Lens

1
LENS
SOLO HERMELIN
Updated: 27.10.07http://www.solohermelin.com

2
Table of Content (continue(
SOLO OPTICS
Plane-Parallel Plate
The Three Laws of Geometrical Optics
Fermat’s Principle (1657)
Prisms
Lens Definitions
Derivation of Gaussian Formula for a Single Spherical Surface Lens
Using Fermat’s Principle
Derivation of Gaussian Formula for a Single Spherical Surface Lens
Using Snell’s Law
Derivation of Lens Makers’ Formula
First Order, Paraxial or Gaussian Optics
Ray Tracing
Matrix Formulation
References

3
SOLO
The Three Laws of Geometrical Optics
1. Law of Rectilinear Propagation
In an uniform homogeneous medium the propagation of an optical disturbance is in
straight lines.
. Law of Reflection
An optical disturbance reflected by a surface has the
property that the incident ray, the surface normal,
and the reflected ray all lie in a plane,
and the angle between the incident ray and the
surface normal is equal to the angle between the
reflected ray and the surface normal:
. Law of Refraction
An optical disturbance moving from a medium of
refractive index n1 into a medium of refractive index
n2 will have its incident ray, the surface normal between
the media , and the reflected ray in a plane,
and the relationship between angle between the incident
ray and the surface normal θi and the angle between the
reflected ray and the surface normal θt given by
Snell’s Law: ti nn θθ sinsin 21 ⋅=⋅
ri
θθ =
“The branch of optics that addresses the limiting case λ0 → 0, is known as Geometrical Optics, since in
this approximation the optical laws may be formulated in the language of geometry.”
Max Born & Emil Wolf, “Principles of Optics”, 6th Ed., Ch. 3
Foundation of Geometrical Optics

4
SOLO Foundation of Geometrical Optics
Fermat’s Principle (1657)
The Principle of Fermat (principle of the shortest optical path( asserts that the optical
length
of an actual ray between any two points is shorter than the optical ray of any other
curve that joints these two points and which is in a certai neighborhood of it.
An other formulation of the Fermat’s Principle requires only Stationarity (instead of
minimal length).
∫
2
1
P
P
dsn
An other form of the Fermat’s Principle is:
Princple of Least Time
The path following by a ray in going from one point in
space to another is the path that makes the time of transit of
the associated wave stationary (usually a minimum).
The idea that the light travels in the shortest path was first put
forward by Hero of Alexandria in his work “Catoptrics”,
cc 100B.C.-150 A.C. Hero showed by a geometrical method
that the actual path taken by a ray of light reflected from plane
mirror is shorter than any other reflected path that might be
drawn between the source and point of observation.

5
SOLO
1. The optical path is reflected at the boundary between two regions
( ) ( )
0
21
21 =⋅







− rd
sd
rd
n
sd
rd
n
rayray 

In this case we have and21 nn =
( ) ( )
( ) 0ˆˆ
21
21 =⋅−=⋅







− rdssrd
sd
rd
sd
rd rayray 

We can write the previous equation as:
i.e. is normal to , i.e. to the
boundary where the reflection occurs.
21
ˆˆ ss − rd

( ) 0ˆˆˆ 2121 =−×− ssn
REFLECTION & REFRACTION
Reflection Laws Development Using Fermat Principle
This is equivalent with:
ri θθ = Incident ray and Reflected ray are in the
same plane normal to the boundary.&

6
SOLO
2. The optical path passes between two regions with different refractive indexes
n1 to n2. (continue – 1)
( ) ( )
0
21
21 =⋅







− rd
sd
rd
n
sd
rd
n
rayray 

where is on the boundary between the two regions andrd
 ( ) ( )
sd
rd
s
sd
rd
s
rayray 2
:ˆ,
1
:ˆ 21

==
Therefore is normal
to .
2211
ˆˆ snsn − rd

Since can be in any direction on
the boundary between the two regions
is parallel to the unit
vector normal to the boundary surface,
and we have
rd

2211
ˆˆ snsn − 21
ˆ −n
( ) 0ˆˆˆ 221121 =−×− snsnn
We recovered the Snell’s Law from
Geometrical Optics
REFLECTION & REFRACTION
Refraction Laws Development Using Fermat Principle
ti nn θθ sinsin 21 = Incident ray and Refracted ray are in the
same plane normal to the boundary.
&

7
SOLO
Plane-Parallel Plate
A single ray traverses a glass plate with parallel surfaces and emerges parallel to its
original direction but with a lateral displacement d.
Optics
( ) ( )irriri
lld φφφφφφ cossincossinsin −=−=
r
t
l
φcos
=






−=
r
i
ri
td
φ
φ
φφ
cos
cos
sinsin
ir
nn φφ sinsin 0
=Snell’s Law






−=
n
n
td
r
i
i
0
cos
cos
1sin
φ
φ
φ
For small anglesi
φ 





−≈
n
n
td i
0
1φ

8
SOLO
Plane-Parallel Plate (continue – 1(
Two rays traverse a glass plate with parallel surfaces and emerge parallel to their
original direction but with a lateral displacement l.
Optics
( ) ( )irriri
lld φφφφφφ cossincossinsin −=−=
r
t
l
φcos
=






−=
r
i
ri
td
φ
φ
φφ
cos
cos
sinsin
ir
nn φφ sinsin 0
=Snell’s Law






−=
n
n
td
r
i
i
0
cos
cos
1sin
φ
φ
φ






−==
r
i
i
n
n
t
d
l
φ
φ
φ cos
cos
1
sin
0
For small anglesi
φ 





−≈
n
n
tl 0
1

9
SOLO
Prisms
Type of prisms:
A prism is an optical device that refract, reflect or disperse light into its spectral
components. They are also used to polarize light by prisms from birefringent media.
Optics - Prisms
2. Reflective
1. Dispersive
3. Polarizing

10
OpticsSOLO
Dispersive Prisms
( ) ( )2211 itti
θθθθδ −+−=
21 it
θθα +=
αθθδ −+= 21 ti
202
sinsin ti
nn θθ =Snell’s Law
10
≈n
( ) ( )[ ]1
1
2
1
2
sinsinsinsin tit
nn θαθθ −== −−
( )[ ] ( )[ ]11
21
11
1
2 sincossin1sinsinsincoscossinsin ttttt nn θαθαθαθαθ −−=−= −−
Snell’s Law 110
sinsin ti
nn θθ =
11
sin
1
sin it
n
θθ =
( )[ ]1
2/1
1
221
2 sincossinsinsin iit n θαθαθ −−= −
( )[ ] αθαθαθδ −−−+= −
1
2/1
1
221
1
sincossinsinsin iii
n
The ray deviation angle is
10
≈n

11
OpticsSOLO
Prisms
( )[ ] αθαθαθδ −−−+= −
1
2/1
1
221
1
sincossinsinsin iii
n

12
OpticsSOLO
Prisms
( )[ ] αθαθαθδ −−−+= −
1
2/1
1
221
1
sincossinsinsin iii
n
Let find the angle θi1 for which the deviation angle δ is minimal; i.e. δm.
This happens when

01
0
11
2
1
=−+=
ii
t
i d
d
d
d
d
d
θ
α
θ
θ
θ
δ
Taking the differentials
of Snell’s Law equations
22
sinsin ti
n θθ =
11
sinsin ti
n θθ =
2222
coscos iitt
dnd θθθθ =
1111
coscos ttii
dnd θθθθ =
Dividing the equations

1
2
1
2
1
1
2
1
2
1
cos
cos
cos
cos
−−
=
i
t
i
t
t
i
t
i
d
d
d
d
θ
θ
θ
θ
θ
θ
θ
θ
2
22
1
22
2
2
2
2
1
2
2
2
1
2
2
2
1
2
sin
sin
/sin1
/sin1
sin1
sin1
sin1
sin1
t
i
t
i
i
t
t
i
n
n
n
n
θ
θ
θ
θ
θ
θ
θ
θ
−
−
=
−
−
=
−
−
=
−
−
1
1
2
−=
i
t
d
d
θ
θ
21 it
θθα +=
1
2
1
−=
i
t
d
d
θ
θ
2
2
1
2
2
2
1
2
cos
cos
cos
cos
i
t
t
i
θ
θ
θ
θ
= 21 ti
θθ =
1≠n

13
OpticsSOLO
Prisms
( )[ ] αθαθαθδ −−−+= −
1
2/1
1
221
1 sincossinsinsin iii n
We found that if the angle θi1 = θt2 the deviation angle δ is minimal; i.e. δm.
Using the Snell’s Law
equations
22
sinsin ti
n θθ =
11
sinsin ti
n θθ = 21 ti
θθ =
21 it
θθ =
This means that the ray for which the deviation angle δ is minimum passes through
the prism parallel to it’s base.
Find the angle θi1 for
which the deviation
angle δ is minimal; i.e.
δm (continue – 1(.

14
OpticsSOLO
Prisms
( )[ ] αθαθαθδ −−−+= −
1
2/1
1
221
1 sincossinsinsin iii n
Using the Snell’s Law 11
sinsin ti
n θθ =
21 it
θθ =
This equation is used for determining the refractive index of transparent substances.
21 it
θθα +=
21 ti
θθ =
mδδ =
2/1 αθ =t
αθδ −= 12 im
( ) 2/1 αδθ += mi
( )[ ]
2/sin
2/sin
α
αδ +
= m
n
Find the angle θi1 for
which the deviation
angle δ is minimal; i.e.
δm (continue – 2(.

15
OpticsSOLO
Prisms
The refractive index of transparent substances varies with the wavelength λ.
( )[ ]{ } αθαθλαθδ −−−+= −
1
2/1
1
221
1
sincossinsinsin iii
n

16
OpticsSOLO
http://physics.nad.ru/Physics/English/index.htm
Prisms
Color λ0 (nm( υ [THz]
Red
Orange
Yellow
Green
Blue
Violet
780 - 622
622 - 597
597 - 577
577 - 492
492 - 455
455 - 390
384 – 482
482 – 503
503 – 520
520 – 610
610 – 659
659 - 769
1 nm = 10-9
m, 1 THz = 1012
Hz
( )[ ]{ } αθαθλαθδ −−−+= −
1
2/1
1
221
1
sincossinsinsin iii
n
In 1672 Newton wrote “A New Theory about Light and Colors” in which he said that
the white light consisted of a mixture of various colors and the diffraction was color
dependent.
Isaac Newton
1542 - 1727

17
SOLO
Dispersing Prisms
Pellin-Broca Prism
Abbe Prism
Ernst Karl
Abbe
1840-1905
At Pellin-Broca Prism an
incident ray of wavelength
λ passes the prism at a
dispersing angle of 90°.
Because the dispersing angle
is a function of wavelength
the ray at other wavelengths
exit at different angles.
By rotating the prism around
an axis normal to the page
different rays will exit at
the 90°.
At Abbe Prism the dispersing
angle is 60°.
Optics - Prisms

18
SOLO
Dispersing Prisms (continue – 1(
Amici Prism
Optics - Prisms

19
SOLO
Reflecting Prisms BED∠−= 
180δ

360=∠+∠+∠+ ABEBEDADEα
1
90 i
ABE θ+=∠ 
2
90 t
ADE θ+=∠ 

3609090 12 =++∠+++ it BED θθα
12180 itBED θθα −−−=∠ 
αθθδ ++=∠−= 21180 tiBED
The bottom of the prism is a reflecting mirror
Since the ray BC is reflected to CD
DCGBCF ∠=∠
Also
CGDBFC ∠=∠
CDGFBC ∠=∠
FBCt ∠−= 
901θ
CDGi ∠−= 
902θ
21 it
θθ =
202
sinsin ti
nn θθ =Snell’s Law
Snell’s Law 110
sinsin ti
nn θθ = 21 ti
θθ = αθδ += 1
2 i
CDGFBC ∆∆ ~
Optics - Prisms

20
SOLO
Reflecting Prisms
Porro Prism Porro-Abbe Prism
Schmidt-Pechan Prism
Penta Prism
Optics - Prisms
Roof Penta Prism

21
SOLO
Reflecting Prisms
Abbe-Koenig Prism
Dove Prism
Amici-roof Prism
Optics - Prisms

22
SOLO
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Polarization can be achieved with crystalline materials which have a different index of
refraction in different planes. Such materials are said to be birefringent or doubly refracting.
Nicol Prism
The Nicol Prism is made up from
two prisms of calcite cemented
with Canada balsam. The
ordinary ray can be made to
totally reflect off the prism
boundary, leving only the
extraordinary ray..
Polarizing Prisms
Optics - Prisms

23
SOLO
Polarizing Prisms
A Glan-Foucault prism deflects polarized light
transmitting the s-polarized component.
The optical axis of the prism material is
perpendicular to the plane of the diagram.
A Glan-Taylor prism reflects polarized light
at an internal air-gap, transmitting only
the p-polarized component.
The optical axes are vertical in the plane of
the diagram.
A Glan-Thompson prism deflects the p-polarized
ordinary ray whilst transmitting the s-polarized
extraordinary ray.
The two halves of the prism are joined with
Optical cement, and the crystal axis are
perpendicular to the plane of the diagram.
Optics - Prisms

24
OpticsSOLO
Lens Definitions
Optical Axis: the common axis of symmetry of an optical system; a line that connects all
centers of curvature of the optical surfaces.
Lateral Magnification: the ratio between the size of an image measured perpendicular
to the optical axis and the size of the conjugate object.
Longitudinal Magnification: the ratio between the lengthof an image measured along
the optical axis and the length of the conjugate object.
First (Front( Focal Point: the point on the optical axis on the left of the optical system
(FFP( to which parallel rays on it’s right converge.
Second (Back( Focal Point: the point on the optical axis on the right of the optical system
(BFP( to which parallel rays on it’s left converge.

25
OpticsSOLO
Definitions (continue – 2(
Aperture Stop (AS(: the physical diameter which limits the size of the cone of radiation
which the optical system will accept from an axial point on the object.
Field Stop (FS(: the physical diameter which limits the angular field of view of an
optical system. The Field Stop limit the size of the object that can be
seen by the optical system in order to control the quality of the image.
A.S. F.S.
IΣ
Aperture and Field Stops
Image
plane
Hecht
"Optics"

26
OpticsSOLO
Entrance Pupil: the image of the Aperture Stop as seen from the object through the
(EnP( elements preceding the Aperture Stop.
Exit Pupil: the image of the Aperture Stop as seen from an axial point on the
(ExP( image plane.
Entrance
pupil
Exit
pupil
A.S.
IΣ
xpE
npE
Chief
Ray
Entrance and Exit pupils
Image
plane
Marginal
Ray
Hecht
"Optics"
Entrance
pupilExit
pupil
A.S. I
Σ
xpE
npE
Chief
Ray
Image
plane
A front Aperture Stop
Hecht
"Optics"
Chief Ray: an object Ray passing through the center of the aperture stop and
(CR( appearing to pass through the centers of entrance and exit pupils.
Marginal Ray: an object Ray passing through the edge of the aperture stop.
(MR(

27
OpticsSOLO
Entrance
pupil
Exit
pupil
A.S.
IΣ
Chief
Ray
Marginal
Ray
Exp
Enp
Image
plane
Hecht
"Optics"

28
OpticsSOLO
Principal Planes: the two planes defined by the intersection of the parallel incident rays
entering an optical system with the rays converging to the focal points
after passing through the optical system.
Principal Points: the intersection of the principal planes with the optical axes.
Nodal Points: two axial points of an optical system, so located that an oblique ray
directed toward the first appears to emerge from the second, parallel
to the original direction. For systems in air, the Nodal Points coincide
with the Principal Points.
Cardinal Points: the Focal Points, Principal Points and the Nodal Points.

29
OpticsSOLO
Relative Aperture (f# (: the ratio between the effective focal length (EFL( f to Entrance
Pupil diameter D.
Numerical Aperture (NA(: sine of the half cone angle u of the image forming ray bundles
multiplied by the final index n of the optical system.
If the object is at infinity and assuming n = 1 (air(:
Dff /:# =
unNA sin: ⋅=
#
1
2
1
2
1
sin
ff
D
uNA =





==

30
OpticsSOLO
Perfect Imaging System
• All rays originating at one object point reconverge to one image point after passing
through the optical system.
• All of the objects points lying on one plane normal to the optical axis are imaging
onto one plane normal to the axis.
• The image is geometrically similar to the object.

31
OpticsSOLO
Lens
Convention of Signs
1. All Figures are drawn with the light traveling from left to right.
2. All object distances are considered positive when they are measured to the left of the
vertex and negative when they are measured to the right.
3. All image distances are considered positive when they are measured to the right of the
vertex and negative when they are measured to the left.
4. Both focal length are positive for a converging system and negative for a diverging
system.
5. Object and Image dimensions are positive when measured upward from the axis and
negative when measured downward.
6. All convex surfaces are taken as having a positive radius, and all concave surfaces
are taken as having a negative radius.

32
OpticsSOLO
Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Fermat’s Principle
Karl Friederich Gauss
1777-1855
The optical path connecting points M, T, M’ is
''lnlnpathOptical ⋅+⋅=
Applying cosine theorem in triangles MTC and M’TC
we obtain:
( ) ( )[ ] 2/122
cos2 βRsRRsRl +−++=
( ) ( )[ ] 2/122
cos'2'' βRsRRsRl −+−+=
( ) ( )[ ] ( ) ( )[ ] 2/122
2/122
cos'2''cos2 ββ RsRRsRnRsRRsRnpathOptical −+−+⋅++−++⋅=
Therefore
According to Fermat’s Principle when the point T
moves on the spherical surface we must have ( ) 0=
βd
pathOpticald
( ) ( ) ( ) 0
'
sin''sin
=
−⋅
−
+⋅
=
l
RsRn
l
RsRn
d
pathOpticald ββ
β
from which we obtain 




 ⋅
−
⋅
=+
l
sn
l
sn
Rl
n
l
n
'
''1
'
'
For small α and β we have ''& slsl ≈≈
and we obtain
R
nn
s
n
s
n −
=+
'
'
'
Gaussian Formula for a Single Spherical Surface

33
OpticsSOLO
Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Snell’s Law
Apply Snell’s Law: 'sin'sin φφ nn =
If the incident and refracted rays
MT and TM’ are paraxial the
angles and are small and we can
write Snell’s Law:
φ 'φ
From the Figure βαφ += γβφ −='
''φφ nn =
( ) ( ) ( ) βγαγββα nnnnnn −=+⇒−=+ '''
For paraxial rays α, β, γ are small angles, therefore '/// shrhsh ≈≈≈ γβα
( )
r
h
nn
s
h
n
s
h
n −=+ '
'
'
or
( )
r
nn
s
n
s
n −
=+
'
'
'
Gaussian Formula for a Single Spherical Surface
1777-1855
Willebrord van Roijen
Snell
1580-1626
( )
φ
φφ
φφ
φ
≈+++=


O`
53
!5!3
sin

34
OpticsSOLO
Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Snell’s Law
for s → ∞ the incoming rays are parallel to optical
axis and they will refract passing trough a common
point called the focus F’.
( )
r
nn
s
n
s
n −
=+
'
'
'
( )
r
nn
f
nn −
=+
∞
'
'
'
r
nn
n
f
−
=
'
'
'
for s’ → ∞ the refracting rays are parallel to optical
axis and therefore the incoming rays passes trough
a common point called the focus F.
( )
r
nnn
f
n −
=
∞
+
'' r
nn
n
f
−
=
'
'' n
n
f
f
=

35
OpticsSOLO
Derivation of Lens Makers’ Formula
We have a lens made of two
spherical surfaces of radiuses r1
and r2 and a refractive index n’,
separating two media having
refraction indices n a and n”.
Ray MT1 is refracted by the first
spherical surface (if no second
surface exists) to T1M’.
( )
111
'
'
'
r
nn
s
n
s
n −
=+
11111
''& sMAsTA ==
Ray T1T2 is refracted by the second spherical surface to T2M”. 2222
""&'' sMAsMA ==
( )
222
'"
"
"
'
'
r
nn
s
n
s
n −
=+
Assuming negligible lens thickness we have , and since M’ is a virtual object
for the second surface (negative sign) we have
21
'' ss ≈
21
'' ss −≈
( )
221
'"
"
"
'
'
r
nn
s
n
s
n −
=+−

36
OpticsSOLO
Derivation of Lens Makers’ Formula (continue – 1)
( )
111
'
'
'
r
nn
s
n
s
n −
=+
Add those equations
( )
221
'"
"
"
'
'
r
nn
s
n
s
n −
=+−
( ) ( )
2121
'"'
"
"
r
nn
r
nn
s
n
s
n −
+
−
=+
The focal lengths are defined by
tacking s1 → ∞ to obtain f” and
s”2 → ∞ to obtain f
( ) ( )
f
n
r
nn
r
nn
f
n
=
−
+
−
=
212
'"'
"
"
Let define s1 as s and s”2 as s”
to obtain
( ) ( )
21
'"'
"
"
r
nn
r
nn
s
n
s
n −
+
−
=+
( ) ( )
f
n
r
nn
r
nn
f
n
=
−
+
−
=
21
'"'
"
"

37
OpticsSOLO
Derivation of Lens Makers’ Formula (continue – 2)
If the media on both sides of
the lens is the same n = n”.






−





−=+
21
11
1
'
"
11
rrn
n
ss






−





−==
21
11
1
'1
"
1
rrn
n
ff
Therefore
"
11
"
11
ffss
==+
Lens Makers’ Formula

38
OpticsSOLO
First Order, Paraxial or Gaussian Optics
In 1841 Gauss gave an exposition in “Dioptrische Untersuchungen”
for thin lenses, for the rays arriving at shallow angles with respect to
Optical axis (paraxial).
1777-1855
Derivation of Lens Formula
From the similarity of the triangles
and using the convention:
( )
''
'
'~'
f
y
s
yy
TAFTSQ =
−+
⇒∆∆
Lens Formula in Gaussian form
( ) ( )
f
y
s
yy
FASQTS
''
~
−
=
−+
⇒∆∆
( ) 0' >− y
Sum of the
equations: ( ) ( ) ( )
'
'
'
''
f
y
f
y
s
yy
s
yy
+
−
=
−+
+
−+
since f = f’
fss
1
'
11
=+
( )
φ
φφ
φφ
φ
≈+++=


O`
53
!5!3
sin

39
OpticsSOLO
First Order, Paraxial or Gaussian Optics (continue – 1)
Gauss explanation can be extended to the first order approximation
to any optical system.
1777-1855
Lens Formula in Gaussian form
fss
1
'
11
=+
s – object distance (from the first principal point to the object).
s’ – image distance (from the second principal point to the image).
f – EFL (distance between a focal point to the closest principal plane).
'y
s 's
M’A F’
M
T
F
'ffx 'x
Q
Q’
'y
y
S
Axis
y

40
OpticsSOLO
Derivation of Lens Formula (continue)
From the similarity of the triangles
and using the convention:
( )
f
y
x
y
FASQMF
'
~
−
=⇒∆∆
Lens Formula in Newton’s form
( )
f
y
x
y
QMFTAF =
−
⇒∆∆
'
'
'''~'
( ) 0' >− y
Multiplication
of the equations:
( ) ( )
2
'
'
'
f
yy
xx
yy −⋅
=
⋅
−⋅
or 2
' fxx =⋅
Isaac Newton
1643-1727
Published by Newton in “Opticks” 1710
'y
s 's
M’A F’
M
T
F
'ffx 'x
Q
Q’
'y
y
S
Axis
y

41
OpticsSOLO
Derivation of Lens Formula (continue)
Lateral or Transverse Magnification
f
x
x
f
s
s
h
h
mT
'''
−=−=−==
Quantity (+) sign (-) sign
s real object virtual object
s’ real image virtual image
f converging lens diverging lens
h erect object inverted object
h’ erect image inverted image
mT erect image inverted image
'y
s 's
M’A F’
M
T
F
'ffx 'x
Q
Q’
'y
y
S
Axis
y

42
OpticsSOLO
Derivation of Lens Formula (Summary)
If the media on both sides of
the lens is the same n = n”.






−





−=+
21
11
1
'
"
11
rrn
n
ss






−





−==
21
11
1
'1
"
1
rrn
n
ff
Therefore
"
11
"
11
ffss
==+
Lens Makers’ Formula
f
x
x
f
s
s
h
h
mT
'''
−=−=−==
Gauss’ Lens Formula
Magnification

45
OpticsSOLO
Ray Tracing
F C
O
I
Object Virtua
l
Image
Convex
Mirror
R/2 R/2
R
FC
O
I
Object
Real
Image
Concave
Mirror
Ray Tracing is a graphically implementation of paralax ray analysis. The construction
doesn’t take into consideration the nonideal behavior, or aberration of real lens.
The image of an off-axis point can be located by the intersection of any two of the
following three rays:
1. A ray parallel to the axis that is
reflected through F’.
2. A ray through F that is reflected
parallel to the axis.
3. A ray through the center C of the
lens that remains undeviated and
undisplaced (for thin lens).

47
OpticsSOLO
Matrix Formulation
The Matrix Formulation of the Ray Tracing method for the paraxial assumption
was proposed at the beginning of nineteen-thirties by T.Smith.
Assuming a paraxial ray entering at some input plane of an optical system at the distance
r1 from the symmetry axis and with a slope r1’ and exiting at some output plane at the
distance r2 from the symmetry axis and with a slope r2’, than the following linear (matrix)
relation applies:






=











=





''' 1
1
1
1
2
2
r
r
M
r
r
DC
BA
r
r






=
DC
BA
Mwhere ray transfer matrix
When the media to the left of the input plane
and to the right of the output plane have the
same refractive index, we have:
1det =⋅−⋅= CBDAM

48
OpticsSOLO
Matrix Formulation (continue -1)
Uniform Optical Medium
In an Uniform Optical Medium of length d no change in ray angles occurs:
''
'
12
112
rr
rdrr
=
+=






=
10
1 d
M
Medium
Optical
Uniform
Planar Interface Between Two Different Media
12 rr =
'' 1
2
1
2
12
r
n
n
r
rr
=
=
Apply Snell’s Law: 2211
sinsin φφ nn =
paraxial assumption: φφφφ ≈=⇒≈ tan'sin r
From Snell’s Law: '' 1
2
1
2
r
n
n
r =






=
21
/0
01
nn
M
Interface
Planar
1det
2
1
≠=
n
n
M
Interface
Planar
1det =
Medium
Optical
Uniform
M
The focal length of this system is infinite and it has
not specific principal planes.

49
OpticsSOLO
A Parallel-Sided Slab of refractive index n bounded on both sides with media of
refractive index n1 = 1
We have three regions:
• on the right of the slab (exit of ray): 











=





'/0
01
' 3
3
124
4
r
r
nnr
r
• in the slab:












=





'10
1
' 2
2
3
3
r
rd
r
r
• on the left of the slab (entrance of ray):












=





'/0
01
' 1
1
212
2
r
r
nnr
r
Therefore:
























=





'/0
01
10
1
/0
01
' 1
1
21124
4
r
r
nn
d
nnr
r












=

















=
21
21
122112 /0
/1
/0
01
/0
01
10
1
/0
01
nn
nnd
nnnn
d
nn
M
mediaentranceslabmediaexit
Slab
Sided
Parallel







=
10
/1 21 nnd
M
Slab
Sided
Parallel
1det =
Slab
Sided
Parallel
M

50
OpticsSOLO
Spherical Interface Between Two Different Media
12 rr =
Apply Snell’s Law: rnin sinsin 21
=
paraxial assumption: rrii ≈≈ sin&sin
From Snell’s Law: rnin 21
=
( )










−
=










−=
2
1
2
1
2
1
12
21
0101
n
n
n
D
n
n
Rn
nnM
Interface
Spherical 1det
2
1
≠=
n
n
M
Interface
Spherical
12
11
'
'
φ
φ
+=
+=
rr
ri
From the Figure:
( ) ( )122111
'' φφ +=+ rnrn
111
/ Rr=φ
( )
12
121
2
11
2
'
'
Rn
rnn
n
rn
r
−
+=
( )
1
12
11
112
2
12
'
'
n
rn
Rn
rnn
r
rr
+
−
=
=
( )
1
12
1
:
R
nn
D
−
=where: Power of the surface If R1 is given in meters D1 gives diopters

51
OpticsSOLO
Thick Lens
We have three regions:
• on the right of the
slab (exit of ray):
















−
=





'
01
' 3
3
1
2
1
2
4
4
r
r
n
n
n
D
r
r
• in the slab:












=





'10
1
' 2
2
3
3
r
rd
r
r
• on the left of the
slab (entrance of ray): 















−
=





'
01
' 1
1
2
1
2
1
2
2
r
r
n
n
n
D
r
r
Therefore:


















−
−










−
=















−















−
=





'
101
'
01
10
1
01
' 1
1
2
1
2
1
2
1
2
1
1
2
1
2
1
1
2
1
2
1
1
2
1
2
4
4
r
r
n
n
n
D
n
n
d
n
D
d
n
n
n
D
r
r
n
n
n
D
d
n
n
n
D
r
r














−





−
+
−
−
=
2
2
21
21
1
21
2
1
2
1
1
1
n
D
d
nn
DD
d
n
DD
n
n
d
n
D
d
M
Lens
Thick
( )
2
21
2
R
nn
D
−
=
( )
1
12
1
:
R
nn
D
−
=














−





−
+
−−
=
−
2
1
21
21
1
21
2
1
2
2
1
1
1
n
D
d
nn
DD
d
n
DD
n
n
d
n
D
d
M
Lens
Thick
1det =
Lens
Thick
M
or
21 DD ⇔

52
OpticsSOLO
Thick Lens (continue -1)
Let use the second Figure where Ray 2 is parallel
to Symmetry Axis of the Optical System that is refracted
trough the Second Focal Point.
























−





−
+
−
−
=










'1
1
' 1
1
2
2
21
21
1
21
2
1
2
1
4
4
r
r
n
D
d
nn
DD
d
n
DD
n
n
d
n
D
d
r
r
We found:
2141 /'&0' frrr −==Ray 2:
By substituting Ray2 parameters we obtain:
1
2
1
21
21
1
21
4
1
' r
f
r
nn
DD
d
n
DD
r −=





−
+
−=
1
21
21
1
21
2
−






−
+
=
nn
DD
d
n
DD
f
frrr /'&0' 414 −==Ray 1:
We found:
























−





−
+
−−
=










'1
1
' 4
4
2
1
21
21
1
21
2
1
2
2
1
1
r
r
n
D
d
nn
DD
d
n
DD
n
n
d
n
D
d
r
r
4
1
4
21
21
1
21
1
1
' r
f
r
nn
DD
d
n
DD
r −=





−
+
= 2
1
21
21
1
21
1 f
nn
DD
d
n
DD
f −=





−
+
=
−

53
OpticsSOLO
Thin Lens
For thick lens we found














−





−
+
−
−
=
2
2
21
21
1
21
2
1
2
1
1
1
n
D
d
nn
DD
d
n
DD
n
n
d
n
D
d
M
Lens
Thick






−
+
=
21
21
1
211
nn
DD
d
n
DD
f
For thin lens we can assume d = 0 and obtain










−
=
1
1
01
f
M
Lens
Thin
1
211
n
DD
f
+
= ( )
2
21
2
R
nn
D
−
=
( )
1
12
1
:
R
nn
D
−
=






−





−=
+
=
211
2
1
21 11
1
1
RRn
n
n
DD
f

54
OpticsSOLO
Thin Lens (continue – 1)
For a biconvex lens we have R2 negative








+





−=
211
2 11
1
1
RRn
n
f
For a biconcave lens we have R1 negative








+





−−=
211
2 11
1
1
RRn
n
f










−
=
1
1
01
f
M
Lens
Thin

55
OpticsSOLO
A Length of Uniform Medium Plus a Thin Lens










−−
=















−
==
+
f
d
f
d
d
f
MMM
Medium
Uniform
Lens
Thin
Lens
Thin
Medium
Uniform
1
1
1
10
1
1
1
01
Combination of Two Thin Lenses












+−−−+−−
−+−
=










−−









−−
==
21
21
2
2
2
1
1
1
21
2
21
1
21
21
2
2
1
1
1
1
22
2
1
11
1
1
1
1
1
1
1
1122
ff
dd
f
d
f
d
f
d
ff
d
ff
f
dd
dd
f
d
f
d
f
d
f
d
f
d
MMMMM
dMedium
Uniform
fLens
Thin
dMedium
Uniform
fLens
Thin
Lenses
Thin
Two
The Focal Length of the Combination of
Two Thin Lenses is:
21
2
21
111
ff
d
fff
−+= Return to
Chromatic Aberration

56
OpticsSOLO
Real Imaging Systems – Aberrations
Departures from the idealized conditions of Gaussian Optics in a real Optical System are
called Aberrations
Monochromatic Aberrations
Chromatic Aberrations
• Monochromatic Aberrations
Departures from the first order theory are embodied in the five primary aberrations
1. Spherical Aberrations
2. Coma
3. Astigmatism
4. Field Curvature
5. Distortion
This classification was done in 1857 by Philipp Ludwig von Seidel (1821 – 1896)
• Chromatic Aberrations
1. Axial Chromatic Aberration
2. Lateral Chromatic Aberration

57
OpticsSOLO
Real Imaging Systems – Aberrations (continue – 5)

58
OpticsSOLO

59
OpticsSOLO

60
OpticsSOLO

61
OpticsSOLO

62
OpticsSOLO
Seidel Aberrations
Consider a spherical surface of radius R, with an object P0 and the image P0’ on the
Optical Axis.
The Chief Ray is P0 V0 P0’ and a
General Ray P0 Q P0’.
The Wave Aberration is defined as
the difference in the optical path
lengths between a General Ray and
the Chief Ray.
( ) [ ] [ ] ( ) ( )snsnQPnQPnPVPQPPrW +−+=−= '''''' 00000000
On-Axis Point Object
The aperture stop AS, entrance pupil EnP,
and exit pupil ExP are located at the
refracting surface.

63
OpticsSOLO
Seidel Aberrations (continue – 1)








−−=−−= 2
2
22
11
R
r
RrRRz
Define:
( ) 2
2
11
2
2
R
r
xxf
R
r
x
−=+=
−=
( ) ( ) 2/1
1
2
1
'
−
+= xxf
( ) ( ) 2/3
1
4
1
"
−
+−= xxf ( ) ( ) 2/5
1
8
3
'"
−
+−= xxf
Develop f (x) in a Taylor series ( ) ( ) ( ) ( ) ( ) ++++= 0"'
6
0"
2
0'
1
0
32
f
x
f
x
f
x
fxf
1
168
11
32
<++−+=+ x
xx
xx 
Rr
R
r
R
r
R
r
R
r
Rz <+++=








−−= 5
6
3
42
2
2
1682
11
On Axis Point Object
From the Figure:
( ) 222
rzRR +−= 02 22
=+− rRzz

64
OpticsSOLO
From the Figure:
( )[ ] [ ]
( )[ ] ( ) 2/1
2
2/12
2
2/12222/122
0
212
2
22





 −
+=+−=
++−=+−=
−=
z
s
sR
sszsR
rsszzrszQP
rzRz
( ) ( )






+
−
−
−
+−≈
<++−+=+


2
4
2
2
1
168
11
2
1
1
32
z
s
sR
z
s
sR
s
x
xx
xx
( ) ( )








+





+
−
−





+
−
+−=
+≈

2
3
42
4
2
3
42
2
82
822
1
82
1
3
42
R
r
R
r
s
sR
R
r
R
r
s
sR
s
R
r
R
r
z
( )[ ] +














−+





−+





−+−≈+−= 4
2
2
22/122
0
11
8
111
8
111
2
1
r
sRssRR
r
sR
srszQP
( )[ ] +














−+





−+





−+≈+−= 4
2
2
22/122
0
1
'
1
'8
11
'
1
8
11
'
1
2
1
''' r
RssRsR
r
Rs
srzsPQ
In the same way:

65
OpticsSOLO
+














−+





−+





−+−≈ 4
2
2
2
0
11
8
111
8
111
2
1
r
sRssRR
r
sR
sQP
+














−+





−+





−+≈ 4
2
2
2
0
1
'
1
'8
11
'
1
8
11
'
1
2
1
'' r
RssRsR
r
Rs
sPQ
Therefore:
( ) ( ) ( )
4
22
2
42
000
11
'
11
'
'
8
1
82
'
'
'
''''
r
sRs
n
sRs
n
R
rr
R
nn
s
n
s
n
snsnQPnQPnrW














−−





−−





+




 −
−−=
+−+=
Since P0’ is the Gaussian image of P0 we have
( ) R
nn
s
n
s
n −
=
−
+
'
'
'
and:
( ) 44
22
0
11
'
11
'
'
8
1
rar
sRs
n
sRs
n
rW S
=














−−





−−=

66
OpticsSOLO
Off-Axis Point Object
Consider the spherical surface of radius R, with an object P and its Gaussian image P’
outside the Optical Axis.
The aperture stop AS, entrance pupil EnP, and
xit pupil ExP are located at the refracting surface.
Using
''~ 00 CPPCPP ∆∆
the transverse magnification
( ) ( )
s
n
s
n
nn
s
s
n
s
n
nn
s
Rs
Rs
h
h
Mt
−
−
+−
−
−
−
=
+−
−
=
−
=
'
'
'
'
'
'
'
''
( )sn
sn
nn
s
s
nn
nn
s
s
nn
Mt
−
=
−+−
+−−
=
'
'
'
'
'
'
'
'

67
OpticsSOLO
The Wave Aberration is defined as the difference
n the optical path lengths between the General
Ray and the Undeviated Ray.
( ) [ ] [ ]
[ ] [ ]{ } [ ] [ ]{ }
( )4
0
4
0 ''''
''
VVVQa
PVPPPVPVPPQP
PVPPQPQW
S −=
−−−=
−=
For the approximately similar triangles VV0C and CP0’P’ we have:
CP
CV
PP
VV
''' 0
0
0
0
≈ ''
'
''
'
0
0
0
0 hbh
Rs
R
PP
CP
CV
VV =
−
=≈
Rs
R
b
−
=
'
:














−−





−−=
22
11
'
11
'
'
8
1
sRs
n
sRs
n
aS

68
OpticsSOLO
Wave Aberration.
( ) [ ] [ ] ( )4
0
4
'' VVVQaPVPPQPQW S −=−=
Define the polar coordinate (r,θ) of the projection of Q in the plane of exit pupil, with
V0 at the origin.
θθ cos'2'cos2 222
0
2
0
2
2
hbrhbrVVrVVrVQ ++=++=
'0 hbVV =
( ) [ ] [ ] ( )
( )[ ]442222
4
0
4
'cos'2'
''
hbhbrhbra
VVVQaPVPPQPQW
S
S
−++=
−=−=
θ
( ) ( )θθθθ cos'4'2cos'4cos'4';, 33222222234
rhbrhbrhbrhbrahrW S ++++=

69
OpticsSOLO
eneral Optical Systems
( ) θθθθ cos''cos'cos'';, 33222222234
rhbCrhbCrhbCrhbCrChrW DiFCAsCoSp ++++=
A General Optical Systems has more than on Reflecting or
Refracting surface. The image of one surface acts as an
bject for the next surface, therefore the aberration is additive.
We must address the aberration in the plane of the exit pupil, since the rays follow
straight lines from the plane of the exit pupil.
The general Wave Aberration Function is:
1. Spherical Aberrations CoefficientSpC
2. Coma CoefficientCoC
3. Astigmatism CoefficientAs
C
4. Field Curvature CoefficientFC
C
5. Distortion CoefficientDi
C
where:

70
OpticsSOLO
( ) ( )θθθθ cos'4'2cos'4cos'4';, 33222222234
rhbrhbrhbrhbrahrW S ++++=

71
OpticsSOLO
( ) θθθθ cos''cos'cos'';, 32222234
rhCrhCrhCrhCrChrW DiFCAsCoSp ++++=

72
OpticsSOLO
nWPP TR /=
Assume that P’ is the image of P.
The point PT is on the Exit Pupil (Exp) and on the
True Wave Front (TWF) that propagates toward P’.
This True Wave Front is not a sphere because of the
Aberration. Without the aberration the wave front
would be the Reference Sphere (RS) with radius PRP.
W (x’,y’;h’) - wave aberration
n - lens refraction index
L’ - distance between Exp and Image plane
ά - angle between the normals to the TWF and RS
at PT.
Assume that P’R and P’T are two points on
RS and TWF, respectively, and on a ray close
to PRPT ray, converging to P’, the image of P.
lPPPP TRTR ∆+=''

73
OpticsSOLO
( )
'
';','
'
'
'
x
hyxW
n
L
x
∂
∂
=∆
( )
'
';','
'
'
'
y
hyxW
n
L
y
∂
∂
=∆
θ
θ
sin'
cos'
ry
rx
=
=
( ) nhyxWPP TR /';','= lPPPP TRTR ∆+=''
α=
∆
∆
=
∆
−
=
∂
∂
→∆→∆ r
l
r
PPPP
x
W
n r
TRTR
r 00
lim
''
lim
1
x
W
n
L
Lr
∂
∂
==∆
'
'α

74
OpticsSOLO
1. Spherical Aberrations
( )
( ) ( )';','''
';,
222
4
hyxWyxC
rChrW
SpSp
SpSp
=+=
=θ
( ) '
'
'
4
'
';','
'
'
' 2
xrC
n
L
x
hyxW
n
L
x Sp
=
∂
=∆
( ) '
'
'
4
'
';','
'
'
' 2
yrC
n
L
y
hyxW
n
L
y Sp
=
∂
=∆
To Update
( ) ( )[ ] 32/122
'
'
4'' rC
n
L
yxr Sp=∆+∆=∆
Consider only the Spherical Wave Aberration Function
The Spherical Wave Aberration is a
Circle in the Image Plane

75
OpticsSOLO
2. Coma
Assume an object point outside the Optical Axis.
Meridional (Tangential) plane is
the plane defined by the object point
and the Optical Axis.
Sagittal plane is the plane normal to
Meridional plane that contains the
Chief Ray passing through the
Object point.

76
OpticsSOLO
2. Coma
Consider only the Coma Wave Aberration Function
( ) ( ) ''''cos'';, 223
xyxhCrhChrW CoCoCo +== θθ
( ) ( ) ( ) ( )θθ 2cos2
'
''
cos21
'
''
''3
'
''
'
';','
'
'
' 2222
+=+=+=
∂
=∆ r
n
Lh
Cr
n
Lh
Cyx
n
Lh
C
x
hyxW
n
L
x CoCoCo
( ) ( ) θ2sin
'
''
''2
'
''
'
';','
'
'
' 2
r
n
Lh
Cyx
n
Lh
C
y
hyxW
n
L
y CoCo ==
∂
=∆
1
'
''
'
2
'
''
'
2
2
2
2
=












∆
+












−
∆
r
n
Lh
C
y
r
n
Lh
C
x
CoCo
( )( ) ( ) ( )222
'2' rRyrRx CoCo =∆+−∆
( ) 2
'
''
: r
n
Lh
CrR CoCo
=

77
OpticsSOLO
2. Coma
We obtained
2
'
''
: MAXCoS r
n
Lh
CC =
( )( ) ( ) ( )222
'2' rRyrRx CoCo =∆+−∆
( ) MAXCoCo rrr
n
Lh
CrR ≤≤= 0
'
''
: 2
Define:
1
2
3 4
P
Image
Plane
O
SC
SC
ST CC 3=
Coma Blur Spot Shape
Tangential
Coma
Sagittal
Coma

30
'h
'x
'y

78
OpticsSOLO
Graphical Explanation of Coma Blur

79
OpticsSOLO
phical Explanation of Coma Blur (continue – 1)

81
OpticsSOLO
3. Astigmatism

82
OpticsSOLO
3. Astigmatism

83
OpticsSOLO
3. Astigmatism

84
OpticsSOLO
Field Curvature

85
OpticsSOLO
5. Distortion

86
OpticsSOLO
hin Lens Aberrations
( ) 2222234
'cos'cos'';, rhCrhCrhCrChrW FCAsCoSp +++= θθθ
ven a thin lens formed by two
faces with radiuses r1 and r2
h centers C1 and C2. PP0 is
object, P”P”0 is the Gaussian
ge formed by the first surface,
’0 is the image of virtual object
”0 of the second surface.
( )
( ) ( ) ( ) 





++
−
+
+−++
−−
−= qpnq
n
n
pnn
n
n
fnn
CSp 14
1
2
123
1132
1 22
3
3
( ) 





−
+
++= q
n
n
pn
sfn
CCo
1
1
12
'4
1
2
( )2
'2/1 sfCAs −=
( ) ( )2
'4/1 sfnnCFC +−=
where:
f
s
OA
C11
r
F”
F
''f
''s
2
r
1=n
n
h
"h
D
0P
P
0'P
0"P
"P
'P
'h
's
CR
AS
EnP
ExP
r
( )θ,rQ
OC2
1=n
( ) [ ] [ ]0000 '', OPPQPPrW −=θ
Coddington shape factor:
Coddington position factor: ss
ss
p
−
+
=
'
'
12
12
rr
rr
q
−
+
=
From:
we find:

87
OpticsSOLO
oddington Position Factor
2R 1R f
1C 2FO 1F
2
C
2n
1n
s 's
'2 sfs ==
2R 1
R f
1
C 2F1F
2
C
2n
1
n
s 's
fss =∞= ',
2R 1R f
1
C 2F1F
2
C
2n
1
n
s 's
fss <> ',0
2R 1
R
f
1
C 2F1F
2
C
2n
1
n
s 's
∞== ', sfs
2R 1Rf
1
C2F1
F 2
C 2n
1
n
s 's
0', << sfs
CRCR
2
R
1
R
f1C
2
F
1F 2
C2n
1
n
s
's
0'0 <<> sfs
2
R
1
R
f1
C
2F1F 2C2n
1n
s 's
fss =∞= ',
1=p
2
R1R f
1C 2F1F
2C
2n
1
n
s
's
∞== ', sfs
1>p
2
R1R f
1C 2F1F
2C
2n
1
n
s 's
0',0 ><< ssf
0=p
2
R1R f
1C 2FO 1F
2C
2n
1
n
s 's
'2 sfs ==
1−=p1−<p
ss
ss
p
−
+
=
'
'
ss
ss
p
−
+
=
'
'
'
111
ssf
+=
'
2
11
2
s
f
s
f
p −=−=

88
OpticsSOLO
Coddington Position Factor
f f2f2− f− 0
Figure Object
Location
Image
Location
Image
Properties
Shape
Factor
Infinity
Principal
focus
'ss
fs 2> fsf 2'<<
fs 2= fs 2'=
fsf 2<< fs 2'>
's
's
s
s
fs = ∞='s
s
s
's
fs < fs <'
Real, inverted
small p = -1
Real, inverted
smaller
-1 < p <0
Real, inverted
same size
p = 0
Real, inverted
larger
0 < p <1
No image p = 1
Virtual, erect
larger
p>1
's
's
0<s fs <' p < -1
Imaginary,
inverted
small

89
OpticsSOLO
oddington Shape Factor
1
02
1
−=
<
∞=
q
R
R
2
R
1
R
2
C 2
n
1
n
Plano
Convex
2
n
1
0,0
21
21
−<
>
<<
q
RR
RR
1
C 2
C
1
n
1
R
2
R
Positive
Meniscus
2
R
1
R f
1
C 2
F1
F 2
C 2
n
1
n
0
0,0
21
21
=
=
<>
q
RR
RR
Equi
Convex
2
R
1
R
1C2n
1n
Plano
Convex
1
0
2
1
=
∞=
>
q
R
R
2
R1
R
f
1
C 2
F 2
C
2
n
1
n
1
0,0
21
21
>
<
>>
q
RR
RR
Positive
Meniscus
12
12
RR
RR
q
−
+
=
2
R
1
R f
2F1
F
2C
2n
1
n
1C
Negative
Meniscus
1
0,0
21
21
−<
>
>>
q
RR
RR
1
0, 21
−=
>∞=
q
RR
Plano
Concave
2
R
1
R
f
2
F1
F
2
C
2n
1
n
2
R1
R f
1
C 2F1
F
2C
2
n
1
n
0
0,0
21
21
=
=
><
q
RR
RR
Equi
Concave
2
R
1
R
f
1F 2F
1C
2
n
1
n
1
,0 21
=
∞=<
q
RR
Plano
Concave
Negative
Meniscus
1
0,0
21
21
>
<
<<
q
RR
RR
2
R
1
R
f
2
F1
F 2C
2n
1
n
1C

90
REFLECTION & REFRACTIONSOLO
http://freepages.genealogy.rootsweb.com/~coddingtons/15763.htm
History of Reflection & Refraction
Reverent Henry Coddington (1799 – 1845) English mathematician and cleric.
He wrote an Elementary Treatise on Optics (1823, 1st
Ed., 1825, 2nd
Ed.). The book
was displayed the interest on Geometrical Optics, but hinted to the acceptance of the
Wave Theory.
Coddington wrote “A System of Optics” in two parts:
1. “A Treatise of Reflection and Refraction of Light” (1829), containing a
thorough investigation of reflection and refraction.
2. “A Treatise on Eye and on Optical Instruments” (1630), where he explained
the theory of construction of various kinds of telescopes and microscopes.
He recommended the ue of the grooved
sphere lens, first described by David
Brewster in 1820 and inuse today as the
“Coddington lens”.
Coddington introduced for lens:
Coddington
Shape Factor:
Coddington
Position Factor:
12
12
rr
rr
q
−
+
=
ss
ss
p
−
+
=
'
'
Coddington Lens
http://www.eyeantiques.com/MicroscopesAndTelescopes/Coddington%20microscope_thick_wood.htm

91
OpticsSOLO
hin Lens Spherical Aberrations
( ) 4
rCrW SpSA =
ven a thin lens and object O on the
ical Axis (OA). A paraxial ray will cross
OA at point I, at a distance s’p from
lens. A general ray, that reaches the lens
distance r from OA, will cross OA at
nt E, at a distance s’r.
( )
( ) ( ) ( ) 





++
−
+
+−++
−−
−= qpnq
n
n
pnn
n
n
fnn
CSp 14
1
2
123
1132
1 22
3
3
where:
Define:
2
R
1
R
1
C
IO
2C
Paraxial
focal plane2
n
1n
s
ps'
E
rs' Long. SA
Lat. SA
φ
Paraxial
Ray
General
Ray
'φ
r
rp ssSALongAberrationSphericalalLongitudin ''. −==
( ) rrp srssSALatAberrationSphericalLateral '/''. −==
We have:

92
OpticsSOLO
12
12
RR
RR
qK
−
+
==
( )
( )
( ) ( ) ( ) 





++
−
+
+−++
−−
−= qpnq
n
n
pnn
n
n
fnn
r
rWSp 14
1
2
123
1132
22
3
3
4
Thin Lens Spherical Aberrations (continue – 1)

93
OpticsSOLO
2
R
1
R
1
C
IO
2C
Paraxial
focal plane2
n
1
n
s
ps'
E
rs' Long. SA
Lat. SA
φ
Paraxial
Ray
General
Ray
'φ
r
12
12
RR
RR
q
−
+
=
F.A. Jenkins & H.E. White, “Fundamentals of Optics”,
4th
Ed., McGraw-Hill, 1976, pg. 157
Lens thickness = 1cm, f = 10cm, n = 1.5, h = 1cm
In Figure we can see a comparison
of the Seidel Third Order Theory
with the ray tracing.

94
OpticsSOLO
We can see that the Thin Lens Spherical Aberration WSp is a parabolic function of the
Coddington Shape Factor q, with the vertex at (qmin,WSp min)
( )
( ) ( ) ( ) 





++
−
+
+−++
−−
−= qpnq
n
n
pnn
n
n
fnn
r
WSp 14
1
2
123
1132
22
3
3
4
Thin Lens Spherical Aberrations (continue -2)
The minimum Spherical Aberration for a given Coddington Position Factor p is obtained
by:
( )
( ) 014
1
2
2
132 3
4
=



++
−
+
−
−=
∂
∂
pnq
n
n
fnn
r
q
W
p
Sp
1
1
2
2
min
+
−
−=
n
n
pq








+
−





−
−= 2
2
3
4
min
2132
p
n
n
n
n
f
r
WSp
The minimum Spherical Aberration is zero for ( )
( )
1
1
2
2
2
>
−
+
=
n
nn
p

95
OpticsSOLO
In order to obtain the radii of the lens for a given focal length f and given Shape Factor
and Position Factor we can perform the following:
Those relations were given by Coddington.
'
2
11
2
s
f
s
f
p −=−= p
f
s
p
f
s
−
=
+
=
1
2
'&
1
2
( )
fRR
n
ss
111
1
'
11
21
=





−−=+
( ) ( )
1
12
&
1
12
21
−
−
=
+
−
=
q
nf
R
q
nf
R
12
12
RR
RR
q
−
+
=
12
1
12
2 2
1&
2
1
RR
R
q
RR
R
q
−
=−
−
=+
( ) ( )12
21
1 RRn
RR
f
−−
=
2
R
1
R
1
C
IO
2
C
Paraxial
focal plane2
n
1
n
s
ps'
E
rs' Long. SA
Lat. SA
φ
Paraxial
Ray
General
Ray
'φ
r

96
OpticsSOLO
hin Lens Coma
( ) ( )
( ) ( ) 



−
+
++
+
=
+==
q
n
n
pn
sfn
xyxh
xyxhCrhChrW CoCoCo
1
1
12
'4
''''
''''cos'';,
2
22
223
θθ
or thin lens the coma factor is given by:
where:we find:
( ) 2
22
2
1
1
12
4
'''
: MAXMAXCoS rq
n
n
pn
fn
h
r
n
sh
CC 



−
+
++==
1
2
3 4
P
Image
Plane
O
SC
SC
ST CC 3=
Coma Blur Spot Shape
Tangential
Coma
Sagittal
Coma

30
'h
'x
'y
( )( ) ( ) ( )222
'2' rRyrRx CoCo =∆+−∆ ( ) MAXCoCo rrr
n
sh
CrR ≤≤= 0
''
: 2
Define:
( ) ( ) ( ) ( )θθ 2cos2
''
cos21
''
''3
''
'
';',''
' 2222
+=+=+=
∂
=∆ r
n
sh
Cr
n
sh
Cyx
n
sh
C
x
hyxW
n
s
x CoCoCo
( ) ( ) θ2sin
''
''2
''
'
';',''
' 2
r
n
sh
Cyx
n
sh
C
y
hyxW
n
s
y CoCo ==
∂
=∆

97
OpticsSOLO
hin Lens
F.A. Jenkins & H.E. White, “Fundamentals of Optics”,
4th
Ed., McGraw-Hill, 1976, pg. 165
Lens thickness = 1cm, f = 10cm, n = 1.5, h = 1cm, y = 2 cm
( ) 2
22
1
1
12
4
'
: MAXS rq
n
n
pn
fn
h
C 





−
+
++=
oma is linear in q
( ) ( )
( )
p
n
nn
qCS
1
112
0
+
−+
−=⇐=
n Figure 800.00 =⇐= qCS
The Spherical Aberration is
arabolic in q
( )
( ) ( ) ( ) 





++
−
+
+−++
−−
−= qpnq
n
n
pnn
n
n
fnn
CSp 14
1
2
123
1132
1 22
3
3
1
1
2
2
min
+
−
−=
n
n
pq








+
−





−
−= 2
2
3min
2132
1
p
n
n
n
n
f
CSp
In Figure
714.0min =q

98
OpticsSOLO

99
OpticsSOLO

100
SOLO Optics
Chromatic Aberrations arise in
Polychromatic IR Systems because
the material index n is actually
a function of frequency. Rays at
different frequencies will traverse
an optical system along different paths.

101
SOLO Optics

102
SOLO Optics
Chester Moor Hall (1704 – 1771) designed in secrecy the achromatic lens.
He experienced with different kinds of glass until he found in 1729 a combination of
convex component formed from crown glass with a concave component formed from
flint glass, but he didn’t request for a patent.
http://microscopy.fsu.edu/optics/timeline/people/dollond.html
In 1750 John Dollond learned from George Bass on Hall achromatic lens and designed
his own lenses, build some telescopes and urged by his son
Peter (1739 – 1820) applied for a patent.
Born & Wolf,”Principles of Optics”, 5th
Ed.,p.176
In 1733 he built several telescopes with apertures of 2.5” and 20”. To keep secrecy
Hall ordered the two components from different opticians in London, but they
subcontract the same glass grinder named George Bass, who, on finding that both
Lenses were from the same customer and had one radius in common, placed them
in contact and saw that the image is free of color.
The other London opticians objected and
took the case to court, bringing Moore-Hall
as a witness. The court agree that Moore-
Hall was the inventor, but the judge Lord
Camden, ruled in favor of Dollond saying:”It
is not the person who locked up his invention
in the scritoire that ought to profit by a
patent for such invention, but he who
brought it forth for the benefit of the public”

103
SOLO Optics
Every piece of glass will separate white light into a spectrum
given the appropriate angle. This is called dispersion. Some
types of glasses such as flint glasses have a high level of
dispersion and are great for making prisms. Crown glass
produces less dispersion for light entering the same angle as
flint, and is much more suited for lenses. Chromatic aberration
occurs when the shorter wavelength light (blue) is bent more
than the longer wavelength (red). So a lens that suffers from
chromatic aberration will have a different focal length for each
color
To make an achromat, two lenses are put together to work as a
group called a doublet. A positive (convex) lens made of high
quality crown glass is combined with a weaker negative
(concave) lens that is made of flint glass. The result is that the
positive lens controls the focal length of the doublet, while the
negative lens is the aberration control. The negative lens is of
much weaker strength than the positive, but has higher
dispersion. This brings the blue and the red light back together
(B). However, the green light remains uncorrected (A),
producing a secondary spectrum consisting of the green and
blue-red rays. The distance between the green focal point and
the blue-red focal point indicates the quality of the achromat.
Typically, most achromats yield about 75 to 80 % of their
numerical aperture with practical resolution

104
SOLO Optics
In addition, to the correction for the chromatic aberration the
achromat is corrected for spherical aberration, but just for green
light. The Illustration shows how the green light is corrected to a
single focal length (A), while the blue-red (purple) is still
uncorrected with respect to spherical aberration. This illustrates the
fact that spherical aberration has to be corrected for each color,
called spherochromatism. The effect of the blue and red
spherochromatism failure is minimized by the fact that human
perception of the blue and red color is very weak with respect to
green, especially in dim light. So the color halos will be hardly
noticeable. However, in photomicroscopy, the film is much more
sensitive to blue light, which would produce a fuzzy image. So
achromats that are used for photography will have a green filter
placed in the optical path.

105
SOLO Optics
As the optician's understanding of optical aberrations improved
they were able to engineer achromats with shorter and shorter
secondary spectrums. They were able to do this by using special
types of glass call flourite. If the two spectra are brought very
close together the lens is said to be a semi-apochromat or flour.
However, to finally get the two spectra to merge, a third optical
element is needed. The resulting triplet is called an apochromat.
These lenses are at the pinnacle of the optical family, and their
quality and price reflect that. The apochromat lenses are
corrected for chromatic aberration in all three colors of light and
corrected for spherical aberration in red and blue. Unlike the
achromat the green light has the least amount of correction,
though it is still very good. The beauty of the apochromat is that
virtually the entire numerical aperture is corrected, resulting in a
resolution that achieves what is theoretically possible as predicted
by Abbe equation.

106
SOLO Optics
With two lenses (n1, f1), (n2,f2) separated by a distance
d we found
2121
111
ff
d
fff
−+=
Let use ( ) ( ) 222111 1/1&1/1 ρρ −=−= nfnf
We have
( ) ( ) ( ) ( ) 22112211 1111
1
ρρρρ −−−−+−= nndnn
f
nF – blue index produced by hydrogen
wavelength 486.1 nm.
nC – red index produced by hydrogen
nd – yellow index produced by helium
Assume that for two colors red and blue we have fR = fB
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) 22112211
22112211
1111
1111
1
ρρρρ
ρρρρ
−−−−+−=
−−−−+−=
FFFF
CCCC
nndnn
nndnn
f

107
SOLO Optics
Let analyze the case d = 0 (the two lenses are in contact)
We have
( ) ( ) ( ) ( ) 22112211 1111
1
ρρρρ −+−=−+−= FFCC nnnn
f
( )
( )
( )
( )1
1
1
1
1
2
1
2
2
1
−
−
−=
−
−
−=
F
F
C
C
n
n
n
n
ρ
ρ ( )
( )CF
CF
nn
nn
11
22
2
1
−
−
−=
ρ
ρ
For the yellow light (roughly the midway between
the blue and red extremes) the compound lens will
have the focus fY:
( ) ( )
YY f
d
f
d
Y
nn
f
21 /1
22
/1
11 11
1
ρρ −+−= ( )
( ) Y
Y
d
d
f
f
n
n
1
2
1
2
2
1
1
1
−
−
=
ρ
ρ
( )
( )
( )
( )
( ) ( )
( ) ( )1/
1/
1
1
111
222
2
1
11
22
1
2
−−
−−
−=
−
−
−
−
−=
dCF
dCF
d
d
CF
CF
Y
Y
nnn
nnn
n
n
nn
nn
f
f

108
SOLO Optics
( ) ( )
( ) ( )1/
1/
111
222
1
2
−−
−−
−=
dCF
dCF
Y
Y
nnn
nnn
f
f
The quantities are called
Dispersive Powers of the two materials forming the lenses.
( )
( )
( )
( )1
&
1 2
22
1
11
−
−
−
−
d
CF
d
CF
n
nn
n
nn
Their inverses are called
V-numbers or Abbe numbers.
( )
( )
( )
( )CF
d
CF
d
nn
n
V
nn
n
V
22
2
2
11
1
1
1
&
1
−
−
=
−
−
=

109
OpticsSOLO
To define glass we need to know more than one index of refraction.
In general we choose the indexes of refraction of three colors:
nF – blue index produced by hydrogen
nC – red index produced by hydrogen
Define:
nF – nC - mean dispersion
CF
d
nn
n
v
−
−
=
1
- Abbe’s Number or v value or V-number
Crowns: glasses of low dispersion (nF – nC small and V-number above 55)
Flints: glasses of high dispersion (nF – nC high and V-number bellow 50)
Fraunhofer
line
color Wavelength
(nm)
Spectacle Crown
C - 1
Extra Dense Flint
EDF - 3
F
d
C
Blue
Yellow
Red
486.1
587.6
656.3
1.5293
1.5230
1.5204
1.7378
1.7200
1.7130V - number
58.8 29.0

110
OpticsSOLO
Refractive indices and Abbe’s numbers of various glass materials

111
OpticsSOLO
Camera Lenses
Hecht, “Optics”
Addison Wesley,
4th Ed., 2002,
pp.218

112
OpticsSOLO
Camera
Lenses
Born & Wolfe, “Principle of Optics”,
Pergamon Press, 5th
Ed., pp.236-237

113
SOLO
References
Lens Design
1. Kingslake, R., “Lens Design Fundamentals”, Academic Press, N.Y., 1978
6. Geary, J. M., “Introduction to Lens Design with Practical ZEMAX Examples”,
Willmann-Bell, Inc., 2002
5. Laikin, M., “Lens Design”, Marcel Dekker, N.Y., 1991
2. Malacara, D., Ed., “Optical Shop Testing”, John Wiley & Sons, N.Y., 1978
7. Kidger, M. J., “Fundamental Optical Design”, SPIE Press., 2002
3. Kingslake, R., “Optical System Design”, Academic Press, N.Y., 1983
4. O’Shea, D.,C., “Elements of Modern Optical Design”, John Wiley & Sons, N.Y.,
1985

114
SOLO
References
OPTICS
1. Waldman, G., Wootton, J., “Electro-Optical Systems Performance Modeling”,
Artech House, Boston, London, 1993
2. Wolfe, W.L., Zissis, G.J., “The Infrared Handbook”, IRIA Center,
Environmental Research Institute of Michigan, Office of Naval Research, 1978
3. “The Infrared & Electro-Optical Systems Handbook”, Vol. 1-7
4. Spiro, I.J., Schlessinger, M., “The Infrared Technology Fundamentals”,
Marcel Dekker, Inc., 1989

115
SOLO
References
[1] M. Born, E. Wolf, “Principle of Optics – Electromagnetic Theory of Propagation,
Interference and Diffraction of Light”, 6th
Ed., Pergamon Press, 1980,
[2] C.C. Davis, “Laser and Electro-Optics”, Cambridge University Press, 1996,
OPTICS

116
SOLO
References
Foundation of Geometrical Optics
[3] E.Hecht, A. Zajac, “Optics ”, 3th
Ed., Addison Wesley Publishing Company, 1997,
[4] M.V. Klein, T.E. Furtak, “Optics ”, 2nd
Ed., John Wiley & Sons, 1986

117
OPTICSSOLO
References Optics Polarization
A. Yariv, P. Yeh, “Optical Waves in Crystals”, John Wiley & Sons, 1984
M. Born, E. Wolf, “Principles of Optics”, Pergamon Press,6th
Ed., 1980
E. Hecht, A. Zajac, “Optics”, Addison-Wesley, 1979, Ch.8
C.C. Davis, “Lasers and Electro-Optics”, Cambridge University Press, 1996
G.R. Fowles, “Introduction to Modern Optics”,2nd
Ed., Dover, 1975, Ch.2
M.V.Klein, T.E. Furtak, “Optics”, 2nd Ed., John Wiley & Sons, 1986
http://en.wikipedia.org/wiki/Polarization
W.C.Elmore, M.A. Heald, “Physics of Waves”, Dover Publications, 1969
E. Collett, “Polarization Light in Fiber Optics”, PolaWave Group, 2003
W. Swindell, Ed., “Polarization Light”, Benchmark Papers in Optics, V.1,
Dowden, Hutchinson & Ross, Inc., 1975
http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi)

January 4, 2015 118
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 – 2013
Stanford University
1983 – 1986 PhD AA

Lens

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Viewers also liked

Viewers also liked (20)

Similar to Lens

Similar to Lens (20)

More from Solo Hermelin

More from Solo Hermelin (20)

Recently uploaded

Recently uploaded (20)

Lens

Editor's Notes