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

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

Wednesday, July 27, 2011
Hold a (linear) polarizing film or lens up to a light, and the light&#8217;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&#8217;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&#8217;ve got light that&#8217;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&#8217;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&#8217;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&#8230;

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?