Theory of paraxial optical skyrmions

The skyrmion field components have a simple general expression in terms of μ: Σ_l = 4ε_lmn/(1+|μ|^2)^2 ∂R(μ)/∂x_m ∂I(μ)/∂x_n = ε_lmn/(1+|μ|^2)^2 (i ∂μ/∂x_m ∂μ*/∂x_n − i ∂μ*/∂x_m ∂μ/∂x_n).

“(3.7) Σ l = 4ε lmn (1 + | μ | 2 ) 2 ∂ℜ(μ) ∂x m ∂ℑ(μ) ∂x n , = ε lmn (1 + | μ | 2 ) 2 i ∂μ ∂x m ∂μ * ∂x n -i ∂μ * ∂x m ∂μ ∂x n .”

A pattern with a skyrmion number n will typically be associated with 2n singular lines or strings.

“A pattern with a skyrmion number n will typically be associated with 2n singular lines or strings.”

A special case exists in which there is a single string, of strength n.

“A special case exists, however, in which there is a single string, of strength n.”

Skyrmion field lines are basis independent in that they are unchanged by a global rotation of the Stokes vector on the Poincaré sphere.

“Second, the skyrmion field lines are basis independent in that they are unchanged by a global rotation of the Stokes vector on the Poincaré sphere.”

The skyrmion field is invariant under a global rotation of the polarization on the Poincaré sphere.

“This statement, which is readily confirmed by direct calculation, means that each optical skyrmion is part of a family with differently oriented polarizations but identical skyrmion fields [13].”

• Also stated in: § 2. Skyrmions and their mathematical properties · plain

  “the skyrmion field is invariant under a global rotation of the polarization on the Poincaré sphere (the same rotation applied to every point).”

• cites [13]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Selecting the two orthogonal spatial modes to be Laguerre-Gaussian modes means that the function μ reduces to f(ρ,z)e^{iΘ(ρ,z)}e^{i(l1-l0)φ}.

“Selecting the modes u 0 (r) and u 1 (r) to be Laguerre-Gaussian modes means that the function μ reduces to the simple form”

The skyrmion field is transverse (divergenceless): ∇·Σ = 0.

“The first thing to note is that the field is transverse, or divergenceless [13]:”

• Also stated in: § 2. Skyrmions and their mathematical properties · hedged

  “It then follows that ∇ ⋅ Σ = 0 everywhere (at least as long as the derivative is well-defined).”

• cites [13]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionRests on an assumption: The polarization varies continuously in space.
• AssumptionThe authors state this as an assumption: The derivative of the Stokes vector S is well-defined everywhere needed for the divergence calculation.

Skyrmion field lines are lines of constant polarization.

“Inspection of numerous polarization patterns leads to the conjecture that skyrmion field lines are lines of constant polarization.That this is indeed the case was proven in [73] but, for completeness, we give a summary of the main points here.”

• Also stated in: § 4. Skyrmion field lines · plain

  “The one remaining non-zero component of the skyrmion field at r 0 is Σ u and it follows, therefore, that the skyrmion field line at any point, r 0 , points along the direction of constant polarization.”

• Also stated in: § 8. Conclusion · plain

  “Skyrmion field lines have significance beyond their association with skyrmions: they are lines of constant polarization [73].”

• cites [73]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

In spite of the 1/2 coefficient, the skyrmion number n is guaranteed to be an integer.

“In spite of the presence of the 1/2 coefficient, n is guaranteed to be an integer.”

The skyrmion number can be expressed topologically as n = 1/2 ∑_j S_z(j) N_j − S̄_z(∞) N_∞, where N are winding numbers.

“(3.11) n = 1 2 ∑ j S z (j) N j -S ¯z (∞) N ∞ ,”

The skyrmion potential simplifies to V = -S_z ∇Φ = -S_z ∇arg(μ).

“(3.9) V = -S z ∇Φ = -S z ∇arg(μ), which is reminiscent of the form of the superfluid velocity in helium 4, which also depends on the gradient of the argument of a complex wavefunction [70,71].”

• cites [70]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [71]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The skyrmion number can be expressed as n = 1/4π (∮_{L∞} V·dl − ∑_j ∮_{L_j} V·dl), where L∞ is a large circular contour and L_j are small contours around singularities.

“(2.23) n = 1 4π L ∞ V ⋅ dl -∑ j L j V ⋅ dl ,”

In a typical skyrmionic optical beam with n = 1, the lines of constant polarization form a twisted mesh of straight lines.

“In a typical skyrmionic optical beam with n = 1, the lines of constant polarization form a twisted mesh of straight lines shown in figure 4b.”

The skyrmion number can be rewritten as a line integral of the skyrmion vector potential, n = 1/4π ∮ V · dl, with singular points omitted from the integration contour.

“We can use Stokes' theorem to rewrite our skyrmion number as a line integral of the skyrmion potential [51]:”

• Also stated in: § 2. Skyrmions and their mathematical properties · plain

  “The singular points in V must be omitted from the region of integration.”

• cites [51]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The volume integral of this divergence over the cylinder equals the surface flux, as given by equation (7.6).

“so that the volume integral of this over our cylinder gives (7.6) ∇ ⋅ Σ dΩ = 8πℓ 1 1 + f 2 (0, z + Δz) - 1 1 + f 2 (0, z) = Σ ⋅ dS ,”

For small ρ, the ratio of the mode amplitudes tends to f(0,z) exp[-i2(p1-p0) tan^{-1}((z-z0)/z_R)] e^{i2lφ}.

“For small values of ρ the ratio of the mode amplitudes tends to the value (7.3) lim ρ → 0 μ = f(0, z) exp -i2(p 1 -p 0 ) tan -1 z -z 0 z e i2ℓϕ .”

On the z-axis, as ρ→0, f(0,z) tends to the value given by equation (7.1).

“(7.1) f(0, z) = p 1 !(2p 0 + | ℓ | )! p 0 !(2p 1 + | ℓ | )! β α w 0 (z) w 1 (z) |ℓ | + 1 .”

It is straightforward to construct paraxial beams with skyrmionic structures by superposing orthogonally polarized Laguerre-Gaussian modes.

“It is straightforward to construct paraxial beams with skyrmionic structures by superposing orthogonally polarized Laguerre-Gaussian modes.”

The associated skyrmion number n(z) = 2l[1/(1+f^2(0,z)) - 1/(1+f^2(∞,z))] can take any desired value.

“It follows that the associated skyrmion number, (7.2) n(z) = 2ℓ 1 1 + f 2 (0, z) -1 1 + f 2 (∞, z) , can take any desired value.”

For small ρ, f(ρ) is proportional to ρ^{|l1|-|l0|}.

“For small values of ρ, however, f(ρ) is proportional to a power of ρ determined by the orbital angular momenta of the two modes: (6.11) f(ρ) ∝ ρ |ℓ 1 | -|ℓ 0 | .”

The flux of the skyrmion field through the surface of a cylindrical volume of radius Δρ and height Δz centred on the z-axis is given by equation (7.4), which depends only on the polarization on the z-axis through f(0,z).

“We find (7.4)Σ ⋅ dS = 4π2ℓ 1 1 + f 2 (0, z + Δz) -1 1 + f 2 (0, z) . This depends, clearly, only on the polarization on the z-axis, through the function f(0, z).”

There is an exception when neither mode is sufficiently dominant at ρ=0, in which case the skyrmion number is noninteger.

“There is one exception, which is when neither mode is sufficiently dominant at ρ = 0 and in this case we find a noninteger skyrmion number.”

The value of f(∞,z) is determined by the dominant spatial mode at large distances from the z-axis.

“The value of f(∞, z) is similarly determined by the dominant spatial mode at large distances from the z-axis.”

A further example given here is the existence of non-integer skyrmions.

“A further example given here is the existence of non-integer skyrmions.”

For Laguerre-Gaussian modes, the skyrmion number is n(z) = (l1-l0)[1/(1+f^2(0,z)) - 1/(1+f^2(∞,z))].

“(6.5) n(z) = 1 4π 0 ∞ ρ dρ 0 2π dϕ Σ z = (ℓ 1 -ℓ 0 ) 1 1 + f 2 (0, z) - 1 1 + f 2 (∞, z)”

In the example of rotating the Stokes vector through -π/2 about the y-axis, the sole singularity at the origin in figure 5a, around which the phase changes by 6π corresponding to a skyrmion number of 3, is replaced by six (2n) strings around each of which the phase changes by 2π.

“We see that the sole singularity at the origin in figure 5a, around which the phase changes by 6π corresponding to a skyrmion number of 3, is replaced by six, that is 2n, strings around each of which the phase changes by 2π.”

For these Laguerre-Gaussian modes, the Stokes components are Sx=2f cos[Θ+(l1-l0)φ]/(1+f^2), Sy=2f sin[Θ+(l1-l0)φ]/(1+f^2), Sz=(1-f^2)/(1+f^2).

“so that the components of S are (6.3) S x = 2f cos[Θ + (ℓ 1 -ℓ 0 )ϕ] 1 + f 2 , S y = 2f sin[Θ + (ℓ 1 -ℓ 0 )ϕ] 1 + f 2 , S z = 1 -f 2 1 + f 2 .”

The skyrmion field can be written as the curl of a skyrmion vector potential V: Σ = ∇ × V.

“The transverse nature of Σ, which we have just established, suggests that we can write it as the curl of a further field, by analogy with the vector potential and magnetic induction for which we write B = ∇ × A. By analogy with the vector potential in electromagnetism, we refer to this new field as the skyrmion vector potential, V:”

The identification of skyrmion field lines with lines of constant polarization is general and holds whether or not the structured light has a non-zero skyrmion number.

“The identification of skyrmion field lines with lines of constant polarization is general and holds whether or not the structured light has a non-zero skyrmion number.”

The skyrmion field, together with the underlying skyrmion vector potential, determines the properties of skyrmions and the polarization structure of every paraxial vector beam.

“The key feature is the skyrmion field, which, together with the underlying skyrmion vector potential, determines the properties of the skyrmions and, more generally, the polarization structure of every paraxial vector beam.”

The simplest vector light beams can exhibit a polarization pattern identifiable with a skyrmionic structure.

“Vector light beams, characterized by a spatially varying polarization, can exhibit localized structures reminiscent of the skyrmions familiar from the study of magnetic media.”

• Also stated in: § 1. Introduction · boosted

  “We have shown, in 2020, that the simplest vector beams can exhibit a polarization pattern that is readily identified with a skyrmionic structure [13].”

• cites [13]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

In various situations, the skyrmion number will be a constant and take the same value in every transverse plane.

“In various situations, the skyrmion number will be a constant and take the same value in every transverse plane.”

An explicit expression for the skyrmion vector potential is V_i = m · ∂n/∂x_i, where m and n are orthogonal unit vector fields with m × n = S.

“The advantage of turning to the superfluid literature is that an explicit expression, due to Mermin & Ho [17,60], exists for the form of the V field, and so our skyrmion potential becomes:”

• cites [17]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [60]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

It is possible to change the skyrmion number on propagation.

“It is possible, however, to change the skyrmion number on propagation as we shall see.”

As lines of constant polarization, skyrmion field lines underlie and map the forms of all structured paraxial beams.

“As such they underlie and map the forms of all structured paraxial beams.”

The skyrmion number depends only on the dominant polarization on the z-axis where the optical vortex is situated and at large distances from the axis.

“It is remarkable that this depends only on the dominant polarization on the z-axis where the optical vortex is situated and at large distances from the axis.”

As the beam propagates, this skyrmion number will change as diffraction causes the ratio of beam waists w0(z)/w1(z) to change.

“As the beam propagates, moreover, this skyrmion number will change as diffraction causes the ratio of beam waists w 0 (z)/w 1 (z) to change.”

This potential provides a simple expression for the skyrmion number in paraxial optical skyrmion fields based solely on topological considerations.

“This potential, although not unique, provides a simple expression for the skyrmion number in paraxial optical skyrmion fields based solely on topological considerations.”

The behavior for small ρ is quite general and applies to modes other than the Laguerre-Gaussians with a charge l vortex.

“Note that the behaviour for small ρ is quite general and applies to modes other than the Laguerre-Gaussians with a charge ℓ vortex.”

We can readily construct analogues of the skyrmions encountered in the study of magnetic media.

“We can readily construct analogues of the skyrmions encountered in the study of magnetic media.”

The situation for non-integer skyrmions is distinct from that produced by a pair of spatial modes focused in different places, as there the skyrmion number changes abruptly at a single plane.

“The situation is clearly distinct from that produced by a pair of spatial modes focused in different places, as there the skyrmion number changes abruptly at a single plane.”

The origin of this effect is the failure of the transversality or divergenceless condition on the beam axis.

“The origin of this effect is the failure of the transversality or divergenceless condition on the beam axis.”

When the relative widths of the modes switch, skyrmion field lines turn away from the propagation direction heading radially outwards.

“When this happens, skyrmion field lines turn away from the propagation direction heading radially outwards.”

The failure of the transversality condition for non-integer skyrmions suggests that a description of these in terms of an underlying skyrmion vector potential may not be helpful.

“The failure of the transversality condition for non-integer skyrmions also suggests that a description of these in terms of an underlying skyrmion vector potential may not be helpful.”

For these beams, the skyrmion number can be determined purely in terms of the polarization on the beam axis and at large distances.

“For these, we can determine the skyrmion number purely in terms of the polarization on the beam axis and at large distances.”