TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS
14. NEWTON'S RINGS
1. Objective
Determining radius of curvature of a long focal length plano-convex lens (large radius of
curvature).
2. Equipment needed
Measuring microscope, plano-convex long focal length lens, monochromatic light source.
3. Theory
Waves reflecting from two (or more) surfaces can interfere constructively and destructively –
energy of light is re-distributed: in some places of space energy increases (light is stronger), in
others energy decreases. This is called interference.
Energy change depends of phase difference of (light) waves interfering. Increase of energy is
maximal if phase difference is zero or integer times 2π . Minimal intensity is observed in
places where incoming waves have contrary (opposite) phases i.e. phase difference is odd
number times π . If phase difference of waves is constant over time in every point of our
experiment, we have coherent waves and image resulting from interference is stable.
Common light sources (incandescent light bulbs etc) do not emit coherent waves. Independent
atoms and molecules emit light intermittently. One act of emission during which one light
„packet“ is generated lastsr about 10 −9 − 10 −8 seconds. After some time another packet with
different angle of vector E and different phase is generated. Interference pattern formed due
to summing such light wave’s changes rapidly – every time one light „packet“ in given point
of space is replaced by another. Stable interference pattern can not form and both
experimenter's eye and electronic light intensity meters register uniform intensity in whole
observed experiment space.
A coherent light packet is generated by one atom in one act of emission. Light source
generates multiple of such rays. Stable interference pattern can be obtained by separating
waves, introducing an optical path difference and then summing them.
Separation can be achieved at least in two practical ways. In case of wave front distribution
method light from source is separated in two beams by letting it pass through closely located
small holes in a screen. Waves passing those holes are coherent. In case of amplitude
distribution method waves from source are distributed by means of partially reflecting and
partially transparent surfaces (mirrors).
Spatially separated beams must be guided to one spot where they interfere and form a pattern.
Phase difference of beams is constant in time if they originate from the same act of emission
of atom. To achieve this, optical path difference of beams must not be too big since then
summing waves can originate from different acts of emission.
Phase difference for light with different wavelengths is different for the same geometric path
difference. Waves of one color (wavelength) can sum constructively and in the same place
waves of another color may sum destructively (delete each other). Distribution of light
intensity in interference pattern is therefore different for different wavelengths. Using broad
spectra of light sharpness of interference pattern is low or pattern is not formed at all – surface
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TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS
is lit uniformly. That's why a monochromatic (one color/wavelength only) source has to be
used to achieve good interference pattern.
A classical example of interference pattern formed by coherent waves is the case of Newton's
rings. Interference pattern is formed in a system consisting of a long focal length planoconvex lens in contact with a plane glass disk (see figure 14.1).
When a long focal length lens is placed on glass (see figure 14.1), a very thin layer of air
resides between contact point of lens and glass. Depth of air layer is comparable to lights
wavelength.
Figure 14.1
The bigger focal length the wider is mentioned super slim air layer. If this system is lit with
coherent light, dark and bright rings can be observed in contact point of lens and glass. Those
circles are called Newton's rings. If white light is used, rings will be colored.
We shall refer to figure 14.1 for explanation. Please note that lens displayed on fig.14.1 has
short focal length and therefore is useless for our experiment. For explanations it will be good
enough.
A monochromatic beam is sent to the lens in the direction of its surface normal. For simplicity
only one ray falling to point B is shown. Part of light reflects back from point B, part of it
passes air gap between lens and glass surface and falls to point C. Here again part of the beam
is reflected back towards the lens and part is refracted to glass plate. If air gap between lens
and plate is short, waves reflected from points B and C are coherent and form an interference
pattern when summing. Knowing that light reflects from points B and C practically in the
same direction (in case of a long focal distance lens!), optical path difference ∆ can be
calculated as follows:
∆ = 2nBC +
λ0
2
,
where λ0 is wavelength of used light source in vacuum and n – refractive index of air.
λ
Constant 0 is added due to fact that reflecting from glass plate (optically denser matter than
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air) changes waves phase 180o which equals to change of path difference for ½ wavelength in
point C. Reflection in point B does not introduce phase change. Marking density of air gap
between points B and C as d and knowing refractive index of air n ≈ 1 , path difference can
be written:
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TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS
λ0
∆ = 2d +
.
2
Light will be deleted (dark zones) on upper surface of air gap in places where light waves
meet in opposite phases. This means that path difference of beams must be odd number of
times of half wavelengths:
∆ = 2d +
λ0
2
= (2k + 1)
λ0
2
,
(1)
where k = 0, 1, 2, ... .
Therefore height of air gap d for minimum light (dark) zones will be:
d =k
λ0
.
(2)
2
Maximums can be observed when beams are summed in the same phase. Path difference of
beams must be integer wavelengths:
∆ = 2d +
λ0
2
= k ′λ0 ,
(3)
where k ′ = 1, 2, 3, ...
Minimum and maximum conditions (1) and (3) show that summary intensity of reflected
beams in our experiment depends on thickness of air gap d between the lens and glass plate.
This means that in places with equal d , brightness is also equal – rings with uniform
brightness can be observed. Arrangements occurring due to interference in places with equal
thickness are called equal thickness ribbons (from geometrical point of view they do not have
to be ribbons). Corresponding interference effect is called equal thickness interference. In
case of a spherical lens equal thickness ribbons are concentric rings centre of which is located
in point of contact between lens and glass plate. Intensity of light in interference pattern varies
together with change of thickness of air gap (in direction of lenses radius). So we have a
pattern of bright and dark concentric rings – Newton's rings.
Let us find a correlation between lenses radius of curvature R and radius r of Newton’s
ring. We will use a triangle AOB (see figure 1). For arm AB = r of this orthogonal triangle
one can write:
r 2 = OB 2 − OA 2 .
Knowing that OB = R and OA = R − d , gives:
r 2 = R 2 − (R − d ) = 2 Rd − d 2 .
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Lens with big radius of curvature has d 2 << 2 Rd . Therefore:
r 2 ≈ 2 Rd .
(4)
A formula for finding radius rk of dark Newton's rings can be expressed by inserting value d
corresponding to minimum condition (2) into formula (4):
rk2 = Rkλ 0 ,
(5)
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TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS
where k = 0, 1, 2, ... .
Maximum condition (3) gives us analogous formula for bright rings:
1

rk2′ = R  k ′ − λ0 ,
2

(6)
where k ′ = 1, 2, 3, ... .
Inserting to formula (5) values k = 0 , meaning that d = 0 , results in r0 = 0 . This means that
in contact point of lens and glass plate we have a minimum (round dark dot). First dark ring is
formed on distance r1 = Rλ 0 , second at r2 = 2 Rλ 0 etc. For bright rings we use formula
(6), and see that first bright ring is formed at distance r1′ =
Rλ 0
2
(k ′ = 1) ,
second one at
3Rλ 0
etc. One can see that r1′ < r1 < r2′ < r2 ... , meaning that dark and bright rings are
2
really alternating.
r2′ =
Objective of the present lab is determining radius of curvature of a plano-convex lens by
measuring radiuses of Newton's rings forming in the system. Lens and glass plate (or a system
of them) is placed on measuring microscope's bed (see figure 14.2). Microscope must be
focused on air gap's upper surface (on spherical surface of lens). A parallel plane glass plate P
is placed between objective of microscope and lens of the experiment to guide light to the
system. Plate P has an angle 45o. Light beams from source will be reflected from plate P to
the lens. Part of beams reflected back from the lens will be headed through plate P to
microscope and we see enlarged image of Newton's rings. Measuring microscope (or its bed)
can be shifted in horizontal direction with corresponding knob giving the experimenter a
chance to measure radiuses of dark or bright rings with great accuracy.
Figure 14.2
According to formulas (5) and (6), lenses radius of curvature R can be calculated as follows:
rk2
R=
kλ0
(for dark ring)
(7)
or:
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TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS
R=
rk2′
.
1

 k ′ − λ 0
2

(for bright ring)
(8)
Given above formulas can be used for calculation of R in case there is a direct contact
between lens and glass plate. Unfortunately we do not know if there is contact or not. This
means that we do not know numeric values of k (or k ′ ) for each distinct ring. We do know
that value of k (or k ′ ) increases by a factor of 1 for each next ring. Let k start increasing
from some start value k1 to m . Increase can be calculated as m − k1 . According to formula
(5) we can write for dark ring with k1 and with order of interference m :
rk21 = k1 Rλ0 ,
rm2 = mRλ 0 .
Subtracting formulas gives:
rm2 − rk21 = (m − k1 )Rλ0 .
From which
rm2 = (m − k1 )Rλ0 + rk21 .
(9)
Let’s assume rings starting from the centre. Although we do not know the order of
interference, its change is equal to ring's number. Suppose we have a dark ring number j1
with order of interference k1 and ring number j with rank m . Obviously m − k1 = j − j1 . So
we can re-write formula (9) as follows:
r j2 = ( j − j1 ) Rλ0 + r j21 .
(10)
One can see that a square of dark ring's radius is a linear function of its number j: r j2 = f ( j ) ,
functions ascent is Rλ 0 . Lenses radius of curvature R can be easily found by determining
mentioned functions ascent graphically and knowing wavelength λ0 . Radius of curvature can
be found directly from formula (10) as well:
R=
r j2 − r j21
λ0 ( j − j1 )
.
(11)
j1 – number of the smallest measured ring, j – number of current ring.
Analogic formulas can be written for bright rings, function r j2 = f ( j ) has an ascent Rλ 0 (j –
number of ring) and formula (11) is true for them as well.
4. Experimental procedure
1. Get acquainted to the measuring microscope and its properties.
2. Switch on light source. Glass plate P (see figure 14.2) must have an angle about 45o to
light beam. If it is not so, adjust it. Criteria: beams reflected from plate should light
microscope's bed right under objective.
3. Focus the image of a cross by turning or pushing microscope's ocular.
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TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS
4. Put a piece of paper on microscopes bed and focus to it by turning objective or moving
microscope's tube. In about the same place you will later observe Newton's rings.
5. Remove paper and put a system of lens and glass plate given by instructor to microscope's
bed. Move the system and find a position with best image of Newton's rings.
6. Position measuring cross at centre of rings by turning shift knob. At the same time check
that crosses’ vertical line moves parallel to an imaginary line crossing rings' centre. Adjust
position of lens/plate if it is not so.
7. Now the experiment stand is adjusted. Ask instructor to check it. Also ask which rings
(dark or bright) should be used in the experiment. You should measure at least 6 rings!
During the whole experiment turn the shift knob in one direction only. This compensates
the mechanical errors in measuring system. For example bring the cross left to all
measured rings. Then, using shift knob, start moving back right. When you have reached
first ring to be measured, take the reading from scale. Continue to next ring (see figure
14.3).
Figure 14.3
8. Moving support only to left place cross to the right side of ring with smallest radius. Take
reading. In the same manner measure other rings taking readings when on ring's right side.
Experiment ends on the right side of the biggest ring.
9. Write results in the table 1 marking also which rings (dark or bright) you measured.
Calculate (from diameters measured) radiuses of Newton's rings and their squares.
(Measuring radiuses would be not accurate since central dot is big and finding exact
centre is complicated).
Table 1
Measuring Newton's rings
Reading on the scale
Order of
Right side
interference Left side
lp
lv
j
rj =
l p − lv
2
rj
2
1
...
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Light sources wavelength λ0 = ...
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TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS
10. Plot a graph of function r j = f ( j ) using your results y-axis: r j , x-axis: j . Add a
2
2
trendline (straight line) to your measurement points. If everything was correct, your points
should be located near the straight trendline. Find ascent Rλ 0 of the trendline using least
squares method with A-type expanded measurement uncertainty on level 95%. (Use a
computer program “Lineaarne regressioon” – linear regression for it. This program is
installed in lab's computers. You can find a short guide for the program from manual of
lab work nr. 6). Calculate lenses radius of curvature using ascent just found. Estimate its
expanded combined measurement uncertainty.
5. Questions and tasks
1. Define optical path difference and coherence of light.
2. What is interference, when can an interference pattern be observed?
3. When we can talk about equal thickness interference and when equal inclination
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
interference?
How are Newton's rings formed?
What depends Newton's rings radius on?
Why is light reflected from upper side of lens and lower side of glass plate not taken into
consideration?
Why should a lens with big radius of curvature be used?
What is interference image of the light which passes through the lens and its base plate?
Why there is a dark dot in the centre of rings? Does it appear always?
Where is density of interference rings greater – in the centre or on sides? Why?
What kind of interference pattern can be observed when white light source is used?
What kind of interference pattern can be observed when water is between the lens and
glass plate?
How does number of formed rings depend on lenses radius?
Name some applications of interference.
6. Literature
1. Halliday, D., Resnick, R., Walker, J. Fundamentals of Physics.–6th ed. New York, 2001,
John Wiley & Sons, Inc., §§ 36-1 – 36-7.
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