Historical Survey
Optical pattern formation gained prominence in the late 1980s and early 1990s, although early indications of spontaneous pattern formation in broad-aperture lasers emerged two decades earlier This connection between laser physics and fluid dynamics was established by simplifying the laser equations for class A lasers, where material variables relax quickly compared to the optical field, leading to the formulation of the complex Ginzburg–Landau (CGL) equation, which is also applicable to superconductors and superfluids.
Given the shared theoretical framework, one might anticipate that the behavior of light in lasers will exhibit similar dynamics to those observed in superconductors and superfluids.
After a decade of neglect, the study of optical patterns in nonlinear resonators regained attention as the optical community shifted focus to spatial effects in the unidirectional mirrorless propagation of intense light beams in nonlinear materials In simpler scenarios, this involves the filamentation of light in a focusing medium, while more complex situations lead to the emergence of bright spatial solitons The renewed interest in spatial patterns within lasers was sparked by recent studies showcasing nontrivial stationary and dynamic transverse mode formations in laser beams, as well as the discovery of vortex solutions in the laser Maxwell–Bloch equations.
Experimental confirmation revealed a relationship among the optical vortices in lasers and the phase defects observed in earlier speckle fields.
Following the groundbreaking research, numerous studies emerged focusing on developing an order parameter equation for lasers and nonlinear resonators This equation aims to succinctly encapsulate the primary spatio-temporal characteristics of laser radiation in a straightforward manner The Ginzburg–Landau equation was derived as a result of these efforts.
The simplest laser patterns can be understood as locked transverse modes within a resonator featuring curved mirrors Initial models for lasers with spatial degrees of freedom led to the development of a more precise order parameter equation, applicable in the red detuning limit, where the atomic resonance frequency is lower than that of the nearest longitudinal mode However, this equation has limitations, as it fails to predict spontaneous pattern formation, which typically occurs in blue-detuned cavities In these blue-detuned resonators, the excitation of higher-order transverse modes or tilted waves can occur, influenced by the cavity aperture.
The derivation of an order parameter equation for lasers was successfully achieved through the complex Swift–Hohenberg (CSH) equation, which includes nonlocal terms that facilitate spatial mode selection and induce pattern formation instability This equation, initially presented in studies [22, 23], was later rederived using a multiscale expansion approach The CSH equation effectively describes the spatio-temporal evolution of the field amplitude, while an order parameter equation for the laser phase was formulated as the Kuramoto–Sivashinsky equation Notably, both the Swift–Hohenberg and Kuramoto–Sivashinsky equations are commonly utilized in hydrodynamic and chemical problem analyses, respectively.
The development of an order parameter equation for lasers represents a major breakthrough, enabling a deeper understanding of the mechanisms behind pattern formation in laser systems This advancement also facilitates the analysis of broad-aperture lasers within the wider framework of pattern-forming systems found in nature.
The exploration of laser patterns has sparked interest in spontaneous pattern formation across various nonlinear resonators, particularly in photorefractive oscillators, which have been extensively studied for their theoretical foundations and complex experimental structures Research has also focused on passive and driven nonlinear Kerr resonators, revealing significant insights into pattern formation Additionally, optical parametric oscillators have garnered attention, with predictions of basic patterns and the derivation of order parameter equations in both degenerate and nondegenerate regimes The relationship between patterns in planar and curved-mirror resonators has been addressed, highlighting an order parameter equation for weakly curved nonlinear optical resonators.
These are just a few examples In the next section, the general character- istics of nonlinear resonators, and the state of the art are reviewed.
Patterns in Nonlinear Optical Resonators
Localized Structures: Vortices and Solitons
The optical vortex is a widely studied transverse structure known for its topological properties, characterized by a zero amplitude in the field and a singularity in the field phase.
Optical vortices have primarily been investigated in nonlinear materials during free propagation; however, some research has explored vortex formation in resonators Early studies on these intriguing phenomena have significantly sparked interest in the broader field of pattern formation The presence of vortices suggests a fascinating analogy between optics and hydrodynamics.
Research indicates that vortices can trigger the onset of defect-mediated turbulence These vortices may appear as stationary isolated structures or be organized into regular vortex lattices Additionally, both single vortices and vortex lattice structures exhibit nonstationary dynamics Recent experiments have successfully observed optical vortex lattices in microchip lasers.
Spatial solitons are a type of localized non-topological structure gaining significant attention in optics due to their potential technological applications These solitons, which can exist in dissipative systems and exhibit bistability, are capable of carrying information As a result, they hold great promise for advancements in parallel storage and information processing technologies.
Cavity solitons, which are spatial solitons excited in optical resonators, can be categorized into two primary types: amplitude solitons (including bright and dark solitons) and phase solitons (specifically dark-ring solitons) The exploration of bright localized structures originated from foundational studies on bistable lasers with saturable absorbers and passive nonlinear resonators.
Amplitude solitons can be generated in subcritical systems characterized by bistability, serving as homoclinic connections between the lower (unexcited) and upper (excited) states These solitons have been observed in various passive nonlinear optical resonators, including degenerate configurations.
Recent studies have focused on both degenerate and nondegenerate optical parametric oscillators, particularly in the context of second-harmonic generation, which is illustrated in Figure 1.2 The phenomenon of bistability in these systems is linked to the presence of nonlinear resonance Additionally, the dynamics and interactions of solitons within these optical systems have been explored, revealing significant insights into their behavior.
1.2 Patterns in Nonlinear Optical Resonators 7
Fig 1.2 Interaction of two moving amplitude solitons in vectorial intracavity second-harmonic generation: (a) central collision, (b) noncentral collision From
74, 75] Resonators containing Kerr media also support amplitude solitons, as a result of either Kerr [76] or polarization (vectorial) [77] instabilities.
In active systems, bright solitons have been demonstrated in photore- fractive oscillators [78, 79, 80] and in lasers containing saturable absorbers
The vertical cavity surface emission laser (VCSEL) is a promising system for practical applications, utilizing a semiconductor as a nonlinear material to create a microresonator Recent experiments have confirmed the theoretically predicted patterns for this system, highlighting its potential in various fields.
Recent studies have shown that stable solitons can occur in cascade lasers without the need for an additional absorbing medium, challenging the traditional requirement of subcriticality achieved through intracavity elements In supercritical resonators, a distinct form of bistable soliton emerges, characterized by broken phase symmetry and only two allowable phase values These phase solitons, which connect two homogeneous solutions of equal amplitude but opposite phase, manifest as dark rings against a bright background This innovative type of optical soliton is garnering significant attention due to its experimental feasibility compared to the bright solitons found in subcritical systems.
The degenerate optical parametric oscillator (DOPO) has been extensively studied in both one-dimensional and two-dimensional configurations Research has also focused on the soliton formation process and its dynamic behavior within these systems.
[101,102] have been analyzed Optical bistability in a passive cavity driven by a coherent external field is another example of a system supporting such phase
Fig 1.3.Phase domains and phase (dark- ring) solitons in a cavity four-wave-mixing experiment From [115], c1999 Optical Society of America
The modulational instability of a straight domain boundary leads to the formation of a finger pattern in a type II degenerate optical parametric oscillator The upper row illustrates the intensity distribution, while the lower row depicts the corresponding phase pattern, as demonstrated in the study from 2001.
The American Physical Society highlights that both Dissipative Optical Parametric Oscillators (DOPO) and optical bistability systems can be described by the real Swift–Hohenberg equation Additionally, systems exhibiting higher nonlinearity, such as vectorial Kerr resonators, have demonstrated the capability to support phase solitons.
Phase solitons can form bound states, resulting in soliton aggregates or clusters [94, 112].
Phase solitons in cavities are significantly easier to excite compared to those in subcritical systems Experimental evidence of these phase solitons has already been observed in degenerate four-wave mixers.
Extended Patterns
In addition to localized patterns, vortices, and solitons, which are the primary focus of the book, extensive research has also been conducted on extended patterns in optical resonators Within these optical resonators, patterns can be categorized into two main types.
1.2 Patterns in Nonlinear Optical Resonators 9 be distinguished One class of patterns appears in low-aperture systems, char- acterized by a small Fresnel number, such as a laser with curved mirrors Since this is the most typical configuration of an optical cavity, this phenomenon was observed in the very first experimental realizations, although a system- atic study was postponed to a later time [16] The patterns of this kind are induced by the boundary conditions, and can be interpreted as a weakly non- linear superposition of a small number of cavity modes of Gauss–Hermite or Gauss–Laguerre type.
Theoretical predictions based on modal expansions of the field [14, 116,
Numerous experiments have confirmed that the unique geometry of the cavity predominantly produces optical patterns Increasing the aperture leads to a greater excitation of cavity modes, resulting in enhanced spatial complexity of the observed patterns.
Large-aperture resonators, such as those configured with plane mirrors in a ring or Fabry–Pérot arrangement, exhibit distinct extended optical patterns Unlike small-aperture systems, the transverse boundary conditions have minimal impact on the dynamics of these systems As a result, the patterns observed are predominantly nonlinear, allowing the system dynamics to be simplified to the evolution of a single field known as the order parameter.
The simplest patterns in laser systems are characterized by a single traveling wave (TW), which represents the fundamental transverse solution However, more complex patterns involving multiple TWs have also been identified Experimental observations of predicted laser TW patterns have been conducted in large-Fresnel-number cavities Additionally, TW solutions are present in passive resonators governed by the same order parameter equation, including nondegenerate optical parametric oscillators (OPOs) Furthermore, the impact of externally injected signals in lasers has been a subject of study.
[130,131], showing the formation of more complex patterns, such as rolls or hexagons.
Roll or stripe patterns frequently occur in various nonlinear passive cavities, including degenerate optical parametric oscillators (OPOs), four-wave mixers, systems exhibiting optical bistability, and cavities that incorporate Kerr media.
[133] Patterns with hexagonal symmetry are also frequently found in such resonators [134, 135] Both types of pattern are familiar in hydrodynamic systems, such as systems showing Rayleigh–B´enard convection.
Optical resonators exhibit a unique traveling solution characterized by spiral patterns, commonly observed in lasers and optical parametric oscillators (OPOs) These structures are frequently associated with chemical reaction-diffusion systems, highlighting their significance in various scientific applications.
Incorporating more complex models that account for additional effects reveals a wider range of patterns, including some that may appear exotic These models serve to generalize previously mentioned frameworks by acknowledging the existence of these diverse patterns.
The introduction explores the competition among various parametric processes, highlighting the differences between scalar and vectorial instabilities It also addresses the walk-off effect caused by birefringence within the medium and considers the impact of external temporal variations in cavity parameters.
Some systems enable the simultaneous excitation of patterns with varying wavenumbers, resulting in unique formations known as quasicrystals and daisy patterns Achieving the experimental conditions for large-aspect-ratio resonators can be challenging, leading most studies to focus on multimode regimes featuring high-order transverse modes These patterns have been successfully observed in lasers and optical parametric oscillators (OPOs), aligning closely with numerical solutions derived from large-aspect-ratio models Additionally, boundary-free, nonlinear patterns were achieved through the use of self-imaging resonators, enabling experiments to reach exceptionally high Fresnel numbers.
Recent advancements have shown the emergence of three-dimensional patterns in optical parametric oscillators (OPOs), which are nonlinear resonators utilizing Kerr media Unlike the previously reviewed two-dimensional patterns, these three-dimensional configurations distribute light in a transverse space that is perpendicular to the resonator axis while also evolving over time.
The article discusses key concepts in nonlinear optics, including optical bistability and second-harmonic generation It also addresses the impact of noise on the pattern formation properties of nonlinear resonators, highlighting the significance of these effects in understanding complex optical behaviors.
Fig 1.5 Experimentally observed hexagonal patterns with sixfold and twelvefold symmetry (quasipatterns), in a nonlinear optical system with continuous rotational symmetry From
Optical Patterns in Other Configurations
Optical Feedback Loops
The feedback loop configuration serves as a hybrid between single feedback mirrors and nonlinear resonators, allowing for continuous manipulation of the field distribution with each round trip This setup enables various two-dimensional transformations, including rotation, translation, scaling, and filtering of the pattern Notably, initial research demonstrated pattern formation by effectively controlling the spatial scale and topology of the transverse interactions.
1.3 Optical Patterns in Other Configurations 13
Fig 1.7.Hexagon formation in a single-feedback-mirror configuration Numerical results from [207], c 1991 American Physical Society
Fig 1.8.Dissipative solitons observed experimentally in sodium vapor with a single feedback mirror From [219], c 2000 American Physical Society
Fig 1.9.Experimental patterns in an optical system with two-dimensional feedback (a) Hexagonal array, (b)– (d) “black-eye” patterns, (e) island of bright localized structures, (f) optical squirms From [224], c 1998 American
Fig 1.10.Quasicrystal patterns with dodecagonal symmetry, with differ- ent spatial scales, together with the corresponding spatial spectra From
In 1998, the American Physical Society explored the light field in a medium characterized by cubic nonlinearity This research involved manipulating the phase of the light field using a spatial Fourier filter and incorporating a medium with a binary-type refractive nonlinear response.
A feedback loop utilizing a liquid-crystal light valve as a phase modulator with Kerr-type nonlinearity offers remarkable versatility The necessary conversion from phase to intensity distribution to complete the feedback loop can occur through two methods: free propagation, known as diffractive feedback, or interference with reflected waves, termed interferential feedback Both methods have led to the theoretical and experimental generation of diverse kaleidoscope-like patterns Additionally, these patterns can be manipulated through nonlocal interactions, including rotation or translation of the signal within the feedback loop, resulting in more complex solutions such as quasicrystals.
In Figure 1.11, the patterns observed in a liquid-crystal light valve under an interferential feedback configuration are depicted, highlighting the effects of increasing translational nonlocality (∆x) The top row illustrates the near field patterns, while the corresponding spectrum is presented in the bottom row, showcasing the relationship between these two elements This data is sourced from [230], published in 1998.
The Contents of this Book
Experimental patterns of crystals and quasicrystals were generated through the rotation of signals in a liquid-crystal light valve feedback loop The first two columns illustrate the near-field distributions, while the third and fourth columns depict the corresponding far fields This research was published by the American Physical Society in 1995.
Fig 1.13.Bound state of spatial solitons in a liquid- crystal light valve interferometer From [236], c 2002
The American Physical Society has reported on the existence of spatial solitons and the experimental observation of bound states of solitons in liquid-crystal light valve systems, as illustrated in Fig 1.13.
1.4 The Contents of this Book
Chapters 2 and 3 present the order parameter equations (OPEs) for broad-aperture lasers and various nonlinear resonators While these sections are predominantly mathematical, the derived OPEs serve as a foundation for further exploration in the field.
In Chapter 2 of the book, the derivation of the Optical Phase Equation (OPE) for class A and class C lasers is presented Two techniques are utilized for this derivation: one involves the adiabatic elimination of fast variables, while the other employs multiscale expansion methods Both approaches ultimately yield the complex Swift–Hohenberg equation as the OPE for lasers This equation captures the spatio-temporal dynamics of the complex-valued order parameter, which is directly related to the envelope of the optical field.
In Chapter 3, the optical parametric oscillators (OPEs) and photorefractive oscillators (PROs) are analyzed, leading to the derivation of the real Swift–Hohenberg equation in the degenerate case, originally identified in hydrodynamics For scenarios with significant pump detuning, a generalized model incorporating nonlinear resonance effects is established Additionally, the CSH equation for PROs is derived using the adiabatic elimination technique The order parameter equations from Chapters 2 and 3 categorize nonlinear optical resonators into distinct classes, facilitating the study of pattern formation phenomena without the need to examine each nonlinear optical system individually.
Chapters 2 and 3 focus on systems described by the CSH equation, including lasers, photorefractive oscillators, and nondegenerate optical parametric oscillators (OPOs) In these systems, localized patterns manifest as optical vortices, which are characterized by zeros in the amplitude of the optical field and act as singularities in the field phase These optical vortices significantly influence system dynamics, particularly in near-resonant conditions where the detuning is minimal The complex Ginzburg-Landau (CGL) equation can be reformulated into a hydrodynamic framework in this near-resonant limit, revealing a striking analogy between laser dynamics and hydrodynamics Consequently, the behavior of laser radiation's transverse distribution resembles that of a superfluid It is demonstrated that optical vortices with identical topological charges rotate around each other, and pairs of such vortices can either translate in parallel through the laser aperture or annihilate, depending on specific parameters.
Chapter 5 explores the dynamics of fields under conditions of large or moderate detuning, where the CSH equation cannot be expressed in a hydrodynamic form Despite this limitation, hydrodynamic interpretations remain effective In scenarios of significant detuning, tilted waves are generated, favoring flows with a consistent velocity but varying directions This dynamic results in counterpropagating flows that are separated by vortex sheets, leading to the formation of optical vortices that are carried along by the mean flow These phenomena are both theoretically analyzed and numerically demonstrated.
A pattern of square symmetry, called a square vortex lattice, consisting of four counterpropagating flows in the form of a cross, is also described and discussed.
Chapter 6 examines the impact of mirror curvature on the resonator's behavior Most theoretical studies on pattern formation in nonlinear optics, which comprise a significant portion of this book, focus on these effects.
This book introduces a model of lasers with curved mirrors, which adds a term to the order parameter equation that reflects the mirrors' total curvature This results in a parabolic phase shift of the order parameter during propagation, enabling the expansion of the resonator's field in terms of eigenfunctions (transverse modes) While this mode expansion is valid for linear resonators, the nonlinearity introduces a weak coupling of the complex amplitudes of the modes, leading to a set of coupled ordinary differential equations This approach provides an alternative method for analyzing the transverse dynamics of lasers, particularly class A lasers and photorefractive oscillators, where phenomena like transverse mode pulling and locking are observed Additionally, the book explores degenerate resonators, such as self-imaging and confocal types, revealing that self-imaging resonators can be treated as planar resonators of zero length, thus paving the way for new experimental opportunities in nonlinear optical systems.
Chapter 7 explores the unique dynamics of class B lasers, which cannot be described by the CSH equation due to the slow population inversion Instead, the order parameter is represented by a system of two coupled equations, similar to those found in excitatory or oscillatory chemical systems, where the slow population inversion acts as the recovery variable and the fast optical field serves as the excitable variable The chapter analyzes the self-sustained spatio-temporal dynamics within class B lasers, highlighting the phenomenon of "restless vortices," which meander unlike the stationary vortices in class A lasers Additionally, vortex lattices in class B lasers exhibit self-sustained oscillatory dynamics, where neighboring vortices can oscillate in antiphase, creating an "optical" mode of oscillation, or the lattice can drift with a defined velocity, resulting in an "acoustic" oscillation mode.
The following chapters, Chaps 8 to 11, are devoted to amplitude and phase domains, as well as amplitude and phase solitons in bistable nonlinear
In Chapter 8, the general theory of subcritical spatially extended optical systems is explored, focusing on two key mechanisms that contribute to subcriticality in optical resonators: the inclusion of a saturable absorber and the presence of a nonlinear resonance The chapter also provides a discussion framed around order parameter equations, enhancing the understanding of these phenomena in optical systems.
Chapter 9 presents a theoretical and experimental analysis of domain dynamics and spatial solitons in lasers with saturable absorbers The study utilizes two resonator configurations: one with coincident nonlinearities on the optical axis and another with nonlinearities at Fourier-conjugated positions In the first configuration, both numerical and experimental results demonstrate the evolution of domains leading to the formation of spatial solitons In contrast, the second configuration explores the competition, interaction, and drift of solitons, supported by both theoretical and experimental findings.
Chapter 10 explores a unique subcriticality mechanism distinct from saturable absorption, focusing on nonlinear resonance caused by nonresonant pumping It examines the order parameter equation from Chapter 3 for a degenerate optical parametric oscillator (OPO) with a detuned pump This nonlinear resonance indicates that the pattern wavenumber is influenced by radiation intensity With suitable detuning values, nonlinear resonance can induce bistability, facilitating the excitation of amplitude domains and spatial solitons The chapter also presents numerical results from the degenerate OPO mean-field model for comparison.
Chapter 11 explores the dynamics of phase domains in supercritical real-valued order parameter systems, specifically focusing on the degenerate Optical Parametric Oscillator (OPO) The study emphasizes that these systems are best described by the real Swift–Hohenberg equation, which captures their complex behavior effectively.
Domain boundaries, which separate regions of opposite phase, can contract or expand based on the resonator detuning value, behaving like elastic ribbons with a variable elasticity coefficient At low detuning values, contracting domains may eventually vanish, while high detuning values lead to expanding domains that form labyrinthine structures For intermediate detuning values, the contraction of domain boundaries halts at a specific radius, resulting in stable rings of domain boundaries known as phase solitons The article also includes experimental confirmation of these predicted phenomena.
Model of a Laser
Our starting point is the semiclassical Maxwell–Bloch (MB) equation system, which describes many types of lasers with transverse degrees of freedom:
The complex fields E(r, t) and P(r, t) represent the envelopes of the electromagnetic field and polarization, respectively, while D(r, t) denotes the real-valued field of population inversion, which equals its unsaturated value D₀(r) in the absence of stimulated radiation The relaxation rate of the optical field in the resonator, denoted by κ, is influenced by the small transmittivity of the mirrors and linear losses Additionally, γ⊥ and γ represent the decay rates of polarization and population inversion, respectively Lastly, ω signifies the resonator detuning, indicating the difference between the resonance frequency of the corresponding longitudinal mode and the center of the gain line.
The optical field E(r, t) is considered to be linearly polarized, with a homogeneously broadened gain line Additionally, it is assumed that only a single longitudinal mode family is excited, eliminating any dependence on the longitudinal coordinate z and focusing solely on the time t and the transverse coordinates r = (x, y).
The system described in (2.1) models a laser exhibiting multiple transverse modes while maintaining a single longitudinal mode, with time evolution occurring at a significantly slower pace than the light's round-trip time in the cavity, ensuring the validity of the single longitudinal mode assumption The nonlocal diffraction term influences the spatial degrees of freedom, coupling the field across the laser's cross section and contributing to the collective behavior of the emitted radiation In the simplest scenario, known as the class A laser, the polarization and population inversion dynamics are rapid relative to the optical field within the resonator This "good cavity limit" allows for the adiabatic elimination of fast material variables, resulting in a simplified order parameter equation, as demonstrated in previous studies While this adiabatic elimination may obscure some pattern-forming characteristics of lasers, it serves as a foundational starting point for further analysis.
In this analysis, it is assumed that material variables decay rapidly, specifically that κ/γ ⊥ = O(ε) and κ/γ = O(ε), where ε represents a smallness parameter Additionally, the temporal derivatives of all variables are considered to have finite values, indicated by ∂E/∂t ∼ ∂P/∂t ∼ ∂D/∂t = O(1) By multiplying both sides of equations (2.1b) and (2.1c) by κ/γ ⊥ and κ/γ, respectively, we find that the left-hand sides of these equations are of order O(ε) By focusing solely on the zero-order terms O(1), we can effectively eliminate the material variables from (2.1b) and (2.1c).
The expressions (2.2) imply that the material variablesP and D follow instantaneously, or adiabatically, the changes of the field variableE Inserting (2.2) into (2.1a) we obtain a single equation for the field,
E , (2.3) whereτ=κtis a slowtime Close to the emission thresholdp= (D 0 −1)
1, the emitted fields are relatively weak, |E| 2 1, which allows a cubic approximation for the nonlinear term in (2.3),
E−E|E| 2 ; (2.4) this is the complex Ginzburg–Landau equation In (2.4), p is the balance between the gain and loss of the laser, and is a criticality parameter of the CGL.
The CGL equation serves as a basic approximation for laser behavior, but it fails to consider the selection of transverse wavenumbers, or transverse modes A linear stability analysis of the nonlasing solution reveals that all components of the spatial spectrum exhibit equal growth exponents This can be demonstrated by substituting a test solution in the form of a tilted wave, E(r, t) = e exp(i k⊥ r + λt), where e represents a small amplitude By linearizing this equation with respect to e, one can calculate the exponent λ, which remains constant and is independent of the transverse wavenumber k, specifically equating to λ = p = D₀⁻¹.
Lasers emit specific transverse modes that are influenced by the resonator's length When the peak of the gain line aligns with a particular family of transverse modes, that mode is enhanced and experiences the fastest growth However, this tunability characteristic of lasers is not captured in the derivation of the relevant equations.
It is not difficult to understand why spatial-frequency selection is absent in(2.4): the derivation assumes, among other things, the conditionγ ⊥ /κ→ ∞,
36 2 Order Parameter Equations for Lasers frequency frequency transverse modes transverse modes gain line gain line a) b)
The transverse modes of a laser resonator can be tuned by adjusting the resonator length, which shifts these modes in relation to the amplifying medium's gain line Increasing the resonator length allows for the excitation of higher-order transverse mode families, indicated by a negative detuning parameter Conversely, decreasing the resonator length excites lower-order modes, resulting in a positive detuning where the gain line becomes infinitely broad This broad gain line prevents effective transverse frequency selection, necessitating a more advanced derivation of the laser's optical properties, which will be explored in the subsequent sections.
To continue with the derivation of a more precise OPE a linear stability analysis of the laser equations is useful.
Linear Stability Analysis
A standard technique is applied here to investigate the stability of the non- lasing solution of (2.1), given by E( r, t) = 0, P( r, t) = 0, D( r, t) = D 0
By perturbing this trivial zero solution by E( r, t) = eexp(i k ãr +λt),
P( r, t) = pexp(i kãr +λt) and D( r, t) = D 0 +dexp(i kãr +λt), insert- ing it into (2.1) and gathering the linear terms with respect toe,pandd, w e obtain λe=−iκ ω+ak 2 e−κe+κp , (2.5a) λp=−γ ⊥ p+γ ⊥ eD 0 , (2.5b) λd=−γ d (2.5c)
The last equation in (2.5) is independent, allowing for a straightforward calculation of one λ-branch, resulting in λ₃ = −γ The solvability condition of the secular equation in (2.5) leads to two additional branches of the growth exponent, expressed as λ₁,₂(∆ω) = −κ + γ⊥ + iκ∆ω.
Figure 2.2 illustrates the real parts of the three Lyapunov growth exponents as a function of the transverse wavenumber for the MB equations (2.1) with parameters κ= 1, γ ⊥ = 1.2, and γ = 0.4 The λ3 branch, represented as a horizontal straight line, indicates the decay of population inversion Additionally, the figure presents the real and imaginary parts of the most significant upper branch of the Lyapunov growth exponents (2.6) as dashed curves, alongside their corresponding Taylor expansions (2.8) depicted by solid curves.
Here ∆ω=ω+ak 2 is proportional to the deviation of the mode with trans- verse wavenumber k from its resonant value, ω res =−ak 2 Figure 2.2 illus- trates the dependence given by (2.6).
From (2.6), the threshold for the laser emission (which occurs whenλ= 0) is
A simplification of (2.1) is possible when only oneλ-branch is relevant to the dynamics of the system This occurs, in particular, for class A and class
In lasers operating near the emission threshold, the two lower eigenvalue branches, characterized by a negative sign and the straight line λ = -γ associated with population inversion relaxation, remain significantly below the zero axis Consequently, the dynamics of these lower branches are dominated by the upper branch By expanding the upper λ-branch in a Taylor series around its maximum, we derive the relationship λ(∆ω) = p - i ∆ω - κ², where κ and γ are parameters influencing the system's behavior.
This is plotted in Fig.2.2b as the solid curves.
The growth rate (2.8) is obtained by assuming that ∆ω is of O(ε) and p= (D 0 −1) is ofO(ε 2 ) In this case the truncation of the Taylor series at
The Taylor expansion under varying smallness assumptions can yield different expressions, as indicated by O(ε^3) leading to equation (2.8) Additionally, linear stability analysis provides insights into the initial evolution of radiation stemming from initial thermal or quantum noise To demonstrate the dynamics described by equation (2.8), the Maxwell–Bloch system outlined in equation (2.1) was integrated.
The 38 2 Order Parameter Equations for lasers demonstrate the numerical evolution of a random field distribution in space, illustrated by plots in Figs 2.3 and 2.4 These figures reveal a clear discrimination against nonresonant components in the spatial spectrum, leading to the emergence of either a central spot or a resonant ring, influenced by the resonator's detuning The radius of the resonant ring is defined by k² = -ω/a, aligning with the linear stability analysis results Consequently, structures with a specific spatial scale develop, directly related to the radius of the resonant ring, expressed as l = 2π/k.
Linear stability analysis enables the formulation of a model equation that characterizes the linear phase of field evolution In this process, the time evolution operator ∂/∂t is replaced by λ, while the Laplace operator ∇² is substituted with -k² Conversely, applying the opposite substitution in the equation leads to the integration of algebraic variables with their respective operators.
A , (2.9) where A( r, τ) is the order parameter related to the optical field in the laser resonator (the relation between A( r, τ) and E( r, t) is obtained in the next section), andτ is a normalized time,τ =tκγ ⊥ /(κ+γ ⊥ ).
The final plots in Figs 2.3 and 2.4 illustrate the nonlinear evolution stage, where wavevectors from the resonant spot and ring begin to compete This leads to a nonlinear broadening of the resonant ring and central spot In the spatial spectral domain, the ring may split into multiple spots representing various wave configurations, including single tilted waves and counterpropagating waves The emergence of these nonlinear patterns, including the cross-roll or square vortex lattice, will be explored further in subsequent chapters Notably, linear theory fails to predict the symmetry of these nonlinear patterns, as the last plots in Figs 2.3 and 2.4 exceed the predictions of the linear order parameter equation While linear stability analysis identifies favored modes based on de-tuning, it does not account for the exponential growth of spatial spectral components or competition between wavevectors.
To accurately capture the nonlinear evolution, it is essential to incorporate saturating nonlinear terms into the linear evolution equation (2.8) This closure process will be explored in the subsequent sections through two distinct methods, both leveraging insights gained from linear stability analysis.
The linear stage of spatial pattern formation for zero detuning is illustrated in Fig 2.3, showcasing the intensity, phase, and spatial Fourier spectra of the optical field across three columns The analysis begins with a random optical field distribution characterized by a broadband spatial spectrum, utilizing parameters such as ω= 0, κ= 1, γ⊥ = 2, γ = 10, and a= 0.0005 Integration is conducted under periodic boundary conditions within a unit-sized region, with time progression displayed from the top to the bottom row At t= 0.5, a speckle field of laser radiation is observed; at t= 2.5, the Fourier domain spot begins to narrow, indicating spatial spectrum filtering; at t= 7.5, the resonant spot continues to narrow, revealing a more distinct vortex structure in the near field; and at t= 25, a nonlinear "vortex glass" structure emerges, with the far field spot stabilization as linear narrowing is countered by nonlinear broadening.
40 2 Order Parameter Equations for Lasers
The linear stage of spatial pattern formation under finite negative detuning (ω = -2) reveals a progression of laser radiation behavior over time Initially, at t = 1, a speckle field is observed, transitioning to the emergence of a resonant ring in the far field by t = 5 By t = 15, the resonant ring narrows, and the near field exhibits a more regular field distribution Finally, at t = 25, competition among different components of the ring becomes evident, with domains of tilted waves and areas of relatively homogeneous distribution starting to form in the near field.