In linear optics, the concept of a mode or eigenmode is well established. Often these modes correspond to a set of fields that are mutually orthogonal with intensity profiles that are invariant as they propagate through a given optical system. More generally, using an eigenmode decomposition, one can define a set of orthogonal modes with respect to an optical measure given that is linear in the intensity of the fields or Hermitian in the fields themselves. However, if the intensity of the light is sufficiently large, the dipole response of an optical medium includes nonlinear terms that cause the eigenmode decomposition to break down. In this work, we introduce the eigenmode decomposition in the presence of these nonlinear source terms by introducing small perturbation fields whose interaction is mediated by some high-intensity background field. Unlike the eigenmodes of linear optics, these novel modes correspond to a set of orthogonal fields that are, in general, distributed across multiple wavelengths. Here, we study the definition and interaction of these eigenmodes for classical electromagnetic fields and multiphoton fields. In the context of classical fields, with our eigenmodes established, we highlight the influence of the high-intensity background field on the symmetry of the eigenmodes. At the multiphoton level, we show that the description of multiphoton fields is simplified by using the propagation eigenmodes while remaining equivalent to the standard approach.
In linear optics, the concept of a mode or eigenmode is well established. Often these modes correspond to a set of fields that are mutually orthogonal with intensity profiles that are invariant as they propagate through a given optical system. More generally, using an eigenmode decomposition, one can define a set of orthogonal modes with respect to an optical measure given that is linear in the intensity of the fields or Hermitian in the fields themselves. However, if the intensity of the light is sufficiently large, the dipole response of an optical medium includes nonlinear terms that cause the eigenmode decomposition to break down. In this work, we introduce the eigenmode decomposition in the presence of these nonlinear source terms by introducing small perturbation fields whose interaction is mediated by some high-intensity background field. Unlike the eigenmodes of linear optics, these novel modes correspond to a set of orthogonal fields that are, in general, distributed across multiple wavelengths. Here, we study the definition and interaction of these eigenmodes for classical electromagnetic fields and multiphoton fields. In the context of classical fields, with our eigenmodes established, we highlight the influence of the high-intensity background field on the symmetry of the eigenmodes. At the multiphoton level, we show that the description of multiphoton fields is simplified by using the propagation eigenmodes while remaining equivalent to the standard approach.
When light scatters from an object, it can impart some physical quantity such as momentum or angular momentum. This can act as a measurement on the photon, which collapses on to an eigenstate of the measurement operator. However the corresponding operator is not the same as that describing the total linear or angular momentum in free space. Optical eigenmodes provide a powerful method to describe this interaction by expanding the field as a linear combination of some basis modes and examining the eigenvalues and eigenvectors of the quadratic measure in question. We extend this to the quantum case by writing the quantum operator corresponding to a given measurement such as energy, momentum or angular momentum as a superposition of creation and annihilation operators for each eigenmode. Upon measurement we find that the possible states of a single photon are simply the classical eigenmodes of the measurement. As an application, we examine the force and torque on a general, possibly anisotropic, material. By looking at eigenvalues of the measurement operator we show that the amount of a given quantity transferred in an interaction with matter is not in general the expected amount which a photon carries in free space, even at the single photon level. In particular the difference in linear or angular momentum from before and after is in general not equal to ~k or ~ which are the eigenvalues of these quantities in free space.
The optical eigenmode representation of light fields defines a natural orthogonal basis of solutions of Maxwell's equations taking into account the geometry and interactions involved in a problem. Formally, the optical eigenmodes are similar to the wave functions in quantum mechanics. Here, I put forward that, in a linear interaction at the single photon level, the electromagnetic field collapses into an optical eigenmode of the interaction in question. This is to satisfy the principle that no linear optical system can distinguish between pure states and their superposition for single photons. To exemplify this statement, we will consider the case of angular momentum and linear momentum transfer in optical scattering.
Nonlinear optics is the study of the response of dielectric media to strong optical fields. The fields are sufficiently strong that the response of the medium is, as its name implies, nonlinear. That is, the polarization, which is the dipole moment per unit volume in the medium, is not a linear function of the applied electric field. In the equation for the polarization, there is a linear term, but, in addition, there are terms containing higher powers of the electric field. This leads to significant new types of behavior, one of the most notable being that frequencies different from that of the incident electromagnetic wave, such as harmonics or subharmonics, can be generated. Linear media do not change the frequency of light incident upon them. The first observation of a nonlinear optical effect was, in fact, second-harmonic generation – a laser beam entering a nonlinear medium produced a second beam at twice the frequency of the original. Another type of behavior that becomes possible in nonlinear media is that the index of refraction, rather than being a constant, is a function of the intensity of the light. For a light beam with a nonuniform intensity profile, this can lead to self-focusing of the beam. Most nonlinear optical effects can be described using classical electromagnetic fields, and, in fact, the initial theory of nonlinear optics was formulated assuming the fields were classical. When the fields are quantized, however, a number of new effects emerge. Quantized fields are necessary if we want to describe fields that originate from spontaneous emission. For example, in a process known as spontaneous parametric down-conversion, a beam of light at one frequency, the pump, produces a beam at half the original frequency, the signal. This second beam is a result of spontaneous emission.
These lectures form a general introduction to the domain of Quantum Optics, with a special emphasis on the notion of field eigenmodes that are necessary to describe the quantum effects in light and define the photons: possibility of changing the basis of field eigenmodes and choosing the most appropriate one, definition of intrinsic properties, independent of the choice of basis. Are successively studied the single mode quantum effects, the quantum correlations existing in bimodal systems, and some interesting features of highly multimode systems such as the optical images.
In a realistic treatment of a three-dimensional nonlinear optical experiment, the complete Maxwell equations in (3 + 1) space-time dimensions should be employed. It is then necessary to utilize a multi-mode Hamiltonian that correctly describes the propagating modes. There is an important difference between these experiments and traditional particle scattering. Quantum field dynamics in nonlinear media is dominated by multiple scattering, which is the reason why perturbation theory is less useful. Nevertheless, it is interesting to make a link to conventional perturbation theory. Accordingly, we start by considering a perturbative theory of propagation in a one-dimensional nonlinear optical system in a χ (3) medium. While this calculation cannot treat long interaction times, it does give a qualitative understanding of the important features. This problem is the ‘hydrogen atom’ of quantum field theory: it has fully interacting fields with exact solutions for their energy levels. This is because a photon in a waveguide is an elementary boson in one dimension. The interactions between these bosons are mediated by the Kerr effect, which in quantum field theory is a quartic potential, equivalent to a delta-function interaction. The quantum field theory involved is the simplest model of a quantum field that has an exact solution. This elementary model is still nontrivial in terms of its dynamics, as the calculation of quantum dynamics using standard eigenfunction techniques would require exponentially complex sums over multi-dimensional overlap integrals. This is not practicable, and accordingly we use other methods including quantum phase-space representations to solve this problem.
The interaction of light and matter can be described by a set of coupled nonlinear evolution equations for the material and electromagnetic fields. In order to account for the effects of vacuum fluctuations, noise forces which cause spontaneous transitions have to be included. In these macroscopic systems various sources of noise besides fundamental quantum fluctuations exist which are of macroscopic origin. These cause fluctuations of the cavity dissipation as well as of the external pumping power. In the course of an adiabatic elimination procedure, these fluctuations appear as multiplicative noise in the equations of motion (e.g. for the field intensity, due to the nonlinearity of the interaction). While fluctuations of the cavity damping create noise linear in the field intensity, randomness in the pump strength causes fluctuations that depend nonlinearly on the light intensity. In this respect, the noise terms enter the equations of motion quite differently and have to be distinguished from spontaneous emission noise which is independent of the field strength, and is the basis of all traditional laser models.