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Introduction Basic Considerations Elements of a Theory Sucking Force Summary References

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Optical phase conjugation involves the use of any of a large variety of nonlinear optical phenomena to exactly reverse the direction of propagation of a light beam [16]. The basis of our hypothesis of biological organization is that living systems are able to display phase conjugation of electromagnetic waves in such a way that they may move always into the nodal planes of the impinging electromagnetic waves in order to provide that at every space- time point the re-emitted wave takes the negative amplitude of the incoming wave. Since phase conjugation suffices for the reemitted electric wave amplitude E(r)_r to become proportional to the complex conjugate E*(r)_i of the incident wave, we simply have to look then for solutions of the form E*(r) = -E(r) which means that only the imaginary (uneven part) of the field amplitude is reflected, while the real part (E*=E) penetrates the biological subject under investigation. In the present paper we show that a double layer of nonlinear polarizability may provide the necessary boundary conditions for getting desctructive interference outside, and constructive interference within the subject. However, in contrast to a nonliving system, the double layer should be movable and the whole system should use the phase information to enable the movement within a time interval that is small compared with the coherence time T into a position where at least one of the nodal planes of the electric field components of the incoming electromagnetic waves matches most efficiently the double layers of the biological system under study. Fig. 5 and 6 display this situation.

Figure 6. Double Layer of Non-Linearly Polarizable Matter.

The classical description of this phenomenon follows the usual derivation of classical electrodynamics which is presented, for instance, by A. Yariv and R.A. Fisher [16]. One starts from the Maxwell equations (1-4)
 

	equ.1
	equ.2
	equ.3
	equ.4

 

where E, B, H and D are the electric field strength, the magnetic induction, the magnetic field strength and the displacement, respectively, r is the charge density, j the current density, t the time.
 
 

By combining Ampere’s and Faraday’s (equs. 3 and 4) laws and by introducing the polarization vector P (= D - e₀ E) in homogeneous, nonmagnetic and non-conducting material without free charges, one gets the well known wave equation (in SI units):
 

equ.5

 

The linear part of the polarization which is proportional to E has been separated since only the nonlinear part P_NL which includes quadratic and higher order dependency on E plays a role for phase conjugation.

µ₀ and e are the permeability and the dielectric constant of the material under consideration, respectively.

The reversal of the direction of the reflected beam into the same optical path of the incident beam as a result of phase conjugation is a consequence of equ.5 . Actually, as soon as a solution of the wave equation including non-linear polarization is known, any backward going wave is also a solution of the Maxwell equation, as its complex amplitude is everywhere the complex conjugate of the incident wave. This means that the transformation of E(k,w ) of the incident wave into aE^*(-k,w ) provides a solution of the reflected wave, with the constant a an arbitrary complex value.

Actually, take E₁(r,t) = ½ J(r) exp(-i(w t-kz)) +c.c. as a solution of equ. 5, then we have after insertion into the homogeneous part of equ.5 (which is always valid outside of the matter under study) the condition
 

equ.6

Now, take instead of equ.6 the complex conjugated
 

equ.7

which is just the same wave equation applied to the wave propagation in the -z direction of the form
 

equ. 8

provided
 

equ.9

where a is any constant.
 
 

In general this does by no means lead to destructive or constructive interference, unless

for any instant t.

We will show now that this further condition may be satisfied under definite boundary conditions which are likely to be fulfilled in biological systems.
 
 

At first we define an optical double layer (Fig. 6) by equs 10a and 10b:
 

	equ.10a
	equ.10b

For the penetrating wave E(z>0)the second time-derivative of non-linear polarization shall be just opposite to that of the reflected wave E(z<0). It is clear that this can be realized by double layers like the cell membranes or the exciplexes (excited complexes) of the DNA [17]. Biological systems are the most suitable candidates for satisfying the boundary condition of equs. 10a and 10b.

We then have by definition
 

equ. 11

where E₁(k,z) shall describe the incident wave (k>0, z<0, z>0), and E₂(k,z) both the reflected wave for k<0, z<0 and the penetrating wave (k>0, z>0).

The energy conservation law requires for the total volume V and the partial volumes V₁(z<0) and V₂(z>0) the equality
 

equ.12

 

Finally, for the double layer under consideration we have to require that all time derivatives of E₁ and E₂ vanish for z=0.
 

equ. 13

This classical boundary condition requires the existence of a double layer where a vacuum of vanishing refraction index is sandwiched between two layers of high, but opposite polarizability. This again can be realized by a small double layer of highly nonlinear polarizability.

At the same time, the gradient of E₁ and E₂ in z.direction has to be rather high at z=0, in order to satisfy the inhomogeneous equation 5:
 

equ. 14

By taking account of these boundary conditions one finds a rather general solution of equ.5 which includes phase conjugation under destructive interference for z<0 and constructive interference for z>0. This solution takes the following form
 

equ. 15

As it can be easily verified, for phase conjugation in the zone z<0. the reflected beam E₂(r,k,w ,z,t)= E(r,-k,w ,z,t) = E^*(r,k,w ,z,t) = - E(r,k,w , z,t). Consequently, for z<0 the solution provides for phase conjugation as well as for perfect destructive interference. It fulfills all the necessary boundary conditions, including E(r,k,w ,z,t) = 0 for z=0 for all k, w , and t and the vanishing time derivatives in z=0.

For z>0, k> 0, E remains unchanged. Consequently, constructive interference takes place. Otherwise, the energy conservation law would be violated.

It is evident that this kernel of a possible solution can be modulated in the form
 

equ. 16

where P and R are slowly varying functions accounting for the penetrating and the reflected part, respectively. They describe possible variations around the ideal stationary state. It is clear that a more detailed calculation requires knowledge about the actual properties of the matter under examination.

However, this is not necessary for understanding the basic mechanism of destructive interference outside and constructive interference within the biological system.

Under these boundary conditions, the general "standing-wave"-solutions (equ. 15) provide phase conjugates in terms of interference strings with destructive interference at the outside of the system and, as a consequence, constructive interference within the system..

It should be noted that there is no confinement to the spatial and temporal pattern outside z=0 which means that every modulation of the wave is based on destructive interference by the special boundary conditions of phase conjugation effects. This means, in addition, that this mechanism is able to serve as the ideal basis of a communication system between both biological systems and the outside world and between and even within living systems, where the receiver has simply to provide the boundary conditions in terms of shifting its double layers into one of the nodal planes of the incoming carrier waves while the language is free for expressing the total information by spatio-temporal modulations of the interference fringes. It is evident that this does not hold only for the optical range but becomes obviously even more likely with increasing wavelengths. In this way, it may represent an evolutionary principle of nature.
 

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