Confinement of Light:
Standing Wave Transformations in a Phase-Locked Resonator
L. J. Reed
Torrance CA 90504
e-mail: larryreed@dphi-dt.net
Abstract/Synopsis:
Electromagnetic resonant wave interactions in a phased-locked resonator at rest and in motion are compared.
The origin of mass and inertia as a standing wave interaction in a phased-locked cavity as demonstrated in work by
Jennison is reviewed and phase relationships illustrated. For matter (composed of resonant EM standing waves) in
motion, the Lorentz contraction is interpreted as a physical wavelength compression due to variation in EM field
energy density as measured by vacuum refractive index KPV. Dipole radiation emitted from a phase-locked
resonator in motion is described. A graphical representation of Ivanov-LaFreniere standing wave transformations
is shown. Experimental possibilities for potential phase conjugate wave phase-locked resonator development are
discussed including inertia modification and propulsion.
Keywords: electromagnetic (EM), standing wave, travelling wave, phase-locked resonator, confined light, Lorentz, mass,
frequency, oscillator, inertia, phase conjugate waves, de Broglie, Doppler, photon, electron
Page 1 of 22
1. Introduction
Standing electromagnetic waves in a phase-locked resonator have been shown to explain the origin of mass and
inertia while standing wave interactions between coupled resonators provide an explanation for the origin of
gravity. In this paper, the focus is on the internal dynamics of an isolated resonator. Properties of light when
confined within a phase-locked cavity are reviewed and graphical representations of Lorentz contracted standing
waves of phase-locked cavity resonators in motion are portrayed. Radiation and propagation characteristics of EM
waves are briefly reviewed and illustrated. Motion of a phase-locked resonator with phase conjugate mirrors is
described. Potential for induced motion of a phase-locked, phase conjugate resonator by energy input of pump
beams of simulated Doppler shifted frequencies is discussed.
Jennison and Drinkwater[1,2,3] have shown that a standing EM wave trapped in a phase-locked cavity exhibits rest
ass a d i t i si i e tia a d lassi all de i ed Ne to s “e o d La F = a a d the Ei stein relation (E = mc2).
For a free-floating wave system consisting of two counter-propagating travelling waves in a phase-locked resonant
cavity, application of an external force results in an imbalance of radiation pressure of Doppler-shifted waves
causing the wave system to move as a whole in a stepwise series of velocity increments. Upon application of an
external force to the motive boundary, the blue-shifted incident wave exerts an excess radiation pressure on the
reflecting wall and the red-shifted reflected wave exerts a decreased radiation pressure on the motive wall
provided the force was applied for an interval equal to or greater than the return of the reflected wave.
Motion of a phase-locked resonator exhibits an oscillatory pulsation in the direction of motion and gives rise to
transverse EM waves. Macken[4] observes that coherent light confined in a reflecting box exhibits many of the
same properties of fermions including inertia (rest mass), kinetic energy, deBroglie waves, phase velocity, Lorentz
contraction, time dilation, etc with the same energy (E = h = mc2). The relativistic contraction due to combination
of two Doppler shifts in two opposing propagating waves produces a net decrease in Compton wavelength by
factor of (1 - v2/c2). An electron at rest corresponds to a confined photon of energy E = 0.511 MeV in a fixed
reference frame.
2. Confined light
A freely-propagating photon in a zero curvature vacuum has zero rest mass, but when trapped between two Bragg
mirrors in a phase-locked cavity resonator, light acquires rest mass. A light beam upon reflection from a mirror
undergoes momentum reversal cancelling the momentum of the incident wave. Jennison and Drinkwater[1] in
9 de i ed Ne to s se o d la fo a phase-locked cavity model of a wave system representing a fundamental
particle. A trapped standing wave exhibits not only rest mass but also intrinsic inertia. This effect is illustrated in
Figure 1. Under acceleration, forward and backward propagating waves interact undergoing Doppler shifts
resulting in an imbalance of radiation pressure. The total energy ET of the system consists of the potential energy
EP required to hold the system together plus the forward and backward wave energy EWF + EWB. At rest, the wave
energy equals the binding energy. Application of an external force F results in an acceleration F = (2/c2)(ET - EP)a =
Eta/c2 = m0a provided that the force is applied for a duration t greater than or equal to the time to complete a
feedback loop otherwise the excess incident radiation is re-radiated back into space.
The inertial force is a function of only half the total energy of the system as the potential energy makes no
contribution. Inertia is the result of internal self-referral dynamics of EM standing waves in an isolated phaselocked cavity oscillator when subjected to an external force. Rest mass is the result of EM momentum transfer at
the wall boundaries of the oscillator during wave reflection resulting from wavefront deceleration. A photon
propagating in an optically dense medium acquires an effective mass as the velocity is slowed (c = c0/n = KPVc0).
Page 2 of 22
Jennison postulated a tentative model of the electron consisting of two orthogonal spinning EM waves in phase
quadrature phase-locked at the Compton wavelength c into a closed-loop system. An equivalent model is that of
a toroid generated by two orthogonal spinors representing a photon helix forming a closed-loop soliton wave.
Figure 1. A phase-locked resonator in motion represented here as a closed-loop soliton wave exhibits
intrinsic inertial properties a d illust ates the e ha is of Ne to s la s of otio .
As illustrated in Figure 1, a phased-locked resonator in motion exhibits an oscillatory rhythmic pulsation and
represents a kind of clock. The oscillatory motion of the boundary walls form, in effect, a Hertzian dipole
antenna generating a dipole radiation field transverse to the direction of motion. The far field radiation pattern
corresponds to that of a half-wave dipole and is similar to that of a magnetic loop antenna.
Macken[4] has elaborated on the mechanism of origin of mass from energy of confined light noting that a massless
photon acquires rest mass when confined in closed-loop reflecting resonator cavity in a moving reference frame.
An illustration of effects of motion on a standing wave system set up by counter-propagating waves in a resonator
cavity formed from a laser mirror system is shown in Figure 2. As shown, the wave medium corresponds to
ordinary positive index of refraction (n>0) resulting in a phase velocity v ph propagating in the same direction as the
group velocity vg. For a negative index of refraction metamaterial, the phase velocity propagates in a direction
Page 3 of 22
opposite the group velocity (anomalous dispersion). For a nondispersive medium, the phase velocity v p (= /k =
c/n) equals the group velocity vg (= d/dk). The Bragg mirrors approximate perfectly reflecting conductor surfaces
and do not reflect all frequencies.
Figure 2. Motion of a confined closed-loop EM wave exhibits inertia characteristics due to self-referral dynamics.
Upon application of an external force applied to the motive boundary, the blue-shifted incident wave
exerts an excess radiation pressure on the reflecting wall and the red-shifted reflected wave exerts a
decreased radiation pressure on the motive wall. Once set in motion, the wave system remains in
motion until acted upon by an external force. The de Broglie wavelength represents a matter wave
generated by the motion of matter.
Relationship of energy, momentum and Inertial characristics of a phase-locked wave system is illustrated in Figure
3. For a freely-propagating photon in vacuo, the photon travelling wave has no rest mass as there is no fixed
reference frame and no defined position operator. In a standing wave resonator, the incident and reflected waves
combine to produce a standing wave with cancellation of momentum for a resonator at rest. Once in motion, a
phase-locked resonator acquires a relativistic increase in mass m = m0 = m0/(1 - v2/c2) and corresponding
increase in energy. A phasor diagram of the relationship of standing wave and travelling wave energy is depicted
in Figure 4. Also illustrated are corresponding equations of photon/electron energy, momentum and associated
electron Compton and de Broglie wavelengths.
Page 4 of 22
Figure 3. Light confined with reflecting walls of a resonator cavity acquires rest mass (m0).
Figure 4. Phasor diagram of relationship of standing wave and travelling wave energy, momentum
and wavelength.
Page 5 of 22
A o je t s otio i spa eti e is des i ed
a geodesi path sho test path i u its of ti e et ee t o
points) in a curved spacetime manifold. In Lorentz wave transformations, a curved spacetime is replaced by a
geodesic path of least resistance describing the wavelength nodal distance (the distance between nodes) in flat
spacetime. For example, as shown in Figure 5, consider a moving body consisting of standing EM wave between
sources at Points P and E co-moving at a constant velocity v = 0.5c relative to an observer. Let the distance
between Points P and E measured simultaneously represent the wavelength of an EM standing wave packet. In
tensor notation, the separation between Points P and E is the line element ds2 = gdxdx = -dt2 + dx2 + dy2 +dz2.
Contravariant position 4-vector x: x = (x0, x1,x2, x3) = (ct, -x, -y, -z)
Covariant position 4-vector: x = x = (x0, -x1, -x2, -x3)
Contravariant 4-velocity: u = dx/d = (c,v)
In Special Relativity (SR), the Lorentz factor (gamma) is given by = 1/(1- v2/c2) which relates transformation of
relative motion in flat space of inertial (non-accelerated) reference frames including Lorentz contraction, time
dilation and relativistic mass increase effects. Moving bodies (standing matter wave nodal distances) are
contracted by a factor of = 1/(1 - 2) in the direction of motion in accordance with the Lorentz transformation.
In normalized coordinates with speed of light c = 1, for the example illustrated, the measured unit length in the xa is i the “ est f a e u de goes a Lo e tz o t a tio of . % i the o t a a ia t “ o i g f a e of efe e e
at = 0.5. The associated frequency reduction is 14.9% as measured in wavelength units in the covariant S frame,
in accordance with the Lorentz Doppler shift.
The physical basis for wavelength contraction may be understood in the context of the polarized vacuum (PV)
model[5] where EM waves undergo a contraction in regions of increased energy density and corresponding change
in the dielectric constant KPV(r, ). The EM wave energy density in uniform motion remains an invariant covariant
physical quantity in the Lorentz transformation. Spacetime curvature is replaced by variable refractive index
KPV(r,M) which is a measure of EM energy density. Gravity is associated with a gradient in the EM Poynting vector
P due to variation in KPV (∇P = k∆KPV2). A standing wave phase synchronization interaction between oscillators in
a nondissipative medium results in a force of attraction equivalent to gravity. The force imbalance is proportional
to the difference in wave energy density and inversely to the wave velocity. The instantaneous wave energy
density, in turn, is proportional to the square of the wave amplitude.
Page 6 of 22
Figure 5. Illustration of Lorentz wavelength contraction of a standing EM wave in motion at v = 0.5c.
Lengths L (projected onto the x-a is a d L p oje ted o to the -axis) are given in wavelength
units. The Lorentz contraction in the direction of motion as measured by a fixed observer is L =
L ∙ (1-v2/c2). Proper time is given by d = 1/c∙ (dxvdxv) = dt/
Page 7 of 22
3. Electromagnetic wave propagation
Relative motion of the two reflecting walls of a phase-locked resonator corresponds to an oscillating dipole which
emits a transverse radiation field orthogonal to the direction of motion of dipole oscillator. As represented in
standard texts, the E and H wavefront vectors are in-phase far from the source dipole oscillator. Within the
evanscent near field region, the E and H vectors are out-of-phase with a longitudinal polarization component. In
the midrange or Fresnel region, the E and H vectors are partially in-phase. In the near field region, dipole field
effects are prominent and drop off as 1/r2 whereas EM field drops off as 1/r. Induction term decays as 1/r2 and
electrostatic field decays as 1/r3 and rapidly decay beyond the evanescent region. The wave phase velocity is
superluminal (vph > c) whereas the group velocity is subluminal (vg < c). Refer to Table 1 summary below.
Table 1. Electromagnetic wave phase propagation
Region
Near field (non-radiative,
reactive) region – Dipole wave
emission
E and H Phase Relationship
Induction E-field lags 90 deg out-of-phase with
magnetic H field. Nea field ≤ ¼ ). Reactive field
0 < r < /2. Radiative to 1 . E, H decays as 1/r2
and 1/r3, respectively. The electric field E is due to
charge dipoles and the magnetic field H is due to
source currents. H0 = E0/Z, Z = (r 0/r 0).
For evanescent waves, phase is independent of
distance, so that phase velocity is superluminal. E
& H are out-of-phase, hence Zo is not related by
377 ohm characteristic vacuum impedance.
Evanescent electromagnetic near-field waves
initially propagate faster than the speed of light
slowing down to the speed of light within about
one wavelength.
Remarks
Longitudinal E-field component in the near field is
partially in the direction of propagation (parallel to
k vector). Close to the antenna the Poynting vector
S = f(r,) is imaginary (reactive), hence energy is
not propagating (non-radiating). Energy stored in
the field volume is detectable capacitively. Field
energy not radiated is alternately returned to the
transmitter.
Evanescent waves exhibit momentum and spin
components orthogonal to the direction of
propagation. The transverse momentum varies in
proportion to helicity while the transverse spin
component does not depend on helicity or
polarization.
Transition (radiative) region
(Fresnel zone)
Dynamic E-field consists of induction and radiation
field components, the sum total out of phase with
the magnetic field. Radiating near-field (Fresnel
Zone) extends from /2< r < 2D2/ where D =
largest antenna dimension. For D2/4 < r < 2D2/,
E, H fields decay as 1/r.
Radiating field begins to dominate. Phase angle (90
< <0 deg).
Far Field (Fraunhofer Zone) –
Planar EM Hertzian wave
Radiation E-field in phase quadrature with
magnetic H field. E, H radiation fields decay as 1/r
and dominate over static near-fields; field pattern
independent of r. Radiating far-field r > 2D2/. The
antenna impedance Z0 = |E|/|H|≈ 376.73
(vacuum free space impedance).
Far field generally regarded to start 2 to 5
wavelengths from source (=2D2/, where D = dipole
element length) or r > 10 for small antennas.
Poynting vector S = f() is real-valued. Radiation
resistance of a Hertizian dipole due to electron
energy loss due to radiation: Rt = (2/3)Z0(l/)2.
The radiation field emitted from a stationary dipole antenna and a rotating dipole antenna are illustrated in Figures
6 and 7, respectively. Phase relationship of the electric field intensity E and magnetic field intensity H as a function
of distance from the dipole antenna is shown in Figure 8.
Page 8 of 22
In the Liénard-Wiechart scalar and magnetic vector potential functions describing the electric and magnetic fields
generated by motion of a point charge, electromagnetic radiation arises as a result of acceleration whereas static
ele t i a d ag eti fields that esult f o the pa ti le s u ifo
otio a e asso iated ith the o -radiative
EM near-field. The static fields point towards the instantaneous (non-retarded) charge position. The electroag eti adiatio appea s to o igi ate at the ha ge s eta ded positio
he e the ha ge as he
accelerated) reflecting delay due to the finite speed of light. Contrary to the widely accepted view that E and B
field create each other to create transverse EM waves, Jefimenko [6] notes that E and B fields are created by charge
density and current density J fluctuations at the source and far-field radiation is due to retarded potentials (r,t)
and A(r,t). The retarded scalar and magnetic vector potentials were derived by Jefimenko for electric and
magnetic fields in terms of charge and current distributions at retarded times.
An electron, In the ansatz model considered here, may be represented as a EM wave trapped in a phase-locked
resonator or, equivalently, a closed-loop, topologically-bound soliton wave. The electron acts as an antenna with
emitted or absorbed photon wave vector k parallel to electron spin axis s. The antenna diameter of the electron
corresponds to the electron Compton wavelength. Virtual photons at the Compton frequency are continuously
emitted/absorbed from an electron in pairs in opposite directions and helicities. The photon EHV dreibein rotates
at a electron Compton angular frequency (c = 7.763 x 1020 rad/s) while the observed photon frequency emitted
during electron acceleration is a measure of the overall oscillatory motion of the electron during emission.
Although small in size, an electron, by resonant phasing, can couple to much larger wavelengths forming, in effect,
a much larger antenna.
Martins and Pinheiro [10] note the induced electrokinetic force Fk = qEk as a function of vector potential A described
by Jefimenko[6] is the source of the inertial mass and the radiation force. The radiation force adds to the inertia by
energy transfer between the field and the source at a retarded time. For an accelerated charge, the induced
electric field generated by the time variation of the vector potential Ek = -dA/dt results in an acceleration of the
electric fields in a direction opposite to acceleration vector. Martins illustrates electric field deformation of a
charged particle subjected to a gravitational force, external force, electric force and inertial force. The flux
patterns illustrate the same asymmetry as described by Ivanov[7,8] for contracted moving standing wave
transformations and associated frequency arrthymia.
Page 9 of 22
Figure 6. Illustration of EM field generation by an oscillating, non-rotating dipole.
Figure 7 Representation of electromagnetic (EM) wave emission from rotating dipole with a tangential
velocity at the speed of light. The pair of wavefront spiral arms represent an entangled state.
Page 10 of 22
Figure 8. Electromagnetic wave propagation illustrating change in electric E-field and magnetic H-field with
distance from source generator.
Page 11 of 22
4. Standing Wave Transformations
Ivanov and LaFreniere[7,8,9] have shown that standing waves undergo wavelength (nodal) contraction in the
direction of motion. An object in motion relative to a fixed observer undergoes a Lorentz contraction (wavelength
compression) in the direction of motion and a Lorentz Doppler shift in frequency (reduction). See Figure 9. The
wavelength compression is a physical result of an increase in the vacuum energy density. Moving clocks which are
made of standing matter waves undergo time dilation as a result. The EM wavelength contraction and frequency
shift in a polarizable vacuum accounts for mass in motion and gravitational effects including the energy change,
deflection of light, gravitational frequency shift and clock slowing. The speed of light c appears invariant in all
inertial frames due to Lorentz contraction of the measurement apparatus. Spacetime remains Euclidean. The
apparent Lorentz space contraction and time dilation are the result of contraction of the nodal distance of the
standing wave(s) which constitute the length of measurement. Time dilation is equivalent to a change in the size
of the units of measurement which are undetectable to an observer as both the object and the co-moving
measurement apparatus undergo Lorentz transformation.
LaFreniere[9] de i ed alpha t a sfo atio s elati g s ste speed , arithmetic mean wavelength am, geometric
mean wavelength gm, Lorentz contraction factor g and wavelength compression . The alpha t a sfo atio s, in
terms of standing wave ratios, ield esults e ui ale t to the Lo e tz eta t a sfo atio i te ms of velocity
ratios. The relation of Lorentz contraction, Doppler shift and relativistic Doppler shift for a contracted standing
wave in motion at v = 0.5c is illustrated in Figure 10. A comparison of the Lorentz contraction of a moving
wavefront as observed in a reference frame at rest is shown together with the relativistic aberration as observed in
a co-moving inertial frame is depicted in Figure 11.
Figure 9. Lorentz contraction of an object in motion in the direction of motion as viewed by a stationary observer.
An object and associated gravitational flux field appears contracted referenced to retarded lengths and
volumes as measured by a stationary observer. Physical contraction of nodal distance of matter waves
occurs in a polarized vacuum as the vacuum dielectric constant KPV is identically equivalent to gamma
which increases with velocity. A co-moving observer will not detect distortion as sensing instruments
undergo like contraction.
Page 12 of 22
Figure 10. Illustration of Doppler and Lorentz Doppler effect for a contracted standing wave moving to the
right in the x-axis direction at a velocity of 0.5c. [Adapted from glafreniere.com]
Page 13 of 22
Figure 11. Lorentz Doppler shift of a moving object emitter at v = 0.5 c as observed in a co-moving reference
frame (primed axes) is compared with the relativistic aberration as observed in a co-moving
reference frame. Distances under relativistic aberration scale by a factor ((1 - )/(1 + )). The
ordinary Doppler shift corresponds to that observed in a reference frame at rest (unprimed axes).
Page 14 of 22
A comparison of the Lorentz-Fitzgerald transformation (applied to spacetime) and the Ivanov-LaFreniere transformations (applied to wavelength) are summarized in Table 2. The unprimed coordinates refers to a stationary
observer reference frame. The primed system coordinates refer to a reference frame moving to the right relative
to the unprimed frame. The x-coordinate denotes number of wavelengths expressed in multiples of gm. The tcoordinate denotes number of waves expressed in multiple of period T.
Table 2. Comparison of Lorentz-Fitzgerald and Ivanov-La Freniere transformations.[10, 11]
Description
Lorentz-Fitzgerald
Ivanov-LaFreniere
Velocity of source
v = vg
Velocity of wave
c = (vpvg)
v = vg = envelope node speed = c∙ =
/k; matter: vdB = vg = c∙
Speed ratio
Lorentz contraction factor
Lorentz factor
c = (vpvg)
= v/c
= v/c = /g
g = (1 - 2) = 1/
g = gm/am = (1 - 2) = 1/=
= 1/(1 - 2) = 1/g
= 1/(gm/am) = 1/(1 - 2) = 1/g
gm = geometric mean wavelength
am = arithmetic mean wavelength
Freniere-Lorentz
contraction factor
On-Axis Wavelength
Stationary observer coord.
relativistic contraction (matter):
g = gm/am = (1 - 2)
-
= (1 - 2) =
=
= am(1 - 2) = g2am= gm
+ t /(1 - 2 =
- t)/g
=
t= t +
=
Proper length ┴ Direction
of motion
=
Proper length ┴ Direction
of motion
z =z
Proper time
/ )/(1 - )
2
2
- vt)/(1 - 2) = (x - vt)
= (x - ct) = (x – vt)/g
- t)/= g - t
t = t + /g = g t + = gt -
= g + t (light)
= gx + t (matter)
z =z
t = t - vx)/(c2))/(1 - 2)
= (t - vx/c ) = (t - x/c)/g
R = a/r
z=z
=y
2
Wavelength ratio
=
=
z=z
Proper length║ Direction of
motion
((1 + )/1- ))
Redshift RR = /Blueshift RB = /
t = gt - x (light)
= gt - x (matter)
R = (1 + )/g = g/(1 - ) = b/f
= (1 + )/(1 - ); RR = /RB = /
Page 15 of 22
Summary equations for Ivanov – LaFreniere transformations (applied to wavelength) are shown in Table 3.
Table 3. Standing Wave Lorentz Transformations[9]
'
am
gm
LDf
LDb
Standing wavelength = am = cT = c/f
Contracted wavelength (on-axis) ' = am(1 - 2) = ((1 + )/(1 - )) = g∙am
Arithmetic mean wavelength am = (b + f)/2 = ct = kt
Geometric mean wavelength gm = (b∙f) = cos
Lorentz Doppler shift forward wavelength = √((1 - )/(1 + )) = (1 - )/g (contracted on-axis)
Lorentz Doppler shift backward wavelength = b = √((1 + )/(1 - )) = (1 + )/g (dilated on-axis)
Standing wave velocity ratio = (R - 1)/(R + 1) = (b - f)/(b + f) = (1 - g)(1 + g)/= v/c = g
Normalized speed ratio = v/c = (R - 1)/(R + 1) = sin = /g
g
Lorentz contraction g = (1 - 2) = 1/ = gm/am = √(1 - 2) = cos= /
r
Doppler redshift r = (1 + )
b
R
aD
rD
aLD
rLD
D
LD
p
c
vg
vp
Lorentz factor = 1/g = 1/(1 - 2) = /g
Doppler blueshift b = (1 - )
Wavelength ratio R = b/f = (1 + )/(1 - );
Redshift RR = / Blueshift RB = /
Doppler approaching (source of light) wavelength aD = (1 - cos)
Doppler receding (source of light) wavelength rD = (1 + cos)
Lorentz Doppler approaching (source of light) wavelength aLD = (1 - cos) = (1 - cos)/g
Lorentz Doppler receding (source of light) wavelength rLD = (1 + cos) = (1 + cos)/g
Average wavelength <> = ½(Red + Blue) = ½(aD + rD)
Doppler wavelength shift D(v/c)Redshift R Blueshift B
Lorentz Doppler wavelength shift LD = r
Phase wavelength p = / = ∙g/
Velocity of light c = co/n = 1/(00) = (vpvg) = KPVco = co/
Group velocity vg = c2/vp = d/dk = vp –(vp/; vg = pc2/E = v = vdB = node speed = ∙c (matter wave)
= 0, k = 0/c
Phase velocity vp = c2/vg = E/p = ∙f = /T = c/ = /k; vp = c/ = mc2/mv (matter wave); p = ħk = mv
z
Wavelength shift/Wavelength z = = (1 + )/(1 - 2) = [(1 + )/(1 - )] - 1
x'
Proper length (on-a is
Lorentz Doppler:
t'
=
= g + t
– vt)/g = gx + t
Wavefront angle =
= g2x + t;
a d
dista e i light-se o d u its, t a d t pe iod i se o ds
a d
dista e i light-se o d u its, t a d t pe iod in seconds)
P ope ti e t = t – x/c)/g = gt -
Lo e tz Dopple : t = gt -
Dopple :
Dopple : t = t – x;
Sin-1(Tan/g)=
Cos-1(gm/am) = /2 - ; Aberration angle = = tan(g/)
Wavelength compression = – = (1 - g)
Page 16 of 22
5. Phase-locked Resonators with Phase Conjugate Wave Reflectors
The reflecting boundaries of a resonator cavity need not be limited to conventional mirrored surfaces. Reflectors
may take several forms, such as magnetic mirrors, retroreflectors, phase conjugate mirrors, etc. In a conventional
mirror, a reflected EM spherical divergent wave from a point source remains divergent. If the wave is reflected
from a phase conjugate mirror (PCM), the wavefront is inverted in a convergent beam back to the source in a timereversed replica. Phase-conjugate wave (PCW) generation may be accomplished via three or four-way mixing
(FWM) in photorefractive crystals, nonlinear optical Kerr media or metamaterials. Four-wave mixing is a nonlinear
effect arising from a third-order optical nonlinearity described by a susceptibility χ(3) coefficient in a Taylor series
expansion resulting in induced polarization (P(E) = e(1)E(t) + e(2)E(t)2 + e(3)E(t)3 + ....) of electric field strength.
Nonlinear phenomena that produce phase conjugation include Brillouin scattering, Raman scattering, Kerr FWM,
resonant FWM, photon echoes, etc. A Bragg reflector is formed from an interference pattern in the overlap zone
of two counter-propagating beams. The effect is similar to conventional Bragg x-ray diffraction from crystals
where atomic lattices form periodic scattering centers. A signal beam incident on the interference pattern results
in a reflected phase conjugate wave propagating back along the path of the signal beam. A diagram illustrating
phase conjugation FWM process is shown in Figure 12 for a case where pump beam frequencies are not equal.
Four-way mixing can be considered as two simultaneous three-way mixing and scattering processes. In optical
mixing, a pump wave mixes with a signal wave generating an interference grating and a second pump wave
scatters off the grating generating a phase conjugate wave.
Figure 12. EM wave reflection/diffraction from Bragg planes formed by interference pattern of EM waves.
Wave interference nodes act as scattering centers of a holographic amplitude grating for incident
EM waves to form a reflected phase conjugate beam. Reflection occurs when the incident
wavelength is comparable to the Bragg plane spacing.
Page 17 of 22
An illustration of motion of a wave and phase conjugate wave in the complex plane is shown in Fig, 13. In the
complex plane, rotation of a phasor A = Aeit + and its conjugate A* at an angular velocity corresponds to a
multiplication. Rotation (CCW) by an angle of 90 degrees, for example, results from multiplication of the complex
number by i = -1. Modulation of the standing wave (or carrier wave) results in the creation of upper and lower
side band frequencies corresponding to harmonics of the modulation frequency centered about the de Broglie
frequency (dB mod). Amplitude modulation produces two sideband signals (S l, Su) whenever the amplitude of a
signal (fc) is modulated at a lower frequency (fm). Sidebands are also produced when the phase or frequency of a
carrier signal is modulated. Sideband generation by carrier signal modulation is illustrated in Fig. 14.
Under FWM, interference of a pump beam A1 and an opposing pump (or signal) beam A2 create a refractive index
grating of alternating grid of variation of refractive index in a nonlinear medium as a result of a Kerr/Pockels
effects. A signal (or probe beam) A3 reflecting off the interference grating is reflected as a counter-propagating
phase conjugate wave A4. The PCW is generated as a third-order nonlinear response the medium. For nondegenerate FWM (NDFWM), a refractive index modulation at the difference frequency occurs in which two input
frequencies 1 and 2 (with 2 > 1) creates two additional frequency components: 3 = 1 − (2 − 1) = 2 1 − 2
and 4 = 2 + (2 − 1) = 22 − 1. The frequency 3 or 4 can be amplified as a result of parametric amplification.
The summation of the A3 signal (or probe beam) and A4 phase conjugate beams forms a standing wave that
oscillates in-place if the field amplitudes are equal. Amplitude varies with incidence angle. If the amplitudes are
unequal there is a net propagation toward the higher amplitude beam.
With pump beams of sufficiently high amplitude, a portion of the energy in the nonlinear standing waves can
transfer to the conjugate wave resulting in amplification. In parametric pumping (3-way mixing), a pump wave at
double frequency (p = 2) and an incident wave of frequency () results in a PCW at frequency pc (= 2- ).
Degenerate four-way mixing (DFWM) phase conjugation involves beams all of the same frequency . In a
1-channel DFWM process when the two pump frequencies coincide, the idler frequency = 21 – 2 where 1 is
the degenerated pump frequency and 2 is the probe frequency. For nearly degenerate FWM with pump waves of
frequency and incident probe beam (+ ), the resultant PCW beam frequency o is a difference frequency
(= - ). In backward NDFWM, the probe beam A1(1) and the signal beam A3(1) have the same frequency 1
while beams A2(2) and A4(2) have a different frequency 2.
Optical phase conjugation has been a subject of intense study and implemented in a wide variety of applications.
Optical resonators with a PCMs have been utilized, for example, in laser oscillators with phase conjugation
feedback, laser amplifiers with multi-pass gain medium, laser target aiming and auto-focusing implemented with
Brillouin enhanced FWM. Phase conjugators provide an alternative to adaptive optics for aberration correction,
target aiming, pointing and targeting, interferometry, lensless imaging and optical computing. An unexplored
potential is modulation of standing waves by synthesized Lorentz-Doppler pump waves to artificially generate de
Broglie matter waves utilizing phase conjugation to effect a change in motion of matter.
Page 18 of 22
Figure 13. An electromagnetic wave and its phase conjugate represented in the complex plane. Motion
corresponds to rotation (multiplication) of the phasor.
Figure 14. Sideband signals produced by amplitude, frequency or phase modulation of a carrier signal.
Page 19 of 22
6. Experimental Potential
in the 1980s, Jennison[1,2] experimentally demonstrated phase-locking effects of free-floating resonators with both
light and microwaves. Using a servoed optical etalon on movable trolleys, a fixed wavelength distance between
source and reflector was demonstrated. The source emitter and reflector wave system moved as if mechanically
connected with a spacing accuracy of > 0.001 wavelength. In another experiment, a travelling EM wave
propagating along an oppositely rotating, circular slow wave transmission line was brought to rest and reversed in
direction without reflection or refraction resulting in a static dipole electric field in a laboratory rest frame[3]. With
on-going research developments in metamaterials, nano structures, microelectromechanical systems (MEMS) and
demonstrations of negative index of refraction, phase conjugation, squeezed/slow light, negative radiation
pressure, patterned wave fronts, cross field/phased array/plasmonic/fractal antennas, inverse Doppler effect, etc,
phased-locked cavity resonators with unusual properties may be realized such as delayed response of the
following wall or synchronization interactions between oscillators of different frequencies. Addition of energy
such as with oscillating wave guide walls or pump waves may allow non-linear response or adjustable gain
characteristics. For example, it may be possible to introduce a nonlinearity in response to an applied external
force with asymmetric ring resonators with graded index of refraction incorporated into the surface of the
following wall. A small additional force component proportional to the index gradient is produced as the absorbed
radiation is reradiated back towards the motive wall emitter. Depending on the orientation of symmetry axis of
the resonators, the force component may be added or subtracted from the radiation reaction force on the
following wall. Force amplification (corresponding to a Fresnel coefficient (|F|>1) may possibly be realized by
parametric pumping of a nonlinear medium at double frequency of the incident wave generating an amplified
phase-conjugate wave in a four-wave mixing process. The inertial reaction force may potentially be counteracted
by application of an opposing vector potential. Phase relation between the signal and pump waves determines
energy flow, i.e., amplification or deamplification of the signal or phase conjugate (PC) wave.
Another potential is the possibility of self-induced motion of a standing wave resonator in response to irradiation
with EM pump waves acting on a phase conjugate mirror (PCM). This is the inverse effect of motion of matter
hi h esults i de B oglie atte
a es. I e se effe ts a e ot ithout p e ede t as, fo e a ple, i e se
Sagnac effect, inverse Doppler effect, inverse piezoelectric effect, etc. See Figure 15. Input conditions are
established by pumping of a PCM nonlinear medium by opposing pump beams A1(1) and A2(2) at frequencies
corresponding to the desired Doppler red-shift frequency (1 = 0 - ) and Doppler blue-shift frequency (2 = 0 +
), respectively. The signal beam A3() corresponds to the standing wave frequency of the resonator at rest
(0). The phase conjugate beam A4(4) corresponds to the difference in frequency of the pump beams (1-3).
Mixing of simulated Doppler blue-shifted wave and red-shifted wave pump beams with signal (standing wave) and
PC beams is predicted to reproduce a modulated wave phase shift generating an unbalanced radiation pressure
resulting in net motion. The impulse imparted includes an alternating push/pull force in the direction of motion.
The induced phase and frequency shifts replicate that produced with application of an external force producing a
wave system velocity > 0. The energy associated with motion is E – E0 = [2F(v/c)2]/[1 – (v/c)2] where F = applied
force. The pump beam energy input provides the kinetic energy for motion.
Page 20 of 22
Figure 15. Conceptual diagram for Induced motion of a phase-locked resonator with a phase conjugate reflector
irradiated by EM pump waves of simulated Doppler shifted wavelengths in four-way mixing with an
internal standing wave. The radiation pressure imbalance results in a net pondermotive force.
Page 21 of 22
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Page 22 of 22
Confinement of Light:
Standing Wave Transformations in a Phase-Locked Resonator
L. J. Reed
Torrance CA 90504
e-mail: larryreed@dphi-dt.net
Abstract/Synopsis:
Electromagnetic resonant wave interactions in a phased-locked resonator at rest and in motion are compared. The origin of mass and inertia as a standing wave interaction in a phased-locked cavity as demonstrated in work by Jennison is reviewed and phase relationships illustrated. For matter (composed of resonant EM standing waves) in motion, the Lorentz contraction is interpreted as a physical wavelength compression due to variation in EM field energy density as measured by vacuum refractive index KPV. Dipole radiation emitted from a phase-locked resonator in motion is described. A graphical representation of Ivanov-LaFreniere standing wave transformations is shown. Experimental possibilities for potential phase conjugate wave phase-locked resonator development are discussed including inertia modification and propulsion.
Keywords: electromagnetic (EM), standing wave, travelling wave, phase-locked resonator, confined light, Lorentz, mass, frequency, oscillator, inertia, phase conjugate waves, de Broglie, Doppler, photon, electron
1. Introduction
Standing electromagnetic waves in a phase-locked resonator have been shown to explain the origin of mass and inertia while standing wave interactions between coupled resonators provide an explanation for the origin of gravity. In this paper, the focus is on the internal dynamics of an isolated resonator. Properties of light when confined within a phase-locked cavity are reviewed and graphical representations of Lorentz contracted standing waves of phase-locked cavity resonators in motion are portrayed. Radiation and propagation characteristics of EM waves are briefly reviewed and illustrated. Motion of a phase-locked resonator with phase conjugate mirrors is described. Potential for induced motion of a phase-locked, phase conjugate resonator by energy input of pump beams of simulated Doppler shifted frequencies is discussed.
Jennison and Drinkwater[1,2,3] have shown that a standing EM wave trapped in a phase-locked cavity exhibits rest mass and intrinsic inertia and classically derived Newton’s Second Law (F = ma) and the Einstein relation (E = mc2). For a free-floating wave system consisting of two counter-propagating travelling waves in a phase-locked resonant cavity, application of an external force results in an imbalance of radiation pressure of Doppler-shifted waves causing the wave system to move as a whole in a stepwise series of velocity increments. Upon application of an external force to the motive boundary, the blue-shifted incident wave exerts an excess radiation pressure on the reflecting wall and the red-shifted reflected wave exerts a decreased radiation pressure on the motive wall provided the force was applied for an interval equal to or greater than the return of the reflected wave.
Motion of a phase-locked resonator exhibits an oscillatory pulsation in the direction of motion and gives rise to transverse EM waves. Macken[4] observes that coherent light confined in a reflecting box exhibits many of the same properties of fermions including inertia (rest mass), kinetic energy, deBroglie waves, phase velocity, Lorentz contraction, time dilation, etc with the same energy (E = hn = mc2). The relativistic contraction due to combination of two Doppler shifts in two opposing propagating waves produces a net decrease in Compton wavelength by factor of (1 - v2/c2). An electron at rest corresponds to a confined photon of energy E = 0.511 MeV in a fixed reference frame.
2. Confined light
A freely-propagating photon in a zero curvature vacuum has zero rest mass, but when trapped between two Bragg mirrors in a phase-locked cavity resonator, light acquires rest mass. A light beam upon reflection from a mirror undergoes momentum reversal cancelling the momentum of the incident wave. Jennison and Drinkwater[1] in 1977 derived Newton’s second law for a phase-locked cavity model of a wave system representing a fundamental particle. A trapped standing wave exhibits not only rest mass but also intrinsic inertia. This effect is illustrated in Figure 1. Under acceleration, forward and backward propagating waves interact undergoing Doppler shifts resulting in an imbalance of radiation pressure. The total energy ET of the system consists of the potential energy EP required to hold the system together plus the forward and backward wave energy EWF + EWB. At rest, the wave energy equals the binding energy. Application of an external force F results in an acceleration F = (2/c2)(ET - EP)a = Eta/c2 = m0a provided that the force is applied for a duration dt greater than or equal to the time to complete a feedback loop otherwise the excess incident radiation is re-radiated back into space.
The inertial force is a function of only half the total energy of the system as the potential energy makes no contribution. Inertia is the result of internal self-referral dynamics of EM standing waves in an isolated phase-locked cavity oscillator when subjected to an external force. Rest mass is the result of EM momentum transfer at the wall boundaries of the oscillator during wave reflection resulting from wavefront deceleration. A photon propagating in an optically dense medium acquires an effective mass as the velocity is slowed (c = c0/n = KPVc0).
Jennison postulated a tentative model of the electron consisting of two orthogonal spinning EM waves in phase quadrature phase-locked at the Compton wavelength lc into a closed-loop system. An equivalent model is that of a toroid generated by two orthogonal spinors representing a photon helix forming a closed-loop soliton wave.
Figure 1. A phase-locked resonator in motion represented here as a closed-loop soliton wave exhibits intrinsic inertial properties and illustrates the mechanism of Newton’s laws of motion.
As illustrated in Figure 1, a phased-locked resonator in motion exhibits an oscillatory rhythmic pulsation and represents a kind of clock. The oscillatory motion of the boundary walls form, in effect, a Hertzian dipole antenna generating a dipole radiation field transverse to the direction of motion. The far field radiation pattern corresponds to that of a half-wave dipole and is similar to that of a magnetic loop antenna.
Macken[4] has elaborated on the mechanism of origin of mass from energy of confined light noting that a massless photon acquires rest mass when confined in closed-loop reflecting resonator cavity in a moving reference frame. An illustration of effects of motion on a standing wave system set up by counter-propagating waves in a resonator cavity formed from a laser mirror system is shown in Figure 2. As shown, the wave medium corresponds to ordinary positive index of refraction (n>0) resulting in a phase velocity vph propagating in the same direction as the group velocity vg. For a negative index of refraction metamaterial, the phase velocity propagates in a direction
opposite the group velocity (anomalous dispersion). For a nondispersive medium, the phase velocity vp (= w/k = c/n) equals the group velocity vg (= dw/dk). The Bragg mirrors approximate perfectly reflecting conductor surfaces and do not reflect all frequencies.
Figure 2. Motion of a confined closed-loop EM wave exhibits inertia characteristics due to self-referral dynamics. Upon application of an external force applied to the motive boundary, the blue-shifted incident wave exerts an excess radiation pressure on the reflecting wall and the red-shifted reflected wave exerts a decreased radiation pressure on the motive wall. Once set in motion, the wave system remains in motion until acted upon by an external force. The de Broglie wavelength represents a matter wave generated by the motion of matter.
Relationship of energy, momentum and Inertial characristics of a phase-locked wave system is illustrated in Figure 3. For a freely-propagating photon in vacuo, the photon travelling wave has no rest mass as there is no fixed reference frame and no defined position operator. In a standing wave resonator, the incident and reflected waves combine to produce a standing wave with cancellation of momentum for a resonator at rest. Once in motion, a phase-locked resonator acquires a relativistic increase in mass m = g m0 = m0/(1 - v2/c2) and corresponding increase in energy. A phasor diagram of the relationship of standing wave and travelling wave energy is depicted in Figure 4. Also illustrated are corresponding equations of photon/electron energy, momentum and associated electron Compton and de Broglie wavelengths.
Figure 3. Light confined with reflecting walls of a resonator cavity acquires rest mass (m0).
Figure 4. Phasor diagram of relationship of standing wave and travelling wave energy, momentum and wavelength.
An object’s motion in spacetime is described by a geodesic path (shortest path in units of time between two points) in a curved spacetime manifold. In Lorentz wave transformations, a curved spacetime is replaced by a geodesic path of least resistance describing the wavelength nodal distance (the distance between nodes) in flat spacetime. For example, as shown in Figure 5, consider a moving body consisting of standing EM wave between sources at Points P and E co-moving at a constant velocity v = 0.5c relative to an observer. Let the distance between Points P and E measured simultaneously represent the wavelength of an EM standing wave packet. In tensor notation, the separation between Points P and E is the line element ds2 = gmndxmdxn = -dt2 + dx2 + dy2 +dz2.
Contravariant position 4-vector xm: xm = (x0, x1,x2, x3) = (ct, -x, -y, -z)
Covariant position 4-vector: xm = hunxn = (x0, -x1, -x2, -x3)
Contravariant 4-velocity: um = dxm/dt = (cg,gv)
In Special Relativity (SR), the Lorentz factor (gamma) is given by g = 1/(1- v2/c2) which relates transformation of relative motion in flat space of inertial (non-accelerated) reference frames including Lorentz contraction, time dilation and relativistic mass increase effects. Moving bodies (standing matter wave nodal distances) are contracted by a factor of g = 1/(1 - b2) in the direction of motion in accordance with the Lorentz transformation. In normalized coordinates with speed of light c = 1, for the example illustrated, the measured unit length in the x-axis in the S rest frame undergoes a Lorentz contraction of 13.4% in the contravariant S’ moving frame of reference at b = 0.5. The associated frequency reduction is 14.9% as measured in wavelength units in the covariant S frame, in accordance with the Lorentz Doppler shift.
The physical basis for wavelength contraction may be understood in the context of the polarized vacuum (PV) model[5] where EM waves undergo a contraction in regions of increased energy density and corresponding change in the dielectric constant KPV(r, w). The EM wave energy density in uniform motion remains an invariant covariant physical quantity in the Lorentz transformation. Spacetime curvature is replaced by variable refractive index KPV(r,M) which is a measure of EM energy density. Gravity is associated with a gradient in the EM Poynting vector P due to variation in KPV (∇P = k∆KPV2). A standing wave phase synchronization interaction between oscillators in a nondissipative medium results in a force of attraction equivalent to gravity. The force imbalance is proportional to the difference in wave energy density and inversely to the wave velocity. The instantaneous wave energy density, in turn, is proportional to the square of the wave amplitude.
Figure 5. Illustration of Lorentz wavelength contraction of a standing EM wave in motion at v = 0.5c. Lengths L (projected onto the x-axis) and L’ (projected onto the x’-axis) are given in wavelength units. The Lorentz contraction in the direction of motion as measured by a fixed observer is L = L’∙ (1-v2/c2). Proper time is given by dt = 1/c∙ (dxvdxv) = dt/g.
3. Electromagnetic wave propagation
Relative motion of the two reflecting walls of a phase-locked resonator corresponds to an oscillating dipole which emits a transverse radiation field orthogonal to the direction of motion of dipole oscillator. As represented in standard texts, the E and H wavefront vectors are in-phase far from the source dipole oscillator. Within the evanscent near field region, the E and H vectors are out-of-phase with a longitudinal polarization component. In the midrange or Fresnel region, the E and H vectors are partially in-phase. In the near field region, dipole field effects are prominent and drop off as 1/r2 whereas EM field drops off as 1/r. Induction term decays as 1/r2 and electrostatic field decays as 1/r3 and rapidly decay beyond the evanescent region. The wave phase velocity is superluminal (vph > c) whereas the group velocity is subluminal (vg < c). Refer to Table 1 summary below.
Table 1. Electromagnetic wave phase propagation
Region
E and H Phase Relationship
Remarks
Near field (non-radiative, reactive) region – Dipole wave emission
Induction E-field lags 90 deg out-of-phase with magnetic H field. Near field (≤ ¼ l). Reactive field 0 < r < l/2p. Radiative to 1 l. E, H decays as 1/r2 and 1/r3, respectively. The electric field E is due to charge dipoles and the magnetic field H is due to source currents. H0 = E0/Z, Z = (mr m0/er e0).
For evanescent waves, phase is independent of distance, so that phase velocity is superluminal. E & H are out-of-phase, hence Zo is not related by 377 ohm characteristic vacuum impedance. Evanescent electromagnetic near-field waves initially propagate faster than the speed of light slowing down to the speed of light within about one wavelength.
Longitudinal E-field component in the near field is partially in the direction of propagation (parallel to k vector). Close to the antenna the Poynting vector S = f(r,q,f) is imaginary (reactive), hence energy is not propagating (non-radiating). Energy stored in the field volume is detectable capacitively. Field energy not radiated is alternately returned to the transmitter.
Evanescent waves exhibit momentum and spin components orthogonal to the direction of propagation. The transverse momentum varies in proportion to helicity while the transverse spin component does not depend on helicity or polarization.
Transition (radiative) region (Fresnel zone)
Dynamic E-field consists of induction and radiation field components, the sum total out of phase with the magnetic field. Radiating near-field (Fresnel Zone) extends from l/2p< r < 2D2/l where D = largest antenna dimension. For D2/4l < r < 2D2/l, E, H fields decay as 1/r.
Radiating field begins to dominate. Phase angle (90 < f <0 deg).
Far Field (Fraunhofer Zone) – Planar EM Hertzian wave
Radiation E-field in phase quadrature with magnetic H field. E, H radiation fields decay as 1/r and dominate over static near-fields; field pattern independent of r. Radiating far-field r > 2D2/l. The antenna impedance Z0 = |E|/|H|≈ 376.73 W (vacuum free space impedance).
Far field generally regarded to start 2 to 5 wavelengths from source (=2D2/l, where D = dipole element length) or r > 10l for small antennas. Poynting vector S = f(q,f) is real-valued. Radiation resistance of a Hertizian dipole due to electron energy loss due to radiation: Rt = (2p/3)Z0(l/l)2.
The radiation field emitted from a stationary dipole antenna and a rotating dipole antenna are illustrated in Figures 6 and 7, respectively. Phase relationship of the electric field intensity E and magnetic field intensity H as a function of distance from the dipole antenna is shown in Figure 8.
In the Liénard-Wiechart scalar and magnetic vector potential functions describing the electric and magnetic fields generated by motion of a point charge, electromagnetic radiation arises as a result of acceleration whereas static electric and magnetic fields that result from the particle’s uniform motion are associated with the non-radiative EM near-field. The static fields point towards the instantaneous (non-retarded) charge position. The electro-magnetic radiation appears to originate at the charge’s retarded position (where the charge was when accelerated) reflecting delay due to the finite speed of light. Contrary to the widely accepted view that E and B field create each other to create transverse EM waves, Jefimenko[6] notes that E and B fields are created by charge density r and current density J fluctuations at the source and far-field radiation is due to retarded potentials f(r,t) and A(r,t). The retarded scalar and magnetic vector potentials were derived by Jefimenko for electric and magnetic fields in terms of charge and current distributions at retarded times.
An electron, In the ansatz model considered here, may be represented as a EM wave trapped in a phase-locked resonator or, equivalently, a closed-loop, topologically-bound soliton wave. The electron acts as an antenna with emitted or absorbed photon wave vector k parallel to electron spin axis s. The antenna diameter of the electron corresponds to the electron Compton wavelength. Virtual photons at the Compton frequency are continuously emitted/absorbed from an electron in pairs in opposite directions and helicities. The photon EHV dreibein rotates at a electron Compton angular frequency (wc = 7.763 x 1020 rad/s) while the observed photon frequency emitted during electron acceleration is a measure of the overall oscillatory motion of the electron during emission. Although small in size, an electron, by resonant phasing, can couple to much larger wavelengths forming, in effect, a much larger antenna.
Martins and Pinheiro [10] note the induced electrokinetic force Fk = qEk as a function of vector potential A described by Jefimenko[6] is the source of the inertial mass and the radiation force. The radiation force adds to the inertia by energy transfer between the field and the source at a retarded time. For an accelerated charge, the induced electric field generated by the time variation of the vector potential Ek = -dA/dt results in an acceleration of the electric fields in a direction opposite to acceleration vector. Martins illustrates electric field deformation of a charged particle subjected to a gravitational force, external force, electric force and inertial force. The flux patterns illustrate the same asymmetry as described by Ivanov[7,8] for contracted moving standing wave transformations and associated frequency arrthymia.
Figure 6. Illustration of EM field generation by an oscillating, non-rotating dipole.
Figure 7 Representation of electromagnetic (EM) wave emission from rotating dipole with a tangential velocity at the speed of light. The pair of wavefront spiral arms represent an entangled state.
Figure 8. Electromagnetic wave propagation illustrating change in electric E-field and magnetic H-field with distance from source generator.
4. Standing Wave Transformations
Ivanov and LaFreniere[7,8,9] have shown that standing waves undergo wavelength (nodal) contraction in the direction of motion. An object in motion relative to a fixed observer undergoes a Lorentz contraction (wavelength compression) in the direction of motion and a Lorentz Doppler shift in frequency (reduction). See Figure 9. The wavelength compression is a physical result of an increase in the vacuum energy density. Moving clocks which are made of standing matter waves undergo time dilation as a result. The EM wavelength contraction and frequency shift in a polarizable vacuum accounts for mass in motion and gravitational effects including the energy change, deflection of light, gravitational frequency shift and clock slowing. The speed of light c appears invariant in all inertial frames due to Lorentz contraction of the measurement apparatus. Spacetime remains Euclidean. The apparent Lorentz space contraction and time dilation are the result of contraction of the nodal distance of the standing wave(s) which constitute the length of measurement. Time dilation is equivalent to a change in the size of the units of measurement which are undetectable to an observer as both the object and the co-moving measurement apparatus undergo Lorentz transformation.
LaFreniere[9] derived ‘alpha’ transformations relating system speed a, arithmetic mean wavelength lam, geometric mean wavelength lgm, Lorentz contraction factor g and wavelength compression l’. The alpha transformations, in terms of standing wave ratios, yield results equivalent to the Lorentz ‘beta’ transformation in terms of b velocity ratios. The relation of Lorentz contraction, Doppler shift and relativistic Doppler shift for a contracted standing wave in motion at v = 0.5c is illustrated in Figure 10. A comparison of the Lorentz contraction of a moving wavefront as observed in a reference frame at rest is shown together with the relativistic aberration as observed in a co-moving inertial frame is depicted in Figure 11.
Figure 9. Lorentz contraction of an object in motion in the direction of motion as viewed by a stationary observer. An object and associated gravitational flux field appears contracted referenced to retarded lengths and volumes as measured by a stationary observer. Physical contraction of nodal distance of matter waves occurs in a polarized vacuum as the vacuum dielectric constant KPV is identically equivalent to gamma g which increases with velocity. A co-moving observer will not detect distortion as sensing instruments undergo like contraction.
Figure 10. Illustration of Doppler and Lorentz Doppler effect for a contracted standing wave moving to the right in the x-axis direction at a velocity of 0.5c. [Adapted from glafreniere.com]
Figure 11. Lorentz Doppler shift of a moving object emitter at v = 0.5 c as observed in a co-moving reference frame (primed axes) is compared with the relativistic aberration as observed in a co-moving reference frame. Distances under relativistic aberration scale by a factor ((1 - b)/(1 + b)). The ordinary Doppler shift corresponds to that observed in a reference frame at rest (unprimed axes).
A comparison of the Lorentz-Fitzgerald transformation (applied to spacetime) and the Ivanov-LaFreniere trans-formations (applied to wavelength) are summarized in Table 2. The unprimed coordinates refers to a stationary observer reference frame. The primed system coordinates refer to a reference frame moving to the right relative to the unprimed frame. The x-coordinate denotes number of wavelengths expressed in multiples of lgm. The t-coordinate denotes number of waves expressed in multiple of period T.
Table 2. Comparison of Lorentz-Fitzgerald and Ivanov-La Freniere transformations.[10, 11]
Description
Lorentz-Fitzgerald
Ivanov-LaFreniere
Velocity of source
v = vg
v = vg = envelope node speed = c∙b = Dw/Dk; matter: vdB = vg = c∙a
Velocity of wave
c = (vpvg)
c = (vpvg)
Speed ratio
b = v/c
b = v/c = a/gg
Lorentz contraction factor
g = (1 - b2) = 1/g
g = lgm/lam = (1 - b2) = 1/g = a/bg
Lorentz factor
g = 1/(1 - b2) = 1/g
g = 1/(lgm/lam) = 1/(1 - b2) = 1/g
lgm = geometric mean wavelength
lam = arithmetic mean wavelength
Freniere-Lorentz contraction factor
-
relativistic contraction (matter): g = lgm/lam = (1 - a2)
On-Axis Wavelength
l’ = l/(1 - b2) = gl
l’ = lam(1 - b2) = g2lam = lgm
= l((1 + b)/1- b))
Stationary observer coord.
x = x’ + vt’/(1 - b2) = (x’ - bt)/g
y = y’
z = z’
t = (t’ + vx’/c2)/(1 - b2)
x = (x’ - at)/g = gx’ - bt’
y = y’
z = z’
t = (t’ + ax)/g = g(t’ + ax) = gt’ - bx’
Proper length║ Direction of motion
x’ = (x - vt)/(1 - b2) = g(x - vt)
= g(x - bct) = (x – vt)/g
x’ = gx + at (light)
= gx + bt (matter)
Proper length ┴ Direction of motion
y’ = y
y’ = y
Proper length ┴ Direction of motion
z’ = z
z’ = z
Proper time
t’ = ((t - vx)/(c2))/(1 - b2)
= g(t - vx/c2) = (t - bx/c)/g
t’ = gt - ax (light)
= gt - bx (matter)
Wavelength ratio
R = la/lr
Redshift RR = l’/l, Blueshift RB = l/l’
R = (1 + b)/g = g/(1 - b) = lb/lf
= (1 + b)/(1 - b); RR = l’/l, RB = l/l’
Summary equations for Ivanov – LaFreniere transformations (applied to wavelength) are shown in Table 3.
Table 3. Standing Wave Lorentz Transformations[9]
l
Standing wavelength = lam = cT = c/f
l'
Contracted wavelength (on-axis) l' = lam(1 - b2) = l((1 + b)/(1 - b)) = g∙lam
lam
Arithmetic mean wavelength lam = (lb + lf)/2 = ct = kwt
lgm
Geometric mean wavelength lgm = (lb∙lf) = lcosq
lLDf
Lorentz Doppler shift forward wavelength = l√((1 - b)/(1 + b)) = l(1 - b)/g (contracted on-axis)
lLDb
Lorentz Doppler shift backward wavelength = lb = l√((1 + b)/(1 - b)) = l(1 + b)/g (dilated on-axis)
a
Standing wave velocity ratio a = (R - 1)/(R + 1) = (lb - lf)/(lb + lf) = (1 - g)(1 + g)/b = vb/c = gbg
b
Normalized speed ratio b = v/c = (R - 1)/(R + 1) = sinq = a/gg
g
Lorentz contraction g = (1 - b2) = 1/g = lgm/lam = √(1 - a2) = cosq = a/bg
g
Lorentz factor g = 1/g = 1/(1 - b2) = a/gb
lr
Doppler redshift lr = l(1 + b)
lb
Doppler blueshift lb = l(1 - b)
R
Wavelength ratio R = lb/lf = (1 + b)/(1 - b); Redshift RR = l’/l, Blueshift RB = l/l’
laD
Doppler approaching (source of light) wavelength laD = l(1 - bcosq)
lrD
Doppler receding (source of light) wavelength lrD = l(1 + bcosq)
laLD
Lorentz Doppler approaching (source of light) wavelength laLD = gl(1 - bcosq) = l(1 - bcosq)/g
lrLD
Lorentz Doppler receding (source of light) wavelength lrLD = gl(1 + bcosq) = l(1 + bcosq)/g
<l>
Average wavelength <l> = ½(lRed + lBlue) = ½(laD + lrD)
DlD
Doppler wavelength shift DlD = (v/c)l; Redshift lR + Dl, Blueshift lB - Dl
DlLD
Lorentz Doppler wavelength shift DlLD = <lr>-l = (g-1)l
lp
Phase wavelength lp = l’/a = l∙g/a
c
Velocity of light c = co/n = 1/(e0m0) = (vpvg) = KPVco = co/G = nl
vg
Group velocity vg = c2/vp = dw/dk = vp – l(Dvp/Dl; vg = pc2/E = v = vdB = node speed = a∙c (matter wave); w = gw0, k = gbw0/c
vp
Phase velocity vp = c2/vg = E/p = l∙f = l/T = c/b = w/k; vp = c/a = gmc2/gmv (matter wave); p = ħk = gmv
z
Wavelength shift/Wavelength z = Dl/l = (1 + b)/(1 - b2) = [(1 + b)/(1 - b)] - 1
x'
Proper length (on-axis) x’ = (x – vt)/g = gx + at Doppler: x’ = g2x + bt;
Lorentz Doppler: x’ = gx + bt (x and x’ distance in light-second units, t and t’ period in seconds)
t'
Proper time t’ = (t – bx/c)/g = gt - ax Doppler: t’ = t – bx;
Lorentz Doppler: t’ = gt - bx (x and x’ distance in light-second units, t and t’ period in seconds)
q
Wavefront angle q = Sin-1(b) = Tan-1(a/g) = Cos-1(lgm/lam) = p/2 - f; Aberration angle = f = tan(g/a)
Dl
Wavelength compression Dl = l – l’ = l(1 - g)
5. Phase-locked Resonators with Phase Conjugate Wave Reflectors
The reflecting boundaries of a resonator cavity need not be limited to conventional mirrored surfaces. Reflectors may take several forms, such as magnetic mirrors, retroreflectors, phase conjugate mirrors, etc. In a conventional mirror, a reflected EM spherical divergent wave from a point source remains divergent. If the wave is reflected from a phase conjugate mirror (PCM), the wavefront is inverted in a convergent beam back to the source in a time-reversed replica. Phase-conjugate wave (PCW) generation may be accomplished via three or four-way mixing (FWM) in photorefractive crystals, nonlinear optical Kerr media or metamaterials. Four-wave mixing is a nonlinear effect arising from a third-order optical nonlinearity described by a susceptibility χ(3) coefficient in a Taylor series expansion resulting in induced polarization (P(E) = ce(1)E(t) + ce(2)E(t)2 + ce(3)E(t)3 + ....) of electric field strength. Nonlinear phenomena that produce phase conjugation include Brillouin scattering, Raman scattering, Kerr FWM, resonant FWM, photon echoes, etc. A Bragg reflector is formed from an interference pattern in the overlap zone of two counter-propagating beams. The effect is similar to conventional Bragg x-ray diffraction from crystals where atomic lattices form periodic scattering centers. A signal beam incident on the interference pattern results in a reflected phase conjugate wave propagating back along the path of the signal beam. A diagram illustrating phase conjugation FWM process is shown in Figure 12 for a case where pump beam frequencies are not equal. Four-way mixing can be considered as two simultaneous three-way mixing and scattering processes. In optical mixing, a pump wave mixes with a signal wave generating an interference grating and a second pump wave scatters off the grating generating a phase conjugate wave.
Figure 12. EM wave reflection/diffraction from Bragg planes formed by interference pattern of EM waves. Wave interference nodes act as scattering centers of a holographic amplitude grating for incident EM waves to form a reflected phase conjugate beam. Reflection occurs when the incident wavelength is comparable to the Bragg plane spacing.
An illustration of motion of a wave and phase conjugate wave in the complex plane is shown in Fig, 13. In the complex plane, rotation of a phasor A = Aeiwt + q and its conjugate A* at an angular velocity w corresponds to a multiplication. Rotation (CCW) by an angle of 90 degrees, for example, results from multiplication of the complex number by i = -1. Modulation of the standing wave (or carrier wave) results in the creation of upper and lower side band frequencies corresponding to harmonics of the modulation frequency centered about the de Broglie frequency (ndB nmod). Amplitude modulation produces two sideband signals (Sl, Su) whenever the amplitude of a signal (fc) is modulated at a lower frequency (fm). Sidebands are also produced when the phase or frequency of a carrier signal is modulated. Sideband generation by carrier signal modulation is illustrated in Fig. 14.
Under FWM, interference of a pump beam A1 and an opposing pump (or signal) beam A2 create a refractive index grating of alternating grid of variation of refractive index in a nonlinear medium as a result of a Kerr/Pockels effects. A signal (or probe beam) A3 reflecting off the interference grating is reflected as a counter-propagating phase conjugate wave A4. The PCW is generated as a third-order nonlinear response the medium. For non-degenerate FWM (NDFWM), a refractive index modulation at the difference frequency occurs in which two input frequencies n1 and n2 (with n2 > n1) creates two additional frequency components: n3 = n1 − (n2 − n1) = 2 n1 − n2 and n4 = n2 + (n2 − n1) = 2n2 − n1. The frequency n3 or n4 can be amplified as a result of parametric amplification. The summation of the A3 signal (or probe beam) and A4 phase conjugate beams forms a standing wave that oscillates in-place if the field amplitudes are equal. Amplitude varies with incidence angle. If the amplitudes are unequal there is a net propagation toward the higher amplitude beam.
With pump beams of sufficiently high amplitude, a portion of the energy in the nonlinear standing waves can transfer to the conjugate wave resulting in amplification. In parametric pumping (3-way mixing), a pump wave at double frequency (np = 2n) and an incident wave of frequency (n) results in a PCW at frequency npc (= 2n - n). Degenerate four-way mixing (DFWM) phase conjugation involves beams all of the same frequency n. In a 1-channel DFWM process when the two pump frequencies coincide, the idler frequency n = 2n1 – n2 where n1 is the degenerated pump frequency and n2 is the probe frequency. For nearly degenerate FWM with pump waves of frequency n and incident probe beam (n + d), the resultant PCW beam frequency no is a difference frequency (= n - d). In backward NDFWM, the probe beam A1(n1) and the signal beam A3(n1) have the same frequency n1 while beams A2(n2) and A4(n2) have a different frequency n2.
Optical phase conjugation has been a subject of intense study and implemented in a wide variety of applications. Optical resonators with a PCMs have been utilized, for example, in laser oscillators with phase conjugation feedback, laser amplifiers with multi-pass gain medium, laser target aiming and auto-focusing implemented with Brillouin enhanced FWM. Phase conjugators provide an alternative to adaptive optics for aberration correction, target aiming, pointing and targeting, interferometry, lensless imaging and optical computing. An unexplored potential is modulation of standing waves by synthesized Lorentz-Doppler pump waves to artificially generate de Broglie matter waves utilizing phase conjugation to effect a change in motion of matter.
Figure 13. An electromagnetic wave and its phase conjugate represented in the complex plane. Motion corresponds to rotation (multiplication) of the phasor.
Figure 14. Sideband signals produced by amplitude, frequency or phase modulation of a carrier signal.
6. Experimental Potential
in the 1980s, Jennison[1,2] experimentally demonstrated phase-locking effects of free-floating resonators with both light and microwaves. Using a servoed optical etalon on movable trolleys, a fixed wavelength distance between source and reflector was demonstrated. The source emitter and reflector wave system moved as if mechanically connected with a spacing accuracy of > 0.001 wavelength. In another experiment, a travelling EM wave propagating along an oppositely rotating, circular slow wave transmission line was brought to rest and reversed in direction without reflection or refraction resulting in a static dipole electric field in a laboratory rest frame[3]. With on-going research developments in metamaterials, nano structures, microelectromechanical systems (MEMS) and demonstrations of negative index of refraction, phase conjugation, squeezed/slow light, negative radiation pressure, patterned wave fronts, cross field/phased array/plasmonic/fractal antennas, inverse Doppler effect, etc, phased-locked cavity resonators with unusual properties may be realized such as delayed response of the following wall or synchronization interactions between oscillators of different frequencies. Addition of energy such as with oscillating wave guide walls or pump waves may allow non-linear response or adjustable gain characteristics. For example, it may be possible to introduce a nonlinearity in response to an applied external force with asymmetric ring resonators with graded index of refraction incorporated into the surface of the following wall. A small additional force component proportional to the index gradient is produced as the absorbed radiation is reradiated back towards the motive wall emitter. Depending on the orientation of symmetry axis of the resonators, the force component may be added or subtracted from the radiation reaction force on the following wall. Force amplification (corresponding to a Fresnel coefficient (|F|>1) may possibly be realized by parametric pumping of a nonlinear medium at double frequency of the incident wave generating an amplified phase-conjugate wave in a four-wave mixing process. The inertial reaction force may potentially be counteracted by application of an opposing vector potential. Phase relation between the signal and pump waves determines energy flow, i.e., amplification or deamplification of the signal or phase conjugate (PC) wave.
Another potential is the possibility of self-induced motion of a standing wave resonator in response to irradiation with EM pump waves acting on a phase conjugate mirror (PCM). This is the inverse effect of motion of matter which results in de Broglie ‘matter’ waves. Inverse effects are not without precedent as, for example, inverse Sagnac effect, inverse Doppler effect, inverse piezoelectric effect, etc. See Figure 15. Input conditions are established by pumping of a PCM nonlinear medium by opposing pump beams A1(n1) and A2(n2) at frequencies corresponding to the desired Doppler red-shift frequency (n1 = n0 - Dn) and Doppler blue-shift frequency (n2 = n0 + Dn), respectively. The signal beam A3(n3) corresponds to the standing wave frequency of the resonator at rest (n0). The phase conjugate beam A4(n4) corresponds to the difference in frequency of the pump beams (n1-n3). Mixing of simulated Doppler blue-shifted wave and red-shifted wave pump beams with signal (standing wave) and PC beams is predicted to reproduce a modulated wave phase shift generating an unbalanced radiation pressure resulting in net motion. The impulse imparted includes an alternating push/pull force in the direction of motion. The induced phase and frequency shifts replicate that produced with application of an external force producing a wave system velocity > 0. The energy associated with motion is E – E0 = [2F(v/c)2]/[1 – (v/c)2] where F = applied force. The pump beam energy input provides the kinetic energy for motion.
Figure 15. Conceptual diagram for Induced motion of a phase-locked resonator with a phase conjugate reflector irradiated by EM pump waves of simulated Doppler shifted wavelengths in four-way mixing with an internal standing wave. The radiation pressure imbalance results in a net pondermotive force.
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