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We propose a volatile static all-optical memory capable of storing phase information of a slowly-varying electric field. The scheme and its realization (a memory circuit) are based on two mutually coupled lasers subject to external optical injection. The proposed circuit has a single optical input for write and hold operations and two opposite-sign outputs for reading the memory. The proposed circuit operates with a single wavelength of light, a single direction of propagation, and without a need to switch the state of polarization. We prove mathematically that the proposed arrangement has equilibrium points that may discreetly quantify and store the phase in a bistable manner. The circuit is studied numerically for solid-state and semiconductor lasers with zero and non-zero linewidth enhancement factors, respectively. Simulations based on a rate equation system confirm the essential findings. Using typical parameters of a semiconductor laser and optimizing for a possibly wide range of operation, the write-read operations were simulated using PRBS-9 at the rate of 1 Gb/s with negligible errors. The proposed circuit will enable integrated memory implementations for future all-optical signal processing and computing systems.

The dissertation concerns inverse problems and dynamical systems. Inverse problems, as a subfield of mathematics, studies the mathematical theory of indirect measurements. It is an active area of research with extensive mathematical theory and numerous applications. Many inverse problems are concerned with physical systems that evolve in time. A mathematical model that describes how a quantity evolves in time is called a dynamical system. X-ray computed tomography is a technique where the inner structure of an object is computed from a number of its X-ray images. In the first publication of the dissertation we consider X-ray computed tomography in a setting where the orientations in which the object was imaged are unknown. This problem is called tomography with unknown view angles; such a problem arises e.g. in cryogenic electron microscopy of viral particles. We show that under general assumptions it is possible to reconstruct the structure of the object in tomography with unknown view angles. Diffusion is the flow of a substance from areas of high concentration to areas of low concentration. In the second publication we consider an inverse problem for the space–time fractional diffusion equation. This equation models diffusion and anomalous diffusion processes, such as those sometimes observed in fractured geological formations. We show that the geometry of the underlying space can be determined by observing the evolution of a solution of the equation in a subset of the space. In the third and fourth publications we consider a dynamical system that models injection locking in a laser. Injection locking is a technique where light from one laser is injected into another laser's cavity (the part of a laser where the emitted light is created) with the intention of altering the laser's properties. The idea is to consider injection locking as a process that can provide the basis for optical computing devices. In the third publication we derive an approximation for the nonlinear relationship between the injected light and the injection-locked emitted light, and we show that it is possible to construct an optical logic gate based on this relationship. In the fourth publication we do a detailed analysis of the dynamical system that models injection locking in lasers, and based on this analysis, we propose a design for an optical neural network. Väitöskirja käsittelee käänteisten ongelmien ja dynaamisten järjestelmien matemaattista teoriaa. Käänteisissä ongelmissa pyritään ymmärtämään ilmiön syy sen seurauksia. Esimerkki käänteisestä ongelmasta on röntgenkuvaus, jossa kappaleen läpäisseiden säteiden vaimenemisesta (seuraus) halutaan päätellä vaimenemisen syy (kappaleen sisäinen rakenne). Usein käänteisen ongelman liittyvä tarkasteltava suure on dynaaminen, eli se muuttuu ajan mukana. Tällaisen ilmiön matemaattista mallia kutsutaan dynaamiseksi järjestelmäksi. Väitöskirjan ensimmäisessä julkaisussa tutkitaan tietokonekerroskuvauksen matemaattista teoriaa. Tietokonekerroskuvauksessa tarkasteltavasta kohteesta kuvataan röntgenkuvia useista eri suunnista, ja näistä kuvista pyritään laskemaan kohteen sisäinen rakenne. Usein suunnat joista kuvat on otettu tunnetaan, mutta tietyissä tilanteissa nämä kuvaussuunnat ovat tuntemattomia. Tällainen tilanne on esimerkiksi virusten kuvantamisessa käytetyssä kryogeenisessa elektronimikroskopiassa. Julkaisussa osoitetaan, että kohteen rakenteen määrittäminen on yleensä mahdollista vaikka kuvaussuunnat olisivatkin tuntemattomia. Lämmön johtumisen matemaattista mallia kutsutaan lämpöyhtälöksi. Lämmön johtuminen aineen sisällä riippuu sekä aineen kyvystä johtaa lämpöä (ns. lämmönjohtavuuskertoimesta), että tarkasteltavan kappaleen muodosta. Väitöskirjan toisessa julkaisussa tutkitaan käänteistä ongelmaa lämpöyhtälölle. Julkaisussa osoitetaan, että tarkastelemalla kappaleen lämpötilan käyttäytymistä osassa kappaletta voidaan päätellä sekä koko kappaleen muoto, että sen lämmönjohtavuuskerroin. Kolmannessa ja neljännessä julkaisussa tutkitaan optiseen laskentaan liittyvän dynaamisen järjestelmän matemaattista teoriaa. Nykyisten puolijohdekomponentteihin perustuvien tietokoneiden toiminta perustuu sähköön, optisessa laskennassa tavoite on rakentaa tietokone jonka toiminta perustuu sähkön sijaan valoon. Julkaisuissa tutkitaan lasereiden synkronoitumiseen liittyvän ilmiön (ns. injektiolukituksen) matemaattisia ominaisuuksia. Julkaisuissa osoitetaan että ominaisuuksiensa puolesta injektiolukitusta voi olla mahdollista hyödyntää optisten transistoreiden tai optisten neuroverkkojen rakentamisessa.

An all-optical computer has remained an elusive concept. To construct a practical computing primitive equivalent to an electronic Boolean logic, one should utilize nonlinearity that overcomes weaknesses that plague many optical processing schemes. An advantageous nonlinearity provides a complete set of logic operations and allows cascaded operations without changes in wavelength or in signal encoding format. Here we demonstrate an all-optical majority gate based on a vertical-cavity surface-emitting laser (VCSEL). Using emulated signal coupling, the arrangement provides Bit Error Ratio (BER) of 10−6 at the rate of 1 GHz without changes in the wavelength or in the signal encoding format. Cascaded operation of the injection-locked laser majority gate is simulated on a full adder and a 3-bit ripple-carry adder circuits. Finally, utilizing the spin-flip model semiconductor laser rate equations, we prove that injection-locked lasers may perform normalization operations in the steady-state with an arbitrary linear state of polarization.

An analytical study of dynamical properties of a semiconductor laser with optical injection of arbitrary polarization is presented. It is shown that if the injected field is sufficiently weak, then the laser has nine equilibrium points, however, only one of them is stable. Even if the injected field is linearly polarized, six of the equilibrium points have a state of polarization that is elliptical. Dependence of the equilibrium points on the injected field is described, and it is shown that as the intensity of the injected field increases, the number of equilibrium points decreases, with only a single equilibrium point remaining for strong enough injected fields. As an application, a complex-valued optical neural network with working principle based on injection locking is proposed., Final version, to appear in SIAM Journal on Applied Dynamical Systems (SIADS)

We consider uniqueness of two-dimensional parallel beam tomography with unknown view angles. We show that infinitely many projections at unknown view angles of a sufficiently asymmetric object determine the object uniquely. An explicit expression for the required asymmetry is given in terms of the object's geometric moments. We also show that under certain assumptions finitely many projections guarantee uniqueness for the unknown view angles. Compared to previous results about uniqueness of view angles, our result reduces the minimum number of required projections to approximately half and is applicable to a larger set of objects. Our analysis is based on algebraic geometric properties of a certain system of homogeneous polynomials determined by the Helgason-Ludwig consistency conditions.

We consider uniqueness of two-dimensional parallel beam tomography in which both the object being imaged and the projection directions are unknown. This problem occurs in certain practical applications. For example, in magnetic resonance imaging there may be uncertainty in the projection directions due to the involuntary motion of the patient. The three-dimensional version of this problem occurs in cryo electron microscopy of viral particles, where the projection directions may be completely unknown due to the random orientations of the particles being imaged. We show that the problem is related to some algebraic geometric properties of a certain system of homogeneous polynomials. We also show that for sufficiently asymmetric objects, the object is uniquely determined up to an orthogonal transformation by the projection data from unknown directions.