Time Reversal and Phase Conjugation

By John Kooiman,

I think that I'm finally beginning to get a handle on some of these "time-reversed" and "phase conjugation" concepts, so I'll start by trying to answer some of your questions. This topic can be very confusing, since the word "conjugate" is used to represent different kinds of "conjugation". To determine if time reversal is involved, I find it helpful to ask: "What is the behavior of the resulting wave? Does it arrive somewhere AFTER it was emitted (or reflected) or BEFORE? Does it DIVERGE or CONVERGE?

A standard silvered mirror is indeed an E-field short, and the E-field is conjugated, but this does not result in time reversal as can be seen by the fact that the reflected wave arrives somewhere after it is reflected and continues to diverge. The reason why can be understood by analogy to a transmission line example.

The voltage on a transmission line is expressed by:

V(x) =3D Vplus*( e^(-jBx) + e^(+jBx) )

And B =3D 2*PI*f/v,

where f =3D frequency, v =3D velocity, x =3D distance.

Or: V(x) =3D Vplus*( e^(-j2*PI*f/v*x) + e^(+j2*PI*f/v*x) )

At first glance you might say: "Wow, one term is positive and one is negative, is this time reversal? The answer is no, this is space reversal instead. The negative sign is associated with the "negative x" direction.

As far as your AM side band theory goes, we can tell from the effects that this is still a time-forward wave, since it arrives somewhere after it is generated and diverges along the way. Otherwise the lower side band would arrive before the upper side band, which doesn't happen. I think that the explanation for this is that the "negative f" in the above equation has the same effect as the "negative x". Regarding beating the carrier down to zero, whenever you mix two frequencies, you always get the sum and difference frequencies. If you mix a frequency with itself, you get a doubled frequency and a DC component. This is how commercially available frequency doublers work. No time reversal here.

To really understand Optical Phase Conjunction, we need to first understand what is the difference between a time-forward and a time-reversed EM wave. To this end, I have prepared the attached graphic, which illustrates my hypothesis of the relationship of the E & M fields in these two different kinds of waves.

Please refer to this graphic to help make sense of the following discussion:

Assuming a linear polarized plane wave, a normal time-forward (or "retarded") wave obeys the "Right Hand Rule" as follows: Extend the index finger of your right hand so that it is pointing straight out. Then, bend your middle finger so that it is at a right angle to your index finger. Now, extend your thumb so that it is at a right angle to both of the above. Align your index finger with the E-field and your middle finger with the H-field and your thumb will now be pointing in the direction of propagation. A time-reversed (or "advanced") wave will obey the corresponding "Left Hand Rule" instead, as illustrated in the attached graphic.

How does this work in the case of a reflection from a short circuit? Upon encountering a short circuit, the E-field is conjugated, as it must be to assure zero volts across the short circuit, and the wave is reflected in the –x direction. However, the combination of the reversed E-field and the reversed direction effectively cancel each other out and the reflected wave continues to follow the "Right Hand Rule", as can be visualized by imagining the time-forward graphic flipped 180=B0 end-for-end in the E plane.

Similarly, in the case of a reflection from an open circuit, the H-field is conjugated, and the wave is reflected in the -x direction, which also cancel each other out, as can be visualized by imagining the time-forward graphic flipped 180=B0 end-for-end in the H plane.

So, what does it take to generate a time-reversed reflection? We need something that will either conjugate BOTH the E & H fields simultaneously, OR NEITHER.

It is also instructive to consider this problem from the particle point of view. In this view, the time-forward wave consists of a stream of photons. When they are reflected from a normal mirror, they are still photons that happen to now be traveling in the opposite direction. However, a time-reversed wave may be considered to consist of a steam of anti-photons. Quantum mechanics tells us that any anti-particle may be viewed simply as the original particle traveling backwards in time. In the case of particles with mass, like an electron, it is apparently impossible to turn a particle into its' anti-particle and if the two should ever meet they result in total annihilation. However, quantum mechanics also tells us that the massless photon is its' own anti-particle and this provides some hope that it may be possible to turn a photon into an anti-photon. This is not necessarily what is happening in the case of the Optical Phase Conjunction Mirror though. Most of these devices seem to rely upon the quantum mechanical effect of an atom absorbing a photon and emitting one or more correlated anti-photons instead, as I will explain below.

In scanning the Internet for information on Phase Conjugated Mirrors, I found that there are a number papers describing this phenomena in all sorts of different media. What most of these papers seem to have in common though, is that they use a media that is suitable for producing a laser beam. That is, they use a media that can be excited, or "pumped", by a pair of cross-fired laser beams, to excite the electrons in the media to higher orbital states, so that when stimulated by the "tickler beam" these excited electrons will drop back down and emit a photon that is correlated to the "tickler beam". These cross-fired laser beams also set up a standing wave in the non-linear media that tends to act as sort of an electromagnetic diffraction grating.

Now, returning to the wave description, one can imagine the "tickler beam" to be the time-forward wave as illustrated in the graphic. From the graphic, one can also see that if the direction of propagation of the time-reversed beam is flipped around, then the E & M fields can be brought into alignment with the time-forward beam. It is my hypotheses that the "tickler beam" imprints its' EM field pattern on the excited atoms and causes them to emit their photons in synchronization with this, while the "electromagnetic diffraction grating" somehow causes the photons to be spit out in the opposite direction, thereby effectively creating a stream of anti-photons that travel backwards in time.

Is it possible to turn a photon into an anti-photon without going through this absorption and emission process? Maybe. I have found an abstract for a paper (Theory of Optical Phase Conjunction in Kerr Media, H.F. Arnoldus and T.F. George, Physical Review A 51 ,1995, p4250) that claims to have solved Maxwell's equations for four wave mixing in a crystal media that not only predicts the time-reversed reflected wave, but also a time-reversed transmitted wave in addition to the time-forward reflected and transmitted waves. When I finally located the body of the paper at the Physical Review web site (http://prola.aps.org/abstract/PRA/v51/i5/p4250_1), the server informed me that I need to first send them \$100 for an annual subscription before I can download the paper. This seems to be a bit of a high price to pay for a single paper that may or may not be helpful in understanding this phenomena. If anybody out there has a subscription to this site, I would appreciate if you could forward a copy of this paper to me (john.kooiman@andrew.com).

Since there is a microwave equivalent to a laser, called a MASER. It would seem that it should at least be possible to reproduce the optical phase conjugation technique using microwaves. The most common Maser uses a chromium doped ruby crystal as its media, biased in a magnetic field to split the energy states of the + and - spin electrons. This is necessary to split their corresponding spectral frequencies, so that it can be pumped at one frequency and emit at a slightly different frequency. It also has to be super cooled to 2=B0K to prevent the thermal noise from overwhelming the electron energy states. Therefore, a likely starting point would be to try four wave mixing in this magnetically biased, chromium doped, ruby crystal media. The frequencies and the strength of the magnetic field all have to be chosen very carefully, in order to get everything to match up correctly with the appropriate electron energy states, to make this work. This would seem to offer the possibility of a backwards in time communication device, but I'm not sure how this would relate to anti-gravity or time distortion.