Optical vortices are ubiquitous in nature. Physical light, which can be de- scribed as a superposition of many plane waves traveling in different di- rections, delineates the perfect situation in which destructive interference can lead to the creation of threads of darkness. Interfering three or more plane waves can create a phase singularity in the laboratory [13]. In an optics experiment it is common to use laser beams as a light source, which are described by a Gaussian intensity profile and have well-behaved and understood properties. Among these are the well defined propagation direction and the high coherence of the output light. However, in con- trast with the broad spectral decomposition of the Hopfion and the torus knot vortex fields that we have studied (see Sec. 3.2), lasers are nearly monochromatic. This constitutes a major implication in the goal to realize them experimentally.
4.1
Spectral modification
To tackle this problem, we attempt to diminish the spectral width of our fields by modifying their Fourier amplitude. This is accomplished by mul- tiplying it with a scalar function of the frequency, or equivalently (switch-
ing back to natural units) the wavenumber k, which we denote as g(k).
This function is designed to have a narrow width while being peaked at a certain valuek0, such that to filter out only a few of the frequency com-
ponents of the original field. The resulting Fourier representation can be generically written as ˜F(k,t) = g(k)F(k,t). Under the constraints of being scalar and time-independent, an arbitrary choice for g(k) will serve for ˜F
4.1 Spectral modification 29
tory while F(k,t) is known to be a solution. This is made more intuitive by considering that (due to linearity) a single or a few components of a multi-colored superposition is still a Maxwell solution.
The spectral decomposition of the resulting field, in the form of Equa- tion 3.4, can be written as the the inverse Fourier transform
˜
F(r,t) =
Z d3k
(2π)3/2F(k,t)g(k)e
i(k·r−ωt), (4.1)
where for simplicity we have chosena>0. Apart from the 3-dimensionality of these integrals, polarization effects render their calculation long and te- dious. However, by expressing them in terms of spherical coordinates, we can exploit the fact that the spectra in Equation 3.7, contain a special combination of the frequency space coordinates which allows the angular part of each of the polarization components to be written as a finite sum of spherical harmonicsYlm. Then, for a number of non-zero coefficients clm,
the jthcomponent can be written in the form
Fj(k,t) = ζ(k)
∑
l
∑
mclmYlm(kˆ),
where ˆk stands for the Fourier angles (θk,φk) and ζ(k) contains the k-
dependent terms including the exponential factore−ik(t−i|a|). For instance, in the Hopfion case (Eq. 3.6) the functionζ takes the explicit formζ(k) =
a2√2π
4 ke
−ik(t−i|a|)and it is the same for all three components. As a common
practice in momentum-space Feynman diagram analysis, we then replace the exponential factor in Eq. 4.1 with its Rayleigh expansion [2]
eik·r = ∞
∑
l=0 l∑
m=−l 4πiljl(kr)Ylm(rˆ)Y m∗ l (kˆ), (4.2)where, as above, ˆr denotes the real space angles and jl is the spherical
Bessel function (of the first kind). Together with the orthonormality condi- tion for spherical harmonics which has the explicit formR dΩkYm
0
l0 (kˆ)Y m∗ l (kˆ) =
δll0δmm0, wheredΩk = sinθkdθkdφkis the solid angle, the jthcomponent of
the integral 4.1 becomes ˜ Fj(r,t) =
∑
l,m clmilYlm(ˆr) Z dkk2ζ(k)g(k)jl(kr). (4.3)Thus the problem now reduces to solving three (one for each j={1, 2, 3}) 1-dimensional integrals in order to fully determine the field ˜F(r,t). Yet,
4.1 Spectral modification 30
due to the fact that the integrands contain a Bessel function together with an exponential, the solution is frequently inaccessible for a general g(k); for an extensive tabulation of Bessel integrals the reader is referred to [31].
There is one particular case for which the k-integral in equation 4.3
becomes trivial. This occurs for the choice of g(k) to be an 1-dimentional delta functionδ(k−k0), which corresponds to converting to the monochro-
matic variants of our fields at the frequency ω0 = k0. The resulting field
˜
F(r,t)essentially represents what we would “see” by looking at the(p,q)
knotted vortex fields through a single-colored glass. The explicit expres- sions (apart from the Hopf field case) resulting from this caclulation, are quite lengthy and therefore we shall not present them here. They are in- cluded in a Mathematica notebook which can be provided by the author on request.
Note that the Hopfion solution does not possess a vortex structure in its original form, and thus we do not expect that its single-colored variant would either. We therefore focus our attention to the case of the simplest torus knot vortex field, corresponding to the trefoil (2, 3). At zero time, several plots of planar intersections of the monochromatic field intensity (considering the electric and magnetic field vectors separately), exhibit in- teresting symmetries owing to its nature as superposition of Bessel func- tions. These plots contain multiple zero-intensity points which might in- stantly form a dark filament. Although a more careful investigation is re- quired to be able to draw a definite conclusion, at a first glance, the knotted vortex structure seems to not persist under the constraint of monochro- maticity. Moreover, the modified fields fail to satisfy the null condition and therefore, even if they did retain a topologically non-trivial structure att=0, it would be lost with time evolution.
A more systematic approach would involve imposing an external con- dition on the functiong(k), which will assure the nullness of the modified field, requiring ˜ F(r,t)·F˜(r,t) = 3
∑
j=1 ˜ Fj(r,t)F˜j(r,t) = 0.Using the fact that the Fourier transform of the product of two functions becomes the convolution integral of their individual Fourier transforms, the above condition, written in frequency space reads
3
∑
j=1
4.2 In a future laboratory 31
However, this expression involves 3-dimensional convolution integrals and therefore does not yield an obvious constraint on g(k) in order to be satisfied.
4.2
In a future laboratory
Due to the short time-span of this project and the aforementioned implica- tions, we unfortunately did not have the opportunity to directly engage in the practical aspects of knotting an electromagnetic vortex in the lab. Nev- ertheless, we devote this section to briefly outline and expose some ideas of how to make this experimentally feasible, assuming monochromaticity. In the laboratory, vortex carrying beams can be (readily) created be means of diffractive optical components. A high NA microscope objective lens can be used to create highly focused beams of light, from an arbitrary incident paraxial field. This was extensively studied by Leutenegger et al. [20], who proposed a numerical technique (fast Fourier transform) with which the amplitude, phase and polarization of the transmitted (focused) field can be fully determined via its plane wave spectrum.
By choosing the suitable input field one can prescribe the knotted topol- ogy in the resulting light configuration. The non-trivial phase and am- plitude information can be imprinted in each component of the incident beam via the combination of a spatial light modulator (SLM) and a spa- tial filter, according to the novel method proposed in [34]. Duplicating this system, can create 2 orthogonal (a horizontal and a vertical) polar- ization components of which the phase and amplitude modulation can be independently controlled, making thus possible to synthesize any de- sired field. Finally, the resulting singularity can be detected via camera scans that display 2-dimensional intersections of the focused fields, each containing a vortex point. By accumulating these planes the complete 3- dimenional singular curve can can be reconstructed [18].