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M.P. Vaughan
PY3101 Optics
Optical instruments
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• Lenses
• Apertures
• The prism
• The human eye
• The magnifying glass
• The refracting telescope
• Chromatic aberration
• The achromatic doublet
• The reflecting telescope
• Coma
Learning objectives
2
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Applications using lenses and mirrors
Photography Telescopes Microscopes
Ophthalmetry
Largely applied using geometrical optics
Lenses: a summary
3
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Convex lens
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Convex lens – transverse magnification
,f
xM i−= ,
ox
fM −= .2
ioxxf =
4
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Concave lens
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Concave lens – transverse magnification
,ox
fM −=,
f
xM i−=
as we had for the convex lens.
.2
ioxxf =
5
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Other types of lenses
plano-convex plano-convex doublet
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Optical power
.1
f=P
The optical power P of a lens is a measure of the degree to which it converges or diverges light.
P is defined as the reciprocal of the focal length.
The SI unit of P is called the dioptre (m-1).
6
Apertures
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Apertures
Aperture StopImage Plane
Field Stop
The larger the aperture the more light collected.
DD.Flux 22 Dr ∝∝
7
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Camera Lenses
The larger the focal length, the larger the image size.
f
f
2Area Image f∝
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Flux density and f number
An exposure time will require a certain amount of optical energy.
Time ExposureDensityFlux Exposure ×=
( )2f/#Time Exposure ∝
f number
2
2
DensityFlux f
D∝
f
D=Aperture Relative
D
f=/#f
8
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Examples (Canon)
IXUS 70
“Maximum” f/#
f/2.8“Maximum” f/#
f/2.8 – f/4.9
EF-S 17-55mm
Prisms
9
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Prisms
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Prism v diffraction grating
Note that, for a prism, blue
light sees the greatest deviation...
... in contrast to a diffraction grating.
10
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2121 )90()90(180 ititA θθθθ +=−−−−=
Prisms – angular deviation
βα +=D
1iθ
1tθ 2iθ2tθ
D
A
n
190 tθ−α β
D is the angular deviation
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• From the angle of the minimum
deviation of a ray light, we can find the
refractive index of the prism
Prism – angular deviation
11
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Prisms – angular deviation
( ) ( ))sin(sinsinsin 1
1
2
1
2 tit Ann θθθ −== −−
AD tiitti −+=−+−= 212211 )()( θθθθθθ
1iθ
1tθ 2iθ2tθ
D
A
n
190 tθ−α β
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Prisms – angular deviation
( )11
1
2 sincoscossinsin ttt AnAn θθθ −= −
( )11
21 sincossin1sinsin tt AnAn θθ −−= −
( )11
221 sincossinsinsin ii AnA θθ −−= −
AD ti −+= 21 θθ
( )11
221
1 sincossinsinsin iii AnAA θθθ −−+−= −
12
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Prism – angular deviation
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The minimum deviation
Also:
Start with:
AD ti −+= 21 θθ
(A is a constant). Minimum where:
01
=id
dD
θ101
1
2
1
2 −=→=+→i
t
i
t
d
d
d
d
θθ
θθ
1102
1
2
1
2
−=→+==i
t
i
t
i d
d
d
d
d
dA
θθ
θθ
θ
21 itA θθ +=
13
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The minimum deviation
111111 coscossinsin ttiiti dndn θθθθθθ =→=
222222 coscossinsin iittit dndn θθθθθθ =→=
22
11
22
11
cos
cos
cos
cos
ii
tt
tt
ii
dn
dn
d
d
θθθθ
θθθθ
=
12
1 −=i
t
d
d
θθ
11
2 −=i
t
d
d
θθ
Use and
2
1
2
1
cos
cos
cos
cos
i
t
t
i
n
n
θθ
θθ
=2
2
1
2
2
2
1
2
sin1
sin1
sin1
sin1
i
t
t
i
n
n
θ
θ
θ
θ
−
−=
−
−→
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The minimum deviation
( )( )22
2
2
1
2
2
2
1
2
sin
sin
sin1
sin1
i
t
t
i
nn
nn
θ
θ
θ
θ
−
−=
−
−→
2
2
1
22
2
2
1
2
sin
sin
sin1
sin1
t
i
t
i
n
n
θθ
θθ
−−
=−−
→
With n > 1, this can only be true if:
12 ti θθ =So
21 ti θθ =
14
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The minimum deviation
,212 ntiA θθθ =+=
.22221 AAD anati −=−=−+= θθθθθ
21 ti θθ = and .12 ti θθ =
Using n for in the prism and a for in the air
Now
So
2
An =θ .
2
ADa
+=θand
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The minimum deviation
+
==→=
2sin
2sin
sin
sinsinsin
A
DA
nnn
ana θ
θθθ
Hence, from the minimum deviation D,
you can calculate n
15
The human eye
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Alhazen
Alhazen
(c. 965 - c. 1040)
Kitab al-Manazir (Book of Optics) (1011 – 1021)
Gave early detailed description of the human eye.
16
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The anatomy of the eye
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• Cornea• Transparent part of front of the eye. Also refracts
light and accounts for about 2/3 of the optical
power of the eye. Typically the optical power of the
cornea is about 43 dioptres.
• Iris• Variable aperture controlling amount of light
entering the eye
• Sclera• The white part of the eye providing a protective
covering
The anatomy of the eye
17
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• Retina• The photosensitive region of the eye onto which
the image is projected
• Choroid• vascular layer of the eye, containing connective
tissue, lying between the retina and the sclera.
• Fovea• fovea centralis – pit near back of eye responsible
for sharp, central vision
The anatomy of the eye
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The lens
The curvature of the lens may be changed by muscle contractions in the eye.
18
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Near and far points
• The near point is the closest distance for
which the lens can focus light on the retina
• Typically at age 10, this is about 18 cm
• It increases with age, ~ 25 cm for an adult
• The far point of the eye represents the
largest distance for which the lens of the
relaxed eye can focus light on the retina
• Normal vision has a far point of infinity
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Farsightedness – hyperopia
Distant objects may be focussed but not nearby objects.
19
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Correcting farsightedness
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Nearsightedness – myopia
• axial myopia
• Lens too far from retina
• refractive myopia
• Lens-cornea system too powerful to focus properly onto the
retina
20
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Correcting nearsightedness
The magnifying glass
21
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Angular size of unaided image
The object at o subtends an angle of αu at the viewing point.
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Aided image
The image i of an object placed at
o within the focal length of a convex mirror subtends an angle of αa at the viewing point.
22
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Angular magnification
The angular magnification Mα is defined as the ratio of the aided and unaided viewing angles
.u
aMαα
α =
We shall employ the paraxial approximation to obtain an expression for this.
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Angular magnification
.,i
i
o
oa
ou
d
h
d
h
D
h=== αα
From the diagrams
So
.ou
a
d
DM ==
αα
α
where D is the distance to the object in the unaided case.
23
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Angular magnification
.fd
h
f
h
i
io
+=
From the diagram
,i
o
i
o
d
d
h
h=
.111
oi ddf=+
From the expression for αa,
Leading to
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Angular magnification
From
,od
DM =α
.11
+=
idfDMα
We then have
Thus the shorter the focal length, the greater the angular magnification.
24
The system matrix
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Analytical ray tracing
We can trace a ray through an optical system using
( ) ( ).ˆˆˆˆnttnii nn ukuk ×=×
Clearly, this just states Snell’s
Law.
25
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Analytical ray tracing
Trace of a light ray through a lens passing through points P1and P2.
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Analytical ray tracing
1111 ttii nn θθ =
( ) ( ).111111 αααα +=+ ttii nn
.1
11R
y≈α
Applying the paraxial approximation, Snell’s Law becomes
or, from the diagram
We also have, from the approximation of the tan function,
26
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Analytical ray tracing
.1
111
1
111
+=
+
R
yn
R
yn ttii αα
Putting these results together,
.111
211
11
−
−=
RRn
nn
f i
it
We shall make use of the lens maker’s formula
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Analytical ray tracing
For a single surface, this reduces to
,1
f=P
We also recall that the optical power of a lens is defined as
.11
11
11
Rn
nn
f i
it −=
27
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Analytical ray tracing
Hence, rearranging
we have
,1
111
1
111
+=
+
R
yn
R
yn ttii αα
.11
111111 y
R
nnnn it
iitt
−−= αα
This is the refraction equation.
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Analytical ray tracing
.1111
1
11 ynyR
nni
it P=
−
In terms of the optical power, for the first surface we may put
.1111111 ynnn iiitt P−= αα
Soa
28
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Matrix analysis of lenses
We can write out our result in the form
,0
,
11
1111111
it
iiiiii
yy
ynnn
+=
−= Pαα
where yi1 and yt1 are the heights of the point P1 immediately on either side of the first lens surface. Then, clearly
.111 yyy it ==
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Matrix analysis of lenses
This becomes useful when we re-write the simultaneous equations in matrix form
.,10
1
1
111
1
11
−=
i
ii
t
ii
y
n
y
n αα P
We can then write this in the form
,111 it xRx =
where R1 is called the refraction matrix.
29
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Analytical ray tracing
We now consider the path P1 to P2.
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Matrix analysis of lenses
.12112 tdyy α+≈
This is the transfer equation. From the diagram, the required simultaneous equations for the ray are
Using the paraxial approximation for tan, we have, for the vertical heights of P1 and P2,
.
,0
11212
1122
tti
tttt
ydy
nn
+=
+=
α
αα
30
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Matrix analysis of lenses
In matrix form, this is
.,1
01
1
11
1212
22
=
t
tt
ti
tt
y
n
ndy
n αα
This can be written in the form
,1212 ti xTx =
where T21 is the transfer matrix.
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Matrix analysis of lenses
Combining
we may put
,1212 ti xTx =
111 it xRx =and
.11212 ii xRTx =
31
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Analytical ray tracing
Now, for the second surface, the lens maker’s formula gives
.11
22
22
Rn
nn
f t
ti −−=
Hence, the optical power is
.1
22
222
Rn
nn
t
it −=P
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The system matrix
The refraction at the second interface my be encapsulated by
where the second refraction matrix is
,222 it xRx =
.10
1 2
2
−=
PR
The system matrix is then defined as
.1212 RTRA =
32
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The system matrix
More generally, the matrix for a system of lenses is
.. 12122111 RTRTRTRA K−−−−−= mmmmmmmm
The system matrix facilitates a ray
tracing analysis of a system of
lenses.
In summary,
Refracting telescopes
33
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• Terrestrial
• Produces upright image
• Employs a concave lens for the eyepiece
• Astronomical
• Inverts the image
• Employs a convex lens for the eyepiece
Types of refracting telescope
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Galileo
Improved Hans Lippershey’s design of refracting telescope
Observes phases of Venus, sunspots and Galilean moons.
Galileo (1564 - 1642)
34
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Galilean (terrestrial) telescope
Gives upright image. Note the focal points coincide.
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Johannes Kepler
Improved Galileo’s design of refracting telescope
However, the Keplerian (astronomical) telescope inverts the viewed image.
Kepler (1571 - 1630)
35
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Keplerian (astronomical) telescope
Gives inverted image. Note the focal points coincide.
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Angular magnification
Galilean telescope Keplarian telescope
From the system matrix of each telescope, both are found to have an angular magnification of
.e
o
f
fM −=α
36
Chromatic aberration
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• An inherent problem of refracting
telescopes is that they suffer from
chromatic aberration
• This is due to different frequencies of
light having different refracting indices
and therefore refracting at different
angles
Chromatic Aberration
37
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Chromatic Aberration
Blue refracts more than red (greater refractive index for normal dispersion
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• Chromatic aberration may be
compensated to some degree via the
use of an achromatic doublet
Chromatic Aberration
38
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Achromatic doublet
An achromatic doublet (achromat) is often used to compensate for the chromatic aberration.
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Achromatic doublet
First lens:
−−=
21
1
1
11)1(
1
RRn
fR
R
−−=
21
1
1
11)1(
1
RRn
fB
B
Second lens:
−−=
32
2
2
11)1(
1
RRn
fR
R
−−=
32
2
2
11)1(
1
RRn
fB
B
(The R and Bsubscripts stand for red and blue
respectively).
39
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Achromatic doublet
Red:
RRR fff 21
111+=
Blue:
BBB fff 21
111+=
fff
111
21
=+
Using the result for adding thin lenses in close combination
we obtain
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Achromatic doublet
−−+
−−=
32
2
21
1
11)1(
11)1(
1
RRn
RRn
fRR
R
Red:
Blue:
−−+
−−=
32
2
21
1
11)1(
11)1(
1
RRn
RRn
fBB
B
Writing these expressions out in full
40
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Achromatic doublet
011
)(11
)(32
12
21
11 =
−−+
−−
RRnn
RRnn RBRB
BR
BRff
ff11
=→=
Now, we require
Thus, we need to choose parameters such that
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Example of the use of an achromatic doublet, using a doublet as the objective.
Achromatic doublet
41
Reflecting telescopes
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Newton
Isaac Newton
(1642 - 1727)
Demonstrated decomposition of white light into the different colours of the rainbow via refraction through a prism.
Built the earliest known functional reflecting telescope in 1668 in a bid to escape achromatic aberration.
42
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• Newton’s telescope focussed the
incoming light via a curved mirror
• This was then reflected to the eyepiece
Newtonian telescope
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Newtonian telescope
43
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Newtonian telescope
.e
o
f
fM −=α
Essentially, we have the same magnification system as for the astronomical telescope.
Hence, the angular magnification is
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• An inherent problem of reflecting
telescopes is that they suffer from a
type of monochromatic aberration
known a coma
• This occurs in off-axis points due to the
transverse magnification being a
function of ray height
• This creates a pattern for a point that
looks like a comet
Coma
44
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Coma
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Coma in a parabolic mirror
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Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• Corrective lenses for Newtonian
telescopes with f numbers less than f/6
have been designed
• These employ a dual lens system of a
plano-convex and a plano-concave lens
fitted into an eyepiece adaptor
• An example of a correction strategy for
coma is Baader Rowe Coma Correction
Coma
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Baader Rowe Coma Correction
Comparison of the coma in an uncorrected f/3.9 Newtonian telescope vs the affects of coma with the Baader Rowe Coma Corrector.
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