Figure 3 - uploaded by Konstantin Bliokh
Content may be subject to copyright.
The three types of optical AM for paraxial light: (i) spin, (ii) intrinsic orbital, and (iii) extrinsic orbital AM. They are associated with circular polarization, an optical vortex, and the motion of the field centroid, respectively. The quantum number σ = ± 1 indicates the helicity of the right-hand and left-hand circular polarizations, while = 0, ± 1, ± 2, . . . indicates the charge of optical vortex. The equations show the normalized values of the AM per photon (we use units h ̄ = 1), where R stands for the radius-vector of the centroid of the optical field, P is the mean momentum (wavevector), and K = P / P is the direction of propagation, see equations (2.21) and (2.23).
Source publication
We consider reflection and transmission of polarized paraxial light
beams at a plane dielectric interface. The field transformations taking
into account a finite beam width are described based on the plane-wave
representation and geometric rotations. Using geometrical-optics
coordinate frames accompanying the beams, we construct an effective
Jones...
Contexts in source publication
Context 1
... K = u z is the direction of propagation of the beam and we took into account that the spin operator in the basis of circular polarizations (2.5) is ( S ˆ ) C = V ˆ S ˆ V ˆ † . Equations (2.23) represent well-known results for the intrinsic orbital and spin AM of the paraxial vortex beams [57, 58] (see figure 3). It is easy to see that beams (2.22) are eigenmodes of the L ˆ z and ( S ˆ z ) C operators with the discrete eigenvalues and σ . Indeed, these operators ...
Context 2
... there are three types of the AM of light: (i) spin, (ii) intrinsic orbital, and (iii) extrinsic orbital AM. For the locally paraxial fields they are associated, respectively, with circular polarization, optical vortices, and transverse beam shifts, figure 3. If the medium possesses rotational symmetry about a certain axis, the corresponding component of the total AM, L + S = L int + L ext + S , is conserved upon evolution of light. However, mutual conversion between different parts of the AM is possible, which signals the spin–orbit (or orbit–orbit) interaction of light. In a similar way, if the medium is stationary and possesses translational symmetry about a certain axis, the energy W and corresponding component of the momentum P must be conserved upon light evolution. Considering paraxial beams propagating along the z -axis, θ 1, we neglect small longitudinal z -components of the field and deal with the transverse ( x , y ) -components denoted by | E ̃ ) ⊥ . As an example, let us consider paraxial circularly polarized Laguerre–Gaussian vortex beams with the azimuthal quantum number = 0, ± 1, ± 2, . . . and radial quantum number p = 0 [58]. The electric field in the laboratory basis of circular polarizations can be written ...
Similar publications
In this work, an explicit formula is deduced for identifying the orbital
angular moment (OAM) of vectorial vortex. Different to scalar vortex, the OAM
of vectorial vortex can be attributed to two parts: the spiral spatial phase
defined by the Pancharatnam connection and geometrical phase induced by the
space-variant state of polarization. With our...
Citations
... Over the past few decades, the photonic spin Hall effect (PSHE) has been a heated topic in modern optics [1][2][3][4][5][6][7][8]. In 2013, Zayats et al. observed a new type of PSHE where the unidirectional propagation of surface plasmon polaritons (SPPs) can be achieved by modulating the polarization of the incident beam [9]. ...
In this paper, we introduce a new method to realize the separation of waveguide modes in photonic crystal waveguides by controlling the position and polarization of the dipole sources. Our study shows that the waveguide modes in the transversal channel and longitudinal channel can be manipulated, respectively, by the dipole sources placed within the corresponding channels. Based on this discovery, we may further adjust the dipole sources in the transversal (longitudinal) channel individually and separate the zeroth order and the first order modes to propagate towards the preassigned channels. Compared with the former schemes to realize the mode separation by changing the structure, our method is more intuitive, convenient, and flexible.
... A comprehensive understanding of Gaussian optical beams is crucial for designing optical systems and harnessing light for a wide range of practical and diversified purposes. Studying the propagation of Gaussian lasers through an optical prism enables the comprehension of optical phenomena, such as angular deviations [7][8][9][10] from the reflection and Snell laws, lateral shifts [11][12][13][14][15] known as Goos-Hänchen shifts-named in honor of the German physicists who first reported experimental evidence in 1947 [16][17][18]-and, last but not least, the phenomenon of transverse symmetry breaking. An additional advantage lies in the swift and cost-effective realization of experiments confirming these optical phenomena [19][20][21]. ...
... Extending the Taylor series expansion of the optical phase to the second order, along with the first-order expansion of the Fresnel coefficients conducted in Section 2, also enables us to obtain integrand functions, see Equations (40), (47) and (54), that can now be analytically solved. This provides the opportunity to rewrite the reflected and transmitted beams as functions of the incident electric field given in Equation (13). The ability to represent the reflected and transmitted fields through analytical formulas allows for an immediate understanding of the terms responsible for phenomena such as angular deviations, lateral displacements, and the breakdown of transverse symmetry. ...
The study of a Gaussian laser beam interacting with an optical prism, both through reflection and transmission, provides a technical tool to examine deviations from the optical path as dictated by geometric optics principles. These deviations encompass alterations in the reflection and refraction angles, as predicted by the reflection and Snell laws, along with lateral displacements in the case of total internal reflection. The analysis of the angular distributions of both the reflected and transmitted beams allows us to understand the underlying causes of these deviations and displacements, and it aids in formulating analytic expressions that are capable of characterizing these optical phenomena. The study also extends to the examination of transverse symmetry breaking, which is a phenomenon observed in the laser beam as it traverses the oblique interface of the prism. It is essential to underscore that this analytical overview does not strive to function as an exhaustive literature review of these optical phenomena. Instead, its primary objective is to provide a comprehensive and self-referential treatment, as well as give universal analytical formulas intended to facilitate experimental validations or applications in various technological contexts.
... Important manifestations of the intrinsic coupling between the spatial and temporal properties of STOVs come to light in the processes of their reflection and refraction at a flat isotropic interface between two media. In this situation, in addition to the conventional Goos-Hänchen and Imbert-Fedorov shifts [107], a number of new spatial shifts and time delays are found, which are controlled by the value and orientation of the intrinsic optical AM [97]. In this approach, due to the special combination of spatial and temporal degrees of freedom in space-time vortices, time delays and spatial shifts occur without the frequency dependence of the reflection/ refraction coefficients, and the "slow" and "fast" propagation of pulses can be realized without the medium dispersion. ...
The review describes the principles and examples of practical realization of diagnostic approaches based on the coherence theory, optical singularities and interference techniques. The presentation is based on the unified correlation-optics and coherence-theory concepts. The applications of general principles are demonstrated by several examples including the study of inhomogeneities and fluctuations in water solutions and methods for sensitive diagnostics of random phase objects (e.g., rough surfaces). The specific manifestations of the correlation-optics paradigms are illustrated in applications to non-monochromatic fields structured both in space and time. For such fields, the transient patterns of the internal energy flows (Poynting vector distribution) and transient states of polarization are described. The single-shot spectral interference is analyzed as a version of the correlation-optics approach adapted to ultra-short light pulses. As a characteristic example of such pulses, uniting the spatio-temporal and singular properties, the spatio-temporal optical vortices are considered in detail; their properties, methods of generation, diagnostics, and possible applications are exposed and characterized. Prospects of further research and applications are discussed.
... In fact, as shown in Appendix D, achieving this level of roundness also requires allowing the STOV to shift laterally by an amount x 0 = ∓λ 0 /14π. This shift of a fraction of a wavelength can be regarded as a spin-orbit effect, similar to other spin-induced transverse shifts in optical beams [48][49][50]. ...
Spatiotemporal optical vortices (STOVs) are short pulses that present a vortex whose axis is perpendicular to the main propagation direction. We present analytic expressions for these pulses that satisfy exactly Maxwell's equation, by applying appropriate differential operators to complex focus pulses with Poisson-like frequency spectrum. We also provide a simple ray picture for understanding the deformation of these pulses under propagation. Finally, we use these solutions to propose a type of pulse with sagittal skyrmionic polarization distribution covering all states of transverse polarization.
... It establishes the coordinates of the beam's center of mass based on the Snell-Fresnel formula and the effective Jones matrix [23]. Consequently, the GH shift is ascertained by the equation [24]: ...
hBN is a natural hyperbolic van der Waals material that can enhance light–matter interactions. In this work, the maximum Goos-Hanchen(GH) shift has a magnitude of −4680 μm with high reflection using the central beam method in the trapezoidal dielectric/hBN grating (TDG) metasurface (MS) where the beam waist w0≥4λ. Analysis of the electromagnetic field distribution indicated that the amplified GH shift was a result of guided mode resonances excited in the waveguide dielectric layer. The frequency and magnitude of GH shifts can be effectively controlled by adjusting the height of the trapezoidal hBN, as well as by manipulating the widths of the top and bottom dielectric layers, the period of the TDG structure, and the height of the multi-layered structure. Moreover, an evaluation of the structure-sensing properties based on the GH shift was conducted. The enhanced and controllable GH shift exhibited by the TDG MS presents promising prospects for applications in optical sensors, optical switches, and optoelectronic detectors.
... IFS usually occurs when an elliptically polarized or circularly polarized beam is incident [21] and has broad utility across various fields, including optical microscopy [22], plasmonics [23], meteorology [24], photonic crystals [25], and metal optics [26]. In addition, IFS also has been reported for various structures such as graphene plasmonics metasurfaces [27], optical beam shifts in graphene [28], metamaterials [29], and at the interface of several non-linear media [30] at the NID chiral-chiral planar interface [31], homogeneous media such as metallic [32], and lossless and lossy dielectric [33]. ...
... Every single radiation of the circularly polarized beam can be assumed made up of two types of waves: TE-wave and TMwave. The non-specular reflection effects are unaffected by the profile of the beam calculated using the Fresnel reflection coefficients, derived from central plane waves in the incident and reflected beams [21]. The central plane waves always have electromagnetic fields that contain a factor exp j k x x + j k i z− j ωt . ...
... The first part of the equation deals with intraband photon-electron scattering processes, while the second part represents electron transitions in the interband. Based on established findings from [21,25], the GHS of TE/TM-waves is determined using ...
In this work, we have theoretically investigated the tunable non-specular effects, specifically Imbert-Fedorov and Goos-Hänchen shifts, on a graphene-based uniaxial hyperbolic crystal geometry. The hyperbolic crystal is made of hexagonal boron nitride and the source of excitation is a circularly polarized light beam. The influence of chemical potential and absolute temperature of the graphene on these effects is examined. Notably, the research reveals that the shifts exhibit complex and significantly varying behavior within and outside the infrared reststrahlen frequency-bands of hyperbolic crystal.
... The momentum space PSHEs are generated mainly by arranging the orientations of anisotropic structure units of liquid crystals [11] and metasurfaces [12][13][14][15] to create spatially-varying PB phases. Since the size of the unit can be down to subwavelength scale, large momentum-space spin separations can be achieved [16,17]. The separation angle between light field components with opposite spin can be up to thousand times diverging angle of the fields [18]. ...
To the best of our knowledge, a novel tunable photonic spin Hall effect is proposed based on a pair of liquid crystal Pancharatnam-Berry (PB) lenses. Owing to the spin-dependent geometric phases, a PB lens focus or defocus the incident light field according to its spin angular momentum. By cascading two PB lenses with a small gap, the focus and defocus effects can be suppressed, and the transmitted light fields with opposite spin will be deflected toward opposite directions when the two PB lenses have a relative lateral displacement. The deflection angles vary linearly with the displacements, thus double-lines two-dimensional continuous beam scanning is achieved with a scanning angle of 39o × 39° and a beam diverging angle of 0.028o × 0.028°. The scanning beam is used to write different patterns on a 200 nm thick gold film. We believe this beam scanning system can find wide applications ranging from laser processing, Lidar, particle manipulation, to free space optical communications.
... In recent years, extensive research has focused on the outof-plane displacement [9][10][11][12][13][14][15][16]33]. The manifestation of in-plane displacement is similar to the Goos-Hänchen shift, but their physical mechanisms are entirely distinct [31,32,34]. The in-plane spin-Hall shift is spin-dependent, whereas the Goos-Hänchen shift is measured on linear-polarization basic. ...
... The in-plane spin-Hall shift is spin-dependent, whereas the Goos-Hänchen shift is measured on linear-polarization basic. Because in-plane displacement is relatively weaker compared to the out-of-plane displacement, it has long been overlooked by researchers, with few exceptions [31,32,34]. However, it is important to recognize that in-plane displacement does indeed affect the direction of spin separation. ...
The photonic spin-Hall effect (PSHE) is a result of the interaction of spin and orbital angular momentum (OAM). However, previous research on the PSHE has primarily focused on out-of-plane (transverse) shifts. In fact, it encompasses both out-of-plane shifts and relatively weaker in-plane shifts, and the latter of which plays a crucial role in controlling the direction of spin separation. Here, we investigate the in-and out-of-plane shifts of the PSHE based on a full-wave theory, revealing that they have the same physical mechanism. The in-and out-of-plane displacements of the PSHE are determined by a momentum-dependent Pancharatnam-Berry (PB) phase, where the in-plane wavevector k x-dependent phase gradient determines the in-plane shift, and the transverse wavevector k y-dependent phase gradient dictates the out-of-plane shift. We discover that the intrinsic OAM (IOAM) and extrinsic OAM (EOAM) carried by incident off-axis vortex beams can respectively convert into the EOAM and IOAM of abnormal modes in the transmitted beam. The OAM generated during the spin-reversal process, together with the OAM carried by the incident beam, collectively determines the in-and out-of-plane displacements. When off-axis vortex beams are incident, the competition between the IOAM and EOAM is more complex than in those without OAM. Our study unifies in-and out-of-plane shifts into a single theoretical framework, which offers an alternative perspective on the spin-orbit interaction of light.
... The aforementioned intriguing phenomena offer novel insights for the advancement of sensor technology and optical encoder design. incident plane, while the IF shift is perpendicular to the incident plane [24]. The GH shift, initially proposed by Newton, refers to a specific displacement between the incident and reflected points of a beam in the incident plane during total reflection. ...
... Likewise, the expression of the GH and IF shifts upon the isotropic media can be obtained with the same method. Consequently, Equations (10) and (15) can be simplified to a well-known result, which was cited in the relevant reference [24]. ...
This investigation focuses on the Goos–Hänchen (GH) and Imbert–Fedorov (IF) shifts on the surface of the uniaxial hyperbolic material hexagonal boron nitride (hBN) based on the biaxial hyperbolic material alpha-molybdenum (α-MoO3) trioxide structure, where the anisotropic axis of hBN is rotated by an angle with respect to the incident plane. The surface with the highest degree of anisotropy among the two crystals is selected in order to analyze and calculate the GH- and IF-shifts of the system, and obtain the complex beam-shift spectra. The addition of α-MoO3 substrate significantly amplified the GH shift on the system’s surface, as compared to silica substrate. With the p-polarization light incident, the GH shift can reach 381.76λ0 at about 759.82 cm⁻¹, with the s-polarization light incident, the GH shift can reach 288.84λ0 at about 906.88 cm⁻¹, and with the c-polarization light incident, the IF shift can reach 3.76λ0 at about 751.94 cm⁻¹. The adjustment of the IF shift, both positive and negative, as well as its asymmetric nature, can be achieved by manipulating the left and right circular polarization light and torsion angle. The aforementioned intriguing phenomena offer novel insights for the advancement of sensor technology and optical encoder design.
... In contrast, since the origination of the IF shift has more sophisticated physics, a lot of investigations have been done, including the theory of the nonspecular reflection [5,6]. Recently, from the spin-orbit interaction, a theoretical approach for IF shift was developed by Bliokh et al [7] and confirmed in experiments conducted by Hosten and Kwiat [8]. The IF shift is considered as an optical spin-Hall effect governed by the conversation law of total angular momentum [9,10]. ...
... Figures 8(b) and (c) present the GH and IF shifts, respectively, to examine the contribution of SPhPs on the spatial shifts. We observed that each GH and IF shift curve jumps The reflectance, GH and IF shifts in f f 1. 7 1.7895 t / < < near the RB-II are provided for comparison. The reflection curves are blunt, hinting that the propagating modes are also observed. ...
The effect of surface phonon polaritons (SPhPs) on Goos-Hänchen (GH) and Imbert-Fedorov (IF) shifts of reflected rotating elliptical Gaussian beams (REGBs) is investigated in the Otto configuration with a naturally hyperbolic material (NHM) substrate. The spatial shifts are enhanced significantly and adjusted by tuning the REGBs and the optical axis orientation of NHM. The magnitude of GH and IF shifts is easily evaluated by timing the elliptical parameter b02 on the basis of the spatial shift at b0=1.0 when η(=a0/b0)≥2.0. The attenuation total reflection (ATR) spectra are simulated for a hexagonal boron nitride (hBN) substrate. To better understand the impact of SPhPs on the GH and IF shifts, a dispersion equation for SPhPs is established. The features of GH and IF shifts around SPhPs can be observed through the frequency scanning along the dispersion curves at a fixed incident angle. The comparison between the ATR spectra and the dispersion curves demonstrates that the enhancement of GH and IF shifts with higher reflection always occurs in the reststrahlen bands (RBs) of NHM once SPhPs are excited. In RB-I, the magnitudes of GH and IF shifts attributable to the SPhPs are located between the peak and dip of a shift curve within a very narrow frequency region. However, in RB-II, the positive maximum values of GH and IF shifts just coincide with the SPhPs. As a result, it is predicted that a relatively large spatial shift will be observed along the frequency scanning line. The large shifts accompanied by the higher reflection may show potential applications in optical or optoelectronic technologies.
