Prof. Vilas Shamrao Patil

1. Department of Physics
B.SC.II
Sem-IV.Paper-VIII
DSC-D2- Waves and Optics-II
Topic-Resolving Power of Optical Instruments

2. Resolving Power of Optical Instruments
Introduction
1.If we are observing two-point objects (or images) close to
each other and if they are seen as two distinct, clear point
objects, then they are said to be well resolved.
2.Now bring the point objects closer and closer to each other
until they are just seen as two clear, separate objects then
they are said to be just resolved.
3.If the distance between two objects is reduced still further,
then we will not be able to distinguish them as two
different objects and then they are said to be not
resolved.
4.The minimum distance between the two objects which
can be just resolved by an optical instrument (including
eye which is also an optical instrument) is called limit of
resolution.
5.Limit of resolution is also defined as the smallest angle
subtended by the two-point objects at the optical
instrument (eye, objective of telescope or microscope etc.)
when they are just resolved.
6.Smaller the limit of resolution higher is the resolving
power of the optical instrument.
7.Therefore, the resolving power of an optical instrument
is equal to the reciprocal of limit of resolution.
8. Resolving power of the instrument adds to finer details
of the object.
9.The objective of the optical instrument produces first
magnification of the object. At this stage if the resolution
is higher, then a magnified clear image will be seen.
10.Any further increase in magnification only will not
increase the resolving power.
11.So, by simply increasing the magnifying power beyond
certain limit is not going to produce clear image or
additional details of the object. So, resolving power
should be large to get good quality images.

3. Resolving Power of Optical Instruments
2.2 Resolution:
Resolution is the ability of an optical instrument to resolve
two point objects very close to each other i.e. two points as
separate structures rather than a single fuzzy dot. Resolution
is the level of details that can be seen using optical
instruments.
2.3 : Resolving Power of Optical Instruments
1. "Resolving power of an optical instrument is defined as
its ability to resolve the images of two nearby objects".
2.It is generally expressed in numerical measure. Magnifying
power of objective lens depends on its focal length. So by
proper choice of focal length, the size of image can be
increased.
3.But beyond a certain limit, we lose the clarity of image i.e.
the images (point images) are not resolved. This infact is due
to wave nature of light.
4.The light from a point source, passing through a slit
produces a diffraction pattern called Fraunhoffer diffraction
at single slit i.e. the image of a point is actually, a bright
circle surrounded by alternate dark and bright rings.
5. So, when the image of two points close to each other is
formed, we get two diffraction patterns overlapping on each
other. Thus, clarity of the images is lost, or the images are
not resolved

4. Resolving Power of Optical Instruments
(i) The geometrical resolution of optical instruments
(telescope, microscope) is expressed in terms of either
linear separation (∆x) or angular separation (∆𝜃) of
two nearby objects.
(ii) The spectral resolution (of prismatic or grating
spectroscopes) is expressed in terms of the difference
in wavelength (∆𝜆) of two neighboring spectral lines.
(iii)The resolving power of spectroscope is,
𝝀
∆𝝀
where 𝜆
is the wavelength region i.e. average wavelength of
two spectral lines.
(iv)In general, the smallest separation ( ∆x, ∆𝜃 or ∆𝜆)
between two neighboring objects (or images) or
spectral lines is known as the limit of resolution and
(v) Reciprocal of limit of resolution is defined
as the resolving power of the optical
instrument.
Resolving power =
1
∆x
or
1
∆𝜃
Spectral resolution =
𝜆
∆𝜆
in the wavelength
region 𝜆

5. Rayleigh's Criterion for the Limit of Resolution
1)We know that when two nearby point objects
(or images) are just resolved i.e. seen as two
distinct objects, for a certain minimum linear (or
angular) separation called limit of resolution.
2)To express the limit of resolution (and hence
resolving power (R.P.) of an optical instrument)
as a numerical value, Rayleigh suggested a
criterion familiarly known as Rayleigh's
criterion.
3) The image of every point object is a
Fraunhoffer diffraction pattern (Fig. 2.1), in
which there is a central bright disc called, Airy's
disc surrounded by dark and bright rings.
4) The first dark ring corresponds to first minima
in the intensity distribution.
5)The Rayleigh's criterion states that, "the two
neighbouring images are just resolved, when the
central maximum of one diffraction pattern falls
on the first minimum of the other diffraction
pattern".
6) According to Fraunhoffer diffraction, the
position of first minima on either side is given by
the condition, a sin α= 𝜆, where 𝜆 is the
wavelength of light used and a is the aperture
width of rectangular slit and α is the angular
separation of first minima from the central
maximum.

6. Rayleigh's Criterion for the Limit of Resolution
Fig. 2.2 Resultant intensity distributions due to
overlapping diffraction patterns due to two
neighboring points A and B.
7)To explain Rayleigh's criterion for limit of
resolution, let us consider the Fraunhoffer diffraction
patterns, due to two neighbouring point objects A and
B, which overlap on one another as shown in Fig. 2.2.
C.
Case (a) If A and B are well separated, then in the resultant
intensity distribution, we observe two distinct peaks i.e. two
clear images are formed and the points A and B are said to be
well resolved (as shown in Fig. 2.2(a)).
Case (b) : For a certain minimum separation between A and
B, there appears a dip in the resultant intensity distribution as
shown in Fig. 2.2(b),
so that we observe two distinct images, when they are said to
be just resolved. In this case the central maximum of one
diffraction pattern coincides with the first minimum of the
other diffraction pattern.
This is the condition for just resolution.
Case (c) : If A and B are very close to each other, so that the
resultant intensity distribution shows only one peak as shown
in Fig. 2.2(c), then images A and B of two points are not
resolved.

7. Rayleigh's Criterion for the Limit of Resolution
Fig. 2.2 Resultant intensity distributions due to
overlapping diffraction patterns due to two
neighboring points A and B.
7)To explain Rayleigh's criterion for limit of
resolution, let us consider the Fraunhoffer diffraction
patterns, due to two neighbouring point objects A and
B, which overlap on one another as shown in Fig. 2.2.
C.
Case (a) If A and B are well separated, then in the resultant
intensity distribution, we observe two distinct peaks i.e. two
clear images are formed and the points A and B are said to be
well resolved (as shown in Fig. 2.2(a)).
Case (b) : For a certain minimum separation between A and
B, there appears a dip in the resultant intensity distribution as
shown in Fig. 2.2(b),
so that we observe two distinct images, when they are said to
be just resolved. In this case the central maximum of one
diffraction pattern coincides with the first minimum of the
other diffraction pattern.
This is the condition for just resolution.
Case (c) : If A and B are very close to each other, so that the
resultant intensity distribution shows only one peak as shown
in Fig. 2.2(c), then images A and B of two points are not
resolved.

8. As examples, let us find the resolving powers of telescope and
microscope using Rayleigh's criterion for resolution.
1.Telescope is used to observe distant objects.
2. Let A and B the be two neighboring point objects which
subtend a small angle (dα) at objective lens of the telescope
with an aperture width of l.
3.The images A' and B' are formed in the focal plane of
objective, such that dα =
𝑟
𝑓
4.where, r is the radius of Airy disc i.e. the distance between
central maximum and the first minimum on either side.
5. Now applying Rayleigh's criterion for just resolution, we get,
D sin (dα) = 𝜆 ... for rectangular aperture.
6.But according to Airy, for a circular aperture,
D sin(dα) =1.22 𝜆
D sin(dα) =1.22 𝜆
sin(dα)≈ dα =
1.22 𝜆
𝐷
-------------from equ.2.1 and 2.2
We get
𝑟
𝑓
=
1.22 𝜆
𝐷
Limit of resolution, r =
1.22 𝜆𝑓
𝐷
-----------2.3
Resolving power of telescope
1
𝑟
=
𝐷
1.22 𝜆𝑓
...........2.4
7.Thus, resolving power of telescope is larger for larger aperture,
smaller focal length and smaller wavelength of light.

9. 1.Microscope is used to observe two neighboring
points (A and B) very close to each other.
2. Let angle subtended by the points A and B at the
objective of microscope be (dα) and the distance
between A and B when they are just resolved be (h).
3.The path difference between extreme rays starting
from A and B is obtained as,
4. ∆ = MA + AN (where BM and BN are normal
drawn on extreme rays AP and AQ)
∆= h sin i + h sin i = 2h sin i
( but sin i=
𝐴𝑁
𝐴𝐵
=
𝐴𝑀
𝐴𝐵
𝑎𝑛𝑑 𝐴𝐵 = ℎ ) -----2.5
5.According to Rayleigh's criterion, the images A' and
B' are just resolved, when the central maximum of one
coincides with the first minimum of the other.
6.Therefore, the path difference between central
maximum and first minimum in the diffraction pattern
is, ∆ =D sin(dα)= 𝜆,-----2.6
7.We get, 2h sin i = 𝜆 ...for rectangular aperture (from
2.5 and 2.6)
8.But for circular aperture of width D,
2h sin i =1.22 𝜆 .... (According to Airy)

10. Limit of resolution, h =
𝟏.𝟐𝟐 𝝀
𝟐𝒔𝒊𝒏 𝒊
=
𝟏.𝟐𝟐 𝝀𝟎
𝟐𝝁𝒔𝒊𝒏 𝒊
----
(𝝁 =
𝝀𝟎
𝝀
) or 𝝀 =
𝝀𝟎
𝝁
------ 2.7
Resolving power of microscope =
2𝜇 𝑠𝑖𝑛 𝑖
1.22 𝜆0
---
-------2.8
where 𝜇 sin i is called numerical aperture.
9.Thus, resolving power of microscope is
higher for larger numerical aperture i.e. for
larger value of refractive index of the medium.
10.So increase the resolving power of
immersed microscopes are used Secondly,
R.P. is higher for smaller wavelengths of light
(or radiation) used to illuminate the objects.
11.Therefore, ultraviolet microscopes have
higher resolved power as compared to optical
microscopes used in visible region.
12. Also Electron microscopes have still
higher resolving power and hence they can be
used to see more details of the object to be
observed.

11. 1.The rays from a point object passing through an aperture
of width (a), produce Fraunhoffer diffraction pattern (Fig.
2.1), in which there is a central maximum at α= 0, and
minima at α = n𝜋 where n = 1, 2, 3, ...
2.Therefore, position of first minimum on either side of
central maximum is given by,
a sinα = 𝜆, where 𝜆, is wavelength of light.
where 𝐼𝑀𝑎𝑥 is maximum intensity at α= 0 (central
maximum).
3.Now referring to Fig. 2.2(b), when two neighboring
images (or objects) are just resolved when the central
maximum of one pattern just falls at the first minimum of
other, according to Rayleigh's criterion for resolution.
4. Therefore, in the resultant intensity distribution, the
intensities at the two central maxima are Imax each, which
correspond to α = 0.
5.The first minimum occurs at α = 𝜋. Thus, two maxima are
at an angular separation of 𝜋. At the middle point of two
peaks i.e. at α =
𝜋
2
, the intensity in the diffraction pattern due
to each point source is given by,
𝐼𝑚𝑖𝑑 = 𝐼𝑚𝑎𝑥(
𝑠𝑖𝑛𝜋
2
𝜋
2
)2
=
4
𝜋2
× 𝐼𝑚𝑎𝑥
6.Due to two overlapping diffraction patterns the total
intensity at the middle of the two peaks where a dip is
observed is given by,

12. 𝐼𝑚𝑖𝑑 = Total
𝐼𝑚𝑖𝑑 = 2 ×
4
𝜋2 𝐼𝑚𝑎𝑥 =
8
𝜋2 𝐼𝑚𝑎𝑥 = 0.81𝐼𝑚𝑎𝑥
7.Thus. intensity at mid-point i.e. at dip is 81% of
intensity at either maxima.
8.This difference in intensity can be distinguished by
the eye.
9.However, at angular separations less than
𝜋
2
, the
two peaks can't be observed as two independent
peaks i.e. the images are not resolved (Fig. 2.2(c)).
10. Therefore, Rayleigh's modified criterion for just
resolution may be stated as,
“Two neighbouring images are just resolved, when the
intensity at the dip (Fdip) is 0.81 times the intensity at
either maxima (Imax.)".
11.This modified Rayleigh's criterion is used to find the
resolving powers of interferometers in which the
intensity at no point actually becomes zero.

13. 1.Resolution is a property by which two nearby objects are
seen as two separate objects.
2.For a given optical device the resolving power depends
on the diameter (D) of the objective lens.
3.With increase in D, the resolving power also increases
4. If the two neighboring objects are resolved, then two see
them as two distinct objects, further magnification is
required.
5.The minimum magnifying power below which the eye
can't resolve the objects which are already resolved by the
optical instrument is called necessary magnifying power.
7.However, excessive magnification is not going to resolve
the objects, if the optical instrument does not resolve the
objects.
8. On the contrary the images get diffused. So the excessive
magnification than the essential value is called empty
magnification.
1.Magnification alone does not gain clarity in the final
image seem So the objective of optical instrument,
should be able to resolve the objects, then alone the
magnification adds to clarity upto a certain limit.
2. Thereafter magnification is futile. So magnification is
meaningful after resolution.
3.Increase in the magnification alone can never resolve
the objects, if they are not resolved already.
4. So quality of optical instrument is decided on its
resolving power.
5.Both resolving power and magnifying power increase
with increase in the aperture width (D) of the
objective, as the light gathering capacity of the
objective also increases.

14. Two nearby spectral images corresponding to
wavelengths , 𝜆 and 𝜆 + d𝜆 are said to be just
resolved according to Rayleigh's criterion if the
central maximum of one diffraction pattern just falls
on the first minimum of the other in the Fraunhoffer
diffraction pattern.
The spectral resolving power of grating (or prism)
spectrograph is given by
R.P. =
𝜆
d𝜆
where d𝜆 is difference in wavelength of two spectral
lines just resolved in wavelength region 𝜆.
Consider a collimated beam of light containing
wavelengths 𝜆, and 𝜆 + d 𝜆, incident on plane
diffraction grating GG',
having total number of lines drawn (slits) on the surface and
grating element e = a + b.
After diffraction, the rays are passed through telescope
objective to form spectral images P1 and P2 in nth order at
angle of diffraction 𝜃n and 𝜃n +d𝜃 corresponding to
wavelengths 𝜆, and 𝜆 +d 𝜆 respectively.
The diffraction patterns due to 𝜆 and 𝜆 +d 𝜆 overlap on each
other as shown in Fig. 2.5. The positions of central maxima
(P1 and P2) are given by,

15. (𝑎 + 𝑏)𝑠𝑖𝑛𝜃𝑛 = 𝑛𝜆 ---------------------2.13 And
(𝑎 + 𝑏)𝑠𝑖𝑛(𝜃𝑛 + 𝑑𝜃) = 𝑛(𝜆 + d 𝜆) ------------------2.14
According to Rayleigh's criterion for resolution, the central
maximum of (𝜆 +d 𝜆) should just fall at first minimum of 𝜆 ,
which is given by the condition,
(𝑎 + 𝑏)𝑠𝑖𝑛(𝜃𝑛 + 𝑑𝜃) = 𝑛𝜆 +
𝜆
𝑁
------------2.15
where
𝜆
𝑁
is the additional path difference. From equ. (2.14)
and (2.15), we get, 𝑛𝜆 +
𝜆
𝑁
= 𝑛(𝜆 + d 𝜆) = 𝑛𝜆 + 𝑛d 𝜆
Resolving power =
𝜆
d𝜆
= N.n------------2.16
Thus, resolving power of a grating is proportional to (i) the
order number (n) and (ii) total number of lines (N) on the
grating surface (i.e. the width of the grating surface).
Now differentiating eqn. (2.13) w.r.t. 𝜆 we get,
(a + b) cos𝜃𝑛.
𝑑𝜃𝑛
𝑑𝜆
=n
Dispersive power =
𝑑𝜃𝑛
𝑑𝜆
=
𝑛
(𝑎+𝑏)cos𝜃𝑛
=
𝑛.𝑁′
cos𝜃𝑛
-----------
2.17
where N' =
1
a+b
is number of lines (or slits) per unit
distance. Dispersive power in similar to magnification
which depends on N', but resolving power is ability to
produce nearby spectral images as separate which
depends on (N).

16. 1. Spectral resolving power of a prism is given by,
𝜆
d𝜆
where (𝜆 )
and (𝜆 + d 𝜆)are two nearby wavelength which are just resolved
according to Rayleigh's criterion
2.A collimated beam of light having wavelengths 𝜆 and 𝜆 +
𝑑𝜆, are incident on the prism.
3. The refracted beam gets dispersed and after passing
through telescope lens (L2), forms the spectral images P1, P2
in the focal plane of L2.
4.The refracting face of the prism limits the beam to a
width (a) of rectangular section.
5.Fraunhoffer diffraction patterns are formed due to
rectangular aperture.
6. The spectral images at P1 and P2 are just resolved when
P2 is at the first minimum of P1 corresponding to
wavelength (𝜆), which undergoes a deviation (𝛿) and
wavelength (𝜆 + 𝑑𝜆) deviates through (𝛿 + 𝑑𝛿) when the
prism
(is adjusted for minimum deviation position for both 𝜆 and
𝜆 + d 𝜆 , as d 𝜆 is very small. Thus when Rayleigh's condition
for resolution is satisfied, we have,
d𝛿 = 𝜆
From the Fig. 2.6,
𝛼 + 𝐴 + 𝛼 + 𝛿 = 𝜋
𝛼 =
𝜋
2
− (
𝐴+𝛿
2
)

17. ∴ 𝑠𝑖𝑛 𝛼 = 𝑠𝑖𝑛
𝜋
2
−
𝐴+𝛿
2
= cos
𝐴+𝛿
2
=
𝑎
𝑙
Also, sin =
𝐴
2
=
𝑡
2
.
1
𝑙
For minimum deviation
position, the prism formula is,
𝜇. 𝑠𝑖𝑛
𝐴
2
= 𝑠𝑖𝑛(
𝐴 + 𝛿
2
)
where 𝜇 and 𝛿 are functions of 𝜆.
Differentiating above equation w.r.t. 𝜆, we
get,
𝑑𝜇
𝑑𝜆
. 𝑠𝑖𝑛
𝐴
2
= 𝑐𝑜𝑠 (
𝐴+𝛿
2
).
𝑑𝛿
𝑑𝜆
.
1
2
𝑑𝜇
𝑑𝜆
.
𝑡
2𝑙
=
𝑎
2𝑙
.
𝑑𝛿
𝑑𝜆
=
𝑎
2𝑙
.
𝜆
𝑎
.
1
𝑑𝜆
Spectral resolving power of prism =
𝜆
d𝜆
= 𝑡.
𝑑𝜇
𝑑𝜆
Thus, the resolving power of prism is proportional to
(1) base length (t) of the prism and (ii) change of
refractive index of the material of the d p prism with
wavelength i.e.
𝑑𝜇
𝑑𝜆