Focus Wave Modes Research Articles

Approximation of the scalar focus wave modes

The physical-realizability issues of scalar focus wave modes that are solutions of the homogeneous wave equation with the boundary data exp(−r2w0−2) given on a characteristic are addressed. The Gaussian beams are solutions of the paraxial equation with the same boundary data in a fixed plane. A comparison of these two situations suggests that the focus wave modes can be approximated by giving the exponential exp(−r2w0−2) on some moving plane. This suggestion is implemented.

Nondispersive waves: interpretation and causality

It is proved that the focus wave modes are only a subclass of a larger class of nondispersive waves. Then these nondispersive waves are interpreted in the framework of Goursat's theory of boundary value problems where data are given on the characteristics of the wave equation. Finally, from the remark that causality requires only that no effect precede the cause, it is proved that there is no casual dilemma for the nondispersive waves.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

PLANE WAVES WITH TRANSVERSE STRUCTURE

We prove that some inhomogeneous scalar waves among which the focus wave modes are nondispersive plane waves with a nonuniform transverse structure.

Embedding of focus wave modes into a wider class of approximate wave equation solutions

Focus wave modes are embedded into a wider class of solutions of the wave equation in the paraxial approximation. The generalizations of focus wave modes are made in regard to the following attributes: (1) a wave-packet envelope is admitted that moves with the group velocity in the direction opposite that of the focus envelope and (2) the velocity of the focus envelope can be chosen arbitrarily, in contrast to the group velocity. The Fourier spectrum of the generalized approximate solutions of the wave equation is calculated. This Fourier spectrum becomes singular on the dispersion surface with a strong correlation between frequencies and longitudinal components of the wave vectors in case of focus wave modes. The well-known Gaussian beam solutions with resting focus appear as another special case of the considered class of approximate solutions of the wave equation.

Generalized Gaussian beam solutions of paraxial optics and their connection to a hidden symmetry

It is shown that the Hermite–Gauss- and Laguerre–Gauss-type solutions of paraxial optics whose corresponding Hermite and Laguerre polynomials have complex arguments are closely related to a hidden symmetry in the parabolic equation. These solutions are generated from the fundamental Gaussian beam solution by applying the powers of the infinitesimal operators of this symmetry group. The Fourier spectrum of these solutions is obtained from the Fourier spectrum of the fundamental Gaussian beam solution in a similar manner. The Gaussian beam solutions containing Hermite polynomials with complex arguments that were derived by Siegman [ J. Opt. Soc. Am.63, 1093 ( 1973)] represent a limiting case of the more-general solutions considered here. The generalized Gaussian beam solutions show a sharp mode picture with exact zeros in the field distribution only in the focal or waist plane. Unconventional applications of the parabolic approximation in optics including focus wave modes are demonstrated.

A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation

A new decomposition of exact solutions to the scalar wave equation into bidirectional, forward and backward, traveling plane wave solutions is described. The resulting representation is a natural basis for synthesizing pulse solutions that can be tailored to give directed energy transfer in space. The development of known free-space solutions, such as the focus wave modes, the electromagnetic directed energy pulse trains, the spinor splash pulses, and the Bessel beams, in terms of this decomposition will be given. The efficacy of this representation in geometries with boundaries, such as a propagation in a circular waveguide, will also be demonstrated.

Focus wave modes: a dilemma with causality

It is demonstrated that the equivalent aperture distribution for a well-collimated FWM (focus wave mode) radiates essentially backward; i.e. the important propagator in the forward-propagating FWM is time-reversed (anticausal). This feature has already been noted for each constituent in various source-free synthesis options, and it is verified here directly, within the context of aperture radiation. The conclusion is that despite their appealing physical properties, these source-free wave fields, when well collimated so that their focusing properties are taken advantages of, are inherently unsuited as elements for direct and efficient modeling of time-dependent causal phenomena.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Solutions of the wave equations with boundary conditions on the hyperplane z−c t=0

The boundary value problem for the wave equation is discussed with data φ(r) on the hyperplane z−ct=0. We give the general form of the solution when φ(r) is an entire function of r2 of order 1 and type 0&amp;lt;τ&amp;lt;∞. As a particular case the focus wave mode solutions of the wave equation are obtained.

Some exotic solutions of the wave equation in unbounded isotropic media

We prove that the wave equation has two kinds of focus wave mode solutions with infinite energy. The splash wave solutions defined as a weighted superposition of the focus wave modes may have a finite energy.

Spinor fields with zero mass in unbounded isotropic media

The Dirac equation for massless fields in unbounded media has solutions similar to the focus wave mode solutions of Maxwell's equations leading to infinite dynamical invariants. We define the splash wave mode solutions as a weighted superposition of the focus wave modes, and discuss the conditions to be fulfilled by the weight functions to make the dynamical invariants bounded. We leave open the physical interpretation of these solutions.

Spectral analysis of focus wave modes

The focus wave mode (FWM), which is a time-dependent beam field that propagates at the speed of light without dispersion and retains its shape in space, is an interesting wave object with possible implications for synthesizing focused fields under transient conditions. To explore this potential, it is necessary to understand fully the properties of this wave field. It is already known that the FWM is a homogeneous solution of the wave equation, which is related in a special way to the field of a source moving on a complex trajectory parallel to the real axis of propagation. This suggests that there may be a connection between the FWM and the conventional free-space Green’s function. It is shown here that the FWM is related, in fact, to a source-free combination of causal and anticausal free-space Green’s functions and that one can formulate a bilateral transform pair relating these solutions. This new representation is then analyzed by using the spectral theory of transients to establish the properties of the FWM in terms of a distribution of transient plane waves. The spectral decomposition in the spatial wave-number domain reveals that the FWM is synthesized by both forward- and backward-propagating plane waves that are restricted to the visible spectrum. Asymptotic considerations show that the dominant mechanism is constructive interference of the backward-propagating waves. Taken together, the Green’s-function and spectral approaches grant further insight into the physical and spectral properties of the FWM. The conclusions cast doubt on the possibility of embedding the FWM within a causal excitation scheme.

Spinor focus wave modes

New solutions of the homogeneous spinor wave equation are obtained. They are similar to the focus wave mode solutions of Maxwell’s equations leading to a Gaussian pulse energy. A weighted superposition of these modes may supply finite energy pulses. The particular case of Bessel weight functions is discussed.

More on focus wave modes in Maxwell equations

New solitary electromagnetic wave solutions of Maxwell’s equations are obtained. Brittingham [J. Appl. Phys. 54, 1179 (1983)] calls them focus wave modes.

A general formulation of focus wave modes

A general formulation is given for electromagnetic pulses which remain localized in a multidimensional space, and which propagate at the speed of light without dispersing (focus wave modes). It is shown that such modes necessarily have infinite electromagnetic energy in the source-free, three-dimensional space. Finite-energy focus wave modes cannot exist without sources. A set of complete focus wave modes with Hermite–Gaussian transverse variation is derived. The relation of focus wave modes to the solutions of the paraxial wave equation is established.

Comment on ‘‘Focus wave modes in homogeneous Maxwell’s equations: Transverse electric mode’’

The recent formulation by Brittingham [J. Appl. Phys. 54, 1179 (1983)] of a solitary electromagnetic wave called a ‘‘focus wave mode’’ in terms of three regions with discontinuities is found not to satisfy Maxwell’s equations across the discontinuities.