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.

a, 08/16/2005

Hero proof is described in M.V.Klein, T.E.Furtak, "Optics", pp.3-5

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


12

Optics SOLO

Prisms

( )[ ] αθαθαθδ −−−+= −1

2/1

1

221



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


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(.

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Optics SOLO

Prisms

( )[ ] αθαθαθδ −−−+= −1

2/1

1

221


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

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


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


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


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


Entrancepupil

Exitpupil

A.S.

IΣ

ChiefRay

MarginalRay

Exp Enp

Imageplane

Hecht"Optics"

Pupil and stops for a three - lens system

28

Optics SOLO


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


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

a, 04/15/2006

Hecht & Zajac, " Optics", 4th Ed., McGraw-Hill, 1979, pp.103-104

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

a, 04/12/2006

Jenkins & White, "Fundamentals of Optics", 4th Ed., McGraw-Hill, 1976, pp.56-57

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 =

a, 04/12/2006


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 −=+−

a, 04/12/2006


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

'"'

"

"

a, 04/12/2006


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

a, 04/12/2006


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).


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.


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


Published by Newton in “Opticks” 1710

'y

s 's

M’A F’M

T

F

'ffx 'x

Q

Q’'y

y

S

Axisy

a, 04/12/2006


41

Optics SOLO

Derivation of Lens Formula (continue)


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

a, 04/12/2006


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

a, 04/12/2006


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

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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.

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Optics SOLO


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

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Optics SOLO


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

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Optics SOLO


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 ⇔

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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 −=

−

+=

−

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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

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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

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Optics SOLO


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

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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

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Optics SOLO

Real Imaging Systems – Aberrations (continue – 5)

58

Optics SOLO


59

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60

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61

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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.

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Optics SOLO


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

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Optics SOLO



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:


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Optics SOLO



+

−+

−+

−+−≈ 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=

−−

−−=


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Optics SOLO



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 −=

−+−

+−−=

'

'

''

'

''

'

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Optics SOLO




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

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Optics SOLO




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



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:

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Optics SOLO



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

71

Optics SOLO



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

72

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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



( )'

';','

'

''

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


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


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


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


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


Graphical Explanation of Coma Blur

79

Optics SOLO


Graphical Explanation of Coma Blur (continue – 1)

81

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3. Astigmatism

82

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3. Astigmatism

83

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3. Astigmatism

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4. Field Curvature

85

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5. Distortion

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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:

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Optics SOLO



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



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



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


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



2R

1R

1C

IO

2C


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


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


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:


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


1n

sps'

E

rs' Long. SA

Lat. SA

φ ParaxialRay

General

Ray

'φ

r

96

Optics SOLO



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



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


99

Optics SOLO


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


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”

a, 06/24/2006

1. Jurgen R. Meyer-Arendt,"Introduction to Classical & Modern Optics",Prentice Hall, 3th Ed.,1989,p.123

103


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


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


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


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


Let analyze the case d = 0 (the two lenses are in contact)


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


( ) ( )( ) ( )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.


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

a, 06/24/2006

Jurgen R. Meyer,"Introduction to Classical and Modern Optics", Prentice Hall, 3th Ed., p.22

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

lens

Science

optical ray

incident ray

thereflected ray

ray of light

actual ray

single ray

reflected path

fermat principle refracted