Optics

Optics

Section of PS12206 Lectures

2 E-grade problemsets

Lecturer: John HoldsworthRoom P104 Physics Building

School of Mathematical and Physical Sciences

Ph: 4921-5436E-mail: [email protected]

Semester 2, 2002 http://maths.newcastle.edu.au/phys1000/ 2

School of Mathematical and Physical Sciences PHYS1220

Light

Light travels as an electromagnetic wave

hfErefractionofindexn

nc

ck

ffk

tkxsinBBB

tkxsinEEE

vacuo

z

y

22

0

0



Spectrum

Optics concerns itself primarily with the visible and near IR region of the spectrum.

X-ray optics is a new and exciting field. IR optics is a mature field due to military use.



Reflections

Reflection from a surface may be diffuse or specular Specular reflected light has the incident ray, the normal to

the surface and the reflected ray in the same plane. The plane of incidence



Basic geometry



Terminology

Object distance d0 Image distance di Real image: the image is

on the same side of the optical surface as the outgoing light.

Virtual image: the image is on the opposite side of the optical surface as the outgoing light.



Spherical aberration and paraxial approximation

A spherical surface is the easiest to manufacture

Parabolic surface is required to focus to a single point

Close to the axis (= paraxial) the spherical and parabolic surfaces do not differ much

Quite a useful approximation in geometric optics



Spherical mirrors

Within the paraxial approximation, rays parallel to the principal axis will focus at the focal point

Important points: Centre of curvature Focal point 2

rf



Image Location Through Ray Diagrams I



Image location through ray diagrams II

i

o

i

o

dd

hh

ffd

hh o

i

o

Define the Lateral Magnification m

Similar triangles OO’A and II’A

Similar triangles OO’F and ABF(within the paraxial approximation) leads to the mirror equation

fddffd

dd

io

o

i

o 111

o

i

o

i

dd

hh

m



Image location through ray diagrams III

Virtual images di negative (behind the

reflecting surface) do positive f and r positive m positive and >1

Convex Optics di negative (behind the

reflecting surface) do positive f and r negative (behind

the reflecting surface) m positive and < 1



Refraction

Regularly observed phenomena Mirages “Bent” teaspoon in water

Snell’s Lawn1 sin1 = n2 sin2

“n” index of refraction is defined by the electro-magnetic properties of a particular material. It varies with light of different wavelengths and is different for most materials



The spectrum of visible light and dispersion



Total internal reflection For light passing from a medium with high index of

refraction one of lower index of refraction, light is bent away from the normal

For light incident at < c some light will be refracted out of the medium

For light incident at > c no light will be refracted out of the medium, it is totally internally reflected

1

2

1

2

2211

90sinsin

sinsin

nn

nnnn

crit



Fibre optics



Lenses and optical instruments

Refraction at a surface will deflect light according to Snell’s Law. A lens refracts light a different amount at different distances from the center of the lens

An array of trapezoids will cause light from a point to refract in 2-D towards another point and serves to illustrate what a smoothly varying lens surface can do in 3-D



Lens: Converging or Diverging?

Light enters from the left, by convention Converging lens are thicker in the middle

than at the edges Diverging lens are thinner in the centre

than at the edges. Many shapes are possible, each with

their own focusing and aberration properties



Paraxial and thin lens approximations

An aspheric surface is required to focus parallel rays of a single wavelength to the smallest point however a spherical surface is the easiest to manufacture. Spherical aberration Chromatic aberration

Close to the axis the spherical and aspherical surfaces do not differ much: Paraxial approximation.

“Thin Lens” approximation allows one to draw a ray through the vertex of the lens and simplify ray tracing. The diameter of the lens is smaller than the radii of curvature of the surfaces in this case.



Focal plane of a lens

The focal point is the image point for an object at infinity on the principal axis.

The distance of the focal point from the centre of the lens is the focal length

The Power of a lens in Diopters is the inverse of the focal length in metres. Power (Diopters) = 1/f(m)



Image location through ray diagrams I

These three rays allow the imaging properties to be determined graphically.

The image is real On the same side of the

lens as the outgoing light The image is inverted The focal length is positive

by convention



Image location through ray diagrams II

The same approach of the three rays can determine the image of a diverging lens.

The image is virtual It is on the other side of the lens from the outgoing light

The image distance is negative It is on the other side of the lens from the outgoing light

The object distance is positive The focal length is negative by convention as this is a diverging

lens



The lens equation

fddffd

dd

io

i

o

i 111

Define the Lateral Magnification: m

Similar triangles FBA and FI’I

Similar triangles OAO’ and IAI’ (within the paraxial approximation)

leads to the lens equation

o

i

o

i

dd

hhm

o

i

o

i

dd

hh

ffd

hh i

o

i



Virtual image location through ray diagrams

Positive lens do positive di negative (opposite

side of the lens from outgoing light)

m positive and >1 f positive

Negative lens do positive di negative (opposite

side of the lens from outgoing light)

f negative by convention

m positive and < 1



Combination of lens

Use two or more lens in combination Telescope Microscope

Image from first lens becomes the object of the second



Example Example: Two lens, both bi-convex converging lens with

f1=20.0cm and f2=25.0cm, are placed 80.0cm apart. An object is placed in front of the first lens so that do = 60.0cm. What is the position di and magnification mtotal of the final image?

cm.cm.cm.dfd oi 0501

0501

0251111

222

50060030

1

11 .

cm.cm.

ddm

o

i

1050050

2

22

cm.cm.

ddm

o

i5021 .mmmtotal

cm.cm.cm.dfd oi 0301

0601

0201111

111



Lensmaker’s equation I How do you make a lens of a particular focal length?

Refraction at a spherical surface

Paraxial approximation: h is small. = + 2 and 1= +

R positive, doand do positive

2211

2211 sinsinθnθn

θnθn

Rnn

dn

dn

dh

Rh

dn

io

io

1221

1 ,,



Thin lens approximation: L<<di L0

Lensmaker’s equation II

11

1221 11R

ndn

dRnn

dn

dn

ioio

Refraction at the second spherical surface

22

111R

nR

ndd

n

io

Ldd io

fRR

ndd io

111111

21

Already included the sign change for R1 and R2 so both radii are entered as positive values.

Given a piece of glass in air with a refractive index of n and radii as shown we can apply the spherical refraction equation twice to arrive at the lensmaker’s equation. Refraction at the first spherical surface

O

OI1 R1

di

do

doL

diI

Ldi

II1

di

do

R2



The human eye, a superb optical instrument.

Light enters through the cornea, where most refraction occurs, passes through the lens and forms an image on the retina.

The eye focuses on items of specific interest by squeezing the perimeter of the lens to form a crisp image on the fovea, the region of central and detailed colour vision.

This ability to accommodate between a relaxed state and nearby objects worsens with age as the lens becomes more crystalline.

Define the near and far points of vision. Far point is, ideally, infinity and near point is 25 cm for adults.


School of Mathematical and Physical Sciences PHYS1220Common problems of the human eye: short-sightedness.

Nearsightedness has the image of an object at infinity forming before the retina. Vision of an object at infinity is corrected by a negative lens forming a virtual image at the person’s far point.

D.)m(fPower

cmcmf

76115

1115

11



Farsightedness has the image of nearby object forming behind the retina. Vision of an object at 25cm, the standard near point, is corrected by a positive lens forming a virtual image at the person’s near point.

Common problems of the human eye: far-sightedness.

D.)m(fPower

cmcmcmf

03133

1100

125

11



Astigmatism: An element of cylindrical correction. The image is distorted in a particular axis. Corrected by placing a rod-like lens at an appropriate angle.

Glaucoma: The pressure in the vitreous humor is too high. This causes shortsightedness and loss of vision through retinal problems.

Macula degeneration: Blood supply in the region of the macula is affected and vision is lost due to nerves being damaged.

Other common problems of the human eye.



Magnifying glass I

Close inspection differs from distant viewing by the size of the image on the retina.

The larger the angle subtended the greater the detail observed.

When direct vision is inadequate, we use a magnifying glass to enhance this further.



Magnifying glass II

Angular Magnification.

NNf

NdN

Nhdh

fcm

fN

NhfhM

o

o at focussed eyefor

at focussed eyefor

11

25



Telescopes

Refracting and reflecting telescopes have been made. The limit to refracting ones is the ability to make large, well corrected glass lens.

For a relaxed eye, fe = do and the distance between the lens is fo + fe.

e

offM



Newtonian and Cassegrainian reflecting telescopes are the styles most of the astronomical telescopes use for the reason that it is possible to make very large reflecting mirrors to high surface quality.

Hale 200 inch (5.08 m) at Mt. Palomar. Keck 10 m effective diameter from 36 segments at Mauna Kea Hawaii.

Telescopes II



Terrestrial telescopes have upright images.

Telescopes III

Galilean

Spyglass folded into binoculars



Microscopes

Object is placed in front of the objective lens and the object distance is just longer than the focal point.

Magnification is a product of the lateral magnification of the objective and angular magnification of the eyepiece.

oe

o

e

eoe

ee

o

e

o

i

o

io

ffNlM

dfl

fNmMM

fNM

dfl

dd

hhm



Common aberrations in lens and mirrors

Spherical aberration Due to reflecting or refracting

surfaces being ground to spherical shapes when the ideal shapes are not spherical. Parabolic for mirrors.

Off-axis called coma.

Chromatic aberration Affects lens due to dispersion in

refractive index. Corrected by making compound lens from materials with slightly different RI and therefore dispersion.

Does not affect mirrors.



Huygen’s principle

Every point on a wavefront may be considered as a source of tiny wavelets that spread out in the forward direction at the speed of the wave itself.

The new wavefront is the summation of all the wavelets and is tangential to individual wavelets.



Diffraction

Huygens principle offers an explanation for the effects of diffraction around objects.

This is supportive of the wave theory of light.

In general diffraction is not observed with large aperture optics like windows but you certainly may observe diffraction with pinholes.



Wavefront refraction

Huygen’s principle is consistent with Snell’s Law of refraction.

2211

22

11

22

11

sinnsinnn

cvncv

ADtvsin

ADtvsin

and

and



Young’s double slit interference

Young’s observation when two slits were uniformly illuminated.

Consistent with interference between wavefronts emerging from each slit.



Young’s double slit interference II

Interference arises due to the path difference between light emerging from each slit varying as d sin.



Young’s double slit interference III

Overall intensity envelope is due to diffraction from each single slit.

,...,,m,msind

,...,,m,msind

210

210

21

ceinterferen eDestructiv

ceinterferen veConstructi



Line spacing for double-slit interference

A screen is 1.2m from two slits 0.100mm apart. 500nm wavelength light is incident upon the slit. What is the separation between bright fringes on the screen?

L

Given: d= 0.100mm = 1x10-4 m, =500x10-9 m, L= 1.20m

mmd

mLx

mmmLxd

m

mmd

00.12220.1

00.61000.520.1

1000.51000.110500sin

1,sin

22

311

13

4

9

1

1



Wavelengths from double-slit interference

White light passes though two slits 0.5mm apart allows the measurement of wavelengths on a screen 2.5m away. The estimate of the violet and red wavelengths may be obtained.

nm

nm

Lx

md

md

mmd

violet

red

4005.2100.2

1100.5

7005.2105.3

1100.5

sin,1,sin

34

34

111

mmm

mmm



Coherence

Interference is only possible if there is a fixed phase relationship between the radiation emerging from each of the two slits.

As the light source for Young’s fringes was a distant source, the sun, and as both slits “sampled” the wavefront at the same time, there was interference between the emerging wavefronts and these slits are then said to be coherent sources.

Most light sources are incoherent. An incandescent light bulb will emit light along the length of the filament. Light emitted at each end of the filament bears no phase relationship to the light emitted at the other. Most lasers are sources of very coherent light. They may have a

phase relationship which extends both across the beam and along the beam. This high degree of coherence is required to improve the contrast of interference fringes and in holography.



Intensity in the double-slit interference pattern The electric field is given by the sum of the fields from each slit. These vary

with angle in terms of a phase difference .

The intensity in the double slit interference pattern has periodic maxima corresponding to d sin =m or, in terms of the phase difference :

The intensity in between these maxima may be determined as a function of angle .

)sin(sin

00 21

21

tEtEEEE

sin2 d



Intensity in the double-slit interference pattern II

A phasor diagram illustrates the phase summation of the two equal fields.

Observe intensity rather than E field

2sin2cos2

2sin)()(2cos2cos2)(

2

0

0

000

021 00

tE

tEE

EEE

EEE ,

sincos

2cos)(

20

20

dI

II



Interference in thin films An interference effect is seen in the reflection of light from water

with a layer of oil on the surface. Rainbow coloured fringes are visible. Light passing from material of lower to higher refractive index undergoes a 180° phase change upon reflection. (Higher to lower has no associated change.)

The additional optical path from A to B to C allows constructive interference when equal to a integer multiple of /noil. (It is assumed that nair > noil > nwater.)

The film thickness for normal incidence corresponding to a bright fringe is (/noil)/2.



Newton’s rings Robert Hooke, a contemporary and

antagonist of Newton, observed and recorded the fringes which now bear Newton’s name.

The ray, reflected in glass at point B, has no phase change.

The ray reflected with a phase change at point C may interfere the light reflected at B. The optical path BCD is (2 x physical distance +½ )

When viewed in reflection Newton’s rings are dark in the centre due to the ½ phase change.

When viewed in transmission Newton’s rings are bright in the centre due to two ½ phase changes.



Wedges and optical testing

Fringes may easily be generated by placing a thin wire or sheet of paper along one edge of a piece of glass resting on an optically flat surface and illuminating with monochromatic light.

Bright bands occur at when 2 t =(m+ ½) . First LH fringe is

Extremely useful in determining the flatness of optical surfaces.



Anti-reflection coating

The deposition of a “quarter-wave” thickness of a mechanically hard material possessing an appropriate refractive index allows the surface reflection from glasses, camera lens and optical surfaces to be minimised.

Usually MgF2 is used as it’s refractive index is almost halfway between air and glass.



Michelson interferometer

Extremely precise measurements of distance are possible using an optical interferometer. Varying the position of M1 by as little as 100nm would alter the fringes of 400nm light from dark to bright.

By sweeping M1 over a large known distance from the beam splitter, it is possible to perform a Fourier analysis on the allowed frequencies and obtain very high resolution spectra.



Diffraction and polarization

Fresnel proposed a wave theory that predicted and explained diffraction and interference with non-plane wavefronts. Fraunhofer explained them with planar wavefronts and this is the more illustrative method to follow.

A counter intuitive aspect of this is that a solid disc will have a bright spot at the centre as a result of diffraction around the edges.



Single slit diffraction

Diffraction is analysed by looking at the path difference in light sourced from each element of the slit.

Minima occur at: 321,sin ,, mma



Intensity in the single slit diffraction pattern I

Minima are when sin = /a The intensity between minima is

given by the vector sum of the wavefronts emerging from each segment of the slit y.

There is a phase difference in the waves so the approach may be by phasor diagrams as with the interference example.



Intensity in the single slit diffraction pattern II There is a phase

difference in the waves from each segment of the slit given by:

siny2

00 ENE

The total amplitude arriving in the centre of the screen, where is zero, and all elements are in phase is :

The amplitude, when is small, between the central maximum and the first minima is given by a similar sum but the phase components of each segment, given below, mean that E < E0 in magnitude.

sinasinyNN 22



Intensity in the single slit diffraction pattern III

The phase difference eventually curls around on itself when = 2.

asin

sinyN

22

The total amplitude reaches a secondary maximum when = 3



Intensity in the single slit diffraction pattern IV As y goes to dy, E0 and E

are related by:

sinasinEE

rE

sinrE

where

and 2

22

2222

0

0

The intensity is the square of the electric field so the analytical expression for the intensity is:

2

0

2

0 22

sina

sinasinIsinII



Diffraction and double-slit interference combined.

22

0 222

cossinII

sindsina and 22

The observed two slit interference pattern incorporates diffraction from the slit as well. The diffraction from slit width a is convolved with the interference from the slits spaced distance d apart.



Airy disc and Rayleigh criterion

Circular apertures exhibit diffraction too. This is the limit to performance of optical systems even when all aberrations have been removed.

In the limit where the angles are small, the minima of the diffraction from a circular aperture are found at:

D.

sin

221

The minimum angular separation that may be resolved corresponds to when the maxima from one point source overlaps with the first minima in the diffraction pattern. This is the Rayleigh criterion.



Telescope resolution

Atmospheric turbulence limits the effective resolution of Earth based telescopes to about 0.5 arc seconds.(1 degree is equivalent to 60 minutes. Each minute is divided into 60 seconds of arc, so 1 degree = 3,600 seconds.)

Hubble is limited by diffraction as it is above the Earth's atmosphere. It has a primary mirror aperture of 2.4m so the resolution limit is:

rad.m.

m.D

.

7

9

108242

10550221

221



Resolution limit of microscopes

Microscopes are diffraction limited in their performance if they are well designed.

It is more useful to show the limiting distance s between two points than the angular resolution. This is the resolving power

2

221

RP

Df.fsRP

fs

It is not possible to resolve detail of objects smaller than the wavelength of the radiation being used.



Resolution limit of the eye

radm

mf

s 42

6103

102106

The diffraction limit ranges from 8-60 x 10-5 rad depending on pupil size and this corresponds to a resolving power between 2 and

15m. Spherical and chromatic aberration contribute a limit about 10m. The limit is really the fovea and this translates to us resolving something about 3mm tall at 10m distance.

When looking at a microscope, the limit is about 500 X magnification. Above this is not really useful due to diffraction.

The eye is principally limited by the spacing of the cones in the fovea. These are about 3m apart and there needs to be one unexcited cone in between two excited cones for us to distinguish them as being excited. The limiting angular resolution is therefore:



Diffraction gratings

A diffraction grating has multiple slits from which interference between components of an incident wavefront and diffraction of each component at each “slit” take place.

The multiple interference narrows the spectrum considerably but the maxima still occur at:

...,,, m d

msin 3210

Gratings may be either reflection or transmission with the number of lines/mm going as high as 4800. Large sizes 400mm X 200mm give extremely narrow lines due to the very large number of “slits” interfering.



Spectra

Spectra of line source and a continuum source are compared above. There is usually an overlap and interference between 2nd order UV and first order deep red / NIR wavelengths.

Third and higher order diffraction also occurs at lower efficiencies.



X-ray diffraction I

Crystals were shown to diffract light by Max von Laue. The theory of this was developed by the Bragg’s. (The father W.H. Bragg and son W.L. Bragg lived in Adelaide for a time.)

The x-rays reflecting from lower atomic planes travel an increased distance:...,, m sindm 3212

Things can get complicated as different planes reflect radiation. Either pure crystals or polycrystalline powders are required.



X-ray diffraction II

Polycrystalline materials or powdered pure crystals produce circular patterns on film.

Sodium acetoacetate in (b) X-ray diffraction is a standard analytical chemical technique for

structure determination in new molecules.



Polarisation

The polarisation of a light wave refers to the plane of oscillation of the E field in the E-M radiation.

Different materials may allow the E field to propagate or to be blocked depending on their molecular structure.

Light from incandescent sources is usually unpolarised while laser light is usually polarised in either a known or a random direction.



Polaroid polarisers Edwin Land invented polaroid,

which preferentially transmits the E field perpendicular to the long molecules comprising the material.

If light is passed through a polariser at an angle to the optical axis, the transmitted fraction is: cosEE 0

The transmitted intensity is (Malus Law):

20 cosII

In general an unpolarised source will be, on average, 50% horizontally polarised and 50% vertically polarised. A polariser will reduce the intensity to 50% of the initial value where the remaining intensity is linearly polarised in the axis of the polariser.



Crossed and angled polarisers

If two polarisers have orthogonal axes then incident light is totally absorbed (in the ideal case).

If there are multiple polarisers then Malus Law is applied successively to the sequence of polarisers.

For example two crossed polarisers have a third polariser inserted between them with the axis oriented at 45° to both.



Polarisation by reflection

Light reflected from any non-metallic surface has a different reflectivity for the horizontal and vertical polarisations.

Glare is predominantly horizontally polarised due to this effect.

Brewster’s Angle

1

2nntan p

Documents

Optics