The Teachings of don James

The Incident in the Mountains

Nothing is more enjoyable than walking through silent, spectacular hills in autumn. Body and spirit merge with the landscape, and all irritating thoughts gradually fade away.

My companion - James Bond - did not appear nearly as relaxed. He suddenly stopped, and his eyes slowly converged, fixing on a point only inches from his nose.

``Feeling dizzy?''

For a while he did not answer. Then he replied in a seemingly unrelated manner:

``Have you read Journey to Ixtlan by Carlos Castaneda?''

``That scandalous mystification?''

``Well, I would not be so critical. In my opinion, it is a rather gifted popularization of ideas from Wittgenstein and other authors whom Castaneda met during his adventurous journey through the library of the University of California.

But that is beside the point. I was just recalling one of his magical tricks: if you disentangle your eyes and look between two diverging images, you may notice something that is normally invisible...''

``Demons, angels, esoteric creatures?''

``I have no idea. I had only just begun to see something before you interrupted me so tactlessly...''

I do not know what James was about to see. His speculations were as obscure as ever. Yet one idea hidden within his strange discourse sounded very familiar to me: inducing interference effects in an image.

Electron Holography

That is precisely the principle behind so-called electron holography.

Suppose we send an electron wave through an object and record its image on an electron detector. If the object absorbs or scatters electrons, we observe its shadow. But what if the object is made of a material that does not absorb electrons and only slightly modifies the phase of the electron wave?

An exotic case? Not at all. Such objects are quite common.

Unfortunately, they are invisible to the detector as only the amplitude, not the phase is registered.

Now let us split the electron wave into two parts. One part passes through the object, while the other travels through vacuum. We then recombine the two waves at the detector.

As a result of interference, the image becomes decorated with periodic intensity variations---interference fringes.

Take a look at such an image, a so-called hologram. Suddenly, the particle becomes visible. The interference fringes are shifted at the location where the particle resides. One of the waves accumulates a phase shift while passing through the object, and the entire interference pattern is displaced accordingly.

Fine. But how do we quantify such an image?

For a one-dimensional phase-shifting feature, the answer is straightforward: simply measure the displacement of the fringes. The fringe shift directly reflects the phase shift.

Things become more complicated in two dimensions. What happens where the particle boundaries intersect the fringes at an angle?

At this point, my secret skills as a data scientist enter the scene.

Compute the FFT of the hologram. There we observe two prominent copies of the same pattern shifted relative each other. These are known as sidebands. The shift between their centres reflects the periodicity of the interference fringes.

The sidebands are perfectly symmetric, so one of them is sufficient for treatment. Draw an aperture around a sideband and perform an inverse reconstruction.

Voilà!

The phase-shifting particle emerges clearly from the background.

I shall spare the reader the rather tedious explanation of why this works. Let us simply regard it as a magic trick à la Castaneda.

Strictly speaking, this is not exactly what James was experimenting with. He was overlapping two images rather than an image and an empty reference wave. Nevertheless, some form of interference effect must have arisen.

What I cannot understand is how he managed to track the resulting interference fringes. The two images were not superimposed on his retina; they were combined somewhere within his brain.

At that point, my knowledge becomes useless. Only God knows what was happening inside the mind of James.

The Python codes can be found in the pdf version of this document: Full Text with Codes.

If you have any comments or suggestions, please email pavel@temdm.com ".

Posted June 8, 2026

Alice (June 8, 2026)

A citation from Wikipedia: "A hologram is a recording of an interference pattern that can reproduce a 3D light field". Where is a 3D object in your example?

Pavel (June 8, 2026)

It is often but not necessarily reconstruction of a 3D objects. The key property of a hologram is the use interference within a 3D light field.