Modern View of Light

Photon = elementary particle

Wave-particle duality

Energy E = h!

h = Planck's constant

= 6.626 x 10-34

J s

! = frequency (1/s)Temporal oscillation frequency of light waves

mass = 0speed c = 3 x 10

8 m/sc = "!

Dispersion relation

" = wavelength (m)spatial period of the light wavesMomentum p = h/"

The concept of a "ray"

t=0

wavefronts(planes of constant phase)

"

The concept of a "ray"

t = 0

"

t = #t

direction of energyflow = ray

In homogeneous media, light propagates in rectilinear paths

Light in matter

vacuum matter

Speed c = 3 x 108 m/s v = c / n

n = index of refracton (or refractive index)

Absorption Coefficient 0 $energy transmitted through length L = exp(-2$L)

Example: Glass has n ~ 1.5, glass fiber has $ ~ 0.0288 /km

Reflection

medium 1

medium 2

incidentlight

reflectedlight

normal

refractedlight

% i %r % i = %r

Normal, incident & reflected rays lie in one plane

Refraction

medium 1

medium 2

incidentlight

reflectedlight

normal

refractedlight

% i

%t

Incident, reflected & refracted rays lie in one plane

n1

n2

n1sin % i n2sin %t=Snell's Law

Total Internal Reflection

medium 1

medium 2

incidentlight

reflectedlight

normal

refractedlight

%c

%t

n1

n2

%c

n1 > n2

% iwhen = %c sin-1n2n1

=

%t = 900

when % i > %call light is reflected

Fustrated Total Internal Reflection(FTIR)

medium 1

medium 2

incidentlight

reflectedlight

normal

% in1

n2

%r

n1 > n2

% i > %c

medium 3n3 = n1

refracted beam is evanescent

Light "tunnels" into 3rd medium

Geometrical Optics

", wavelength is small

Wave effects such as interference & diffraction ignored

Simple analysis, yet sufficient for many situations

. . . as opposed to

Physical Optics" is non-zero

Huygen's Principle

point source

primary wavefront

Huygen's Principle

point source

primary wavefront

secondary point source

Huygen's Principle

point source

primary wavefront

secondary point source

secondary wavefront

Huygen's Principle

generalwavefront

Each point on a wavefrontacts as a secondary point sourceemanating spherical wavelets.

The wavefront after a short propagation distance is thesuperposition of all the sphericalwavelets.

Why Imaging Systems are Necessary?

Each point on an object scatters incident illumination into a sphericalwavelet according to Huygen's Principle.

At a short distance from the obect, the wavelets from all the pointsget entangled and object details are delocalized.The objective of imaging is to relocalize the object details by assigning ("focusing") rays from a single object point to a single "image" point.

Lens: Main Instrument for Image Formation

The curved surface makes the rays bend at angles proportional to their distance from the optical axis, according to Snell's law. Thus a diverging wavefront becomes converging on the output side.

pointsource(object)

pointimage

opticalaxis

air airglass

Analyzing Lenses: Ray Tracing

pointsource(object)

pointimage

opticalaxis

air airglass

free spacepropagation

in airrefraction

at air-glass interface free space

propagationin glass

refractionat glass-airinterface

free spacepropagation

in air

Paraxial Approximation

pointsource(object)

pointimage

opticalaxis

air airglass

&

Only rays close to the optical axis are considered

& << 1 rad

1st order Taylor approximations applysin & & tan & & cos & 1

Valid for & upto 10-30 degrees

Paraxial Approximation

pointsource(object)

air glass

&

Apply Snell's law assuming refraction occured at the intersection of the optical axis and the lens

optical axis

Ignore the distance between theactual off-axis ray intersection& the optical-axis intersectionwith the lens

Valid for small curvatures & thin optical elements

Example: 1 Sperical Surface

pointsource(object)

medium 1index = n

medium 2index = n'

& optical axis

center of sphericalsurface

R R = radius of curvature

off-axis rayparaxial approx.gives large error

free-space propagation refraction

free-space propagation

Example: 1 Sperical Surface

n n'

optical axis

R

R = radius of curvatureD01 D12

x0 x1 x2$0

$2

$1=

Example: 1 Sperical Surface

n n'

optical axis

R = radius of curvatureD01 D12

x0 x1x2$0

$2

$1=

Starting Location: Position x0 Direction $0

x'1

$'1=

Propagation through distance D01x1 = x0 + D01$0$1 $0={

Refraction at spherical interfacex'1 = x1

$'1 = nn' $1 + n - n'

n'Rx1{

Example: 1 Sperical Surfacen n'

optical axis

R = radius of curvatureD01 D12

x0 x1x2$0

$2

$1= x'1

$'1=

Propagation through distance D12x2 = x1 + D12$'1$2 $'1={

Putting together . . .

Example: 1 Sperical Surfacen n'

optical axis

R = radius of curvatureD01 D12

x0 x1x2$0

$2

$1= x'1

$'1=

x2 = n - n'n'

D12R

+ 1( ) x0 + D01( + D12nn'

n - n'n'

D12R

+ D01 ) $0

$2n - n'n'R

= ( ) x0 + ( nn' + n - n'

n'D10R ) $0

Sign Conventions

Light travels from left to right

Radius of curvature is positive when surface is convex towards left

Longitudinal distances are positive if pointing to the right

Lateral distances are positive when pointing up

Ray angles are positive if ray direction is obtained by rotating theoptical axis (+z) counter-clockwise through an acute angle

+

On-axis Image Formation

$0

$2

n n'

D01 D12

S P

All rays emanating from S converge to P irrespective of angle, $0

x0 = 0 x2( ) ( )

x2$0

= 0

n'D12

+ nD01

= n' - nR

"power" of spherical surface(units= Diopters; 1D = 1/m)

Image of Point Object at Infinity

$2

n n'

D01 D12

Px2( )

D12 = f' = n' - nn' R

Image Focal Length

Point Object Imaged at Infinity

D01 = f = n' - nn R

Object Focal Length

$0

n n'

D01 D12

Sx0 = 0( )

Matrix Formulationnout $out

xout=

M11 M12

M21 M22

nin $ in

x in

Translation through Uniform Medium

n $

x1

1=

1 0D01n 1

n $

x0

0

Refraction by Spherical Surface

n' $'

x'1

1=

1

1

n $

x1

1

n' - nR

( )-

0

power

Example: 1 Sperical Surfacen n'

optical axis

R = radius of curvatureD01 D12

x0 x1x2$0

$2

$1= x'1

$'1=

x2 = n - n'n'

D12R

+ 1( ) x0 + D01( + D12nn'

n - n'n'

D12R

+ D01 ) $0

$2n - n'n'R

= ( ) x0 + ( nn' + n - n'

n'D10R ) $0

n' $'

x'2

2=

Translationthrough

D01

XRefraction

at sphericalinterface

XTranslation

throughD12

Xn $

x0

0

Thin Lens

n = 1n'

optical axis

R = radius of curvature

$'out

x'out= X X

$in

x in

n = 1

Refraction at 1st spherical

interface

Refraction at 2nd spherical

interface

Thin Lensn = 1

n'

optical axis

$'out

x'out= X

$in

x in

n = 1

1

1

1 - n'R'

( )-

0X

radius ofcurvature = R

radius ofcurvature = R'

=1

1

n' - 1R

( )-

0

+ 1 - n'R' Pthin-lens = (n' - 1) 1

R1R'-( )

Lens-maker's formula

1

1

n' - 1R

( )-

0

Power of SurfacesPositive power bends rays "inwards"

R > 0

Plano-convex lensR > 0 R < 0

Bi-convex Lens

Negative power bends rays "outwards"

R < 0

Plano-concave lens

R < 0 R > 0

Bi-concave Lens

Power of Surfaces

Matrix Formulation

Power & Focal Length

Power = - M12

f = 1 / power

Thick Lens: Principal Planes

1st focalplane 1st principal

plane

2nd principalplane 2nd focal

plane

optical axis

Note: In paraxial approximation, principal and focal planes are flat,whereas in reality these are curved surfaces (not spherical).

Thick Lens: Focal Lengths

1st focalplane 1st principal

plane

2nd principalplane 2nd focal

plane

optical axisFFL

EFL

BFL

EFL

FFL = Front Focal LengthBFL = Back Focal Length

EFL =Effective Focal Length

Significance of Principal Planes

2nd principalplane

optical axiscomplex optical system

optical axis

2nd focus

Significance of Principal Planes

1st principalplane

optical axiscomplex optical system

optical axis

1st focus

Thick Lens: Matrix Transformation

optical axis

Dl

refraction(radius of curvature

= R1)propagation through Dl

refraction(radius of curvature

= R2)

n=1 n' n=1

$'out

x'out=

1

1

1 - n'R2

( )-

0

1

1

0

Dln'

1

1

n' - 1R1

( )-

0

$in

x in

Thick Lens: Matrix Transformation

optical axis

Dl

refraction(radius of curvature

= R1)propagation through Dl

refraction(radius of curvature

= R2)

n=1 n' n=1

$'out

x'out=

$in

x in

Dln'

n' - 1R2

( )1 + - (n' - 1){ 1R1

- 1R2

+ (n' - 1) Dln'R1R2

Dln'

Dln'

n' - 1R1

( )1 -

}

Thick Lens: Matrix Transformation

optical axis

Dl

refraction(radius of curvature

= R1)propagation through Dl

refraction(radius of curvature

= R2)

n=1 n' n=1

(n' - 1){ 1R1

- 1R2

+ (n' - 1) Dln'R1R2

EFL = f 1f = }

Imaging Condition: Ray Tracing

1st principalplane

optical axis

1st focus 2nd principalplane

2nd focusobject

image

Image point is located at the common intersection of all rays emanating from the corresponding object point.

Rays passing through the two focal points (focii), and the chief ray can be ray-traced directly.

chief ray

Imaging Condition: Matrix Form

1st principalplane

optical axis

1st focus 2nd principalplane

2nd focusobject

imagechief ray

S S'

systemmatrix

n n'

1 0S'/n' 1

1 -P0 1

1 0S/n 1 =

1 - PS/n -P

S'/n' + S/n - PSS'/nn' 1 - PS'/n'

Imaging Condition: Matrix Form

1st principalplane

optical axis

1st focus 2nd principalplane

2nd focusobject

imagechief ray

S S'

n n'

=1 - PS/n -P

S'/n' + S/n - PSS'/nn' 1 - PS'/n'

x

x'

n $'

x'

n $

x

0 x' must be independent of $

Imaging Condition: Matrix Form

1st principalplane

optical axis

1st focus 2nd principalplane

2nd focusobject

imagechief ray

S

n n'

x

x'

Imaging Condition nS

n'S'+ = P

System immersed in air 1S

1S'+ = 1

ff = EFL

Lateral Magnification

1st principalplane

optical axis

1st focus 2nd principalplane

2nd focusobject

imagechief ray

S

n n'

x

x'

When Imaging Condition is satisfied,

mx = x'x = 1 - PS'/n'

Angular Magnification

1st principalplane

optical axis

1st focus 2nd principalplane

2nd focus

chief ray

S

x

x'

When Imaging Condition is satisfied,

ma = = n/n'(1 - PS'/n')

#$

#$'

#$'#$

Generalized Imaging Conditions

n $'

x'

n $

x

M11 M12

M21 M22=

image systemmatrix

object

Power P = -M12

Imaging Condition M21 = 0

Lateral Magnification mx = M22

Angular Magnification ma = (n/n')M11

Aperture Stop & Field Stop

optical axis

AS FS

AS limits the amount of light energy reaching each image point

FS limits the number of image points (extent of the image)

Entrance Pupil

optical axis

AS

Entrance Pupil is the image of the AS seen from the object side.

entrance pupil

Exit Pupil

optical axis

AS

Exit Pupil is the image of the AS seen from the image side.

exit pupil

Mirrors & prisms

• Last time: optical elements,

– Lenses• Basic properties of spherical surfaces• Ray tracing• Image formation• Magnification

• Today: more optical elements,– Prisms– Mirrors

Lens: main instrument for 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

Cardinal Planes and Points

• Rays generated from axial point at infinity (i.e., forming a ray bundle parallel to the optical axis) and entering an optical system intersect the optical axis at the Focal Points.Focal Points.

• The intersection of the extended entering parallel rays and the extended exiting convergent rays forms the Principal SurfacePrincipal Surface (PlanePlane in the paraxial approximation.)

• The extension of a ray which enters and exits the optical system with the same angle of propagation intersects the optical axis at the Nodal Nodal Points.Points.

2nd PS

1st PS

2nd FP1st FP 2nd NP1st NP

n n´

Recap of lens-like instruments

• Cardinal Points and Focal Lengths

• Imaging conditions

2nd PS1st PS 2nd FP

1st FP EFL

BFLFFL

EFL

!!"

#$$%

&!!"

#$$%

&=!!"

#$$%

&

!!!

xn

MMMM

xn ""

2221

1211

M12 # 0

P = –M12 # 0

M21 = 0

22Mmx =

11a Mnn

m!

=

Magnification

lateral

angular

Matrix formulation

Prisms

glassair air

TIR TIR

glass

air air

airair

Refracting Prism

Da a' b' b

Assume a symmetric case,

a = ba' = b'

!

a' = !2

a = ! + D2

n = sin(D+ !

2)

sin(!2

)

From Snell's law,

n

Prism Equation

Dispersion

Refractive index n is function of the wavelength

white light(all visible

wavelengths)

Newton’s prism

red

green

blue

glassair

Dispersion measures

Reference color linesC (H- $=656.3nm, red), D (Na- $=589.2nm, yellow), F (H- $=486.1nm, blue)

Crown glass has

52933.1F =n 52300.1D =n 52042.1C =n

1D

CF

%%=

nnn

V

CF

D 11nn

nV

v%%==

Dispersive power

Dispersive index

Mirrors: the law of reflection

P

P’

O

P !!mirror

&

&

Plane Mirrors

Plane Mirrors have zero power.

Image is always virtual

Images are laterally inverted

R R

RR

Sign conventions for reflection

• Light travels from left to right before reflection and from right to left after reflection

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

• Longitudinal distances before reflection are positive if pointing to the right; longitudinal distances after reflection are positive if pointing to the left

• Longitudinal 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

Example: spherical mirror

In the paraxial approximation,It (approximately) focuses anIncoming parallel ray bundle(from infinity) to a point.

R

Reflective optics formulae

RDD211

0112

%=+

2R

f %=

01

12

DD

mx %=12

01

DD

m %="

Imaging condition

Focal length

Magnification

Paraboloid mirror: perfect focusing

s(x) x

z

(e.g. satellite dish)

What should the shape function s(x) be in order for the incoming parallel ray bundle to come to perfect focus?

Aberrations

Deviation of the wavefront from its ideal spherical shape due to the imperfectrefraction/reflection by the optical elements.

perfect spherical wavefront focused to a point.

aberrated wavefrontdoes not come to a focus. Image is blurred.

Optical elements (lenses, mirrors) produce perfect spherical wavefronts onlyin the paraxial approximation (i.e. for small angles of propagation with respectto the optical axis).

At larger angles, Seidel (or primary) aberrations occur.

Chromatic Aberration

Index of refraction varies with wavelenght (index for blue > index for red)

Blue light comes to focus closer to the lens than red light. The horizontaldistance between the two images is called longitudinal chromatic aberration.

whitelight

longitudinal chromatic aberration

Images formed by different wavelengths also have different transverse (lateral)magnifications. This is called lateral chromatic aberration.

Image formed by blue light is smaller and closer to the lens than that formed by redlight.

Correcting Chromatic Aberration

Using a combination of two kinds of glasses, crown and flint toform a dichromat. Crown can have more +ve power and has moderate dispersion,while flint can have lower -ve power and has high dispersion.

crown(1)

flint(2)

V = ne "1nF " nC

P1

P2

= V1

V2

P = (n1 "1)( 1R1

" 1R2

) + (n2 "1)( 1R1

" 1R2

)

P = P1 + P2

Set the powers equal for the wavelengths,F, C and e.

(dispersion factoror Abbe's factor)

P1 = P V1

V1 "V2

P2 = "P V2

V1 "V2

Summary of Aberrations

Aberration Character Correction

Sphericalaberration

monochromatic, on- and off-axis, image blur

Aspherics, doublet,high-index

Coma monochromatic, off-axis only,blur

spaced doublet with central stop

Oblique astigmatism monochromatic, off-axis, blur spaced doublet with stop

Curvature offield monochromatic, off-axis spaced doublet

Distortion monochromatic, off-axis spaced doublet with stop

Chromaticaberration

polychromatic, on- and off-axis, blur

contact doublet, spaceddoublet

Optical Systems

Light gathering power (capacity to form a bright image)

Magnification

Resolving power (capacity to form sharp images of small detail)

Others such as physical size, weight, cost etc.

General criterion while designing an optical system:

Astronomical Telescope

lightfrom distantstar

objective eyepiece

f0 fe

Angular Magnification of Telescope

lightfrom distantstar

objective eyepiece

f0 fe

! !'

M = angular size of imageangular size of object

=tan !tan !'

y'

=y'/-fe

y'/f0= "

f0fe

Angular Magnification of Telescope

objective eyepiece

f0 fe

M = "f0fe

entrance pupil exit pupil

diameter of entrance pupil diameter of exit pupilf0 -fe

=

=diameter of entrance pupildiameter of exit pupil

Angular Resolution of Telescope

objective eyepiece

f0 fe

y'

Lateral resolution = y'

Angular resolution =

D

y'f0

= 0.61 #D/2

(Rayleigh resolution)

Telescope: Matrix Formulation

objective eyepiece

f0 fe

y'D

system matrix = [ [thin lens (eyepiece) [ [propagation

through d [ [thin lens (objective)

1 " 1fe

0 1

$

%

& &

'

(

) )

1 0d 1

$

% &

'

( )

1 " 1f 0

0 1

$

%

& &

'

(

) ) =

1" dfe

df 0 fe

" 1fe

" 1f 0

d 1" df 0

$

%

& & & &

'

(

) ) ) )

=" f 0fe

0

d " fef 0

$

%

& & & &

'

(

) ) ) )

=

power = 0

angularmagnification

d = f0+fe

Galilean Telescope

objective eyepiece

f0

fe

Newtonian Telescope

imageplane parabolodial primary

mirror