Stimulated Brillouin scattering is a fundamental interaction between light and travelling acoustic waves and arises primarily from electrostriction and photoelastic effects, with an interaction strength several orders of magnitude greater than that of other relevant non-linear optical processes. Here we report an experimental demonstration of Brillouin-scattering-induced transparency in a high-quality whispering-gallery-mode optical microresonantor. The triply resonant Stimulated Brillouin scattering process underlying the Brillouin-scattering-induced transparency greatly enhances the light–acoustic interaction, enabling the storage of light as a coherent, circulating acoustic wave with a lifetime up to 10 μs. Furthermore, because of the phase-matching requirement, a circulating acoustic wave can only couple to light with a given propagation direction, leading to non-reciprocal light storage and retrieval. These unique features establish a new avenue towards integrated all-optical switching with low-power consumption, optical isolators and circulators. Stimulated Brillouin Scattering (SBS) in fibres and waveguides has attracted strong interest in a variety of photonic applications during past decades, such as light storage, slow light, lasers and optical isolators.
SBS can be incorporated into photonic integrated circuits, where the tight confinement of the optical fields greatly enhances the SBS interaction. Giant enhancement of SBS in the subwavelength scales due to radiation pressures or boundary-induced non-linearities has been demonstrated recently. SBS has also been realized in optical microresonators, such as silica microspheres and disks, crystalline cylinders. In these whispering gallery resonators, optical and acoustic waves circulate along the equatorial surface, forming optical and mechanical whispering-gallery modes (WGMs) with ultrahigh-quality ( Q) factor. The SBS process can become triply resonant when the control field, the Stokes or anti-Stokes field and the acoustic wave are all resonant with the relevant optical and mechanical modes. The triply resonant SBS process, along with the ultrahigh Q-factors and the small mode volume, provides new opportunities for exploring coherent light–acoustic interactions. Over the past few years, low-threshold Brillouin lasers, Brillouin optomechanics, and Brillouin cooling, have been reported in such triply resonant WGMs.
Here we report the experimental demonstration of Brillouin-scattering-induced transparency (BSIT) and non-reciprocal light storage in a silica microsphere resonator. In contrast to the optomechanically induced transparency that has been observed in a variety of optomechanical systems, two optical modes couple resonantly to an acoustic mode via forward Brillouin scattering in BSIT. A strong optical control field near the lower-frequency optical WGM drives a coherent interaction between the acoustic and the higher-frequency optical WGMs, inducing a transparency window for a probe field near the higher-frequency optical WGM.
A number of remarkable coherent optical phenomena and potential applications are possible, such as light storage, dark modes and frequency conversions. In addition, due to the phase-matching requirement for travelling waves in SBS, the coherent photon–phonon conversion is only allowed for waves propagating in certain directions. We have taken advantage of this property to demonstrate the non-reciprocal storage and retrieval of a coherent optical field. These results indicate that SBS is an excellent candidate for applications in photonic integrated circuits. With the acoustic vibrations cooled to their motional ground states, applications in a quantum regime, such as single-photon storage and frequency conversion, also become possible. Stimulated Brillouin scattering In a silica microsphere resonator, there are optical and acoustic WGMs that propagate along the surface.
Both optical and acoustic WGMs are characterized by the orbital angular momentum mode number, m. When the acoustic WGM ( a) and two optical WGMs ( c and d) satisfy the energy and momentum conservations, such that ω a= ω d− ω c and m a= m d− m c, photons can be scattered between the two optical resonances through Brillouin scattering. In this work, we focus on the forward SBS, in which m c and m d have the same sign and both optical modes are coupled to travelling waves in the same direction through the tapered fibre, as depicted in. As schematically illustrated in (more detailes in and ), the SBS process, for which the control laser pumps on the lower-frequency optical mode, leads to phonon absorption and anti-Stokes photon generation, whereas the Stokes process is inhibited. ( a) Schematic illustration of the light–acoustic wave interaction in a microsphere. The optical modes of the microsphere are excited by the control and probe lasers through the tapered fibre.
These modes interact with the travelling acoustic wave via the forward Brillouin scattering. ( b) Spectrum diagram of the coherent photon–phonon interaction: the control laser is near resonance with the lower-frequency mode, and the probe field is on resonance with another cavity mode, whereas the Stokes process is suppressed. ( c) The experimental set-up. EOM, electro-optical modulator; FPC, fibre polarization controller. ( d) The spectrum of a typical acoustic mode at 42.3 MHz in the microsphere when only the control laser was fixed on resonance with ω c.
Inset: simulated deformation of microsphere at the equator for acoustic WGM with m a=6. Triply resonant SBS was observed for a control laser wavelength near 1,562 nm in a silica microsphere with a radius of 98 μm.
We detected the scattered anti-Stokes light by measuring its beating signal with the control laser field. The corresponding power density spectrum was monitored with a spectrum analyser, as shown in. The Lorentzian lineshape observed indicates an acoustic WGM with a frequency of ω a/2 π=42.3 MHz and a linewidth of γ a/2 π=4 kHz (acoustic Q a≈10,600). The acoustic WGM was also further verified by measuring directly the scattered Stokes light. Using the acoustic velocity in silica, we determined the orbital angular momentum mode number of the acoustic WGM to be m a=6 (inset of ). Brillouin-scattering-induced transparency The triply resonant Brillouin scattering discussed above can lead to coherent optical phenomena, such as BSIT, which is an analogue to the well-known electromagnetically induced transparency in atomic system. In a simple and intuitive picture, the control and probe induce a coherent vibration of the acoustic mode.
BSIT arises from the destructive interference between the probe field and the anti-Stokes optical field generated by scattering of the control laser from the coherent acoustic vibration. For our system, this destructive interference prevents the excitation of the higher-frequency optical mode.
For a detailed theoretical description of BSIT, we consider the system Hamiltonian including the non-linear photon–phonon interaction where a is Boson operator of the acoustic mode and c, d are Boson operators of the optical modes (see ), g is the single-photon Brillouin coupling rate, which is non-zero only when the phase-matching condition is satisfied. The energy diagram of the system is schematically illustrated in, where the energy levels are described by phonon and photon Fock states n a, n c, n d›, where n a( c, d) is the number of phonons (photons) and Brillouin scattering induces transitions between n a, n c, n d +1› and n a+1, n c+1, n d›. For a strong control laser such as, the energy diagram can be simplified as, which has resemblance to the Λ-type system in atomic electromagnetically induced transparency.
( a– b) Energy diagram for the triply resonant Brillouin scattering discussed in the text. ( c– e) Emission power from the higher-frequency optical mode obtained when we scanned the probe frequency by sweeping the modulation frequency Ω of the control laser for several values of control detuning, Δ=1.3,0,−1.3 MHz. Although the peak of the cavity resonance shifts with Δ, the sharp BSIT dip was always observed at δ=0. The incident control power used was 0.3 mW. The insets show the spectral response near δ=0 with an expanded frequency scale. The short-dashed lines are the results of calculations using the parameters k d/2 π=3.5 MHz, γ a/2 π=0.004 MHz and, 0.05, 0.03 MHz, respectively.
( f) The emission power from the higher-frequency optical mode near δ=0 for five different powers of the control beam from 0.18 mW up to 0.35 mW. ( g) The spectral linewidth and depth of the BSIT dip as a functions of the cooperativity C, as derived from f. The solid lines in g represent the theoretically expected values. Under the strong control laser driving, the Hamiltonian is effectively simplified to (see ) where the acoustic mode and the higher-frequency optical mode (mode d) are coupled linearly, with the effective coupling rate determined by the photon number of lower-frequency optical mode. Here, ε p denotes the weak probe field with a frequency of ω p coupling to mode d (see ), Δ= ω a+ ω l− ω d and δ= ω p− ω l− ω a are relevant frequency detunings, where ω l is the frequency of the control laser.
The steady-state intracavity power spectrum is given by, The intracavity power of mode d is modified by the coherent photon–phonon interaction, giving rise to significant changes in the intracavity power when is comparable to κ d. It is convenient to introduce the cooperativity defined as, a dimensionless parameter that characterizes the relative strength of the coherent photon–phonon interaction. C also characterizes quantitatively the BSIT process. To observe BSIT in our system, we probed the intracavity power spectrum in the higher-frequency optical mode using a heterodyne detection technique, with the control laser serving as the local oscillator.
We generated the probe field by phase modulating the control laser with an EOM, with modulation frequency Ω. Using a network analyser, the intracavity power can be extracted from the heterodyne signal with modulation frequency Ω. As shown in, we investigated the dependence of the BSIT response on detuning Δ by adjusting ω l with a fixed control power of P=300 μW. The transparency window was observed when the two-photon detuning is near the mechanical resonance frequency, ω p− ω l≈ ω a. The lineshapes of the BSIT response, as shown in the expanded spectra in, demonstrate directly the optical interference between the intracavity probe field and the anti-Stokes field. A theoretical calculation of the BSIT response based on equation (3) leads to estimated values of C=5.6, 0.71 and 0.26 for Δ=1.3, 0 and −1.3 MHz, respectively. The triply resonant Brillouin Scattering system enables us to achieve a cooperativity near 6 with a control laser power about 300 μW.
Much greater control powers are needed for the achievement of similar cooperativity in WGM-type optomechanical resonantors that do not satisfy the triple resonant conditions. The strongest cooperativity was not observed at Δ=0, because the triply resonant condition was not exactly satisfied in the experiment, as we estimated ω d− ω c− ω a≈1.3 MHz. Further studies of the BSIT dip at Δ=0 are presented in, for which the control laser power was varied from 180 to 350 μW. BSIT dips of increasing depths and widths were observed, which can be fitted numerically with a simple Lorentzian lineshape.
As shown in equation (3), the linewidth of the BSIT dip is given by (1+ C) γ a and the relative depth of the BSIT dip is given. However, we found that C derived from the numerical fits was not proportional to the input control power because of the thermal effect, which slightly changes the triply resonant condition. Non-reciprocal light storage Owing to the coherent Brillouin interaction, the coherent conversion between photons and acoustic phonons can be used for light storage.
In contrast to the previously studied optomechanics driven by radiation pressure and gradient forces, in which the mechanical vibrations can couple to a variety of optical modes, Brillouin scattering is possible only for specific modes that satisfy the energy and momentum conservation conditions. The input control laser coupling to clockwise (CW) optical mode only permits interactions between CW acoustic phonons and photons, whereas the counter-clockwise (CCW) probe field or acoustic wave is decoupled from the control field. Therefore, the requirement of phase matching for the Brillouin scattering process gives rise to the breaking of the relevant reversal symmetry, leading to non-reciprocal non-linear optical processes that are switchable by the control laser. To demonstrate this non-reciprocity, we studied the light storage of CW or CCW signal fields with a fixed CW control laser. As shown in, the control and signal fields are switched and frequency shifted by two acousto-optic modulators (AOM 1 and 2), with frequency difference matching ω a. The control laser pulses, including writing and readout, are coupled to the CW mode only. The signal pulse is coupled to either CW or CCW mode by moving a flip mirror.
The sequences of writing, readout and signal pulses are shown in the inset of. When the triply resonant condition is satisfied, the CW signal pulse will be converted to a CW acoustic wave coherently via the beam-splitter-type Brillouin coupling shown in equation (2). The coherence nature of the interconversion between optical and mechanical excitations have been demonstrated in an earlier study of mechanical breathing modes. As shown in, the CW signal (black line) was stored during the writing pulse and retrieved during the readout pulse after a delay of 3 μs, which is much longer than the cavity photon lifetime. The exponential decay of the signal intensity during the writing pulse results from the underlying dynamical BSIT process.
A similar phenomenon has also been reported for mechanical breathing modes in a silica microsphere in ref. The decay time of the converted signal is 14 μs, as determined from an exponential fit (green dashed line in ). In addition, the bandwidth for light storage is determined by the transparency bandwidth of BSIT, (1+ C) γ a, where C is proportional to the input control power (see ). ( a) The experimental set-up for the non-reciprocal light storage. The CW signal is combined with the CW control laser through a beam splitter (BS2). The CCW signal is launched into the fibre through a circulator. ( b) The measured intracavity signal power during the storage and retrieval processes.
The black and blue lines correspond to the input signal with different input directions. Inset: the pulse sequences for writing, readout and signal, with the arrows indicating the propagation direction. ( c) Schematic diagram of single-bit quantum memory based on circulating acoustic phonons. PBS, polarization beam splitter; PR, polarization rotator. In the previous section, we provided experimental evidences of coherent and non-reciprocal photon–phonon interconversion arising from Brillouin scattering in a silica microsphere, though the experiments at room temperature are affected by the thermal acoustic phonons.
Since the interaction between the signal field and the acoustic wave with a strong control field is in the form of (equation (2)), the coherent interconversion should persist even when the signal field is at the single-photon level. In particular, single-photon storage becomes possible when the relevant acoustic modes are cooled down to their ground states. For ω a/2 π=1 GHz and at T=0.5 K, the thermal phonon occupation is near 10. In this case, the ground state cooling requires a cooperativity C 10, which is achievable for silica microspheres, given the reduced thermal bistability and improved mechanical Q factor at the low temperature. Note that the state transfer between a motional state and a microwave field in the quantum regime has already been demonstrated recently in an electromechanical system. Because of the momentum conservation required for SBS, the degenerate CW and CCW acoustic modes can be written and readout independently by the corresponding CW and CCW control laser.
Assuming degenerate CW and CCW optical WGMs in a high-quality whispering gallery microresonator, we can treat a pair of WGMs as one bit of binary information carriers. Here we also proposed a design of quantum memory for polarization-encoded photon state. As illustrated in, with the use of the polarization beam splitter and the polarization rotator, horizontally and vertically polarized input photon states can individually couple to CW and CCW optical modes. These states can be stored as the CW and CCW acoustic waves and be readout, separately. In principle, the input state, a superposition of polarization state α H›+ β V›, can be stored as a superposition of circulating acoustic state α ›+ β ›. In the reversal of the storage process, the states can be readout by being converted to the photon state α V›+ β H› after tens of microseconds. Brillouin scattering in whispering-gallery microresonators is promising in several aspects.
(a) Brillouin scattering enables the optical coupling to acoustic waves with frequency ranging from a few MHz to 11 GHz, thereby providing a diverse platform for coherent light-matter interactions. (b) The triply resonant configuration can greatly enhance the Brillouin scattering, thereby reducing the power consumptions. (c) Phase matching for the travelling waves enables non-reciprocal optical processes, thus offering potential application in an all-optical integrated isolator and circulator devices. Our studies pave the way towards the coherent coupling between photons and acoustic phonons and should stimulate further investigations of non-reciprocity and memory at the quantum level. During the preparation of this manuscript, a similar work has been reported on the arXiv.
Experiment method The experimental set-up is schematically illustrated in. All experiments were performed at room temperature and at atmospheric pressure. Silica microspheres were fabricated by melting a tapered fibre with a CO 2 laser. Optical WGMs in the microsphere were excited through the evanescent field of a tapered optical fibre with a tunable narrow-linewidth (.
Kobyakov, A., Sauer, M. & Chowdhury, D. Stimulated Brillouin scattering in optical fibers. Eggleton, B., Poulton, C.
Communication
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Aperture 3.6 Released for Mac OS X Yosemite Compatibility. Assuming not, please open iPhoto and also Image Capture and see if you see the entire card there. Dec 21, 2010 This update adds new email options to iPhoto '11. It also improves overall stability and addresses a number of other minor issues. Specific fixes include. OS X Yosemite for Mac, free and safe. Meanwhile iPhoto has disappeared and is replaced by a faster slicker app called Photo. ITunes has also been through an. Iphoto yosemite скачать. Now that OS X Yosemite is rolling out to everyone we're seeing app updates galore start to roll in. Apple's own iMovie. IMovie for Mac gets dressed up for OS X. If you're already running Yosemite, you can download Photos for Mac by checking for updates in the Mac App Store. IPhoto is Apple's flagship application for managing and viewing photos on your Mac. As a competitor to Picasa it packs a powerful punch as a slick OS X image management app that's fully integrated into iCloud, Maps and more.
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The minimum system requirements for Java Virtual Machine are as follows: • Windows 8/7/Vista/XP/2000 • Note: Downloading and installing of Java will only work in Desktop mode on Windows 8. Verify the system requirements for Java Virtual Machine Verify the system requirements for Java Virtual Machine before installing it on your computer. 64-Bit Browser. • Windows Server 2008/2003 • Intel and 100% compatible processors are supported • Pentium 166 MHz or faster processor with at least 64 MB of physical RAM • 98 MB of free disk space Download and install the latest Java Virtual Machine in Internet Explorer 1. If you are using the Start screen, you will have to switch it to Desktop screen to run Java.
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For best results, use the separate Authors field to search for author names. Author name searching:. Use these formats for best results: Smith or J Smith. Use a comma to separate multiple people: J Smith, RL Jones, Macarthur. Note: Author names will be searched in the keywords field, also, but that may find papers where the person is mentioned, rather than papers they authored. Abstract We present a detailed overview of stimulated Brillouin scattering (SBS) in single-mode optical fibers. The review is divided into two parts.
In the first part, we discuss the fundamentals of SBS. A particular emphasis is given to analytical calculation of the backreflected power and SBS threshold (SBST) in optical fibers with various index profiles. For this, we consider acousto-optic interaction in the guiding geometry and derive the modal overlap integral, which describes the dependence of the Brillouin gain on the refractive index profile of the optical fiber. We analyze Stokes backreflected power initiated by thermal phonons, compare values of the SBST calculated from different approximations, and discuss the SBST dependence on the fiber length. We also review an analytical approach to calculate the gain of Brillouin fiber amplifiers (BFAs) in the regime of pump depletion. In the high-gain regime, fiber loss is a nonnegligible effect and needs to be accounted for along with the pump depletion. We provide an accurate analytic expression for the BFA gain and show results of experimental validation.
Finally, we review methods to suppress SBS including index-controlled acoustic guiding or segmented fiber links. The second part of the review deals with recent advances in fiber-optic applications where SBS is a relevant effect.
In particular, we discuss the impact of SBS on the radio-over-fiber technology, enhancement of the SBS efficiency in Raman-pumped fibers, slow light due to SBS and SBS-based optical delay lines, Brillouin fiber-optic sensors, and SBS mitigation in high-power fiber lasers, as well as SBS in multimode and microstructured fibers. A detailed derivation of evolutional equations in the guided wave geometry as well as key physical relations are given in appendices. © 2009 Optical Society of America OSA Recommended Articles.
The reciprocal lattices (dots) and corresponding first Brillouin zones of (a) and (b). In and, the first Brillouin zone is a uniquely defined in. In the same way the is divided up into in the real lattice, the is broken up into Brillouin zones. The boundaries of this cell are given by planes related to points on the.
The importance of the Brillouin zone stems from the description of waves in a periodic medium, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone. The first Brillouin zone is the of points in reciprocal space that are closer to the origin of the reciprocal lattice than they are to any other reciprocal lattice points (see the derivation of the ). Another definition is as the set of points in k-space that can be reached from the origin without crossing any. Equivalently, this is the around the origin of the reciprocal lattice. There are also second, third, etc., Brillouin zones, corresponding to a sequence of disjoint regions (all with the same volume) at increasing distances from the origin, but these are used less frequently. As a result, the first Brillouin zone is often called simply the Brillouin zone. (In general, the n-th Brillouin zone consists of the set of points that can be reached from the origin by crossing exactly n − 1 distinct Bragg planes.) A related concept is that of the irreducible Brillouin zone, which is the first Brillouin zone reduced by all of the symmetries in the of the lattice (point group of the crystal).
The concept of a Brillouin zone was developed by (1889–1969), a French physicist. First Brillouin zone of, a, showing symmetry labels for high symmetry lines and points Several points of high symmetry are of special interest – these are called critical points.
Not to be confused with. Information theory studies the, and of. It was originally proposed by in 1948 to find fundamental limits on and communication operations such as, in a landmark paper entitled '. Applications of fundamental topics of information theory include (e.g. And ), and (e.g. Its impact has been crucial to the success of the missions to deep space, the invention of the, the feasibility of, the development of the, the study of and of human perception, the understanding of, and numerous other fields.
A key measure in information theory is '. Entropy quantifies the amount of uncertainty involved in the value of a or the outcome of a. For example, identifying the outcome of a fair (with two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a (with six equally likely outcomes). Some other important measures in information theory are, and. The field is at the intersection of, and. The theory has also found applications in other areas, including, human vision, the evolution and function of molecular codes , in statistics, and. Important sub-fields of information theory include, and measures of information.
Contents. Overview Information theory studies the transmission, processing, extraction, and utilization of information. Abstractly, information can be thought of as the resolution of uncertainty. In the case of communication of information over a noisy channel, this abstract concept was made concrete in 1948 by in his paper ', in which 'information' is thought of as a set of possible messages, where the goal is to send these messages over a noisy channel, and then to have the receiver reconstruct the message with low probability of error, in spite of the channel noise.
Shannon's main result, the showed that, in the limit of many channel uses, the rate of information that is asymptotically achievable is equal to the, a quantity dependent merely on the statistics of the channel over which the messages are sent. Information theory is closely associated with a collection of pure and applied disciplines that have been investigated and reduced to engineering practice under a variety of throughout the world over the past half century or more:, along with of many descriptions. Information theory is a broad and deep mathematical theory, with equally broad and deep applications, amongst which is the vital field of. Coding theory is concerned with finding explicit methods, called codes, for increasing the efficiency and reducing the error rate of data communication over noisy channels to near the. These codes can be roughly subdivided into (source coding) and (channel coding) techniques. In the latter case, it took many years to find the methods Shannon's work proved were possible. A third class of information theory codes are cryptographic algorithms (both and ).
Concepts, methods and results from coding theory and information theory are widely used in and. See the article for a historical application. Information theory is also used in, and even in. Historical background. Main article: The landmark event that established the discipline of information theory and brought it to immediate worldwide attention was the publication of 's classic paper ' in the in July and October 1948. Prior to this paper, limited information-theoretic ideas had been developed at, all implicitly assuming events of equal probability. 's 1924 paper, Certain Factors Affecting Telegraph Speed, contains a theoretical section quantifying 'intelligence' and the 'line speed' at which it can be transmitted by a communication system, giving the relation W = K log m (recalling ), where W is the speed of transmission of intelligence, m is the number of different voltage levels to choose from at each time step, and K is a constant.
's 1928 paper, Transmission of Information, uses the word information as a measurable quantity, reflecting the receiver's ability to distinguish one from any other, thus quantifying information as H = log S n = n log S, where S was the number of possible symbols, and n the number of symbols in a transmission. The unit of information was therefore the, which has since sometimes been called the in his honor as a unit or scale or measure of information. In 1940 used similar ideas as part of the statistical analysis of the breaking of the German second world war ciphers. Much of the mathematics behind information theory with events of different probabilities were developed for the field of by and. Connections between information-theoretic entropy and thermodynamic entropy, including the important contributions by in the 1960s, are explored in. In Shannon's revolutionary and groundbreaking paper, the work for which had been substantially completed at Bell Labs by the end of 1944, Shannon for the first time introduced the qualitative and quantitative model of communication as a statistical process underlying information theory, opening with the assertion that 'The fundamental problem of communication is that of reproducing at one point, either exactly or approximately, a message selected at another point.' With it came the ideas of.
the and of a source, and its relevance through the;. the, and the of a noisy channel, including the promise of perfect loss-free communication given by the;.
the practical result of the for the channel capacity of a; as well as. the —a new way of seeing the most fundamental unit of information. Quantities of information.
Kelly, Jr., 'A New Interpretation of Information Rate' Bell System Technical Journal, Vol. 35, July 1956, pp. 917–26. Landauer, 'Information is Physical' Proc. Workshop on Physics and Computation PhysComp'92 (IEEE Comp. Sci.Press, Los Alamitos, 1993) pp. 1–4. Landauer, 'Irreversibility and Heat Generation in the Computing Process' IBM J. 3, 1961.
Timme, Nicholas; Alford, Wesley; Flecker, Benjamin; Beggs, John M. 'Multivariate information measures: an experimentalist's perspective'.:. Textbooks on information theory. Information Measures, Information and its Description in Science and Engineering (Springer Series: Signals and Communication Technology), 2004,. Ash, RB. Information Theory. New York: Interscience, 1965.
New York: Dover 1990. Information Theory and Reliable Communication.
New York: John Wiley and Sons, 1968. Goldman, S. Information Theory. New York: Prentice Hall, 1953.
New York: Dover 1968, 2005.; Thomas, Joy A. Elements of information theory (2nd ed.)., Korner, J. Information Theory: Coding Theorems for Discrete Memoryless Systems Akademiai Kiado: 2nd edition, 1997. Cambridge: Cambridge University Press, 2003.
Mansuripur, M. Introduction to Information Theory. New York: Prentice Hall, 1987. The Theory of Information and Coding'. Cambridge, 2002. Pierce, JR.
'An introduction to information theory: symbols, signals and noise'. Dover (2nd Edition).
1961 (reprinted by Dover 1980). An Introduction to Information Theory. New York: McGraw-Hill 1961. New York: Dover 1994. Chapter 1 of book, University of Sheffield, England, 2014. Kluwer Academic/Plenum Publishers, 2002.
Springer 2008, 2002. Other books. Leon Brillouin, Science and Information Theory, Mineola, N.Y.: Dover, 1956, 1962 2004., New York: Pantheon, 2011. Khinchin, Mathematical Foundations of Information Theory, New York: Dover, 1957. Rex, Editors, Maxwell's Demon: Entropy, Information, Computing, Princeton University Press, Princeton, New Jersey (1990). What is Information?
- Propagating Organization in the Biosphere, the Symbolosphere, the Technosphere and the Econosphere, Toronto: DEMO Publishing. Tom Siegfried, The Bit and the Pendulum, Wiley, 2000. Charles Seife, Viking, 2006. Jeremy Campbell, Touchstone/Simon & Schuster, 1982,. Henri Theil, Economics and Information Theory, Rand McNally & Company - Chicago, 1967.
Escolano, Suau, Bonev, Springer, 2009. MOOC on information theory Wikiquote has quotations related to: about Information theory. Raymond W. Yeung, ' External links Wikiquote has quotations related to: about Information theory., ed. (2001) 1994, Springer Science+Business Media B.V. / Kluwer Academic Publishers,.
Lambert F. (1999), ', Journal of Chemical Education. and.
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