Thursday, December 22, 2011

Refraction

Light waves are part of the EM wave spectrum. When moving through an optical medium (i.e. air, glass, etc. …), the E field of the wave excites the electrons within the medium, causing them to oscillate, as a result, the light wave slows down slightly due to the loss of some of its kinetic energy. Its new speed is always less than that of the speed of light in a vacuum (v<c). Materials are characterized by their ability to bend as well as slow down light, which is known as optical refractive index (n).

```         c
n = -
v
speed of light in a vacuum
= ----------------------------
speed of light in the medium```
```n = 1 in a vacuum
n = more than 1 in all other media```

Refraction itself occurs when light passes across an interface between two media with different indices of refraction. As a general rule (which can be derived by Snell’s law below), light refracts towards the normal when passing to a medium with a higher refractive index, and away from the normal when moving to a medium of lower refractive index.

Snell’s Law:

n₁sinα = n₂sinβ

where n₁ is the refractive index of the first medium

Reflection

One of the properties of a boundary between optical media is that some of the light that’s approaching the interface at the angle of incidence (α) is reflected back into the first medium, while the rest continues on into the second medium at the angle of refraction (β).

Angle of incidence = Angle of Reflection

Tuesday, August 30, 2011

Imagine you had a function P that upon swallowing a subset E of a universal set Ω will return a number x from the real number line. Keep imagining that P must also obey the following rules:

1. If P can eat the subset, it will always return a nonnegative number.
2. If you give P the universe Ω, it will give you back 1.
3. If you collected together disjoint subsets and gave them to P to process, the result would be the same as feeding P each subset individually and adding the answers.

Simple, if odd out of context.

Mathematicians have a curious way of pulling magic of out simplicity.

~

Probability today is studied as a mathematical science based on the three axioms (flavored by set theory) stated above. These are the “first principles” from which many other, derivative propositions have been speculated and proved. The results of the modern study of probability fuel many branches of engineering, including signals processing in electrical and computer engineering, the insurance and finance industries, which translate probabilities into economic movement, and many other enterprises. Along the way it borrowed from the other giants of mathematics, analysis and algebra, and goes on generating new research ideas for itself and other fields. This is the way of math: set down a bunch of rules (preferably simple to start) and see how their consequences play out.

But what is probability? If it is quantitative measure, what is it measuring? How valid is that measure and how could it be checked? Even these are rich questions to probe. A working qualitative description for practitioners might be that probability quantifies uncertainty. It answers with some degree of success such questions as “What is the chance?” or “How likely is this?” If a system contains uncertainty, probability provides the model for handling it, and data gathered from the system can validate or improve the probability model.

According to Wikipedia, there are three main interpretations for probability:

1. Frequentists talk about probabilities only when dealing with experiments that are random and well-defined. The probability of a random event denotes the relative frequency of occurrence of an experiment’s outcome, when repeating the experiment. Frequentists consider probability to be the relative frequency “in the long run” of outcomes.
2. Subjectivists assign numbers per subjective probability, i.e., as a degree of belief.
3. Bayesians include expert knowledge as well as experimental data to produce probabilities. The expert knowledge is represented by a prior probability distribution. The data is incorporated in a likelihood function. The product of the prior and the likelihood, normalized, results in a posterior probability distribution that incorporates all the information known to date.

~

So let’s reinterpret the math.

Let Ω be the sample space, the set of all possible outcomes, be Ei be subsets of Ω which denote different events for different i, and 𝔹 be the set of all events. Then a probability map P is defined as any function from 𝔹 → satisfying

1. P(Ei) ≥ 0
All probabilities are non-negative.
2. P(Ω) = 1
It is certain that one of the outcomes of Ω will happen.
3. Ei ∩ Ej = ∅ if i≠j ⇔ P(∑i Ei) = ∑iP(Ei)
Probabilities of disjoint events can be added to get the probability of any of them happening.

Image generated by Rene Schwietzke using POV-Ray, a raytracing freeware that creates 3D computer graphics.

A First Course in Probability (8th ed., 2010), Sheldon Ross.
Probability and Statistics (4th ed., 2010), Mark J. Schervish and Morris H. Degroot.

Wednesday, July 27, 2011

Hold a (linear) polarizing film or lens up to a light, and the light’s intensity will decrease by half.

Most polarizing lenses are made of parallel chains of polymers, so when electromagnetic radiation hits this sheet of chains, it excites electrons in the polymers to vibrate along the chains, and the electric field that gets transmitted is the component that is parallel to the chains.

From an unpolarized light source where the incident radiation has electric field vectors oriented in random directions, there will be approximately equal amounts of vector components oriented parallel and perpendicular to the polarizer’s axis (the direction of the polymer chains). Thus, only half of the total amount of electric field per unit of time through the area of the sheet gets transmitted, and this quantity - electric field x time x area - corresponds to the intensity of light.

But once you’ve got light that’s polarized along one axis, what if you pass it through another linear polarizer? If this second polymer sheet is oriented in the same direction as the first, then everything that got through the first one will get through the second one. It’ll be as if there were only one polarizer where the incident light is concerned. Introduce an angle between the two axes of polarization, and now the linearly polarized light from the first polarizer will get split into components parallel and perpendicular to the axis of the second, and only the parallel components will get transmitted. Intensity is proportional to the square of the electric field amplitude, so if θ is the angle between the polarization axes, then

Itransmit = (E||/Etotal)2 Iincident = (Etotal cos(θ)/Etotal)2 Iincident = cos2(θ)I_incident

This can be observed experimentally if you take two linear polarizers and rotate one against the other - you will see the light go from about half intensity of the surroundings to completely dark and back again. (Watch at :50)

At theta = π/2, the polarizing axes are perpendicular to each other, and the light that is transmitted through the first polarizer has no chance of passing through the second one. But if you insert a third polarizer between these two, it is possible to transmit light through all three polarizers. This is because as long as the radiation that is incident on the last polarizer is not polarized perpendicularly to the axis of polarization, light will be transmitted because there will be a component of the phase vector that is parallel to the polarizing axis. Since the middle polarizer can be oriented to change the polarization of the incoming light (any orientation besides totally perpendicular or totally parallel will work), you will be able to pass light through three polarizers even when the first and last and oriented so that alone they would completely block the transmission of light.

Experiments such as these demonstrate the wave nature of light; the wave model for electromagnetic radiation is based on Maxwell’s equations (or more compactly the Riemann-Sommerfeld equation) which today forms the basis of the classical optics. It was a major development after geometrical (ray-based) optics, which is still a useful model for studying light propagation through lenses, and is juxtaposed with/a precursor to quantum optics and nonclassical light, which employs the particle nature of light (as photons).

Linear polarization of light also serves as an illuminating analogy for the Stern-Gerlach experiment of quantum mechanics…

Tuesday, July 19, 2011

Light becomes polarized because of its wave nature.

As electromagnetic radiation (in the visible spectrum), a light wave is composed of both electric and magnetic field components, and is usually represented by a phase vector that encodes information about just the electric field. The phase vector points in the direction of the electric field, and its magnitude denotes the electric field strength. Because the phase vector can be decomposed into orthogonal components that oscillate sinusoidally, light is called a wave, and the phase of the wave at a certain place and time refers to its place along the sine curve.

Examine a planar wave, which travels along the z axis with electric and magnetic fields in the x-y plane. Let E(t,z) be the phase vector with orthogonal components x(t,z) and y(t,z) that are time- and space-dependent. If the x and y field components oscillate with amplitudes Ex and Ey, then

Ex2 + Ey2 = ||E||2 for all t, z
x(z,t) = Excos(kz-ωt1)
y(z,t) = Eycos(kz-ωt2)

where the following are constants:

ω  is the frequency, in units of Hertz (Hz) or radians/second,
k  is the (angular) wavenumber, in units of radians/meter, and
φ1, φ2 are the phases, in units of radians.

Polarization is the phenomenon of light waves having the same spatial orientation of their phase vectors (if it happens in nature), or the restriction of the phase vectors to certain orientations (by experiment). Ordinary sunlight is generally unpolarized because the direction of the individual phase vectors are aligned randomly with each other as they oscillate. By passing unpolarized light through a linear polarizing filter, waves result whose phase vectors only oscillate along a particular axis, say the horizontal (x) axis. One has “filtered out” the vertical (y) electric field components from every wave that passed through the linear polarizer. Thus Ex = E is the amplitude for the whole wave, and

x(t,z) = E cos(kz-ωt)
y(t,z) = 0

If you projected the endpoint of the phase vector onto a cross-section of the travelling wave (a picture called a Lissajous curve), you would see a line - hence the name, linear polarization.

Other polarizations are possible where the direction rather than the magnitude of the phase vector changes. If one takes the horizontally polarized light, tilts it 45 degrees to halfway between horizontal and vertical (like the animation), and then passes it through a quarter wave plate that slows down electric field along the horizontal axis by a quarter phase, one obtains circularly polarized light. The horizontal component of the phase vector now oscillates a quarter phase (2π/4 = π/2) behind the vertical components, resulting in the parametric equations

x(t,z) = (1/√2) E cos(kz-ω(t-π/2))
y(t,z) = (1/√2) E cos(kz-ωt)

which, keeping z constant, represent a circle (the 1/√2 comes from sin 45 deg and cos 45 deg). You can calculate that the phase φis

φ1 = ωπ/2

If the polarizing plate inserts a different phase in the x field component, then you’ll get elliptically polarized light.

How can you tell what polarization light has, or if it’s polarized at all? Unless you have a sensory organ that detects the direction of the electric or magnetic fields around you, the only way change you can see after polarizing a light source is a reduction (usually) in the intensity of the light, which is proportional to the square of the electric field amplitude. For example, the intensity of ordinary sunlight is approximately halved when you put on Polaroid sunglasses.

What happens when you pass light through two linear polarizers? Three linear polarizers?