Transcript spie07chiral

The Sagnac Effect and the
Chirality of Space Time
Prof. R. M. Kiehn, Emeritus
Physics, Univ. of Houston
www.cartan.pair.com
[email protected]
Dimdim December 5 2009
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
1a.
Fringes vs. Beats
1 = e i(k1• r - 1 t)
2 = e i(k2• r - 2 t)
Superpose two outbound waves k1  k2, 1  2
1a.
Fringes vs. Beats
1 = e i(k1• r - 1 t)
Two outbound waves superposed:
1 + 2 ~
2 = e i(k2• r - 2 t)
k = k1 - k2  = 1 - 2
exp (k•r/2 - ω•t/2) 1
Fringes vs. Beats
1 = e i(k1• r - 1 t)
Two outbound waves superposed:
1 + 2 =
2 = e i(k2• r - 2 t)
k = k1 - k2  = 1 - 2
2 cos(k•r/2 - ω•t/2) • 1
Fringes are measurements of wave vector
variations k
(t = constant, r varies)
Fringes vs. Beats
1 = e i(k1• r - 1 t)
Two outbound waves superposed:
1 + 2 =
2 = e i(k1• r - 2 t)
k = k1 - k2  = 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 vs. Group velocity
Phase Velocity = /k = C/n
C = Vacuum Speed
n = index of refraction
Phase vs. Group velocity
Phase Velocity = /k = C/n
C = Lorentz Speed
n = index of refraction
Group Velocity = ///k ~ /k
Phase Velocity =
C/n  /k = Group Velocity
4 Propagation Modes
Outbound Phase
1 = e i(k1• r - 1 t)
k =
2 = e i(- k2• r + 2 t)
k=
4 Propagation Modes
Outbound Phase
1 = e i(k1• r - 1 t)
k =
2 = e i(- k2• r + 2 t)
k=
Note opposite orientations of Wave and phase vectors
4 Propagation Modes
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)
k =
4 = e i(- k4• r - 4 t)
k =
Note opposite orientations of wave and phase vectors
4 Propagation Modes
Mix Outbound phase pairs
or Inbound phase pairs
for Fringes and Beats.
4 Propagation Modes
Mix Outbound phase pairs
or Inbound phase pairs
for Fringes and Beats.
Mix Outbound with Inbound
phase pairs
to produce Standing Waves.
4 Propagation Modes
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.
Viewing Fringes.
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
4 Polarized beams –CWLH, CCWLH, CWRH, CCWRH
Sagnac Ring with internal laser light source
Dual Polarized
Dual Polarized
Ring Laser
Polarization
beam
splitters
Ring laser - Early design
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.
More modern design of Ring Laser
Hogged out Quartz monolithic design
Ring Laser gyro built from 2-beam Ring lasers on 3 axes
These aircraft use
(or will use)
Ring Laser gyros
These missiles use Ring Laser gyros
Aerospace devices use ring Laser gyros
Under water devices use ring Laser gyros
2. The Sagnac effect and Dual
polarized Ring lasers.
Dual Polarized Ring Lasers
Non-reciprocal
measurements with a
Q = ~ 1018
Better than Mossbauer
Dual Polarized Ring Lasers
Non-reciprocal
measurements with a
Q = ~ 1018
Better than Mossbauer
This technology has had little
exploitation !!!
Non-Reciprocal Media.
As this is a meeting of those who like a bit of heresy, and Optical Engineers,
who know that the speed of light in media can be different for different
states of polarization, let me start out with the first, little appreciated,
heretical statement:
Non-Reciprocal Media.
As this is a meeting of those who like a bit of heresy, and Optical Engineers,
who know that the speed of light in media 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.
Non-reciprocal Media
Faraday rotation or Fresnel-Fizeau
Consider Linearly polarized light passing through
Faraday
or Optical Active media
Non-reciprocal 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 Sagnac 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.
More modern design of dual polarized Ring Laser
Technique
Tune to a single mode.
If no intra Optical Cavity effects,
then get a single beat frequency
due to Sagnac Rotation.
Tune to a single mode.
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.
Conclusion
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.
The 4 modes correspond to:
1. Outbound LH polarization
2. Outbound RH polarization
3. Inbound LH polarization
4. Inbound RH polarization
Fundamental PDE’s of Electromagnetism
A review
Maxwell Faraday PDE’s
Maxwell Ampere PDE’s
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.
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 = 0 generates Maxwell Faraday PDE’s
dG = J generates Maxwell Ampere PDE’s
For any 4D system of base variables
EM from a Topological Viewpoint.
dF = 0 generates Maxwell Faraday PDE’s
dG = J generates Maxwell Ampere PDE’s
A differential ideal (if J=0) for any 4D system of base variables
EM from a Topological Viewpoint.
dF = 0 generates Maxwell Faraday PDE’s
dG = J generates Maxwell 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
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
This Chiral Constitutive Equation
Explains the Dual Polarized
Sagnac ring laser
Sagnac Effect Fresnel Surface
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
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 ?
The Lorentz Vacuum
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.
The Simple Chiral Vacuum
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,
kxH+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
 =  + i 
= 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
 = Optical Activity diagonal imaginary part
(“rotation”)
(“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?
And now some heresy
Zero Admittance  infinite impedance
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
Some Examples from Maple
Real f = 0, ω = 0 Imag  = 1/3,  = 0
Real f = 0, ω = 0 Imag  = 0,  = 2
Real f = 0, ω = 0 Imag  = 1/3,  = 1/3
The 4-mode Sagnac Effect - with No center of symmetry
Real ω = 1/3, f = 0, Imag  = 1/6,  = 0
Ebooks – Paperback, or Free pdf
http://www.lulu.com/kiehn
or
http://www.cartan.pair.com
email: [email protected]