Polarization entangled photon pairs

Throughout his life, Albert Einstein was never reconciled to quantum theory being an essentially indeterministic description of natural processes, even though he himself contributed fundamental ideas to its development. “God does not play dice” was his inner conviction. In his opinion, quantum theory was only makeshift. His doubts about the completeness of the quantum mechanical description were expressed concisely in a paper published jointly with Podolsky and Rosen (Einstein, Podolsky and Rosen, 1935). This paper analyzes a sophisticated Gedanken experiment, now famous as the Einstein–Podolsky–Rosen paradox, which has excited theoreticians ever since.

The Gedanken experiment was recently realized in a laboratory. The analyzed objects are photon pairs – and this is what has motivated us to dedicate a chapter to this problem which has bearing upon the foundations of quantum mechanics. The photon pairs are formed by two photons generated in sequence (in a so-called cascade transition, as shown in Fig. 11.1). Due to the validity of the angular momentum conservation law (discussed in Section 6.9) for the elementary emission process, the two photons exhibit specifically quantum mechanical correlations, which are incompatible with the classical reality concept, as will be discussed in detail below.

How do the correlations appear in detail? Let us assume the initial state of the atom to be a state with angular momentum (spin) J = 0, the intermediate state to have angular momentum J = 1, and the final state to have again J = 0.

How can we construct a picture of the photon from the wealth of observation material available to us? The photon appears to have a split personality: it is neither a wave nor a particle but something else which, depending on the experimental situation, exhibits a wave- or a particle-like behavior. In other words, in the photon (as in material particles such as the electron) the particle–wave dualism becomes manifest. Whereas classically the wave and the particle pictures are separate, quantum mechanics accomplishes a formal synthesis through a unified mathematical treatment.

Let us look first at the wave aspect familiar from classical electrodynamics, which seems to be the most natural description. It makes all the different interference phenomena understandable, such as the “interference of the photon with itself” on the one hand and the appearance of spatial and temporal intensity correlations in a thermal radiation field on the other (which are obviously brought about by superposition of elementary waves emitted independently from different atoms). It might come as a surprise (at least for those having quantum mechanical preconceptions) that the classical theory is valid down to arbitrarily small intensities: the visibility of the interference pattern does not deteriorate even for very small intensities – the zero point fluctuations of the electromagnetic field advocated by quantum mechanics do not have a disturbing effect – and is valid not only for conventional interference experiments but also for interference between independently generated light beams (in the form of laser light).

Fundamentals of cryptography

The essence of the Einstein–Podolsky–Rosen experiment analyzed in the preceding chapter is our ability to provide two observers with unpolarized light beams, consisting of sequences of photons, which are coupled in a miraculous way. When both observers choose the same measurement apparatus – a polarizing prism with two detectors in the two output ports, whereby the orientation of the prism is set arbitrarily but identically for both observers – their measurement results are identical. The measurement result, characterized, say, by “0” and “1”, is a genuine random sequence – the quantum mechanical randomness rules unrestricted – from which we can form a sequence of random numbers using the binary number system. The experimental setup thus allows us to deliver simultaneously to the two observers an identical series of random numbers. This would be, by itself, not very exciting. Mathematical algorithms can be used to generate random numbers, for example the digit sequence of the number π, which can be calculated up to an arbitrary length. Even though we cannot be completely sure that such a sequence is absolutely random, such procedures are sufficient for all practical purposes. The essential point of the Einstein–Podolsky–Rosen experiment is that “eavesdroppers” cannot listen to the communication without being noticed by the observers. When eavesdroppers perform an observation on the photons sent, they inevitably destroy the subtle quantum mechanical correlations, and this damage is irreparable.

The electromagnetic field and its energy

The conclusion by Maxwell, based on theoretical considerations, that light is, by its character, an electromagnetic process, is surely a milestone in the history of optics. By formulating the equations bearing his name, Maxwell laid the foundations for the apparently precise description of all optical phenomena. The classical picture of light is characterized by the concept of the electromagnetic field. At each point of space, characterized by a vector r, and for each time instant t, we have to imagine vectors describing both the electric and the magnetic field. The time evolution of the field distribution is described by coupled linear partial differential equations: the Maxwell equations.

The electric field strength has a direct physical meaning: if an electrically charged body is placed into the field, it will experience a force given by the product of its charge Q and the electric field strength E. (To eliminate a possible distortion of the measured value by the field generated by the probe body itself, its charge should be chosen to be sufficiently small.) Analogously, the magnetic field strength H, more precisely the magnetic induction B = μH (where μ is the permeability), describes the mechanical force acting on a magnetic pole (which is thought of as isolated). Also, the field has an energy content, or, more correctly (because in a precise field theory we can think only about energy being distributed continuously in space), a spatial energy density.

Light absorption

Whereas receiving radio waves is a macroscopic process and hence belongs to the area of classical electrodynamics – in a macroscopic antenna an electric voltage is induced whereby a large number of electrons follow the electric field strength of the incident wave, in a kind of collective motion – the detection of light, so far as the elementary process is concerned, takes place in microscopic type objects such as atoms and molecules. As a consequence, the response of an optical detector is determined by the microstructure of matter. In particular, it is impossible – due to the enormously high frequency of light (in the region of 1015 Hz) – to measure the electric field strength. What is in fact detectable is the energy transfer from the radiation field to the atomic receiver, and this allows us to draw conclusions about the (instantaneous) intensity of light.

We might ask what we can say about the above-mentioned absorption process from an experimentalist's point of view. Among the basic experiences that provide an insight into the structure of the micro-cosmos is the resonance character of the interaction between light and an atomic system. The atomic system, when hit by light, behaves like a resonator with certain resonance frequencies; i.e. it becomes excited (takes up energy) only when the light frequency coincides with a value that is characteristic for the particular atom. Hence, an incident light wave with an initial broadband frequency spectrum that has passed through a gas exhibits in its spectrum dark zones, the so-called absorption lines.

In the preface I think it is better if I abandon the formality of the text and address you the reader, directly.

As I hope you will have gathered from the title, this is a book that attempts to lay out the basis for the design of analog optical links. Let me give an example that should drive home this point. It is customary in books on lasers to start with an extensive presentation based on the rate equations (do not worry at this point if you do not know what these are). In this book we also discuss lasers, but the rate equations are relegated to an appendix. Why? Because in over 15 years of link design, I have never used the rate equations to design a link! So why all the emphasis on the rate equations in other texts? Probably because they are targeted more to, or at least written by, device designers. The view in this book is that you are a user of devices, who is interested in applying them to design of a link. Of course to use a device most effectively, or even to know which device to choose for a particular link design, requires some knowledge of the device, beyond its terminal behavior. To continue the laser example, it is important to know not only what the laser frequency response is, but also how it changes with bias.

Introduction

In this chapter we develop the small-signal relationships between the RF and optical parameters for the most common electro-optic devices used in intensity modulation, direct detection links. There are numerous device parameters we could use for this task; we concentrate here – as we will throughout this book – on those parameters that can be measured and selected by the link designer – as opposed to those parameters that can only be measured and controlled by the device designer.

To provide the basis for comparing these and future devices, we develop a figure of merit for optical modulators and detectors: the RF-to-optical incremental modulation efficiency for modulation devices and its converse the optical-to-RF incremental detection efficiency for photodetection devices. These efficiencies are useful in link design because they provide a single parameter for evaluating device performance in a link that represents the combined effects of a device's optical and electrical parameters. Further, by using the same parameter for both direct and external modulation devices, we begin the process – which will carry on through much of the book – of using a single set of tools for evaluating both types of links.

The most common electro-optic devices in use for links today are the in-plane diode laser, both Fabry–Perot and DFB, for direct modulation, the Mach–Zehnder modulator for external modulation and a photodiode for photodetection. Thus on a first reading, one may want to focus on these devices.

Introduction

The device slope efficiencies that we developed in Chapter 2, and that were cascaded to form links in Chapter 3, explicitly ignored any frequency dependence. In this chapter we remove that restriction. As we shall see, virtually all modulation and photodetection devices have an inherently broad bandwidth. Digital links require broad bandwidth, which is one of the reasons for the numerous applications of fiber optic links to digital systems. A few analog link applications also require the full device bandwidth. However, it is far more common for analog links to need only a portion of the devices' inherent bandwidth. Consequently most analog link designs include some form of RF pre- or post-filtering to reduce the bandwidth.

For completeness we address bandpass and broad bandwidth impedance matching for three electro-optic devices: PIN photodiode, diode laser and Mach–Zehnder modulator. We then combine the bandpass impedance matched cases to form both direct and external modulation links. However, the same analytical approach is used for both impedance matching methods and both modulation techniques. Therefore those readers desiring a less exhaustive treatment can obtain a complete introduction to the subject by studying only one of the impedance matching methods and one of the modulation techniques.

One may be tempted to ask: why bother with bandwidth reduction, since this adds components and complicates the design? There are at least two key reasons for implementing bandwidth reduction.

Background

Optical communication links have probably been around for more than a millennium and have been under serious technical investigation for over a century, ever since Alexander Graham Bell experimented with them in the late 1800s. However, within the last decade or so optical links have moved into the communications mainstream with the availability of low loss optical fibers. There are of course many reasons for this, but from a link design point of view, the reason for fiber's popularity is that it provides a highly efficient and flexible means for coupling the optical source to a usually distant optical detector. For example, the optical loss of a typical terrestrial 10-km free-space optical link would be at least 41 dB (Gowar, 1983), whereas the loss of 10 km of optical fiber is about 3 dB at wavelengths of ~1.55 μm. To put the incredible clarity of optical fibers in perspective, if we take 0.3 dB/km as a representative loss for present optical fibers, we see that they are more transparent than clear air, which at this wavelength has an attenuation of 0.4 to 1 dB/km (Taylor and Yates, 1957).

Today the vast majority of fiber optic links are digital, for telecommunications and data networks. However, there is a growing, some might say exploding, number of applications for analog fiber optic links. In this case, the comparison is not between an optical fiber and free space but between an optical fiber and an electrical cable.

Introduction

In Chapter 5 we explored one type of extraneous signals in links – noise – that because of its random nature is characterized byits statistical properties. In this chapter we investigate the other type of extraneous signals in links– distortion. Unlike noise however, distortion signals are deterministic. A further distinctionbetween noise and distortion is the fact that while noise is always present, independent ofwhether there are any signals present, distortion is only present when at least one signal is present. We continue in this chapter a theme of this book by using one model to describe the distortionof both direct and external modulation, although the detailed nature of the distortion will dependon the particular modulation method that is used.

The discussion that begins this chapter is general in that the results apply to all devices with some non-linearity. The general results include the frequencies at which distortionproducts occur, the measures of distortion and the conversions among them. We then apply thesetools to the characterization of the distortion produced by the modulation and photodetection devicesthat we have been studying throughout this book. For some applications the distortion levels areunacceptably high. This has led to the development of a variety of linearization techniques. The chapterconcludes with an examination of two linearization techniques.

An optical link as defined in this book consists of linear passive electrical andoptical components as well as modulation and photodetection devices.

Introduction

In Chapters 2 through 4 we have shown how a single formalism can be used to describe the gain and frequency response of both direct and external modulation links. We continue with that same approach in this chapter. However, we will see that because different noise sources dominate in each type of link, the specific form of the link noise model depends on the type of link.

Up to this point all the signal sources we have dealt with were deterministic, in the sense that we could express their output voltage at any instant of time in terms of a known function of time, say v(t). In the case of the noise sources discussed here, there are – at present – no known expressions for any of the noise sources that give the noise source output as a deterministic function of time. Consequently we are forced to use the next best description, which is to describe the noise source output in terms of its statistical properties.

There are many statistical descriptors that could be used; by far the most common one for describing noise sources in electrical and optical applications is the mean-square value. There are primarily two bases for the popularity of the mean-square value. One is that it can be derived from the statistical distribution for the noise source, without ever knowing the underlying deterministic function. The other reason is that the mean-square value corresponds to the heating effect generated by the noise source.

Introduction

Up to this point we have discussed each of the primary measures of link performance – gain, bandwidth, noise figure and dynamic range – in as complete isolation from the other parameters as possible. While such an approach permitted us to focus on the various aspects of each parameter, it did miss the effects of interactions among the parameters to a large extent. Clearly when designing a link, one needs to take into account such interactions; in fact one might argue that maturity in link design is gauged by the link designer's ability to balance often conflicting requirements to meet a given combination of link parameters.

As one might expect, there are myriad potential interactions among link parameters. Therefore in this chapter we can only offer a sampling of these interactions. We begin by exploring interactions among the primary parameters of the intrinsic link. In general the best link designs usually result from attaining the required performance via optimization of the intrinsic link.

However, there are situations where despite a link designer's best efforts, the intrinsic link performance falls below the requirements. In some of these cases electronic pre- and/or post-amplification may be used to improve performance. Consequently we expand our interaction space to include a sampling of tradeoffs between amplifier and link parameters.

Tradeoffs among intrinsic link parameters

Direct modulation

Diode laser bias current

In Fig. 2.2 we saw that the slope efficiency of a directly modulated link is highest just above threshold and decreases as the bias current is increased above threshold – slowly at first and then more rapidly as the bias current is increased further.

Introduction

The devices discussed in Chapter 2 are rarely used individually. More commonly a modulation device – either a diode laser or an external modulator – is combined with a photodetection device to form a link. In this chapter we begin to examine the performance of complete links by developing expressions for the gain of a link in terms of the modulation and photodetection device parameters. In subsequent chapters we develop analogous expressions for link frequency response, noise figure and dynamic range.

Recall from Chapter 1 that we defined a link as comprising all the components necessary to convey an electrical signal over an optical carrier. Since the definition of available power requires an impedance match, we expand the link definition slightly to include those passive electrical components needed to impedance match the modulation and photodetection devices to the electrical signal source and load, respectively. The impedance matching function is also required by the definitions of some of the link parameters we will be discussing. A more detailed version of the link block diagram is shown in Fig. 3.1.

Although the models we develop have applicability at any frequency, we choose to focus on relatively low frequencies here where lumped-element RLC passive elements are appropriate. This permits us to get the important concepts across without their being obscured by the myriad detailed effects that microwave models require.