Optical angular momentum derivation and evolution from vector field superposition

Liang Fang; Jian Wang

doi:10.1364/OE.25.023364

1. Introduction

Recently, extensive attention and academic interest have been increasingly attracted by photon’s angular momentum, including spin angular momentum (SAM) and orbital angular momentum (OAM). The well-known longitudinal SAM and OAM with the same direction as the optical axis are associated with polarization states and spiral phase distribution of light beams, respectively, which are the salient intrinsic features of light [1–3]. Photon’s longitudinal OAM, as a new degree of freedom since pioneered by Allen and associates [4], has open a door to a large research fields with numerous applications in optical communication [5,6], quantum optics and information [7], optical tweezers and micromechanics [8,9], and even astronomy [10,11]. Apart from the longitudinal angular momentum, the classical transverse OAM as an extrinsic property of structured light beam is associated with beam trajectory dependent of transverse coordinates of the beam centroid, similar to mechanical AM of classics particles [12,13]. Furthermore, the new-discovered transverse SAM has attracted a rapidly growing interest [14–16]. It arises as a result of the prominent longitudinal electric field, and exhibits unique features in sharp contrast to the usual longitudinal SAM, such as the effects of spin-momentum locking and lateral forces [17–19].

It is well known that optical vector fields are characterized by spatially inhomogeneous state of polarization, and manifest the full vectorial nature of electromagnetic wave [20,21]. They can be potentially applied to particle acceleration, microscopy, and sensing because of its unique properties [22–24]. Optical vector fields are supplied with higher-order solutions of vector Helmholtz equation [25–27]. Under weakly guiding condition, fiber-guided modes carrying integer longitudinal SAM and OAM per photon can be regarded as the superposition of odd and even vector modes (i.e. eigenmodes) of optical fiber waveguides with a phase shift of $π / 2$ [28,29], uniformly spinning and spiraling the electric-field vector. However, when arbitrarily superposing two spatial vector fields, the resultant fields may exhibit distinct features on polarization state and spatial phase distribution. As a result, the longitudinal SAM and OAM as mean values are not confined to integers [30–32]. Non-integer OAM can be also generated by using non-integer spiral phase plates [33,34], or with differential operators [35] and astigmatic elements [36], etc. Especially, in high-contrast-index optical waveguides or in the case of non-paraxial propagation, there exists a non-negligible longitudinal angular momentum shift from SAM to OAM. It is analogous to the spin-to-vortex conversion of a paraxial beam in uniaxial anisotropic crystal or the polarization dependence of both SAM and OAM in non-paraxial case due to Berry-phase shift [37–39]. Non-integer OAM and spin-orbit shift provide a discrete multi-dimensional state space for photons, which may find applications in quantum information processing, encryption and quantum digital spiral imaging [40–43]. Furthermore, in the case of non-paraxial fields, the transverse SAM increases sharply., because of the large longitudinal electric-field. The dominant ability of transverse spin is to achieve spin-controlled unidirectional propagation of light in nanofiber, surface plasmon-polaritons, and photonic-crystal waveguide [44–46], connected with the quantum spin Hall effect of light [47].

In this article, beyond paraxial approximation, we employ fully vectorial formulation derivation to comprehensively study the derivation and evolution of all optical angular momentum components from arbitrary superposition of spatial vector fields. We visualize the superposition to get arbitrary no-integer longitudinal SAM and OAM, and present the inherent spin-orbit shift and transverse SAM as a result of large longitudinal electric fields. We also show the local helix manifestation of whole spin flow as a combination of longitudinal and transverse SAM in the non-paraxial orthogonal superposition fields, and discuss the handedness of spin flow and the property of spin-momentum locking of transverse SAM.

2. Angular momentum expression

Firstly, we derive the optical angular momentum components of general optical fields in fully vectorial formulation. Based on the canonical momentum expression [48], we deduce the longitudinal OAM density in the cylindrically symmetric coordinate systems, as follows,

L_{z} = - \frac{i ε_{0}}{2 ω} (\begin{array}{l} ψ_{r}^{*} \frac{\partial}{\partial ϕ} ψ_{r} + ψ_{ϕ}^{*} \frac{\partial}{\partial ϕ} ψ_{ϕ} + ψ_{z}^{*} \frac{\partial}{\partial ϕ} ψ_{z} \\ - ψ_{r}^{*} ψ_{ϕ} + ψ_{ϕ}^{*} ψ_{r} \end{array}) {\overset{⇀}{e}}_{z},

where

ω

is the angular frequency of light,

ε_{0}

stands for permittivity in vaccum. In general,

ψ_{r}

,

ψ_{ϕ}

, and

ψ_{z}

are the radial, azimuthal, and longitudinal field components of vector fields, respectively. It represents the intrinsic vortex-dependent OAM that depends upon spatially varying phase distribution. As for the transverse OAM density, it can be given by

L_{ϕ} = - \frac{i ε_{0} r}{2 ω} (ψ_{r}^{*} \frac{\partial}{\partial z} ψ_{r} + ψ_{ϕ}^{*} \frac{\partial}{\partial z} ψ_{ϕ} + ψ_{z}^{*} \frac{\partial}{\partial z} ψ_{z}) {\overset{⇀}{e}}_{ϕ},

which is origin-dependent and belongs to the extrinsic OAM.

The SAM density with three components that is along the radial, azimuthal and longitudinal direction, respectively, can be written as

S = - \frac{i ε_{0}}{2 ω} [\begin{array}{l} (ψ_{ϕ}^{*} ψ_{z} - ψ_{z}^{*} ψ_{ϕ}) {\overset{⇀}{e}}_{r} + (ψ_{z}^{*} ψ_{r} - ψ_{r}^{*} ψ_{z}) {\overset{⇀}{e}}_{ϕ} \\ + (ψ_{r}^{*} ψ_{ϕ} - ψ_{ϕ}^{*} ψ_{r}) {\overset{⇀}{e}}_{z} \end{array}] .

The longitudinal SAM component

S_{z}

is well-known commonplace, of which the direction is determined by the polarization degrees of freedom. However, the unusual transverse SAM

S_{r}

and

S_{ϕ}

are independent on the polarization of beam, but are determined by the longitudinal electric-field. Equations (1) and (3) indicate that the longitudinal OAM has an intrinsic longitudinal-SAM-dependence, associated with the origin of spin-orbital shift. Additionally, the time averaged energy density per unit length can be given by

W = \frac{1}{2} ε ({| ψ_{r} |}^{2} + {| ψ_{ϕ} |}^{2} + {| ψ_{z} |}^{2}),

where

ε

is permittivity of waveguide material.

3. Angular momentum derivation and evolution

When superposing two azimuthal-dependent vector fields with a phase shift of $π / 2$ and a normalized energy allocation, one can get the arbitrary resultant vector field that propagates along the + z direction. It can be formulated by

ψ_{m n} (φ_{1}, φ_{2}, γ) = [\cos γ \cdot V_{m n} (φ_{1}) + i \sin γ \cdot V_{m n} (φ_{2})] \exp [i (ω t + β z)]

with

V_{m n} (φ_{k}) = E_{r} \cos (m ϕ - φ_{k}) \cdot {\overset{⇀}{e}}_{r} + E_{ϕ} \sin (m ϕ - φ_{k}) \cdot {\overset{⇀}{e}}_{ϕ} + i E_{z} \cos (m ϕ - φ_{k}) \cdot {\overset{⇀}{e}}_{z} (k = 1, 2),

where

V_{m n} (φ)

represents the azimuthal-dependent vector field that is the solution of vector Helmholtz equation [25–27],

(r, ϕ, z)

describes the cylindrical coordinate,

m

denotes the mode order or winding number, and

n

the radial order. Here we just consider the vector fields with first radial order, i.e.

n = 1

.

φ_{1}

and

φ_{2}

indicate the initial azimuthal orientation of two vector fields, respectively, which determine the spatial dislocation between two fields producing longitudinal angular momentum.

E_{r}

,

E_{ϕ}

and

E_{z}

correspond to the radial, azimuthal and longitudinal electric-field distribution functions of vector fields, respectively. They fulfill Maxwell equations and we assume they are only dependent on radial positions.

β

is the propagation constant, defined by

β = 2 π \cdot n_{e f f} / λ

with

λ

being wavelength and

n_{e f f}

being effective refractive index (

n_{e f f} = 1

in free-space). The coefficients

\cos γ

and

\sin γ

(0 \leq γ \leq π / 2)

allocate the normalized energy to two vector field components. Especially, when

m = 0

and

φ_{1, 2} = 0

, the vector field corresponds to the radially polarized mode TM₀₁; when

m = 0

and

φ_{1, 2} = π / 2

, it corresponds to the azimuthal polarized mode TE₀₁.

By substituting three scalar electric-field components $ψ_{r}$ , $ψ_{ϕ}$ and $ψ_{z}$ of Eq. (5) along $(r, ϕ, z)$ directions into Eqs. (1) and (3), beyond paraxial approximation, one can express the longitudinal OAM density

L_{z} = \frac{ε_{0} \sin 2 γ \cdot \sin Δ φ}{4 ω} [m (E_{r}^{2} + E_{ϕ}^{2} + E_{z}^{2}) +2 E_{r} E_{ϕ}],

and the classical longitudinal SAM density

S_{z} = - \frac{ε_{0}}{2 ω} \sin 2 γ \cdot \sin Δ φ \cdot E_{r} E_{ϕ} .

The transverse SAM density can be written by,

S_{ϕ} = - \frac{ε_{0}}{ω} E_{r} E_{z} [\cos^{2} γ \cos^{2} (m ϕ - φ_{1}) + \sin^{2} γ \cos^{2} (m ϕ - φ_{2})],

S_{r} = \frac{ε_{0}}{2 ω} E_{ϕ} E_{z} [\cos^{2} γ \sin 2 (m ϕ - φ_{1}) + \sin^{2} γ \sin 2 (m ϕ - φ_{2})] .

Compared with longitudinal and transverse angular momentum density from Eqs. (7)-(10), the direction of longitudinal angular momentum is determined by spatial orientation dislocation

Δ φ = φ_{2} - φ_{1}

, i.e. the polarization handedness. However, the direction of transverse SAM does not depend upon it. In the specific orthogonal superposition cases, i.e.

Δ φ = \pm π / 2

, the radial transverse SAM density is counteracted so that

S_{r} = 0

, while the azimuthal SAM density is independent of azimuthal orientation and manifest uniform in azimuthal positions, i.e.

S_{ϕ} = - ε_{0} E_{r} E_{z} / ω

. The direction of transverse SAM only depends upon the propagation direction of light inherently regulating the phase correlation of radial and longitudinal electric-field components. It gives rise to the phenomenon of spin-momentum locking, belonging to the intrinsic property of transverse SAM, which makes sense that it enables spin-controlled unidirectional propagation of light [44–46].

The mean SAM and OAM content can be written as the ratio of the integral angular momentum to the averaged energy in the Minkowski expression form [49], as follows,

\frac{n_{i}^{2} 〈 S_{z} 〉}{〈 W 〉} = \frac{n_{i}^{2} \iint S_{z} r d r d ϕ}{\iint W r d r d ϕ} = - \frac{ς}{ω} \sin 2 γ \cdot \sin Δ φ \cdot (1 - ξ),

for SAM, and

\frac{n_{i}^{2} 〈 L_{z} 〉}{〈 W 〉} = \frac{n_{i}^{2} \iint L_{z} r d r d ϕ}{\iint W r d r d ϕ} = \frac{1}{ω} \sin 2 γ \cdot \sin Δ φ \cdot [m + ς (1 - ξ)],

for OAM, where

n_{i}

is the refractive index of optical medium, subscript

i

corresponds to different waveguide layer, and

ξ

represents the amount of spin-orbit shift in orthogonal superposition state, and given by

ξ = 1 - ς \frac{2 \int E_{r} E_{ϕ} r d r}{\int ({| E_{r} |}^{2} + {| E_{ϕ} |}^{2} + {| E_{z} |}^{2}) r d r} .

For HE_m1 modes, $ς = - 1$ , and the signs of $E_{r}$ and $E_{ϕ}$ are opposite; whereas for EH_m1 modes (including TM₀₁ and TE₀₁ modes with $m = 0$ ), $ς = 1$ , and $E_{r}$ and $E_{ϕ}$ have the same signs. In weakly guiding approximation or paraxial propagation, generally, $E_{z} ≃ 0$ , $| E_{r} | ≃ | E_{ϕ} |$ , and thus $ξ ≃ 0$ when $Δ φ = \pm π / 2$ and $γ = π / 4$ . The topological charges of OAM and SAM are $l = \pm (m - 1)$ and $s = \pm 1$ for orthogonal superposition of HE_m1 modes, whereas $l = \mp (m + 1)$ and $s = \pm 1$ for orthogonal superposition of EHm1 modes $(m \neq 0)$ . In particular, $l = \mp 1$ and $s = \pm 1$ for $T M_{01} \pm i T E_{01}$ modes $(m = 0)$ .

From the above expressions, the values of mean OAM can be taken within a continuous range, and the mean SAM can take arbitrary values from $- 1$ to $+ 1$ . The angular momentum has an intrinsic spin-orbit tangle for non-planar optical wave. The spin-orbit shift becomes noticeable in the non-paraxial case, because of the non-negligible longitudinal electric-field $E_{z}$ . The amount of spin-orbit shift can be quantified by Eq. (13). The ratio of the sum of integral longitudinal SAM and OAM to the averaged energy can be given by

\frac{n_{i}^{2} (〈 S_{z} 〉 + 〈 L_{z} 〉)}{〈 W 〉} = \frac{1}{ω} m \sin 2 γ \cdot \sin Δ φ .

It is only dependent on the superposition states and the winding order

m

. In any superposition states, it is conserved for the total longitudinal angular momentum.

Analogously, we give the expressions in terms of the ratio of the integral transverse OAM and SAM to the averaged energy, as follows,

\frac{n_{i}^{2} 〈 L_{ϕ} 〉}{〈 W 〉} = \frac{r β}{ω},

for transverse OAM,

\frac{n_{i}^{2} 〈 S_{ϕ} 〉}{〈 W 〉} = - \frac{π}{ω} \frac{\int E_{r} E_{z} r d r}{\int ({| E_{r} |}^{2} + {| E_{ϕ} |}^{2} + {| E_{z} |}^{2}) r d r},

for azimuthal SAM,

\frac{n_{i}^{2} 〈 S_{r} 〉}{〈 W 〉} = 0,

for radial SAM that has no contribution to the integral transverse SAM.

4. Angular momentum visualization

To visually illustrate the resultant vector fields in arbitrary superposition states on the basis of vector beams, the superposition state $ψ_{m n} (Δ φ, γ)$ can be described with an unit semi-sphere. Here the azimuth-dependent angle $φ_{2}$ is fixed as $φ_{2} = π / 2$ , ( $V_{m n} (π / 2)$ corresponds to the odd vector field), and $0 \leq Δ φ < 2 π$ , $0 \leq γ < π / 2$ . For instance, we plot the polarization states and longitudinal OAM values in arbitrary superposition states on the basis of HE₂₁ mode, as shown in Fig. 1. The spin-orbit shift in this case is set as $ξ = 0.12$ . Three projections of the superposition states in Cartesian coordinate system can be derived by

ψ_{21}^{1} = \cos γ \cdot \cos Δ φ \cdot ψ_{21},

ψ_{21}^{2} = \cos γ \cdot \sin Δ φ \cdot ψ_{21},

ψ_{21}^{3} = \sin γ \cdot ψ_{21},

and

{| ψ_{21}^{1} |}^{2} + {| ψ_{21}^{2} |}^{2} + {| ψ_{21}^{3} |}^{2} = {| ψ_{21} |}^{2},

where

ψ_{21}^{1}

,

ψ_{21}^{2}

,

ψ_{21}^{3}

indicate the vector mode components

{HE}_{21}^{o}

,

{HE}_{21}^{e}

, and

i {HE}_{21}^{o}

, respectively.

Fig. 1 Arbitrary polarization states and OAM evolution from arbitrary superposition states $ψ_{21} (Δ φ, γ)$ on the basis of HE₂₁ mode with the amount of spin-orbit shift $ξ = 0.12$ . Positions A and B correspond to superposition states $ψ_{21} (π / 6, π / 4)$ and $ψ_{21} (π / 2, π / 6)$ , respectively.

Download Full Size | PDF

On the semi-sphere, the yellow circle on the bottom denotes uniform vector field $V_{21} (φ_{1})$ with an unit amplitude. The yellow semi-circle across vertical axis indicates the superposed vector field of $\cos γ \cdot {HE}^{o} + i \sin γ \cdot {HE}^{o}$ . These superposition states do not carry SAM and OAM. Beyond them, any other states on the semi-spherical surface would spin and spiral the spatial electric-field vector. It gives rise to spatially-variant asymmetry of polarization ellipticity, analogous to the combination of hybrid states of polarization [50,51]. It thereby induces intriguing arbitrary non-integer values of mean SAM and OAM per photon. The colour map on semi-spherical surface represents different mean OAM values per photon. The yellow semi-circle divides the semi-spherical surface into two parts. The polarization orientation of the first quadrant is characterized by left handedness, and that of the second quadrant is right handedness. In the center of two quadrants, i.e. $ψ_{21} (π / 2, π / 4)$ and $ψ_{21} (- π / 2, π / 4)$ as orthogonal superposition states, the polarization states display uniform ellipse distribution with an azimuthal symmetry. This non-paraxial superposed vector field carrying non-integer OAM can be found in high-contrast-index waveguides, for instance, hollow ring-core fiber [52–54], where exists considerable spin-orbit shift. It degrades into the fully circular polarization carrying integer SAM and OAM in the weakly guiding fibers due to the nearly identical radial and azimuthal field components. In reverse, when $γ = 0$ or $π / 2$ , and $η = \pm 1$ , the resultant vector fields can describe fiber-guided vector modes and can be combined by two purely circularly polarized OAM modes [55,56]. Note that when $φ_{2}$ varies, it changes the azimuthal angle of the overall superposition fields, but does not affect the investigation on SAM and OAM. In Fig. 2, we present the mean longitudinal angular momentum (including SAM, OAM and total angular momentum (TAM)) value evolution per photon with arbitrary superposition states on the basis of HE₂₁ and EH11 modes, respectively. It is worth pointing out that there is a noticeable difference between the resultant fields by Eq. (5) and the fields described on higher-order Poincaré sphere [55,56].

Fig. 2 Mean longitudinal angular momentum per photon versus $γ$ and $Δ φ$ on the basis of vector modes HE₂₁ (a) and EH₁₁ (b), both with $ξ = 0.12$ .

Download Full Size | PDF

It is worth pointing out that there is a noticeable difference between the resultant fields by Eq. (5) and the fields described on higher-order Poincaré sphere [55,56]. On higher-order Poincaré sphere, two OAM modes with opposite SAM and OAM states serve as the mode bases. Nevertheless, there is only one azimuthal orientation dependence linked to two mode bases. However, the resultant vector fields here are on the basis of two vector modes, both of them are azimuthal-orientation dependent. It can be equivalent to presentation of higher-order Poincaré sphere if two azimuthal angles $φ_{1}$ and $φ_{2}$ here are constrained with $Δ φ = π / 2$ in Eq. (5). The case of superposition based on two uniform vector fields in this article has an additional degree of freedom in terms of optical polarization states. It can give rise to periodically no-uniform polarization ellipses in different azimuthal positions for transverse fields, such as superposition states $ψ_{21} (π / 6, π / 4)$ and $ψ_{21} (π / 2, π / 6)$ in Fig. 1, while the polarization ellipses are uniform on the higher-order Poincaré sphere.

If ignoring the non-dominant longitudinal field, the arbitrary superposition field written as Eq. (5) can be expanded to eight uniform field components by Jones vector carrying integer OAM and SAM with different amplitude weight and initial phases, as follows,

\begin{array}{l} ψ_{m n} (φ_{1}, φ_{2}, γ) = \frac{(1 + η)}{2 \sqrt{1 + η^{2}}} (\cos γ e^{- i φ_{1}} + i \sin γ e^{- i φ_{2}}) e^{i (m + 1) ϕ} σ^{-} + \\ \frac{(1 - η)}{2 \sqrt{1 + η^{2}}} (\cos γ e^{- i φ_{1}} + i \sin γ e^{- i φ_{2}}) e^{i (m - 1) ϕ} σ^{+} + \frac{(1 - η)}{2 \sqrt{1 + η^{2}}} (\cos γ e^{i φ_{1}} + i \sin γ e^{i φ_{2}}) e^{- i (m - 1) ϕ} σ^{-}, \\ + \frac{(1 + η)}{2 \sqrt{1 + η^{2}}} (\cos γ e^{i φ_{1}} + i \sin γ e^{i φ_{2}}) e^{- i (m + 1) ϕ} σ^{+} \end{array}

where

η

is the average ratio of the azimuthal to radial field component related to electric amplitude, and can be defined by

η ≃ \pm i \sqrt{\frac{\int_{0}^{\infty} \int_{0}^{2 π} E_{ϕ} \cdot H_{r} r d r d ϕ}{\int_{0}^{\infty} \int_{0}^{2 π} E_{r} \cdot H_{ϕ} r d r d ϕ}},

where

H_{r}

and

H_{ϕ}

are the radial and azimuthal magnetic field functions of vector fields.

σ^{\pm} = {[1, \pm i]}^{T}

indicates Jones matrix where superscript

T

means transpose, corresponding to left and right circular polarization carrying SAM of + 1 and −1

ℏ

per photon, respectively. Especially, for example, when

γ = 0

or

π / 2

, and

η = \pm 1

, the arbitrary superposition field degrades into vector field that can be combined by two uniform fields carrying integer OAM and SAM with equal energy [56].

Finally, we investigate the whole SAM of the orthogonal superposition fields $ψ_{m n} (\pm π / 2, π / 4)$ in the non-paraxial case. For non-paraxial fields, TM₀₁ and TE₀₁ modes are split, whereas even and odd HE/EH modes are degenerated in terms of the effective refractive index. Considering the combination of longitudinal and transverse SAM density, we plot the local spin flow trajectories of uniform vector field TM₀₁ that is a closed loop as shown in Fig. 3(a). Note that the TE₀₁ vector field does not have transverse SAM, because of no longitudinal electric field, as shown in Fig. 3(b). We further present the local spin flow of orthogonal superposed fields $ψ_{m n} (\pm π / 2, π / 4)$ on the basis of HE₁₁ and HE₂₁ modes, as shown in Figs. 4(a) and 4(b), and Figs. 5(a) and 5(b), respectively. In contrast, we also give the spatial phase distribution of these fields that reflect optical OAM states. These local spin trajectories are characterized by helix as a combination of longitudinal and transverse SAM. It arises as a result of phase offset of transverse SAM lobes along the propagation around the optical axis based on the definition of SAM from Eqs. (3) and (9), which is also discussed in our previous work [57]. The fold number of local spin trajectories equals to the azimuthal order $m$ of vector fields. The local spin trajectories of orthogonal superposed fields with $ψ_{m n} (+ π / 2, π / 4)$ , $(m \neq 0)$ are characterized by right-handed helix with the same direction as the propagation direction aligned with the linear momentum $k$ , as shown in Figs. 4(a) and 5(a). However, for superposed fields with $ψ_{m n} (- π / 2, π / 4)$ , $(m \neq 0)$ , the local spin trajectories manifest left-handed helix with the reverse direction as the propagation direction, opposite to the linear momentum $k$ , as shown in Figs. 4(b) and 5(b). Significantly, in all above cases, the direction of transverse SAM remains the same, and is locked to the linear momentum $k$ , which is the intrinsic property of transverse SAM [17–19].

Fig. 3 Local spin flow trajectories of (a) TM₀₁ mode and (b) TE₀₁ mode.

Download Full Size | PDF

Fig. 4 Contrast between OAM and SAM. Spatial phase distribution and local spin flow trajectories of superposed fields for (a) $ψ_{11} (+ π / 2, π / 4)$ and (d) $ψ_{11} (- π / 2, π / 4)$ .

Download Full Size | PDF

Fig. 5 Contrast between OAM and SAM. Spatial phase distribution and local spin flow trajectories of superposed fields for (a) $ψ_{21} (+ π / 2, π / 4)$ and (d) $ψ_{21} (- π / 2, π / 4)$ .

Download Full Size | PDF

5. Conclusion

In conclusion, the resultant field by arbitrarily superposing vector fields has been comprehensively investigated in terms of various optical angular momentum, including the longitudinal and transverse, spin and orbital components. For a non-paraxial superposed field, there are noticeably inherent spin-orbit shift and transverse SAM due to the large longitudinal electric-field. We can obtain arbitrary non-integer OAM and fractional SAM by arbitrarily superposing the vector fields. The whole local spin flow of TM₀₁ is characterized by closed loop, and that of higher-order superposing fields manifest multiple-fold helical trajectories, of which the fold number equals the azimuthal order. Such helical local spin can enrich optical force and torque as a potential new degree of freedom that enables the exploitation of the next generation photonic traps. Our studies provide a systematic physical insight into optical angular momentum derived from vector field superposition, including various angular momentum components, spin-orbit shift and evolution, and the property of transverse SAM. It may facilitate the development of optical vector fields and optical angular momentum in fundamental studies and various applications, such as optical communications, optical manipulation, and quantum application, etc.

Funding

National Natural Science Foundation of China (NSFC) (11574001, 11274131); National Basic Research Program of China (973 Program, 2014CB340004); National Program for Support of Top-Notch Young Professionals; Program for New Century Excellent Talents in University (NCET-11-0182).

References and links

1. M. J. Padgett, J. Courtial, and L. Allen, “Light’s orbital angular momentum,” Phys. Today 57(5), 35–40 (2004). [CrossRef]

2. S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. 2(4), 299–313 (2008). [CrossRef]

3. A. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011). [CrossRef]

4. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]

5. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015). [CrossRef]

6. J. Wang, “Advances in communications using optical vortices,” Photon. Res. 4(5), B14–B28 (2016). [CrossRef]

7. J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010). [CrossRef] [PubMed]

8. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef] [PubMed]

9. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011). [CrossRef]

10. G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101(10), 100801 (2008). [CrossRef] [PubMed]

11. M. Harwit, “Photon orbital angular momentum in astrophysics,” Astrophys. J. 597(2), 1266–1270 (2003). [CrossRef]

12. M. A. Player, “Angular momentum balance and transverse shift on reflection of light,” J. Phys. Math. Gen. 20(12), 3667–3678 (1987). [CrossRef]

13. V. G. Fedoseyev, “Conservation laws and transverse motion of energy on reflection and transmission of electromagnetic waves,” J. Phys. Math. Gen. 21(9), 2045–2059 (1988). [CrossRef]

14. K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014). [CrossRef] [PubMed]

15. K. Y. Bliokh and F. Nori, “Transverse spin of a surface polariton,” Phys. Rev. A 85(6), 1577–1581 (2012). [CrossRef]

16. Z. Wang, B. Chen, R. Wang, S. Shi, Q. Qiu, and X. Li, “New investigations on the transverse spin of structured optical fields,” https://arxiv.org/abs/1605.03103 (2016).

17. S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 3307 (2014). [PubMed]

18. F. Kalhor, T. Thundat, and Z. Jacob, “Universal spin-momentum locked optical forces,” Appl. Phys. Lett. 108(6), 061102 (2016). [CrossRef]

19. T. Van Mechelen and Z. Jacob, “Universal spin-momentum locking of evanescent waves,” Optica 3(2), 118–126 (2016). [CrossRef]

20. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–67 (2009). [CrossRef]

21. N. Radwell, R. D. Hawley, J. B. Götte, and S. Franke-Arnold, “Achromatic vector vortex beams from a glass cone,” Nat. Commun. 7, 10564 (2016). [CrossRef] [PubMed]

22. W. Kimura, G. Kim, R. Romea, L. Steinhauer, I. Pogorelsky, K. Kusche, R. Fernow, X. Wang, and Y. Liu, “Laser Acceleration of Relativistic Electrons Using the Inverse Cherenkov Effect,” Phys. Rev. Lett. 74(4), 546–549 (1995). [CrossRef] [PubMed]

23. A. F. Abouraddy and K. C. Toussaint Jr., “Three-Dimensional Polarization Control in Microscopy,” Phys. Rev. Lett. 96(15), 153901 (2006). [CrossRef] [PubMed]

24. S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2(10), 864–868 (2015). [CrossRef]

25. C. Tsao, Optical Fibre Waveguide Analysis (Oxford University, 1992); Part 3.

26. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

27. D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21(1), 9–11 (1996). [CrossRef] [PubMed]

28. P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Phys. Rev. Lett. 96(4), 043604 (2006). [CrossRef] [PubMed]

29. S. Ramachandran and P. Kristensen, “Optical vortices in fiber,” Nanophotonics 2(5-6), 455–477 (2013). [CrossRef]

30. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001). [CrossRef] [PubMed]

31. R. Zambrini and S. M. Barnett, “Angular momentum of multimode and polarization patterns,” Opt. Express 15(23), 15214–15227 (2007). [CrossRef] [PubMed]

32. J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16(2), 993–1006 (2008). [CrossRef] [PubMed]

33. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6(2), 259–268 (2004). [CrossRef]

34. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004). [CrossRef]

35. I. Martinez-Castellanos and J. C. Gutiérrez-Vega, “Vortex structure of elegant Laguerre-Gaussian beams of fractional order,” J. Opt. Soc. Am. A 30(11), 2395–2400 (2013). [CrossRef] [PubMed]

36. A. M. Nugrowati, W. G. Stam, and J. P. Woerdman, “Position measurement of non-integer OAM beams with structurally invariant propagation,” Opt. Express 20(25), 27429–27441 (2012). [CrossRef] [PubMed]

37. E. Brasselet, Y. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Dynamics of optical spin-orbit coupling in uniaxial crystals,” Opt. Lett. 34(7), 1021–1023 (2009). [CrossRef] [PubMed]

38. K. Y. Bliokh, M. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 13289 (2010). [CrossRef]

39. K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems,” Opt. Express 19(27), 26132–26149 (2011). [CrossRef] [PubMed]

40. S. S. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95(24), 240501 (2005). [CrossRef] [PubMed]

41. A. Vaziri, G. Weihs, and A. Zeilinger, “Superpositions of the orbital angular momentum for applications in quantum experiments,” J. Opt. B Quantum Semiclassical Opt. 4(2), S47–S51 (2002). [CrossRef]

42. S. S. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92(21), 217901 (2004). [CrossRef] [PubMed]

43. L. Chen, J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light Sci. Appl. 3(3), e153 (2014). [CrossRef]

44. J. Petersen, J. Volz, and A. Rauschenbeutel, “Chiral nanophotonic waveguide interface based on spin-orbit interaction of light,” Science 346(6205), 67–71 (2014). [CrossRef] [PubMed]

45. D. O’Connor, P. Ginzburg, F. J. Rodríguez-Fortuño, G. A. Wurtz, and A. V. Zayats, “Spin-orbit coupling in surface plasmon scattering by nanostructures,” Nat. Commun. 5, 5327 (2014). [CrossRef] [PubMed]

46. B. le Feber, N. Rotenberg, and L. Kuipers, “Nanophotonic control of circular dipole emission,” Nat. Commun. 6, 6695 (2015). [CrossRef] [PubMed]

47. K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348(6242), 1448–1451 (2015). [CrossRef] [PubMed]

48. K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 4, 592 (2015).

49. F. L. Kien, V. I. Balykin, and K. Hakuta, “Angular momentum of light in an optical nanofiber,” Phys. Rev. A 73(5), 053823 (2006). [CrossRef]

50. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007). [CrossRef]

51. H.-T. Wang, X.-L. Wang, Y. Li, J. Chen, C.-S. Guo, and J. Ding, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18(10), 10786–10795 (2010). [CrossRef] [PubMed]

52. S. Golowich, “Asymptotic theory of strong spin-orbit coupling in optical fiber,” Opt. Lett. 39(1), 92–95 (2014). [CrossRef] [PubMed]

53. L. Fang and J. Wang, “Mode conversion and orbital angular momentum transfer among multiple modes by helical gratings,” IEEE J. Quantum Electron. 52(8), 6600306 (2016). [CrossRef]

54. Z. Zhang, J. Gan, X. Heng, Y. Wu, Q. Li, Q. Qian, D. Chen, and Z. Yang, “Optical fiber design with orbital angular momentum light purity higher than 99.9%,” Opt. Express 23(23), 29331–29341 (2015). [CrossRef] [PubMed]

55. G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011). [CrossRef] [PubMed]

56. S. Chen, X. Zhou, Y. Liu, X. Ling, H. Luo, and S. Wen, “Generation of arbitrary cylindrical vector beams on the higher order Poincaré sphere,” Opt. Lett. 39(18), 5274–5276 (2014). [CrossRef] [PubMed]

57. L. Fang and J. Wang, “Intrinsic transverse spin angular momentum of fiber eigenmodes,” Phys. Rev. A 95(5), 053827 (2017). [CrossRef]

Optical angular momentum derivation and evolution from vector field superposition

Abstract

1. Introduction

2. Angular momentum expression

3. Angular momentum derivation and evolution

4. Angular momentum visualization

5. Conclusion

Funding

References and links

Cited By

Figures (5)

Equations (23)

Optics Express