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Lenses and Imaging (Part I)

Why is imaging necessary: Huygens principle

Spherical & parallel ray bundles, points at infinity

Refraction at spherical surfaces (paraxial approximation) Optical power and imaging condition

Matrix formulation of geometrical optics

The thin lens

Surfaces of positive/negative power

Real and virtual images

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Parabloid mirror: perfect focusing

s(x) x

z

(e.g. satellite dish)

( )f

xxs4

2=

f

f: focal length

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Lens: main instrument for image

formation

Point source(object) Pointimage

The curved surface makes the rays bend proportionally to their distance

from the optical axis, according to Snells law. Therefore, the divergent

wavefront becomes convergent at the right-hand (output) side.

air glass air

optical

axis

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Why are focusing instruments

necessary?

Ray bundles: spherical waves and plane waves Point sources and point images

Huygens principle and why we can see around us

The role of classical imaging systems

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Ray bundles

point

source

spherical

wave(diverging)

point

source

very-veryfar awayplane

wave

wave-front

rays

wave-front

rays

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Huygens principle

optical

wavefronts

Each point on the wavefront

acts as a secondary light source

emitting a spherical wave

The wavefront after a short

propagation distance is the

result of superimposing allthese spherical wavelets

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necessary?

objectobject

...

is scattered by

the object

incident light

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necessary?

objectobject

...

is scattered by

the object

incident light

imageimage

need optics to undothe effects of scattering,i.e. focus the light

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Ideal lens

Point sources(object) Point images

Each point source from the object planeobject plane focuses onto a point

image at the image planeimage plane; NOTENOTE the image inversionthe image inversion

air glass air

optical

axis

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Summary:Why are imaging systems needed?

Each point in an object scatters the incident illumination into a

spherical wave, according to the Huygens principle. A few microns away from the object surface, the rays emanating from

all object points become entangled, delocalizing object details.

To relocalize object details, a method must be found to reassign

(focus) all the rays that emanated from a single point object intoanother point in space (the image.)

The latter function is the topic of the discipline of Optical Imaging.

?

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The ideal optical imaging system

opticalelements

Ideal imaging system:

each point in the objectobject is mappedonto a single point in the imageimage

objectobject imageimage

Real imaging systems introduce blur ...

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Focus, defocus and blur

Perfect focus Defocus

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Focus, defocus and blur

Perfect focus Imperfect focus

sphericalaberration

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Why optical systems dodo notnot focusfocus

perfectlyperfectly

DiffractionDiffraction

AberrationsAberrations

However, in the paraxial approximation to Geometrical

Optics that we are about to embark upon, optical systemsdo focus perfectlydo focus perfectly

To deal with aberrations, we need non-paraxialGeometrical Optics (higher order approximations)

To deal with diffraction, we need Wave Optics

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Ideal lens

Point sources(object) Point images

Each point source from the object planeobject plane focuses onto a point

image at the image planeimage plane

air glass air

optical

axis

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Refraction at single spherical surface

medium 1

index n,

e.g. airn=1

01D 12D

medium 2

index n,

e.g. glass n=1.5

R

center of

spherical

surface

R: radius of

spherical

surface

for each ray, must calculate

point of intersection with sphere

angle between ray and normal to

surface

apply Snells law to find direction of

propagation of refracted ray

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Paraxial approximation /1 In paraxial optics, we make heavy use of the following approximate

(1st order Taylor) expressions:

where is the angle between a ray and the optical axis, and is a small

number ( 1 rad). The range of validity of this approximationtypically extends up to ~10-30 degrees, depending on the desireddegree of accuracy. This regime is also known as Gaussian optics or

paraxial optics.

Note the assumption of existence of an optical axis (i.e., perfectalignment!)

tansin

2

111 ++

1cos

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Paraxial approximation /2

Apply Snells law as if

ray bending occurred at

the intersection of the

axial ray with the lens

Ignore the distance

between the location

of the axial ray

intersection and theactual off-axis ray

intersection

axial ray

off-axi

sray

Valid for small curvatures

& thin optical elements

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Refraction at spherical surface

optical axis

off-ax

isray

Refraction at positive spherical surface: ( )111

11

xRn

nn

n

n

xx

=

=

1

1

1

x1

x

x: ray height

: ray direction

n n

R: radius

of curvaturePowerPower

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Propagation in uniform space

off-axisr

ay

Propagation through distance D:

1

1

1x

1xn

optical axis

11

111

=

+= Dxx

D

x: ray height

: ray direction

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Paraxial ray-tracing

air glass air

Free-space

propagation

Free-space

propagation

Refraction at

air-glass

interface

Refraction at

glass-air

interface

Free-space

propagation

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Example: one spherical surface,translation+refraction+translation

medium 1

index n,

e.g. airn=1

Paraxial rays

(approximation

valid)

Non-paraxial ray

(approximation

gives large error)

01D 12D

medium 2

index n,

e.g. glass n=1.5

R

center of

spherical

surface

R: radius of

spherical

surface

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Translation+refraction+translation /1

01D 12D

0x1x 2x

10 =21 =

Starting ray: location direction0x 0

Translation through distance (+ direction):01D01

00101

=

+= Dxx

Refraction at positive spherical surface: ( )111

11

x

Rn

nn

n

n

xx

=

=

1x

n n

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01D 12D

0x1x 2x

10 =

Translation through distance (+ direction):12D

12

11212

=

+= Dxx

Put together:

1x

21 =n n

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( ) ( )0

120112010

122 1

+

++

+

=

Rn

DDnn

n

nDDx

Rn

Dnnx

( )0

0102

++

= Rn

Dnn

n

nxRn

nn

01D 12D

0x1x 2x

10 =

1x

21 =n n

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Sign conventions for refraction Light travels from left to right

A radius of curvature is positive if the surface is convex towards the

left Longitudinal distances are positive if pointing to the right

Lateral distances are positive if pointing up

Ray angles are positive if the ray direction is obtained by rotating the

+z axis counterclockwise through an acute angle

optical axis +z

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On-axis image formation

Point

objectobject

Point

imageimage

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All rays emanating at arrive at

irrespective of departure angle0x 2x

00

0

2 =

x

01D 12D

00 =x 2x0

2n n

[ ]( )

0120112

0102

++=

Rn

DDnn

n

nDDxx "

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All rays emanating at arrive at

irrespective of departure angle0x 2x

00

0

2 =

x

Rnn

Dn

Dn =+

0112Power of the spherical

surface [units: dioptersdiopters, 1D=1 ]-1m

01D 12D

00 =x 2x0

2n n

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Off-axis image formation

Point

objectobject(off-axis)

Point

imageimage

opticalaxis

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Magnification: lateral (off-axis), angle

01D 12D

00 =x

2x

02

n n

01

1212

0

2 .....1' D

D

n

n

n

D

R

nn

x

xmx

==+

==

0

2

12

01

0

2

D

Dm =

=

LateralLateral

AngleAngle

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Object-image transformation

01D 12D

00 =x

2x

02

n n0

2

02 xmx x=

002 1 mxf

+=

Ray-tracing transformation

(paraxial) between

object and image points

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Image of point object at infinity

Point

imageimage

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Image of point object at infinity

=01D 12D

0x02 =x

00 =

0x

2n n

lengthfocalimage:1212

fnn

RnD

R

nn

D

n

=

=

object

at

image

Note:nn

Rnnn

Rnf=

ambient refractive index

at space of point image1/Power

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Point object imaged at infinity

Point

objectobject

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Point object imaged at infinity

=12D01D

00 =x2x

02 =0n n

lengthfocalobject:0101

fnn

RnD

R

nn

D

n

=

=

object

image

at

Note:nn

Rnnn

Rnf=

ambient refractive index

at space of point object1/Power

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Image / object focal lengths

Point

objectobject

Point

imageimage

Object

at Image

at

Power

1 =

=

n

nn

Rnf

Power

1 =

=

n

nn

Rnf

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Matrix formulation /1

01

00101

=

+= Dxx

( ) 111

11

xRn

nnnn

xx

+=

=

02 xmx x=

002

1

mx

f+

=

refraction by

surface with radius

of curvature R

ray-tracing

object-image

transformation

translation by

distance01D

form common to all

in22in21out

in12in11out

xMMx

xMM

+=

+=

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Matrix formulation /2

in22in21out

in12in11out

xMMx

xMM

+=

+=

=

in

in

2221

1211

out

outout

x

n

MM

MM

x

n

=

1

1

1

1

10

1x

nR

nnx

n ( )111

11

xRn

nn

n

n

xx

+

=

=

01

00101

=

+= Dxx

=

0

001

1

1

1

01

x

n

n

Dx

n

Refraction by spherical surface

Translation through uniform medium

Power

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Translation+refraction+translation

( ) ( )0

120112010

122 1

+

++

+

=

Rn

DDnn

n

nDDx

Rn

Dnnx

( )0

0102

++

=

R

Dnnnx

R

nnn

01D 12D

0x1x 2x

10 =

1x

21 =n n

=

0

0

01122

2

by

ntranslatio

r.curv.by

refraction

by

ntranslatio

x

n

DRDx

n

result

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Thin lens in air

R

R)(

n=1 n=1

in

in

x

out

out

x

=

in

in

out

out

??

??

xx

ObjectiveObjective: specify input-output relationship

n

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Thin lens in air

R

R)(

++

Model: refraction from firstfirst (positive) surface

+ refraction from secondsecond (negative) surface

IgnoreIgnore space in-between (thin lens approx.)

n=1 n=1

n n

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Thin lens in air

R

R)(

++

in

in

x

1

1

x

n

1

1

x

n

out

out

x

=

in

in

1

1

10

11

xR

n

x

n

=

1

1

out

out

10

11

x

nR

n

x

n n

n=1 n=1

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Thin lens in air

R

R)(

n=1 n=1

in

in

x

out

out

x

( )

=

=

in

in

out

out

in

in

out

out

10

11

11

10

1

1

10

1

1

x

RRn

x

xR

n

R

n

x

n

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Thin lens in air

R

R)(

n=1 n=1

in

in

x

out

out

x

( )

=

in

in

out

out

10

11

11

xRR

n

x

n

R

nP

1=

R

nP

=1PowerPower of the

first surface

PowerPower of the

second surface

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Thin lens in air

n=1 n=1

in

in

x

out

out

x

=

in

inlensthin

out

out

10

1

x

P

x

( )

=

+

=+=

RRn

R

n

R

nPPP

111

11lensthin

Lens-makers

formula

n

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Thin lens in air

n=1 n=1

in

in

x

out

out

x

=

in

inlensthin

out

out

10

1

x

P

x

n

inout

inlensthininout

xx

xP

=

= Ray bending isproportionalproportionalto the distanceto the distance from the axis

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The power of surfaces Positive power bends rays inwards

Negative power bends rays outwards

+ ++ N

Plano-convex

lens

Bi-convex

lens

Plano-concave

lens

Bi-concave

lens

N

+

Simple spherical

refractor (positive)

1 n

R>0R>0

R

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