IB Physics/Optics HL
H.6 Ray Optics
Circular mirrors do not focus all their light to a single point (that's what parabolas do). Close to the center, the shape of a circle and a parabola (on it's side) are very similar, and so paraxial rays (close to the center) will be focused to a point, but the further away from the center we go, the further away from the original focus the rays become (they come closer to the mirror as we move further out). This effect is known as spherical aberration, and causes a blurry image. It can be overcome with parabolic mirrors, but these are much more expensive to produce.
Optical fibres are (as mentioned in the SL section) composed of an optically dense core surrounded by a less dense coating. They rely on total internal reflection, and as such there is a critical angle beyond which the light will escape from the fibre. This angle can be calculated with the following equation.
sin Ø = ± √( n12 - n22 )
n1 is the refractive index of the core, and n2 the index of the coating. Hopefully that's obvious because the number we're taking the root of needs to be positive. This equation is in the data book.
Rainbows occur when the sun is behind the viewer, and there are raindrops in front. The sunlight enters the raindrops, and is totally internally reflected on the back, and then comes out and towards the viewer. The light is also dispersed within the rain drops, resulting in red being aimed the lowest, and violet the highest.
To draw the diagram, two drops must be drawn as large circles, one somewhat above the other. Sunlight enters at the same angle into each, and is immediately split into red and violet rays, where the violet ray is diffracted more towards the normal than the red ray. The rays are totally internally reflected by different angles from the curved back of the drop, and then cross over before leaving with red aimed down the most and violet up the most. (Note: this is the reverse order compared with the diffraction through a triangular prism.) From the top drop, the red beam should reach the viewpoint (illustrated by an eye) and violet from the bottom drop. These two rays are then extended back from the drops to produce a virtual image of the rainbow behind the drops (with red at the top, and violet at the bottom, and orange, yellow, green blue and indigo between).
The lens makers equation is 1/f = (n-1)(1/R1 + 1/R2).
f is the focal length of the lens, n is the refractive index of the lens material. R1 is the radius of the front side, and R2 the radius of the back (these radii are negative if the lens is concave on that side, positive if it's convex).
A large aperture lens will produce spherical aberration as a result of it's large diameter. This will cause the resultant image to focus, not at a single point, but rather over a line, which may cause problems in optical instruments.
Chromatic aberration results from the dispersion effect when light changes media. Because the lens has a large aperture, the light travels a greater distance in the glass, perspex or whatever, and so this dispersion becomes more significant than in a thin lens.
Both of these can be greatly reduced by the use of two lenses (ie a convex lens followed by a concave one).
How thin lenses can rectify some eye problems:
Myopia (Nearsightedness) : This is when an eye which can only focus on close objects. Parallel rays entering the eye focus before the back of the eye, crossing over as thus producing a blurry image. This can be rectified by using a diverging lens (a concave one) as this makes the light diverging rather than parallel as it strikes the lens of the eye, and so it focuses further back.
Hypermetropia (Farsightedness) : This is when an eye which is unable to focus on close objects, as it focuses too far back, the rays crossing behind the back of the eye. This can be corrected by using a converging lens (convex) which brings light rays from a close object closer to parallel to the eye's lens can focus them on the back of the eye.
Presbyopia : This is the decreasing ability of the eye to focus on close objects as the eye ages, and so it can be corrected in the same way as Hypermetropia above.
H.7 Wave optics
Two things limit the resolution of an image produced by a lens, aberration and diffraction. The aberration, discussed above, is ignored for the Rayleigh criterion. The Rayleigh criterion basically says that two images are just separable when the central peak of one is at the first minimum of the other (or further away from it). The smallest angular separation of two objects in the distance which can be resolved by a given lens (assuming no aberration) can be calculated with the following two formulas.
λ = asinØ where λ is the wavelength of the light being used, and a is the aperture of the (in this case) slit and Ø is the angular separation.
The more common one is for a spherical lens Ø = (1.22 x λ)/D where D is the diameter of the lens (in a multi lens system, for the objective lens).
Note the angular separation is in radians. π radians = 180 degrees.
The second equation is given in the data book and the first is just the same as the equation for calculating the position of the first minimum in single slit diffraction.
Resolving power : This is generally used for microscopes, where the object is placed close to the focal point of the lens, and is used to calculate the minimum distance between two objects for them to be resolvable. RP = s = fØ (where f is the lens focal length, and s is the minimum distance).
This equation is not in the data book. Resolving power can also be expressed as (1.22 x λ)/(2 sin a) where a is the angle formed if you take a line from the centre of the lens along the principal axis to the focal point, and then up to the top of the lens.
H.8 Electromagnetic Optics
Linear, circular and elliptical states of polarisation, and Malus' law.
Malus' Law : I = Iocos2Ø. This law relates the intensity of light to the difference in angle between two polaroids. If the two polaroids are aligned so they both polarise in exactly the same direction, then there is an angle of zero between them, thus cos2 0 = 1, and the initial intensity, Io = I, the resultant intensity.
This relation ship can be derived by considering the vector components of the amplitude of the light waves. First, against a horizontal and vertical axis, draw a line going up and to the right from the origin. The angle Ø is the angle between this line, and the vertical axis. If we then take the length of the diagonal line to be Ao, then simple trig shows us that the vertical component is AocosØ. Since intensity is proportional to the square of amplitude, we get the relationship I = Iocos2Ø.
We need someone to volunteer to dig up some information about the states of polarisation.
Retarding plates: Can someone fill this in?
H.9 Corpuscular optics
Stimulated emission is characterised by the fact that is produces highly focused, coherent and monochromatic light. Spontaneous emission, however, produces light in all direction (so intensity decreases rapidly) and is not coherent. Light going in different directions may be completely out of phase (or not, you just don't know).
Stimulated emissions generally occur when electrons within atoms are excited to a higher electron shell, and then fall back of their own accord (usually very quickly). Light stimulated emission occurs when an atom which is already in an excited state is hit by a photon of exactly the same energy. This causes the electrons to immediately fall beck to its ground state, and produce a photon. Thus, we have two photons, the original one and the produced one, and they are both exactly in phase, and travelling in the same direction. Under normal circumstances this would be very rare, but in the production of laser light it is intentionally achieved.
Production of laser (Light Amplification by Stimulated Emission) light : As seen above, focused, coherent light can be produced by stimulated emission, however this is very rare in normal circumstances. For a laser to operate, the atoms within it must be an inverted population (there must be more atoms in the excited state than the ground state). This excited state must also be a metastable state, which means the electrons stay in this excited position for longer than usual before spontaneously falling back to the ground state. How the metastable inverted population is actually achieved varies depending on the type of laser (and I assume it's not required here, but it's usually done with flashes of light or an electric current). The actual device consists of these atoms contained between two mirrors at either end of a tube (the sides of which absorb light). The mirror at the end which you want the laser to come out of is partially transparent, but the majority of light hitting it will be reflected. After a while (actually a very short time), some of the electrons start falling back to the ground state, and thus produce random photons in any direction. Eventually, one of these will be travelling along the tube, and strike an excited atom, producing two photons. These two will strike two more atoms, producing 4 and so on, thus the photons bounce back and forward between the mirrors continually increasing in number. Some of the photons leak out the partially transparent mirror and leave to form the actual 'beam' of the laser.
The importance of lasers in optics stems primarily from the fact that they can be far more precisely controlled than ordinary light. Thus they are far more precise to be used in calculations. Because they are monochromatic, their can be no chromatic aberration, and because they are highly focused, they produce very sharp points rather than a general dot you might get with ordinary light. In addition to this, they do not lose intensity even over long distances, and so they are useful for taking measurements over long distances.
H.10 Contemporary optics
Different methods for producing holograms, and seeing them in coherent light (are we referring there to laser light, or to light from a coherent point source ??? )
Producing holograms (basically a duplicate of the SL section)
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. The only variation I know of on this method is to use, rather than flat film, a thick emulsion. In this emulsion, the interference pattern is recorded in 3D, producing what is known as a volume hologram, which can be seen in white light (though best with a single, small point source).
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. Volume holograms works similarly, bit it is possible the use 3 different 'coloured' lasers, red, blue and green to produce a hologram which can then be seen in full colour under white light.