The concept of a "ray"

t=0

wavefronts(planes of constant phase)

λ

The concept of a "ray"

t = 0

λ

t = ∆t

direction of energyflow = ray

In homogeneous media, light propagates in rectilinear paths

Reflection

medium 1

medium 2

incidentlight

reflectedlight

normal

refractedlight

θ i θrθ i = θr

Normal, incident & reflected rays lie in one plane

Refraction

medium 1

medium 2

incidentlight

reflectedlight

normal

refractedlight

θ i

θt

Incident, reflected & refracted rays lie in one plane

n1

n2

n1sin θ i n2sin θt=Snell's Law

Total Internal Reflection

medium 1

medium 2

incidentlight

reflectedlight

normal

refractedlight

θc

θt

n1

n2

θc

n1 > n2

θ iwhen = θc sin-1n2n1

=

θt = 900

when θ i > θc

all light is reflected

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

ε

Only rays close to the optical axis are considered

ε << 1 rad

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

Valid for ε upto 10-30 degrees

Matrix Formulationnout αout

xout=

M11 M12

M21 M22

nin α in

x in

Translation through Uniform Medium

n α

x1

1=

1 0D01n 1

n α

x0

0

Refraction by Spherical Surface

n' α'

x'1

1=

1

1

n α

x1

1

n' - nR

( )-

0

power

Thin Lens

n = 1n'

optical axis

R = radius of curvature

_'out

x'out= X X

_in

x in

n = 1

Refraction at 2nd spherical

interface

Refraction at 1st spherical

interface

Thin Lensn = 1

n'

optical axis

α'out

x'out= X

αin

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 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

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

α'out

x'out=

1

1

1 - n'R2

( )-

0

1

1

0

Dln'

1

1

n' - 1R1

( )-

0

αin

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

α'out

x'out=

αin

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 = }

Generalized Imaging Conditions

n α'

x'

n α

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

Prisms

glassair air

TIR TIR

glass

air air

airair

Refracting Prism

Da a' b' b

Assume a symmetric case,

a = ba' = b'

θ

a' = θ2

a = θ + D2

n = sin(D+ θ

2)

sin(θ2

)

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

Crown glass has

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

1D

CF

−−=

nnnV

CF

D 11nn

nV

v−−==

Dispersive power

Dispersive index

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

−=+

2Rf −=

01

12

DDmx −=

12

01

DDm −=α

Imaging condition

Focal length

Magnification

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