The Poynting vectors, spin and orbital angular momentums of uniformly polarized cosh-Pearcey-Gauss beams in the far zone

LIAO Sai，CHENG Ke，HUANG Hong-wei，YANG Ceng-hao，LIANG Meng-ting，SUN Wang-xuan

（College of Optoelectronic Engineering, Chengdu University of Information Technology,Chengdu 610225, China）

* Corresponding author，E-mail: ck@cuit.edu.cn

Abstract: We propose cosh-Pearcey-Gauss beams with uniform polarization, which are mainly modulated by a hyperbolic cosine function (n, Ω) and the angles related to uniform polarization (α, δ).Based on angular spectrum representation and the stationary phase method, the Poynting vector, Spin Angular Momentums(SAM) and Orbital Angular Momentums (OAMs) in the far zone are studied.The results show that a larger n or Ω in the hyperbolic cosine function can partition the longitudinal Poynting vectors, SAMs and OAMs into more multi-lobed parabolic structures.Different polarizations described by (α, δ) can distinguish their Poynting vectors and angular momentums between the TE and TM terms, though this does not affect the patterns of the whole beam.Furthermore, the weight of the left and right sides of longitudinal Poynting vectors,SAMs and OAMs in TE and TM terms can be modulated by left-handed or right-handed elliptical polarization, respectively.The results in this paper may be useful for information storage and polarization imaging.

Key words: cosh-Pearcey-Gauss beams; spin angular momentum; orbital angular momentum; Poynting vector

1 Introduction

Pearcey function with non-convergence or infinite energy is the second type of catastrophic function.It can be expressed by the integral formula of oscillating structure[1].In 2012, Ringet al.introduced the Gaussian envelope to confine its energy,wherein the Pearcey beams were theoretically proposed and experimentally demonstrated[2].It was found that the Pearcey beams also have self-healing and auto-focusing properties that are analogous to the Airy beams in real space propagation.Half-Pearcey beams[3], Bi-Pearcey beams[4], symmetric odd-Pearcey beams[5]and chirped Pearcey beams[6]have also been extensively studied.For example, Denget al.found that the tunable circular Pearcey beams with an annular spiral-zone phase possess the distinctive ability to trap particles[7].Zhaoet al.showed that the properties of autofocusing performance and the inversion effect could be perfectly inherited without being affected by spatial coherence[8].

For optical catastrophe beams, their vectorial structures can be analyzed by using vector angular spectrum and stationary phase methods.For example, the intensity profiles of the Generalized Humbert-Gaussian beams were obtained by the above methods and it was noted that the side-lobes decrease with a decrease in the topological charge of the helical axicon[9].Jiaet al.found that topological charge, truncation parameters and ellipticity angles have a great influence on the far-field vectorial structures of a Laguerre-Gaussian beam diffracted by a circular aperture[10].Furthermore, our previous work also showed that the symmetry and direction of the far-field vectorial structures of a Pearcey-Gauss vortex beam can be modulated by topological charges and noncanonical strength[11].On the other hand, Spin Angular Momentum (SAM) is associated with polarization.In a scalar field, SAM can be described by its Third Stokes parameters and Poincaré sphere[12].However, for the vector field, the vectorial SAM and Orbital Angular Momentum(OAM) should be considered.In addition, the introduction of the cosh function will bring more degrees of freedom for the Pearcey beams.

This paper is focused on the influence of the initial polarization state and cosh function on the Poynting vector, SAM and OAM of cosh-Pearcey-Gauss beams (cPeG) in the far zone.Our proposed beams with different polarizations can have varying weights of OAM densities in the TE and TM terms.The results obtained are different from those of Pearcey-Gauss vortex beams and provide potential applications in information storage and polarization imaging[13-15].

2 Uniformly polarized cosh-Pearcey-Gauss (cPeG) beams

Assume the uniformly polarized cPeG beam propagates in free space with an electric permittivity ofεand a magnetic permeability ofμ.In the Cartesian coordinate system, the electric field of the cPeG beams with uniform polarization at the source planez=0 is expressed by[2,16-22]

whereαis the azimuthal angle in polarization,δdenotes a phase retardation angle between thex- andycomponents of the electric field, andΩandnare the modulation parameter and the order of the hyperbolic cosine function, respectively.The background functionPeG(x,y) of the cPeG beams in Eq.(1) is given by

wherew0is the beam waist width of the Gaussian term, andPe(x/x0,y/y0) is the Pearcey function withx0andy0of scaling lengths in thex- andy- axes, respectively.The vector angular spectrums ofA(p,q)of cPeG beams are given by[18-21]

where the wave numberk=2π/λis related to the wavelengthλ.Substituting Eqs.(1)-(2) into Eq.(3),one can obtain the analytical expressions of the vector angular spectra as

with

From Eqs.(4)-(5), one can find that the angular spectrum is composed of a Pearcey function and a Gaussian term withΩandnin the cosh part.Fig.1(color online) shows the initial intensity and angular spectrum of cPeG beams forn=1, 2 and 4.From Fig.1, it can be seen that the central zone in the initial intensity darkened and the main lobe that originates from the Pearcey factor and gradually fades with the increasingn.The amplitudes in the angular spectra exhibit a parabolic structure due to the Pearcey factor with a lower-ordern=1 in Fig.1 (d).These parabolic spectra can be partitioned into more complex structures by increasing the order of the cosh part’snas shown in Figs.1 (e)-(f).A higher-ordern(n=4) leads to the appearance of a dark zone of angular spectrum in the center as shown in Fig.1(f),and the maximal values are located on the side lobe of the parabola.One can find that the angular spectrum can be modified with a higher-order hyperbolic cosine function, i.e.Ωandn, and its far-zone vec-torial structures in SAM and OAM is worthy of further analysis.The theoretical design of the cPeG beams by encoding the amplitude and phase of their Fourier spectrums onto a spatial light modulator, as shown in Fig.2 (color online), where the uniformly linearly, circularly or elliptically polarized light can be produced by wave plates.

Fig.1 The initial intensities and angular spectra of the cPeG beams with different initial polarization states for n=1, 2 and 4.(a), (d): α=π/4, δ=π/8; (b), (e): α=0, δ=0; (c), (f): α=π/4, δ=-3π/4.The phase distribution and polarization states are described in the top form left to right in Figs.1 (a)-(c).Blue: left-handed elliptical polarization; Red: right-handed elliptical polarization; Black: linear polarization.The parameters are x0=y0=100 μm and Ω=1.4

Fig.2 The theoretical design scheme of the cPeG beams with uniform polarization.BS: Beam Splitter; SLM: Spatial Light Modulator

3 Far-zone vectorial structure of cPeG beams

An arbitrary vectorial electromagnetic field can be separated into two orthogonal terms of Transverse Electric (TE) and Transverse Magnetic (TM)fields, and it can be as the sum of TE and TM terms and defined asE=ETE+ETMandH=HTE+HTMwhereEandHdenote the whole electric and magnetic fields, respectively[10,18,20].The far-zone behavior of the electromagnetic field, i.e.EandHcan be further analyzed byETE,ETM,HTEandHTM, respectively.These electric and magnetic fields can be obtained by vector angular spectra and the stationary phase method in the far zone[18-23].Our attention is directed to the dependence of the Poynting vector,spin and orbital angular momentum densities of the cPeG beams in uniform polarization (α,δ).Furthermore, the influences of the parameters and the order of the hyperbolic cosine functionΩandnon the vectorial structures ofETE,ETM,HTEandHTMin the far zone are also investigated in detail.The calculation parameters areλ=633 nm,x0=y0=100 μm (unless otherwise stated),w0=1mm andz0=ky02.

3.1 Poynting vector

The magnitude and direction of energy flow during electromagnetic wave propagation can be described by the Poynting vector.The longitudinal component of the Poynting vector is proportional to the light intensity, and its transversal component shows the magnitude and direction of the complex power flow in a fixed plane.The transversal and longitudinal components of the separated TE and TM terms are given by[22]

whereσ=TE or TM, Re represents the real part and the asterisk is complex conjugation.

The effect of uniform polarization (α,δ) on the Poynting vectors of the cPeG beams, including TE,TM terms and whole beam, atz=50z0in free space is shown in Fig.3, where the magnitude and direction of the transverse Poynting vectors are shown by arrows.One can see that the patterns in Poynting vectors are parabolic structures due to the Pearcey factor, and the energy flow moves from the top to the side of the parabola.Although the parabolic patterns of a whole beam are unaffected by initial polarizations, the symmetry of longitudinal Poynting vectors in the TE and TM terms can be modulated by uniform polarization (i.e.,α,δ).The parabolic patterns of the TE and TM terms in cPeG beams are symmetric about horizontally linear polarization(α=δ=0) as shown in Figs.3 (b) and (e) (color online), and other polarizations regulate parabolic patterns in the TE and TM terms.For example, the left side of the TE term is more influential in the lefthanded (LH) elliptical polarization (α,δ)=(π/4, π/8)in Fig.3 (a) (color online), but the right one dominates the TE term in the right-handed (RH) elliptical polarization (α,δ)=(π/4, -3π/4) in Fig.3 (g) (color online).Furthermore, the topological charge in the TE and TM terms will change if the uniform polarizations (α,δ) are modulated, and the net topological charge is always zero under uniform polarization(α,δ) due to the topological charges of the two terms being opposite.Fig.4 (color online) shows the Poynting vectors of cPeG beams in free space for differentnandΩatz=50z0, where the LH elliptical polarization is given by (α,δ)=(π/4, π/8).One can find that the TE, TM and whole terms present multilobed parabolic structures in longitudinal Poynting vectors, and the number of partitioned lobes increases with an increase innorΩ.The intensities of the dark zones on the parabolic patterns are nonzero.

Fig.3 Normalized longitudinal Poynting vector (backgrounds) and transversal Poynting vector (arrows) of cPeG beams for different uniform polarizations (α, δ) at z=50z0.(a), (d), (g): TE term; (b), (e), (h): TM term; (c), (f), (i): whole beam.The parameters are n=2 and Ω=1.4.The red point symbolizes topological charge l=+1, and the white point denotes l=-1

Fig.4 Normalized longitudinal Poynting vector (backgrounds) and transverse Poynting vector (arrows) of cPeG beams for different n and Ω at z=50z0, where the uniform polarization is (α, δ)=(π/4,π/8)

3.2 Spin angular momentum

Angular momentum carried by the electromagnetic field, in tandem with its mechanical properties,is comprised of spin and orbital components.The intrinsic spin angular momentum is associated with left- and right-handed circular polarizations, while the orbital component originates from the spatially helical phase of the wavefield.The total angular momentum of the electromagnetic field can be written as the sum of the spin and orbital components[13,24-25]

whereσ=TE or TM,εis the permittivity of the vacuum,ωis the angular frequency, and the first and second terms on the right side of Eq.(9) denote SAM and OAM, respectively.

Figs.5 and 6 (color online) give the dependencies of the SAM of the TE term, TM term and whole beam with uniform polarization (α,δ), cosh parametersΩand the order numbernin free space, respectively.For linear polarization, the transversal and longitudinal SAMs do not exist.For LH or RH elliptical polarization, the longitudinal SAMs are less or larger than zero, respectively.The longitudinal SAM in the left side of TE terms has an advantage over that on the right side, but for TM terms, the opposite is true.The superposition of the left side in the TE term and the right side in the TM term results in the multilobed parabolic structures of the whole beam in SAM densities, and the peak values of their SAM densities are located on two sides rather than the vertex of their parabolas.From Fig.6,it is clear that the SAM densities can be partitioned by larger cosh parametersΩand order numbersn.In addition, the transverse SAM of a whole beam is determined by the TM term owing to thez-component of the TE term not existing, and these vectorial arrows present different directions.The arrows in the right side point to the left in Figs.5 (c), but those in the left side point to the right as shown in Fig.5 (f).More details on the evolution of transversal SAM in whole beams are provided in Supplementary Videos 1 and 2 (see the article online), wherein one can find that an alternative variation of transverse SAMs is presented.Its direction in the left side varies from right to left for LH polarization where 0<α<π/2 andδ=π/8 as shown in Video 1, whereas its change direction is opposite for the RH polarization with π/2<α<π andδ=π/8 in Video 2, which indicates that the direction of transversal SAMs is closely related to LH or RH polarization.

Fig.5 Normalized longitudinal SAM (3D and 2D) and transverse SAM (arrows) of cPeG beams at z=50z0.(a)-(c): α=π/4,δ=π/8; (d)-(f): α=π/4, δ=-3π/4.The other parameters are the same as those in Fig.3

Fig.6 Normalized longitudinal (3D and 2D) SAM and transverse SAM (arrows) of cPeG beams (n=1, 4) at z=50z0.(a)-(c):n=1, Ω=3; (d)-(f): n=4, Ω=1.The other parameters are the same as those in Figs.5 (α=π/4, δ=π/8)

3.3 Orbital angular momentum

The dependencies of longitudinal and transverse OAMs of TE, TM and whole beams on different polarization in free space are shown in Fig.7(color online).Although the initial polarization cannot affect longitudinal and transversal OAM densities in whole beams, it can vary the distribution of longitudinal OAM in TE and TM terms.For example, their longitudinal OAMs with parabolic structures both appear in the left and right sides for linear polarization (α=0,δ=0) in Figs.7 (d)-(f),whereas for other polarizations, the left side of the TE term or the right side of the TM term can only be presented.The results indicate that the initial polarization can be used to distinguish the distribution of OAM densities in the TE and TM terms.For whole beams, their longitudinal OAM densities with multibranched structures are equivalent in the positive and negative directions, which results in a zero value of the net OAM.However, the sum of OAM densities in the TE or TM terms is non-zero because of their asymmetry aboutx=0.

Fig.7 Normalized longitudinal OAM (3D and 2D) and transverse OAM (arrows) of cPeG beams (Ω=1.4) at z=50z0.(a)-(c):α=π/4, δ=π/8; (d)-(f): α= 0, δ= 0; (g)-(i): α=π/4, δ=-3π/4; The other parameters are the same as those in Fig.3.(n=2)

Fig.8 (color online) further describes the influence of the cosh parameter and order number on longitudinal and transversal OAMs.In Fig.8, one can see that theLzwith a largerΩorntends to partition more branches along parabolic structures,which is analogous with the multi-lobed structures in Fig.4.In Fig.8 (a), one can see that the left side ofLzin the TE term is asymmetric with its right side.The sum of the negative values is larger than that of the positive values, which leads to a negativeLzin the TE term.The behavior is also verified by the topological charge TC=-1 in Fig.4 (a).A similar case is also found in TM terms with a positiveLz.However, for whole beams, the superimposed OAM of the left side in the TE term and the right side in the TM term constitute their parabolic patterns with a zero-valueLzdue to the fact that the sum of negative values in the TE term equal the sum of positive ones in the TM term, which can be easily explained by the non-vortex at the input plane.

Fig.8 Normalized longitudinal and transverse (arrows) OAM of cPeG beams (n=1, 4).(a)-(c): n=1, Ω=3; (d)-(f): n=4, Ω=1 at z=50z0.The other parameters are the same as those in Fig.4

4 Conclusion

By introducing the hyperbolic cosine function to Pearcey beams and the Poynting vector, the SAM and OAM of cPeG beams with uniform polarization are studied by using angular spectrum representation and the stationary phase method.Although the vortex is not embedded at the original plane, the farzone TE and TM terms can still carry topological charge with a spiral phase owing to polarization.The topological charges in the TE and TM terms are opposite, which results in the net topological charge of the whole beams always being zero.A largernorΩin the embedded hyperbolic cosine function can partition the far-zone Pearcey pattern into more multi-lobed parabolic structures.The longitudinal SAMs in the left side of TE terms have distinct advantages over those in the right side, but the opposite holds true for TM terms.The sums of SAMs in TE terms, TM terms and whole beams are all less or larger than zero due to the LH or RH elliptical polarization at the original plane, respectively.In addition, the evolution of direction in a transversal SAM is significantly altered by LH or RH polarization as shown in Videos 1 and 2.The OAMs in the whole terms are not affected by initial polarization, but their longitudinal OAMs in the TE and TM terms can be changed.The results indicate that the initial polarization can be used to distinguish the distribution of OAM densities in its TE and TM terms.The left and right sides ofLzis asymmetric owing to the TE or TM wavefields with topological charges.The results obtained in this paper should be useful for understanding the vectorial structure of cPeG beams in the far zone, and have potential applications in information storage and polarization imaging.

Supplementary Material

Supplementary Video 1:The evolution of transverse SAM of cPeG (n=2) beams atα=0～π/2 in far zone.

Supplementary Video 2:The evolution of transverse SAM of cPeG (n=2) beams atα=π/2～π in far zone.