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.

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 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.

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, B_{1}
and B_{2}, 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 B_{1} and B_{2} 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 K_{L} 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 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.

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 **K _{1}** and

**K _{1}** +

or equivalently,

**K _{2}**

The lattice vector acts in opposite directions on **K _{1}**
and

**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 B_{2} were shut off, then
beam B_{1 }together with the crystal would recreate B_{2} by
Bragg diffraction. C: Conversely, if beam B_{1} were shut off, then
beam B_{2 }together with the crystal would recreate B_{1} by
Bragg diffraction.

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,

**K _{L}** =

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 B_{1} refracted by the functionally equivalent crystal
lattice to produce a reflected beam in the direction of the original beam B_{2},
and Figure 2 C shows beam B_{2 }refracted by the equivalent crystal
lattice to recreate the original beam B_{1}. 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 B_{1 }is of higher amplitude than B_{2},
then the interference pattern between B_{1} and B_{2 }reflects
some of the energy of B_{1} in the direction of B_{2}, as in
Figure 2 A, whereas if B_{2} is of higher amplitude than B_{1},
some of the energy of B2 is reflected in the direction of B_{1}, as in
Figure 2 B. In fact, whether the two beams are of equal amplitude or not, some
portion of B_{1} is always lost to B_{2 }through the crystal,
while some portion of B_{2 }is
lost to B_{1}, 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.

To create a phase conjugate mirror we add a third *probe* beam, B_{3},
to intersect with the other two beams in the nonlinear optical element as shown
in Figure 3 A. This creates a fourth *signal* beam B_{4} 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,

**K _{1}** +

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 **K _{1}**
and

There are two ways that this phenomenon can be understood
intuitively. We can say that probe beam B_{3} interferes with pumping
beam B_{1} to produce an interference pattern as shown in Figure 3 C
along their angular bisector, then beam B_{2} reflects off that
interference pattern to produce the signal beam B_{4}. (angle of
reflection equals angle of incidence) Alternatively we can say that the probe
beam B_{3} interferes with other pumping beam B_{2} to produce
an interference pattern as shown in Figure 3 D, then beam B_{1}
reflects off that interference pattern to produce the signal beam B_{4}.
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 B_{3} is directed into the intersection of the other
beams, which produces a fourth beam B_{4.}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 B_{3 }and B_{1},
followed by a reflection of B_{2 }by that pattern to create B_{4},
or alternatively it can be seen as D: an interference between B_{3 }and
B_{2}, followed by a reflection of B_{1 }by that interference
pattern to create B_{3}.

All we need to do to complete the phase conjugate mirror is to
orient beams B_{1} and B_{2} anti-parallel to each other, so
that in vector terms **K _{1}** +

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 B_{1 }and B_{2} that cross in opposite
directions in the nonlinear optical element. When a third probe beam B_{3}
is projected into the mirror from any direction, a phase conjugate beam B_{4}
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 K_{1} + _{2}_{3} + K_{4 }also equals 0, and thus B_{4}
must be the phase conjugate of B_{3}.

Here is an improved figure courtesy of Henri Hodara showing (in blue) anti-parallel pumping beams B_{1}
and B_{2} whose interference creates the alternating solid and dashed lines, representing standing wave peaks and
troughs respectively. The signal beam B_{3} (in red) interferes with that standing wave along the green
horizontal lines, whose spacing determines the length of difference vectors K_{d} and -K_{d}
in the reciprocal lattice wave representation, resulting in the conjugate wave B_{4} (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.