MIT 2.71/2.71009/14/05 wk2-b-1

Lenses and Imaging (Part I)• Review: paraboloid reflector, focusing• Why is imaging necessary: Huygens’ principle

– Spherical & parallel ray bundles, points at infinity• Refraction at spherical surfaces (paraxial approximation)

– Optical power of surfaces: positive, negative surfaces• Paraxial ray-tracing

– Matrix formulation of paraxial geometrical optics• Imaging condition• Object/image at infinity

MIT 2.71/2.71009/14/05 wk2-b-2

Parabloid mirror: perfect focusing

s(x) x

z

(e.g. satellite dish)

( )f

xxs4

2

=f

f: focal length

MIT 2.71/2.71009/14/05 wk2-b-3

Lens: main instrument of image formation

Point source(object)

Point image

The curved surface makes the rays bend proportionally to their distance from the “optical axis”, according to Snell’s law. Therefore, the divergent

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

air glass air

opticalaxis

MIT 2.71/2.71009/14/05 wk2-b-4

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

MIT 2.71/2.71009/14/05 wk2-b-5

Ray bundles

pointsource

sphericalwave

(diverging)

… …∞

pointsource

very-veryfar away

planewave

wave-front⊥ rays

wave-front⊥ rays

MIT 2.71/2.71009/14/05 wk2-b-6

Huygens principle

opticalwavefronts

Each point on the wavefrontacts as a secondary light sourceemitting a spherical wave

The wavefront after a shortpropagation distance is theresult of superimposing allthese spherical wavelets

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Why are focusing instruments necessary?

objectobject

...is scattered by

the object

incident light

MIT 2.71/2.71009/14/05 wk2-b-8

Why are focusing instruments necessary?

objectobject

...is scattered by

the object

incident light

imageimage

⇒need optics to “undo”the effects of scattering,

i.e. focus the light

MIT 2.71/2.71009/14/05 wk2-b-9

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

opticalaxis

MIT 2.71/2.71009/14/05 wk2-b-10

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 into another point in space (the “image.”)

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

?

MIT 2.71/2.71009/14/05 wk2-b-11

The ideal optical imaging system

opticalelements

Ideal imaging system:each point in the objectobject is mapped

onto a single point in the imageimage

objectobject imageimage

Real imaging systems introduce blur ...

MIT 2.71/2.71009/14/05 wk2-b-12

Focus, defocus and blur

Perfect focus Defocus

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Focus, defocus and blur

Perfect focus Imperfect focus“sphericalaberration”

MIT 2.71/2.71009/14/05 wk2-b-14

Why optical systems do do notnot focus focus perfectlyperfectly

•• DiffractionDiffraction•• AberrationsAberrations

• However, in the paraxial approximation to Geometrical Optics that we are about to embark upon, optical systems do focus perfectlydo focus perfectly

• To deal with aberrations, we need non-paraxial Geometrical Optics (higher order approximations)

• To deal with diffraction, we need Wave Optics

MIT 2.71/2.71009/14/05 wk2-b-15

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

opticalaxis

MIT 2.71/2.71009/14/05 wk2-b-16

Refraction at single spherical surface

medium 1index n,

e.g. air n=1

01D 12D

medium 2index n´,

e.g. glass n´=1.5

R

center ofsphericalsurface

R: radius ofsphericalsurface

for each ray, must calculate• point of intersection with sphere• angle between ray and normal to surface• apply Snell’s law to find direction of propagation of refracted ray

MIT 2.71/2.71009/14/05 wk2-b-17

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

(1st order Taylor) expressions:

where ε is the angle between a ray and the optical axis, and is a smallnumber (ε ⟨⟨1 rad). The range of validity of this approximation typically extends up to ~10-30 degrees, depending on the desired degree of accuracy. This regime is also known as “Gaussian optics” or “paraxial optics.”

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

εεε tansin ≈≈

εε2111 +≈+

1cos ≈ε

MIT 2.71/2.71009/14/05 wk2-b-18

Paraxial approximation /2

Apply Snell’s 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 the actual off-axis ray

intersection

axial ray

off-axis ra

yValid for small curvatures

& thin optical elements

MIT 2.71/2.71009/14/05 wk2-b-19

Refraction at spherical surface

optical axis

off-axis ra

y

Refraction at positive spherical surface: ( )111

11

xRn

nnnn

xx

⎥⎦⎤

⎢⎣⎡

′−′

−′

=′

=′

αα

1α

1α′

1x 1x′

x: ray heightα: ray direction

n n’

R: radiusof curvature

MIT 2.71/2.71009/14/05 wk2-b-20

Refraction at spherical surface

optical axis

off-axis ra

y

Refraction at positive spherical surface: ( )111

11

xRn

nnnn

xx

⎥⎦⎤

⎢⎣⎡

′−′

−′

=′

=′

αα

1α

1α′

1x 1x′

x: ray heightα: ray direction

n n’

R: radiusof curvature

PowerPower

MIT 2.71/2.71009/14/05 wk2-b-21

Propagation in uniform space

off-axis ray

Propagation through distance D:

1α1α′

1x

1x′n

optical axis

11

111

ααα

=′+=′ Dxx

D

x: ray heightα: ray direction

MIT 2.71/2.71009/14/05 wk2-b-22

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

MIT 2.71/2.71009/14/05 wk2-b-23

The power of surfaces

• Positive power bends rays “inwards” (converging bundle)

• Negative power bends rays “outwards” (diverging bundle)

Simple sphericalrefractor (positive)

1 n

R>0

R<0

Simple sphericalrefractor (negative)

1 n

–

( ) ( )( ) 0

1leftright >++−

=−

≡R

nR

nnP

( ) ( )( ) 0

1leftright <−+−

=−

≡R

nR

nnP

+

MIT 2.71/2.71009/14/05 wk2-b-24

The power of surfaces

• Positive power bends rays “inwards” (converging bundle)

• Negative power bends rays “outwards” (diverging bundle)

+

n 1

R<0

Simple sphericalrefractor (negative)

–

( ) ( )( ) 0

1leftright >−−−

=−

≡R

nR

nnP

( ) ( )( ) 0

1leftright <+−−

=−

≡R

nR

nnP

R>0

n 1

Simple sphericalrefractor (positive)

MIT 2.71/2.71009/14/05 wk2-b-25

Paraxial ray-tracing

air glass air

Free-spacepropagation

Free-spacepropagation

Refraction atair-glassinterface

Refraction atglass-airinterface

Free-spacepropagation

MIT 2.71/2.71009/14/05 wk2-b-26

Example: one spherical surface,translation+refraction+translation

medium 1index n,

e.g. air n=1

Paraxial rays(approximation

valid)

Non-paraxial ray(approximation

gives large error)

01D 12D

medium 2index n´,

e.g. glass n´=1.5

R

center ofsphericalsurface

R: radius ofsphericalsurface

MIT 2.71/2.71009/14/05 wk2-b-27

Translation+refraction+translation /1

01D 12D

0x 1x′ 2x10 αα =

21 αα =′

Starting ray: location direction0x 0α

Translation through distance (+ direction):01D01

00101

ααα

=+= Dxx

Refraction at positive spherical surface: ( )111

11

xRn

nnnn

xx

⎥⎦⎤

⎢⎣⎡

′−′

−′

=′

=′

αα

1x

n n´

MIT 2.71/2.71009/14/05 wk2-b-28

Translation+refraction+translation /2

01D 12D

0x 1x′ 2x10 αα =

Translation through distance (+ direction):12D12

11212

ααα

′=

′+= Dxx

Put together:

1x

21 αα =′n n´

MIT 2.71/2.71009/14/05 wk2-b-29

Translation+refraction+translation /3

( ) ( )0

120

120112012 1 x

RnDnn

RnDDnn

nnDDx ⎥⎦

⎤⎢⎣⎡ +

′′−

+⎥⎦⎤

⎢⎣⎡

′′−

+′

+= α

( )00

012 x

Rnnn

RnDnn

nn

⎥⎦⎤

⎢⎣⎡

′′−

+⎥⎦⎤

⎢⎣⎡

′′−

+′

= αα

01D 12D

0x 1x′ 2x10 αα =

1x

21 αα =′n n´

MIT 2.71/2.71009/14/05 wk2-b-30

(Paraxial) Ray-tracing in general

inxoutx

inαoutαnin nout

OPTICALSYSTEM

For an arbitrary ray entering with• lateral departure displacement xin wrt the optical axis• angle of departure αin wrt the optical axis• in a departure space of refractive index nin

determine the ray’s• lateral arrival displacement xout wrt the optical axis• angle of arrival αout wrt the optical axis• in an arrival space of refractive index nout

upon exiting the optical system

MIT 2.71/2.71009/14/05 wk2-b-31

Matrix formulation /1

01

00101

ααα

=+= Dxx

( )111

11

xRnnn

nn

xx

⎥⎦⎤

⎢⎣⎡

′′−

+′

=′

=′

αα

02 xmx x=

0021 αα αmxf

+′

−=

refraction bysurface with radius

of curvature R

ray-tracingobject-image

transformation

translation bydistance 01D

form common to all

in22in21out

in12in11out

xMMxxMM

+=+=

ααα

MIT 2.71/2.71009/14/05 wk2-b-32

Matrix formulation /2

in22in21out

in12in11out

xMMxxMM

+=+=

ααα

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

in

inin

2221

1211

out

outout

xn

MMMM

xn αα

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −′−=⎟⎟

⎠

⎞⎜⎜⎝

⎛′′′

1

1

1

1

10

1x

nR

nn

xn αα( )

111

11

xRnnn

nn

xx

⎥⎦⎤

⎢⎣⎡

′′−

+′

=′

=′

αα

01

00101

ααα

=+= Dxx

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

0

001

1

1

101

xn

nD

xn αα

Refraction by spherical surface

Translation through uniform medium

Power

MIT 2.71/2.71009/14/05 wk2-b-33

Translation+refraction+translation

( ) ( )0

120

120112012 1 x

RnDnn

RnDDnn

nnDDx ⎥⎦

⎤⎢⎣⎡ +

′′−

+⎥⎦⎤

⎢⎣⎡

′′−

+′

+= α

( )00

012 x

Rnn

RDnnnn ⎥⎦

⎤⎢⎣⎡ ′−

+⎥⎦⎤

⎢⎣⎡ ′−

+=′ αα

01D 12D

0x 1x′ 2x10 αα =

1x

21 αα =′n n´

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ′

0

0

01122

2

by ntranslatio

r.curv.by refraction

by ntranslatio

xn

DRDxn αα

result…

MIT 2.71/2.71009/14/05 wk2-b-34

Matrix ray-tracing in general

inxoutx

inαoutαnin nout

OPTICALSYSTEM

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

in

inin

2221

1211

out

outout

xn

MMMM

xn αα

⎟⎟⎠

⎞⎜⎜⎝

⎛=

2221

1211

MMMM

M is the “system matrix,” found as the product, in reverse order, of the matricesof the elements (free space, spherical surfaces)comprising the system

MIT 2.71/2.71009/14/05 wk2-b-35

On-axis image formation

Pointobjectobject

Pointimageimage

MIT 2.71/2.71009/14/05 wk2-b-36

On-axis image formation

All rays emanating at arrive at irrespective of departure angle

0x 2x

0α00 21

0

2 =⇔=∂∂ Mxα

01D 12D

00 =x 2x0α

2αn n´

( ) [ ] 00120112

012 xRn

DDnnn

nDDx L+⎥⎦⎤

⎢⎣⎡

′−′

−′

+= α

Imaging conditionImaging condition

MIT 2.71/2.71009/14/05 wk2-b-37

On-axis image formation

Rnn

Dn

Dn −′

=+′

0112“Power” of the spherical

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

01D 12D

00 =x 2x0α

2αn n´

All rays emanating at arrive at irrespective of departure angle

0x 2x

0α00 21

0

2 =⇔=∂∂ Mxα

Imaging conditionImaging condition

MIT 2.71/2.71009/14/05 wk2-b-38

Off-axis image formation

Pointobjectobject

(off-axis)

Pointimageimage

opticalaxis

MIT 2.71/2.71009/14/05 wk2-b-39

Magnification: lateral (off-axis), angle

01D 12D

00 =x

2x

0α2α

n n´

01

1212

0

2 ..... 1' D

Dnn

nD

Rnn

xxmx ′

−==+′−

==

0α∆

2α∆

12

01

0

2

DDm −=

∆∆

=αα

α

LateralLateral

AngleAngle

MIT 2.71/2.71009/14/05 wk2-b-40

Object-image transformation

01D 12D

00 =x

2x

0α2α

n n´0α∆

2α∆

02 xmx x=

0021 xf

m′

−= αα α Paraxial ray-tracingtransformation betweenobject and image points

condition) (imaging 0 subject to 21 =M

MIT 2.71/2.71009/14/05 wk2-b-41

Matrix ray-tracing: summary

inxoutx

inαoutαnin nout

OPTICALSYSTEM

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

in

inin

2221

1211

out

outout

xn

MMMM

xn αα

• Power• Imaging condition• Lateral magnification• Angular magnification

12M021 =M

01121=M

M021

21=MM

MIT 2.71/2.71009/14/05 wk2-b-42

Image of point object at infinity

Pointimageimage

MIT 2.71/2.71009/14/05 wk2-b-43

Image of point object at infinity

∞=01D 12D

0x02 =x

00 =α

0x

2αn n´

length focal image : 1212

fnn

RnDR

nnDn ′≡

−′′

=⇒−′

=′

objectat ∞

image

Note: nn

Rnnn

Rnf−′

×′≡−′

×′=′

ambient refractive indexat space of point image

1/Power

MIT 2.71/2.71009/14/05 wk2-b-44

Point object imaged at infinity

Pointobjectobject

MIT 2.71/2.71009/14/05 wk2-b-45

Point object imaged at infinity

∞=12D01D

00 =x2x

02 =α0αn n´

length focalobject : 01

01

fnn

RnDR

nnDn

≡−′

=⇒−′

=

object

imageat ∞

Note: nn

Rnnn

Rnf−′

×≡−′

×=

ambient refractive indexat space of point object

1/Power

MIT 2.71/2.71009/14/05 wk2-b-46

Image / object focal lengths

Pointobjectobject

Pointimageimage

Objectat ∞

Imageat ∞

Power1

×=

−′×=

n

nnRnf

Power1 ×′=

−′×′=′

n

nnRnf