
The Sagnac Effect and the Chirality of Space Time Prof. R. M. Kiehn, Emeritus Physics, Univ. of Houston www.cartan.pair.com rkiehn2352@aol.com SPIE, San Diego Aug 25-30, 2007 This presentation consists of several parts 1. Fringes vs. Beats
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The Sagnac Effect and the Chirality of Space Time
Prof. R. M. Kiehn, Emeritus
Physics, Univ. of Houston
www.cartan.pair.com
rkiehn2352@aol.com
SPIE, San Diego Aug 25-30, 2007
This presentation consists of several parts
1. Fringes vs. Beats
This presentation consists of several parts
1. Fringes vs. Beats
2. The Sagnac effect and the
dual Polarized Ring Laser
This presentation consists of several parts
1. Fringes vs. Beats
2. The Sagnac effect and the
dual Polarized Ring Laser
3. The Chirality of the Cosmos
(And if there is time – a bit of heresy)
4. Compact domains of Constitutive
properties that lead to non-radiating
“Electromagnetic Molecules”
(And if there is time – a bit of heresy)
4. Compact domains of Constitutive
properties that lead to non-radiating
“Electromagnetic Molecules”
with infinite Radiation Impedance ?!
(And if there is time – a bit of heresy)
4. Compact domains of Constitutive
properties that lead to non-radiating
“Electromagnetic Molecules”
with infinite Radiation Impedance ?!
Or why an orbiting electron does not radiate
1 = e i(k1• r - 1 t)2 = e i(k2• r - 2 t)
Superpose two outbound waves k1 k2, 1 2
1 = e i(k1• r - 1 t)2 = e i(k2• r - 2 t)
Two outbound waves superposed: k = k1 - k2 = 1 - 2
1 + 2~2 cos(k•r/2 - ω•t/2)1
1 = e i(k1• r - 1 t)2 = e i(k2• r - 2 t)
Two outbound waves superposed: k = k1 - k2 = 1 - 2
1 + 2=2 cos(k•r/2 - ω•t/2)1
Fringes are measurements of wave vector variations k(t = constant, r varies)
1 = e i(k1• r - 1 t)2 = e i(k1• r - 2 t)
Two outbound waves superposed: k = k1 - k2 = 1 - 2
1 + 2=2 cos(k•r/2 - ω•t/2)1
Fringes are measurements of wave vector variations k(t = constant, r varies)
Beats are measurements of frequency variations: ω(r = constant, t varies)
Phase Velocity = /k = C/n
C = Lorentz Speed
n = index of refraction
Group Velocity = d/dk ~ /k
C/n /k
Outbound Phase
1 = e i(k1• r - 1 t)2 = e i(- k2• r + 2 t)
k =
k =
Note opposite orientations of Wave and phase vectors
Outbound Phase
1 = e i(k1• r - 1 t)2 = e i(- k2• r + 2 t)
k =
k =
Inbound Phase
3 = e i(k3• r + 3 t)4 = e i(- k4• r - 4 t)
k =
k =
Note opposite orientations of wave and phase vectors
Mix Outbound phase pairs
or Inbound phase pairs
for Fringes and Beats.
Mix Outbound with Inbound phase pairs
to produce Standing Waves.
Mix all 4 modes for
“Phase Entanglement”
Each of the phase modes has a 4 component
isotropic spinor representation!
1b. The Michelson Morley interferometer.
The measurement of Fringes
Most people with training in Optics know about the
Michelson-Morley interferometer.
The fringes require that the optical paths are equal to within a coherence length of the photons.
L = C • decay time ~ 3 meters for Na light
Many are not familiar with the use of
multiple path optics (1887).
1c. The Sagnac interferometer.
With the measurement of fringes (old)
The Sagnac interferometer encloses a finite area,
The M-M interferometer encloses ~ zero area.
The Sagnac interferometer responds to rotation
The M-M interferometer does not.
1d. The Sagnac Ring Laser interferometer.
With the measurement of Beats (modern)
Has any one measured beats in a M M interferometer ??
Two beam (CW and CCW linearly polarized)
Sagnac Ring with internal laser light source
Linear Polarized
Ring Laser
Polarization fixed by Brewster windows
Ring Laser
Dual Polarized
Polarization beam splitters
4 Polarized beams –CWLH, CCWLH, CWRH, CCWRH
Sagnac Ring with internal laser light source
Brewster windows for single linear polarization state
Rotation rate of the earth produces a beat signal of about 2-10 kHz depending on enclosed area.
Non-reciprocal measurements with a
Q = ~ 1018
Better than Mossbauer
This technology has had little exploitation !!!
As this is a conference for Optical Engineers, who know that the speed of light can be different for different states of polarization, let me start out with the first, little appreciated, heretical statement:
As this is a conference for Optical Engineers, who know that the speed of light can be different for different states of polarization, let me start out with the first, little appreciated, heretical statement:
In Non-Reciprocal media,
the Speed of light not only depends upon polarization, but also depends upon the direction of propagation.
Faraday rotation or Fresnel-Fizeau
Consider Linearly polarized light passing through
Faraday
or Optical Active media
Faraday rotation or Fresnel-Fizeau
Consider Linearly polarized light passing through
Faraday
or Optical Active media
Exact Solutions given by E. J. Post 1962
These concepts stimulated a search for apparatus which could measure the effects of gravity on the polarization of an EM wave,
These concepts stimulated a search for apparatus which could measure the effects of gravity on the polarization of an EM wave,and ultimately to practical applications of a dual polarized ring laser.
Every one should read
E. J. Post
“The Formal Structure of Electromagnetics”
North Holland 1962 or Dover 1997
The Faraday Ratchet can accumulate tiny phase shifts from multiple to-fro reflections.
The hope was that such a device could capture the tiny effect of gravity on the polarization of the PHOTON.
The Faraday Ratchet can accumulate tiny phase shifts from multiple to-fro reflections.
The hope was that such a device could capture the tiny effect of gravity on the polarization of the PHOTON.
It was soon determined that classical EM theory would not give an answer to EM - gravity polarization interactions.
Tune to a single mode.
If no intra Optical Cavity effects,
then get a single beat frequency
due to Sagnac Rotation.
If no intra Optical Cavity effects,
then get a single beat frequency
due to Sagnac Rotation.
If A.O. and Faraday effects
are combined in the Optical Cavity,
then get 4 beat frequencies.
The 4 different beams have
4 different phase velocities,
dependent upon
polarization and
propagation direction.
Experiments conducted by V. Sanders and R. M. Kiehn in 1977, using dual polarized ring lasers verified that the speed of light can have a 4 different phase velocities depending upon direction and polarization. The 4-fold Lorentz degeneracy can be broken.
Such solutions to the Fresnel Maxwell theory, subject to a gauge constraint, were published first in 1979. After patents were secured, the full theory of singular solutions to Maxwell’s equations without gauge constraints was released for publication in Physical Review in 1991.
R. M. Kiehn, G. P. Kiehn, and B. Roberds,
Parity and time-reversal symmetry breaking, singular solutions and Fresnel surfaces,
Phys. Rev A 43, pp. 5165-5671, 1991.
Examples of the theory are presented in the next slides, which shows the exact solution for the Fresnel Kummer singular wave surface for combined Optical Activity and Faraday Rotation.
Generalized Fresnel Analysis of Singular Solutions to Maxwell’s Equations (propagating photons)
Generalized Fresnel Analysis of Singular Solutions to Maxwell’s Equations (propagating photons)
Theoretical existence of 4-modes of photon propagation
as measured in the dual polarized Ring Laser.
1. Outbound LH polarization
2. Outbound RH polarization
3. Inbound LH polarization
4. Inbound RH polarization
Lorentz Constitutive Equations -- The Lorentz vacuum
Substitute into PDE,s get vector wave equation
Phase velocity
EM from a Topological Viewpoint.
USE EXTERIOR DIFFERENTIAL FORMS, NOT TENSORS
Exterior differential forms, A, F and G, carry topological information.
They are not restricted by tensor diffeomorphisms
For any 4D system of base variables
EM from a Topological Viewpoint.
USE EXTERIOR DIFFERENTIAL FORMS, NOT TENSORS
Exterior differential forms, A, F and G, carry topological information.
They are not restricted by tensor diffeomorphisms
F is an exact and closed 2-Form, A is a 1-form of Potentials.
G is closed but not exact, 2-Form. J = dG, is exact and closed.
EM from a Topological Viewpoint.
USE EXTERIOR DIFFERENTIAL FORMS, NOT TENSORS
Exterior differential forms, A, F and G, carry topological information.
They are not restricted by tensor diffeomorphisms/
F is an exact and closed 2-Form, A is a 1-form of Potentials.
G is closed but not exact, 2-Form. J = dG, is exact and closed.
Topological limit points are determined by exterior differentiation
dF = 0generatesMaxwell Faraday PDE’s
dG = J generatesMaxwell Ampere PDE’s
For any 4D system of base variables
EM from a Topological Viewpoint.
dF = 0generatesMaxwell Faraday PDE’s
dG = J generatesMaxwell Ampere PDE’s
A differential ideal (if J=0) for any 4D system of base variables
EM from a Topological Viewpoint.
dF = 0generatesMaxwell Faraday PDE’s
dG = J generatesMaxwell Ampere PDE’s
A differential ideal (if J=0) for any 4D system of base variables
Find a phase function 1-form: =kmdxm dt
Such that the intersections of the 1-form,,and the 2-forms vanish
^F = 0 ^G = 0
Also require that J =0.
^F = 0 ^G = 0 In Engineering Format become:
k × E − ωB = 0, k · B = 0,
k × H + ωD = 0, k · D = 0,
Six equations in 12 unknowns. !!
Need 6 more equations
The Constitutive Equations
Constitutive Equation examples
Lorentz vacuum is NOT chiral, = 0
Constitutive Equation examples
Generalized Complex Constitutive Matrix
Constitutive Equation examples
Generalized Complex Constitutive Matrix
Generalized Complex Constitutive Equation
Chiral Constitutive Equation Examples
Generalized Chiral Constitutive Equation
[ ] 0
[ ] Gamma is a complex matrix.
Chiral Constitutive Equation Examples
Diagonal Chiral Constitutive Equation
Gamma is complex
The real part of Gamma represents Fresnel-Fizeau effects.
The Imaginary part of Gamma represents Optical Activity
Chiral Constitutive Equation Examples
Diagonal Chiral Constitutive Equation
The Wave Speed does not depend upon Fresnel Fizeau “Expansions”
(the real diagonal part).
The Wave Speed depends upon OA “expansions”,
(the imaginary diagonal part).
The Radiation Impedance depends upon both “expansions”.
Chiral Constitutive Equation Examples
Fresnel-Fizeau “rotation” + diagonal Chiral “expansions”
Chiral Constitutive Equation Examples
Fresnel-Fizeau “rotation” + diagonal Chiral “expansions”
Combination of Fresnel-Fizeau “rotation”, , about z-axis
and
Diagonal Optical Activity + Fresnel-Fizeau “expansion”, .
Chiral Constitutive Equation Examples
Fresnel-Fizeau “rotation” + diagonal Chiral “expansions”
Combination of Fresnel-Fizeau “rotation”, , about z-axis
and
Diagonal Optical Activity + Fresnel-Fizeau “expansion”, .
WILL PRODUCE 4 PHASE VELOCITIES
depending on POLARIZATION and K vector
The index of refraction has 4 distinct values depending upon direction and polarization.
Z axis: Index of refraction 4 roots =1/3 - 1/2
3. The Chirality of the Cosmos
Definition of a chiral space
A chiral space is an electromagnetic system
of fields E, B, D, H
constrained by a complex 6x6 Constitutive Matrix
which admits solubility for a real phase function that satisfies both the Eikonal and the Wave equation.
3. The Chirality of the Cosmos
Definition of a chiral space
A chiral space is an electromagnetic system
of fields E, B, D, H
constrained by a complex 6x6 Constitutive Matrix
which admits solubility for a real phase function that satisfies both the Eikonal and the Wave equation.
Hence any function of the phase function is a solution to the wave equation.
3. The Chirality of the Cosmos
Definition of a chiral Vacuum
The chiral Vacuum is a chiral space
which is free from charge and current densities.
J = 0, = 0
3. The Chirality of the Cosmos
Definition of a chiral Vacuum
The chiral Vacuum is a chiral space
which is free from charge and current densities.
Can the Cosmological Vacuum be Chiral ?
3. The Chirality of the Cosmos
Definition of a chiral Vacuum
The chiral Vacuum is a chiral space
which is free from charge and current densities.
Can the Cosmological Vacuum be Chiral ?
Can the chirality be measured ?
For the Lorentz vacuum, it is straight forward to show that there is no Charge-Current density and the fields satisfy the vector Wave Equation.
For the Lorentz vacuum, it is straight forward to show that there is no Charge-Current density and the fields satisfy the vector Wave Equation.
Use Maple to solve more complicated cases:
Six equations 12 unknowns
k x E - B = 0, k x H + D = 0
Use Constitutive Equation to yield 6 more equations
Define
Technique: Use constitutive equations to eliminate, say, D and B
This yields a 6 x 6 Homogenous matrix in 6 unknowns.
The determinant of the Homogeneous matrix must vanish
The determinant can be evaluated in terms of the 3 x 3 sub matrices of the 6 x 6 complex constitutive matrix and the anti-symmetric 3 x 3 matrix, [ n x ] composed of the vector, n = k /ω.The determinant formula is:
The general constitutive matrix can lead to tedious computations. A Maple program takes away the drudgery.
Conformal off-diagonal chiral matrices
Simplified (diagonal )
Constitutive matrix for a chiral Vacuum
= 1 = 1
Conformal + Rotation chiral matrices
Simplified (diagonal + Fresnel rotation ) Constitutive matrix for a chiral Vacuum
Leads to Sagnac 4 phase velocities
Semi-Simplified Constitutive Matrix with Conformal + Rotation chiral submatrices
f = Fresnel Fizeau diagonal real part (“conformal expansion”)
ω = Fresnel Fizeau antisymmetric real part (“rotation”)
= Optical Activity antisymmetric imaginary part (“rotation”)
= Optical Activity diagonal imaginary part (“conformal expansion”)
The Wave Phase Velocity and the
Reciprocal Radiation Impedance
depend upon
the anti-symmetric rotations,
and the conformal factors of the
complex chiral (off diagonal) part
of the Constitutive Matrix.
The Wave Phase Velocity and the
Reciprocal Radiation Impedance
depend upon
the anti-symmetric rotations,
and the conformal factors of the
complex chiral (off diagonal) part
of the Constitutive Matrix.
(All isotropic conformal + rotation chiral matrices have a center of symmetry, unless the Fresnel rotation, ω, is not zero)
As an example of the algebraic complexity, the HAMILTONIAN and ADMittance determinants are shown above for the semi-simplified case.
Fresnel Fizeau Conformal f does not effect phase velocity
AO Conformal modifies phase velocity
Fresnel Fizeau Rotation modifies phase velocity
AO rotation modifies phase velocity
Fresnel Fizeau Conformal f does not effect phase velocity
AO Conformal modifies phase velocity
Fresnel Fizeau Rotation modifies phase velocity
AO rotation modifies phase velocity
All factors give an effect on chiral admittance (cubed):
Fresnel Fizeau Conformal f does not effect phase velocity
AO Conformal modifies phase velocity
Fresnel Fizeau Rotation modifies phase velocity
AO rotation modifies phase velocity
All factors give an effect on chiral admittance (cubed):
IN fact it is possible for the admittance ADM to be ZERO,
But this implies the radiation impedance
Z goes to infinity (not 376.73 ohms) !!!
The idea that chiral effects could cause the Admittance to go to Zero is startling to me.
Zero Admittance infinite Radiation Impedance, Z !
The idea that chiral effects could cause the Admittance to go to Zero is startling to me.
Zero Admittance infinite Radiation Impedance, Z !
Can this idea impact antenna design?
Zero Admittance infinite impedance
What would be the effects of a chiral universe on Cosmology ???
?
Zero Admittance infinite impedance
What would be the effects of a chiral universe on Cosmology ???
Is the Universe Rotating as well as Expanding ?
Zero Admittance infinite impedance
What would be the effects of a chiral universe on Cosmology ???
Is the Universe Rotating as well as Expanding ?
Could the chiral effect be tied to dark matter -- where increased radiation impedance causes compact composites to bind together more than would be expected ??
Zero Admittance infinite impedance
What would be the effects of a chiral universe on Cosmology ???
Is the Universe Rotating as well as Expanding ?
Could the chiral effect be tied to dark matter -- where increased radiation impedance causes compact composites to bind together more than would be expected ??
-- Could the infinite radiation impedance be tied to compact composites such as molecules and atoms which do not Radiate ?
Hopefully these questions will be addressed on Cartan’s Corner
Optical Black Holes in a swimming pool
http://www.cartan.pair.com
The 4-mode Sagnac Effect - with No center of symmetry
Real ω = 1/3, f = 0, Imag = 1/6, = 0