Spie Press BookSelected Papers on Phase-Space Optics
Markus E. Testorf, Jorge Ojeda-Castaneda, Adolf W. Lohmann
The Foundations of Phase-Space Optics
Phase-Space Tools for Optics
Analysis and Design of Optical Systems and Devices
Partially Coherent Signals and Systems
The Discrete Wigner Transform
Optical Display of Phase Space
Optical Metrology and Signal Recovery
Phase-Space Representation of Optical Signals and Systems
Progress in optical sciences is often promoted by nonconventional and convenient mathematical tools to describe the propagation of light through optical systems.
The terms "phase space optics," "Wigner optics," "local frequency spectrum," and "joint time-frequency transformation" all refer to mathematical tools that continue to influence optical sciences. The purpose of these tools is to develop a description of signals and systems to address the inevitable trade off between two optical variables that form a Fourier transform pair. More specifically, the signal is mapped into a configuration space, called the phase space, where each variable corresponds to one coordinate axis. In recent years this type of signal representation has proved of ever-increasing importance not only to optics, but to many areas of science and engineering.
The term "phase space optics" is often perceived as an exotic branch of optics that came into existence only recently. Indeed, it is only for the past three decades that scientists and engineers have started to explore systematically the use of phase space representations for analyzing and synthesizing optical signals. However, for at least 1000 years it has been common practice to represent music as a musical score - which, in fact, is a representation of sound in terms of an instantaneous frequency spectrum: the horizontal axis represents time, while the vertical axis corresponds to the logarithm of the frequency (Fig. 1).
In a growing number of publications, the authors explicitly take into account this trade-off between two variables that form a Fourier transform pair. This trend is widening in scope, effectively creating a new branch of optics. In recent years, phase space methods have been applied successfully to gain insights and to design novel imaging devices that, for instance, extend depth of field. Additional applications include the reinterpretation of radiometry and the fractional Fourier transform as well as contributions to phase retrieval and signal recovery, and to the analysis of short time optical pulses.
The increase in the number of publications on phase space optics, joint time frequency transforms, the Wigner distribution function, and the ambiguity function makes it necessary for anybody working on related problems to become acquainted with the basics of phase space optics. Several good review articles and book chapters provide a useful introduction to the subject [2B4]. However, it is of equal importance to obtain a balanced cross section of key publications as well as quick access to those publications, since they form the foundations of phase space optics and often contain a wealth of information that is barely reflected in more recent studies. We have tried to address this need by selecting papers for this volume that describe the basic principles of phase space methods and sketch some of the many new interesting applications of this fascinating branch of optics.
In optics, phase space methods can be traced back to early contributions to classical geometrical optics, where it was shown that the position and the angle of propagation of an optical ray are interrelated by an invariant that is known as the Smith Helmholtz Lagrange invariant . This concept finds useful applications when describing the light-gathering power of optical instruments  or representing the two most relevant rays in an optical system in a diagram . In Fourier spectroscopy, the same concept is applied to specify the "etendue" [8, 9].
In the context of wave optics, the position and the angle of diffraction are similarly interrelated by an invariant known as the space-bandwidth product, which - loosely speaking - specifies the information capacity of an optical device. The position angle invariant and the space frequency invariant may be used as figures of merit for the performance of optical systems. The space-bandwidth product also specifies the set of signals that may be transmitted without any loss by that optical device. This concept is depicted in one dimension in Fig. 2.
Digital cameras, which are commonly specified by a certain number of megapixels, present an example of the space-bandwidth concept. By estimating the bandwidth of spatial frequencies, it is apparent that the number of pixels is identical to the space-bandwidth product of the camera system. In other words, the phase-space volume defined by the product of signal extension and plane wave spectrum is quantized into irreducible cells, each of which corresponds to one independent mode (Fig. 2). It is yet another remarkable feature of the phase pace representation that the concept of irreducible cells can be interrelated to Heisenberg's uncertainty principle, which emerges from the Fourier transform relationship between position and momentum in quantum mechanics.
While phase-space concepts and joint transforms of Fourier reciprocal variables are widely used in science and engineering, their application in optics is unique in various aspects. In part this is due to the fact that optics is both a subject of research as well as an enabling technology. On the one hand, optics is concerned with the physics of electromagnetic wave propagation, which phase space descriptions incorporate most elegantly into the signal representation. On the other hand, optical signals are carriers of information that often must be extracted from the signal by means of elaborate processing methods. In this context, the full potential of joint transform signal processing can be invoked, independent of any direct physical interpretation.
It is a unique strength of phase space optics to be able to describe the local frequency content of an optical signal, or of an optical system, in terms of simple mathematical tools with an associated insightful diagram that provides a rather intuitive interpretation of many optical phenomena. Phase space optics allows one to understand different models of describing optical signals and systems with one single platform. For instance, phase space representations can be usefully exploited to link the viewpoints of geometrical and physical optics, since the formalism provides a mechanism for assigning "amplitudes," or weights, to geometrical optics rays. Another application for this unifying mathematical tool is the description of coherent light, as well as of partially coherent light, and indeed noncoherent illumination.
It is not surprising that the universal nature of phase space optics finds its expression in a large number of publications on the subject. Therefore, for this collection we had to settle for a rather narrow scope. With few exceptions, only contributions relevant for describing the spatial properties of optical signals in terms of phase space were considered. In addition, we limited our attention to classical optical fields. Publications that did not fit these criteria were included only if necessary for a basic understanding of phase space methods, or to provide additional context. In this way, we deliberately omitted from our selection valuable contributions on digital signal processing, classical mechanics, statistical physics, and quantum optics. However, the vast amount of literature suggested that each of these topics deserved their own special attention. In this context we merely refer the interested reader to a selection of textbooks [10-19].
Despite this limitation in scope, the present selection contains a number of unique and insightful contributions (many of which are not easy to find), and which we believe provides a valuable cross section of studies on the subject.
In Section One, we selected papers that set the foundations of phase space optics. The papers come from several independent efforts starting with interferometry, geometrical optics, and optical processing (Steel; Delano; Lohmann). Other papers were generated within the context of quantum physics (Wigner; de Brujin). These are complemented by papers from the field of signal analysis (Gabor; Ville; Claasen and Mecklenbrauker; Flandrin and Hlawatsch). This section ends with classical contributions to radar processing (Woodward and Davies; Lee, Cheatham, and Wiesner).
In Section Two, we collected the pioneering efforts to incorporate phase space concepts into the analytical tools for describing optical wave fields and optical systems. The section starts with the seminal contributions on the ambiguity function (Papoulis; Dutta and Goodman; Guigay), followed by three papers related to the introduction of the Wigner distribution function into optics (Bastiaans; Bartelt, Brenner, and Lohmann; Szu and Blodgett).
In Section Three, we assembled papers that describe the use of phase space representations to analyze and design imaging devices. This includes the application to first-order optics (Bastiaans), to the nonconventional design of a fake zoom lens (Lohmann), and to the design of an achromatic optical processor (Wang, Pe'er, Lohmann, and Friesem). In addition, selected contributions discuss the influence of wave aberrations (Lohmann, Ojeda Castaneda, and Streibl), a polar display of the OTF (Brenner, Lohmann, and Ojeda Castaneda), the analysis of the Strehl ratio (Ojeda Castaneda, Andres, and Montes), and work on synthesizing the axial behavior of optical beams (Ojeda Castaneda, Berriel, and Montes; Dowski and Cathey; Castro and Ojeda Castaneda; Zalvidea and Sicre; Furlan, Zalvidea, and Saavedra).
In Section Four, we collected key papers on the use of phase space representations to understand radiometry and partially coherent light (Walther; Wolf; Bastiaans; Brenner and Ojeda Castaneda; Bastiaans; Agarwal, Foley, and Wolf), as well as an effort to relate phase space representations to bilinear optical systems (Ojeda Castaneda and Sicre).
In Section Five, we included four useful papers on the discrete Wigner distribution (Claasen and Mecklenbrauker; Brenner; Claasen and Mecklenbrauker; O'Neill, Flandrin, and Williams). While deviating slightly from our main scope, we are convinced that these papers will be valued by readers interested in doing numerical simulations of phase space representations. In addition, these publications highlight some key developments, in our opinion, as well as difficulties regarding the discrete representation of the Wigner distribution functions.
In Section Six, we included the most representative works on the optical implementation of phase space representations. The section starts with the pioneering effort to display optically the ambiguity function (Marks and Hall), followed by contributions on the optical generation of the 1D Wigner distribution function (Bastiaans; Brenner and Lohmann), as well as of its 2D version (Bamler and Glunder; Conner and Li; Subotic and Saleh). In addition, we selected a paper (Easton, Ticknor, and Barrett) that describes a method for generating the Wigner distribution optoelectronically, which is of particular interest because it is a rather early account of the idea to synthesize the Wigner function with tomographic methods.
In Section Seven, we collected papers on optical metrology and signal recovery. The section starts with the pioneering work on image classification and nonconventional spatial filtering (Bescos, Cristobal, and Santamaria; Gonzalo, Bescos, Berriel Valdos, and Artal). The location of point scatters in 3D (Onural and Ozgen) is an example of applying the phase space methods to more generic problems of inverse scattering. A more recent effort has been the application of phase space methods to the phase retrieval problem. To provide the reader with a representative cross section of work related to this topic, we included three contributions that outline the development of phase space tomography (Smithey, Beck, Raymer, and Faridani; McAlister, Beck, Clarke, Mayer, and Raymer; D. Dragoman, M. Dragoman, and Brenner). These papers are good examples of how the phase space representation of optical signals can trigger new and unconventional approaches to solve generic problems. Our selection on phase space tomography is complemented with two contributions, one describing a method for characterizing optical short pulses (Trebino and Kane), and the other describing deterministic phase retrieval in phase space (Alieva, Bastiaans, and Stankovic).
In Section Eight, we selected papers that translate the description of optical signals and systems into the framework of phase space representations. Most contributions in this section make use of phase space diagrams to provide an intuitive approach to phase space optics. Our collection contains work on the longitudinal periodicities of free and bounded wave fields (Ojeda Castaneda and Sicre), the analysis of the fractional Talbot effect (Testorf and Ojeda Castaneda), the space-bandwidth product (Lohmann, Dorsch, Mendlovic, Zalevsky, and Ferreira; Mendlovic and Lohmann; Mendlovic, Lohmann, and Zalevsky), as well as a description of the fractional Fourier transformation in Wigner space (Lohmann) and its relationship with the Radon Wigner transformation (Lohmann and Soffer). Additional contributions relate holography to the Wigner distribution function (Wolf and Rivera; Lohmann, Testorf, and Ojeda Castaneda), and reinterpret the formation of moire patterns (Testorf) as well as mode coupling of light in optical fibers (Onciul). The section concludes with the important but rarely discussed problem of representing rotationally symmetric optical systems in terms of the Wigner function (Bastiaans and van de Mortel).
Finally, in Section Nine we included four papers on time-domain application and the manipulation of short-time pulses (D. Dragoman and M. Dragoman; Paye and Migus; Sutoh, Yasuno, Harada, Itoh, Yatagai and Mori; Olson, Healy, and Osterberg). Within our selection, which focuses on the spatial evolution of optical signals, these applications highlight the universal nature of phase space analysis as well as some features that are unique to time domain processing.
Without question, this selection of publications on phase space optics is biased by our own experience and interest. The number of important contributions that we did not include outnumbers by far the number we were able to select. Thus, we apologize if some of our readers would have selected papers differently or if there are colleagues who feel underrepresented by our selection. However, our goal was not an absolute importance ranking of papers on the subject. Rather, our intention was to convey the background we have acquired through our work on phase space optics in the form of a set of publications that represents this body of knowledge in a unique way. With this goes our hope that this selection will lead readers to explore the exciting field of phase space representations in optics by themselves.
Markus E. Testorf
Hanover, NH, U.S.A.
Jorge Ojeda Castaneda
Puebla, Pue., Mexico
Adolf W. Lohmann