Basic optics

• Geometrical optics and images

• Interference

• Diffraction

• Diffraction integral

we use simple models

that say a lot!

more rigorous approach

Basic optics

• Geometrical optics and images

• Interference

• Diffraction

• Diffraction integral

Images

Édouard Manet: A Bar at the Folies-Bergère (1882)

http://epod.usra.edu/blog/2012/10/inferior-mirage-on-a-desert-road.html

Mirage

light rays are bent to

produce a displaced image

of distant objects or the sky

Plane mirrors

convention:

- light entering from the left

- positive distances: O, I on the left

- real image: i > 0

- virtual image: i < 0

object distance image distance

Point objects

(here, we have

virtual image)

Mirror

ImageObject

p > 0 i < 0

• virtual image

• the same orientation and

size (height) as object

Extended objects

Plane mirrors

Head

Eye

Foot

Mirror

r, f > 0

r, f < 0

Spherical mirrors

convex

concaveReal

focus

Central axis

Virtual

focus

Central axis

Focal lengthRadius

Axis

Mirror

Image formation

Spherical mirrors

4 raysRay tracing

Spherical mirrors

- real image: i > 0

- virtual image: i < 0

Concave mirror:

p > f : real image

p = f : image at infinity

p < f : virtual image

Spherical mirrors

4 raysRay tracing

- real image: i > 0

- virtual image: i < 0

Convex mirror:

image is always

- virtual

- erect

- minified

Magnification m

erect image: m > 0

inverted image: m < 0

Spherical mirrors

- real image: i > 0

- virtual image: i < 0triangles ABV and DEV are similar

Spherical refracting surfaces

- real image: i > 0

- virtual image: i < 0

r < 0

r < 0

r < 0

r > 0

r > 0

r > 0

Real

image

Real

image

Virtual

image

Virtual

image

Virtual

image

Virtual

image

Axis

Thin lens

0 (thin lens)(thick lens)

for both refracting surfaces

Air

Glass

Axis

p i

Thin lens

< 0

f > 0

> 0

> 0< 0

f < 0

converging lens

diverging lens

Extensions

f > 0

3 rays

Ray tracing: converging lens

Ray tracing: diverging lens

f < 0

3 rays

diverging lensconverging lens

Thin lens (bottom line)

Thin lens: magnification m

Simple magnifier

angular magnification:

To distant virtual image

Compound microscope

Objective

Eyepiece

Parallel

rays

To distant virtual image

The lateral

magnification produced

by the objective lens

The overall

magnification

Refracting telescope

Objective

Eyepiece

Parallel

rays

To distant

virtual image

Parallel

rays from

distant

object

(angular magnification

of the telescope)

Aberrations (image errors)

- aberrations can be balanced

- image fidelity is limited only by diffraction

examples

Basic optics

• Geometrical optics and images

• Interference

• Diffraction

• Diffraction integral

Interference

What will happen if we add waves?

Double-slit experiment (Young’s experiment, 1801)

Incident

wave

An interference

pattern

Superposition

of waves

u – a suitable component

of E- or H- vector

assume

Incident

wave

Path length difference

(maxima)

(minima)

Different phases due to

different paths

(maxima)

(minima)

(two coherent sources)

(two incoherent

sources)

(one source)

(m for maxima)

(m for minima)

Inte

nsi

ty

Intensity in double-slit experiment

(missing – sign)

particles

http://www.feynmanlectures.caltech.edu/III_01.html

waves

Double-slit experiment with ...

Interference from thin films

Ray reflected at A

Ray reflected at B

Incident wave

For simplicity we assume

1. Normal incidence

2. Double beam interference

Phase difference

phase shift arising

from reflection at Bphase shift arising

from reflection at A

Ray reflected at A

Ray reflected at B

Incident wave

Example: air - n2 - air

(maxima)

(minima)

(maxima)

(minima)

Example: air - n2 - air

(Newton rings)

Example: glass - air - glass

Incident

light

Glass

Glass

Air

Temporal coherence

coherence length coherence time

Define:

monochromatic wave -

- perfectly coherent

pulse (wave-packet) -

- less coherent

white light -

- incoherent

Interference and temporal coherence

coherence length coherence time

Interference and temporal coherence

Spatial and temporal coherence

Michelson interferometer

Movable

mirror

incident wave

...

...

transmitted

wave

reflected

wave

Interference from thin films (again)

Now we consider

1. Arbitrary incident angle

2. Multiple beam interference

incident wave

...

...

transmitted

wave

reflected

wave

geometrical series

The amplitude of the resultant transmitted wave

Interference from thin films

incident wave

...

...

transmitted

wave

reflected

wave

geometrical series

The amplitude of the resultant reflected wave

Interference from thin films

incident wave

reflected wave transmitted wave

For simplicity assume symmetric structure

and pure real numbers

Interference from thin films

(relative transmitted intensity)

integer

Spectral response (thin film, FP etalon)

(relative transmitted intensity)

The free spectral range, FSR

Spectral response (thin film, FP etalon)

(relative transmitted intensity)

with loss

Spectral response (thin film, FP etalon)

Spectral analyzer

0

const.

Basic optics

• Geometrical optics and images

• Interference

• Diffraction

• Diffraction integral

Huygens-Fresnel principle

Every point of a wavefront at a given instant in time,

serves as a source of spherical secondary waves. The

amplitude of the optical field at any point beyond is

the superposition of all these wavelets.

A wavefront

at t = 0

The new wavefront

at t = ∆t

Huygens-Fresnel principle

Every point of a wavefront at a given instant in time,

serves as a source of spherical secondary waves. The

amplitude of the optical field at any point beyond is

the superposition of all these wavelets.

Diffraction

Every point of a wavefront at a given instant in time,

serves as a source of spherical secondary waves. The

amplitude of the optical field at any point beyond is

the superposition of all these wavelets.

Diffraction

Every point of a wavefront at a given instant in time,

serves as a source of spherical secondary waves. The

amplitude of the optical field at any point beyond is

the superposition of all these wavelets.

Incident

wave

Diffracted

wave

Screen

Diffraction from a single slit

z

x

z

radiates a wavelet

The superposition of all these wavelets:

a source at x

x

?

some “constant”

Incident

wave

Screen

Path length difference

in the Fraunhofer region (far-field region)

Amplitude of diffracted wave

Fourier transform and diffraction

The amplitude of diffracted wave is proportional

to the Fourier transform of the field distribution

across the aperture ( = the aperture function).

z

x

?Incident

wave

Screen

aperture function

(we will prove it later)

... back to diffraction from a single slit

z

x

?Incident

wave

Screen

Diffraction from a single slit (results)

z

x

?Incident

wave

Screen(minima)

Diffraction from a single slit (results)

(minima)

Diffraction from a circular aperture

diameter

Airy rings

first minimum

Diffraction from a circular aperture

Resolution of imagining systems

Rayleigh’s criterion for the minimum

resolvable angular separation

Diffraction from a double slit

z

x

substitution

Diffraction factor – due to the

diffraction by a single slit

Interference factor – due to the

interference between two slits

Diffraction factor – due to the

diffraction by a single slit

Interference factor – due to the

interference between two slits

Diffraction from a double slit

diffraction by

a single slit

interference

between two slits

diffraction from

a double slit

Diffraction factor – due to the

diffraction by a single slit

Interference factor – due to the

interference between two slits

Diffraction from a double slit

diffraction by a

single slit

interference fringes for

a double slit system

Diffraction gratings (multiple slits)

(maxima)

Path length difference

(grating orders)

Diffraction gratings (multiple slits)

Path length difference

Diffraction factor – due to the

diffraction by a single slit

Interference factor – due to the

interference from N slits

Diffraction gratings (multiple slits)

Diffraction factor – due to the

diffraction by a single slit

Interference factor – due to the

interference from N slits

X-ray diffraction

(Bragg’s law)

Incident

x rays

Davisson, C. J., "Are Electrons Waves?,"

Franklin Institute Journal 205, 597 (1928)

Electron diffraction

(Bragg’s law)

Incident

electron beam

Basic optics

• Geometrical optics and images

• Interference

• Diffraction

• Diffraction integral

Angular spectrum representation

arbitrary wave = superposition of plane waves

=

in homogeneous medium

+z

Angular spectrum representation (more details)

Plane wave

Wave function:

real complex

Superposition of plane waves:

For

(IFT)

(FT)

we choose + sign, i.e., we

assume propagation in +z

(possible reflections are

neglected)

for EM waves – scalar

approximation

Propagation of waves

+z

?

paraxial approximation

known

Propagation of waves

paraxial approximation

(calculation of the integral)

Fresnel-Kirchhoff diffraction formula

Fraunhofer approximation:

only for

+z

?known

Diffraction integral