# An Intuitive Explanation of Phase Conjugation

### steven lehar

#### Phase Conjugation

Phase conjugation is a fascinating phenomenon with very unusual characteristics and properties. It operates somewhat like holography, but it is a dynamic hologram, whose "holographic plate" is defined by interfering wave fronts in a nonlinear optical medium, rather than etched as a static pattern on a glass plate. In this page I provide an intuitive explanation of the essential principles behind phase conjugation.

#### Phase Conjugate Mirror

Let us begin with the properties of a phase conjugate mirror. A phase conjugate mirror is like a mirror, in that it reflects incident light back towards where it came from, but it does so in a different way than a regular mirror.

In a regular mirror, light that strikes the mirror normal to its surface, is reflected straight back the way it came (A). This is also true of a phase conjugate mirror (B). When the light strikes a normal mirror at an angle, it reflects back in the opposite direction, such that the angle of incidence is equal to the angle of reflection. (C) In a phase conjugate mirror, on the other hand, light is always reflected straight back the way it came from, no matter what the angle of incidence. (D)

This difference in the manner of reflection has significant consequences. For example if we place an irregular distorting glass in the path of a beam of light, the parallel rays get bent in random directions, and after reflection from a normal mirror, each ray of light is bent even farther, and the beam is scattered. With a phase conjugate mirror, on the other hand, each ray is reflected back in the direction it came from. This reflected conjugate wave therefore propagates backwards through the distorting medium, and essentially "un-does" the distortion, and returns to a coherent beam of parallel rays travelling in the opposite direction. How does the phase conjugate mirror DO that?

In linear optics, light waves pass through each other transparently, as if the other waves were not there, and the same is true of the ripples on a pond that also pass through each other totally unaffected after they cross. But almost any optical, or other wave phenomenon, will go non-linear if the amplitude is sufficiently high, and that is also true of water waves, to help our intuition. When waves in a ripple tank are driven too strongly, they lose their perfect sinusoidal shape, and form sharper peaks between wide valleys, like wind-driven waves on the ocean. A most extreme nonlinear wave is seen in breaking waves on the beach, whose towering crests carry with them a slug of moving water. Waves of this sort do not pass through each other transparently, but they collide and rebound energetically like colliding billiard balls. In reality, nonlinear waves exhibit both linear and nonlinear components, so that colliding waves will simultaneously pass mostly through each other unaffected, and at the same time some portion of those waves collide with, and rebound off each other, creating reflections in both directions.  This concept of waves colliding and rebounding provides the key insight into understanding the otherwise mysterious phenomenon of phase conjugation. This antiparallel rebounding of a ray of light in nonlinear optics, along with Huygen’s principle of wave propagation, are sufficient to explain some of the bizarre time-reversed reconstruction principles in phase conjugation, which is the principle that mirrors an observed property of perceptual reification.

### Huygen’s Principle

Huygen’s principle states that a wave front is mathematically equivalent to a line of point sources all along that front, because the outward-radiating rays from adjacent point sources along the front eventually cancel by destructive interference, leaving only the component traveling in a direction normal to the local orientation of the front.  This principle has an interesting spatial consequence, that if the flame front has a shape, whether curved convex or concave, or a zigzag or wavy line pattern, the shape of that wave front has a profound influence on the pattern of propagation of that front.

### Two-Wave Mixing

The interactions between nonlinear waves is illustrated by the phenomenon of two-wave mixing, performed by projecting two laser beams to cross in the volume of a nonlinear optical medium. Figure 1 A shows two laser beams, B1 and B2, that intersect through some volumetric region of space. In the volume of their zone of intersection, a pattern of standing waves emerges in the form of parallel planes, oriented parallel to the bisector of the angle between the two beams, as shown in Figure 1 A. Figure 1 B shows in two dimensions how the wave fronts from the two beams intersect to produce high amplitude by constructive interference along the vertical lines in the figure, interleaved with planar nodes of low or zero amplitude in between, due to destructive interference. Figure 1 A: Two laser beams B1 and B2 that cross, create an interference pattern in their zone of intersection. B: Constructive interference creates a pattern of high amplitude in parallel planes, parallel to the angular bisector of the two beams, with planes of low amplitude in between.  C: The wave vector diagram for the crossing beams, including a new lattice vector  KL that corresponds to the difference vector between the crossing beams.

In linear optics this interference pattern is a transient phenomenon that has no effect on anything else. However if the crossing of laser beams occurs in the transparent volume of a nonlinear optical medium,  as suggested by the rectangular block in Figure 1 A, and if the amplitude of the beams is sufficiently large, the interference pattern will cause a change in the refractive index of the nonlinear medium in the shape of those same parallel planes, due to the optical Kerr effect.  The alternating pattern of higher and lower refractive index in parallel planes behaves like a Bragg diffractor.

### Bragg Diffraction

Bragg diffraction is distinct from regular diffraction by the fact that the diffracting element is not a two-dimensional grating  of lines etched on a flat sheet, as in standard diffraction, but a solid volume containing parallel planes of alternating refractive index. Bragg diffraction was first observed in x-ray crystallography as a sharp peak of reflection at a particular angle of incidence to the crystal lattice planes. The crystal layers behave much like a stack of partially-silvered mirrors, each plane passing most of the light straight through undiminished, but reflecting a portion of that light like a mirror, with the angle of reflection equal to the angle of incidence.  However because of interference between reflections from successive layers at different depths, Bragg diffraction is stronger at those angles of incidence that promote constructive interference between reflected rays, but weakens or disappears altogether at other angles where the various reflected beams  cancel by destructive interference.  Maximal diffraction occurs at angles that meet the Bragg condition, that is,

2 d sin q = n l

where q is the angle of the incident ray to the plane of the reflecting surface, d is the distance between adjacent planes, l is the wavelength of light, and n is an integer. In words, Bragg reflection occurs at angles of reflection where the path length difference between reflections from adjacent planes differ by an integer number of wavelengths.

### Reciprocal Lattice Wave Vector Representation

The phase matching constraint enforced by the Bragg condition can be seen most easily in a Fourier space called the reciprocal lattice representation. Each beam is represented by a wave vector whose direction is normal to the wave fronts of the corresponding beam, and whose magnitude is proportional to the inverse of the wavelength, or spacing between successive wave fronts of the beam. This is a Fourier representation in that the magnitude of the wave vectors is proportional to the frequency of the corresponding wave. Mathematically, the magnitude k of the wave vector of a wave of wavelength l is given by

k = 2p / l

The convenience of this representation is that the wave vectors of waves that are phase matched so as to be in a mutually constructive relationship, form closed polygons in this space, and this can be used to determine whether the Bragg condition is met.

Figure 1 C shows the wave vector representation for the crossing laser beams depicted in Figure 1 A. The wave vectors K1 and K2 are oriented parallel to their corresponding beams B1 and B2.  The parallel planes of a Bragg diffractor, such as a crystal composed of parallel planes, can also be expressed as a wave vector because it behaves very much like a beam of coherent light to an incident beam that strikes it. As with wave vectors, the direction of this lattice vector KL is normal to the planes of the grating, and the vector magnitude is proportional to the inverse of the spacing between lattice planes.  Figure 2 A shows the nonlinear optical element replaced by a functionally equivalent crystal with lattice planes parallel to those of the standing wave. The vector diagram of Figure 1 C shows the lattice vector KL that would be required for the phase matching relation dictated by the Bragg condition to hold. In terms of wave vectors in the reciprocal lattice representation, the Bragg condition holds when

K1 + KL = K2,

or equivalently,

K2 - KL = K1.

The lattice vector acts in opposite directions on K1 and K2, which is why it is added to one but subtracted from the other. Note how the lattice vector is oriented normal to the planes of the lattice, which are parallel to the angular bisector of the two beams, as required for the angle of incidence to equal the angle of reflection. For example if the lattice spacing were somewhat larger than that dictated by the Bragg condition, that would make the lattice vector shorter, and the three vectors would no longer form a closed triangle, and thus little or no Bragg refraction would be expected to occur with that crystal, that is, the light would pass through with little or no reflection. Bragg refraction could be restored, however, by re-aligning either or both beams to make their wave vectors meet the shorter lattice vector. Figure 2. A: Nonlinear optical element replaced by functionally equivalent crystal with lattice planes parallel to the original standing waves, and with the same spacing as the standing waves. B: If beam B2 were shut off, then beam B1 together with the crystal would recreate B2 by Bragg diffraction. C: Conversely, if beam B1 were shut off, then beam B2 together with the crystal would recreate B1 by Bragg diffraction.

### Magical Reification

The magic of nonlinear optics is that when laser beams cross in the volume of a nonlinear optical medium, as depicted in Figure 1 A, the wave vector of the resultant nonlinear standing wave pattern automatically takes on the configuration required by the Bragg condition, no matter what the angle of intersection of the two beams. So although Bragg reflection occurs off a crystal only for certain specific angles that meet the Bragg condition, laser beams that cross in a nonlinear optical medium create a standing wave whose lattice vector is automatically equal to the difference between the  two crossing beams, or,

KL = K2 - K1.

This is a remarkable constructive, or generative function of nonlinear optics, creating a whole new waveform out of whole cloth, equal to the difference between two parent wave forms. This magical act of creation can be understood as a property of the fundamental resonances in the nonlinear optical material set up by the passage of high amplitude laser beams.  The laser beam sets up a resonance in the electrons that are attached to the molecules in the optical material, that makes them vibrate in sympathy with the passing wave. The difference in nonlinear optics is that this resonance takes energy to establish, as if the electron had a certain momentum to be overcome, or a capacitor that must absorb a certain charge, so that the optical material does not react instantaneously to the passing light, but  with a certain energetic time lag, that borrows energy from the wave when the wave first turns on, and repays that energy debt when the wave is shut off again, like a capacitor discharging through a resistor, or a mass-and-spring system returning to center after wave passage. This is what makes nonlinear optics automatically balance the vector equation. If one wave vector deflects the electron this way, and another deflects it that way, the electron needs to return back to center before it can start the next cycle, and that returning back to center is what closes the wave vector diagram.

If the pattern of standing waves were somehow frozen as a fixed pattern of alternating refractive index, as in a layered crystal, as suggested in Figure 2 A, then this crystal would behave like a hologram that can restore the pattern of light if one of the input beams is removed. For example Figure 2 B shows beam B1 refracted by the functionally equivalent crystal lattice to produce a reflected beam in the direction of the original beam B2, and Figure 2 C shows beam B2 refracted by the equivalent crystal lattice to recreate the original beam B1. The reification in two-wave mixing has created a difference vector that has created a redundancy in the representation that allows either one of the input signals to be removed without loss of information.

If another analogy might be helpful, consider water flowing over sand, and creating little rippling dunes, and the rippling dunes in turn force the water to ripple over them, the flowing water and the rippling sand modulating each other by conforming to each other energetically. You can see the dunes eroding constantly from their flow-ward side, and building back up again on their leeward side, causing the little sand dunes to advance slowly to leeward, all in lock step with each other and with the corresponding ripples in the water. If you could instantly smooth the sand flat, but preserve the rippling pattern in the water flow, it would immediately re-establish the ripples in the sand, by allowing sand to accumulate in the stagnant parts of the flow. In fact, the rippling pattern would automatically re-establish itself naturally anyway, due to the fundamental dynamics of the water/sand interaction. Likewise, if the sand were frozen to a static plaster cast of the ripple pattern, that pattern would coerce any water flowing over it to conform to its pattern of ripples, which the water would happily comply with, if the ripples are of the right natural frequency.

The nonlinear standing wave establishes an energy coupling between the two intersecting waves, such that one wave can “pump” or amplify the other. For example if B1 is of higher amplitude than B2, then the interference pattern between B1 and B2 reflects some of the energy of B1 in the direction of B2, as in Figure 2 A, whereas if B2 is of higher amplitude than B1, some of the energy of B2 is reflected in the direction of B1, as in Figure 2 B. In fact, whether the two beams are of equal amplitude or not, some portion of B1 is always lost to B2 through the crystal, while some portion of B2  is lost to B1, as suggested in Figure 2 A, so the net energy transfer always flows from the higher amplitude beam toward the lower. That is, the two waves are intimately coupled through the nonlinear standing wave, energy-wise, and this energy coupling is what allows phase conjugation to produce an amplified reflection.

### Degenerate Four-Wave Mixing

To create a phase conjugate mirror we  add a third probe beam, B3, to intersect with the other two beams in the nonlinear optical element as shown in Figure 3 A. This creates a fourth signal beam B4 which will eventually be our phase conjugate beam after one last modification. This configuration is known as degenerate four-wave mixing. (The word degenerate refers to the fact that the frequencies of all four beams are equal, as required for the simplest form of phase conjugation exemplified here) The direction of that fourth beam can be computed from the vector diagram shown in Figure 3 B, by the principle that the fourth beam will exactly cancel or balance the sum of the other three vectors, or,

K1 + K2 + K3 + K4 = 0.

Again, this is dictated by the phase matching constraint, whereby the only waves that will emerge are those that reinforce each other constructively, and the reciprocal wave vector diagram helps identify the conditions under which that constraint is met. If the pumping beams K1 and K2 remain fixed, then whichever way the probe beam wave vector K3 is directed from the point (K1 + K2) in the vector diagram, the conjugate beam will always return back to the origin, as shown in Figure 3 B.

There are two ways that this phenomenon can be understood intuitively. We can say that probe beam B3 interferes with pumping beam B1 to produce an interference pattern as shown in Figure 3 C along their angular bisector, then beam B2 reflects off that interference pattern to produce the signal beam B4. (angle of reflection equals angle of incidence) Alternatively we can say that the probe beam B3 interferes with other pumping beam B2 to produce an interference pattern as shown in Figure 3 D, then beam B1 reflects off that interference pattern to produce the signal beam B4. It is more accurate however to think of all four beams as interlocked in a four-way energy coupling consummated by the newly created signal beam that appears so as to balance the vector equation and maintain phase coherence between all four beams. In other words, both interference patterns of Figure 3 C and D, co-exist simultaneously along with the original pattern of Figure 1 A, interlocking the four beams in a mutually interdependent energy relation. Figure 3. A: A third beam B3 is directed into the intersection of the other beams, which produces a fourth beam B4.The angle of that new beam can be calculated from the wave vector diagram as shown in B. This can be seen intuitively as C: an interference that forms between B3 and B1, followed by a reflection of B2 by that pattern to create B4, or alternatively it can be seen as D: an interference between B3 and B2, followed by a reflection of B1 by that interference pattern to create B3.

### Phase Conjugate Mirror

All we need to do to complete the phase conjugate mirror is to orient beams B1 and B2 anti-parallel to each other, so that in vector terms K1 + K2 = 0. This in turn means that K3 + K4 = 0, which means that the reflected beam B4 must be the phase conjugate of the probe beam B3. Figure 4 A shows the configuration required for phase conjugation. Pumping beams B1 and B2 are projected into the nonlinear optical element from opposite directions where they interfere to form a nonlinear standing wave. The probe beam B3 can now be projected into the mirror from any direction, and this will produce the phase conjugate beam B4 superimposed on B3 but traveling in the opposite direction as a “time-reversed” reflection. The summation of B3 and B4 traveling in opposite directions converts the two waves into a standing wave that oscillates without propagation if they are of equal amplitude, otherwise there will be a net propagation in the direction of the higher amplitude beam. Figure 4 B shows the wave vector diagram showing how if  K1 + K2 = 0, then K3 + K4 also equals 0 no matter what angle the probe beam enters the mirror, and thus B4 must be the phase conjugate of B3.

If the pumping beams are provided at high amplitude, then the energy built up in the nonlinear standing wave can spill over to the conjugate wave, creating an amplified reflection of the incoming wave back outward in the direction from whence it came. This is the phase conjugate mirror produced by degenerate four-wave mixing. Figure 4. A: A phase conjugate mirror is produced by anti-parallel pumping beams B1 and  B2 that cross in opposite directions in the nonlinear optical element. When a third probe beam B3 is projected into the mirror from any direction, a phase conjugate beam B4 will appear as a time-reversed reflection of the probe beam in the direction from whence it came. B: The wave vector diagram shows how if  K1 + K2 = 0, then K3 + K4 also equals 0, and thus B4 must be the phase conjugate of B3.

Here is an improved figure courtesy of Henri Hodara showing (in blue) anti-parallel pumping beams B1 and B2 whose interference creates the alternating solid and dashed lines, representing standing wave peaks and troughs respectively. The signal beam B3 (in red) interferes with that standing wave along the green horizontal lines, whose spacing determines the length of difference vectors Kd and -Kd in the reciprocal lattice wave representation, resulting in the conjugate wave B4 (red) propagating anti-parallel to the signal beam. For a paper relating the principles of phase conjugate mirrors to human visual perception, see

The Constructive Aspect of Visual Perception.