Extraordinary spin density and energy back-flow under interference

Zhen-Lai Wang; Dan-Dan Lian; Xiang-Song Chen

doi:10.1364/OE.26.033712

1. Introduction

Light, or more generally an electromagnetic wave, can carry energy, momentum and angular momentum, and all these properties are well confirmed by light-matter interactions. Now it is well-known that a light beam can carry two distinct forms of angular momentum. One, called spin angular momentum, is related with the polarization of light; the other, called orbital angular momentum, is related with light’s wave-front shape [1–4]. However, in the theoretical framework, it is not exactly clear how to handle the two different forms of angular momentum. One familiar expression to describle the optical angular momentum in classical electrodynamics [5] is written as ∫

J = \int d^{3} x [r \times (E \times B)],

and therefore the total optical angular momentum density j_poy = r ×(E × B) = r × p is expressed in an orbital structure with the Poynting vector p which is treated as the energy flux density of light:

p = E \times B .

Another expression of the optical angular momentum is commonly used in laser optics

J = \int d^{3} x E \times A + \int d^{3} x E^{i} (r \times \nabla) A^{i},

which splits the total optical angular momentum into spin and orbital angular momentum, where A is the vector potential of magnetic field. Here the index i runs over the three spatial coordinates. The expression given by Eq. (3) can be derived from Noether’s theorem in virtue of the space-rotation symmetry and is called canonical version of optical angular momentum. As Eq. (3) manifestly shows, each term is gauge-dependent and so not measurable in principle. A gauge-invariant version can be yielded by introducing the gauge-invariant part A_⊥ of the vector potential A [6–8]:

J = \int d^{3} x E \times A_{⊥} + \int d^{3} x E^{i} (r \times \nabla) A_{⊥}^{i},

Thus, Eq. (4) defines the total angular momentum density by j_gic = s_gic + l_gic with the gauge-invariant canonical (GIC) spin density

s_{gic} = E \times A_{⊥}

and the GIC orbital angular momentum density

l_{gic} = E^{i} (r \times \nabla) A_{⊥}^{i} = r \times p_{gic},

where the GIC momentum density reads

p_{gic} = E^{i} \nabla A_{⊥}^{i} .

For the free electromagnetic field, the integrated angular momentum of light in Eqs. (1) and (4) are identical, but the angular momentum density j_poy differs from j_gic by a surface term. Recently, based on the measurability of the local spin and orbital angular momentum density, a measurement scheme was proposed by E. Leader [9] for testing which one between j_poy and j_gic is the effective expression of optical angular momentum density. Analysing a circularly polarised Laguerre-Gaussian beam in the paraxial approximation, he found that j_poy and j_gic can yield different spin angular momentum densities. For j_poy its ${\bar{s}}_{poy, z}$ , the time average of the component of “spin density” along the propagation direction, can display two different signs as function of the distance from the optical axis, while for j_gic its ${\bar{s}}_{gic, z}$ possesses a single sign.

In this work, we analyse the interference of three plane waves propagating in the same plane with positive momentum in their common propagation direction (z-axis). The GIC spin density ${\bar{s}}_{gic, z}$ can also display two different signs locally under interference of plane waves even if all of the plane waves possess the same polarization state. In other words, ${\bar{s}}_{gic, z}$ can locally reverse its orientation. Moreover, the interference field might possess a transverse spin density (out-of-plane) which is perpendicular to the propagation plane and have more rich relations with the polarization. More interestingly, the Poynting vector can not only produce counter-intuitive back-flow, but also form a circular motion in the propagation plane, which implies a transverse orbital angular momentum (out-of-plane) related to the polarization.

2. Extraordinary spin density and energy back-flow under interference

Throughout this paper, we focus on monochromatic plane waves in free space. Specifically, we consider the interference of three plane waves propagating in the x-z plane. The electric and magnetic fields of the three waves are, respectively

E_{a} (r, t) = Re [ℰ_{a} (r) e^{- i ω t}], (a = 1, 2, 3)

ℰ_{a} (r) = E_{0} e^{i ϕ_{a} (r)} χ_{a}, ℬ_{a} (r) = \frac{k_{a}}{ω} \times ℰ_{a} (r)

{\hat{k}}_{a} \cdot e_{y} = 0, χ_{a} = [e_{y} + i σ_{a} ({\hat{k}}_{a} \times e_{y})] / \sqrt{1 + σ_{a}^{2}}

Here (E_a, B_a) represent the real electric and magnetic fields, expressed in terms of complex fields (ℰ_a, ℬ_a), the phase factors ϕ_a(r) = k_a · r with k_a the wave vectors, ${\hat{k}}_{a} = k_{a} / | k_{a} |$ is the corresponding unit wave vectors, E₀ is the real electric-field amplitude, e_y (e_z, e_x) is the unit vector of y (x, z) axis, and σ ∈ [−1, 1] determines the polarization state. For simplicity, we set k₃ in the z axis and k_1,2 to be distributed mirror-symmetrically about k₃ in the x-z plane, as depicted in Fig. 1:

k_{1, 2} = k (\cos θ e_{z} \pm \sin θ e_{x}), k_{3} = k e_{z},

where k = ω/c = ω (taking c = 1) is the wave number, and θ denotes the angle between k₁ (or k₂) and k₃(θ ∈[0, π/2]). The resulting real fields are

E = \sum_{a = 1}^{3} E_{a} (r, t)

and

B = \sum_{a = 1}^{3} B_{a} (r, t)

, and the resulting complex fields

ℰ = \sum_{a = 1}^{3} ℰ_{a} (r)

and

ℬ = \sum_{a = 1}^{3} ℬ_{a} (r)

.

Fig. 1 Interference of three plane waves all propagating in the x-z plane with equal amplitudes and different wave vectors k₁, k₂ and k₃. The angle θ denotes the angle between k_1,2 and k₃.

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For a monochromatic-wave case, one has the time averages of the Poynting vector, the GIC momentum density and the GIC spin density from Eqs. (2), (5) and (7):

\bar{p} = \bar{E \times B} = \frac{1}{2} Re [ℰ^{*} \times ℬ],

{\bar{p}}_{gic} = \bar{E^{i} \nabla A_{⊥}^{i}} = \frac{1}{2 ω} Im [ℰ_{i}^{*} \nabla ℰ_{i}],

{\bar{s}}_{gic} = 2 {\bar{s}}_{e} = \frac{1}{2 ω} Im [ℰ^{*} \times ℰ] .

Recently, the dual-symmetry between electric and magnetic fields has re-attracted considerable attention and the optical spin density was suggested in the so-called dual-symmetric form [10, 11]:

{\bar{s}}_{dua} = {\bar{s}}_{e} + {\bar{s}}_{m} = \frac{1}{4 ω} Im [ℰ^{*} \times ℰ + ℬ^{*} \times ℬ] .

Here, ${\bar{s}}_{e}$ is the so-called electric spin density and ${\bar{s}}_{m}$ is the magnetic spin density. One can find that ${\bar{s}}_{gic}$ is only determined by the electric field but ${\bar{s}}_{dua}$ is composed of two parts determined separately by electric and magnetic fields.

We now insert the resulting complex electric and magnetic fields (ℰ, ℬ) into Eqs. (11)–(14) and obtain

\bar{p} = \sum_{a, b = 1}^{3} \frac{E_{0}^{2}}{2 \sqrt{(1 + σ_{a}^{2}) (1 + σ_{b}^{2})}} [(1 + σ_{a} σ_{b}) \cos (δ_{a b}) {\hat{k}}_{a} - σ_{a} \sin (δ_{a b}) [({\hat{k}}_{a} \times {\hat{k}}_{b}) \cdot e_{y}] e_{y}],

{\bar{p}}_{gic} = \sum_{a, b = 1}^{3} \frac{E_{0}^{2}}{2 \sqrt{(1 + σ_{a}^{2}) (1 + σ_{b}^{2})}} (1 + σ_{a} σ_{b} {\hat{k}}_{a} \cdot {\hat{k}}_{b}) \cos (δ_{a b}) {\hat{k}}_{a},

{\bar{s}}_{e} = \sum_{a, b = 1}^{3} \frac{E_{0}^{2}}{4 ω \sqrt{(1 + σ_{a}^{2}) (1 + σ_{b}^{2})}} [2 σ_{a} \cos (δ_{a b}) {\hat{k}}_{a} - σ_{a} σ_{b} \sin (δ_{a b}) [({\hat{k}}_{a} \times {\hat{k}}_{b}) \cdot e_{y}] e_{y}],

{\bar{s}}_{m} = \sum_{a, b = 1}^{3} \frac{E_{0}^{2}}{4 ω \sqrt{(1 + σ_{a}^{2}) (1 + σ_{b}^{2})}} [2 σ_{a} \cos (δ_{a b}) {\hat{k}}_{b} - \sin (δ_{a b}) [({\hat{k}}_{a} \times {\hat{k}}_{b}) \cdot e_{y}] e_{y}] .

Here δ_ab = ϕ_a − ϕ_b are the corresponding phase differences, all of which vary in the x-z plane.

As we can see from Eqs. (15)–(18), the time averages of the Poynting vector, the electric and magnetic spin densities have y-component while the resulting fields propagate entirely in the x-z plane. Note that the transverse y-component of the Poynting vector ${\bar{p}}_{y}$ depends upon the polarization state, while the transverse y-component of the GIC momentum density ${\bar{p}}_{gic, y} = 0$ . More interestingly, the y-components of the electric and magnetic spin densities ${\bar{s}}_{e, y}$ and ${\bar{s}}_{m, y}$ exhibit a strong electric-magnetic asymmetry: ${\bar{s}}_{e, y}$ is dependent on the polarization, while ${\bar{s}}_{m, y}$ is independent of the polarization. Under the electromagnetic duality transformation on the resulting fields, ${\bar{s}}_{e, y}$ (or ${\bar{s}}_{gic, y}$ ) will become independent of the polarization but ${\bar{s}}_{m, y}$ dependent on the polarization. In [12], the transverse optical spin density yielded by interference of two plane waves might also show the electric-magnetic asymmetry, in which case the transverse electric and magnetic spin densities actually exhibit different local distributions, but are both independent of the helicity parameter σ. However, the strong electric-magnetic asymmetry presented here is due to the fact that the transverse electric and magnetic spin densities can have different relations with the polarization state. Typically, the spin angular momentum of light is longitudinal and polarization-dependent, but here we show a transverse (out-of-plane) polarization-dependent and polarization-independent spin density. On the topic of transverse optical spin density, a good review paper was written by Bliokh and Nori [13].

More remarkably, from Eqs. (15)–(18) one can get that the Poynting vector $\bar{p}$ and the spin density can reverse their direction along the main propagating direction. For example, if all σ_a = 1, the Poynting vector $\bar{p}$ , the electric and magnetic spin densities ${\bar{s}}_{e}$ and ${\bar{s}}_{m}$ have the same structure. For clarity, we plot the components of $\bar{p}$ , ${\bar{s}}_{e}$ , ${\bar{s}}_{m}$ and ${\bar{p}}_{gic}$ projected in the x-z plane when all σ_a = 1 in Fig. 2. Seen from Fig. 2(a), it is clear that ${\bar{p}}_{z}$ and ${\bar{s}}_{z}$ can be negative in some regions, telling that the Poynting vector $\bar{p}$ could be opposite to the main propagation direction of the three incident waves and that the spin density can experience the transition of sign locally.

Fig. 2 $\bar{p}$ , $\bar{s}$ (short for ${\bar{s}}_{gic}$ , ${\bar{s}}_{e}$ and ${\bar{s}}_{m}$ ) and ${\bar{p}}_{gic}$ projected in the x-z plane when all σ_a = 1. (a) If all σ_a = 1, the Poynting vector and spin density have the same structure; omitting their constant factors and plotting the distribution of the 2-D vectors of $({\bar{p}}_{x}, {\bar{p}}_{z})$ and $({\bar{s}}_{x}, {\bar{s}}_{z})$ with θ = 1.4 and k = 2π. The red and green points are the singularities of $({\bar{p}}_{x}, {\bar{p}}_{z})$ and $({\bar{s}}_{x}, {\bar{s}}_{z})$ . The circular motions take place around the red points. (b) The distribution of the 2-D vector of $({\bar{p}}_{gic, x}, {\bar{p}}_{gic, z})$ with θ = 1.4 and k = 2π.

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Figure 2(a) shows vividly that the Poynting vector $\bar{p}$ could form a circle near the region of negative Poynting vector. In fact, the circle is yeilded around a point where $\bar{p} = 0$ , and the point is the so-called Poynting singularity [14, 15]. Naturally, the Poynting vector $\bar{p}$ might change from a forward to a reversed flow at the region of Poynting singularity. For the Poynting version of the optical angular momentum density ${\bar{j}}_{poy} = r \times \bar{p}$ , the circular motion of Poynting vector $\bar{p}$ suggests the existence of transverse y-component angular momentum density. It is clear from Eq. (15) that, when all σ_a = 0, the components of the Poynting vector $({\bar{p}}_{x}, {\bar{p}}_{z})$ do keep the same structure as these components do when all σ_a = 1, so the same circular motion of Poynting vector $\bar{p}$ yields the same transverse y-component angular momentum density. On the other hand, when all σ_a = 0, from Eqs. (15) and (16), one can get $\bar{p} = {\bar{p}}_{gic}$ , which hints that the transverse y-component angular momentum density does not contain spin ingredient but fully belongs to “orbital” angular momentum density. Thus, when all σ_a = 1, j_poy implies a new kind of “orbital” angular momentum density which is polarization-dependent. However, when all σ_a = 1, the GIC momentum density ${\bar{p}}_{gic}$ cannot produce vortex in the x-z plane, as depicted in the Fig. 2(b). Therefore, for ${\bar{j}}_{gic} = r \times {\bar{p}}_{gic} + s_{gic}$ , only the spin part ${\bar{s}}_{gic}$ leads to its y-component angular momentum density but the orbital part do not.

Also, Fig. 2(a) informs us that the circular motion (vortex) always emerges in pairs with opposite rotation directions. They mutually cancel out each other and make the integrated transverse angular momentum vanish. If all σ_a = 1, the electric and magnetic spin densities ${\bar{s}}_{e}$ and ${\bar{s}}_{m}$ have the same structure as the Poynting vector $\bar{p}$ , and they of course form spin vortexes in the x-z plane. The reversed Poynting vector $\bar{p}$ , the electric and magnetic spin densities ${\bar{s}}_{e}$ and ${\bar{s}}_{m}$ appear in the middle region of a pair of vortexes. All of these counter-intuitive phenomena are the direct result of wave interference effect.

To observe the backflow and circulation of the Poynting vector (or momentum density) and the extraordinary spin density, we need a detection scheme sensitive to local energy flow (or momentum density) and spin density. One possible experiment could involve optical forces and torque on small particles. Usually, an absorptive and nonmagnetic particle (Rayleigh particle) with dimensions much smaller than the wavelength of the scattered light is well described by an electric-dipole [12, 16–23]. From thise ffective model, one can get the time averaged optical force f and torque τ on the particle:

f = \frac{Re (α)}{2} \nabla \bar{E^{2}} + ω Im (α) {\bar{p}}_{gic}, τ = ω Im (α) {\bar{s}}_{gic} .

The first term in the optical force expression stands for the gradient force and the second one the scattering force. Here the complex polarizability α of the particle depends on the nature of the particle. Therefore, the extraordinary spin density and Poynting vector $\bar{p}$ (is equal to ${\bar{p}}_{gic}$ if all σ_a = 0.) can be directly measured by the local interaction between light and the test-particle.

3. Conclusion

In conclusion, we have shown the “local optical spin density” of the GIC expression also might be counter-intuitively opposite to the integrated spin orientation under interference of plane waves. Further analysis shows that the interference fields can produce an extraordinary Poynting vector, which can not only be negative (the energy flux is opposite to the propagation direction), but also form a circular motion (vortex) and result in local transverse (out-of-plane) “orbital” angular momentum related to the polarization. Furthermore, this simple optical system also acquires a transverse spin density which shows strong electric-magnetic asymmetry; for the transverse electric and magnetic spin densities, one of them is polarization-dependent, while the other is polarization-independent. All of these counter-intuitive findings are the result of wave interference effect, which should have useful implications in fundamental studies and applications of light.

Funding

National Natural Science Foundation of China (11535005, 11275077).

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Extraordinary spin density and energy back-flow under interference

Abstract

1. Introduction

2. Extraordinary spin density and energy back-flow under interference

3. Conclusion

Funding

References

Cited By

Figures (2)

Equations (20)

Optics Express