The Computational Dipole
31 May 2018

The computational dipole is an alternate representation of 1-bit states, similar to the computational sphere (sometimes called the Poincare or Bloch sphere). However, unlike the computational sphere, the computational dipole manifests as a simple physical optical system existing in normal 3-d space. This system, or field, consists of polarized light, such that every (x, y, z) location contains a different 1-bit optical state, and (almost) every state on the sphere is repeated at multiple points in the unbounded 3-d volume of the dipole. The polarized field is generated by two point sources of light separated by a short distance (hence the name "dipole"), where the two sources are polarized orthogonally to one another (i.e. they are states located at antipodal points on the sphere). All other points in the field are linear combinations of these two primary states, with respect to magnitude and phase, just like on the computational sphere. The two light sources thus form a basis for the sphere (i.e. an axis). Replacing the two sources with other antipodal points (e.g. H-V, D-A, R-G, etc...) effects a change of basis of the sphere. The light field of the computational dipole is radially symmetric, owing to the arrangement of the two sources. Therefore all information in the field can be obtained in a single half-plane passing through the two sources, and then rotated about the axis between them.

For a brief tutorial on the diffraction and interference of polarized light, please see these pages.

The following images show one half-section through the computational dipole in the XZ plane. The two light sources are at left and are separated by 5 wavelengths (λ = 635 nm). The entire square is 10x10 wavelengths. In this case the upper source is H polarized, and the lower source is V polarized. The plot at left shows the overall magnitude of the illumination, while the plot at right shows the phase difference between the V component and the H component of the light at each point:

The next plot at left shows the relative magnitudes of the H (blue) and V (red) components of light at each point. The plot at right shows the resulting ellipses defined by the relative magnitudes and phase difference at each point. Green ellipses have a clockwise sense, while red ellipses are anti-clockwise. Yellow line segments are linearly polarized:

The next plots show the coverage of 1-bit states on the computational sphere for the HV basis. The ellipse at every (x, z) point in the plot above (including those not shown) is represented by a single point on the sphere. A location on the sphere is given by its {psi, chi} angles, which correspond to longitude and latitude. The plot at right shows these points in a simple rectangular projection (psi = x axis, chi = y axis). Click on the image for a full-size image of the sphere. You will also have to fully zoom the resulting image in your browser (e.g. click again) to avoid drawing artifacts:

The coverage of states is not quite complete for this basis, or for any other set of orthogonal points. This is because:

  1. The dipole is computed using a grid of points with a finite resolution. As the grid gets finer, and contains more points (and takes longer), the coverage of the sphere becomes more continuous.
  2. The two light sources are never entirely "pure", in that each is slightly contaminated by light from the other source. Therefore the axis states cannot be exactly reached, except when the two sources are moved infinitely apart.
  3. The great circle equidistant between the source states is only reached at one point. This is because, for equal intensity sources with zero phase difference, the intensity of both are equal only on the perpendicular plane equidistant between them, and everywhere on this plane the two components of the state remain in phase. All points in this plane have the same (linear) state.

There are other parameters in addition to the basis axis that can be varied in this optical system, including the phase difference between the two sources, the intensity difference between the sources, the separation between the sources, and the opacity (optical depth) of the propagating medium, and these will alter the resulting coverage of the sphere. Some of these parameters will be explored below. But, by varying the illumination states or other parameters, all points of the sphere can be reached. In general the limiting case for any particular set of light sources and parameters will be the sphere minus (a) the two source polarizations (antipodal points) and (b) the great circle equidistant between the source states, except for one point connecting the two hemispheres.

The following montage shows how every radial cross-section of the dipole is an identical pattern of polarization:

Here are images of the dipole as a grid of (x, y, z) points rather than sections:

The computational dipole can be based on any of an infinite number of axes. For example, here is the state at {psi = pi/8, chi = pi/8}:

Here are the radial polarization field and relative intensities of H and V components for a dipole created by this state and its orthogonal complement:

Here is the coverage of the {psi, chi} sphere and plane for this basis:

Here is this dipole in 3-d:

In general, every covered point is reached multiple times in the infinite volume of the dipole, although at considerably diminished intensity as the distance from the light sources increases. Here are plots of the locations of points that are close to the states {psi = pi/8, chi = pi/8} (in green), and its orthogonal counterpart (in red), in the HV dipole:

Here are plots of the HV dipole cross section and sphere coverage for light source separations of 1 wavelength (left) and 10 wavelengths (right). Note that as the separation increases, so does the coverage of the sphere and the number of times each state is reached:

Here are plots of varying the phase of the lower light source. At left the phase has been increased by pi/4, and at right it has been decreased by pi/4. The effect of this is to rotate the computational sphere by theta about the X axis. This is a unitary Z operation (go figure), sometimes called a "phase gate":

Here are plots of varying the intensity of the lower light source. At left the intensity is 1/16 that of the upper source, and at right it is 4 times that of the upper source. The effect is to distort the sphere toward one or the other of the basis states:

The default opacity of the propagating medium is set to optical depth at 1 wavelength. This means that the intensity of light falls to 1/e (~0.3679) one wavelength away. The magnitude falls as the square root of the intensity:

medium: 'optical depth 1 wavelength', 6.35e-07 0.606531
  0.000000000 : 1.000000 : ********************************************************************************
  0.000000032 : 0.998588 : ********************************************************************************
  0.000000063 : 0.994442 : ********************************************************************************
  0.000000095 : 0.987697 : *******************************************************************************
  0.000000127 : 0.978488 : ******************************************************************************
  0.000000159 : 0.966952 : *****************************************************************************
  0.000000190 : 0.953222 : ****************************************************************************
  0.000000222 : 0.937435 : ***************************************************************************
  0.000000254 : 0.919726 : **************************************************************************
  0.000000286 : 0.900230 : ************************************************************************
  0.000000317 : 0.879082 : **********************************************************************
  0.000000349 : 0.856417 : *********************************************************************
  0.000000381 : 0.832372 : *******************************************************************
  0.000000413 : 0.807081 : *****************************************************************
  0.000000444 : 0.780680 : **************************************************************
  0.000000476 : 0.753304 : ************************************************************
  0.000000508 : 0.725087 : **********************************************************
  0.000000540 : 0.696167 : ********************************************************
  0.000000571 : 0.666677 : *****************************************************
  0.000000603 : 0.636753 : ***************************************************
> 0.000000635 : 0.606531 : *************************************************
  0.000000667 : 0.577648 : **********************************************
  0.000000698 : 0.551392 : ********************************************
  0.000000730 : 0.527418 : ******************************************
  0.000000762 : 0.505442 : ****************************************
  0.000000794 : 0.485225 : ***************************************
  0.000000825 : 0.466562 : *************************************
  0.000000857 : 0.449282 : ************************************
  0.000000889 : 0.433236 : ***********************************
  0.000000921 : 0.418297 : *********************************
  0.000000952 : 0.404354 : ********************************
  0.000000984 : 0.391310 : *******************************
  0.000001016 : 0.379082 : ******************************
  0.000001048 : 0.367594 : *****************************
  0.000001079 : 0.356783 : *****************************
  0.000001111 : 0.346589 : ****************************
  0.000001143 : 0.336961 : ***************************
  0.000001175 : 0.327854 : **************************
  0.000001206 : 0.319227 : **************************
  0.000001238 : 0.311041 : *************************
  0.000001270 : 0.303265 : ************************

Here are plots of varying the optical depth of the medium from 1/4 wavelength (left) to 4 wavelengths (right). Note that although the sphere covering changes considerably, only the near field of the dipole which is close to the light sources (at left in ellipse plots) changes. The far field is virtually identical, indicating that much of the diversity of states is created near to the light sources, and at sub-wavelength distances:

As a final amusement, here are a couple of paintings I did several years ago. Precognition?

©Sky Coyote 2018.