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Canonical momentum, angular momentum, and helicity of circularly polarized Airy beams
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Canonical momentum, angular momentum, and helicity of circularly polarized Airy beams Yuanfei Hui, Zhiwei Cui ∗ , Pan Song, Yiping Han, Wenjuan Zhao School of Physics and Optoelectronic Engineering, Xidian University, Xi’an, 710071, China
a r t i c l e
i n f o
Article history: Received 1 September 2019 Received in revised form 18 December 2019 Accepted 17 January 2020 Available online xxxx Communicated by V.A. Markel Keywords: Airy beams Momentum Spin angular momentum Orbital angular momentum Helicity
a b s t r a c t We report a study of the momentum, angular momentum, and helicity of circularly polarized Airy beams propagating in free space. By using the vector angular spectrum representation, the explicit analytical expressions for the electric and magnetic field components of circularly polarized Airy beams are derived in detail. To overcome the drawbacks of classical kinematics formulae when applied to structured light beams, a general canonical approach is introduced to describe the momentum, angular momentum and helicity of Airy beams. Numerical simulation results for the spatial distributions of the canonical momentum, spin and orbital angular momentum, as well as the helicity densities are presented and discussed. This study may provide useful insights into the dynamical properties of Airy beams that may be important in several applications, including the optical control, micromanipulation, and information processing. © 2020 Elsevier B.V. All rights reserved.
1. Introduction In the recent years, structured light beams with non-Gaussian, customized intensity profiles and spatially variant state of polarization have attracted intensive attention [1,2]. As a typical type of structured light beams, the Airy beams, which are solutions of the Schrödinger equation, exhibit a number of highly desirable features, such as non-diffraction [3], self-acceleration [4], and self-healing [5]. These properties led to many potential applications, such as particles manipulation [6–10], light-sheet microscopy [11], vacuum electron acceleration [12], light bullets [13], and so on. Motivated by their features and applications, extensive theoretical and experimental studies on the properties, generation, and manipulation of the Airy beams have been conducted [14]. Among which, the study of some fundamental properties, including the intensity, phase, and polarization distributions of Airy beams is the basis of their generation, manipulation, and application. On the other hand, Airy beams, as a special type of electromagnetic waves, carry energy, momentum, and angular momentum (AM). In addition, Airy beams themselves can have a helical structure: left- and right-circularly polarized beams trace out helices with opposite handedness. As the main dynamical quantities, energy, momentum, AM, and helicity are crucial for understanding the properties of Airy beams and their interactions with matter. During the past decade, several theoretical studies of the energy, momentum, and AM of Airy beams have been reported [15–17]. Sztul and Alfano were the first ones to describe and analyze the evolution of the energy flow, i.e. Poynting vector, and AM of the Airy beams [15]. Afterward, Deng et al. further examined the energy flow and AM of nonparaxial Airy beams [16]. In these two studies, the AM was determined by the Poynting vector, which meant that the adopted momentum was the well-known Poynting momentum, also known as kinetic momentum density [18]. Although it is widely accepted that the momentum density of a light in free-space can be well described by Poynting vector, this formalism looses its clear physical meaning in the case of structured light beams [18,19]. Moreover, the Poynting formalism does not describe separately the spin and orbital AM, which are widely seen as independent degrees of freedom [18,19]. The difficulties encountered in the separation of the total AM of light into its orbital and spin parts have been emphasized by some authors [18–23]. Bialynicki-Birula and Bialynicka-Birula proposed a canonical separation of AM of light into its orbital and spin parts [20]. Leader and Lorce [21,22] addressed this problem and showed that it is the gauge invariant version of the canonical AM which agree with the results of a host of laser optics experiments. Bliokh et al. [18,19,23] also showed that the canonical AM densities are consistent with
*
Corresponding author. E-mail address: [email protected] (Z. Cui).
https://doi.org/10.1016/j.physleta.2020.126284 0375-9601/© 2020 Elsevier B.V. All rights reserved.
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directly observable properties of structured optical fields. In this paper we follow the canonical approach for describing of the momentum, SAM, and OAM densities proposed in Ref. [23] to study the issues we concern. With this approach, the momentum density corresponds to the local gradient of the phase of the field, the spin AM (SAM) density is proportional to the local ellipticity of the field polarization, and the orbital AM (OAM) density is defined via the canonical momentum density. Based on the canonical separation of AM, Kim has studied the transverse SAM of Airy beams [17]. Nevertheless, to the best of our knowledge, the canonical momentum and OAM of the Airy beams have not been reported. Moreover, the previous studies of the momentum and AM of Airy beams concentrated on the state of linear polarization. In this paper, we report a study of the momentum, SAM and OAM of circularly polarized Airy beams based on the canonical approach. As we know, the polarization is one of the most important properties of light beam. When a circularly polarized Airy beam propagates in an inhomogeneous medium, there are some interesting polarization-dependent effects phenomena induced by the spin-orbit coupling [24]. To reveal the mechanism of spin-orbit interaction within the circularly polarized Airy beams, it is necessary to study their momentum, SAM and OAM. Further insight into the physical properties of Airy beams can be obtained by considering an additional dynamical characteristic of the light, i.e., the helicity, which can be regarded as the projection of the spin angular momentum onto the linear momentum direction [25–29]. 2. Formulation 2.1. Electric and magnetic field components of Airy beams Airy beams, a typical class of non-diffracting beams, are solutions of scalar wave equation under paraxial approximation or its quantummechanics analog, i.e. the Schrödinger equation. Like other non-diffracting beams, the ideal Airy beams are not realizable in practice since they carry infinite energy and propagate in the free space without distortion. In order to describe the propagation of the Airy beams in a more realistic way, usually a truncation of this beams family by an exponential term is adopted to obtain the finite Airy beams, which are expressed as a product of the exact Airy mode and the linear exponential function. In this paper, the Airy beams refer to the finite Airy beams. In Cartesian coordinates, assuming that the beams propagate along the z-direction, the amplitude distribution of the Airy beams in the initial plane z = 0 can be expressed as [3,5]
E (x, y , z = 0) = A i
x
Ai
x0
y
exp a0
y0
x
exp a0
x0
y
(1)
y0
in which
Ai (x) =
1 2π
∞ exp
iu 3
−∞
3
+ ixu du
(2)
is the Airy function, 0 ≤ a0 < 1 in the exponential function is a decay parameter that determines the beam propagation distance, x0 and y 0 are the scaled parameters in the x and y direction, respectively. To explore the momentum, SAM, OAM, and helicity of Airy beams, the electric and magnetic field components of Airy beams need to be determined firstly. There are several approaches, such as vector potential method, angular spectrum representation and RayleighSommerfeld diffraction integrals, to obtain expressions for the electric and magnetic field components of Airy beams. In the present work, we derive the explicit analytical expressions for the electric and magnetic field components of Airy beams by using the vector angular spectrum representation. The basic idea of such approach is to expand any a structured light beam in terms of lots of plane waves with variable amplitudes and propagation directions by Fourier transform. Specifically, the angular spectrum amplitude of a given structured light beam in the initial plane z = 0 can be obtained by taking the two-dimensional Fourier transform of the electric field in the plane z = 0, as follows
1 E kx , k y ; z = 0 = 2 4π
∞ ∞
E (x, y , z = 0) exp −i kx x + k y y
dxdy
(3)
−∞ −∞
where x and y are the Cartesian transverse coordinates, while k x and k y are the corresponding spatial frequencies coordinates. Substituting Eq. (1) into Eq. (3), and recalling the following integral formula
+∞
Ai (x) exp (ax) dx = exp a3 /3
(4)
−∞
we obtain the angular spectrum amplitude of the Airy beams
2 x0 y 0 i 2 2 2 2 3 3 3 3 3 2 − E kx , k y = exp a k x + k y exp k x + k y 3a k x + k y + − a 0 x 0 y 0 x 0 y 0 x 0 y 0 0 0 2 4π
3
3
(5)
Once the angular spectrum amplitude is obtained, the vector angular spectrum of Airy beams can be expressed as [30,31]
1 E kx , k y ; z = α xˆ + β yˆ − αkx + β k y zˆ E kx , k y ; z kz
(6)
where α and β satisfying σ = i (α β ∗ − α ∗ β) are the polarization √ parameters with√σ = ±1 for left-handed and right-handed circularly polarized light beam. In particular, the values of (α , β) with (1, i ) / 2 and (1, −i ) / 2 correspond to the left-circular polarization (L-CP)
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and right-circular polarization (R-CP), respectively. The vector angular spectrum of Airy beams in an arbitrary plane z = const can be
obtained by E kx , k y ; z = E kx , k y ; z = 0 e ik z z , where k z =
k2 − k2x − k2y refers to the Airy beams propagating into the half-space z > 0,
and k = 2π /λ is the wave number with λ being the wavelength of the beams. Then, we can calculate the electric field of the Airy beams in an arbitrary plane z = const by the inverse Fourier transform of the angular spectrum
∞ ∞ E (x, y , z) =
E kx , k y exp i kx x + k y y + k z z dkx dk y
(7)
−∞ −∞
In a similar way, we can also represent the magnetic field as
∞ ∞ H (x, y , z) =
H kx , k y exp i kx x + k y y + k z z dkx dk y
(8)
−∞ −∞
in which
H kx , k y = √
E kx , k y 1 k × Z
(9)
k
with Z = μ/ε being the wave impedance in homogeneous medium characterized by permittivity ε and permeability μ. As has been mentioned, the Airy beams are solutions of the paraxial wave equation. Such beams propagate along a certain direction z and spread out only slowly in the transverse direction. In this case, the wavevectors k are almost parallel to the z-axis and the transverse wavenumbers kx , k y are small compared to k. Therefore, we can expand k z in a series as
kz =
k2 − k2x − k2y ≈ k −
k2x + k2y
(10)
2k
Under the above paraxial approximation, Eqs. (7) and (8) can be expressed as
E (x, y , z) H (x, y , z)
∞ ∞ k2x + k2y E kx , k y = exp (ikz) exp i kx x + k y y − dkx dk y z H kx , k y 2k
(11)
−∞ −∞
In addition, for the expressions of vector angular spectrum, i.e. Eqs. (6) and (9), k z can be further simplified as k z ≈ k. Consequently, the angular spectrum components can be represented as
1 E x = α E, E y = β E, E z = − αk x − β k y E
(12)
1 1 11 β k x − αk y H x = − β E, H y = α E, Hz = E
(13)
k
Z
Z
Zk
Substituting Eq. (5) into Eqs. (12) and (13), using the paraxial vector angular spectrum representation given by Eq. (11) and the following integral formulae [32]
+∞
Ai (x) exp bx2 + cx dx =
2 π c c − exp − + 2 − b
−∞
+∞
x Ai (x) exp bx2 + cx dx = −∞
4b
8b
4b
8b
Ai
96b3
2 π c c − exp − + 2 − b
1
1 16b2
−
c
(14)
2b
1 96b3
c 1 1 c 1 1 c × − + 2 Ai − − − Ai 2 2 2b
8b
2b
16b
2b
16b
(15)
2b
we obtain the explicit analytical expressions for the electromagnetic field components of circularly polarized Airy beams as follows
E x = α A i ( T x ) A i T y exp ( M x ) exp M y exp (ikz)
(16)
E y = β A i ( T x ) A i T y exp ( M x ) exp M y exp (ikz)
(17)
i
Ez = α
+β H x = −β
2kx20 a0 + iz 2k2 x30
i 2ky 20 a0 + iz 2k2 y 30 1 Z
Ai ( T x ) +
Ai T y +
i kx0
Ai ( T x ) exp ( M x ) A i T y exp M y exp (ikz)
i ky 0
Ai T y
(18)
A i ( T x ) exp ( M x ) exp M y exp (ikz)
A i ( T x ) A i T y exp ( M x ) exp M y exp (ikz)
(19)
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Hy = α
1 Z
H z = −β
+α
i 2kx20 a0 + iz
1
2k2 x30
Z
i 2ky 20 a0 + iz
1
A i ( T x ) A i T y exp ( M x ) exp M y exp (ikz) Ai ( T x ) +
2k2 y 30
Z
i kx0
Ai T y +
i
Ai ( T x ) exp ( M x ) A i T y exp M y exp (ikz)
ky 0
(20)
Ai T y
(21)
A i ( T x ) exp ( M x ) exp M y exp (ikz)
where the parameters T x , T y , M x and M y are defined by
x
Tx =
−
x0 y
Ty =
y0
My =
4k2 x40
+
z2
−
a0 x
Mx =
z2
4k2 y 40
−
a0 z 2
ia0 z
+
ia0 z
−
x0 2k2 x40 a0 y a0 z 2 − y0 2k2 y 40
(22)
kx20
(23)
ky 20 iz3 12k3 x60 3
−
+
ia20 z
+
2kx20 ia2 z iz + 02 6 2ky 0 12k3 y 0
ixz 2kx30
+
iyz 2ky 30
(24)
(25)
2.2. Canonical momentum, AM, and helicity of Airy beams After obtaining the electric and magnetic field components, we now consider the momentum, SAM and OAM of Airy beams. Following the canonical approach for characterization of the dynamical characteristics of structured light beams [18,19,22], the momentum, SAM and OAM densities of the Airy beams in homogeneous medium characterized by permittivity ε and permeability μ can be expressed as
P= S=
1 4ω 1
Im Im
εE∗ · (∇) E + μH∗ · (∇) H
4ω L=r×P
εE∗ × E + μH∗ × H
(26) (27) (28)
where ω is the angular frequency, Im [·] denotes the imaginary parts, superscript “∗” represents the complex conjugate, and the notation A · (∇) B = A x ∇ B x + A y ∇ B y + A z ∇ B z is used. From the definitions, we can see that the momentum, SAM and OAM densities are vectors, including longitudinal and transverse components. Here, the longitudinal component is the z-component of these quantities, while the transverse components include the contributions of x- and y-components. It should be noted that Eq. (26) is the linear momentum (LM), which contributes to OAM, as can be seen from Eq. (28). In addition to the LM, SAM and OAM, the helicity is also an important dynamical quantity, which characterizes the intrinsic rotational content of electromagnetic waves [33]. In free space, the helicity is a conserved quantity. It is closely connected to the SAM of electromagnetic waves. For the simplest case of plane-wave, the helicity is equivalent to the projection of the SAM in the direction of propagation of the wave. However, for the case of structured Airy beams, a special type of electromagnetic waves, the helicity is different from the SAM. In particular, the helicity is a scalar whereas the SAM is a vector. The SAM is closely connected to the helicity flow. Note that, the helicity and its corresponding flow stem from the conservation equation [33,34]. Mathematically, the helicity density in homogeneous medium characterized by permittivity ε and permeability μ can be written as [28]
ϑ=
1 √ 2ω
εμ Im H∗ · E
(29)
Equations (26)-(29) are the central expressions for description of the momentum, SAM, OAM, and helicity of Airy beams. It is worthwhile to note that we adopt the canonical approach to describe these dynamical quantities. Compared with the kinetic momentum density defined by the Poynting vector, the canonical momentum density has a clear physical interpretation, namely it is proportional to the local gradient of the phase of the field. Moreover, the canonical description allows for the spin-orbital momentum decomposition, and the canonical AM densities have been demonstrated to be consistent with the results of a host of laser optics experiments [23]. 3. Numerical results and discussion In this section, we performed some numerical calculations to explore the canonical momentum, SAM, OAM, and helicity of circularly polarized Airy beams propagating in free space. In the calculations we set the free space wavelength of Airy beams λ = 632.8 nm, the decay parameter a0 = 0.2, and the transverse scaled parameters x0 = y 0 = 0.15 mm. For the sake of convenience in the numerical analysis, we introduce the Rayleigh distance z R = kx20 /2. Hereafter, the parameters used are the same as mentioned above, unless otherwise stated. Fig. 1 presents the LM density distributions of the circularly polarized Airy beams, where Figs. 1(a1) and (b1) show the distributions of the total LM density with different decay parameters a0 , Figs. 1(a2) and (b2) display the transverse LM density distributions in the x– y plane (z = z R ) with a0 = 0.2, and Figs. 1(a3) and (b3) correspond to the longitudinal LM density distributions. As we can see, the change of the state of polarization (from L-CP to R-CP, or vice versa) has no effect on the total and longitudinal LM densities, but leads to a redistribution of the transverse LM density. From Figs. 1(a1) and (b1), it is found that the magnitudes of the total LM density of Airy
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Fig. 1. LM density distributions of Airy beams with the state of polarization (a1)-(a3): L-CP, and (b1)-(b3): R-CP. (a1) and (b1) are the total LM density distributions, where the transverse coordinates are set as x = −0.15 mm and y = −0.25 mm. (a2), (a3) and (b2), (b3) are the transverse and longitudinal LM density distributions, respectively, where the white arrows represent the orientations of transverse LM density. (For interpretation of the colors in the figures, the reader is referred to the web version of this article.)
beams gradually decrease with the increase of the decay parameter a0 when z < 3z R . Simultaneously, the total LM density of the Airy beams will decline slowly with the propagation distance increasing. When z > 6z R , the decay parameter has little effect on momentum density. Comparing Fig. 1(a2) with Fig. 1(a3) and Fig. 1(b2) with Fig. 1(b3), it can be seen from the range of colorbar that the longitudinal LM density is dominant for Airy beams with circular polarization. Further observation, we find that the transverse LM density of main peak exhibits the crescent pattern, as shown in Figs. 1(a2) and (b2). The above analyses indicate that the state of polarization and beam parameters of the Airy beams have a much stronger effect on the LM density. Fig. 2 shows the transverse and longitudinal SAM density distributions of the circularly polarized Airy beams with the decay parameter a0 = 0.05, where the white arrows stand for the orientations of transverse SAM density. All the SAM densities are normalized by the maximum of their respective total SAM density. From the illustrations of this Figure, we can see that the change of the polarization state has a significant effect on both the transverse and longitudinal SAM densities. To be specific, when L-CP changes to R-CP, the transverse SAM density is redistributed, as shown in Figs. 2(a1) and (b1), and the direction of the longitudinal SAM density reverses with its amplitude stays the same, as shown in Figs. 2(a2) and (b2). Meanwhile, we note that the directions of longitudinal SAM densities in the x–z plane for Airy beams with L-CP and R-CP are opposite, as shown in Figs. 2(a3) and (b3). In addition, comparing Fig. 2(a1) with Fig. 2(a2) and Fig. 2(b1) with Fig. 1(b2), we can easily find that the SAM density is predominated by the longitudinal part because the beams with L-CP and R-CP carry SAM along the propagation direction. Fig. 3 illustrates the transverse and longitudinal OAM density distributions of the circularly polarized Airy beams in the x– y plane (z = z R ), where the white arrows show the orientations of transverse OAM densities. Obviously, the change of the state of polarization (from L-CP to R-CP, or vice versa) has no effect on the transverse OAM density, but leads to a reversal of the longitudinal OAM, as shown in Figs. 3(a3) and (b3). Comparison between the colorbar range of transverse and longitudinal OAM densities indicates that the OAM density of the circularly polarized Airy beams is predominated by the transverse part. It is worthwhile to note that although the longitudinal OAM density is weak, it has a non-zero contribution. This arises from the fact that the OAM is not an independent dynamical characteristic of the field because it is defined via the canonical momentum density and dependent on the choice of the coordinate origin. Finally, Fig. 4 gives the helicity density distributions of the circularly polarized Airy beams, where Figs. 4(a) and (c) present the side views and the cross sections of the helicity density distributions, and Figs. 4(b) and (d) display three-dimensional distributions of the helicity density in the x– y plane. It can be observed that the state of polarization changes from the L-CP to R-CP, the helicity densities obviously exhibit a negative value while their amplitude stay the same, which can be explained from the definition of the helicity density given by Eq. (29). It is also observed that the main peak of the helicity density in the x– y plane moves along the line x = y, i.e. points at
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Fig. 2. Transverse and longitudinal SAM density distributions of the Airy beams with the state of polarization (a1)-(a3): L-CP, and (b1)-(b3): R-CP. (a1), (b1) and (a2), (b2) are the transverse and longitudinal SAM density distributions in the x– y plane (z = z R ), and (a3) and (b3) are the longitudinal SAM density distributions in x–z plane with y = 0.
Fig. 3. Transverse and longitudinal OAM density distributions of circularly polarized Airy beams in the x– y plane (z = z R ). (a1), (a2) and (b1), (b2) are the transverse and longitudinal OAM densities of L-CP and R-CP Airy beams, respectively. (a3) and (b3) are the transverse and longitudinal OAM density distribution of circularly polarized Airy beams versus y, where x = −0.1 mm.
45◦ , as the propagation distance z increases. In addition, comparing Fig. 4 with Fig. 2, we find that the distribution of helicity density is very similar to that of longitudinal SAM density. It is worth noting that the helicity is different from the SAM, which is usually associated with the helicity flow. Specifically, the helicity is a scalar whereas the SAM is a vector. The helicity and its flow fulfill a conservation equation. 4. Conclusion To summarize, we have investigated the momentum, SAM and OAM, as well as the helicity of circularly polarized Airy beams. The scalar expression of Airy beams was given and its angular spectrum amplitude was deduced by using Fourier transform. Under the paraxial approximation, the explicit analytical expressions for the electric and magnetic field components of circularly polarized Airy beams were derived in detail by using the vector angular spectrum representation. With the aid of canonical LM, SAM, OAM, and helicity densities, we performed some numerical calculations and the results showed that the state of polarization, decay parameter, and propagation distance of the Airy beams have a significant effect on the distributions of these quantities. For circularly polarized Airy beams, the polarization state changes from the L-CP to R-CP has no effect on the longitudinal LM and transverse OAM densities, but leads to a redistribution of the transverse LM density and a reversal of the longitudinal OAM density. At the same time, when L-CP changes to R-CP, the values of longitudinal SAM and helicity densities change from positive to negative, and the transverse SAM density is redistributed. The results
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Fig. 4. The helicity density distributions of the Airy beams with the state of polarization (a) and (b): L-CP, and (c) and (d): R-CP. (a) and (c) present the side views (x–z plane) and the cross sections (x– y plane) of the helicity density distributions. (b) and (d) display three-dimensional distributions of the helicity density in the x– y plane (z = 2z R ).
obtained in this paper may provide new insights into the mechanism of spin-orbit interaction within the circularly polarized Airy beams, and find potential applications in the fields of optical micromanipulation control, and information processing. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This research was supported by the National Natural Science Foundation of China (61675159). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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[24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]
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H. Li, J. Wang, M. Tang, X. Li, J. Opt. Soc. Am. A 34 (2017) 1114–1118. I. Fernandez-Corbaton, X. Zambrana-Puyalto, G. Molina-Terriza, Phys. Rev. A 86 (2012) 042103. K.Y. Bliokh, A.Y. Bekshaev, F. Nori, New J. Phys. 15 (2013) 033026. I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M.L. Juan, G. Molina-Terriza, Phys. Rev. Lett. 111 (2013) 060401. F. Alpeggiani, K.Y. Bliokh, F. Nori, L. Kuipers, Phys. Rev. Lett. 120 (2018) 243605. M.F. Picardi, K.Y. Bliokh, F.J. Rodríguez-Fortuño, F. Alpeggiani, F. Nori, Optica 5 (2018) 1016–1026. C.G. Chen, P.T. Konkola, J. Ferrera, R.K. Heilmann, M.L. Schattenburg, J. Opt. Soc. Am. A 19 (2002) 404–412. W. Lu, H. Chen, S. Liu, Z. Lin, Opt. Express 25 (2017) 23238–23253. V. Olivier, S. Manuel, Airy Functions and Applications to Physics, Imperial College Press, 2004. M. Fernández-Guasti, Phys. Lett. A 383 (2019) 3180–3186. M. Fernández-Guasti, J. Mod. Opt. 66 (2019) 1265–1271.
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Physics Letters A www.elsevier.com/locate/pla
Canonical momentum, angular momentum, and helicity of circularly polarized Airy beams Yuanfei Hui, Zhiwei Cui ∗ , Pan Song, Yiping Han, Wenjuan Zhao School of Physics and Optoelectronic Engineering, Xidian University, Xi’an, 710071, China
a r t i c l e
i n f o
Article history: Received 1 September 2019 Received in revised form 18 December 2019 Accepted 17 January 2020 Available online xxxx Communicated by V.A. Markel Keywords: Airy beams Momentum Spin angular momentum Orbital angular momentum Helicity
a b s t r a c t We report a study of the momentum, angular momentum, and helicity of circularly polarized Airy beams propagating in free space. By using the vector angular spectrum representation, the explicit analytical expressions for the electric and magnetic field components of circularly polarized Airy beams are derived in detail. To overcome the drawbacks of classical kinematics formulae when applied to structured light beams, a general canonical approach is introduced to describe the momentum, angular momentum and helicity of Airy beams. Numerical simulation results for the spatial distributions of the canonical momentum, spin and orbital angular momentum, as well as the helicity densities are presented and discussed. This study may provide useful insights into the dynamical properties of Airy beams that may be important in several applications, including the optical control, micromanipulation, and information processing. © 2020 Elsevier B.V. All rights reserved.
1. Introduction In the recent years, structured light beams with non-Gaussian, customized intensity profiles and spatially variant state of polarization have attracted intensive attention [1,2]. As a typical type of structured light beams, the Airy beams, which are solutions of the Schrödinger equation, exhibit a number of highly desirable features, such as non-diffraction [3], self-acceleration [4], and self-healing [5]. These properties led to many potential applications, such as particles manipulation [6–10], light-sheet microscopy [11], vacuum electron acceleration [12], light bullets [13], and so on. Motivated by their features and applications, extensive theoretical and experimental studies on the properties, generation, and manipulation of the Airy beams have been conducted [14]. Among which, the study of some fundamental properties, including the intensity, phase, and polarization distributions of Airy beams is the basis of their generation, manipulation, and application. On the other hand, Airy beams, as a special type of electromagnetic waves, carry energy, momentum, and angular momentum (AM). In addition, Airy beams themselves can have a helical structure: left- and right-circularly polarized beams trace out helices with opposite handedness. As the main dynamical quantities, energy, momentum, AM, and helicity are crucial for understanding the properties of Airy beams and their interactions with matter. During the past decade, several theoretical studies of the energy, momentum, and AM of Airy beams have been reported [15–17]. Sztul and Alfano were the first ones to describe and analyze the evolution of the energy flow, i.e. Poynting vector, and AM of the Airy beams [15]. Afterward, Deng et al. further examined the energy flow and AM of nonparaxial Airy beams [16]. In these two studies, the AM was determined by the Poynting vector, which meant that the adopted momentum was the well-known Poynting momentum, also known as kinetic momentum density [18]. Although it is widely accepted that the momentum density of a light in free-space can be well described by Poynting vector, this formalism looses its clear physical meaning in the case of structured light beams [18,19]. Moreover, the Poynting formalism does not describe separately the spin and orbital AM, which are widely seen as independent degrees of freedom [18,19]. The difficulties encountered in the separation of the total AM of light into its orbital and spin parts have been emphasized by some authors [18–23]. Bialynicki-Birula and Bialynicka-Birula proposed a canonical separation of AM of light into its orbital and spin parts [20]. Leader and Lorce [21,22] addressed this problem and showed that it is the gauge invariant version of the canonical AM which agree with the results of a host of laser optics experiments. Bliokh et al. [18,19,23] also showed that the canonical AM densities are consistent with
*
Corresponding author. E-mail address: [email protected] (Z. Cui).
https://doi.org/10.1016/j.physleta.2020.126284 0375-9601/© 2020 Elsevier B.V. All rights reserved.
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directly observable properties of structured optical fields. In this paper we follow the canonical approach for describing of the momentum, SAM, and OAM densities proposed in Ref. [23] to study the issues we concern. With this approach, the momentum density corresponds to the local gradient of the phase of the field, the spin AM (SAM) density is proportional to the local ellipticity of the field polarization, and the orbital AM (OAM) density is defined via the canonical momentum density. Based on the canonical separation of AM, Kim has studied the transverse SAM of Airy beams [17]. Nevertheless, to the best of our knowledge, the canonical momentum and OAM of the Airy beams have not been reported. Moreover, the previous studies of the momentum and AM of Airy beams concentrated on the state of linear polarization. In this paper, we report a study of the momentum, SAM and OAM of circularly polarized Airy beams based on the canonical approach. As we know, the polarization is one of the most important properties of light beam. When a circularly polarized Airy beam propagates in an inhomogeneous medium, there are some interesting polarization-dependent effects phenomena induced by the spin-orbit coupling [24]. To reveal the mechanism of spin-orbit interaction within the circularly polarized Airy beams, it is necessary to study their momentum, SAM and OAM. Further insight into the physical properties of Airy beams can be obtained by considering an additional dynamical characteristic of the light, i.e., the helicity, which can be regarded as the projection of the spin angular momentum onto the linear momentum direction [25–29]. 2. Formulation 2.1. Electric and magnetic field components of Airy beams Airy beams, a typical class of non-diffracting beams, are solutions of scalar wave equation under paraxial approximation or its quantummechanics analog, i.e. the Schrödinger equation. Like other non-diffracting beams, the ideal Airy beams are not realizable in practice since they carry infinite energy and propagate in the free space without distortion. In order to describe the propagation of the Airy beams in a more realistic way, usually a truncation of this beams family by an exponential term is adopted to obtain the finite Airy beams, which are expressed as a product of the exact Airy mode and the linear exponential function. In this paper, the Airy beams refer to the finite Airy beams. In Cartesian coordinates, assuming that the beams propagate along the z-direction, the amplitude distribution of the Airy beams in the initial plane z = 0 can be expressed as [3,5]
E (x, y , z = 0) = A i
x
Ai
x0
y
exp a0
y0
x
exp a0
x0
y
(1)
y0
in which
Ai (x) =
1 2π
∞ exp
iu 3
−∞
3
+ ixu du
(2)
is the Airy function, 0 ≤ a0 < 1 in the exponential function is a decay parameter that determines the beam propagation distance, x0 and y 0 are the scaled parameters in the x and y direction, respectively. To explore the momentum, SAM, OAM, and helicity of Airy beams, the electric and magnetic field components of Airy beams need to be determined firstly. There are several approaches, such as vector potential method, angular spectrum representation and RayleighSommerfeld diffraction integrals, to obtain expressions for the electric and magnetic field components of Airy beams. In the present work, we derive the explicit analytical expressions for the electric and magnetic field components of Airy beams by using the vector angular spectrum representation. The basic idea of such approach is to expand any a structured light beam in terms of lots of plane waves with variable amplitudes and propagation directions by Fourier transform. Specifically, the angular spectrum amplitude of a given structured light beam in the initial plane z = 0 can be obtained by taking the two-dimensional Fourier transform of the electric field in the plane z = 0, as follows
1 E kx , k y ; z = 0 = 2 4π
∞ ∞
E (x, y , z = 0) exp −i kx x + k y y
dxdy
(3)
−∞ −∞
where x and y are the Cartesian transverse coordinates, while k x and k y are the corresponding spatial frequencies coordinates. Substituting Eq. (1) into Eq. (3), and recalling the following integral formula
+∞
Ai (x) exp (ax) dx = exp a3 /3
(4)
−∞
we obtain the angular spectrum amplitude of the Airy beams
2 x0 y 0 i 2 2 2 2 3 3 3 3 3 2 − E kx , k y = exp a k x + k y exp k x + k y 3a k x + k y + − a 0 x 0 y 0 x 0 y 0 x 0 y 0 0 0 2 4π
3
3
(5)
Once the angular spectrum amplitude is obtained, the vector angular spectrum of Airy beams can be expressed as [30,31]
1 E kx , k y ; z = α xˆ + β yˆ − αkx + β k y zˆ E kx , k y ; z kz
(6)
where α and β satisfying σ = i (α β ∗ − α ∗ β) are the polarization √ parameters with√σ = ±1 for left-handed and right-handed circularly polarized light beam. In particular, the values of (α , β) with (1, i ) / 2 and (1, −i ) / 2 correspond to the left-circular polarization (L-CP)
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and right-circular polarization (R-CP), respectively. The vector angular spectrum of Airy beams in an arbitrary plane z = const can be
obtained by E kx , k y ; z = E kx , k y ; z = 0 e ik z z , where k z =
k2 − k2x − k2y refers to the Airy beams propagating into the half-space z > 0,
and k = 2π /λ is the wave number with λ being the wavelength of the beams. Then, we can calculate the electric field of the Airy beams in an arbitrary plane z = const by the inverse Fourier transform of the angular spectrum
∞ ∞ E (x, y , z) =
E kx , k y exp i kx x + k y y + k z z dkx dk y
(7)
−∞ −∞
In a similar way, we can also represent the magnetic field as
∞ ∞ H (x, y , z) =
H kx , k y exp i kx x + k y y + k z z dkx dk y
(8)
−∞ −∞
in which
H kx , k y = √
E kx , k y 1 k × Z
(9)
k
with Z = μ/ε being the wave impedance in homogeneous medium characterized by permittivity ε and permeability μ. As has been mentioned, the Airy beams are solutions of the paraxial wave equation. Such beams propagate along a certain direction z and spread out only slowly in the transverse direction. In this case, the wavevectors k are almost parallel to the z-axis and the transverse wavenumbers kx , k y are small compared to k. Therefore, we can expand k z in a series as
kz =
k2 − k2x − k2y ≈ k −
k2x + k2y
(10)
2k
Under the above paraxial approximation, Eqs. (7) and (8) can be expressed as
E (x, y , z) H (x, y , z)
∞ ∞ k2x + k2y E kx , k y = exp (ikz) exp i kx x + k y y − dkx dk y z H kx , k y 2k
(11)
−∞ −∞
In addition, for the expressions of vector angular spectrum, i.e. Eqs. (6) and (9), k z can be further simplified as k z ≈ k. Consequently, the angular spectrum components can be represented as
1 E x = α E, E y = β E, E z = − αk x − β k y E
(12)
1 1 11 β k x − αk y H x = − β E, H y = α E, Hz = E
(13)
k
Z
Z
Zk
Substituting Eq. (5) into Eqs. (12) and (13), using the paraxial vector angular spectrum representation given by Eq. (11) and the following integral formulae [32]
+∞
Ai (x) exp bx2 + cx dx =
2 π c c − exp − + 2 − b
−∞
+∞
x Ai (x) exp bx2 + cx dx = −∞
4b
8b
4b
8b
Ai
96b3
2 π c c − exp − + 2 − b
1
1 16b2
−
c
(14)
2b
1 96b3
c 1 1 c 1 1 c × − + 2 Ai − − − Ai 2 2 2b
8b
2b
16b
2b
16b
(15)
2b
we obtain the explicit analytical expressions for the electromagnetic field components of circularly polarized Airy beams as follows
E x = α A i ( T x ) A i T y exp ( M x ) exp M y exp (ikz)
(16)
E y = β A i ( T x ) A i T y exp ( M x ) exp M y exp (ikz)
(17)
i
Ez = α
+β H x = −β
2kx20 a0 + iz 2k2 x30
i 2ky 20 a0 + iz 2k2 y 30 1 Z
Ai ( T x ) +
Ai T y +
i kx0
Ai ( T x ) exp ( M x ) A i T y exp M y exp (ikz)
i ky 0
Ai T y
(18)
A i ( T x ) exp ( M x ) exp M y exp (ikz)
A i ( T x ) A i T y exp ( M x ) exp M y exp (ikz)
(19)
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Hy = α
1 Z
H z = −β
+α
i 2kx20 a0 + iz
1
2k2 x30
Z
i 2ky 20 a0 + iz
1
A i ( T x ) A i T y exp ( M x ) exp M y exp (ikz) Ai ( T x ) +
2k2 y 30
Z
i kx0
Ai T y +
i
Ai ( T x ) exp ( M x ) A i T y exp M y exp (ikz)
ky 0
(20)
Ai T y
(21)
A i ( T x ) exp ( M x ) exp M y exp (ikz)
where the parameters T x , T y , M x and M y are defined by
x
Tx =
−
x0 y
Ty =
y0
My =
4k2 x40
+
z2
−
a0 x
Mx =
z2
4k2 y 40
−
a0 z 2
ia0 z
+
ia0 z
−
x0 2k2 x40 a0 y a0 z 2 − y0 2k2 y 40
(22)
kx20
(23)
ky 20 iz3 12k3 x60 3
−
+
ia20 z
+
2kx20 ia2 z iz + 02 6 2ky 0 12k3 y 0
ixz 2kx30
+
iyz 2ky 30
(24)
(25)
2.2. Canonical momentum, AM, and helicity of Airy beams After obtaining the electric and magnetic field components, we now consider the momentum, SAM and OAM of Airy beams. Following the canonical approach for characterization of the dynamical characteristics of structured light beams [18,19,22], the momentum, SAM and OAM densities of the Airy beams in homogeneous medium characterized by permittivity ε and permeability μ can be expressed as
P= S=
1 4ω 1
Im Im
εE∗ · (∇) E + μH∗ · (∇) H
4ω L=r×P
εE∗ × E + μH∗ × H
(26) (27) (28)
where ω is the angular frequency, Im [·] denotes the imaginary parts, superscript “∗” represents the complex conjugate, and the notation A · (∇) B = A x ∇ B x + A y ∇ B y + A z ∇ B z is used. From the definitions, we can see that the momentum, SAM and OAM densities are vectors, including longitudinal and transverse components. Here, the longitudinal component is the z-component of these quantities, while the transverse components include the contributions of x- and y-components. It should be noted that Eq. (26) is the linear momentum (LM), which contributes to OAM, as can be seen from Eq. (28). In addition to the LM, SAM and OAM, the helicity is also an important dynamical quantity, which characterizes the intrinsic rotational content of electromagnetic waves [33]. In free space, the helicity is a conserved quantity. It is closely connected to the SAM of electromagnetic waves. For the simplest case of plane-wave, the helicity is equivalent to the projection of the SAM in the direction of propagation of the wave. However, for the case of structured Airy beams, a special type of electromagnetic waves, the helicity is different from the SAM. In particular, the helicity is a scalar whereas the SAM is a vector. The SAM is closely connected to the helicity flow. Note that, the helicity and its corresponding flow stem from the conservation equation [33,34]. Mathematically, the helicity density in homogeneous medium characterized by permittivity ε and permeability μ can be written as [28]
ϑ=
1 √ 2ω
εμ Im H∗ · E
(29)
Equations (26)-(29) are the central expressions for description of the momentum, SAM, OAM, and helicity of Airy beams. It is worthwhile to note that we adopt the canonical approach to describe these dynamical quantities. Compared with the kinetic momentum density defined by the Poynting vector, the canonical momentum density has a clear physical interpretation, namely it is proportional to the local gradient of the phase of the field. Moreover, the canonical description allows for the spin-orbital momentum decomposition, and the canonical AM densities have been demonstrated to be consistent with the results of a host of laser optics experiments [23]. 3. Numerical results and discussion In this section, we performed some numerical calculations to explore the canonical momentum, SAM, OAM, and helicity of circularly polarized Airy beams propagating in free space. In the calculations we set the free space wavelength of Airy beams λ = 632.8 nm, the decay parameter a0 = 0.2, and the transverse scaled parameters x0 = y 0 = 0.15 mm. For the sake of convenience in the numerical analysis, we introduce the Rayleigh distance z R = kx20 /2. Hereafter, the parameters used are the same as mentioned above, unless otherwise stated. Fig. 1 presents the LM density distributions of the circularly polarized Airy beams, where Figs. 1(a1) and (b1) show the distributions of the total LM density with different decay parameters a0 , Figs. 1(a2) and (b2) display the transverse LM density distributions in the x– y plane (z = z R ) with a0 = 0.2, and Figs. 1(a3) and (b3) correspond to the longitudinal LM density distributions. As we can see, the change of the state of polarization (from L-CP to R-CP, or vice versa) has no effect on the total and longitudinal LM densities, but leads to a redistribution of the transverse LM density. From Figs. 1(a1) and (b1), it is found that the magnitudes of the total LM density of Airy
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Fig. 1. LM density distributions of Airy beams with the state of polarization (a1)-(a3): L-CP, and (b1)-(b3): R-CP. (a1) and (b1) are the total LM density distributions, where the transverse coordinates are set as x = −0.15 mm and y = −0.25 mm. (a2), (a3) and (b2), (b3) are the transverse and longitudinal LM density distributions, respectively, where the white arrows represent the orientations of transverse LM density. (For interpretation of the colors in the figures, the reader is referred to the web version of this article.)
beams gradually decrease with the increase of the decay parameter a0 when z < 3z R . Simultaneously, the total LM density of the Airy beams will decline slowly with the propagation distance increasing. When z > 6z R , the decay parameter has little effect on momentum density. Comparing Fig. 1(a2) with Fig. 1(a3) and Fig. 1(b2) with Fig. 1(b3), it can be seen from the range of colorbar that the longitudinal LM density is dominant for Airy beams with circular polarization. Further observation, we find that the transverse LM density of main peak exhibits the crescent pattern, as shown in Figs. 1(a2) and (b2). The above analyses indicate that the state of polarization and beam parameters of the Airy beams have a much stronger effect on the LM density. Fig. 2 shows the transverse and longitudinal SAM density distributions of the circularly polarized Airy beams with the decay parameter a0 = 0.05, where the white arrows stand for the orientations of transverse SAM density. All the SAM densities are normalized by the maximum of their respective total SAM density. From the illustrations of this Figure, we can see that the change of the polarization state has a significant effect on both the transverse and longitudinal SAM densities. To be specific, when L-CP changes to R-CP, the transverse SAM density is redistributed, as shown in Figs. 2(a1) and (b1), and the direction of the longitudinal SAM density reverses with its amplitude stays the same, as shown in Figs. 2(a2) and (b2). Meanwhile, we note that the directions of longitudinal SAM densities in the x–z plane for Airy beams with L-CP and R-CP are opposite, as shown in Figs. 2(a3) and (b3). In addition, comparing Fig. 2(a1) with Fig. 2(a2) and Fig. 2(b1) with Fig. 1(b2), we can easily find that the SAM density is predominated by the longitudinal part because the beams with L-CP and R-CP carry SAM along the propagation direction. Fig. 3 illustrates the transverse and longitudinal OAM density distributions of the circularly polarized Airy beams in the x– y plane (z = z R ), where the white arrows show the orientations of transverse OAM densities. Obviously, the change of the state of polarization (from L-CP to R-CP, or vice versa) has no effect on the transverse OAM density, but leads to a reversal of the longitudinal OAM, as shown in Figs. 3(a3) and (b3). Comparison between the colorbar range of transverse and longitudinal OAM densities indicates that the OAM density of the circularly polarized Airy beams is predominated by the transverse part. It is worthwhile to note that although the longitudinal OAM density is weak, it has a non-zero contribution. This arises from the fact that the OAM is not an independent dynamical characteristic of the field because it is defined via the canonical momentum density and dependent on the choice of the coordinate origin. Finally, Fig. 4 gives the helicity density distributions of the circularly polarized Airy beams, where Figs. 4(a) and (c) present the side views and the cross sections of the helicity density distributions, and Figs. 4(b) and (d) display three-dimensional distributions of the helicity density in the x– y plane. It can be observed that the state of polarization changes from the L-CP to R-CP, the helicity densities obviously exhibit a negative value while their amplitude stay the same, which can be explained from the definition of the helicity density given by Eq. (29). It is also observed that the main peak of the helicity density in the x– y plane moves along the line x = y, i.e. points at
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Fig. 2. Transverse and longitudinal SAM density distributions of the Airy beams with the state of polarization (a1)-(a3): L-CP, and (b1)-(b3): R-CP. (a1), (b1) and (a2), (b2) are the transverse and longitudinal SAM density distributions in the x– y plane (z = z R ), and (a3) and (b3) are the longitudinal SAM density distributions in x–z plane with y = 0.
Fig. 3. Transverse and longitudinal OAM density distributions of circularly polarized Airy beams in the x– y plane (z = z R ). (a1), (a2) and (b1), (b2) are the transverse and longitudinal OAM densities of L-CP and R-CP Airy beams, respectively. (a3) and (b3) are the transverse and longitudinal OAM density distribution of circularly polarized Airy beams versus y, where x = −0.1 mm.
45◦ , as the propagation distance z increases. In addition, comparing Fig. 4 with Fig. 2, we find that the distribution of helicity density is very similar to that of longitudinal SAM density. It is worth noting that the helicity is different from the SAM, which is usually associated with the helicity flow. Specifically, the helicity is a scalar whereas the SAM is a vector. The helicity and its flow fulfill a conservation equation. 4. Conclusion To summarize, we have investigated the momentum, SAM and OAM, as well as the helicity of circularly polarized Airy beams. The scalar expression of Airy beams was given and its angular spectrum amplitude was deduced by using Fourier transform. Under the paraxial approximation, the explicit analytical expressions for the electric and magnetic field components of circularly polarized Airy beams were derived in detail by using the vector angular spectrum representation. With the aid of canonical LM, SAM, OAM, and helicity densities, we performed some numerical calculations and the results showed that the state of polarization, decay parameter, and propagation distance of the Airy beams have a significant effect on the distributions of these quantities. For circularly polarized Airy beams, the polarization state changes from the L-CP to R-CP has no effect on the longitudinal LM and transverse OAM densities, but leads to a redistribution of the transverse LM density and a reversal of the longitudinal OAM density. At the same time, when L-CP changes to R-CP, the values of longitudinal SAM and helicity densities change from positive to negative, and the transverse SAM density is redistributed. The results
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Fig. 4. The helicity density distributions of the Airy beams with the state of polarization (a) and (b): L-CP, and (c) and (d): R-CP. (a) and (c) present the side views (x–z plane) and the cross sections (x– y plane) of the helicity density distributions. (b) and (d) display three-dimensional distributions of the helicity density in the x– y plane (z = 2z R ).
obtained in this paper may provide new insights into the mechanism of spin-orbit interaction within the circularly polarized Airy beams, and find potential applications in the fields of optical micromanipulation control, and information processing. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This research was supported by the National Natural Science Foundation of China (61675159). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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