1. Introduction
In recent years, optical phase conjugation (OPC) has been an important research subject in the field of lasers and nonlinear optics. OPC defines a link between two coherent optical beams propagating in opposite directions with reversed wave front and identical transverse amplitude distributions. The distinctive characteristic of a pair of phase-conjugate beams is that the aberration influence imposed on the forward beam passed through an inhomogeneous or disturbing medium can be automatically removed for the backward beam passed through the same disturbing medium. There are three main approaches that are efficiently able to produce the backward phase-conjugate beam. The first one is based on the degenerate (or partially degenerate) four-wave mixing processes (FWM), the second is based on a variety of backward simulated (e.g. Brillouin, Raman or Kerr) scattering processes, and the third is based on one-photon or multi-photon pumped backward stimulated emission (lasing) processes. Among these different approaches, there is a common physical mechanism in generating a backward phase-conjugate beam, which is the formation of the induced holographic grating and the subsequent wave-front restoration via a backward reading beam. In most experimental studies, certain types of resonance enhancements of induced refractive-index changes are desirable for obtaining higher grating-refraction efficiency. OPC-associated techniques can be effectively utilized in many different application areas: such as high-brightness laser oscillator/amplifier systems, cavity-less lasing devices, laser target-aiming systems, aberration correction for coherent-light transmission and reflection through disturbing media, long distance optical fiber communications with ultra-high bit-rate, optical phase locking and coupling systems, and novel optical data storage and processing systems (see Ref. [1] and references therein).
The power performance of a phase conjugated laser oscillator can be significantly improved introducing intracavity nonlinear elements, e.g. Eichler et al. [2] and O'Connor et al. [3] showed that a stimulated-Brillouin-scattering (SBS) phase conjugating cell placed inside the resonator of a solid-state laser reduces its optical coherence length, because each axial mode of the phase conjugated oscillator experiences a frequency shift at every reflection by the SBS cell resulting in a multi-frequency lasing spectrum, that makes the laser insensitive to changing operating conditions such as pulse repetition frequency, pump energy, etc. This ability is very important for many laser applications including ranging and remote sensing. The intracavity cell is also able to compensate optical aberrations from the resonator and from thermal effects in the active medium, resulting in near diffraction limited output [4], and eliminate the need for a conventional Q-switch as well, because its intensity-dependent reflectivity acts as a passive Q-switch, typically producing a train of nanosecond pulses of diffraction limited beam quality. One more significant use of OPC is a so-called short hologram, which does not exhibit in-depth diffraction deformation of the fine speckle pattern of the recording fields [5]. A thermal hologram in the output mirror was recorded by two speckle waves produced as a result of this recording a ring Nd:YAG laser [6]. Phase conjugation by SBS represents a fundamentally promising approach for achieving power scaling of solid-state lasers [7, 8] and optical fibers [9].
There are several theoretical models to describe OPC in resonators and lasers. One of them is to use the SBS reflection as one of the cavity mirrors of a laser resonator to form a so-called linear phase conjugate resonator [10], however ring-phase conjugate resonators are also possible [11]. The theoretical model of an OPC laser in transient operation [12] considers the temporal and spatial dynamic of the input field the Stokes field and the acoustic-wave amplitude in the SBS cell. On the other hand the spatial mode analysis of a laser may be carried out using transfer matrices, also know as ABCD matrices, which are a useful mathematical tool when studying the propagation of light rays through complex optical systems. They provide a simple way to obtain the final key characteristics (position and angle) of the ray. As an important example we could mention that transfer matrices have been used to study self-adaptive laser resonators where the laser oscillator is made out of a plane output coupler and an infinite nonlinear FWM medium in a self-intersecting loop geometry [13].
In this chapter we put forward an approach where the intracavity element is presented in the context of an iterative map (e.g. Tinkerbell, Duffing and Hénon) whose state is determined by its previous state. It is shown that the behavior of a beam within a ring optical resonator may be well described by a particular iterative map and the necessary conditions for its occurrence are discussed. In particular, it is shown that the introduction of a specific element within a ring phase-conjugated resonator may produce beams described by a Duffing, Tinkerbell or Hénon map, which we call “Tinkerbell, Duffing or Hénon beams”. The idea of introducing map generating elements in optical resonators from a mathematical viewpoint was originally explored in [14-16] and this chapter is mainly based on those results.
This chapter is organized as follows: Section 2 discusses the matrix optics elements on which this work is based. Section 3 presents as an illustration some basic features of Tinkerbell, Duffing and Hénon maps, Sections 4,5 and 6 show, each one of them, the main characteristics of the map generation matrix and Tinkerbell, Duffing and Hénon Beams, as well as the general case for each beams in a ring phase conjugated resonator. Finally Section 6 presents the conclusions.
2. ABCD matrix optics
Any optical element may be described by a 2×2 matrix in paraxial optics. Assuming cylindrical symmetry around the optical axis, and defining at a given position
For any optical system, one may obtain the total [
2.1. Constant ABCD elements
For passive optical elements such as lenses, interfaces between two media, reflections, propagation, and many others, the elements
2.2. Non constant ABCD elements
However, for active or non-linear optical elements the
2.2.1. Curved interface with a Kerr electro-optic material
Due to the electro-optic Kerr effect the refraction index of an optical media
Having vacuum (
Clearly the elements
2.2.2. Phase conjugate mirror
A second example is a phase conjugate mirror. The process of phase conjugation has the property of retracing an incoming ray along the same incident path [7]. The ideal ABCD phase conjugate matrix is
One may notice that the determinant of this particular matrix is not 1 but -1. The ABCD matrix of a real phase conjugated mirror must take into account the specific process to produce the phase conjugation. As already mentioned, typically phase conjugation is achieved in two ways; Four Wave Mixing or using a stimulated scattering process such as Brillouin, i.e. SBS. However upon reflection on a stimulated SBS phase conjugated mirror, the reflected wave has its frequency
Furthermore, since in phase conjugation by SBS a light intensity threshold must be reached in order to have an exponential amplification of the scattered light, the above ideal matrix (4) must be modified. The scattered light intensity at position
where
The modeling of a real stimulated Brillouin scattering phase conjugate mirror usually takes into account a Gaussian aperture of radius
where the aperture
2.3. Systems with hysteresis
At last, as third example we may consider a system with hysteresis. It is well known that such systems exhibit memory. There are many examples of materials with electric, magnetic and elastic hysteresis, as well as systems in neuroscience, biology, electronics, energy and even economics which show hysteresis. As it is known in a system with no hysteresis, it is possible to predict the system's output at an instant in time given only its input at that instant in time. However in a system with hysteresis, this is not possible; there is no way to predict the output without knowing the system's previous state and there is no way to know the system's state without looking at the history of the input. This means that it is necessary to know the path that the input followed before it reached its current value. For an optical element with hysteresis the
3. Dynamic maps
An extensive list of two-dimensional maps may be found in Ref. [20]. A few examples are Tinkerbell, Duffing and Hénon maps. As will be shown next they may be written as a matrix dynamical system such as the one described by Eq. (1) or equivalently as
3.1. Tinkerbell map
The Tinkerbell map [21, 22] is a discrete-time dynamical system given by the equations:
where
It should be noted that these coefficients are not constants but depend on the state variables
3.2. Hénon map
The Hénon map has been widely studied due to its nonlinear chaotic dynamics. Hénon map is a popular example of a two-dimensional quadratic mapping which produces a discrete-time system with chaotic behavior. The Hénon map is described by the following two difference equations [23, 24]:
Following similar steps as those of the Tinkerbell map, this map may be written as a dynamic matrix system:
where
And the corresponding eigenvalues are
3.3. Duffing map
The study of the stability and chaos of the Duffing map has been the topic of many articles [26-27]. The Duffing map is a dynamical system which may be written as follows:
where
Therefore as an ABCD matrix system the Duffing map may be written as:
4. Maps in a ring phase-conjugated resonator
In this section an optical resonator with a specific map behavior for the variables
For this system, the total transformation matrix [
The above one round trip total transformation matrix is
As can be seen, the elements of this matrix depend on the elements of the map generating matrix device [
It should be noticed that the results given by equations (28) and (29) are only valid for
Therefore the round trip total transformation matrix is:
Matrix (29) describes a simplified ideal case whereas matrix (31) describes a general more complex and realistic case. These results will be widely used in the next three sections.
5. Tinkerbell beams
This section presents an optical resonator that produces beams following the Tinkerbell map dynamics; these beams will be called “Tinkerbell beams”. Equation (29) is the one round trip total transformation matrix of the resonator. If one does want a particular map to be reproduced by a ray in the optical resonator, each round trip described by (
Equations (32-35) define a system for the matrix elements
The introduction of the above values for the
5.1. Tinkerbell beams: General case
To obtain the Eqs. (36-39)
From Eqs. (16) and (31) we obtain the following system of equations for the matrix elements
The solution to this new system is written as:
where:
and
It should be noted that if one takes into account the thickness of the map generating element, the equations complexity is substantially increased. Now only
6. Duffing beams
This section presents an optical resonator that produces beams following the Duffing map dynamics; these beams will be called “Duffing beams”. Equation (29) is the one round trip total transformation matrix of the resonator. If one does want a particular map to be reproduced by a ray in the optical resonator, each round trip described by (
Equations (48-51) define a system for the matrix elements of
As can be seen these matrix elements depend on the Duffing parameters
6.1. Duffing beams: General case
The results given by Eqs. (52-55) are valid only when the
The solution to this system is given by:
As we may see, taking into account the thickness of the map generating element device described by matrix [
7. Hénon beams
This section presents an optical resonator that produces beams following the Hénon map dynamics; these beams will be called “Hénon beams”. Equation (29) is the one round trip total transformation matrix of the resonator. If one does want a particular map to be reproduced by a ray in the optical resonator, each round trip described by (
The solution for the Hénon chaos matrix elements [
As can be seen the matrix elements depend on the Hénon parameters α and
7.1. Hénon beams: General case
In an analogous way to the two previous cases, using expression (18) and (31) we obtain for the general Hénon chaos matrix elements [
8. Conclusions
This chapter presents a description of the application of non-constant ABCD matrix in the description of ring optical phase conjugated resonators. It is shown how the introduction of a particular map generating device in a ring optical phase-conjugated resonator can generate beams with the behavior of a specific two dimensional map. In this way beams that behave according to the Tinkerbell, Duffing or Henon Maps which we call “Tinkerbell, Duffing or Henon Beams”, are obtained.
In particular, this chapter shows how Tinkerbell beams can be produced if a particular device is introduced in a ring optical phase-conjugated resonator. The difference equations of the Tinkerbell map are explicitly introduced in an
Also, it is explicitly shown how the difference equations of the Duffing map can be used to describe the dynamic behavior of what we call Duffing beams i.e. beams that behave according to the Duffing map. The matrix elements a, b, c, e of a map generating device are found in terms of α and β, the Duffing parameters, the state variable θn and the resonator parameter d.
Finally it is shown that the difference equations of the Hénon map can be used to describe the dynamical behavior of Hénon beams. The matrix elements a, b, c, e of a chaos generating device are found in terms of α and β the Hénon parameters, and d the resonator parameter.
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