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### Optics

Section of PS1220

6 Lectures

2 E-grade problemsets

Lecturer: John Holdsworth

Room P104 Physics Building

School of Mathematical and Physical Sciences

Ph: 4921-5436

E-mail: [email protected]

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.

http://maths.newcastle.edu.au/phys1000/

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

http://maths.newcastle.edu.au/phys1000/

Basic geometry

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

Spherical aberration andparaxial 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

http://maths.newcastle.edu.au/phys1000/

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

http://maths.newcastle.edu.au/phys1000/

Image Location Through Ray Diagrams I

http://maths.newcastle.edu.au/phys1000/

Image location through ray diagrams II

- Similar triangles OO’A and II’A

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

- Define the Lateral Magnification m

http://maths.newcastle.edu.au/phys1000/

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

http://maths.newcastle.edu.au/phys1000/

Refraction

- Regularly observed phenomena
- Mirages
- “Bent” teaspoon in water

- Snell’s Law
n1 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

http://maths.newcastle.edu.au/phys1000/

The spectrum of visible light and dispersion

http://maths.newcastle.edu.au/phys1000/

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 > cnolight will be refracted out of the medium, it is totally internally reflected

http://maths.newcastle.edu.au/phys1000/

Fibre optics

http://maths.newcastle.edu.au/phys1000/

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

http://maths.newcastle.edu.au/phys1000/

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

http://maths.newcastle.edu.au/phys1000/

Paraxial and thin lensapproximations

- 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.

http://maths.newcastle.edu.au/phys1000/

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)

http://maths.newcastle.edu.au/phys1000/

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

http://maths.newcastle.edu.au/phys1000/

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

http://maths.newcastle.edu.au/phys1000/

The lens equation

- Similar triangles FBA and FI’I

- Similar triangles OAO’ and IAI’
(within the paraxial approximation) leads to the lens equation

- Define the Lateral Magnification: m

http://maths.newcastle.edu.au/phys1000/

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

http://maths.newcastle.edu.au/phys1000/

Combination of lens

- Use two or more lens in combination
- Telescope
- Microscope

- Image from first lens becomes the object of the second

http://maths.newcastle.edu.au/phys1000/

Example

- Example: Two lens, both bi-convex converging lens withf1=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 positiondi and magnification mtotal of the final image?

http://maths.newcastle.edu.au/phys1000/

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, doanddo positive

http://maths.newcastle.edu.au/phys1000/

I

di

do

L

I1

O

R1

do

di

I1

I

R2

di

di

L

do

Lensmaker’s equation II- 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 atthe first spherical surface

- Refraction atthe second spherical surface

- Thin lens approximation: L<<diL0

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

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

Common 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.

http://maths.newcastle.edu.au/phys1000/

Common problems of the human eye: far-sightedness.

- 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.

http://maths.newcastle.edu.au/phys1000/

Other common problems of the human eye.

- 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.

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

Telescopes II

- 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.

http://maths.newcastle.edu.au/phys1000/

Telescopes III

- Terrestrial telescopes have upright images.

- Galilean

- Spyglass folded into binoculars

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

Wavefront refraction

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

http://maths.newcastle.edu.au/phys1000/

Young’s double slit interference

- Young’s observation when two slits were uniformly illuminated.

- Consistent with interference between wavefronts emerging from each slit.

http://maths.newcastle.edu.au/phys1000/

Young’s double slit interference II

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

http://maths.newcastle.edu.au/phys1000/

Young’s double slit interference III

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

http://maths.newcastle.edu.au/phys1000/

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?

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

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

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 dsin =mlor, in terms of the phase difference :
- The intensity in between these maxima may be determined as a function of angle .

http://maths.newcastle.edu.au/phys1000/

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

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

Single slit diffraction

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

http://maths.newcastle.edu.au/phys1000/

Intensity in the single slit diffraction pattern I

- Minima are when sin q=l/a
- The intensity between minima is given by the vector sum of the wavefronts emerging from each segment of the slit Dy.
- There is a phase difference in the waves so the approach may be by phasor diagrams as with the interference example.

http://maths.newcastle.edu.au/phys1000/

Intensity in the single slit diffraction pattern II

- There is a phase difference in the waves from each segment of the slit given by:

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

- The amplitude, when q 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
Eq < SDE0 in magnitude.

http://maths.newcastle.edu.au/phys1000/

Intensity in the single slit diffraction pattern III

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

- The total amplitude reaches a secondary maximum when b = 3p

http://maths.newcastle.edu.au/phys1000/

Intensity in the single slit diffraction pattern IV

- As Dy goes to dy, E0 and Eq are related by:

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

http://maths.newcastle.edu.au/phys1000/

Diffraction and double-slit interference combined.

- 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.

http://maths.newcastle.edu.au/phys1000/

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:

- 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.

http://maths.newcastle.edu.au/phys1000/

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:

http://maths.newcastle.edu.au/phys1000/

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

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

http://maths.newcastle.edu.au/phys1000/

Resolution limit of the eye

- The eye is principally limited by the spacing of the cones in the fovea. These are about 3mm 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:

- 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 15mm.
- Spherical and chromatic aberration contribute a limit about 10mm.
- 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.

http://maths.newcastle.edu.au/phys1000/

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:

- 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.

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

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:

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

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

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:

- The transmitted intensity is (Malus Law):

- 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.

http://maths.newcastle.edu.au/phys1000/

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.

http://maths.newcastle.edu.au/phys1000/

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

http://maths.newcastle.edu.au/phys1000/

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