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H.6 Ray Optics

H.6.1

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 mover 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.

H.6.2

Optical fibers are (as mentioned previously) 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 fiber. this angle can be calculated with the equation...

• where the refractive index of the core, and

• the index of the coating - which is obvious because the number we're square rooting must be positive :). This equation is in the data book.

H.6.3

How rainbows are made...well, first you need a pot of gold, and then...

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 two rays, one for red, one for violet. both are totally internally reflected on the back, and then leave with red aimed down the most and violet up the most. 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)...not that there's any such colors as indigo and violet, they just split purple into two so there'd be seven colors...cause seven's a lucky number...isn't that interesting :)

H.6.4

The lens makers equation is:

.

• is the focal length of the lens,

• is the refractive index of the material it's made of,

• is the radius of the front side, and

• the radius of the back.

These radii are negative if the lens is concave on that side, positive if it's convex.

H.6.5

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).

H.6.6 : How thin lenses can rectify some eye problems

Myopia (Nearsightedness) - This refers to an eye which can only focus on close objects...ie, 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 refers to 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 refers to 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

H.7.1

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.

wavelength

• where wavelength is the wavelength of the light being used,

• is the aperture of the, in this case, slit and

• is the angular separation.

The more common one is for a spherical lens:

where is the diameter of the lens (in a multi lens system, for the objective lens).

Nb: this answer will be given in RADIANS... radians = 180 degrees...(this equation is given in the data book...and the other is just the same a the equation for calculating the position of the first minimum in single slit diffraction.

H.7.2 : Resolving power

This is generally used for microscopes, where the object is place close to the focal point of the lens, and is used to calculate the minimum distance between two objects for them to be resolvable.

(where is the lens focal length, and is the minimum distance).

This quation is not in the data book.

Resolving power can also be expressed as where a is the angle formed if you take a line from the center of the lens along the principle axis to the focal point, and then up to the top of the lens.

H.8 Electromagnetic Optics

H.8.1

Linear, circular and elliptical states of polarisation, and Malus' law...

Malus' Law - .

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 , and the initial intensity, , 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 is the angle between this line, and the vertical axis. If we then take the length of the diagonal line to be , then simple trig shows us that the vertical component is . Since intensity is proportional to the square of amplitude, we thus get the relationship: .

Still needs stuff about the states of polarisation...

H.8.2 : Retarding plates=

H.9 Corpuscular optics

H.9.1

Stimulated emission is characterized 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 cause the electrons to immediately fall beck to it's 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 traveling in the same direction. Under normal circumstance this would be very rare, but in the production of laser light, this 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...that is, 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 flashed of light or passing electric current through them). The actual laser 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 traveling 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...but decreasing also because some 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

H.10.1

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...Well, this is basically what I said in the SL bit...to recap...

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, this 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, plus, it is possible the use 3 different 'colored' lasers, red, blue and green to produce a hologram which can then be seen in full color under white light.

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