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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Holographic Grids

Walter Schempp

Published Oct 25, 1988• 6 citations
Publisher

Abstract

The holographic transform H is thought of as the sesquilinear mapping which assigns to the elements of the tensor product vector space L2(R) @ L2(R) the matrix coefficient function of the linear Schrodinger representation of the three-dimensional real Heisenberg nil-potent Lie group. - It is the purpose of the present paper to study the quantum mechanical approach to optical holography including optical phase conjugation, electron holography, and computerized tomographic methods in high resolution imaging radar, for instance synthetic-aperture imaging radar like side-looking airborne SAR, and space shuttle SAR. The aspect common to these techniques is the geometric encoding of the two-wave mixing u @ v by the interference pattern H(u,v;.,.) in the holographic plane R @ R performed simultaneously for the amplitude and the phase by the holographic transform H.

DOI
10.1117/12.968944

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The holographic transform H is thought of as the sesquilinear mapping which assigns to the elements of the tensor product vector space L2(R) @ L2(R) the matrix coefficient function of the linear Schrodinger representation of the three-dimensional real Heisenberg nil-potent Lie group. - It is the purpose of the present paper to study the quantum mechanical approach to optical holography including optical phase conjugation, electron holography, and computerized tomographic methods in high resolution imaging radar, for instance synthetic-aperture imaging radar like side-looking airborne SAR, and space shuttle SAR. The aspect common to these techniques is the geometric encoding of the two-wave mixing u @ v by the interference pattern H(u,v;.,.) in the holographic plane R @ R performed simultaneously for the amplitude and the phase by the holographic transform H.
Quantum holography is a well established theory of mathematical physics based on harmonic analysis on the Heisenberg Lie group G. The geometric quantization is performed by projectivization of the complexified coadjoint orbit picture of the unitary dual Ĝ of G in order to achieve a geometric adjustment of the quantum scenario to special relativity theory. It admits applications to various imaging modalities such as synthetic aperture radar (SAR) in the microwave range, and, most importantly for the field of non-invasive medical diagnosis, to the clinical imaging modality of magnetic resonance tomography (MRI) in the radio frequency range. Quantum holography explains the quantum teleportation phenomemon through Einstein-Podolsky-Rosen (EPR) channels which is a consequence of the non-locality of phase coherent quantum field theory, the concept of absolute simultaneity of special relativity theory which provides the Einstein equivalence of energy and Fitzgerald-Lorentz dilated mass, and the perfect quantum holographic replication process of molecular genetic information processing. It specifically reveals what was before unobservable in quantum optics, namely the interference phenomena of matter wavelets of Bose-Einstein condensates, and what was before unobservable in special relativity, namely the light in flight (LIF) recording processing by ultrafast laser pulse trains. Finally, it provides a Lie group theoretical approach to the Kruskal coordinatized Schwarzschild manifold of relativistic cosmology with large scale applications to general relativity theory such as gravitational instanton symmetries and the theory of black holes. The direct spinorial detection of gravitational wavelets emitted by the binary radio pulsar PSR 1913+16 and known only by anticipatory system computation so far will also be based on the principles of quantum holography applied to very large array (VLA) radio interferometers.
Quantum holography is a well established theory of mathematical physics based on harmonic analysis on the Heisenberg Lie group G. The geometric quantization is performed by projectivization of the complexified coadjoint orbit picture of the unitary dual G of G in order to achieve a geometric adjustment to special relativity theory. It admits applications to various imaging modalities such as synthetic aperture radar (SAR), and most importantly for the field of non-invasive medical diagnosis, to the clinical imaging modality of magnetic resonance tomography (MRI). Quantum holography explains the quantum teleportation phenomenon through Einstein-Podolsky-Rosen (EPR) channels which is a consequence of the non-locality of quantum physics. It specifically reveals what was before unobservable in special relativity, namely the light in flight (LIF) recording processing by ultra fast laser pulse trains. Finally, it provides a Lie group theoretical approach to the Kruskal coordinatized Schwarzschild manifold of quantum cosmology with large scale applications to general relativity theory such as gravitational instanton symmetries and the theory of black holes. The direct spinorial detection of gravitational wavelets emitted by the binary radio pulsar PSR 1913+16 will also be based on the principles of quantum holography.
Coherent wavelets form a unified basis of the multichannel perfect reconstruction analysis-synthesis filter bank of high resolution radar imaging and clinical magnetic resonance imaging (MRI). The filter bank construction is performed by the Kepplerian temporospatial phase detection strategy which allows for the stroboscopic and synchronous cross sectional quadrature filtering of phase histories in local frequency encoding multichannels with respect to the rotating coordinate frame of reference. The Kepplerian strategy and the associated filter bank construction take place in symplectic affine planes which are immersed as coadjoint orbits of the Heisenberg two-step nilpotent Lie group GGG into the foliated three-dimensional real projective space $\mbox{\bf P}\bigl(\mbox{\bf R}\times\mbox{\rm Lie}(G)^{\star}\bigr)$. Due to the factorization of transvections into affine dilations of opposite ratio, the Heisenberg group GGG under its natural sub-Riemannian metric acts on the line bundle realizing the projective space $\mbox{\bf P}\bigl(\mbox{\bf R}\times \mbox{\rm Lie}(G)^{\star}\bigr)$. Its elliptic non-Euclidean geometry without absolute quadric, associated to the unitary dual G^\hat{G}G^, governs the design of the coils inside the bore of the MRI scanner system. It determines the distributional reproducing kernel of the tracial read-out process of quantum holograms excited and coexisting in the MRI scanner system. Thus the pathway of this paper leads from Keppler's approach to projective geometry to the Heisenberg approach to the sub-Riemannian geometry of quantum physics, and finally to the enormously appealing topic of ensemble quantum computing.
Summary form only given. In this tallk I will try to supplement what I learned in the field. Here are the topics to be touched on. The order is not historical, but serves a tutorial purpose. 1. Time reversibility (TR) of optical wave propagation. 2. Mathematical description of TR in terms of complex amplitudes and complex conjugate amplitudes. 3. Static holography by real and conjugate images. 4.Static holography: illumination by counterpropagating reference waves at the recording and reconstruction stages. 5. Dynamic holography/four-wave mixing for phase conjugation (R. Hellwarth, A. Yariv, Yariv-Pepper instability. 6. Delayed and hidden holograms, photon echoes, electro-acoustic echoes. 7. Two-photon coherence, fast nonlinearities and three-wave mixing via X(2) nonlinearity. 8. Yet unrealized: Bragg (periodic domain). Time-domain phase conjugation: reversal of chirp and compensation of the dispersion spread in fibers. 10. Why a laser may (and may not) oscillate at a single transverse mode: discrimination of gain/loss. 11. Phase conjugation via stimulated Brillouin scattering (V.V. Ragulsky; V.V. Shkunov). 12. Brillouin-enhanced four-wave mixing for phase conjugation, ten to the tenth power amplification of the image (G.A. Pasmanik). 13. Photorefractive crystals: nonlinear response at low intensity at a cost of long build-up time. 14. Double phase conjugation 15. Feinberg's cat and cat conjugator.
Coherent wavelets form a unified basis of the multichannel perfect reconstruction analysis-synthesis filter bank of high resolution radar imaging and clinical magnetic resonance imaging (MRI). The filter bank construction is performed by the Kepplerian temporospatial phase detection strategy which allows for the stroboscopic and synchronous cross sectional quadrature filtering of phase histories in local frequency encoding multichannels with respect to the rotating coordinate frame of reference. The Kepplerian strategy and the associated filter bank construction take place in symplectic affine planes which are immersed as coadjoint orbits of the Heisenberg two-step nilpotent Lie group GGG into the foliated three-dimensional real projective space $\mbox{ P}\bigl(\mbox{ R}\times\mbox{Lie}(G)^{\star}\bigr)$. Due to the factorization of transvections into affine dilations of opposite ratio, the Heisenberg group GGG under its natural sub-Riemannian metric acts on the line bundle realizing the projective space $\mbox{ P}\bigl(\mbox{ R}\times \mbox{ Lie}(G)^{\star}\bigr)$. Its elliptic non-Euclidean geometry without absolute quadric, associated to the unitary dual G^\hat{G}G^, governs the design of the coils inside the bore of the MRI scanner system. It determines the distributional reproducing kernel of the tracial read-out process of quantum holograms excited and coexisting in the MRI scanner system. Thus the pathway of this paper leads from Keppler's approach to projective geometry to the Heisenberg approach to the sub-Riemannian geometry of quantum physics, and finally to the enormously appealing topic of ensemble quantum computing.
The concept of holography has lured philosophers of science for decades, and is becoming more and more popular on several fronts of science, e. g. in the physics of black holes. In this paper we try to understand things as if the visible universe were a reading of a low-dimensional hologram generated in hyperspace. We performed the whole process of creating and reading the hologram of a point in virtual space by using computer simulations. We claim that the fuzzieness in quantum mechanics, in statistical physics and thermodynamics is due to the fact that we do not see the real image of the object, but a holographic projection of it. We found that the projection of a point particle is a de Broglie-type wave. This indicates that holography could be the origin of the wave nature of a particle. We have also noted that one cannot stabilize the noise (or fuzzieness) in terms of the integration grid-points of the hologram, it means that one needs to give the grid-points a physical significance. So we further claim that the space is quantized, which supports the basic assumption of quantum gravity. Our study in the paper, although it is more qualitative, yet gives a smoking gun hint of a holographic basis of physical reality.