Anomalous behavior of spectra near phase singularities of focused waves

Evolution behavior of the coherent vortex dipole in oceanic turbulence

The propagation of a Gaussian Schell-model (GSM) beam with a coherent vortex dipole (CVD) through oceanic turbulence is studied in detail. The creation and annihilation of the CVDs occur with propagation, which is similar to the case of atmospheric turbulence. However, the appearance and vanishment of a coherent vortex may occur by varying the propagation distance, oceanic turbulence parameters, or beam parameters, which is different from the case of atmospheric turbulence. The stronger the oceanic turbulence, the faster the evolution process of the CVD carried by GSM beams.

Spectral switch anomalies in a Sagnac interferometer with respect to a Galilean frame

We report the spectral switch shift around spectral anomalies in a gyroscopic Sagnac interferometer, which is normally used to calibrate the angular momentum of a gyroscope. The spectral shift in the rotating gyroscope is explained with respect to the longitudinal Doppler shift of the counterpropagating beams in the Sagnac interferometer.

Ultrashort Vortex Pulses with Controlled Spectral Gouy Rotation

Recently, the spatio-spectral propagation dynamic of ultrashort-pulsed vortex beams was demonstrated by 2D mapping of spectral moments. The rotation of characteristic anomalies, so-called “spectral eyes”, was explained by wavelength-dependent Gouy phase shift. Controlling of this spectral rotation is essential for specific applications, e.g., communication and processing. Here, we report on advanced concepts for spectral rotational control and related first-proof-of-principle experiments. The speed of rotation of spectral eyes during propagation is shown to be essentially determined by angular and spectral parameters. The performance of fixed diffractive optical elements (DOE) and programmable liquid-crystal-on silicon spatial light modulators (LCoS-SLMs) that act as spiral phase gratings (SPG) or spiral phase plates (SPP) is compared. The approach is extended to radially chirped SPGs inducing axially variable angular velocity. The generation of time-dependent orbital angular momentum (self-torque) by superimposing multiple vortex pulses is proposed.

Simple Wave-Optical Superpositions as Prime Number Sieves

We encode the sequence of prime numbers into simple superpositions of identical waves, mimicking the archetypal prime number sieve of Eratosthenes. The primes are identified as zeros accompanied by phase singularities in a physically generated wave field for integer valued momenta. Similarly, primes are encoded in the diffraction pattern from a simple single aperture and in the harmonics of a single vibrating resonator. Further, diffraction physics connections to number theory reveal how to encode all Gaussian primes, twin primes, and how to construct wave fields with amplitudes equal to the divisor function at integer spatial frequencies. Remarkably, all of these basic diffraction phenomena reveal that the naturally irregular sequence of primes can arise from trivially ordered wave superpositions.

Self-imaging of tailored vortex pulse arrays and spectral Gouy rotation echoes

Self-imaging of femtosecond pulses with orbital angular momentum is studied in spectral domain by illuminating the orthogonal arrays of spiral gratings. Spectral Gouy rotation, i.e., the characteristic circulation of extremal regions near phase singularities in spatial spectral maps, is found to partially reappear at discrete distances. The self-imaging of co-rotating and counter-rotating vortices is compared in intensity and spectral behavior. High-selectivity pattern recognition from weakly modulated spectral maps is demonstrated by analyzing spectral moments. By our experiments, the classical Talbot effect is extended to polychromatic pulsed vortex arrays with controlled maps of rotation sign.

Spectral anomalies and Gouy rotation around the singularity of ultrashort vortex pulses

Spectral anomalies of femtosecond pulses with orbital angular momentum were studied in the vicinity of singularities. Bessel-Gauss (BG) beams were generated with mode-locked Ti:sapphire oscillators and dispersion-compensated diffractive axicons acting as spiral phase plates (SPPs). High-resolution two-dimensional spectral mapping was performed with a scanning fiber probe. Progressive rotation of the most pronounced features, known as "spectral eyes", in the maps of spectral moments was found at increasing propagation distance. The phenomenon is explained by a wavelength-dependent Gouy phase shift of interfering spectral components in the twisted wavefront. Spatial "spectral switching" was detected for few-cycle pulses. Possible improvements of selectivity are proposed.

Spatial-spectral (space-wavenumber) correspondence relationship and Fresnel zone spectra

A correspondence relationship between space and wavenumber for fully spatially coherent uniform monochromatic and polychromatic light in near- and far-field diffraction is fully illustrated, and it is used to study a phenomenon called the Fresnel zone spectra, which is verified experimentally. The spectra can be filtered or manipulated by moving the detection position relative to Fresnel zone plate along the optical axis.

Experimental observations of spectral changes produced by individual microscopic spheres

Spectral changes in individual micrometer-sized scatters are experimentally measured. Using an interferometric microscope equipped with a wavelength-swept illumination, the optical fields diffracted from the individual scatters are measured precisely, from which 2D light scattering spectra were retrieved as functions of the wavelength and scattering angle. In the measured scattering spectra of individual scatters, spectral shifts were clearly observed, which is also directly explained using the Mie scattering theory.

Spectral anomalies by superposition of polychromatic Gaussian beam and Gaussian vortex beam

We study the spectral property of a composite field superposed by a polychromatic Gaussian beam and a polychromatic Gaussian beam with an embedded mth-order vortex. It is shown that, in the overall spectral shift distribution, there exist m small areas where sharp spectral anomaly takes place, similarly and respectively, which are related with the ratio of the respective amplitudes of the two composite beams and the relative phase between them. Detailed investigation reveals that, for each small area, there exists a "main line", along which spectral switch can be observed.

Experimental observation of the effect of generic singularities in polychromatic dark hollow beams

This Letter presents the essence of our recent experimental study on generic singularities carrying spatially partially coherent, polychromatic dark hollow beams (PDHBs). To the best of our knowledge, this is the first experimental demonstration of generic singularities-induced wavefront tearing in focused polychromatic beams.

Influence of generalized focusing of few-cycle Gaussian pulses in attosecond pulse generation

In contrast to the case of quasi-monochromatic waves, a focused optical pulse in the few-cycle limit may exhibit two independent curved wavefronts, associated with phase and group retardations, respectively. Focusing optical elements will generally affect these two wavefronts differently, thus leading to very different behavior of the pulse near focus. As limiting cases, we consider an ideal diffractive lens introducing only phase retardations and a perfect non-dispersive refractive lens (or a curved mirror) introducing equal phase and group retardations. We study the resulting diffraction effects on the pulse, finding both strong deformations of the pulse shape and shifts in the spectrum. We then show how important these effects can be in highly nonlinear optics, by studying their role in attosecond pulse generation. In particular, the focusing effects are found to affect substantially the generation of isolated attosecond pulses in gases from few-cycle fundamental optical fields.

Spectral shifts and spectral switches produced by the scattering system of two anisotropic particles in different distance

We investigate the scattering of polychromatic plane light wave incident upon a system formed with two anisotropic particles in different distance. The analytical expression for the spectrum of the scattered field is derived. Numerical examples show the phenomena of spectral shifts and spectral switches of the scattered field. The influences of the scattering direction and the difference of the particles on the spectral switch are illustrated.

Spectral coherence anomalies

We describe the anomalous behavior of the spectral degree of coherence as a function of light frequency in the vicinity of phase singularities of partially spatially coherent, polychromatic wave fields. We distinguish the discovered spectral coherence anomalies from conventional spectral anomalies realized with fully spatially coherent optical fields. We also demonstrate how the previously reported spectral anomalies can be engineered in partially spatially coherent fields.

Spectral changes of stochastic beams scattered on a deterministic medium

It is well known that scattering of a polychromatic plane wave by a random medium, i.e., by a medium whose refractive index varies randomly with position, may produce frequency shifts of spectral lines. It has been a common perception that a random medium is required for generation of such spectral shifts by scattering. In this Letter we show that such a phenomenon occurs even when the refractive index of the medium is a deterministic function of position. We also show that this phenomenon may be used to obtain information about the structure of a deterministic medium.

All optical spectral switches

It is shown theoretically that the nonlinear optical Kerr effect can be used to build an all optical configuration controlling spectral switches. The main advantage in it is that it can greatly simplify the control method compared to previous schemes using the aperture mechanism or electro-optical one.

Tunable spectral shifts and spectral switches by controllable phase modulation

In this work, we offer a novel and flexible approach of spectral switches which can be handled more simply by controlling the phase of the diffracted light field of a completely spatially coherent incident beam with spectral profile from a one-dimensional phase step. This scheme has the benefit of easy implementation by simply varying the height of a one-dimensional phase step which causes spectral switches to occur when the step height reaches certain critical values without modulating any properties of the light source. To illustrate this effect, an explicit and analytical expression at an observation point corresponding to the step edge is obtained and some numerical examples are given and examined experimentally. Finally, based on the obtained results, it is shown that this method with the capability of very short response time can be easily applied to information encoding and transmission.

Phase singularities and spectral changes of spectrally partially coherent higher-order Bessel-Gauss pulsed beams

Phase singularities and spectral changes of spectrally partially coherent higher-order Bessel-Gauss pulsed beams in free-space propagation are studied, and the analytical expressions for the spectral degree of coherence and spectrum are derived. It is shown that there always exists an optical vortex at the center of the beam and the topological charge is conserved during the propagation. Some circular edge dislocations appear and the spectrum changes on propagation. A rapid spectral transition takes place. Numerical illustrative examples are given and the results are explained physically.

Enhanced degree of temporal coherence through temporal and spatial phase coupling within a focused supercontinuum

In the diffraction of a supercontinuum source, a redistribution of amplitude and phase at the focal region is incurred by the coupling between the supercontinuum and the spatial phase caused by the lens diffraction, making it extremely difficult to predict the focal behaviour. We show that the coupling between the temporal phase of a SC source and the spatial phase from the diffraction by a low numerical aperture (NA) lens causes dramatic alterations in the spectra and the temporal coherence near the focal region, and that this effect is maximized in points of singularity. Furthermore, we show that such an enhancement in temporal coherence can be controlled by the pulse evolution through the photonic crystal fiber, in which nonlinear and disperive effects such as the soliton fission process provides the key phase evolution necessary for dramatically changing the coherence time of the focused electromagnetic wave.

Redshift and blueshift in the spectra of lights coherently and diffusely scattered from random rough interfaces

We show theoretically and experimentally that the spectrum of coherently scattered light from a randomly rough interface in reflection and transmission is redshifted with a shrinkage in spectral width. In reflection mode the amounts of the redshift and the shrinkage depend on interface roughness, incident angle, and the spectral width of the illuminating light. In transmission mode they also depend on the refractive indices of the surrounding media. The redshift and width shrinkage increase with decrease of the coherently scattered light intensity. This study shows that the spectrum of the diffusely scattered light is blueshifted in the specular direction and in directions with small scattering angles only in situations with appreciable intensity of the coherently scattered light. With decrease of the latter intensity the blueshift reduces and turns into redshift. Also, the redshift and blueshift decay with increase of the scattering angle. An experimental investigation has been carried out, on sheet glasses with different roughness on one side, in reflection and transmission modes. The experimental results and theoretical predictions are quite consistent.

Lattice spectroscopy

It is shown that the structure information for a lattice or periodic pattern can be obtained by measuring the diffracted spectrum of a broadband light source at one spatial position in the far field.

Optical diffractometry

Interference of light has numerous metrological applications because the optical path difference (OPD) can be varied at will between the interfering waves in the interferometers. We show how one can desirably change the optical path difference in diffraction. This leads to many novel and interesting metrological applications including high-precision measurements of displacement, phase change, refractive index profile, temperature gradient, diffusion coefficient, and coherence parameters, to name only a few. The subject fundamentally differs from interferometry in the sense that in the latter the measurement criterion is the change in intensity or fringe location, while in the former the criterion is the change in the visibility of fringes with an already known intensity profile. The visibility can vary from zero to one as the OPD changes by a half-wave. Therefore, measurements with the accuracy of a few nanometers are quite feasible. Also, the possibility of changing the OPD in diffraction allows us to use Fresnel diffraction in Fourier spectrometry, to enhance or suppress diffracted fields, and to build phase singularities that have many novel and useful applications.

Spectroscopic properties unique to nano-emitters

The spectral position of light emission from an individual carbon nanotube is shown to depend on the location of the nanotube within the focal spot, while no such effect is present for macroscopic emitters. In addition, in contrast to macroscopic emitters, the measured line width from the nanotube emitter is independent of spectrometer entrance slit width. The effects are general for any nanoscale optical emitter with at least one dimension smaller than the optical diffraction limit.

Information exchange in free space using spectral switching of diffracted polychromatic light: possibilities and limitations

Spectral switching is now a well-known phenomenon. Several research groups have studied it theoretically and experimentally for different optical systems. Recently, its potential applications have been demonstrated for information encoding and transmission in free space. In this paper we report experimental observations in the far field for spectral switching of polychromatic light passing through an astigmatic aperture lens. An information encoding scheme, a free-space optical link, and related boundary conditions are studied in detail. In addition, on the basis of experimental and theoretical studies carried out so far for spectral switching, we explore possibilities of information transmission in free space and analyze experimental limitations of spectral-switching-based free-space optical communication.

Determination of height distribution on a rough interface by measuring the coherently transmitted or reflected light intensity

In this work it is shown theoretically and examined experimentally that the measurement of coherently transmitted or reflected monochromatic light intensity from a randomly rough interface as a function of incident angle provides the height distribution on the interface. It is also shown that the spectrum of coherently transmitted or reflected light from a rough interface is modified and the modified spectrum yields the height distribution. The experimental results obtained by applying both methods, in transmission and reflection, on rough surfaces prepared by roughening the sheet glasses by powders of different grain sizes are quite consistent. In addition, the effect of the surrounding medium's refractive index on the roughness measurement is studied by immersing the samples into liquids of different refractive indices. Also, the application range and limitations of the introduced methods are discussed.

Spectral anomalies near phase singularities in reflection at Brewster's angle and colored castastrophes

The anomalous behavior of a polychromatic beam reflected from the interface of two homogeneous dielectric media at the neighborhood of Brewster's angle is investigated theoretically and examined experimentally. An explicit expression is derived for the spectrum at an arbitrary observation point for a given incident spectrum. By introducing a modifier function it is shown that the spectrum at each point sensitively depends on the observation angle just in the immediate neighborhood of Brewster's angle and wavelength. Finally, the obtained results are applied to account for the horizontal parts of the supernumerary rainbows.

Spectral anomalies due to temporal correlation in a white-light interferometer

We present what we believe to be the first experimental demonstration of anomalous spectral behavior such as spectral shifts and spectral switches due to temporal correlation around the intensity minima in a white-light interferometer. Unusual behavior in the number of spectral fringes, measured within the source bandwidth, as a function of path delay between the interfering beams is also reported. Experimental observations match well with the spectra calculated by using the interference law in the spectral domain.

Spatial correlation properties of focused partially coherent vortex beams

The spatial correlation properties in the geometrical focal region of a converging, partially coherent vortex wave field are analyzed. Expressions are derived for a pair of points on the axis of symmetry and for a pair of points in the focal plane. It is found that the longitudinal and transverse coherence lengths in the focal region change with the variation of the topological charge and the normalized coherence length of the vortex field. In addition, the degree of coherence is shown to possess phase singularities.

Polychromatic vectorial vortex formed by geometric phase elements

We propose the use of a geometric phase, obtained by spatial polarization state manipulations, for the formation of polychromatic vectorial vortices. Experimental demonstration is obtained by using Pancharatnam-Berry phase optical elements formed by a space-variant subwavelength grating etched on a GaAs wafer. We further demonstrate formation of scalar and unpolarized polychromatic vortices.

Debye representation of dispersive focused waves

We report a matrix-based diffraction integral that evaluates the focal field of any diffraction-limited axisymmetric complex system in the paraxial regime. This diffraction formula is a generalization of the Debye integral, here accommodated to broadband problems. The Fresnel number is reformulated to guarantee that the focal region is entirely within the region of validity of the Debye approximation when this parameter largely exceeds unity. Several examples are examined in detail, one of them exhibiting in-focal-plane compensation of the spatial dispersion. This simple formalism opens the door for the analysis and design of focused beams with arbitrary angular dispersion.

Temporal effects in ultrashort pulsed beams focused by planar diffracting elements

The pulse envelope of an ultrashort pulsed beam is evaluated on the focal points of a Fresnel zone plate. The description of the field dynamics is given in terms of a diffraction-induced pulse train. Within these terms we follow an analytical procedure to characterize the temporal broadening observed at the principal focus, which is significant if the number of Fresnel zones exceeds the number of cycles in the pulse. For Gaussian-type envelopes, the focal field may be accurately expressed in a simple closed form. This expression has a flat-top shape at the principal focus and other odd-order foci, and a two-peak envelope in the case of a low-integer even-order focus. Finally, extremely high orders present a time-domain evolution that emulates a train of uniform pulses with temporal characteristics equivalent to those of the incident beam.

Achromatic optical vortex lens

A vortex lens is a useful optical device having applications ranging from astronomy to microscopy. Current vortex masks operate across a narrow bandwidth. Two design schemes are proposed for creating a vortex across a bandwidth exceeding 100 nm in the visible region of the spectrum.

Spectral density of polychromatic electromagnetic waves

We investigate theoretically how the vectorial nature of polychromatic electromagnetic fields results in polarization components with different spectral characteristics, thus leading to redshifts, blueshifts, and spectral distributions with multiple peaks. We discuss how these effects can be used to design spatially localized spectra with tailored spectral densities.

Strongly focused polarized light pulse

We investigate theoretically the electric field of a focused light pulse carrying an inhomogeneous polarization distribution. It is found that the spectra of the polarization components are in general different, thus leading to a spatial spectral distribution that differs from the scalar case.

Broadband nulling of a vortex phase mask

A pulse transmitted through a helical vortex phase mask undergoes a temporal Hilbert transform. The fluence transmitted into the unfavorable plane wave mode is found to increase as the square of the bandwidth and, to first order, is independent of the topological charge.

Spectral anomalies in focused waves of different Fresnel numbers

Light propagation induces remarkable changes in the spectrum of focused diffracted beams. We show that spectral changes take place in the vicinity of phase singularities in the focal region of spatially coherent, polychromatic spherical waves of different Fresnel numbers. Instead of the Debye formulation, we use the Kirchhoff integral to evaluate the focal field accurately. We find that as a result of a decrease in the Fresnel number, some cylindrical spectral switches are geometrically transformed into conical spectral switches.

Spectral switches of partially coherent light focused by a filter-lens system with chromatic aberration

It is shown that when partially coherent polychromatic light is focused by a filter-lens system with chromatic aberration, a spectral shift exists in the focused field, and a spectral switch that is defined as a sharp transition of the spectral shift also takes place at some positions of the focused field. The influence of the chromatic aberration of the lens, the coherence of the partially coherent light in the filter (a circular aperture), the radius of the aperture, and the spectral width of the partially coherent light in the aperture on the spectral shift and the spectral switch are investigated in detail. The numerical results show that these parameters affect the spectral shift and the spectral switch significantly. Potential applications of the spectral shift and the spectral switch of the partially coherent light are discussed.

Spectral changes and spectral switches of partially coherent beams focused by an aperture lens

Starting from the propagation law of partially coherent polychromatic light in the space-frequency domain, detailed numerical results and physical analysis are given to elucidate spectral changes and spectral switches at the geometrical focal plane of Gaussian Schell-model beams focused by an aperture lens. It is found that, in contrast to the aperture-induced spectral anomalies of spatially fully coherent polychromatic light, for partially coherent polychromatic light aperture diffraction plays an important role in spectral switching, but the truncation parameter, spectral correlation, and bandwidth all affect its spectral behavior.

Effect of spectral correlations on spectral switches in the diffraction of partially coherent light

The subject is the spectral characteristics of partially coherent light whose spectral degree of coherence satisfies or violates the scaling law in diffraction by a circular aperture. Three kinds of spectral correlations of the incident light are considered. It is shown that no matter whether the partially coherent light satisfies or violates the scaling law, a spectral switch defined as a rapid transition of spectral shifts is always found in the diffraction field. Different spectral correlations of the incident field in the aperture result in different points at which the spectral switch occurs. With an increment in the correlations, the position at which the spectral switch takes place moves toward the point at which the phase of the center frequency component omega0 becomes singular for illumination by spatially fully coherent light. For light that satisfies the scaling law, the spectral switch is attributed to the diffraction-induced spectral changes; for partially coherent light that violates the scaling law, the spectral switch is attributed to both the diffraction-induced spectral changes and the correlation-induced spectral changes.

Lissajous singularities

Coherent optical Lissajous states are easily created by nonlinear processes such as second-harmonic generation (SHG). Singular properties of such states are discussed and illustrated theoretically with non-phase-matched SHG of an ellipse field containing a C point.

Phenomenon of spectral switches as a new effect in singular optics with polychromatic light

It is shown that the recently discovered phenomenon of so-called spectral switches has a natural interpretation in the framework of singular optics with polychromatic light and that it should be regarded as being primarily a manifestation of diffraction-induced spectral changes rather than correlation-induced spectral changes as was suggested in the original papers [the first one appearing in Opt. Commun. 162, 57 (1999)] reporting this effect.

Singular behavior of the spectrum in the neighborhood of focus

In a recent paper [Phys. Rev. Lett. 88, 013901 (2002)] it was shown that when a convergent spatially coherent polychromatic wave is diffracted at an aperture, remarkable spectral changes take place on axis in the neighborhood of certain points near the geometrical focus. In particular, it was shown that the spectrum is red-shifted at some points, blueshifted at others, and split into two lines elsewhere. In the present paper we extend the analysis and show that similar changes take place in the focal plane, in the neighborhood of the dark rings of the Airy pattern.

Spectral anomalies in a Fraunhofer diffraction pattern

We show that spectacular spectral changes take place in the vicinity of the dark rings of the Airy pattern formed with spatially coherent, polychromatic light diffracted at a circular aperture.

Spatial coherence and information entropy in optical vortex fields

We show how a vortex structure manifests itself in the one-dimensional projection of a vortex field. We calculate the extent of spatial coherence and entropy of such projections. We quantify the spatial coherence and discuss the properties of the Wigner functions for the projected field.

Spectral anomalies at wave-front dislocations

In a recent Letter, Gbur, Visser, and Wolf [Phys. Rev. Lett. 88, 013901 (2002)] predict that remarkable spectral changes can take place in the neighborhood of phase singularities of a diffracted focused wave. We report here the experimental observation of this anomalous spectral behavior and show that this is a general characteristic of optical vortices. Using a high-resolution interferometric technique, we are able to measure directly both the spectral intensity distribution and the spectral phase at the singular points of optical waves.