IB Physics/Optics SL

H.1 Ray Optics

H.1.1

Specular reflection : This is illustrated by mirrors, where a reflected image is formed as a result. This is caused by very smooth surfaces, which reflect light back evenly across their entire surface.

Diffuse reflection : This is the more common example of light bouncing of anything else to form an image off that object (i.e. when light bounces off a piece of paper, we see the paper not a reflection). This is caused by the fact that the surface of the paper is not smooth, but rather light is reflected in all directions.

If you look at a mirror from different angles and you see different reflections, where as the paper looks the same from all angles. Diffuse reflection allows us to see things, since otherwise we could only see objects emitting light. As a result, we can see anything which isn't pitch black (i.e. absorbs all light) when there's enough light around.

H.1.2

Formation of virtual images in plane mirrors

Light travels out in every direction form every point on the object, but it's impossible to draw them all, so a ray diagram is used. It is usual to have the object as an arrow on a slope, pointing up, though anything which will show both rotation and inversion will be sufficient.

Draw two rays from a point, slightly diverging (separating) which then strike the mirror, and then reflecting (angle of refraction = angle of incidence).

Draw a line perpendicular to the mirror, and the angle of entry one one side equals the angle of exit on the other). These two lines then move away from the mirror and reach the 'eye' (just draw something to represent it, it isn't an art competition).

The two rays reaching the eye are then extended back up 'inside' the mirror to the point where they meet (should be on the same level as the original object). This point is where the image is formed. It should be upright but 'flipped' around a vertical axis, and is a virtual image.

Finally, draw the image inside the mirror.

H.1.3

Mirrors are used in optical instruments, mostly to let us see things which aren't actually in front of us (or viable normally).

Bike mirrors allow us to see the car about to run us down.
Periscopes allow submarines to see, and then blow up ships on the surface.
A mirror on the moon was used to measure it's distance from earth by bouncing a laser off it and measuring the time for it to return.

H.1.4

Radius of curvature : If the curve of a lens was extended into a full circle, the radius of curvature is the radius of that circle. Radius of curvature is only relevant for spherical lenses, but they're the only ones we consider anyway. The focal length of a spherical mirror will be half the radius of curvature, or the radius of curvature will be 2F, depending on how you want to look at it.

Principal axis : The principal axis is a line perpendicular to the curved surface of the lens at its centre (i.e. a line running horizontally through the center of the lens).

Principal focus : This value is related to the focal length of the lens. The principal focus is always one focal length away from the centre of the lens (therefore there are two principal foci, one on either side, usually designated F on the far side, and F' on the same side as the object). Parallel rays of light entering the lens either converge towards the principal focus (convex lenses) or diverge from it (concave lenses).

Focal length : Focal length is the shortest distance between the principal axis and the principal focus.

Paraxial rays : Paraxial rays are those that are close to the principal axis and parallel to it.

Magnification : The magnification is usually defined as ^{image height}/_{object height}, representing in effect how much bigger the image is. This is also directly related to ^{image distance}/_{object distance}.

H.1.5

Virtual image : Defined as and image one which doesn't actually have any light rays running through it, only 'virtual' rays (i.e. the ones extended inside the mirror).

Real images : Those which do have light rays running through them (those which can be shown on a screen).

As for drawing ray diagrams to help analyze this, I suppose that falls under the 'if light rays go through it' definition. Real images are generally used in projection systems, where the image is being displayed on a screen (or cast directly into the eye). Virtual images are often used, for example, in magnifying lenses, where the image must be visible from different angles.

H.1.6

The mirror equation : ¹/_{d_o} + ¹/_{d_i} = ¹/_f. Based on two of image distance, object distance and focal length, the third can be found. This equation is in the data book under the Optics section.

H.1.7

Thin lenses : Used to converge or diverge light within optical systems. Telescopes, for example, focus the light from stars onto our eyes, at the end of the telescope, and magnifying glasses diverge the light, forming a magnified virtual image on the other side.

Acoustic lenses : focus sound in much the same way an optical lens focuses light. Snell's law describes the refraction of sound as it passes through an interface between two materials of differing sound speed. An acoustic lens provides the appropriate material thicknesses that focus a parallel wavefront of sound to a single focal point.

Microwave lenses : Presumably something to do with microwaves, which are really just another part of the em spectrum, so they will do the exact same thing with lenses as viable light. I suppose in microwave ovens you use diverging lenses to scatter the microwaves around, but someone else may be able to elaborate.

Drawing ray diagrams

First draw in the principal axis, and the lens (convex or concave as appropriate) in the centre. Mark points on the principal axis on both sides for the focal point and 2x the focal point (F and 2F).
Draw the object (again usually an arrow) on the left.
Draw two rays from the top of the arrow. The first runs parallel to the principal axis until it hits the lens, then goes in a straight line down through the focal point (for a convex lens) or goes in a straight line away for the focal point on the object's side (for a concave lens). The second ray runs from the top of the object to the centre of the lens, where it meets the principal axis, and continues straight through.
If these two rays meet on the other side, then this is where the image will be formed (i.e. where you should put the screen). If they diverge, then trace both line back from the lens to where they meet on the other side. This forms a virtual image on the same side as the object, which must be viewed without a screen from the opposite side of the lens. If the object was on the focal point, then the two rays will be parallel, and no image will be formed.

Sign conventions : Basically, heights above the principal axis are positive, below are negative. The focal length of a converging lens is considered positive, and diverging NEGATIVE (So if you studied it to be positive, just remember it's NEGATIVE for a DIVERGING lens focal point). The object distance is positive if it is on the same side the light is coming from (almost always the case) and the image distance is positive if it is on the opposite side to where the light is coming from (in general, positive for real images, negative for virtual).

H.1.8

Deriving the lens equation is done using the similar triangles formed by the ray passing through the centre of the lens. Triangle #1 is formed by the object, the principal axis and this ray, and triangle #2 is formed by the image, the principal axis and the ray. This allows us to get the relationship ^H_i/_{H_o} = ^D_i/_{D_o}, since all sides of the triangles are in proportion.

The second pair of triangles are formed by the ray going down from the lens through the focal point (as in drawing the diagrams). This forms a triangle #1 with the lens (height is equal to Ho), the axis and the ray. Triangle #2 is the image, the axis and the ray. This allows us to get the relationship ^H_i/_{H_o} = ^(D_i-f)/_f. Note, the top term is (f-D_i) for a diverging lens, but this is eliminated because f is negative.

These two equations can then be equated and simplified to form ¹/_{d_o} + ¹/_{d_i} = ¹/_f, the lens equation (yes, it's the same as the mirror equation). This is much clearer with diagrams, so perhaps someone would like to add some.

As can be seen also from the first equation, D_i and D_o are also directly related to the ratio of heights. i.e. the magnification, M=^H_i/_{H_o}=^D_i/_{D_o}.

H.1.9

Simple magnifier (i.e. a magnifying glass) : This uses a convex lens, where the object is placed closer than F to the lens. This results in a magnified, virtual image behind the lens (actually further than the object, but it's bigger to compensate). This allows the object to appear bigger than it is. This is used, for example, to read words in small print.

Compound microscope : Produces a magnified, upside down image by using two lenses. The first lens, close to the object, is called the objective, while the second is called the eyepiece. The object is placed further than F_objective away form the objective lens. This then converges the light to form an image closer than F_{eye piece} to the eyepiece lens. this results in the light diverging out of the eyepiece, and so by tracing back, we can find the position of the magnified virtual image.

Astronomical telescope : This is virtually identical to a compound microscope, except this time the light is entering not from a close object, but in parallel lines at an angle to the lens (since it is coming from so far away). The objective lens focuses the light at a point closer than F_e which then diverges it, and forms a magnified, upside down image behind the lens.

H.1.10

Total internal reflection prisms and optical fibres

(See also section 11.1.4)

Total internal reflection (TIR) : Total internal reflection is a result of Snell's law, and occurs when the angle of refraction is greater than 90. Light is traveling in a substance when it meets a boundary with a less dense medium on the other side. The light will be refracted away from the normal. There comes a point, however, where the resulting angle of refraction is 90. SinØ_c = ⁿ²/_n1 (since sin 90 = 1, it can be cancelled out). If the angle of incidence is equal to Ø_c, then the light will travel along the surface of the more dense medium, however if it is greater then all the light is reflected back into the more dense medium.

TIR prisms : When light travelling inside a prism reaches a boundary and attempts to leave, it is subject to total internal reflection if the angle of incidence is greater than Ø_c. As such, a (right angle triangular) prism can be designed where light enters on the vertical side, is totally internally reflected against the hypotenuse, and then leaves through the bottom side. This can be used to form a sort of periscope without using mirrors, which means 100% of the light is transmitted (mirrors tend to lose some). Also, such a prism could be designed so that, when placed with it's hypotenuse down, light enters one side and is refracted downward, TIRs on the bottom side and then leaves the other side. This means that rays entering the prism higher up, come out further down (because they strike the bottom further along). This can be used, therefore, to invert an image (as in the telescope described above).

Optical fibers : Optical fibers consist basically of a very thin 'wire' of a material with a high refractive index, into which light is sent. Since it has a high index, even when the fiber is bent, the critical angle will still not be exceeded, and all light will be totally internally reflected. This can then be used to transmit light very easily around corners etc. Since the light travels very quickly without resistance, it is preferable to using electric current in wires, and is often used in telecommunications. These fibres can also be used to carry light into the human body, and as in an endoscope, to allow us to examine the inside of the human body without cutting it open ( or doing any other gross biology stuff ). Decorative lamps are a rather trivial use, but you can use them for that to.

Note, in the first two applications, it is important that fibres next to each other are optically insulated, so no light can possibly jump from one to another. This is usually done by coating them in a material of lower refractive index than the fibre.

H.1.11

Thanks to Arkadiusz Staekiewicz.

Determine the minimum deviation angle for dispersion in a prism and explain how the dispersion effect is used in spectroscopes.

A prism separates white light into a rainbow of colours, because the index of refraction of a material depends on the wavelength. White light is a mixture of all visible wavelengths, and when incident on a prism, the different wavelengths are bent to varying degrees. Because the index of refraction is greater for the shorter wavelength, violet light is bent the most and red the least as indicated. This spreading of white light into the full spectrum is called dispersion.

A spectroscope is a device to measure wavelengths accurately using a diffraction grating or a prism to separate different wavelengths of light. A prism, because of dispersion, bends light of different wavelengths into different angles. A prism has the disadvantage that it produces less sharp lines and is less able to separate closely spaced lines. But it has the advantage of deflecting more light (and so is more useful for dim sources) than a typical diffraction grating, since with the latter most of the light passes through to the central peak. An important use of any of these devices is for the identification of atoms or molecules. When a gas is heated or a large electric current is passed through it, the gas emits a characteristic line spectrum. That is, only certain wavelengths of light are emitted, and these are different for different elements and compounds.

Minimum deviation by a prism occurs when the refraction is symmetrical. The minimum deviation D appears in the equation: n=[sin1/2(A+D)]/[sin1/2A] , where A is the principal refracting angle of the prism i.e. the angle made by the refracting faces in a principal section perpendicular to the refracting edge, n is the refractive index. For narrow-angle prisms, D=(n-1)A (approximately), and over a moderate range of angle of incidence there is an approximately constant deviation.

H.2 Wave Optics

H.2.1

interference and diffraction in various apparatus.

In each case, monochromatic, coherent light is assumed to be striking the apparatus in question.

Single slit : Light is assumed to strike a thin slit in a medium, and as a result, form an infinite number of point sources along that slit, each radiating light in a semi-circular pattern. We then consider light traveling from these points at a number of critical angles.

Light traveling straight through remains in phase, and so produces a bright band on the screen level with the slit.
Light traveling out at such an angle that the light from the top source must travel exactly 1 wavelength further than the bottom one to reach the screen. This means that there must be a source in the center which must travel 1/2 of a wavelength further than the bottom one. As a result, this 1/2 wavelength and the top ray are completely out of phase, and so they destructively interfere. The same can be said about the ray immediately below the top, and the one immediately below the middle, and so we continue down until all the rays have destructively interfered, causing a dark band on the screen at this angle ( sinØ = ^λ/_{slit length} ).
Now consider the point where the top ray has ³/₂x λ further to go. The bottom and middle thirds of the slit all destructively interfere as in 2, but the top third goes on to the screen, causing another (though smaller) light band ( sinØ = ^{3 x λ}/_{2 x slit length} ).
The point where the top ray has 2 x λ further to go. All rays destructively interfere ( sinØ = ^{2 x λ}/_{slit length} ) and so on. This causes a series of bright and dark bands, with the bright bands becoming smaller and smaller as the angle from the center becomes greater.

Double slit : This is effectively the same as above, only simpler as we have two point sources, rather than a row of them. When the difference in distance from the screen to the two slits is an even number of wavelengths, then a bright band occurs, when it has a ¹/₂λ on the end, then it causes a dark band. In general, both follow the equation D sinØ = m x λ, where D is either the distance between the two slits, or the length of the slit.

Diffraction grating : Again similar to the double silt, only this time we have many slits. This produces a far sharper set of peaks, because with two slits, when the angle is slightly away from a bright band, the two sources are only slightly out of phase, and so will still produce a reasonably bright band. With many slits, a slight change in angle will produce a far greater drop in intensity since one slit can be completely out of phase with another several hundred slits away. This results in the bright bands being very sharp, and makes a diffraction grating far more precise for calculating wavelength than double slits.

Phase change (this definition is needed for next two) : Whenever light changes mediums, some of it will be reflected by the boundary. If the light enters a medium of higher refractive index than the one it was travelling in, then a phase change will occur (i.e. in a transverse wave model, peaks become troughs and vice versa). This sometimes complicates the calculation of when light will destructively interfere, as any phase changes must be taken into account along with the path difference.

Thin films : The classic example of this is a thin layer of oil (assumed to have lower refractive index than water) floating on top of water. This produces a sort of rainbow effect in the right light conditions. When light enters the oil, some of it is reflected (with a phase change). The remaining light continues down and some is reflected of the oil-water boundary (again with a phase change, meaning the two can be ignored in this case. If the film is like a soap bubble, only one phase change will occur, and it must be accounted for). This means that if the film is a certain thickness, certain wavelengths will be reinforced will others will destructively interfere (this is how they make those sun glasses which look red from the outside etc.). Note that the light is always assumed to enter and leave vertically, though it will be easier to draw at an angle, this should be noted with any diagram. It may be necessary to think of the angle involved if the question wants fringes on the film rather than certain wavelengths being reinforced/destructively interfering though.

Newton's rings : In Newton's rings, there is a flat glass surface with a curved glass plate (think of the bottom part of a sphere being cut off) placed on top of it. This means the gap between the two pieces of glass increases going further out from the centre. Light is reflect off the bottom of the curved plate (with no phase change) and off the top of the base plate (with a phase change). This means that to reinforce, the actual difference between the two distance travelled must be (k+1/2) x λ (where k is some whole number). Note that this means that at the very center there will be a dark spot, not a bright spot (as with the various slit ones above).

H.2.2

For single slits, the minima will be defined by m x λ = a sin Ø (where m is a whole number which changes depending on the order of the fringe, and a is the width of the slit). For double slits, or gratings, m x λ = d sin Ø = ^xd/_D (where d is the distance between the center of two consecutive slits, x is the bandwidth (distance between consecutive bright bands on the screen) and D is the distance to the screen).

H.3 Electromagnetic optics

H.3.1

Light normally travels in all planes (i.e. one wave goes up and down, another goes left and right and at every angle in between). Polarised light is light which only travels in one of these planes (i.e. only up and down). This can occur due to a variety of stuff (discussed in the next section), and can usually be analysed using a polaroid (like the stuff they make sun glasses out of) because when it is rotated, the intensity of light passing through will change.

H.3.2

The different ways of getting polarised light.

By reflection : As was discussed above, when light strikes a boundary between two media, some will be reflected and some will enter the second media. If we think about it though, light moving parallel to the surface is more likely to be reflected than light moving perpendicular (i.e. up and down across the boundary. This occurs completely when the difference between the angle of the reflected light and the refracted light is exactly 90 degrees. You can work it out, but this means that tanØ_p=n (assuming we're going from a medium with refractive index 1 to one with n) where Ø_p is the angle at which light must enter (angle of incidence) to be fully polarised.

By double refraction : Before the idea of refraction was even understood, it was known that some crystals (such as Iceland spar) would refract two separate rays of light, and so they were called doubly refracting. When another type of crystal (tourmaline) was rotated, the light from this would either be transmitted or blocked, depending on orientation. It was only much later that this effect was understood as polarisation.

By selective absorption : Certain materials, known as polaroids, are capable of polarising light when it passes through them. They are created by long chains of molecules arranged such that they are all parallel, leaving small slits between. The axis of the polaroid is the direction of the slits, and when unpolarised light passes through, light moving in the plane of the lists is transmitted completely, while light in the perpendicular plane is completely absorbed. Vector diagrams can be used to find the component of light in the appropriate plane, and hence the amount transmitted.

By scattering : As light travels through the atmosphere, the waves will interact with some air molecules (specifically O₂ and N₂). The electric field associated with the light is absorbed, causing the electric charges within the molecule to oscillate. Oscillating electric charges cause electromagnetic radiation to be produced perpendicular to the direction of oscillation. As a result, an observer looking directly up will see polarised light, where the plane of polarisation is perpendicular to the direction of oscillation (and the direction the light was initially traveling). From other viewing angles, the light will be partially polarised. This can be extended to explain why the sky is blue (because short wavelengths are affected more) and why sunsets are red (because almost all the blue has already been wiped out), but I don't think it's necessary to know that.

H.4 Corpuscular optics

H.4.1

Experiments which confirm the wave-particle duality of light:

Interference : This is the double slit thing from above. This is a wave property, as particles can not destructively interfere (or interfere at all really).

Diffraction : Again as above. Particles would not diffract when passing through a thin slit, but waves would.

Polarisation : Particles would not be oscillating, and so could not be polarised, but light can be, so it must be a wave (maybe).

The Photoelectric effect (See also 13.5.1) : When electromagnetic radiation above a certain frequency shines on a metal surface, a current is produced. The wave theory of light predicts that energy is carried within the light in a continuous form (as it is in all waves). This suggests that any frequency of electromagnetic radiation will eventually cause the emission of electrons (and so produce a current) if it is applied to the metal for long enough, however this does not occur. Rather, nothing can make electromagnetic radiation below this frequency cause emission (not increases in intensity, leaving it on for long periods or anything else). Below f_o (the 'certain' frequency), an increase in intensity increases the number of electrons given off per second, but the maximum E_k of the electrons is not changed by anything except frequency. All this is measured by using a variable voltage opposing the current created by the photoelectron emission, and using a galvanometer to measure when the current is exactly zero.

All this information suggests that the energy within light is quantized, and thus light is broken into individual particles called photons, which carry a certain amount of energy related to their frequency. Each atom can therefore accept only one photon, not two added together. This also explains why the E_k of the electrons is determined by the wavelength, because the E_k comes from the energy remaining after the electron has been excited out of the atom. Therefore light is a particle, but also a wave (and so on).

Cromption effect : This effect deals with the collision of short wavelength (high frequency) light such as X-rays with electrons. Crompton used the photon model of light to predict that the electron would be scattered by this collision, and the x-rays remaining afterwards would have less energy. This was demonstrated mathematically using the laws of conservation of energy and momentum, and practically by experimental results. The wave theory predicts that no shift in the energy of the resulting x-rays would occur, because the electromagnetic radiation of frequency f would be absorbed by the electrons, causing it to oscillate with frequency f and re-emit the x-rays with the same frequency. Once again, this is evidence for the particle nature of light.

H.5 Contemporary optics

H.5.1

Uses of lasers :

Technology : Bar code scanners (relies on the reflection or absorption of laser light), laser disks (light will either be reflected or scatters by 'pits' on the surface of the disk, allowing data to be read from it.

Industry : Surveying (because the beam is very intense, it can travel over large distances without losing intensity, so can be used to measure distances etc.), welding (intense light is very hot, and also able to be focused very sharply to a point), ditto for machining metals and drilling tiny holes.

Medicine : Burning away tissue in small areas (very hot and precise), breaking up gall + kidney stones (can easily be carried inside an optical fiber, and so can be inserted in the body), attaching the retina, cauterising lymph vessels and capillaries (again, precise, focused heat).

H.5.2

Holography is the making of holograms, which are 3 dimensional images embedded in a 2 dimensional surface. This process is performed using lasers (due to the fact that they are focused, coherent sources, as opposed to normal light). A wide laser beam shines on a half silvered mirror, so half the light goes through to the film, while the other half is reflected down on to the object to go into the hologram. Light from every point on the object, as a result, strikes every point on the film, and the interference of the two beams allow the film to record both the intensity and the phase of the light reaching it. If we think about it, the intensity bit is just like a normal photograph. The phase relates directly to the 'depth' (because the phase changes over distance). The interference between the rays from the object and the rays going straight through the mirror allows this phase difference to be found, and so recorded on the film.

To view the hologram a laser must once again be used. After the film is developed, the hologram is placed in laser light of the same wavelength as is was produced with. The hologram acts as a sort of diffraction grating, producing a real 2D image of the hologram (on the opposite side to the laser), and a virtual 3D image on the same side as the laser, thus producing the 3D effect.