Primer on Optics and Optical Refractive Synchronization
Continuing from posts #358 and #459: (definitions taken from Encyc. Brit., McGraw-Hill Encyc. of Sci & Tech, Weisstein's Physics)
Index of Refraction
The index of refraction is defined by
n = c / v
where c is the speed of liqht and phase velocity, v = w / k, w is the angular frequency and k, the wavenumber.
It gives the amount of refraction which takes place for light passing from one medium to another. A complex index of refraction can be defined for substances which absorb as well as refract.
n^2 = (c*k/w)^2 = eps + 4Pi*s*i/w
n = sqrt(eps*mu)
where eps is the permitivity of the medium and mu is the permeability of the medium in units watts per area per distance, s is the Stefan-Boltzmann constant.
The following table gives indices of refraction for various substances at lambda = 589 nm (the sodium D line)
medium index
air (STP) 1.100029
water (20'0 C) 1.33
crown glass 1.52
flint glass 1.65
diamond 2.43
Ordinary Ray
The ray in a uniaxial crystal which vibrates in the basal plane (perpendicular to the c axis) and has a spherical ray velocity surface. The index of refraction along this axis is denoted, "no"
Extraordinary Ray
The ray in a uniaxial crystal which vibrates parallel to the c axis and has an ellipsoidal ray velocity surface. The Index of Refraction along this axis is denoted, "ne".
Uniaxial
Crystals have two different indices of refraction along crystallographic axes. Light entering such a crystal is broken up into a so-called Ordinary Ray and an Extraordinary Ray. Depending on the relative size of no and ne, uniaxial crystals are described as positive uniaxial or negative uniaxial.
Birefringence
The division of light into two components (an "Ordinary" ray and an "Extraordinary Ray" found in materials which have two different Indices of Refraction in different directions. The birefringence is defined as
dn = ne - no
Crystals possessing birefringence include hexagonal (such as calcite), tetragonal, and trigonal crystal classes exhibit birefringence, and are known as uniaxial. Orthorhombic, monoclinic, triclinic exhibit three Indices of Refraction. They are therefore trireffingent and are known as biaxial. Birefringent prisms include the Nicol Prism, Glan-Foucault Prism, Glan-Thompson Prism, and Wollaston Prism. (sign of the Birefringence called the Optical Sign)
Pockels Effect
An electro-optical effect in which the application of an electric field produces a birefringence which is proportional to the field. Only crystals which lack a center of symmetry (20 out of the 32 classes) may show this effect. (These are, incidentally, the same classes which are Piezoelectric). The Pockels cell is used in ultrafast shutters. For the longitudinal arrangement, the phase shift is
phase shift = no^3*r*V/L
where r is the electro-optical constant measued in m / V, meters per volt, V is volts and L is lambda wavelength in meters.
Kerr Effect
The development of birefringence when an isotropic transparent substance is placed in an electric field. It is used in constructing Kerr cells, which function as variable wave plates with an extremely fast response time, and find use in high-speed camera shutters.
dn = L*K*E^2
where K is the Kerr constant in cm Statvolts^-2 and E is the applied electric field in volts per meter. 1 statvolt = 300 volts about.
Electro-optic Phenomena
The index of refraction n of a transparent substance is related to its electric polarizability and given by
n^2 = 1 + chi/eta
where chi is the electric susceptibility of a medium, and the equation P = chi * E relates the polarization of the medium to the applied electric field. For most matter, chi is not a constant independent of the value of the electric field, but rather depends to a small degree on the value of the field. Thus, the index of refraction can be changed by applying an external electric field to a medium. In liquids, glasses, and crystals that have a centre of symmetry, the change is usually very small. Called the Kerr effect (for its discoverer, the Scottish physicist John Kerr), it is proportional to the square of the applied electric field. In noncentrosymmetric crystals, the change in the index of refraction n is generally much greater, it depends linearly on the applied electric field.
A varying electric field applied to a medium will modulate its index of refraction. This change in the index of refraction can be used to modulate light and make it carry information. A crystal widely used for its Pockels effect is potassium dihydrogen phosphate, which has good optical properties and low dielectric losses even at microwave frequencies.
An unusually large Kerr effect is found in nitrobenzene, a liquid with highly 'acentric" molecules that have large electric dipole moments. Applying an external electric field partially aligns the otherwise randomly oriented dipole moments and greatly enhances the influence of the field on the index of refraction. The length of the path of light through nitrobenzene can be adjusted easily because it is a liquid.
When a liquid is placed in an electric field, it behaves optically like a uniaxial crystal with the optical axis parallel to the electric lines of force. The Kerr effect is usually observed by passing light between two capacitor plates inserted in a glass cell containing die liquid. Such a device is known as a Kerr cell. There are two principal indices of refraction, no, and ne , and the substance is called a positively or negatively birefringent substance, depending on the optical sign. Light passing through the medium normal to the electric lines of force (that is, parallel to the capacitor plates) is split into two linearly polarized waves traveling with the velocities c/no and c/ne respectively, where c is the velocity of light, and with the electric vector vibrating perpendicular and parallel to the lines of force.The difference in propagation velocity causes a phase difference delta between the two waves, which, for monochromatic light of wavelength L, is
delta = (no - ne)*x/L
where x is the length of the light path in the medium.
The theory of the Kerr effect is based on the fact that individual molecules are not electrically isotropic but have permanent or induced electric dipoles. The dielectric field tends to orient these dipoles, while the normal agitation tends to destroy the orientation. The balance that is struck depends on the size of the dipole moment, the magnitude of the electric field, and the temperature. This theory accounts well for the observed properties of the Kerr effect.
A so-called ac Kerr effect or optical Kerr effect has also been observed and put to use in connection with lasers. When a powerful plane-polarized laser beam propagates through a liquid, it induces a birefringence through a mechanism that is very similar to that of the ordinary, or dc, Kerr effect. In this case, it is the ac electric field of the laser beam (oscillating at a frequency of several hundred terahertz) which lines up the molecules. By using laser pulses with durations of only a few picoseconds (I ps = 10^-12 s), extremely fast optical Kerr shutters have been built in the laboratory.
Polarized Light
Interaction of plane-polarized beams. Fresnel and Arago, using an apparatus based on Young's experiment, investigated the conditions under which two beams of plane polarized light may produce interference fringes. They found that: (1) two beams polarized in mutually perpendicular planes never yield fringes; (2) two beams polarized in the same plane interfere and produce fringes, under the same conditions as two similar beams of unpolanized light, provided that they are derived from the same beam of polarized light or from the same component of a beam of unpolarized light; (3) two beams of polarized light, derived from perpendicular components of the same beam of unpolanized light and afterwards rotated into the same plane (e.g., by using some device such as an optically active plate) do not interfere under any conditions.
Result (1) is to be expected because two displacements in perpendicular planes cannot annul one another, and result (2) is also easily understood. Result (3) shows that mutually perpendicular components of unpolarized light in a beam are non-coherent. Their phase difference vanes in time in an irregular way. Unpolarized light has a randomness, or lack of order, as compared with polarized light (implying an entropy difference). This order (or lack of order), rather than the azimuthal property, is the most fundamental difference between polarized and unpolarized light. Perfectly monochromatic fight is perfectly coherent and completely polarized.
Superposition of polarized beams.
Two coherent beams of plane polarized light may be thought of as propagated in the Oz direction, one with its vector along Ox and the other with the electric vector along Oy where O is the axis origin: i.e., the two vibrations are at right angles to each other as well as to the direction of propagation. If the beams have amplitudes a(x), and a(y), and phases e(x), and e(y), then, in general, the resultant vibration R(x), R(y) , and R(z) may be represented in magnitude and polarization by a vector, or arrow, the tall of which touches the axis of propagation O(z) while the point moves round the ellipse. It goes around once when the phase angle psi changes by 2Pi, i.e., at any given place when t changes by y or for any one time when z changes by L. The beam is said to be elliptically polarized. If the phase difference is Pi/2, then the axes of the ellipse are equal to a(x) and a(y) and are along Ox and Oy
Elliptically polarized light may be regarded as the most general type of polarized light. If the amplitudes of the two waves are equal, a(x) = a(y) and the phase difference is still Pi/2, the ellipse becomes a circle and the light is said to be circularly polarized. If the phase difference e, is not equal to Pi/2, the resultant is still elliptically polarized light, but the axes of the ellipse no longer coincide with the axes of coordinates. If the phase difference e(x)(y) = 0 or Pi, the ellipse shrinks to a straight line and the light is said to be plane-polarized. In the above analysis, elliptically polarized light is regarded as the resultant of two beams plane-polarized in perpendicular planes. Conversely, it is possible to regard plane-polarized light as the resultant of two beams of elliptically (or circularly) polarized light of the same wavelength, provided that the ellipses are similar in orientatiion and eccentricity, but one beam is right-handed and the other left-handed.
Double refraction.
In the 17th century Bartholin showed that a ray of unpolarized light incident on a plate of calcite, unlike glass or water, is split into two rays. One ray, called the ordinary ray, is in the plane containing the incident ray and the normal to the surface. If the angle of incidence is varied, this ray is found to obey Snell's law of sines, equation. The other ray, called the extraordinary ray, is not in general coplanar with the incident ray and the normal; also, for it, the ratio of sines is not constant. The fact that Snell's law is not obeyed in certain directions implies that the velocity of light in such a medium, called anisotropic, depends on the direction of travel In it. The two rays are polarized in mutually perpendicular planes. This is known as double refraction, or birefringence.
In order to apply Huygens method of constructing wavefronts it is necessary to assume that, in an anisotropic medium, the wave surface from a point source consists of two sheets, or surfaces. The observation that one ray obeys both laws of refraction implies that one sheet must be a sphere, like the wave surface in an isotropic medium. Huygens assumed that the other sheet is an ellipsoid of revolution that touches the sphere either internally or externally. There is one velocity of propagation for the direction defined by the line through the two points of contact (called the optic axis) and two velocities for any other direction corresponding to light polarized in two mutually perpendicular planes.
Crystals where the ellipsoidal wave surface is contained in the spherical wave surface are said to be positive uniaxial crystals, and coversely to negative uniaxial crystals. Huygens thought all crystals were uniaxial, but later observations showed that the general form of the wave surface is more elaborate and is biaxial.
When a parallel beam of plane-polarized light is incident normally upon a thin crystal plate and the crystal is rotated about an axis normal to the plate, two orientations will be found in which a single beam of plane-polanized light emerges; for one orientation, the light is called the ordinary ray, and for the other orientation, the extraordinary ray. Two lines may be drawn on the plate (or on its mount) to indicate the direction of the electric vector of the incident beam when the orientation is such that planepolarized light emerges. The directions of these lines are called privileged directions for the given anisotropic plate; they are perpendicular to one another.
Can double refraction be extended to multiple refraction? If waves travel through a medium having a continuously varying index of refraction, the rays follow smooth curves with no abrupt changes of direction. Assume that n = n(y) and that the incident ray lies in the xy plane. If q is the angle between the direction of the ray and the y axis, then Snell's law can be written in differential form in a differential equation:
dq/dn = -tan q / n
whose solution gives the path of the ray.
As outlined above in the Pockels effect the index of refraction can be changed by the application of an electric field to a medium of appropriate characteristics. The Pockels effect can also be used to modulate a light beam as is seen in the Pockels cell modulator which is used to measure the waveform in a waveguide by comparing the phase difference between a fast clock no and refracted waveform carrier ne. When an input ac electric field as was described above in the Kerr effect, "a varying electric field applied to a medium will modulate its index of refraction. This change in the index of refraction can be used to modulate light and make it carry information.", is applied to a specially doped crystal, Kerr effect response times are realizable which are necessary for high throughput. The doped crystal is prepared with the critical anisotropic dq/dn so that radio frequency input electric field changes can be distributed discretely across the clock carrier beam lambda spectrum. Before being input the RF is modulated with a signature frequency unique to that input. The clock carrier has to be especially stable laser beam and Palmer's patent DFB laser device created a beam with the necessary properties to accommodate broad E-field embedding modulation. Refractive mixing occurs when the beam carrier is differentially refringed in the crystal and driven by concurrent E-field effects of RF inputs. |
