Abstract Any light that isn’t perfectly monochromatic (and that’s every kind of light) necessarily has some degree of randomness. In Chapter 9 we described this randomness, how it originates and how (within limits) it may be controlled. However, there were obvious inadequacies: having no quantity to represent a ‘degree of coherence’, we could not give a precise condition for coherence to disappear or for fringes to be so indistinct that they should be considered absent. Moreover, we found ourselves using awkward wordings: ‘obtaining knowledge’; ‘permits us to predict’; ‘so clumsy that interference fails to manifest itself’; ‘a wavetrain is the whole wave defined for all t’. The author of this book used words with more than usual care in Chapter 9, in an attempt at giving a correct impression and excluding common misconceptions; but the need for that care was itself an indication that something better was called for.
Abstract Any light that isn’t perfectly monochromatic (and that’s every kind of light) necessarily has some degree of randomness. In Chapter 9 we described this randomness, how it originates and how (within limits) it may be controlled. However, there were obvious inadequacies: having no quantity to represent a ‘degree of coherence’, we could not give a precise condition for coherence to disappear or for fringes to be so indistinct that they should be considered absent. Moreover, we found ourselves using awkward wordings: ‘obtaining knowledge’; ‘permits us to predict’; ‘so clumsy that interference fails to manifest itself’; ‘a wavetrain is the whole wave defined for all t’. The author of this book used words with more than usual care in Chapter 9, in an attempt at giving a correct impression and excluding common misconceptions; but the need for that care was itself an indication that something better was called for.
Introduction In this chapter we will put the concept of coherence on a more mathematical basis. This will involve the formal definition of a number of functions that are used to describe the coherence properties of optical fields. These include the analytic signal, various correlation functions, and the degree of coherence for describing both temporal and spatial coherence. We shall see that the degree of temporal coherence is quantitatively related to the lineshape function and that the degree of spatial coherence between two points is determined by the size, intensity distribution, and location of an illuminating source. We will use the wave equation, and special solutions to the wave equation called Green's functions, to show how spatial coherence varies from point to point. The chapter will conclude with a discussion of how the intensity fluctuations of a source depend on its coherence properties, and we will examine a specific scheme, namely the Hanbury Brown–Twiss experiment, by means of which this relationship is studied. This discussion will involve a discussion of “photon statistics,” namely the time variation of the “detection” of photons from a source. In a quantum-mechanical context, square-law detectors respond to these quantized excitations of the optical field, which we call photons. In classical coherence theory it is advantageous to represent the real electromagnetic field by a complex quantity, both for its mathematical simplicity and because it serves to emphasize that coherence theory deals with phenomena that are sensitive to the “envelope” or average intensity of the field.
Abstract The modern theory of coherence is based on correlation functions. A generic example could be written 〈 V ∗ ( t 1 ) V ( t 2 ) 〉 , denoting an average of products of the values of a signal V ( t ) at two specified times. Here we infer that t is a degree of freedom that the signal depends on. Typically, physical variables depend on more than one degree of freedom, and recognition of this has prompted attention to some interesting questions for the correlation functions and the several coherences that can be attributed to the same optical field. We examine some of the questions arising from the standpoint of experimental contexts. Degree of polarizability and degree of entanglement (classical non-separability) can serve as starting points for quantitative assignments.
However we have shown in Eqs.(18.29) and (18.30) that the square of the absolute value of a complex scalar wave function Ψ is a measure of the wave intensity, which is the quantity of primary interest for the theories of interference and diffraction for either mechanical or electromagnetic waves. The periodic spatial variation of intensity throughout a region of superposition of waves which originate in the same source is known as interference. Its theory is based essentially on the linearity of the wave equation, which allows a linear combination of partial waves to constitute the actual wave at any given point. Consider first the superposition of two monochromatic plane waves, having the complex representation (16.25), that is (18.1) Since we are usually interested in the relative variation of the intensity only, in a medium 0 ,Z and not in its absolute value, it is common practice to adopt simplified wave intensities, compared to those in Eqs.(18.29) or (18.30), which read ( ) ( )2 21 1 2 2, , ,I r t I r= Ψ = Ψ tG G (18.2) The superposition of monochromatic waves (18.1) results in an intensity ( ) ( ) ( )2 21 1 2, , ,tG ( )I r t r t r= Ψ = Ψ +ΨG G ( ) ( ) ( )2 2 *1 2 1 2, , 2Re ,r t r t r t r t,⎡ ⎤= Ψ + Ψ + Ψ Ψ⎣ ⎦G G G G ( )1 2 1 2 1 2 1 22 cosI I I I k r k r ϕ ϕ= + + ⋅ − ⋅ + −G GG G (18.3) where the last term is called the interference term. If the phase difference, given by ( )1 2 1k k r 2δ ϕ ϕ= − ⋅ + −G G G (18.4) is a constant, the two waves are said to be mutually coherent, and the interference term indicates a sinusoidal spatial variation between maxima and minima, of intensity max 1 2 1 22 ,I I I I I m2δ π= + + = (18.5) ( )min 1 2 1 22 , 2I I I I I m 1δ π= + − = + B A SS 21 Figure 18.1. Interference pattern produced by two monochromatic waves originating from sources 1 2andS S Hence, the stationary interference pattern consists of a succession of surfaces, defined by the condition Nδ π= where .12or2 += mmN It is usually produced by superposition of waves originating in two point sources , which can be assumed, for simplicity, to be of equal phase constants 21 and SS 1 2 0.ϕ ϕ− = If the two sources are far enough from a given point, at distances 1 2 1 2,r r r r>> − ,G G the incident waves at the point can be regarded as plane waves, as we shall show in Chapter 19. Under this assumption Eq.(18.4) reduces to ( ) ( )1 2 1 2 1 2or 2k k r k r r N r r N Nkπ λδ π= − ⋅ ≅ − = − = =G G G (18.6) Both the antinodal and nodal surfaces ( mN 2= ) ( )12 += mN are defined by Eq.(18.6) to be hyperboloids of revolution. Their intersection with a plane will produce alternate nodal and antinodal lines called fringes of the form of either hyperbolas or circular rings if the plane is parallel (A) or normal (B) to the line joining the sources (Figure 18.1). The interference fringes are not observed if the phase difference 1 2ϕ ϕ− has a random variation with time, since the interference term in Eq.(18.3) is averaged out. In this event the waves are said to be mutually uncoherent and do not interfere by superposition. Mutually coherent waves that originate in the same source can be obtained by splitting a single beam, so that on recombining partial waves interference effects occur.
The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The nth order function expresses the correlation of values of the fields at 2n different points of space and time. Certain values of these functions are measurable by means of n-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.
Introduction The protocol we described in Chapter 3 is perfectly correct and perfectly private, but it cannot tolerate that any of the parties deviate from the protocol. In later chapters we will present protocols that are correct and private even if some parties do not follow the protocol. Such protocols are called robust. Before we can prove that these protocols are robust, we need a good definition of what we mean by that. In this section we will describe a security model for cryptographic protocols known as the UC model. Here UC stands for universally composable . The name was adopted because a protocol proven secure in this model remains secure regardless of the context in which it is used. In other words, it can be “universally” composed with any set of other protocols. Our formulation of the UC model differs in several respects from the way it was originally formalized in the literature. We will hint at these differences as we go; some additional comments can be found in the Notes section of this chapter. Before we look at the formal details, let us start by discussing the basic ideas of how to define privacy and robustness of a protocol and last, but not least, why and how these two concepts should be defined via one common definition because they are closely entangled. Defining Privacy It is convenient to first look back on how we defined privacy in Chapter 3. Informally, we defined a protocol to be private (against corruptions of size t ) as follows: first, pick an input ( x 1 ,…, x n )for the protocol, and make a run of the protocol on this input. Then pick some C ⊂{P 1 ,…,P n } with | C | ≤ t and consider the values {view j }P j∈C , where view j is the view of party P j in the execution. The values {view j }P j ∈ C constitute exactly the information leaked to the corrupted parties, C , during an execution. In the following we therefore call the values {view j }P j ∈ C the leaked values. We then want to say that the leaked values do not allow the corrupted parties to learn anything that they should not learn. It is clear that the corrupted parties necessarily must learn their own inputs and outputs; in fact, this is the entire purpose of the protocol. We therefore call the values { x j , y j }P j ∈ C the allowed values.
The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The nth order function expresses the correlation of values of the fields at 2n different points of space and time. Certain values of these functions are measurable by means of n-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.