OPTICAL ECHO HOLOGRAPHY
Eugene I. Shtyrkov
Abstract: This paper discusses an approach in holography based upon the effect of interference of the atomic coherent superposition states that are induced in the resonant medium by time-separated object and reference pulse optical fields.
First of all, let’s give a definition to the phenomenon of the echo-holography. We shell understand it as an effected that involves recording and reconstruction of the optical wave fronts when the reference, object and reconstructed waves are not coincident in time. In such a situation, contrary to the conventional dynamic holography in resonant media , the delay time between the reference and object pulses is too long enough for their fields to overlap in a space. Therefore, there is no direct interference between these optical fields. It has been shown , however, that the information about the wave front of the object field can be reconstructed after a certain time after its action on the resonant recording medium. Under certain conditions of the resonant excitation of the atomic system, this system produces a coherent radiation response. The wave front of this optical response depends not only on the wave front shape of the object and reference fields but also on the order of application of these fields to the recording medium. This fact enables the realization of the 4-dimensional, space-time holography.
This type of holography is based on two conditions. First is the presence in the atomic medium of the superposition states induced by a coherent excitation of the ensemble of atoms. Second is that, after the excitation, this collection of atoms has to have a long enough “memory” of this excitation.
Let’s discuss the first condition in more detail. We will start with considering a single atom. It is known that an atom can be not only in its stationary states, i.e., at its so called energy levels, but also in a linear superposition of these states. The stationary energy states of an atom in absence of external forces are, as well known, solutions of the stationary Schrödinger equation for the wave function of the atom and, in theory, they are stable. In practice, all these energy levels, excluding the lowest one, are quasi-stable. That is, a smallest influence from the environment (such as vacuum fluctuations, electric or magnetic fields, phonon oscillations, collisions, etc.) is sufficient to perturb this state of the atom so it begins relaxing to the ground state. During this transient process, the atom passes through a continuum of the superposition states that correspond to this energy transition. Usually, such states of an atom are described by the wave function that is a solution of the non-stationary Schrödinger equation and depends on time
where are the wave eigenfunctions that correspond to the stationary energy levels. The probability amplitudes , that are the harmonic functions here, determine the temporal behavior of the dipole moment acquired in this superposition state.
The fundamental property of a superposition state is its instability in principle. An atom, that by some means happened to be in such state, immediately starts transitioning to a lower stationary state because of inherent causes, despite all external influence channels are turned off (e.g., external fields are absent and various causes of relaxation are suppressed). This instability of the superposition states becomes very important when we consider coherent oscillations of a ensemble of atoms as a whole. In practice, one naturally deals with a large number of atoms, and, in holography specifically, we are interested in coherent properties of an ensemble of particles. If such an ensemble is excited coherently, every atom in it can be transferred into the same superposition state. Such a collective state of the ensemble is a pure coherent superposition state, and its further dynamics will stay coherent within some time interval. Exactly because of the instability of a superposition state of a separate atom, all atoms start to return to equilibrium simultaneously. Another words, because the atoms in a pure superposition state have the same dipole moment, all dipoles will have the same initial phase. Naturally, this could not be achieved if all atoms were transferred in one of the stationary states even in a coherent way. From these states, atoms can make transition down only spontaneously due to random external forces and only into a mixed superposition state. That is why the initial phase of oscillations of the separate dipoles can not be the same for all. Therefore, a pure superposition state of the ensemble can not take place. Accordingly, the interference of the atomic coherent superposition states can not occur at such a situation.
The second necessary condition is the “memory” property of the recording medium. When a matter is irradiated by time-separated (successive) short pulses of coherent light, interaction between these waves becomes possible only via the medium and, naturally, only if the matter possesses long enough phase memory. In such a setting, each pump pulse transfers the information about its wave characteristics to the medium where it is stored until the next pump pulse arrives. Therefore, this so called non-simultaneous interaction does not mean breaking the laws of causality, despite the optical pump waves themselves never overlap and can not interfere directly. In this case, the recording medium plays a role of a certain wave bridge between the pump fields with their wave characteristics.
Pure coherent superposition states of the medium, obtained by irradiating it with a coherent light pulse, gradually decay during the transient time interval due to various relaxation processes. That is, the ensemble of atoms gradually transfer to a mixed state. This is associated with an irreversible loss of the phase memory. Despite each atom in such state has a certain dipole moment, the macroscopic polarization of the medium, caused by coherent collective oscillations, vanishes, and only spontaneous polarization remains. The intensity of spontaneous radiation of the system of atoms in this case is N times lower than the intensity of radiation due to coherent polarization (N is the number of active centers in the sample). In various materials, the concentration of active centers is sufficiently high (108 - 1020 cm-3). Therefore, it is easy to imagine how effective is the interaction of light with matter on that stage of the transition process where the phase memory of the system of atoms is not much lost. In such situation, various transient quantum optical phenomena can take place such as self-induced transparency , optical nutation , super-radiation , photon echo , generation of spatial gratings by time-separated optical fields [7,8] and other .
Let us focus on the causes of loss of memory of the medium about its coherent excitation. If the ensemble consists of atoms with the same resonant radiation frequency, then, in absence of external causes, the oscillation phase difference between individual dipoles is preserved in time over the entire transient process until all atoms come to the equilibrium state. However, there always are some external influences that brake the phase of oscillation of individual dipoles by means of statistical random change of the instantaneous value of the dipole oscillation frequency. These changes are irreversible and result in a homogeneous broadening of the spectrum of collective radiation, which is the same for all atoms in the ensemble. This broadening is associated with the decay of the macroscopic polarization of the medium due to the gradual decrease of the number of atoms remaining in a coherent radiating group. Due to this, the phase memory loss time can be characterized with the inverse value of the homogeneous broadening of the specific atomic transition (the so-called transversal relaxation time Т2). It is known that the homogeneous broadening is a combination of the natural radiative broadening and another part due to various external influences. That is why suppression of such external causes is needed to decrease the homogeneous broadening, i.e. to prolong the phase memory. For example, cooling the crystal from the room to low temperatures (tens K) allows to prolong the phase memory of the homogeneously broadened ensemble from 10-13s to 10-7s. In gases, the homogeneous broadening is usually caused by collisions and is not so significant. Therefore, the phase memory time in gases, especially in molecular ones, is sufficiently long. With this, the time of the irreversible decay of the macroscopic polarization can reach hundreds microseconds.
However, much more frequent situation is realized in practice when the ensemble has also an inhomogeneous broadening (for example, in crystals due to local inhomogeneities of the crystalline field or in gases due to Doppler’s effect). In this case, the de-phasing of the collective oscillations between the pump pulses happens not only due to irreversible random changes of the resonant frequency. It is also caused by the phase mismatching over time due to the different frequencies of isochromats in the spectrum. The process of such de-phasing of the macroscopic polarization is reversible in principle, and the polarization can be restored at some time point. Exactly this reversibility property is used for the generation of coherent responses of an atomic system, such as the optical transients .
Recording, reconstruction, and transformation of light wave fronts can be realized in ensembles of atoms with both homogeneous and inhomogeneous broadening, by means of coherent responses of the system. With this, the information about the phase perturbation of the object wave is recorded both in the waves of the non-equilibrium polarization and in the pattern of local curving of layers of the periodic structure of a population density (transient induced gratings).
In the first situation, the information about the object can be reconstructed in the signals of self-diffraction , free polarization, and photon echo [2, 11]. Two-pulse or so-called primary photon echo is a response that occurs after the second pump pulse, and its delay time coincides with the time interval between the two pump pulses. The time of the information storage is limited here by the transversal relaxation parameter Т2. Using the population gratings, one can realize a longer storage of the information. The decay of the inverse grating is determined by the decay time of the longitudinal component of Bloch’s vector (longitudinal relaxation time Т1). This time is much grater than Т2 in solids. In this case, the information can be reconstructed even after complete decay of the macroscopic polarization induced in the medium by the object and reference fields, i.e., beyond the phase memory, in the signals of linear diffraction or stimulated photon echo (three-pulse echo). This coherent response of the system to the application of three successive pulses occurs after the third pulse, with delay time also equaled to the time interval between the first two pulses.
The possibility of recording and reconstructing of light wave fronts in the setting of a photon echo experiment was first considered in the work  where the term “echo holography” was introduced. Similar suggestions were later made in other works [12, 13].
Let us focus on the specifics of such holography in detail. In many practical situations, optical transitions can be studied in the framework of the two-level model of the atom, and fruitfulness of such an approach has been proven by many examples. Let a two-level system of atoms with the transition frequency be irradiated by two successive short pulses of monochromatic light with frequency, pulse areas and , and wave vectors and . If the delay between the pulses is less than time Т2 of irreversible phase relaxation, then, as shown in works [7, 8, 14], they induce a spatial holographic grating of population with the modulation depth depending on the degree of the phase memory loss by the time of the second pulse arrival. The simplest picture is observed at exact resonance at the planar waves pumping a the duration of pulses is small, such that the relaxation during their interaction with matter is negligible. In this situation, the stationary spatial grating of population in the two-level system is formed during the application of the second pulse and decays afterwards within the time of longitudinal relaxation Т1  :
where No is the concentration of atoms, and is the grating vector, One can see that such a grating is formed only if atoms are put in a superposition state. If one of the pulses either inverts the population of the system (at or equal to ) or returns it to the ground state (at or equal to ), the spatial modulation depth of the grating is zero. The most effective grating is formed in situation where each of the pump pulses brings the system into the state with the maximum dipole moment.
In more general situation the atomic system has inhomogeneous broadening, i.e., consists of many packets of two-level atoms, and the pump pulses contain information about an object. In this case a series of frequency-shifted propagating isochromatic waves of polarization is induced in the medium, and the character of the spatial modulation becomes more complex. For example, the two-level system population inversion represents with a set of dynamic holograms with parameters depending on the frequency detuning where denotes frequency of isochromatic waves:
where , , are background , modulation depth and four-dimensional phase, accordingly. The parameters , at the , where is the phase information contained in a weak perturbation of the wave front of the first pulse (assuming that the second pulse has a plane wave front), depend on the conditions of pumping in a complex way. The analytical expressions for and were obtained in  by solving Bloch’s equations modified for optical region without other limitations except the assumptions of rectangular pulse shape and large saturation parameter (, where is the Rabi frequency). Characteristics of these holograms are changing in a complex way during pumping (gratings emerge, vanish, emerge again, propagate in space). After pumping is finished, the propagation of the gratings stops, and a complex interference pattern remains frozen in the medium. However, we need to point that this pattern, while standing still in space, decays with time. Over a time interval t > Т2, all collective phase relations become averaged, and the decay of the gratings on this stage is determined only by the energy characteristics of the transition. Such a decay of holograms happens within the time Т1 of longitudinal relaxation that determines the system tendency towards equilibrium. The details of conditions for the generation of the transient induced gratings in systems with phase memory can be found in the review work .
In both homogeneous and inhomogeneous broadening situations, information can be reconstructed from such holograms by means of the probe wave, similarly to the regular holography. With this, the system with inhomogeneous broadening is able to generate a coherent response by itself, allowing reconstruction of the phase information in the signals of primary or stimulated photon echo. This is possible because the dynamics of the hologram forming is closely associated with the polarization induced in the medium in the form of traveling perturbation waves. Under certain conditions, these waves can be phased in space and time. In the general situation of multi-pulse pumping, the system evolves in such a way that any polarization wave existing in the medium before the time of arrival of the next pulse becomes a source of new waves of polarization. The number of waves and gratings grows in a nonlinear way with the increase of the number of pump pulses. After the application of the nth -pulse with the wave vector at time point , such waves have the following form:
A joint multi-step use of formulas (3) and (4) for an arbitrary number of pump pulses allows to trace the system evolution, including creation and vanishing of the polarization waves and the population gratings, the excitation pumping from one wave into another, etc. The evolution of such a system under a multi-pulse pumping was studied in detail in work , showing that, after the application of n pulses, polarization waves and population inversion gratings are formed in the medium in the general situation. Figure 1 shows an example of the system evolution under a three-pulse pumping.
ig.1. Evolution of the system of atoms under a resonant pumping with three short light pulses (dash line represents a traveling wave of polarization, step-like element represent a standing grating of population inversion).
Beside the regular gratings, the inter-modulation gratings can also be formed that are the result of multi-wave interaction (e.g., the grating with the wave vector in Fig. 1).
Under certain conditions, such a mechanism allows to generate polarization waves with a large wave vector magnitude. This provides a unique possibility to form gratings with ultra-short period (even less than half wavelength of the pump radiation applied) . Under the condition of phase synchronization, the polarization waves become a source of the electromagnetic wave response of the system. For example, after the application of the second pulse with the delay time , three polarization waves are generated in the medium beside the population inversion grating (see Fig.1 and Eq. (4)). The third term in Eq. (4) at n=2 represents the result of interference of the superposition states,
This polarization is the source of the primary photon echo. Indeed, the first factor in Eq. (5) is , i.e. it represents a wave conjugated to the polarization wave that was induced at the previous stage by the first pump pulse and contained the phase information about its wave front. As seen in Eq. (4) with the account of Eq. (5), the dependence on the frequency vanishes completely at the time point . This means that all of the isochromatic waves within the hole-burning interval should have the same phase at this time point. This result of co-phasing of the oscillations of all dipoles causes emergence of the super-polarization at this time point. With this, the amplitude of the macroscopic polarization, after averaging over a statistical ensemble, has the form of the pulse of a primary photon echo , radiated in the direction , with the deviation from the planar wave front . Under the condition of phase synchronizm , the optical field generated by this polarization contains the phase information about the wave fronts of both pump pulses. That is more, the form of the echo wave front depends on the order of the pulse application. To see this consider , for example, two situations.
Case 1: The first pulse is the object wave with the weak phase perturbation and the second pulse is a planar reference wave, i.e., .
In this case, we obtain for the phase of the echo wave , i.e., we have the wave front of the echo to be phase-conjugated to the object one.
Case 2: ^
is applied, then the object wave with the information .
In this situation, the primary echo has a phase perturbation doubled as compared with the object wave: . This fact can be used to increase the sensibility of interferometry measurements of weak phase inhomogeneities (for example, in gas flows at low pressure), similarly to work  where the wave front with double curvature was formed by diffraction of the second order at the reconstruction of the hologram obtained by non-linear exposition.
The character of the reconstruction of the wave fronts in the signals of the primary echo can be interpreted with simple geometric considerations. Let us assume that the reference and object waves propagate in the same direction (). Then, the echo will propagate in the same direction (). Let the object wave be applied earlier than the reference wave (Fig. 2a). Approaching the recording medium, the wave excites the atoms of the sample in different parts of the medium at slightly different times. For example, all particles located near point A are excited earlier than particles near point B (Fig. 2b). As pumping of the A-region particles ends also earlier, the de-phasing process of dipole oscillations in this region starts also earlier than in the vicinity of point B. At the moment when the planar wave arrives at points A and B simultaneously (Fig. 2b), dipoles located at these points will have different phase shift of oscillations.
Fig.2. Inversion of curvature of the wave front of primary photon echo at collinear pumping of the resonant medium with successive object and reference pulse fields (a and b – before pumping, c and d – after pumping).
After the passing of the wave, the process of co-phasing of dipoles starts at points A and B simultaneously. On account of the additional shift the dipoles at point A will need more time to co-phase and produce the echo signal. That is why the echo will occur at point B earlier (Fig. 2c). This causes the wave front curving to the opposite direction. There will be three waves, , , , behind the hologram now (Fig. 2d).
et us consider a reversed situation when the planar wave E is applied first, as illustrated in Fig 3.
Fig.3. Twofold increase of the wave front curvature in the primary photon echo at collinear pumping of the resonant medium with a sequence of the reference and object fields.
Here, the de-phasing will start in all parts of the sample simultaneously after the passing of the planar reference wave, and the start of co-phasing process at point B will be delayed by time compared to point A. That is why the echo will occur at point A by time earlier than at point B that will cause doubling of the wave front perturbation. Other situations can be interpreted in a similar way. For example, if the wave fronts of the pump pulses are chosen to be complex conjugated, i.e., and , then the echo wave front will have tripled perturbation of the inverse curvature compared with the first pulse, .
isadvantages of two-pulse echo holography include limitations in correct reconstruction of complex wave fronts. This is because of the fact that the condition of the wave synchronism is not fulfilled for higher spatial Fourier components of the object wave. A correct reconstruction of the wave fronts in the case of primary echo is possible only at angles between the reference wave and higher spatial harmonics of the object wave not greater than 3° [2,8]. Such limitation is lifted only in a case of the multi-wave nonsimultaneous interaction, i.e., when the medium are subjected action of more than two pump pulses. With such interactions, the shape of the wave front is also determined by the order of application of pulses. For example, exact reconstruction of the wave front of the object wave is possible in the case of the three-pulse (stimulated) photon echo (Fig. 4).
Fig.4. Reconstruction of the wavefront in the signal of the stimulated photonic echo with collinear pumping (a – before pumping, b – after the excitation of the medium).
In more general situation (non-collinear pumping) each spatial Fourier component of the object wave propagates in the direction of . Here, the exact reconstruction of the wave front can be reached at pumping by succession of pulses: the planar (reference) wave /the object wave with an arbitrary wave front / the planar probe wave, propagating in the same direction as the first one (). In this case, each component of spatial Fourier spectrum is reconstructed without distortion. Thus in echo holography, we deal with a more general principle of recording and reconstruction of wave fronts than traditional holography. Such a hologram records 4-dimensional information (the wave fronts, the order of application of the object and reference waves as well as the delay time of the object wave with respect to the reference one). The stimulated echo, as well as primary one, allows obtaining a complex conjugated replica of the object wave. This is possible when the object wave is applied first, and then two planar waves are applied in sequence. With this, a complete inversion of the wave front can be obtained if the second and third waves propagate in exactly opposite directions (). In such situation, similarly to a simultaneous four-wave interaction, stimulated echo in the presence of a standing wave will have both a direction and wave front inversed with respect to the object wave. The only difference will be that the inverse wave is delayed in time. The wave front inversion in the setting of photon echo was first observed experimentally in work  and then was investigated more in detail in work . The resonant medium was a ruby crystal with a Cr3+ concentration of 0.03 at%. The sample was a cube with a 10mm edge and it was cooled to 2oK in order to increase the duration of the phase memory of the medium. A resonance at the transition (λ=6934 Å at 2oK ) was excited by a ruby laser which was cooled to 770K and generated the second component of the same doublet (λ=6933,97 Å). Single-pulse emission was ensured by active Q switching at the resonator cavity, which made it possible to suppress the stray optical coupling between the resonator elements and generate pulses of 10ns duration and about 1MW power. The sample was pumped approximately along the optic axes of the crystal with two successive pulses 1 and 2. Polarizations of all the beams were perpendicular to the plane of their incidence on the sample. The angle between first two pulses was 5o which ensured a satisfactory spatial decoupling between the pump and echo signals. A weaker longitudinal static magnetic field (up to 250 Oe) was applied to the sample by Helmholtz coils and it lifted partly the degeneracy of the ground state 4A2 so that the signal echo became stronger. The backward echo was observed in direction 4 (fig.5) at the third pulse p
roduced by reflection of the (2)-pulse from the plane mirror М.
Fig.5. Experimental set.
A system of conjugate lenses with f = 50cm made it possible to increase the intensity of the pump field and, on the other hand, to combine automatically the caustics of all the beams in the same region. In fact, the reversed echo beam 4 had reverse divergence relative to the first beam. The quality of the wave front reversal was checked by introducing deliberate aberrations into the signal beam I by inserting an etched phase plate directly in front of the sample. Figures 6a and 6b show examples of compensation of the aberrations in the reversed echo signal.,
Fig.6. Angular divergence of the beam after reflecting by: a) plane mirror located in the plane of the sample; b) reversing echo mirror; 1) with a phase plate, 2) without the phase plate
Clearly, in the case when the object wave had a wider spatial frequency spectrum (in the presence of phase plate) the echo (1b) reproduced more accurately the pump beam (2a) geometry. This was clearly associated with the more favorable mixing of the spatial frequencies, a consequence of which was a more homogeneous excitation of the sample. The wave front reversal occurred due to a characteristic time reversal, i.e. at those points in the sample where the vibrations of the dipoles where advanced in phase during the application of the first nonplanar wave, and exactly the same phase lag appeared after the action of the pulse III. In accordance with the general principle of holography, this produces a pseudoscopic image of the object.
In the general case the moment of phase matching depends on many conditions, for example, there is a shift of the echo in the case of pumping by long pulses when it is no longer possible to neglect de-phasing of the dipoles during the action of a pump pulse . This may result in distortion of the wave front of the complex-conjugate replica of the signal wave if the intensity of this wave has a strongly inhomogeneous spatial distribution, so that in some parts of the sample there may be an additional induced de-phasing of dipoles (local lag and advance). Distortion of the wave fronts can be ignored only in the case of excitation by short pulses discussed above.
In conclusion, we can point at some prospective applications of this approach to induction of dynamic gratings. First of all, the problem of forming interference patterns by waves separated in time has itself a scientific importance. The reason for this is that, up to now, the interferometry theory in classical optics has considered coherent properties of optical fields only, without taking into account the phase memory of the recording medium. It seems that taking the latter into account will allow to expand and generalize the theory of interferometry significantly, that, in turn, will assist the development of techniques of the interferometry research. A method of forming dynamic echo holograms in systems of multi-level atoms opens greater possibilities to perform various transformations of the light wave fronts. This becomes important for the wavelength conversion, for example, from optical waves to acoustic waves and vice versa. Acousto-optical transformation of wave fronts in echo holography allows visualization of information contained in an acoustic wave. For example, in the situation where an acoustic plain wave is applied first, then an acoustic object wave is applied, and then a plain light wave is applied, the wave front of the stimulated photonic echo repeats the shape of the acoustic wave front . Another property of echo holograms that has a promising outlook is the possibility of the reconstruction of the forward and back wave fronts of the pulse object field separately . Echo holography also enables realization recording and complete reconstruction of information about dynamic of fast processes. For example, CW emission with fast scanning of the beam over the surface of the sample can be used as one of the pump beam. In this case, the direct or reverse scan track, depending on the pumping order, can be reconstructed in photon echo. Also of interest is the application of transient gratings for the investigation of the relaxation and spectroscopic characteristics of atomic transitions (dipole momentum, transversal and longitudinal relaxation times, weak splitting of atomic levels, etc.) to study the fundamental laws of interaction between radiation and matter. At present a new wave of interest to photon echo and time-space holography is stimulated by the problems of quantum information science [24-26].
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1. /RFC-862 -- ECHO Protocol.pdf