Lens
-
Upload
solo-hermelin -
Category
Science
-
view
797 -
download
5
Transcript of 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 PrincipleDerivation 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 instraight lines.
2. 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 thesurface normal is equal to the angle between thereflected ray and the surface normal:
3. Law of Refraction
An optical disturbance moving from a medium ofrefractive index n1 into a medium of refractive indexn2 will have its incident ray, the surface normal betweenthe 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 thereflected 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
2121 =⋅
− rd
sd
rdn
sd
rdn rayray
In this case we have and21 nn =( ) ( ) ( ) 0ˆˆ
2121 =⋅−=⋅
− 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
2121 =⋅
− rd
sd
rdn
sd
rdn rayray
where is on the boundary between the two regions andrd ( ) ( )
sd
rds
sd
rds 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 −21ˆ −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 itsoriginal direction but with a lateral displacement d.
Optics
( ) ( )irriri lld φφφφφφ cossincossinsin −=−=
r
tl
φcos=
−=
r
iritd
φφφφ
cos
cossinsin
ir nn φφ sinsin 0=Snell’s Law
−=
n
ntd
r
ii
0
cos
cos1sin
φφφ
For small anglesiφ
−≈
n
ntd i
01φ
8
SOLO
Plane-Parallel Plate (continue – 1(
Two rays traverse a glass plate with parallel surfaces and emerge parallel to theiroriginal direction but with a lateral displacement l.
Optics
( ) ( )irriri lld φφφφφφ cossincossinsin −=−=
r
tl
φcos=
−=
r
iritd
φφφφ
cos
cossinsin
ir nn φφ sinsin 0=Snell’s Law
−=
n
ntd
r
ii
0
cos
cos1sin
φφφ
−==
r
i
i n
nt
dl
φφ
φ cos
cos1
sin0 For small anglesiφ
−≈
n
ntl 01
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
Optics SOLO
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
1sin 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
Optics SOLO
Prisms
( )[ ] αθαθαθδ −−−+= −1
2/1
1
221
1 sincossinsinsin iii n
12
Optics SOLO
Prisms
( )[ ] αθαθαθδ −−−+= −1
2/1
1
221
1 sincossinsinsin iii n
αθθδ −+= 21 ti
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
θα
θθ
θδ
Taking the differentials of Snell’s Law equations
22 sinsin tin θθ =
11 sinsin ti n θθ =
2222 coscos iitt dnd θθθθ =
1111 coscos ttii dnd θθθθ =
Dividing the equations1
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
θθ
θθ
θθ
θθ
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
θθ
θθ
θθ
θθ
−−
=−−
=−−
=−−
11
2 −=i
t
d
d
θθ
21 it θθα +=
12
1 −=i
t
d
d
θθ
2
2
1
2
2
2
1
2
cos
cos
cos
cos
i
t
t
i
θθ
θθ
= 21 ti θθ =1≠n
13
Optics SOLO
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 tin 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
Optics SOLO
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 +=
αθθδ −+= 21 ti
21 ti θθ =
mδδ =2/1 αθ =t
αθδ −= 12 im( ) 2/1 αδθ += mi
( )[ ]2/sin
2/sin
ααδ += mn
Find the angle θi1 for which the deviation angle δ is minimal; i.e. δm (continue – 2(.
15
Optics SOLO
Prisms
The refractive index of transparent substances varies with the wavelength λ.
( )[ ]{ } αθαθλαθδ −−−+= −1
2/1
1
221
1 sincossinsinsin iii n
16
Optics SOLO
http://physics.nad.ru/Physics/English/index.htm
Prisms
Color λ0 (nm( υ [THz]
RedOrangeYellowGreenBlueViolet
780 - 622622 - 597597 - 577577 - 492492 - 455455 - 390
384 – 482482 – 503503 – 520520 – 610610 – 659659 - 769
1 nm = 10-9m, 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 thatthe white light consisted of a mixture of various colors and the diffraction was color dependent.
Isaac Newton1542 - 1727
17
SOLO
Dispersing PrismsPellin-Broca Prism
Abbe Prism
Ernst KarlAbbe
1840-1905
At Pellin-Broca Prism an incident ray of wavelength λ passes the prism at a dispersing angle of 90°. Because the dispersing angleis a function of wavelengththe ray at other wavelengthsexit at different angles.By rotating the prism aroundan axis normal to the pagedifferent 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
190 iABE θ+=∠
290 tADE 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 ∠−= 901θCDGi ∠−= 902θ21 it θθ =
202 sinsin ti nn θθ =Snell’s Law
Snell’s Law 110 sinsin ti nn θθ = 21 ti θθ = αθδ += 12 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 ofrefraction 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 lighttransmitting the s-polarized component. The optical axis of the prism material isperpendicular to the plane of the diagram.
A Glan-Taylor prism reflects polarized lightat an internal air-gap, transmitting onlythe 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 areperpendicular to the plane of the diagram.
Optics - Prisms
24
Optics SOLO
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
Optics SOLO
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 beseen by the optical system in order to control the quality of the image.
A.S. F.S.
IΣ
Aperture and Field Stops
Imageplane
Hecht"Optics"
26
Optics SOLO
Definitions (continue – 2(
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.
Entrancepupil
Exitpupil
A.S.
IΣ
xpEnpE
ChiefRay
Entrance and Exit pupils
Imageplane
MarginalRay
Hecht"Optics"
EntrancepupilExit
pupil
A.S. IΣ
xpE
npE
ChiefRay
Imageplane
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
Optics SOLO
Definitions (continue – 2(
Entrancepupil
Exitpupil
A.S.
IΣ
ChiefRay
MarginalRay
Exp Enp
Imageplane
Hecht"Optics"
Pupil and stops for a three - lens system
28
Optics SOLO
Definitions (continue – 1(
Principal Planes: the two planes defined by the intersection of the parallel incident raysentering an optical system with the rays converging to the focal pointsafter 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
Optics SOLO
Definitions (continue – 3(
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 bundlesmultiplied 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
1sin
ff
DuNA =
==
30
Optics SOLO
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
Optics SOLO
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
Optics SOLO
Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Fermat’s Principle
Karl Friederich Gauss1777-1855
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 −+−+=
( ) ( )[ ] ( ) ( )[ ] 2/1222/122 cos'2''cos2 ββ RsRRsRnRsRRsRnpathOptical −+−+⋅++−++⋅=Therefore
According to Fermat’s Principle when the point Tmoves on the spherical surface we must have ( )
0=βd
pathOpticald
( ) ( ) ( )0
'
sin''sin =−⋅−+⋅=l
RsRn
l
RsRn
d
pathOpticald βββ
from which we obtain
⋅−⋅=+
l
sn
l
sn
Rl
n
l
n
'
''1
'
'
For small α and β we have ''& slsl ≈≈
and we obtainR
nn
s
n
s
n −=+ '
'
'
Gaussian Formula for a Single Spherical Surface
33
Optics SOLO
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 raysMT and TM’ are paraxial theangles 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
hnn
s
hn
s
hn −=+ '
''
or ( )
r
nn
s
n
s
n −=+ '
'
'
Gaussian Formula for a Single Spherical SurfaceKarl Friederich Gauss
1777-1855
Willebrord van Roijen Snell
1580-1626
( )
φφφφφ
φ
≈+++=
O
53
!5!3sin
34
Optics SOLO
Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Snell’s Law
for s → ∞ the incoming rays are parallel to opticalaxis and they will refract passing trough a commonpoint called the focus F’.
( )r
nn
s
n
s
n −=+ '
'
'
( )r
nn
f
nn −=+∞
'
'
'r
nn
nf
−='
''
for s’ → ∞ the refracting rays are parallel to opticalaxis and therefore the incoming rays passes trough a common point called the focus F.
( )r
nnn
f
n −=∞
+ '' rnn
nf
−='
'' n
n
f
f =
35
Optics SOLO
Derivation of Lens Makers’ Formula
We have a lens made of twospherical surfaces of radiuses r1
and r2 and a refractive index n’,separating two media havingrefraction indices n a and n”. Ray MT1 is refracted by the firstspherical surface (if no secondsurface 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 objectfor the second surface (negative sign) we have
21 '' ss ≈21 '' ss −≈
( )221
'"
"
"
'
'
r
nn
s
n
s
n −=+−
36
Optics SOLO
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” ands”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
Optics SOLO
Derivation of Lens Makers’ Formula (continue – 2)
If the media on both sides of the lens is the same n = n”.
−
−=+
21
111
'
"
11
rrn
n
ss
−
−==
21
111
'1
"
1
rrn
n
ff
Therefore
"
11
"
11
ffss==+
Lens Makers’ Formula
38
Optics SOLO
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 toOptical axis (paraxial).
Karl Friederich Gauss1777-1855
Derivation of Lens Formula
From the similarity of the trianglesand using the convention:
( )''
''~'
f
y
s
yyTAFTSQ =−+⇒∆∆
Lens Formula in Gaussian form
( ) ( )f
y
s
yyFASQTS
''~
−=−+⇒∆∆
( ) 0' >− y
Sum of the equations: ( ) ( ) ( )
'
'
'
''
f
y
f
y
s
yy
s
yy +−=−++−+
since f = f’ fss
1
'
11 =+
( )
φφφφφ
φ
≈+++=
O
53
!5!3sin
39
Optics SOLO
First Order, Paraxial or Gaussian Optics (continue – 1)
Gauss explanation can be extended to the first order approximationto any optical system.
Karl Friederich Gauss1777-1855
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
Axisy
40
Optics SOLO
Derivation of Lens Formula (continue)
From the similarity of the trianglesand using the convention:
( )f
y
x
yFASQMF
'~
−=⇒∆∆
Lens Formula in Newton’s form
( )f
y
x
yQMFTAF =−⇒∆∆
'
''''~'
( ) 0' >− y
Multiplication of the equations: ( ) ( )
2
'
'
'
f
yy
xx
yy −⋅=⋅−⋅
or 2' fxx =⋅
Isaac Newton1643-1727
First Order, Paraxial or Gaussian Optics (continue – 2)
Published by Newton in “Opticks” 1710
'y
s 's
M’A F’M
T
F
'ffx 'x
Q
Q’'y
y
S
Axisy
41
Optics SOLO
Derivation of Lens Formula (continue)
First Order, Paraxial or Gaussian Optics (continue – 3)
Lateral or Transverse Magnification
f
x
x
f
s
s
h
hmT
''' −=−=−==
Quantity (+) sign (-) signs 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
Axisy
42
Optics SOLO
Derivation of Lens Formula (Summary)
If the media on both sides of the lens is the same n = n”.
−
−=+
21
111
'
"
11
rrn
n
ss
−
−==
21
111
'1
"
1
rrn
n
ff
Therefore
"
11
"
11
ffss==+
Lens Makers’ Formula
f
x
x
f
s
s
h
hmT
''' −=−=−==
Gauss’ Lens Formula
Magnification
43
Optics SOLO
44
Optics SOLO
45
Optics SOLO
Ray Tracing
F CO
I
Object Virtual
Image
ConvexMirror
R/2 R/2R
FCO
I
Object
RealImage
ConcaveMirror
Ray Tracing is a graphically implementation of paralax ray analysis. The constructiondoesn’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 thefollowing three rays:
1. A ray parallel to the axis that isreflected through F’.
2. A ray through F that is reflectedparallel to the axis.
3. A ray through the center C of thelens that remains undeviated andundisplaced (for thin lens).
46
Optics SOLO
47
Optics SOLO
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 distancer1 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
rM
r
r
DC
BA
r
r
=
DC
BAMwhere ray transfer matrix
When the media to the left of the input planeand to the right of the output plane have thesame refractive index, we have:
1det =⋅−⋅= CBDAM
48
Optics SOLO
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 dM
MediumOpticalUniform
Planar Interface Between Two Different Media
12 rr =
'' 1
2
12
12
rn
nr
rr
=
=
Apply Snell’s Law: 2211 sinsin φφ nn =
paraxial assumption: φφφφ ≈=⇒≈ tan'sin r
From Snell’s Law: '' 1
2
12 r
n
nr =
=
21 /0
01
nnM
InterfacePlanar
1det2
1 ≠=n
nM
InterfacePlanar
1det =MediumOpticalUniformM
The focal length of this system is infinite and it hasnot specific principal planes.
49
Optics SOLO
Matrix Formulation (continue -2)
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
nnM
mediaentranceslabmediaexit
SlabSidedParallel
=
10
/1 21 nndM
SlabSidedParallel
1det =SlabSidedParallelM
50
Optics SOLO
Matrix Formulation (continue -3)
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
nnMInterfaceSpherical 1det
2
1 ≠=n
nM
InterfaceSpherical
12
11
'
'
φφ
+=+=
rr
ri From the Figure:
( ) ( )122111 '' φφ +=+ rnrn
111 / Rr=φ
( )12
121
2
112
''
Rn
rnn
n
rnr
−+=
( )1
12
11
1122
12
''
n
rn
Rn
rnnr
rr
+−=
=
( )1
121 : R
nnD
−=where: Power of the surface If R1 is given in meters D1 gives diopters
51
Optics SOLO
Matrix Formulation (continue -4)
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
Dr
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
Dr
r
Therefore:
−
−
−=
−
−=
'
101
'
01
10
101
' 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
nd
n
Dd
n
n
n
Dr
r
n
n
n
Dd
n
n
n
Dr
r
−
−
+−
−
=
2
2
21
21
1
21
2
1
2
1
1
1
n
Dd
nn
DDd
n
DD
n
nd
n
Dd
MLensThick
( )2
212 R
nnD
−=
( )1
121 : R
nnD
−=
−
−
+
−−
=−
2
1
21
21
1
21
2
1
2
2
1
1
1
n
Dd
nn
DDd
n
DD
n
nd
n
Dd
MLensThick
1det =LensThickM
or21 DD ⇔
52
Optics SOLO
Matrix Formulation (continue -5)Thick Lens (continue -1) Let use the second Figure where Ray 2 is parallelto Symmetry Axis of the Optical System that is refractedtrough the Second Focal Point.
−
−
+−
−
=
'1
1
' 1
1
2
2
21
21
1
21
2
1
2
1
4
4
r
r
n
Dd
nn
DDd
n
DD
n
nd
n
Dd
r
r We found:
2141 /'&0' frrr −==Ray 2:
By substituting Ray2 parameters we obtain:
1
2
1
21
21
1
214
1' r
fr
nn
DDd
n
DDr −=
−+−=
1
21
21
1
212
−
−
+=
nn
DDd
n
DDf
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
Dd
nn
DDd
n
DD
n
nd
n
Dd
r
r
4
1
4
21
21
1
211
1' r
fr
nn
DDd
n
DDr −=
−+=
2
1
21
21
1
211 f
nn
DDd
n
DDf −=
−
+=
−
53
Optics SOLO
Matrix Formulation (continue -6)
Thin Lens For thick lens we found
−
−
+−
−
=
2
2
21
21
1
21
2
1
2
1
1
1
n
Dd
nn
DDd
n
DD
n
nd
n
Dd
MLensThick
−+=
21
21
1
211
nn
DDd
n
DD
f
For thin lens we can assume d = 0 and obtain
−=
11
01
f
MLensThin
1
211
n
DD
f
+= ( )2
212 R
nnD
−=( )1
121 : R
nnD
−=
−
−=+=
211
2
1
21 111
1
RRn
n
n
DD
f
54
Optics SOLO
Matrix Formulation (continue -7)
Thin Lens (continue – 1)
For a biconvex lens we have R2 negative
+
−=
211
2 111
1
RRn
n
f
For a biconcave lens we have R1 negative
+
−−=
211
2 111
1
RRn
n
f
−=
11
01
f
MLensThin
55
Optics SOLO
Matrix Formulation (continue -8)
A Length of Uniform Medium Plus a Thin Lens
−−=
−==
+f
d
f
dd
f
MMMMediumUniform
LensThin
LensThinMediumUniform 1
1
1
10
1
11
01
Combination of Two Thin Lenses
+−−−+−−
−+−=
−−
−−==
21
21
2
2
2
1
1
1
21
2
21
1
2121
2
2
1
1
1
1
22
2
111
1
11
1
11
1
1122
ff
dd
f
d
f
d
f
d
ff
d
ff
f
dddd
f
d
f
d
f
d
f
d
f
d
MMMMMdMedium
UniformfLens
ThindMedium
UniformfLens
Thin
LensesThinTwo
The Focal Length of the Combination of Two Thin Lenses is:
21
2
21
111
ff
d
fff−+= Return to
Chromatic Aberration
56
Optics SOLO
Real Imaging Systems – Aberrations
Departures from the idealized conditions of Gaussian Optics in a real Optical System arecalled 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
Optics SOLO
Real Imaging Systems – Aberrations (continue – 5)
58
Optics SOLO
Real Imaging Systems – Aberrations (continue – 5)
59
Optics SOLO
Real Imaging Systems – Aberrations (continue – 5)
60
Optics SOLO
Real Imaging Systems – Aberrations
61
Optics SOLO
Real Imaging Systems – Aberrations
62
Optics SOLO
Real Imaging Systems – Aberrations (continue – 1)
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 aGeneral Ray P0 Q P0’.
The Wave Aberration is defined asthe 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
Optics SOLO
Real Imaging Systems – Aberrations (continue – 2)
Seidel Aberrations (continue – 1)
−−=−−=
2
222 11
R
rRrRRz
Define:
( )2
2
112
2
R
rxxf
R
rx
−=+=−=
( ) ( ) 2/112
1' −+= xxf
( ) ( ) 2/314
1" −+−= xxf ( ) ( ) 2/51
8
3'" −+−= xxf
Develop f (x) in a Taylor series ( ) ( ) ( ) ( ) ( ) ++++= 0"'6
0"2
0'1
032
fx
fx
fx
fxf
1168
1132
<++−+=+ xxx
xx
RrR
r
R
r
R
r
R
rRz <+++=
−−=
5
6
3
42
2
2
168211
On Axis Point Object
From the Figure:
( ) 222 rzRR +−= 02 22 =+− rRzz
64
Optics SOLO
Real Imaging Systems – Aberrations (continue – 3)
Seidel Aberrations (continue – 2)
From the Figure:
( )[ ] [ ]( )[ ] ( ) 2/1
2
2/122
2/12222/122
0
212
222
−+=+−=
++−=+−=−=
zs
sRsszsR
rsszzrszQP
rzRz
( ) ( )
+−−−+−≈
<++−+=+
24
2
2
1168
11
2
11
32
zs
sRz
s
sRs
xxx
xx
( ) ( )
+
+−−
+−+−=
+≈
2
3
42
4
2
3
42
2
82
822
1
821
3
42
R
r
R
r
s
sR
R
r
R
r
s
sRs
R
r
R
rz
( )[ ] +
−+
−+
−+−≈+−= 4
2
2
22/122
0
11
8
111
8
111
2
1r
sRssRRr
sRsrszQP
( )[ ] +
−+
−+
−+≈+−= 4
2
2
22/122
0
1
'
1
'8
11
'
1
8
11
'
1
2
1''' r
RssRsRr
RssrzsPQ
In the same way:
On Axis Point Object
65
Optics SOLO
Real Imaging Systems – Aberrations (continue – 4)
Seidel Aberrations (continue – 3)
+
−+
−+
−+−≈ 4
2
2
2
0
11
8
111
8
111
2
1r
sRssRRr
sRsQP
+
−+
−+
−+≈ 4
2
2
2
0
1
'
1
'8
11
'
1
8
11
'
1
2
1'' r
RssRsRr
RssPQ
Therefore:
( ) ( ) ( )4
22
2
42
000
11
'
11
'
'
8
1
82
'
'
'
''''
rsRs
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
1rar
sRs
n
sRs
nrW S=
−−
−−=
On Axis Point Object
66
Optics SOLO
Real Imaging Systems – Aberrations (continue – 5)
Seidel Aberrations (continue – 4)
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 exit pupil ExP are located at the refracting surface. Using ''~ 00 CPPCPP ∆∆ the transverse magnification
( ) ( )
s
n
s
nnn
s
s
n
s
nnn
s
Rs
Rs
h
hM t
−
−+−
−
−−
=+−
−=−
=
'
'''
''
'
''
( ) sn
sn
nns
snn
nns
snn
M t −=
−+−
+−−=
'
'
''
'
''
'
67
Optics SOLO
Real Imaging Systems – Aberrations (continue – 6)
Seidel Aberrations (continue – 5)
Off-Axis Point Object
The Wave Aberration is defined as the difference in the optical path lengths between the General Ray and the Undeviated Ray.
( ) [ ] [ ][ ] [ ]{ } [ ] [ ]{ }
( )404
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
00 hbh
Rs
RPP
CP
CVVV =
−=≈
Rs
Rb
−='
:
−−
−−=
2211
'
11
'
'
8
1
sRs
n
sRs
naS
68
Optics SOLO
Real Imaging Systems – Aberrations (continue – 7)
Seidel Aberrations (continue – 6)
Off-Axis Point Object
Wave Aberration.
( ) [ ] [ ] ( )404'' VVVQaPVPPQPQW S −=−=
Define the polar coordinate (r,θ) of the projection of Q in the plane of exit pupil, withV0 at the origin.
θθ cos'2'cos2 222
0
2
0
22
hbrhbrVVrVVrVQ ++=++=
'0 hbVV =
( ) [ ] [ ] ( )( )[ ]442222
4
0
4
'cos'2'
''
hbhbrhbra
VVVQaPVPPQPQW
S
S
−++=
−=−=
θ( ) ( )θθθθ cos'4'2cos'4cos'4';, 33222222234 rhbrhbrhbrhbrahrW S ++++=
69
Optics SOLO
Real Imaging Systems – Aberrations (continue – 8)
Seidel Aberrations (continue – 7)
General Optical Systems
( ) θθθθ cos''cos'cos'';, 33222222234 rhbCrhbCrhbCrhbCrChrW DiFCAsCoSp ++++=
A General Optical Systems has more than on Reflecting orRefracting surface. The image of one surface acts as anobject 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 Coefficient SpC
2. Coma CoefficientCoC
3. Astigmatism Coefficient AsC
4. Field Curvature Coefficient FCC
5. Distortion Coefficient DiC
where:
70
Optics SOLO
Real Imaging Systems – Aberrations (continue – 7)
Seidel Aberrations (continue – 6)
( ) ( )θθθθ cos'4'2cos'4cos'4';, 33222222234 rhbrhbrhbrhbrahrW S ++++=
71
Optics SOLO
Real Imaging Systems – Aberrations (continue – 7)
Seidel Aberrations (continue – 6)
( ) θθθθ cos''cos'cos'';, 32222234 rhCrhCrhCrhCrChrW DiFCAsCoSp ++++=
72
Optics SOLO
Real Imaging Systems – Aberrations (continue – 9)
Seidel Aberrations (continue – 8)
nWPP TR /=
Assume that P’ is the image of P.
The point PT is on the Exit Pupil (Exp) and on theTrue Wave Front (TWF) that propagates toward P’.This True Wave Front is not a sphere because of theAberration. Without the aberration the wave front would be the Reference Sphere (RS) with radius PRP.
W (x’,y’;h’) - wave aberrationn - 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 onRS and TWF, respectively, and on a ray closeto PRPT ray, converging to P’, the image of P.
lPPPP TRTR ∆+=''
73
Optics SOLO
Real Imaging Systems – Aberrations (continue – 9)
Seidel Aberrations (continue – 8)
( )'
';','
'
''
x
hyxW
n
Lx
∂∂=∆
( )'
';','
'
''
y
hyxW
n
Ly
∂∂=∆
θθ
sin'
cos'
ry
rx
==
( ) nhyxWPP TR /';','= lPPPP TRTR ∆+=''
α=∆∆=
∆−=
∂∂
→∆→∆ r
l
r
PPPP
x
W
n r
TRTR
r 00lim
''lim
1
x
W
n
LLr
∂∂==∆ '
'α
74
Optics SOLO
Real Imaging Systems – Aberrations (continue – 1)
1. Spherical Aberrations
( )( ) ( )';','''
';,222
4
hyxWyxC
rChrW
SpSp
SpSp
=+=
=θ
( )'
'
'4
'
';','
'
'' 2xrC
n
L
x
hyxW
n
Lx Sp=
∂=∆
( )'
'
'4
'
';','
'
'' 2 yrC
n
L
y
hyxW
n
Ly Sp=
∂=∆
To Update
( ) ( )[ ] 32/122
'
'4'' rCn
Lyxr Sp=∆+∆=∆
Consider only the Spherical Wave Aberration Function
The Spherical Wave Aberration is aCircle in the Image Plane
75
Optics SOLO
Real Imaging Systems – Aberrations (continue – 2)
2. ComaAssume an object point outside the Optical Axis.
Meridional (Tangential) plane isthe plane defined by the object point and the Optical Axis.
Sagittal plane is the plane normal toMeridional plane that contains theChief Ray passing through theObject point.
76
Optics SOLO
Real Imaging Systems – Aberrations (continue – 2)
2. ComaConsider only the Coma Wave Aberration Function
( ) ( ) ''''cos'';, 223 xyxhCrhChrW CoCoCo +== θθ
( ) ( ) ( ) ( )θθ 2cos2'
''cos21
'
''''3
'
''
'
';','
'
'' 2222 +=+=+=
∂=∆ r
n
LhCr
n
LhCyx
n
LhC
x
hyxW
n
Lx CoCoCo
( ) ( ) θ2sin'
''''2
'
''
'
';','
'
'' 2r
n
LhCyx
n
LhC
y
hyxW
n
Ly CoCo ==
∂=∆
1
'
'''
2
'
'''
2
2
2
2
=
∆+
−∆
rn
LhC
y
rn
LhC
x
CoCo
( )( ) ( ) ( ) 222 '2' rRyrRx CoCo =∆+−∆
( ) 2
'
'': r
n
LhCrR CoCo =
77
Optics SOLO
Real Imaging Systems – Aberrations (continue – 2)
2. ComaWe obtained
2
'
'': MAXCoS r
n
LhCC =
( )( ) ( ) ( ) 222 '2' rRyrRx CoCo =∆+−∆
( ) MAXCoCo rrrn
LhCrR ≤≤= 0
'
'': 2
Define:
1
2
3 4
P
ImagePlane
O
SC
SC
ST CC 3=
Coma Blur Spot Shape
TangentialComa
SagittalComa
30
'h
'x
'y
78
Optics SOLO
Real Imaging Systems – Aberrations (continue – 2)
Graphical Explanation of Coma Blur
79
Optics SOLO
Real Imaging Systems – Aberrations (continue – 2)
Graphical Explanation of Coma Blur (continue – 1)
81
Optics SOLO
Real Imaging Systems – Aberrations (continue – 3)
3. Astigmatism
82
Optics SOLO
Real Imaging Systems – Aberrations (continue – 3)
3. Astigmatism
83
Optics SOLO
Real Imaging Systems – Aberrations (continue – 3)
3. Astigmatism
84
Optics SOLO
Real Imaging Systems – Aberrations (continue – 4)
4. Field Curvature
85
Optics SOLO
Real Imaging Systems – Aberrations (continue – 5)
5. Distortion
86
Optics SOLO
Real Imaging Systems – Aberrations (continue – 8)
Seidel Aberrations (continue – 7)
Thin Lens Aberrations
( ) 2222234 'cos'cos'';, rhCrhCrhCrChrW FCAsCoSp +++= θθθ
Given a thin lens formed by twosurfaces with radiuses r1 and r2
with centers C1 and C2. PP0 is the object, P”P”0 is the Gaussian image formed by the first surface,P’P’0 is the image of virtual objectP’P”0 of the second surface.
( ) ( ) ( ) ( )
++
−++−++
−−−= qpnq
n
npnn
n
n
fnnCSp 14
1
2123
1132
1 223
3
( )
−+++= qn
npn
sfnCCo 1
112
'4
12
( )2'2/1 sfCAs −=( ) ( )2'4/1 sfnnCFC +−=
where:
f
s
OAC11r
F”
F
''f
''s
2r
1=nn
h
"h
D
0P
P
0'P 0"P
"P'P
'h
's
CR
ASEnPExP
r
( )θ,rQ
OC2
1=n
( ) [ ] [ ]0000 '', OPPQPPrW −=θ
Coddington shape factor:
Coddington position factor: ss
ssp
−+='
'
12
12
rr
rrq
−+=
From:
we find:
87
Optics SOLO
Real Imaging Systems – Aberrations (continue – 8)
Seidel Aberrations (continue – 7)
Coddington Position Factor
2R 1R f
1C 2FO 1F
2C
2n
1n
s 's
'2 sfs ==
2R 1R f
1C 2F1F
2C
2n
1n
s 's
fss =∞= ',
2R 1R f
1C 2F1F
2C
2n
1n
s 's
fss <> ',0
2R1R
f
1C 2F1F
2C
2n
1n
s 's
∞== ', sfs
2R 1Rf
1C2F1F 2C 2n
1n
s 's
0', << sfs
CRCR
2R1Rf1C
2F1F 2C2n
1n
s's
0'0 <<> sfs
2R1Rf1C
2F1F 2C2n
1n
s 's
fss =∞= ',
1=p
2R1R f
1C 2F1F
2C
2n
1n
s's
∞== ', sfs
1>p
2R1R f
1C 2F1F
2C
2n
1n
s 's
0',0 ><< ssf
0=p
2R1R f
1C 2FO 1F
2C
2n
1n
s 's
'2 sfs ==
1−=p1−<p
ss
ssp
−+='
'
ss
ssp
−+='
'
'
111
ssf+=
'
211
2
s
f
s
fp −=−=
88
Optics SOLO
Coddington Position Factor
f f2f2− f− 0
Figure ObjectLocation
ImageLocation
ImageProperties
ShapeFactor
InfinityPrincipalfocus
'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, invertedsmall p = -1
Real, invertedsmaller
-1 < p <0
Real, invertedsame size
p = 0
Real, invertedlarger
0 < p <1
No image p = 1
Virtual, erectlarger
p>1
's
's0<s fs <' p < -1
Imaginary,invertedsmall
89
Optics SOLO
Real Imaging Systems – Aberrations (continue – 8)
Seidel Aberrations (continue – 7)
Coddington Shape Factor
1
02
1
−=<
∞=
q
R
R
2R
1R
2C 2n
1n
PlanoConvex
2n
1
0,0
21
21
−<>
<<
q
RR
RR
1C 2C1n
1R
2R
PositiveMeniscus
2R1R f
1C 2F1F 2C 2n
1n
0
0,0
21
21
==
<>
q
RR
RR
EquiConvex
2R
1R
1C2n
1n
PlanoConvex
1
0
2
1
=∞=
>
q
R
R
2R1Rf
1C 2F 2C2n
1n
1
0,0
21
21
><
>>
q
RR
RR
PositiveMeniscus
12
12
RR
RRq
−+=
2R1R f
2F1F
2C
2n
1n
1C
NegativeMeniscus
1
0,0
21
21
−<>
>>
q
RR
RR
1
0, 21
−=
>∞=
q
RR
PlanoConcave
2R1R
f
2F1F
2C
2n1n
2R1R f
1C 2F1F
2C
2n1n
0
0,0
21
21
==
><
q
RR
RR
EquiConcave
2R1Rf
1F 2F
1C
2n1n
1
,0 21
=
∞=<
q
RR
PlanoConcave
NegativeMeniscus
1
0,0
21
21
><
<<
q
RR
RR
2R
1R
f
2F1F 2C2n
1n
1C
90
REFLECTION & REFRACTION SOLO
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 theWave 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
rrq
−+=
ss
ssp
−+='
'Coddington Lens
http://www.eyeantiques.com/MicroscopesAndTelescopes/Coddington%20microscope_thick_wood.htm
91
Optics SOLO
Real Imaging Systems – Aberrations (continue – 8)
Seidel Aberrations (continue – 7)
Thin Lens Spherical Aberrations
( ) 4rCrW SpSA =
Given a thin lens and object O on theOptical Axis (OA). A paraxial ray will crossthe OA at point I, at a distance s’p from the lens. A general ray, that reaches the lensat a distance r from OA, will cross OA at point E, at a distance s’r.
( ) ( ) ( ) ( )
++
−++−++
−−−= qpnq
n
npnn
n
n
fnnCSp 14
1
2123
1132
1 223
3
where:
Define:
2R
1R
1C
IO
2C
Paraxialfocal plane2n
1n
sps'
E
rs' Long. SA
Lat. SA
φ ParaxialRay
General
Ray
'φ
r
rp ssSALongAberrationSphericalalLongitudin ''. −==
( ) rrp srssSALatAberrationSphericalLateral '/''. −== We have:
92
Optics SOLO
Real Imaging Systems – Aberrations (continue – 1)
12
12
RR
RRqK
−+==
( ) ( ) ( ) ( ) ( )
++
−++−++
−−−= qpnq
n
npnn
n
n
fnn
rrWSp 14
1
2123
113222
3
3
4
Thin Lens Spherical Aberrations (continue – 1)
93
Optics SOLO
Real Imaging Systems – Aberrations (continue – 1)
Thin Lens Spherical Aberrations (continue – 3)
2R
1R
1C
IO
2C
Paraxialfocal plane2n
1n
sps'
E
rs' Long. SA
Lat. SA
φ ParaxialRay
General
Ray
'φ
r
12
12
RR
RRq
−+=
F.A. Jenkins & H.E. White, “Fundamentals of Optics”, 4th Ed., McGraw-Hill, 1976, pg. 157Lens thickness = 1cm, f = 10cm, n = 1.5, h = 1cm
In Figure we can see a comparisonof the Seidel Third Order Theorywith the ray tracing.
94
Optics SOLO
Real Imaging Systems – Aberrations (continue – 1)
We can see that the Thin Lens Spherical Aberration WSp is a parabolic function of theCoddington Shape Factor q, with the vertex at (qmin,WSp min)
( ) ( ) ( ) ( )
++
−++−++
−−−= qpnq
n
npnn
n
n
fnn
rWSp 14
1
2123
113222
3
3
4
Thin Lens Spherical Aberrations (continue -2)
The minimum Spherical Aberration for a given Coddington Position Factor p is obtained by:
( ) ( ) 0141
22
132 3
4
=
++
−+
−−=
∂∂
pnqn
n
fnn
r
q
W
p
Sp
1
12
2
min +−−=
n
npq
+
−
−−= 2
2
3
4
min 2132p
n
n
n
n
f
rWSp
The minimum Spherical Aberration is zero for ( )( ) 1
1
22
2 >−+=
n
nnp
95
Optics SOLO
Real Imaging Systems – Aberrations (continue – 1)
In order to obtain the radii of the lens for a given focal length f and given Shape Factorand Position Factor we can perform the following:
Thin Lens Spherical Aberrations (continue – 3)
Those relations were given by Coddington.
'
211
2
s
f
s
fp −=−= p
fs
p
fs
−=
+=
1
2'&
1
2
( )fRR
nss
1111
'
11
21
=
−−=+
( ) ( )1
12&
1
1221 −
−=+
−=q
nfR
q
nfR
12
12
RR
RRq
−+=
12
1
12
2 21&
21
RR
Rq
RR
Rq
−=−
−=+
( ) ( )12
21
1 RRn
RRf
−−=
2R
1R
1C
IO
2C
Paraxialfocal plane2n
1n
sps'
E
rs' Long. SA
Lat. SA
φ ParaxialRay
General
Ray
'φ
r
96
Optics SOLO
Real Imaging Systems – Aberrations (continue – 8)
Seidel Aberrations (continue – 7)
Thin Lens Coma
( ) ( )( ) ( )
−++++=
+==
qn
npn
sfn
xyxh
xyxhCrhChrW CoCoCo
1
112
'4
''''
''''cos'';,
2
22
223 θθ For thin lens the coma factor is given by:
where:we find:
( ) 2
22
2
1
112
4
''': MAXMAXCoS rq
n
npn
fn
hr
n
shCC
−+++==
1
2
3 4
P
ImagePlane
O
SC
SC
ST CC 3=
Coma Blur Spot Shape
TangentialComa
SagittalComa
30
'h
'x
'y
( )( ) ( ) ( ) 222 '2' rRyrRx CoCo =∆+−∆ ( ) MAXCoCo rrrn
shCrR ≤≤= 0
'': 2
Define:
( ) ( ) ( ) ( )θθ 2cos2''
cos21''
''3''
'
';',''' 2222 +=+=+=
∂=∆ r
n
shCr
n
shCyx
n
shC
x
hyxW
n
sx CoCoCo
( ) ( ) θ2sin''
''2''
'
';',''' 2r
n
shCyx
n
shC
y
hyxW
n
sy CoCo ==
∂=∆
97
Optics SOLO
Real Imaging Systems – Aberrations (continue – 8)
Seidel Aberrations (continue – 7)
Thin Lens
F.A. Jenkins & H.E. White, “Fundamentals of Optics”, 4th Ed., McGraw-Hill, 1976, pg. 165Lens thickness = 1cm, f = 10cm, n = 1.5, h = 1cm, y = 2 cm
( ) 2
22 1
112
4
': MAXS rq
n
npn
fn
hC
−+++=
Coma is linear in q
( ) ( )( ) pn
nnqCS 1
1120
+−+−=⇐=
In Figure 800.00 =⇐= qCS
The Spherical Aberration is parabolic in q
( ) ( ) ( ) ( )
++
−++−++
−−−= qpnq
n
npnn
n
n
fnnCSp 14
1
2123
1132
1 223
3
1
12
2
min +−−=
n
npq
+
−
−−= 2
2
3min 2132
1p
n
n
n
n
fCSp
In Figure
714.0min =q
98
Optics SOLO
Real Imaging Systems – Aberrations (continue – 5)
99
Optics SOLO
Real Imaging Systems – Aberrations (continue – 5)
100
SOLO Optics Chromatic Aberration
Chromatic Aberrations arise inPolychromatic IR Systems because
the material index n is actuallya function of frequency. Rays atdifferent frequencies will traverse an optical system along different paths.
101
SOLO Optics Chromatic Aberration
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 designedhis own lenses, build some telescopes and urged by his sonPeter (1739 – 1820) applied for a patent.
Born & Wolf,”Principles of Optics”, 5th Ed.,p.176
Chromatic Aberration
In 1733 he built several telescopes with apertures of 2.5” and 20”. To keep secrecyHall ordered the two components from different opticians in London, but they subcontract the same glass grinder named George Bass, who, on finding that bothLenses were from the same customer and had one radius in common, placed themin 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 Chromatic Aberration
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 Chromatic Aberration
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 Chromatic Aberration
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 Chromatic Aberration
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 11111 ρρρρ −−−−+−= nndnnf
nF – blue index produced by hydrogen wavelength 486.1 nm.
nC – red index produced by hydrogen wavelength 656.3 nm.
nd – yellow index produced by helium wavelength 587.6 nm.
Assume that for two colors red and blue we have fR = fB
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) 22112211
22112211
1111
11111
ρρρρ
ρρρρ
−−−−+−=
−−−−+−=
FFFF
CCCC
nndnn
nndnnf
107
SOLO Optics Chromatic Aberration
Let analyze the case d = 0 (the two lenses are in contact)
nd – yellow index produced by helium wavelength 587.6 nm.
We have
( ) ( ) ( ) ( ) 22112211 11111 ρρρρ −+−=−+−= FFCC nnnnf
( )( )
( )( )1
1
1
1
1
2
1
2
2
1
−−−=
−−−=
F
F
C
C
n
n
n
n
ρρ ( )
( )CF
CF
nn
nn
11
22
2
1
−−−=
ρρ
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
dY
nnf
21 /1
22
/1
11 111 ρρ −+−= ( )
( ) Y
Y
d
d
f
f
n
n
1
2
1
2
2
1
1
1
−−=
ρρ
( )( )
( )( )
( ) ( )( ) ( )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 Chromatic Aberration
( ) ( )( ) ( )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
nV
nn
nV
22
22
11
11
1&
1
−−=
−−=
109
Optics SOLO
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 wavelength 486.1 nm.
nC – red index produced by hydrogen wavelength 656.3 nm.
nd – yellow index produced by helium wavelength 587.6 nm.
Define:nF – nC - mean dispersion
CF
d
nn
nv
−−
=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)
Fraunhoferline
color Wavelength(nm)
Spectacle CrownC - 1
Extra Dense FlintEDF - 3
FdC
BlueYellow
Red
486.1587.6656.3
1.52931.52301.5204
1.73781.72001.7130V - number
58.8 29.0
110
Optics SOLO
Refractive indices and Abbe’s numbers of various glass materials
111
Optics SOLO
Camera Lenses Hecht, “Optics”Addison Wesley,
4th Ed., 2002,pp.218
112
Optics SOLO
CameraLenses
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
TechnionIsraeli Institute of Technology
1964 – 1968 BSc EE1968 – 1971 MSc EE
Israeli Air Force1970 – 1974
RAFAELIsraeli Armament Development Authority
1974 – 2013
Stanford University1983 – 1986 PhD AA