The Sagnac Effect and the Chirality of Space Time Prof. R. M. Kiehn, Emeritus Physics, Univ. of Houston Dimdim December

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 dual Polarized Ring Laser

This presentation consists of several parts 1. Fringes vs. Beats 2. The Sagnac effect and the dual Polarized Ring Laser 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 properties that lead to non-radiating “Electromagnetic Molecules” “Electromagnetic Molecules”

(And if there is time – a bit of heresy) 4. Compact domains of Constitutive properties that lead to non-radiating properties that lead to non-radiating “Electromagnetic Molecules” “Electromagnetic Molecules” with infinite Radiation Impedance ?! with infinite Radiation Impedance ?!

(And if there is time – a bit of heresy) 4. Compact domains of Constitutive properties that lead to non-radiating properties that lead to non-radiating “Electromagnetic Molecules” “Electromagnetic Molecules” with infinite Radiation Impedance ?! with infinite Radiation Impedance ?! Or why an orbiting electron does not radiate

Superpose two outbound waves k1  k2,  1   2 1a. Fringes vs. Beats  1 = e i(k1 r -  1 t)  2 = e i(k2 r -  2 t)

Two outbound waves superposed:  k = k1 - k2  =  1 -  2 1a. Fringes vs. Beats  1 = e i(k1 r -  1 t)  2 = e i(k2 r -  2 t)  1 +  2 ~ exp (  kr/2 -  ωt/2)  1

Two outbound waves superposed:  k = k1 - k2  =  1 -  2 Fringes vs. Beats Fringes are measurements of wave vector variations  k (t = constant, r varies)  1 = e i(k1 r -  1 t)  2 = e i(k2 r -  2 t)  1 +  2 = 2 cos(  kr/2 -  ωt/2)  1

Two outbound waves superposed:  k = k1 - k2  =  1 -  2 Fringes vs. Beats Fringes are measurements of wave vector variations  k (t = constant, r varies) Beats are measurements of frequency variations:  ω (r = constant, t varies)  1 = e i(k1 r -  1 t)  2 = e i(k1 r -  2 t)  1 +  2 = 2 cos(  kr/2 -  ωt/2)  1

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)  2 = e i(- k2 r +  2 t) k =

4 Propagation Modes Outbound Phase  1 = e i(k1 r -  1 t)  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) Inbound Phase  3 = e i(+k3 r +  3 t)  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 ??

Linear Polarized Ring Laser Polarization fixed by Brewster windows Two beam (CW and CCW linearly polarized) Sagnac Ring with internal laser light source

Dual Polarized Ring Laser Polarization beam splitters 4 Polarized beams –CWLH, CCWLH, CWRH, CCWRH Sagnac Ring with internal laser light source

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 = ~ Better than Mossbauer

Dual Polarized Ring Lasers Non-reciprocal measurements with a Q = ~ Better than Mossbauer This technology has had little exploitation !!!

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.

In Non-Reciprocal media, the Speed of light not only depends upon polarization, but also depends upon the direction of propagation. 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:

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 Non-reciprocal Media Faraday rotation or Fresnel-Fizeau 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

Tune to a single mode. If no intra Optical Cavity effects, then get a single beat frequency due to Sagnac Rotation. Technique

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 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 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 , 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)

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.

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:  =  k m dx m   dt Such that the intersections of the 1-form, , and the 2-forms vanish  ^F = 0  ^G = 0 Also require that J =0.

k × E − ωB = 0, k · B = 0, k × H + ωD = 0, k · D = 0, In Engineering Format become:  ^F = 0  ^G = 0 In Engineering Format become: 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.

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

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 Chiral Constitutive Equation Examples Fresnel-Fizeau  “rotation” + diagonal Chiral  “expansions”

This Chiral Constitutive Equation Explains the Dual Polarized Sagnac ring laser

Z axis: Index of refraction 4 roots  =1/3  - 1/2 Sagnac Effect Fresnel Surface The index of refraction has 4 distinct values depending upon direction and polarization.

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 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 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 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 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, 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.

Simplified (diagonal  ) Constitutive matrix for a chiral Vacuum  =  + i   =  1  =  1 Conformal off-diagonal chiral matrices

Simplified (diagonal  + Fresnel rotation  ) Constitutive matrix for a chiral Vacuum Leads to Sagnac 4 phase velocities Conformal + Rotation chiral matrices

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 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

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

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

Real ω = 1/3, f = 0, Imag  = 1/6,  = 0 The 4-mode Sagnac Effect - with No center of symmetry

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