Theoretical chapter devoted to the detailed description of continuous variable (CV) systems by consideringthe "phase space," that is spanned by position and momentum for massive particles, quadratures for a quantum electromagnetic field, and phase and charge for electrical circuits. It introduces tools like the Glauber, Husimi, or Dirac phase–space functions, and in more details the Wigner function, that are convenient to describe CV quantum states and their time evolution using the Moyal equation. The chapter gives examples of Wigner functions and their time evolution in the presence of dissipation. It then defines symplectic quantum maps that are simple and important cases of Hamiltonian evolution and are simply related to the covariance matrix containingvariances and correlations. It details the characterization of the quantum processes using the Williamson reduction and Bloch–Messiah decomposition. It discusses Gaussian and non-Gaussian states and the specific measurement procedures for CV states, such as homodyne and double homodyne detection. It introduces the EPR entangled state and, finally, describes how to characterize entanglement and unconditionally teleport Gaussian quantum states.

Experimental chapter describing different experiments allowing us to accurately measure physical parameters, such as very small concentrations of atomic species by intensity monitoring, the observation of gravitational waves using giant interferometers, transverse positioning of light beams, transition frequencies of metrological interest using laser frequency combs, and magnetometry of ultra-small magnetic fields using superconducting quantum interference devices (SQUIDs).

Appendix K: this appendix is an introduction to another very active and promising domain of quantum physics, named circuit quantum electrodynamics (cQED), dealing with the quantum properties of macroscopic objects consisting of superconducting electrical circuits. In an LC circuit the energy is quantized, and charge and flux are two quantum canonical conjugate quantities that do not commute. Their quantum fluctuations are bound by a Heisenberg inequality. A Josephson junction inserted in the circuit introduces strong nonlinearities in the system, which breaks the equidistance between the energy levels and makes the circuit look like a qubit, called a transmon. Cooling at mK temperatures is necessary to have quantum effects dominate over thermal effects. The circuit is embedded in a resonant cavity, and the system bears many analogies with cavity QED and Jaynes–Cummings formalism for coupled photons and atoms. One can perform nondestructive read-out and control of the transmon, as well as phase-sensitive, quantum-limited amplification, with nonlinearities that are much stronger than the ones used in quantum optics.

Film-based holography employs the use of high-resolution films such as the use of photopolymers or photorefractive materials for recording. These materials, while having high resolution, have a couple of drawbacks. The film-based techniques are typically slow for real-time applications and difficult to allow direct access to the recorded hologram for manipulation and subsequent processing. With recent advances in high-resolution solid-state 2-D sensors and the availability of ever-increasing power of computers and digital data storage capabilities, holography coupled with electronic/digital devices has become an emerging technology with an increasing number of applications such as in metrology, nondestructive testing, and 3-D imaging. While electronic detection of holograms by a TV camera was first performed by Enloe et al. in 1966, hologram numerical reconstruction was initiated by Goodman and Lawrence. In digital holography, it has meant that holographic information of 3-D objects is captured by a CCD, and reconstruction of holograms is subsequently calculated numerically. Nowadays, digital holography means the following situations as well. Holographic recording is done by an electronic device, and the recorded hologram can be numerically reconstructed or sent to a display device (called a spatial light modulator) for optical reconstruction. Or, hologram construction is completely numerically simulated. The resulting hologram is sent subsequently to a display device for optical reconstruction. This aspect of digital holography is often known as computer-generated holography.

In photography, the intensity of a 3-D object is imaged and recorded in a 2-D recording medium such as a photographic film or a charge-coupled device (CCD) camera, which responds only to light intensity. Since there is no interference during recording, the phase information of the wave field is not preserved. The loss of the phase information of the light field from the object destroys the 3-D characteristics of the recorded scene, and therefore parallax and depth information of the 3-D object cannot be observed by viewing a photograph. Holography is a technique in which the amplitude and phase information of the light field of the object are recorded through interference. The phase is coded in the interference pattern. The recorded interference pattern is a hologram. It is reminiscent of Young’s interference experiment in which the position of the interference fringes depends on the phase difference between the two sources. Once the hologram of a 3-D object has been recorded, we can reconstruct the 3-D image of the object by simply illuminating the hologram or through digital reconstruction. We record the complex amplitude of the 3-D object in coherent holography, whereas in incoherent holography, we record the intensity distribution of the 3-D object. In this chapter, we discuss the principles of coherent holography.

To have some basic understanding of optical coherence, we discuss temporal coherence and spatial coherence quantitatively in the beginning of the Chapter. We then concentrate on spatial coherent image processing, followed by spatially incoherent image processing. While spatial coherent imaging systems lead to the concept of coherent point spread function and coherent transfer function, spatially incoherent imaging system introduces intensity point spread function and optical transfer function. Scanning image processing is also covered in the chapter, illustrating an important aspect in that a mask in front of the photodetector can change the coherence properties of the optical system. Finally, two-pupil synthesis of optical transfer functions is discussed, illustrating bipolar processing in incoherent imaging systems.

In modern optical processing and display applications, there are increased needs of a real-time device and such a device is called a spatial light modulator (SLM). Typical examples of SLMs are acousto-optics modulators (AOMs), electro-optic modulators (EOMs) and liquid crystal displays. In this chapter, we will concentrate on these types of modulators and discuss their uses, such as phase modulation and intensity modulation, in information processing.

We have introduced Gaussian optics and used a matrix formalism to describe light rays through optical systems in Chapter 1. Light rays are based on the particle nature of light. Since light has a dual nature, light is waves as well. In 1924, de Broglie formulated the de Broglie hypothesis, which relates wavelength and momentum. In this chapter, we explore the wave nature of light, which accounts for wave effects such as interference and diffraction.

The purpose of this chapter is twofold. We will first discuss basic aspect of signals and linear systems in the first part. As we will see in subsequent chapters that diffraction as well as optical imaging systems can be modelled as linear systems. In the second part, we introduce the basic properties of Fourier series, Fourier transform as well as the concept of convolution and correlation. Indeed, many modern optical imaging and processing systems can be modelled with the Fourier methods, and Fourier analysis is the main tool to analyze such optical systems. We shall study time signals in one dimension and signals in two dimensions will then be covered. Many of the concepts developed for one-dimensional (1-D) signals and systems apply to two-dimensional (2-D) systems. This chapter also serves to provide important and basic mathematical tools to be used in subsequent chapters.

This chapter contains Gaussian optics and employs a matrix formalism to describe optical image formation through light rays. In optics, a ray is an idealized model of light. However, in a subsequent chapter, we will also see a matrix formalism can also be used to describe, for example, a Gaussian laser beam under diffraction through the wave optics approach. The advantage of the matrix formalism is that any ray can be tracked during its propagation though the optical system by successive matrix multiplications, which can be easily programmed on a computer. This is a powerful technique and is widely used in the design of optical element. In this chapter, some of the important concepts in resolution, depth of focus, and depth of field are also considered based on the ray approach.

Two selected Exercises are solved in full: a question relevant to the twin paradox and a method using special relativity for deriving the Biot–Savart equation.