Many phenomena observable in our everyday life indicate that
light propagates rectilinearly.
Rectilinear propagation is one of the most
apparent properties of light. It serves as an argument that
light is a stream of particles.
However, some optical phenomena and
experiments indicate that the law of rectilinear propagation of light
does not hold. They can be satisfactorily explained only on the assumption
that light is a wave .
Historically, the diffraction effects are associated
with violation of the rectilinear propagation of light.
The strong diffraction
effects appear if the transverse dimensions of the beam of light are
comparable to the wavelength. They are
best appreciable for long waves such as sound or water waves.
In optics, the diffraction effects are less apparent. They are responsible
for the beam divergence in the free propagation and for penetration
of light into the region of the geometric shadow.
In the modern treatment, diffraction effects are not connected with
light transmission through apertures and obstacles, only.
Diffraction is examined as a natural
property of wavefield with the nonhomogeneous transverse intensity
distribution. It commonly appears even if the beam is transversally
The Gaussian beam is the best known example.
In optics, nondiffracting propagation of the beam-like fields
can be obtained in convenient media such as waveguides or nonlinear
materials. The beams then propagate as waveguide modes and
spatial solitons, respectively. In 1987, the term
appeared also in relation to the free-space propagation.
Since that time, nondiffracting beams have been intensively investigated
from both the theoretical and experimental point of view.
A number of practical applications have also been proposed.
In present time, optics of nondiffracting beams represents
an important part of modern classical optics. The particular research
activities are directed to the following
Physical origin of the diffraction phenomena and concept
of the nondiffracting propagation.
Description and properties of the nondiffracting beam-like fields.
Experiments on the nondiffracting beams.
Physical applications of the nondiffracting
Technical applications of the nondiffracting beams.
The contemporary research of nondiffracting beams
follows up the previous research activities supported by
the Grant J14/98 of the Ministry of Education and by the Grant
202/00/0142 of the Grant Agency of the Czech Republic.
Attention is focused on both the theoretical and experimental
problems of the nondiffracting propagation and self-reconstruction
of light beams. The particular results can be classified into the
The nondiffracting beams exhibit interesting properties
important for applications. Some of them are briefly reviewed.
Energetic properties of the vectorial nondiffracting beams
can be described applying the Poynting vector P.
It can be decomposed into the transversal and longitudinal components
Though the intensity profile of the beam remains unchanged
during propagation the transversal energy flow can be in general
nonzero. It is restricted by the condition div PT
=0. This condition can be simply interpreted: The transversal energy flow
is source-free, or, equivalently, the lines of constant transversal energy
flow are closed curves. The transversal energy flow can be conveniently
analyzed as a superposition of the radial and azimuthal components
For the one-mode nondiffracting beam
The electromagnetic energy then flows along the direction of
propagation or has the spiral character.
During last decade
an increasing attention has been given to the wavefields possessing
the line, spiral or combined wavefront dislocations.
In optics, such fields are known as optical vortices.
Some types of nondiffracting beams can also exhibit the wavefront
singularities. The simplest nondiffracting vortices are
Their complex amplitude is given by
u=Jm exp[i m arctan(y/x) - kzz ],
where Jm is the m-th order
Bessel function of the first kind and m and kz
denotes the topological charge and the propagation
constant of the beam, respectively.
The helical wavefronts of the optical vortices with
topological charges 1 and 2 are illustrated in Fig.2a and 2b.
The phase singularities and wavefront dislocations can be identified
by the interferometric methods. The interference of the optical
vortex results in the spiral patterns. The numerical simulation
for the optical vortex with the topological charge m=1
is illustrated in Fig. 3a. The pattern obtained by
interference of the optical
vortex with the plane wave possesses the typical fork-like form.
It is shown in Fig. 3b for the topological charge m=1.
Transversality of the electric and magnetic fields
An ideal homogeneous electromagnetic plane wave with the wave vector
k is the electromagnetically transversal wave. Its transversality
follows directly from divE=0 and divH=0 and can be
expressed as k.E=0 and k.H=0.
The amplitude of the real beam is not homogeneous so that the pure
electromagnetic transversality cannot be achieved. Nevertheless,
the electromagnetic field of the common beams, such as Gaussian beams,
is nearly transversal.
On the contrary, the nondiffracting beams
can possess the longitudinal component of the electric field
comparable with the transversal one. This property is a result of the
special composition of the angular spectrum of the nondiffracting beams
and can be applied to design of the electron accelerators.
Robustness of the nondiffracting beam
An important property of the nondiffracting beam is its
resistance against amplitude and phase distortions.
The transverse intensity profile of the nondiffracting beam
disturbed by the nontransparent obstacle regenerates during
free-propagation behind the obstacle. After certain distance,
the initial transverse intensity profile is restored.
The theoretical explanation of the effect based on application of
Babinet's principle has been proposed and its experimental
verification has been realized. The results of the experiment
are presented in Fig.4. The pseudo-nondiffracting beam with
the transverse intensity profile approximately corresponding to the
zero-order Bessel function J0 is disturbed by the
nontransparent rectangular obstacle (Fig. 4a). During free-propagation
of the beam behind the obstacle its transverse intensity profile
regenerates (Fig. 4b-d). As is obvious from Fig. 4d, the initial
Bessel-like profile is restored with a very good fidelity.
The angular spectrum of the ideal nondiffracting beam consists of
the single radial frequency f0. It can be related to the
propagation constant kz of the beam. The complex
amplitude of the nondiffracting beam then can be written as
u(x,y,z)=U(x,y)exp(-ikzz). The various nondiffracting
modes can propagate with different propagation constants.
appears in the coherent superposition of the nondiffracting modes
if their propagation constants are conveniently coupled.
The effect represents the spatial analog of the mode-locking realized in
the temporal domain. Due to the interference of the modes, the
transverse intensity profile of the beam reappears periodically
at the planes of the constructive interference and vanishes at
the planes of the destructive interference. The intensity distribution
of the two-mode field propagating along the z-axis is illustrated
in Fig. 5.
In recent time it has been shown
that the coherent light field with an arbitrary
transverse amplitude profile can be transformed
into the state possessing an ability of the periodical reconstruction.
The transverse intensity profile appearing at the planes of reconstruction
then represents a very good approximation of the initial intensity
profile. The transformation required for appearance of the
self-reconstruction effect is based on the
decomposition of the initial field into the set of nondiffracting modes with
conveniently chosen propagation constants. It can be simply realized in the
4-f optical system (Fig. 7)
applying the convenient Fourier filter. It is realized as an amplitude
mask transparent only at the set of concentric annular rings.
The field to be periodicly reconstructed is decomposed into the set
of nondiffracting modes if its spatial spectrum is modified by the mask
at the Fourier plane. The proposed method was verified
experimentally. The obtained results are illustrated
in Fig. 8.
The concept of the nondiffracting propagation of the stationary
fields can also be generalized to the pulse propagation.
The wideband pulslike nondiffracting wave packets are of
particular interest for their applications in the ultrasound
medical diagnostics. Optical applications of the
nondiffracting pulses have also attracted attention.
The dispersive temporal spread of ultrashort pulses complicates
their application to the femtosecond spectroscopy and related
fields. During propagation of the nondiffracting pulses
the material dispersion can be eliminated due to the convenient
composition of the spatio-temporal spectrum. It is formed by means of
the generator whose dispersion is adapted to the material dispersion
of the propagation medium. The nondiffracting pulses can be generated
applying the holographic element, the so-called Lensacon.
The required couplings of the spatial- and temporal-frequency spectra
resulting in the vectorial nondiffracting pulses
have also been analysed. This topic is open to further research.
The self-reconstruction concept applied to the monochromatic field
has been generalized to the
spatio-temporal self-reconstruction of the
wave packets fulfilling the wave equation. The effect is based on
the sampling of the spatial spectrum of the wave function and on its
coupling to the temporal frequency spectrum. The predetermined
spatio-temporal profile of the wave function is approximately
reconstructed at the periodic spatial intervals along the propagation
direction. If the temporal frequency spectrum of the field is not
continuous but discrete, the periodicity appearing in both
the spatial and temporal evolutions can be achieved.
The reconstructed field then simultaneously exhibits propagation
properties known as the self-imaging and the mode-locking.
The nondiffracting light propagation is an actual problem
of modern classical optics. It is not only important
for better understanding of the nature of the electromagnetic
field and of the diffraction effects but has also important
consequences for both the physical and technical applications.
The nondiffracting beams are considered to be used in
experiments testing the transfer of the light angular momentum to
the nano-particles, in electron accelerators and in experiments
of nonlinear optics. They were successfully used in acoustics
and metrology. At RCO, the particular attention will be focused
on the following theoretical and experimental problems:
Application of the uncertainty relations and the ambiguity
function to the pseudo-nondiffracting beams.
Theoretical description and experimental realization of the
Analysis of the propagation properties and of the spatio-temporal
couplings of the nondiffracting pulses.
Application of the nondiffracting modes to the direct image
transmission in optical fibers.
The transverse intensity profiles of the nondiffracting beams
depend on their spatial coherence. Numerical simulation of the
nondiffracting beams generated by means of the Gauss-Shell source
illustrates the intensity profiles for the degree of spatial
coherence changing from 1 (left upper figure) to 0 (right bottom figure).
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