436
(neglecting the absorption by the surface) and then the ratio of amplitudes of the two
interfering waves in the reflected part will be 1:0·19, while the ratio of amplitudes of the
two interfering waves in the transmitted part will be 1:0-81. This is done in
and Fabry-Perot's interferometer where fringes are Michelson's
observed in the transmitted part.
2.11. Interference in Thin Films
Interference in thin films occurs by the division of amplitude, We have read
when light falls on a thin film, we get the interfering waves in thal
the reflected part as well
as in the transmitted part due to the successive
partial reflection and partial transmission
from the upper and lower surfaces of the film. If the reflection
film is less (e.g., in case of pair of water film in coefficient of the surfaces 01
between the two glass plates), the distin
interference pattern is seen in the refilected part and not in the transmitted part. Butifthe
reflection coeficient of each surtace of film is increased by
interference pattern is seen in the transmitted part and not in thesilvering them, the distinu
reflected part.
(a) Phase difference due to reflection from a denser
mnedium (Stokes law)
According to the Stokes' law, when a light wave coming from ararer mediut
reflected from the surface of a denser medium, an additional phase n is introduced in it.
, Interference of Light 101
In Fig. 2. I8, PQ is a boundary surface separating the rarer and denser medium. Let
rav of light AB passes from the rarer medium to the denser mcdum and is incident on
the boundary surface at the point B. The ray of light suffers partial reflection and partial
transmission at the surface PQ. The reflected ray is BC and the transmitted ray is BD. If the
ampitude of the incident ray AB is a and for the upper surface, reflection coefficient is r.
transmission coefhcient is , then theamplitude of the reflected ray BC will be ar and the
amplitude of the transmitted ray BD will be at [Fig. 2.18(a).
But if these rays travel in opposite direction then in Fig. 2.18(b), for the incident ray
CB. the reflected ray is BA and the transmitted ray is BE. Similarly, in Fig. 2.18 (b), for
the incident ray DB, the reflected ray is BE and the transmitted ray is BA. Ifr' and / be the
A
att
ar. arr ar RARER
RARER
MEDIUM
MEDIUM
P B B/
art at DENSER
DENSER at MEDIUM
MEDIUM atr
D E D
(a) REFRACTION AND REFLECTION (b) PARTIAL REFRACTION AND
FROM THE RARER MEDIUM TO REFLECTION FROM DENSER
DENSER MEDIUM MEDIUM TO RARER MEDIUM
two media
Fig. 2.18. Partial reflection and refraction on the boundary of
the
coefficients of reflection and transmission respectively for the lower surface, then since
the upper
amplitude of the ray CB is ar, hence the amplitude of the ray BA reflected from
amplitude of
surface willbe arr and that of the transmitted ray BE willbe art. Similarly, the will be
from the lower surface
the ray DB is at, hence the amplitude of the ray BE reflected
amplitudes of the ray BA
atr and that of the transmitted ray BA willbe att. But the sum of surface
the upper
transmitted from the lower surface and that of the ray BA reflected from
must be equal to the amplitude of the incident ray AB, i.e.,
att'+ arr = a
But by conservation of energy,
.2.28)
Hence
from the lower surface and
Similarly, the sum of amplitudes of the ray BE reflected
zero, i.e.,
that of the ray BE transmitted from the upper surface must be
art + atr' = 0
..2.29)
or
medium sutters refiection from the
Thus when a ray of light incident from a rarer
occurs in it, but there is no
Surface of a denser medium, an additional phase change of n
phase change in the ray suffering the transmission.
interfering waves
(b)Path Difference between the two coherent
vary much,can be treated
Asmall part of a thinfilm of which the thickness does not RS.
parallel surfaces PO and
as a parallel film. Fig. 2.19 shows a film in between the two u.
The thickness of the film is tand the refractive index of medium is
