In this chapter, the main foundations for the conception, design, and the project of optical sensors that explore the effects of nonlinear and quantum optics are presented. These sensors have a variety of applications from the design of waveguides with self-selection of propagation modes to signal processing and quantum computing. The chapter seeks to present formal aspects of applied modern optics in a detailed, sequential, and concise manner.
- optical sensor
- nonlinear optics
- quantum optics
- optical fiber
- optical signal processing
Classical electrodynamics is the basis for the analysis and formulation of electromagnetic waves. From the equations of Maxwell, it is possible to obtain the equation of the movement of the electric and magnetic fields whose solution describes the propagation of the electromagnetic wave. This formal treatment was originally developed by Maxwell , who verified that the electromagnetic wave propagated with the speed of light provided that the optics could be described from the electromagnetism. The medium through which the electromagnetic wave propagates responds in various ways to the electromagnetic field. This response depends on how the atoms and molecules are arranged spatially composing the constituent medium and how the interaction or scattering of the electromagnetic wave through the medium will occur. In other words, the way the medium responds to the electromagnetic excitation is contained in the middle polarization due to the propagation of the electromagnetic wave. It is in this context that some recent analyses have discovered some solutions from the nonlinear response of the medium to the propagation of the electromagnetic wave which may lead to an approach of some quantum effects from a nonlinear treatment of electromagnetism in the material medium.
The propagation of optical pulses through waveguides such as optical fibers can give rise to nonlinear optical effects and quantum effects. The appropriate modeling of these effects can be used for the development of sensors to the optical fiber whose resolution can be regulated properly. In addition, the method allows the selection of propagation modes by selecting the desired modes by knowing the band gap of the waveguide or the photonic crystal.
The development of sensors to the optical fiber is based on the propagation of optical pulses through waveguides like optical fibers and photonic crystals. The propagation of the pulses through waveguides can generate nonlinear and quantum effects as Raman and Brillouin effects. We have analytically modeled these effects from the Maxwell equations on dielectric media describing the propagation of these optical pulses by developing a model that can be implemented computationally for the processing and propagation of these optical signals. The optical sensor can select the modes of propagation of the optical beams through the natural conduction band of the photonic crystal since the function of our model, the so-called optical potential, describes the optical light scattering through the crystal .
2. Modeling the optical lattice
The modeling of an optical system is very important for the design and development of many applications of optics in electronics, photonics, integrated optics, and an array of devices based on the light. Nonlinear effects from the interaction of light with matter in waveguides and photonic crystals may be suitable for a variety of optical applications [3, 4, 5]. Quantum effects such as the Raman effect and the Brillouin effect can be conveniently dealt with by exploring the nonlinear aspects of optical wave propagation in waveguides. Effects, e.g., self-focusing and low dispersion, of a guided beam can be applied and exploited in several technologies that use waveguides and sensors to the optical fiber [6, 7]. Consider an optical field given by
In Eq. (1),
Considering the Maxwell equations and the medium polarization, one can obtain the generalized nonlinear Schrödinger (NLS) equation that describes soliton solutions :
In Eq. (2),
We will sequentially show how quantum effects from the interaction of an optical beam with the constituents of the waveguide or the crystal lattice through which it propagates can be described in the context of nonlinear electrodynamics. In other words, we will show an equivalence to quantum optics and nonlinear electrodynamics characterized by nonlinear polarization of the medium. We will demonstrate the equivalence between its properties and the properties of solutions from a dynamical action. This action can map optical systems, and the method is based on the variational principle whose solutions give the same from that of Eq. (2).
Considering the optical rays equation
which is Fermat’s principle for the paths of light rays. So the scalar wave equation in optical context can be written as
that is satisfied by a plane wave solution
The wave number
So Eq. (6) can be written as
and it is assumed that is in the
and the quantities and are real functions.
Now we can establish a relation between this description of optical pulses, the NLS equation and the Schrödinger equation. This equivalence, in context of the classical level at least, between NLS equation solutions and the method that we propose in this work, can be constructed considering the relation
In this sense, we are led to conclude that the energy
The wave length and the frequency are related by
and, now, considering that for a light ray one propagating in a certain medium with speed
and another simple relation
So as an important consequence of this approach, we can write the wave Eq. (5) as
that is the time-independent wave equation. So it is perfectly acceptable that a certain optical field
and doing the identifications
3. Brief introduction on optical solitons
3.1 Nonlinear Schrödinger equation
The equation that describes optical fields in a nonlinear medium is known as the nonlinear Schrödinger equation. In this section, we succinctly present the origin of the NLS equation for a CW beam propagating inside a nonlinear optical medium. From Maxwell equations in a nonlinear medium, one gets the wave equation for the electric field [7, 10]:
are the first- and third-order susceptibility tensors. A general solution of Eq. (21) will be
where and are the propagation constants with the wavelength . The beam diffracts and self-focuses along the two transverse directions X and Y where X, Y, and Z are the spatial coordinates associated with
Introducing the following variables
where ω0 is a transverse scaling parameter related to the input beam width and is the diffraction length; Eq. (26) takes the form of a NLS equation:
Now one can consider the NLS equation in the form
and the y independent form of it
4. Nonlinear and quantum optical sensor principles
Nonlinear effects in optical fibers are common when, for example, increasing the power of the optical source. In this case, optical noise such as Raman effect and Kerr effect originate the interaction of optical fields with matter. These effects, depending on their technological application, may be undesirable. In our work, we discuss one set of solutions for optical fields whose nonlinear effects can be used to suppress certain propagation modes harmful to technological applications. On propagation of signals, for example, the Kerr effect can be a factor representing the loss of optical signal, and in this sense, using the solutions presented in this work, an optical network can be specially designed to suppress the distortion of the optical signal by the Kerr effect. The fiber or group of optical fibers can be designed so that the distortion of optical signals through nonlinear effects is eliminated. In another perspective, our method can be combined with space-division multiplexing (SDM) and nonlinear cancelation methods that offer the opportunity to reverse the effect of Kerr distortion . The method developed in this work can be implemented to identify patterns of nonlinear modes which contribute to the distortion of the optical signal in the transmission system .
In this case, the propagation modes of the optical beam by the system will be conveniently selected and processed in the transmission link as shown in Figure 1. In this section, we will mathematically demonstrate how any optical signal can be transformed conveniently into appropriate optical pulse. In other words, any sign optical can be mapped to a field originally known.
It is appropriate to point out that the mathematical approach is necessary to detail the practical implementation of the recognition and processing of optical signals that can reach the optical beam level in the waveguide or any optical network.
4.1 Optical systems
The discovery of non-Hermitian observables with real spectrum in optical systems may provide an important and useful relationship between the crystal structure of an optical system and parity-time (
In this point it is important to note that the optical system can be mapped by the band structure of the optical lattice or waveguide through the propagation constant λ. In fact the eigenvalue (λ) and the eigenmode
whose solutions can be write in the form
From Eq. (35) result important properties of optical systems described by symmetry
4.2 Mapping optical systems
In order to show how the
whose solution can be given by
5. New non-Hermitian optical systems
The application of non-Hermitian optical systems in the modeling of the optical lattices and waveguides  can be performed by the variational method proposed by . In this approach, a generalization of the NLS equation (29)
can be obtained by a properly Lagrangian density
In this approach, a mechanism to describe nonlinear and non-Hermitian optical systems can be obtained starting from an adequate action given by Eq. (40) with a Lagrangian
and solved as follows
On the other hand, these same solutions emerge from the energy calculation of optical system, which can be written as
The configuration that minimizes the energy of system, given by Eq. (19), is obtained by the following differential equation:
5.1 New models and sensors
Nonlinear and non-Hermitian optical systems can be modeled now by the method based on Lagrangian. The motion equations will describe the propagation of the optical field through of a generic optical lattice.
Consider a model that is based on nonlinear and non-Hermitian optical potential
where and the energy has real spectrum that can be calculated by
Another nonlinear and non-Hermitian optical potential is given by
whose solution for the optical field is
where whose field configuration is plotted in Figure 4.
In general one can rewrite the above equation simply as
getting the non-Hermitian potential
with the correspondent solution
It is important to note that the optical field solution
In this chapter, we describe how nonlinear and quantum optical effects can be applied to the development of optical sensors. From Maxwell’s equations, one can obtain an equation that describes the nonlinear behavior of an optical pulse that propagates through an optical system such as a waveguide. The nonlinear behavior of an optical pulse can be understood here as optical noise, the propagation of an optical pulse through a general optical system, or even quantum optical effects that can be properly described by a nonlinear second-order equation known as generalized nonlinear Schrödinger equation as described in Section 2. This equation has simple solutions called optical solitons. In Sections 3 and 4, we show how the term optical potential
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