Monochromatic knots and other unusual electromagnetic disturbances: light localised in 3D
MMonochromatic knots and other unusualelectromagnetic disturbances: light localised in 3D
Robert P Cameron
SUPA and Department of Physics, University of Strathclyde, Glasgow G4 0NG, UKSchool of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UKE-mail:
Abstract.
We introduce and examine a collection of unusual electromagneticdisturbances. Each of these is an exact, monochromatic solution of Maxwell’s equationsin free space with looped electric and magnetic field lines of finite extent and alocalised appearance in all three spatial dimensions. Included are the first explicitexamples of monochromatic electromagnetic knots. We also consider the generationof our unusual electromagnetic disturbances in the laboratory, at both low and highfrequencies, and highlight possible directions for future research, including the use ofunusual electromagnetic disturbances as the basis of a new form of three-dimensionaldisplay. a r X i v : . [ phy s i c s . c l a ss - ph ] J a n onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D
1. Introduction
Light is usually thought of as being localised in two spatial dimensions but not three:one might imagine a ray of sunshine or a laser beam, for example. A pulse of light,which is necessarily polychromatic, can be fully localised in space at a given time, butspans an extended region as it propagates.In the present paper we show that it is, in fact, possible for monochromaticlight to appear fully localised in free space at a fixed location and, furthermore, thatsuch light can take on a variety of remarkable forms. Specifically, we introduce andexamine a collection of ‘unusual electromagnetic disturbances’, each of which is anexact, monochromatic solution of Maxwell’s equations in free space [1, 2, 3, 4] withlooped electric and magnetic field lines of finite extent and a localised appearance inall three spatial dimensions. Included are the first explicit examples of monochromaticelectromagnetic knots (see below). We also consider the generation of our unusualelectromagnetic disturbances in the laboratory, at both low and high frequencies,and highlight possible directions for future research, including the use of unusualelectromagnetic disturbances as the basis of a new form of three-dimensional display.Within optics the closest field of research is, perhaps, that of so-calledelectromagnetic knots, which has its origins in the work of Ra˜nada [5, 6, 7, 8, 9, 10] andis now beginning to grow, rapidly [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,26, 27, 28, 29, 30, 31, 32]. The term ‘electromagnetic knot’ was introduced by Ra˜nadato refer to electromagnetic disturbances for which “the electromagnetic field as a wholeis tied to itself” [7] in that “any pair of magnetic lines or any pair of electric lines area link” [9]. It has since been used more broadly (and accurately), however, to describe,in addition, electromagnetic disturbances for which the electric and magnetic field linesexhibit legitimately knotted topologies. We refer to any electromagnetic disturbancefor which the electric and / or magnetic field lines exhibit non-trivial topologies asan electromagnetic knot. Note that our focus here is not upon threads of darkness[33, 34, 35, 36, 37] or polarisation singularities [37, 38, 39, 40, 41], which can alsoexhibit non-trivial topologies.The present paper represents a new contribution to the field of electromagneticknots in that some of our unusual electromagnetic disturbances might be regarded asthe first explicit examples of monochromatic electromagnetic knots: the electromagneticknots introduced explicitly by others to date are polychromatic [5, 6, 7, 8, 9, 10,11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32].An obvious advantage of monochromaticity over polychromaticity is that it facilitatesgeneration in the laboratory using lasers. It should be noted, however, that our unusualelectromagnetic disturbances are qualitatively distinct from the electromagnetic knotsintroduced by others [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,24, 25, 26, 27, 28, 29, 30, 31, 32]. In particular, some of our unusual electromagneticdisturbances can be thought of as (exotic) electromagnetic standing waves, which donot propagate, whereas most of the electromagnetic knots introduced by others can be onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D regarded as electromagnetic pulses, propagating at the speed of light whilst distorting[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,30, 31, 32].The total energy and other such properties of all monochromatic solutions toMaxwell’s equations in free space diverge. Our unusual electromagnetic disturbancesare no exception: for each disturbance the electric and magnetic fields exhibit a ‘1 / | r | ’fall off as one moves away from the region of primary interest (as described in section3.1.2), but this is not sufficient to render the total energy and other such propertiesof the disturbance finite. The total energy and other such properties can be renderedfinite without dramatically altering the basic character of the disturbance, however,by introducing a distribution of frequencies with a small spread, in which case thedisturbance is quasi-monochromatic rather than strictly monochromatic. We considerthese subtleties in more detail in section 4.We work in the classical domain, imagining ourselves to be in an inertial frame ofreference described by right-handed Cartesian coordinates x , y and z with associatedunit vectors ˆ xxx , ˆ yyy and ˆ zzz and time t . E is an electric field strength, ω = 2 π/T = ck =2 πc/λ is an angular frequency, l and l (cid:48) are integers, φ is a geometrical angle and δ isa phase. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D
2. Motivation and overview
In the strict absence of charge, the electric field E and magnetic field B obey Maxwell’sequations in the form [1, 2, 3, 4] ∇∇∇ · E = 0 , (1) ∇∇∇ · B = 0 , (2) ∇∇∇ × E = − ˙ B (3) ∇∇∇ × B = (cid:15) µ ˙ E , (4)where ∇∇∇ is the del operator with respect to r = ˆ xxxx + ˆ yyyy + ˆ zzzz and an overdot indicatespartial differentiation with respect to t . Gauss’s law (1) together with the divergencetheorem [42] states that the flux of E through any closed surface S at any given time iszero [3, 4]: (cid:90) S E · d r = 0 . (5)This permits two distinct possibilities: an electric field line must extend indefinitely orelse form a closed loop of finite extent. Similarly for B .Electromagnetic disturbances with indefinitely extending electric (and magnetic)field lines are well known, a plane electromagnetic wave being a clear example. Theelectric field E (cid:63) of the linearly polarised plane wave E (cid:63) = E ˆ xxx cos( k z − ω t ) (6) B (cid:63) = E c ˆ yyy cos( k z − ω t ) (7)is depicted in Fig. 1 to illustrate this: in any given plane of constant z at any giventime t (cid:54) = z/c + (1 / l ) T / x axis.Electromagnetic disturbances with looped electric (and magnetic) field lines offinite extent are less well understood: many interesting results have been presented,in particular with regards to electromagnetic knots [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], but there remains muchto be done. The author’s desire to help explore this avenue led to the present paper.The basic structure of the present paper is as follows. In section 3 we introduceour unusual electromagnetic disturbances and describe some of their properties. Insection 4 we make some general observations with regards to the energies and temporaldependencies of our unusual electromagnetic disturbances. In section 5 we consider thegeneration of our unusual electromagnetic disturbances in the laboratory. In section 6we highlight some possible directions for future research. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Figure 1.
The electric field E (cid:63) of a linearly polarised plane electromagneticwave (section 2), depicted as a three-dimensional vector plot at an instantof time: each blue arrow represents an electric field vector, colour-coded andscaled by magnitude. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D The key features common to our unusual electromagnetic disturbances are asfollows. • Each of our unusual electromagnetic disturbances is an exact solution of Maxwell’sequations (1)-(4) and can thus exist as an entity unto itself. • Each of our unusual electromagnetic disturbances is monochromatic . Owing to thescale invariance of (1)-(4) [43], our unusual electromagnetic disturbances are validin any region of the electromagnetic spectrum: they can be thought of equally in theradiowave domain [44] or the gamma-ray domain [45, 46, 47], for example. Our un-usual electromagnetic disturbances are not electromagnetic pulses. Polychromaticvariants of our unusual electromagnetic disturbances can be constructed. Save forthe discussions in section 4, we refrain from pursuing these further in the presentpaper, however. • Each of our unusual electromagnetic disturbances has looped electric (and mag-netic) field lines of finite extent. Our unusual electromagnetic disturbances are notmerely regions of heightened ‘intensity’ as in a caustic [48], for example. Indeed,they cannot be understood within the framework of ray optics: interference is vitalto the formation of our unusual electromagnetic disturbances, with the wavevec-tors, amplitudes, phases and polarisations of the plane electromagnetic waves thatcomprise a given disturbance all playing important roles. • Each of our unusual electromagnetic disturbances appears to be well localised inall three spatial dimensions in that the electric (and magnetic) field strengths inthe regions described are significantly larger than those found anywhere else. Ourunusual electromagnetic disturbances are not propagating beams of structured light[37], for example.We centre our explicit discussions upon the electric field rather than the magneticfield as the electric field is often of greater importance when one comes to considerbasic interactions between light and matter. Moreover, it is the looped character ofthe (free) electric field lines rather than magnetic field lines that is novel: ∇∇∇ · E (cid:54) = 0in the presence of charge whereas ∇∇∇ · B = 0 in general, it seems [3, 4, 49]. As (1)-(4)place B on equal footing with E , the electric and magnetic properties of a disturbancecan nevertheless be interchanged by performing a duality transformation: E → c B , B → − E /c [4, 50, 51, 52, 53]. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D We use the following superscript labels to distinguish between our explicit solutionsof (1)-(4):LABEL NAME (cid:63) plane electromagnetic wave (section 2) A electric ring (section 3.1.3) á electric globule (section 3.1.4) Á electromagnetic tangle (section 3.1.5) â electric loop (section 3.2.2) @ electric link (section 3.2.3) S straight electric line (section 3.2.4) ä curved electric line (section 3.2.4) H electric torus knot (section 3.2.5) Ì electric non-torus knot (section 3.2.5) J electromagnetic cloud (section 3.2.6) (cid:63)(cid:63) random superposition of plane electromagnetic waves (section 3.2.6)and the superscript label LF to denote the radiation generated by our electric-ringantenna (section 5). Magnetic versions of our electric ring, our electric globule, ourelectric loop, our electric link, our electric lines and our electric knots can be obtainedvia duality transformations. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D
3. Unusual electromagnetic disturbances
In the present subsection we introduce our unusual electromagnetic disturbances of thefirst kind. Any one of these can be regarded as a monochromatic superposition of planeelectromagnetic waves propagating in every direction, each of equal amplitude, equalphase at the origin and with an equivalent polarisation, as described in more detailbelow.In section 3.2 we will employ our unusual electromagnetic disturbances of the firstkind as building blocks for our unusual electromagnetic disturbances of the second kind.
First, we recall that any electromagnetic disturbance can beregarded as a superposition of plane electromagnetic waves: the general solution toMaxwell’s equations (1)-(4) can be cast as E = (cid:60) (cid:20) (cid:90) π (cid:90) π (cid:90) ∞ (ˆ ϕϕϕ ˜ B + ˆ ϑϑϑ ˜ A )ei ( k ˆ kkk · r − ωt ) k d k sin ϑ d ϑ d ϕ (cid:21) (8) B = (cid:60) (cid:20) c (cid:90) π (cid:90) π (cid:90) ∞ (ˆ ϕϕϕ ˜ A − ˆ ϑϑϑ ˜ B )ei ( k ˆ kkk · r − ωt ) k d k sin ϑ d ϑ d ϕ (cid:21) , (9)where ϕ , ϑ and k = ω/c are spherical coordinates in reciprocal space;ˆ ϕϕϕ = − ˆ xxx sin ϕ + ˆ yyy cos ϕ, (10)ˆ ϑϑϑ = ˆ xxx cos ϕ cos ϑ + ˆ yyy sin ϕ cos ϑ − ˆ zzz sin ϑ (11)ˆ kkk = ˆ xxx cos ϕ sin ϑ + ˆ yyy sin ϕ sin ϑ + ˆ zzz cos ϑ (12)are associated unit vectors and ˜ A and ˜ B are complex functions of ϕ , ϑ and k . A particulardisturbance is determined by specifying ˜ A and ˜ B , which is equivalent to specifying the so-called normal variables: ‘˜ ααα = − √ π (cid:15) (ˆ ϕϕϕ ˜ B + ˆ ϑϑϑ ˜ A )e − i ωt / √ (cid:126) ω ’ according to the formalismdescribed in [2], for example. An alternative expansion of the electromagnetic field, interms of multipolar rather than plane waves, is outlined in Appendix A. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Next, we specialise to monochromatic disturbances by taking ˜ A = E ˜ A (cid:48) δ ( k − k ) / πk and ˜ B = E ˜ B (cid:48) δ ( k − k ) / πk without further loss of generality, where ˜ A (cid:48) and ˜ B (cid:48) are complex functions of ϕ and ϑ only and δ ( K ) is the Dirac delta function: each planewave has the same angular frequency ω = ck . (8) and (9) thus reduce to E = (cid:60) (cid:20) E π (cid:90) π (cid:90) π (ˆ ϕϕϕ ˜ B (cid:48) + ˆ ϑϑϑ ˜ A (cid:48) )ei k ˆ kkk · r sin ϑ d ϑ d ϕ e − i ω t (cid:21) (13) B = (cid:60) (cid:20) E πc (cid:90) π (cid:90) π (ˆ ϕϕϕ ˜ A (cid:48) − ˆ ϑϑϑ ˜ B (cid:48) )ei k ˆ kkk · r sin ϑ d ϑ d ϕ e − i ω t (cid:21) . (14)A particular monochromatic disturbance is determined by specifying ˜ A (cid:48) and ˜ B (cid:48) .Finally, we obtain our unusual electromagnetic disturbances of the first kind bytaking ˜ A (cid:48) = ˜ A (cid:48) and ˜ B (cid:48) = ˜ B (cid:48) , where ˜ A (cid:48) and ˜ B (cid:48) are complex constants, independent of ϕ and ϑ : plane waves propagating in every direction, each of equal amplitude, equal phaseat the origin and with an equivalent polarisation relative to ˆ ϕϕϕ and ˆ ϑϑϑ . (13) and (14) thusreduce to E = (cid:60) (cid:26) E (cid:104) i˜ B (cid:48) ˆ φφφf + ˜ A (cid:48) (ˆ sssg + ˆ zzzh ) (cid:105) e − i ω t (cid:27) (15) B = (cid:60) (cid:26) E c (cid:104) i˜ A (cid:48) ˆ φφφf − ˜ B (cid:48) (ˆ sssg + ˆ zzzh ) (cid:105) e − i ω t (cid:27) (16)with the modulating scalar fields f = (cid:90) π J ( k sin ϑs ) cos ( k cos ϑz ) sin ϑ d ϑ, (17) g = − (cid:90) π J ( k sin ϑs ) sin ( k cos ϑz ) sin ϑ cos ϑ d ϑ (18) h = − (cid:90) π J ( k sin ϑs ) cos ( k cos ϑz ) sin ϑ d ϑ, (19)where J α ( X ) is the Bessel function of the first kind of order α and φ , s and z arecylindrical coordinates defined such that x = s cos φ (20) y = s sin φ (21)with associated unit vectorsˆ φφφ = − ˆ xxx sin φ + ˆ yyy cos φ (22)ˆ sss = ˆ xxx cos φ + ˆ yyy sin φ. (23)A particular unusual electromagnetic disturbance of the first kind is determined byspecifying ˜ A (cid:48) and ˜ B (cid:48) . We give explicit examples in section 3.1.3, section 3.1.4 andsection 3.1.5. f , g and h The forms of our unusualelectromagnetic disturbances of the first kind (and, by extension, our unusualelectromagnetic disturbances of the second kind (section 3.2)) derive from those of the onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D modulating scalar fields f , g and h as seen in (15) and (16). We therefore examine f , g and h here for reference in what follows. f , g and h are cylindrically symmetric: they do not depend on φ . Furthermore, itcan be seen by inspecting (17), (18) and (19) that f is even in z : f ( z ) = f ( − z ), g is oddin z : g ( z ) = − g ( − z ) and h is even in z : h ( z ) = h ( − z ). More generally, the variation of f , g and h with s and z can be explored by numerical integration. The results obtainednear the origin for f are depicted in Fig. 2, those for g are depicted in Fig. 3 and thosefor h are depicted in Fig. 4. Note the following features. • The z axis is a nodal line of f : f ( s = 0 , z ) = 0. The highest peak of | f | can befound at s = (0 . . . . ) λ and z = 0, where f = 1 . . . . . The second highest peaksof | f | can be found at s = (0 . . . . ) λ and z = ± (0 . . . . ) λ , where f = − . . . . .The third highest peak of | f | can be found at s = (0 . . . . ) λ and z = 0, where f = 0 . . . . . • The z axis is a nodal line of g and the z = 0 plane is a nodal plane: g ( s = 0 , z ) = g ( s, z = 0) = 0. The highest peaks of | g | can be found at s = (0 . . . . ) λ and z = ± (0 . . . . ) λ , where g = ∓ . . . . . The second highest peaks of | g | can befound at s = (0 . . . . ) λ and z = ± (0 . . . . ) λ , where g = ± . . . . . The thirdhighest peaks of | g | can be found at s = (1 . . . . ) λ and z = ± (1 . . . . ) λ , where g = ∓ . . . . . • The highest peak of | h | can be found at s = 0 and z = 0, where h = − . . . . .The second highest peak of | h | can be found at s = (0 . . . . ) λ and z = 0, where h = 0 . . . . . The third highest peak of | h | can be found at s = (1 . . . . ) λ and z = 0, where h = − . . . . .Further numerical investigation reveals that f , g and h each tend towards a form that isat least qualitatively similar to a sinusoidal undulation modulated by a 1 / | r | fall off as | r | → ∞ in any non-trivial direction. This is more dramatic than the analogous 1 / √ X fall off inherent to each of the J α ( X ) seen in the integrands of (17), (18) and (19). f , g and h are related by − k f = ∂g∂z − ∂h∂s , (24) − k g = − ∂f∂z (25) − k h = 1 s f + ∂f∂s , (26)in accord with the Faraday-Lenz law (3) and the Amp`ere-Maxwell law (4). onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Figure 2.
A two-dimensional density plot of the modulating scalar field f ingreyscale, with contours for which f > f = 0 overlaid in blue and contours for which f < onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Figure 3.
A two-dimensional density plot of the modulating scalar field g ingreyscale, with contours for which g > g = 0 overlaid in blue and contours for which g < onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Figure 4.
A two-dimensional density plot of the modulating scalar field h ingreyscale, with contours for which h > h = 0 overlaid in blue and contours for which h < onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D3.1.3. Electric ring Taking ˜ A (cid:48) = 0 and ˜ B (cid:48) = − i in (15) and (16) corresponds to havingeach plane wave linearly polarised parallel to ˆ ϕϕϕ and gives our ‘electric ring’: E A = E ˆ φφφf cos ( ω t ) (27) B A = E c (ˆ sssg + ˆ zzzh ) sin ( ω t ) , (28)so-named because when t (cid:54) = (1 / l ) T / E A of ourelectric ring is depicted in Fig. 5 and Fig. 6. Note that the magnetic field B A of ourelectric ring is equivalent to the electric field E á of our electric globule (section 3.1.4)in that B A = E á /c .Our electric ring is qualitatively similar to the electromagnetic disturbancedescribed in [54] in that both are monochromatic and exhibit azimuthally directedelectric field vectors. The two differ, however, in their spatial dependence and onemight argue that the particularly prominent ring-like feature of our electric ring is morepronounced than the analogous feature in [54]. See also [55]. Taking ˜ A (cid:48) = i and ˜ B (cid:48) = 0 in (15) and (16) corresponds to havingeach plane wave linearly polarised parallel to ˆ ϑϑϑ and gives our ‘electric globule’: E á = E (ˆ sssg + ˆ zzzh ) sin ( ω t ) (29) B á = − E c ˆ φφφf cos ( ω t ) , (30)so-named because when t (cid:54) = lT / E á of our electric globule is depicted in Fig. 7 and Fig. 8. Note that the magneticfield B á of our electric globule is equivalent to the electric field E A of our electric ring(section 3.1.3) in that B á = − E A /c .Our electric globule is qualitatively similar to a duality-transformed version of theelectromagnetic disturbance described in [54]. See also [55]. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Figure 5.
The electric field E A of our electric ring (section 3.1.3), depicted as athree-dimensional vector plot at an instant of time: each blue arrow representsan electric field vector, colour-coded and scaled by magnitude. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Figure 6.
The electric field E A of our electric ring (section 3.1.3), depicted as atwo-dimensional vector plot at an instant of time: each blue cross ( E A · ˆ φφφ >
0) orcircle ( E A · ˆ φφφ <
0) represents an electric field vector, colour-coded by magnitude. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D
Figure 7.
The electric field E á of our electric globule (section 3.1.4), depicted asa three-dimensional vector plot at an instant of time: each blue arrow representsan electric field vector, colour-coded and scaled by magnitude. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Figure 8.
The electric field E á of our electric globule (section 3.1.4), depicted asa two-dimensional vector plot at an instant of time: each blue arrow representsan electric field vector, colour-coded by magnitude. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D3.1.5. Electromagnetic tangle Taking ˜ A (cid:48) = 1 / √ B (cid:48) = i σ/ √ σ = ± E Á = E √ − σ ˆ φφφf + ˆ sssg + ˆ zzzh ) cos ( ω t ) (31) B Á = E √ c (cid:104) ˆ φφφf − σ (ˆ sssg + ˆ zzzh ) (cid:105) sin ( ω t ) , (32)so-named because the electric and magnetic field lines encode a ‘tangle’ of knots asdescribed in more detail below. Our electromagnetic tangle is chiral [57]: σ = 1and σ = − E Á of the σ = 1 form of ourelectromagnetic tangle is depicted in Fig. 9 and that of the σ = − B Á of our electromagnetic tangle is proportionalto a time-translated version of E Á in that B Á ( t ) = σ E Á ( t + T / /c . Moreover, ourelectromagnetic tangle can be regarded as a superposition of our electric ring (section3.1.3) and a time-translated version of our electric globule (section 3.1.4): E Á ( t ) = 1 √ (cid:104) − σ E A ( t ) + E á ( t + T / (cid:105) (33) B Á ( t ) = 1 √ (cid:104) − σ B A ( t ) + B á ( t + T / (cid:105) , (34)as was suggested to the author by Stephen M Barnett and J¨org B G¨otte.The basic form of the electric field lines of our electromagnetic tangle, which arenon-vanishing when t (cid:54) = (1 / l ) T /
2, can be appreciated by considering the following,with reference to the chart in Fig. 11. In a plane with φ = φ , contours for which f = 0 delimit crescent-shaped areas. The sign of f is constant within the i th crescent( i ∈ { , , . . . } ) and differs between the i th and ( i + 1)th crescent. Within the i thcrescent there is one point a distance s i along the s axis at z = 0 about which thetwo-dimensional vector field ˆ sssg + ˆ zzzh circulates ( g ( s = s i , z = 0) = h ( s = s i , z = 0) = 0),counter-clockwise if f > f <
0. In three dimensions, the crescentsdefine tori (obtained by revolving them about the z axis). We now confine our attentionwithin the i th such torus, without loss of generality: electric field lines do not passbetween tori. There is one circular electric field line in the z = 0 plane: E Á ( s = s i , z = 0) = − √ φφφσf ( s = s i , z = 0) E cos ( ω t ) , (35)which points in the + φ direction if − σf ( s i ,
0) cos( ω t ) > − φ direction if − σf ( s i ,
0) cos( ω t ) <
0. The other electric field lines follow this line azimuthally(because f has the same sign everywhere) whilst twisting around it (because of thecirculation inherent to ˆ sssg + ˆ zzzh ) to form right-handed helices if σ = 1 or left-handedhelices if σ = −
1. Analogous observations can be made for the magnetic field lines, ofcourse, which are non-vanishing when t (cid:54) = lT / t (cid:54) = (1 / l ) T / onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D numerically integrating the streamline equationd r ( τ )d τ = E Á [ r ( τ )] | E Á [ r ( τ )] | , (36)giving the trajectory of the j th electric field line as r j ( τ ) = r j (0) + (cid:90) τ E Á [ r ( τ (cid:48) )] | E Á [ r ( τ (cid:48) )] | d τ (cid:48) , (37)where τ is an arc length which increases from 0 as one follows the line from the seedposition r j (0). We find that certain electric field lines resemble familiar torus knots(with the circular electric field lines given by (35) being unknots, of course) [58]. Aselection of these lines are depicted separately in Fig. 12 and together in Fig. 13.Again, analogous observations can be made for the magnetic field lines.Interestingly, there is a sense in which the chirality [57] of the helical field lines in ourelectromagnetic tangle is opposite to that of the plane waves that comprise the tangle:for σ = ± monochromatic electromagnetic knot of toroidal character. Persistent polychromatic electromagnetic knots of toroidal character are described in [21]. Knotted threads ofdarkness within monochromatic electromagnetic fields are described in [34, 35, 36, 37],for example. Loosely speaking, our electromagnetic tangle might be regarded as acomplimentary structure: here we have knotted ‘threads of brightness’ (specifically,non-vanishing electric and magnetic field lines) rather than darkness. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Figure 9.
The electric field E Á of the σ = 1 form of of our electromagnetictangle (3.1.5), depicted as a three-dimensional vector plot at an instant of time:each blue arrow represents an electric field vector, colour-coded and scaled bymagnitude. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Figure 10.
The electric field E Á of the σ = − onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Figure 11.
A cross section of the i = 1 , , . . . tori defined by f = 0, with regionsfor which f < f > g ( s = s i , z = 0) = h ( s = s i ,
0) = 0 indicated by black crosses and thesense of circulation of ˆ sssg + ˆ zzzh about these points indicated by light-blue arrows. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D
Figure 12.
A selection of torus-knotted electric field lines from the i = 1 torusof the σ = 1 form of our electromagnetic tangle (section 3.1.5) and their ( p, q )designations [58], with each depicted as a three-dimensional streamline plot atan instant of time. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Figure 13.
The electric field lines depicted in Fig. 12, shown together. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D3.2. The second kind
In the present subsection we introduce our unusual electromagnetic disturbances of thesecond kind. Any one of these can be regarded as a superposition of translated andperhaps rotated versions of our unusual electromagnetic disturbances of the first kind(section 3.1).Our unusual electromagnetic disturbances of the second kind are emphatically‘new’: electromagnetic disturbances akin to our electric loop (section 3.2.2), our electriclink (section 3.2.3), our electric lines (section 3.2.4), our electric knots (section 3.2.5)and our electromagnetic cloud (section 3.2.6), for example, do not appear to have beendescribed before.At the level of theory employed in the present paper, there is no obvious upperlimit to the ‘size’ of an unusual electromagnetic disturbance of the second kind: one isfree to imagine an x-ray [60] electric loop large enough to encircle a star or an infrared[61, 62, 63] electromagnetic cloud as big as a house, for example.
We can construct a valid electromagnetic disturbance bysuperposing translated and perhaps rotated versions of our unusual electromagneticdisturbances of the first kind, as Maxwell’s equations (1)-(4) are linear, homogeneousand isotropic [43].To realise such a superposition and thus obtain our unusual electromagneticdisturbances of the second kind, let us introduce N ≥ n th system ( n ∈ { , . . . , N } ) described by right-handed Cartesian coordinates x n , y n and z n orientated such that the associated unit vectors ˆ xxx n , ˆ yyy n and ˆ zzz n are givenin terms of Euler angles φ n , θ n and χ n asˆ xxx n = ˆ xxx(cid:96) (11) n + ˆ yyy(cid:96) (12) n + ˆ zzz(cid:96) (13) n , (38)ˆ yyy n = ˆ xxx(cid:96) (21) n + ˆ yyy(cid:96) (22) n + ˆ zzz(cid:96) (23) n (39)ˆ zzz n = ˆ xxx(cid:96) (31) n + ˆ yyy(cid:96) (32) n + ˆ zzz(cid:96) (33) n (40)with the origin x n = y n = z n = 0 located at x = X n , y = Y n and z = Z n , where (cid:96) (11) n = cos φ n cos θ n cos χ n − sin φ n sin χ n , (41) (cid:96) (12) n = − sin φ n cos χ n − cos φ n cos θ n sin χ n , (42) (cid:96) (13) n = cos φ n sin θ n , (43) (cid:96) (21) n = sin φ n cos θ n cos χ n + cos φ n sin χ n , (44) (cid:96) (22) n = cos φ n cos χ n − sin φ n cos θ n sin χ n , (45) (cid:96) (23) n = sin φ n sin θ n , (46) (cid:96) (31) n = − sin θ n cos χ n , (47) (cid:96) (32) n = sin θ n sin χ n (48) (cid:96) (33) n = cos θ n (49)are direction cosines. It follows from the above together with (15) and (16) that our onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D unusual electromagnetic disturbances of the second kind are given by E = (cid:60) (cid:40) N (cid:88) n =1 E (cid:104) i˜ B (cid:48) n ˆ φφφ n f n + ˜ A (cid:48) n (ˆ sss n g n + ˆ zzz n h n ) (cid:105) e − i ω t (cid:41) (50) B = (cid:60) (cid:40) N (cid:88) n =1 E c (cid:104) i˜ A (cid:48) n ˆ φφφ n f n − ˜ B (cid:48) n (ˆ sss n g n + ˆ zzz n h n ) (cid:105) e − i ω t (cid:41) (51)with the modulating scalar fields f n = (cid:90) π J ( k sin ϑs n ) cos ( k cos ϑz n ) sin ϑ d ϑ, (52) g n = − (cid:90) π J ( k sin ϑs n ) sin ( k cos ϑz n ) sin ϑ cos ϑ d ϑ (53) h n = − (cid:90) π J ( k sin ϑs n ) cos ( k cos ϑz n ) sin ϑ d ϑ, (54)where the ˜ A (cid:48) n and ˜ B (cid:48) n are complex constants and the φ (cid:48) n , s n and z n are cylindricalcoordinates defined such that x n = s n cos φ (cid:48) n (55) y n = s n sin φ (cid:48) n (56)with associated unit vectorsˆ φφφ n = − ˆ xxx n sin φ (cid:48) n + ˆ yyy n cos φ (cid:48) n (57)ˆ sss n = ˆ xxx n cos φ (cid:48) n + ˆ yyy n sin φ (cid:48) n . (58)A particular unusual electromagnetic disturbance of the second kind is determined byspecifying N , the φ n , the θ n , the χ n , the X n , the Y n , the Z n , the ˜ A (cid:48) n and the ˜ B (cid:48) n . We giveexplicit examples in section 3.2.2, section 3.2.3, section 3.2.4, section 3.2.5 and section3.2.6.It is also possible to realise (50) and (51) via appropriate choices of ˜ A (cid:48) and ˜ B (cid:48) in (13)and (14), of course. The forms required turn out to be particularly lengthy, however,and we refrain, therefore, from reproducing them explicitly in the present paper. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D3.2.2. Electric loop An unusual electromagnetic disturbance in which some of theelectric field lines form a particularly prominent loop-like feature (or features) at certaintimes can be constructed by superposing spatially translated and perhaps rotatedversions of our electric ring (section 3.1.3). Various forms are possible.Consider for example the triangular ‘electric loop’ ‡ specified by taking N = 21 and n φ n θ n χ n X n /λ Y n /λ Z n /λ ˜ A (cid:48) n ˜ B (cid:48) n √ d/ − i2 0 0 0 − d √ d/ − i3 0 0 0 d √ d/ − i4 0 0 0 − d √ d/ − i5 0 0 0 0 √ d/ − i6 0 0 0 2 d √ d/ − i7 0 0 0 − d −√ d/ − i8 0 0 0 − d −√ d/ − i9 0 0 0 d −√ d/ − i10 0 0 0 3 d −√ d/ − i11 0 0 0 − d − √ d/ − i12 0 0 0 − d − √ d/ − i13 0 0 0 0 − √ d/ − i14 0 0 0 2 d − √ d/ − i15 0 0 0 4 d − √ d/ − i16 0 0 0 − d − √ d/ − i17 0 0 0 − d − √ d/ − i18 0 0 0 − d − √ d/ − i19 0 0 0 d − √ d/ − i20 0 0 0 3 d − √ d/ − i21 0 0 0 5 d − √ d/ − iin (50) and (51), where it is to be understood that d = 0 .
33: the area bounded by ourelectric loop is tiled with electric rings such that the electric fields of the rings largelycancel within the boundary of the loop whilst those on the loop connect. The formationof the loop is thus analogous to the formation of bound surface currents in magneticmedia [3, 4]. The electric field E â of our electric loop is depicted in Fig. 14.Electric loops, including non-planar electric loops, can also be realised as closedelectric lines (section 3.2.4). ‡ In the present paper ‘electric loop’ is not to be confused with ‘looped electric field line’. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D
Figure 14.
The electric field E â of our electric loop (section 3.2.2), depicted as athree-dimensional vector plot at an instant of time: each blue arrow representsan electric field vector, colour-coded and scaled by magnitude. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D3.2.3. Electric link An unusual electromagnetic disturbance in which some of theelectric field lines form a particularly prominent link-like feature (or features) at certaintimes can be constructed by superposing electric loops (section 3.1.3). Various formsare possible.Consider for example the ‘electric link’ specified by taking N = 32 and n φ n θ n χ n X n /λ Y n /λ Z n /λ ˜ A (cid:48) n ˜ B (cid:48) n − d/ − (3 d + D ) / − i2 0 0 0 − d/ − ( d + D ) / − i3 0 0 0 − d/ d − D ) / − i4 0 0 0 − d/ d − D ) / − i5 0 0 0 − d/ − (3 d + D ) / − i6 0 0 0 − d/ − ( d + D ) / − i7 0 0 0 − d/ d − D ) / − i8 0 0 0 − d/ d − D ) / − i9 0 0 0 d/ − (3 d + D ) / − i10 0 0 0 d/ − ( d + D ) / − i11 0 0 0 d/ d − D ) / − i12 0 0 0 d/ d − D ) / − i13 0 0 0 3 d/ − (3 d + D ) / − i14 0 0 0 3 d/ − ( d + D ) / − i15 0 0 0 3 d/ d − D ) / − i16 0 0 0 3 d/ d − D ) / − i17 0 π/ − (3 d − D ) / − d/ − i18 0 π/ − (3 d − D ) / − d/ − i19 0 π/ − (3 d − D ) / d/ − i20 0 π/ − (3 d − D ) / d/ − i21 0 π/ − ( d − D ) / − d/ − i22 0 π/ − ( d − D ) / − d/ − i23 0 π/ − ( d − D ) / d/ − i24 0 π/ − ( d − D ) / d/ − i25 0 π/ d + D ) / − d/ − i26 0 π/ d + D ) / − d/ − i27 0 π/ d + D ) / d/ − i28 0 π/ d + D ) / d/ − i29 0 π/ d + D ) / − d/ − i30 0 π/ d + D ) / − d/ − i31 0 π/ d + D ) / d/ − i32 0 π/ d + D ) / d/ − iin (50) and (51), where it is to be understood that d = 0 .
75 and D = 1 .
50: our electriclink is comprised of two square electric loops, suitably orientated and positioned. Theelectric field E @ of our electric link is depicted in Fig. 15. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Our electric link might be regarded as another explicit example of a monochromatic electromagnetic ‘knot’.
Figure 15.
The electric field E @ of our electric link (3.2.3), depicted as a three-dimensional vector plot at an instant of time: each blue arrow represents anelectric field vector, colour-coded and scaled by magnitude. Each of the electricrings comprising the link has been plotted for | z n | ≤ λ only, for the sake ofclarity. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D3.2.4. Electric lines An unusual electromagnetic disturbance in which some of theelectric field lines form a particularly prominent line-like feature (or features) at certaintimes can be constructed by superposing spatially translated and perhaps rotatedversions of our electric globule (section 3.1.4). Various forms are possible.As our first example consider the ‘straight electric line’ § specified by taking N = 15and n φ n θ n χ n X n Y n Z n ˜ A (cid:48) n ˜ B (cid:48) n − d i 02 0 0 0 0 0 − d i 03 0 0 0 0 0 − d i 04 0 0 0 0 0 − d i 05 0 0 0 0 0 − d i 06 0 0 0 0 0 − d i 07 0 0 0 0 0 − d i 08 0 0 0 0 0 0 i 09 0 0 0 0 0 d i 010 0 0 0 0 0 2 d i 011 0 0 0 0 0 3 d i 012 0 0 0 0 0 4 d i 013 0 0 0 0 0 5 d i 014 0 0 0 0 0 6 d i 015 0 0 0 0 0 7 d i 0in (50) and (51), where it is to be understood that d = 0 .
25: spatially translated versionsof our electric globule are equally spaced along the z axis like totems in a totem polesuch that the straight electric line which results has a continuous appearance. Theelectric field E S of our straight electric line is depicted in Fig. 16. § In the present paper ‘electric line’ is not to be confused with ‘electric field line’. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D
Figure 16.
The electric field E S of our straight electric line (section 3.2.4),depicted as a three-dimensional vector plot at an instant of time: each bluearrow represents an electric field vector, colour-coded and scaled by magnitude. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D As our second example consider the ‘curved electric line’ specified by taking N = 15and n φ n θ n χ n X n /λ Y n /λ Z n /λ ˜ A (cid:48) n ˜ B (cid:48) n π d cos θ d (sin θ −
1) i 02 π π/
14 0 d cos θ d (sin θ −
1) i 03 π π/ d cos θ d (sin θ −
1) i 04 π π/
14 0 d cos θ d (sin θ −
1) i 05 π π/ d cos θ d (sin θ −
1) i 06 π π/
14 0 d cos θ d (sin θ −
1) i 07 π π/ d cos θ d (sin θ −
1) i 08 π π/ d cos θ d (sin θ −
1) i 09 π π d cos θ d (sin θ + 1) i 010 π π/
14 0 d cos θ d (sin θ + 1) i 011 π π/ d cos θ d (sin θ + 1) i 012 π π/
14 0 d cos θ d (sin θ + 1) i 013 π π/ d cos θ d (sin θ + 1) i 014 π π/
14 0 d cos θ d (sin θ + 1) i 015 π π/ d cos θ d (sin θ + 1) i 0in (50) and (51), where it is to be understood that d = 2 .
00: an S-shaped curve underliesour curved electric line and is populated with electric globules much as a string might bepopulated with beads, the globules being orientated and translated such that the electricfield vectors at the origin of each globule in isolation are tangential with the underlyingcurve and the curved electric line which results has a continuous appearance. Theelectric field E ä of our curved electric line is depicted in Fig. 17. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Figure 17.
The electric field E ä of our curved electric line (section 3.2.4),depicted as a three-dimensional vector plot at an instant of time: each bluearrow represents an electric field vector, colour-coded and scaled by magnitude. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D3.2.5. Electric knots An unusual electromagnetic disturbance in which some of theelectric field lines form a particularly prominent feature resembling a torus or non-torusknot [58] at certain times can be realised as a closed electric line (section 3.2.4) withthe appropriate topology. Various forms are possible.As our first example consider the ‘electric torus knot’ specified by taking N = 80and φ n = atan2(ˆ v y , ˆ v x ) ,θ n = arccos ˆ v z ,χ n = 0 ,X (cid:48) n = d [sin µ n + 2 sin(2 µ n )] ,Y (cid:48) n = d [cos µ n − µ n )] ,Z (cid:48) n = − d sin(3 µ n ) , ˜ A (cid:48) n = i˜ B (cid:48) n = 0 , where atan2( Y, X ) is the four-quadrant inverse tangent function, µ n = 2 π ( n − /N andit is to be understood thatˆ v x = cos µ n + 4 cos(2 µ n ) (cid:112)
35 + 8 cos(3 µ n ) + 18 cos(6 µ n ) , ˆ v y = {− sin µ n + 4 sin(2 µ n ) } (cid:112)
35 + 8 cos(3 µ n ) + 18 cos(6 µ n ) , ˆ v z = − µ n ) (cid:112)
35 + 8 cos(3 µ n ) + 18 cos(6 µ n )and d = 0 .
90: our electric torus knot is based upon a ( p, q ) = (2 , −
3) torus knot, alsoknown as a left-handed trefoil knot [58]. The electric field E H of our electric torus knotis depicted in Fig. 18.Our electric torus knot might be regarded as another explicit example of a monochromatic electromagnetic knot of toroidal character, following our electromagnetictangle (section 3.1.5). onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Figure 18.
The electric field E H of our electric torus knot (section 3.2.5),depicted as a three-dimensional vector plot at an instant of time: each bluearrow represents an electric field vector, colour-coded and scaled by magnitude.Each of the electric rings comprising the knot has been plotted for | s n | ≤ λ and | z n | ≤ λ only, for the sake of clarity and to render the task numericallytractable. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D As our second example consider the electric non-torus knot specified by taking N = 200 and φ n = atan2(ˆ v y , ˆ v x ) ,θ n = arccos ˆ v z ,χ n = 0 ,X (cid:48) n = d [2 + cos(2 µ n )] cos(3 µ n ) ,Y (cid:48) n = d [2 + cos(2 µ n )] sin(3 µ n ) ,Z (cid:48) n = 2 d sin(4 µ n ) , ˜ A (cid:48) n = α n i˜ B (cid:48) n = 0with the amplitudes α n as described below, where it is to be understood thatˆ v x = − µ n ) cos(3 µ n ) − µ n )] sin(3 µ n ) (cid:112) [149 + 72 cos(2 µ n ) + 5 cos(4 µ n ) + 64 cos(8 µ n )] / , ˆ v y = − µ n ) sin(3 µ n ) + 3[2 + cos(2 µ n )] cos(3 µ n ) (cid:112) [149 + 72 cos(2 µ n ) + 5 cos(4 µ n ) + 64 cos(8 µ n )] / , ˆ v z = 8 cos(4 µ n ) (cid:112) [149 + 72 cos(2 µ n ) + 5 cos(4 µ n ) + 64 cos(8 µ n )] / d = 1 .
33: our electric non-torus knot is based upon a figure-of-eight knot [58].The electric field E Ì of our electric non-torus knot is depicted in Fig. 19. The α n areobtained by first calculating E Ì with the α n = 1, then taking α n = E / | E Ì ( x = X n , y = Y n , z = Z n , t (cid:54) = lT / | for the final E Ì . This iteration gives the final electric non-torusknot a ‘cleaner’ form than is obtained with the α n = 1.Our electric non-torus knot might be regarded as the first explicit example of a monochromatic electromagnetic knot of non-toroidal character. A method by which toconstruct polychromatic electromagnetic knots of non-toroidal character (at a particulartime) is described in [28].We note here once more, as in § onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Figure 19.
The electric field E Ì of our non-torus electric knot (section 3.2.5),depicted as a three-dimensional vector plot at an instant of time: each bluearrow represents an electric field vector, colour-coded and scaled by magnitude.Each of the electric rings comprising the knot has been plotted for | s n | ≤ λ and | z n | ≤ λ only, for the sake of clarity and to render the task numericallytractable. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D3.2.6. Electromagnetic cloud Random choices of the parameters defining our unusualelectromagnetic disturbances of the second kind can yield unusual electromagneticdisturbances with prominent electric (and magnetic) features that resemble clouds.Consider for example the ‘electromagnetic cloud’ specified by taking N = 20 and n φ n θ n χ n X n /λ Y n /λ Z n /λ ˜ A (cid:48) n ˜ B (cid:48) n .
20 1 .
60 5 .
60 0 .
00 0 .
00 0 .
69 0 . . i 0 . . i2 1 .
40 0 .
70 6 . − .
64 0 .
15 0 .
15 0 . . i 0 . . i3 4 .
90 0 .
00 0 . − .
63 0 . − .
30 1 . . i 0 . . i4 3 .
00 1 .
20 2 . − .
91 0 . − .
15 0 . . i 0 . . i5 5 .
20 3 .
10 5 . − . − . − .
48 0 . . i 1 . . i6 0 .
80 2 .
80 3 . − . − .
09 0 .
36 0 . . i 1 . . i7 2 .
20 2 .
30 4 . − . − .
42 0 .
12 0 . . i 0 . . i8 3 .
70 0 .
80 1 . − . − .
60 0 .
39 1 . . i 1 . . i9 2 .
40 1 .
40 4 . − .
94 0 . − .
33 0 . . i 0 . . i10 4 .
30 1 .
90 0 . − .
46 0 .
45 0 .
06 0 . . i 0 . . i11 1 .
20 2 .
80 4 .
50 0 .
70 0 .
15 0 .
66 0 . . i 0 . . i12 0 .
30 1 .
60 1 .
10 0 . − .
18 0 .
24 0 . . i 1 . . i13 1 .
80 1 .
30 5 .
30 1 .
24 0 .
48 0 .
30 0 . . i 0 . . i14 4 .
50 0 .
90 6 .
10 1 .
15 0 . − .
18 1 . . i 0 . . i15 6 .
00 0 .
10 2 .
30 1 . − . − .
39 0 . . i 0 . . i16 0 .
40 1 .
80 2 .
20 0 . − .
24 0 .
51 0 . . i 0 . . i17 1 .
40 3 .
10 3 .
20 1 .
48 0 . − .
21 0 . . i 1 . . i18 3 .
20 2 .
80 1 .
80 1 .
12 0 . − .
09 0 . . i 0 . . i19 1 .
90 1 .
50 3 .
60 0 . − .
30 0 .
45 0 . . i 0 . . i20 5 .
10 2 .
40 0 .
90 0 .
64 0 .
21 0 .
66 0 . . i 1 . . iin (50) and (51): our electromagnetic cloud can be regarded as a modest superpositionof spatiotemporally translated and rotated versions of our electric ring (3.1.3) and ourelectric globule (3.1.4), with the translations and rotations assigned randomly (withinappropriate ranges). The well localised appearances of the rings and globules see thecloud itself exhibit a well localised appearance. The random choices nevertheless givethe cloud an amorphous form, with the phase differences between the rings and globulesgiving the cloud a more intricate temporal dependence within each cycle than thatexhibited by each of the rings and globules individually. The electric field E J of ourelectromagnetic cloud is depicted at different times in Fig. 20, Fig. 21 and Fig. 22.The magnetic field B J of our electromagnetic cloud is similarly cloud-like.Let us emphasise that our electromagnetic cloud is not merely a randomsuperposition of plane electromagnetic waves (although such superpositions areinteresting in their own right [65]). In general such superpositions do not yield welllocalised electromagnetic disturbances. Rather, our electromagnetic cloud is a randomlychosen superposition of electric rings and electric globules, each of which is a carefullyonochromatic knots and other unusual electromagnetic disturbances: light localised in 3D chosen superposition of plane waves. The electric field E (cid:63)(cid:63) of a random superposition ofplane electromagnetic waves is depicted in Fig. 23. Comparing this with Fig. 20, Fig. 21and Fig. 22, it can be seen immediately that there is indeed a basic distinction betweenour electromagnetic cloud and such superpositions. E (cid:63)(cid:63) was generated by taking ˜ A (cid:48) =2 π (cid:80) Nn =1 ˜ A (cid:48) n δ ( ϕ − ϕ n ) δ ( ϑ − ϑ n ) / sin ϑ n and ˜ B (cid:48) = 2 π (cid:80) Nn =1 ˜ B (cid:48) n δ ( ϕ − ϕ n ) δ ( ϑ − ϑ n ) / sin ϑ n in (13) and (14), together with the values of N , the φ n , the θ n , the ˜ A (cid:48) n and the ˜ B (cid:48) n listedabove for our electromagnetic cloud (which we recycle here for the sake of brevity). onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Figure 20.
The electric field E J of our electromagnetic cloud (section 3.2.6) for t = lT , depicted as a three-dimensional vector plot: each blue arrow representsan electric field vector, colour-coded and scaled by magnitude. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Figure 21.
The electric field E J of our electromagnetic cloud (section 3.2.6) for t = (1 / l ) T , depicted as a three-dimensional vector plot: each blue arrowrepresents an electric field vector, colour-coded and scaled by magnitude. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Figure 22.
The electric field E J of our electromagnetic cloud (section 3.2.6) for t = (2 / l ) T , depicted as a three-dimensional vector plot: each blue arrowrepresents an electric field vector, colour-coded and scaled by magnitude. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Figure 23.
The electric field E (cid:63)(cid:63) of a random superposition of planeelectromagnetic waves (section 3.2.6), depicted as a three-dimensional vectorplot at an instant in time: each blue arrow represents an electric field vector,colour-coded and scaled by magnitude. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D
4. Energy and temporal dependence
In the present section we make some general observations with regards to the energiesand temporal dependencies of our unusual electromagnetic disturbances.Some of our unusual electromagnetic disturbances (such as our electric ring (section3.1.3), our electric globule (section 3.1.4), our electromagnetic tangle (section 3.1.5), ourelectric loop (section 3.2.2), our electric link (section 3.2.3), our electric lines (section3.2.4) and our electric knots (section 3.2.5)) can be regarded as (exotic) electromagneticstanding waves: for both the electric field E and the magnetic field B , the spatial andtemporal dependencies factorise, with the temporal dependence consisting simply ofa sinusoidal oscillation. Our other unusual electromagnetic disturbances (such as ourelectromagnetic cloud (section 3.2.6)) cannot be regarded as standing waves, however,due to their more intricate temporal dependencies.For each of our unusual electromagnetic disturbances that can be regarded as astanding wave, one can say that there is no flow of energy on average in any directionat any position, in that the integral of the familiar electromagnetic energy flux density[3, 4, 43] 1 µ E × B (59)over one period T vanishes everywhere for any initial time t :1 T (cid:90) t + T t µ E × B d t = 0 . (60)One can also say, however, that there is an oscillation of energy, back and forth betweenthe electric field, which exhibits a time dependence of the form E ∝ cos( ω t + δ ), andthe magnetic field, which exhibits a time dependence of the form B ∝ sin( ω t + δ ), thusdiffering in phase from the electric field by a quarter cycle. When t = ( l − δ /π ) T / | E | takes its maximum value everywhere whereas B = 0, in which case the familiarelectromagnetic energy density [2, 3, 4, 43]12 (cid:18) (cid:15) | E | + 1 µ | B | (cid:19) (61)reduces to (cid:15) | E | , (62)indicating that energy is stored in the electric field. When t = (1 / l − δ /π ) T / E = 0 whereas | B | takes its maximum value everywhere, in which case the familiarelectromagnetic energy density reduces to12 µ | B | , (63)indicating that energy is stored in the magnetic field instead. Such oscillations are,perhaps, a hallmark of electromagnetic standing waves and are sustained throughoutthe disturbance by electromagnetic induction. For each of our unusual electromagnetic onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D disturbances that cannot be regarded as a standing wave the situation is different inthat 1 T (cid:90) t + T t µ E × B d t (cid:54) = 0 (64)in general and there is thus a sense in which the disturbance is propagating, at leastlocally.The above allows leads us to draw a simple distinction between each of our unusualelectromagnetic disturbances that can be regarded as a standing wave and the regionof heightened ‘intensity’ found in a tightly focussed laser beam, for example: one cansay that the former do not transport energy on average whereas a beam of light doestransport energy on average, in particular through planes perpendicular to its directionof propagation.As highlighted in section 2, each of our unusual electromagnetic disturbancesappears to be well localised in all three spatial dimensions in that the electric (andmagnetic) field strengths in the regions described are significantly larger than thosefound anywhere else. It should be noted, however, that the total energy of each of ourunusual electromagnetic disturbances (as we have defined them thus far in the presentpaper) is infinite in that the integral of the familiar electromagnetic energy density overall space diverges: (cid:90) ∞−∞ (cid:90) ∞−∞ (cid:90) ∞−∞ (cid:18) (cid:15) | E | + 1 µ | B | (cid:19) d x d y d z = ∞ . (65)Similar observations can be made of other, well-known solutions of Maxwell’s equations(1)-(4): a monochromatic Bessel beam has an infinite energy per unit length [66]; amonochromatic Gaussian beam has a finite energy per unit length but also an infinitetotal energy, as seen when integrating the energy per unit length along the directionof propagation. Good approximations to these solutions are nevertheless generatedroutinely within finite regions of space in the laboratory [67]. A subtlety is that theseapproximations are only quasi-monochromatic, as they exist over finite time intervals.This leads us to recognise that the total energies of our unusual electromagneticdisturbances can be rendered meaningful by introducing suitable regularisations intothe frequency spectra of the disturbances. For each of our unusual electromagneticdisturbances of the first kind (section 3.1), one such regularisation consists of taking˜ A = rect[( k − k ) / δk ]˜ A (cid:48) / πk δk and ˜ B = rect[( k − k ) / δk ]˜ B (cid:48) / πk δk rather than˜ A = E δ ( k − k )˜ A (cid:48) / πk and ˜ B = E δ ( k − k )˜ B (cid:48) / πk in (13) and (14), where rect( K )is the rectangular function (with rect ( K ) = rect( K )) and 0 < δk (cid:28) k is a range ofwavenumbers. This renders the total energy finite and time-independent at finite times( | t | < ∞ ), as desired: (cid:90) ∞−∞ (cid:90) ∞−∞ (cid:90) ∞−∞ (cid:18) (cid:15) | E | + 1 µ | B | (cid:19) d x d y d z (66)= (cid:90) π (cid:90) π (cid:90) ∞ π (cid:15) (cid:0) | ˜ A | + | ˜ B | (cid:1) k d k sin ϑ d ϑ d ϕ onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D = 2 π (cid:15) ( k − δk ) δk (cid:0) | ˜ A (cid:48) | + | ˜ B (cid:48) | (cid:1) . One can appreciate this regularisation as follows. Recall from section 3.1.2 that themodulating scalar fields f , g and h each tend towards a form that is at least qualitativelysimilar to a sinusoidal undulation modulated by a 1 / | r | fall off as | r | → ∞ in any non-trivial direction. Let us therefore consider1 | r | cos ( k | r | ) cos ( ck t ) (67)for the sake of concreteness. Our regularisation corresponds to replacing this with12 δk (cid:90) k + δkk − δk | r | cos ( k (cid:48) | r | ) cos ( ck (cid:48) t ) d k (cid:48) , (68)which for finite times tends towards1 δk | r | [cos( k | r | ) cos( ck t ) sin( δk | r | ) cos( δkct ) (69) − sin( k | r | ) sin( ck t ) sin( δkct ) cos( δk | r | )]as | r | → ∞ . The 1 / | r | fall off seen in (69) is more dramatic than the 1 / | r | fall off seenin (67), as desired. More potent regularisations might be required to render quantitieslike the total angular momentum [2, 3, 4, 43] (cid:90) ∞−∞ (cid:90) ∞−∞ (cid:90) ∞−∞ (cid:15) r × ( E × B )d r (70)and the total boost angular momentum [43] t (cid:90) ∞−∞ (cid:90) ∞−∞ (cid:90) ∞−∞ (cid:15) ( E × B )d r (71) − c (cid:90) ∞−∞ (cid:90) ∞−∞ (cid:90) ∞−∞ r (cid:18) (cid:15) | E | + 1 µ | B | (cid:19) d r meaningful, due to the presence of r in the integrands.The author acknowledges that densities and flux densities are not necessarily unique[64]. The ‘familiar’ forms considered above enjoy something of a privileged status,however, in that they seem to be singled out by gravitational interactions.Let us conclude here by noting with interest that each of our unusualelectromagnetic disturbances can be set translating with any speed < c in any directionby actively boosting it [3, 4, 68]. We are free to do this because (1)-(4) are Lorentz-invariant [43]. For speeds (cid:28) c , this does not significantly alter the basic form ofthe disturbance. It is thus possible to have an ultraviolet [69] electromagnetic tangletranslating at 10m.s − , for example. Each of the plane electromagnetic waves comprisinga boosted version of one of our unusual electromagnetic disturbances propagates withspeed c [2, 3, 4, 44, 68, 70]. It is the differences in the frequencies of these waves that givesrise to the apparent translation of the disturbance as a whole: the boosted version of amonochromatic electromagnetic disturbance is polychromatic in general, its constituentplane waves having been Doppler shifted [4, 71] by different amounts depending upontheir direction of propagation with respect to the direction of the boost. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D
5. Generation
In the present section we consider the generation of our unusual electromagneticdisturbances in the laboratory. Different methods of generation will be requireddepending upon the form of disturbance sought and the frequencies required. In section5.1 we focus upon the use of an antenna to generate an electric ring (section 3.1.3) in theradiowave or microwave domain [44]. In section 5.2 we focus upon the use of cylindricallypolarised vector beams to generate an electric ring or electric globule (section 3.1.4) inthe visible domain.Works to date on electromagnetic knots are largely silent on the question ofgeneration [5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,27, 28, 30, 31, 32, 54, 55], although some basic discussions have been presented [12, 29].
Consider a collection of M electrically conducting rings, with each ring concentric withthe surface of a sphere of radius R centred upon the origin; that section of each ring with ϕ ∈ ( − ∆ / , ∆ /
2) removed; the ends of each ring connected to an alternating electriccurrent source of (central) angular frequency ω ; θ m the polar angle of the m th ring( m ∈ { , . . . , M } ) and I = (cid:60) (cid:16) ˜ I e − i ω t (cid:17) (72)the (identical) current in each ring, directed in the ˆ φφφ direction for I >
0. The basicingredients of our electric-ring antenna are depicted in Fig. 24. Ignoring interactionsbetween rings, taking each ring to be of negligible cross-section, neglecting the radiationproduced by the elements that connect the rings to the current source and assumingthe surrounding medium to be transparent with phase refractive index n p , we find thatthe electric field radiated by the rings is essentially E LF = (cid:60) (cid:16) ˜ E LFe − i ω t (cid:17) (73)with ˜ E LF a = i µ Rω ˜ I π M (cid:88) m =1 (cid:90) π − ∆ / / ei k n p | r − r (cid:48) m | | r − r (cid:48) m | (cid:88) b ∈{ x,y,z } (74) (cid:20) δ ab (cid:18) k n p | r − r (cid:48) m | − k n p | r − r (cid:48) m | (cid:19) − ( r − r (cid:48) m ) a ( r − r (cid:48) m ) b | r − r (cid:48) m | (cid:18) k n p | r − r (cid:48) m | − k n p | r − r (cid:48) m | (cid:19)(cid:21) ˆ ϕ b sin ϑ m d ϕ, where a ∈ { x, y, z } , δ ab is the Kronecker delta function and r (cid:48) m = R (ˆ xxx cos ϕ sin ϑ m + ˆ yyy sin ϕ sin ϑ m + ˆ zzz cos ϑ m ) (75)is the position of an element of the m th ring: E LF is a sum over contributions due tosuch elements, with each element treated as an oscillating electric dipole [3, 4]. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D
Suppose now that there is a small gap and many rings such that (cid:90) π − ∆ / / d ϕ → (cid:90) π d ϕϑ m → ϑ, M (cid:88) m =1 → Mπ (cid:90) π d ϑ. For regions within the sphere ( | r | < R ) that satisfy the far-field condition( k n p ( R − | r | ) (cid:29) E LF a ≈ i µ Rω ˜ IM π (cid:90) π (cid:90) π ei k n p | r − r (cid:48) | | r − r (cid:48) | (76) (cid:20) δ ab − ( r − r (cid:48) ) a ( r − r (cid:48) ) b | r − r (cid:48) | (cid:21) ˆ ϕ b sin ϑ d ϑ d ϕ where r (cid:48) = R (ˆ xxx cos ϕ sin ϑ + ˆ yyy sin ϕ sin ϑ + ˆ zzz cos ϑ ) . (77)If, moreover, the distance from the origin is small ( | r | (cid:28) R ) such thatei k n p | r − r (cid:48) | | r − r (cid:48) | ≈ ei k n p R e − i k n p r (cid:48) · r /R R (78) (cid:20) δ ab − ( r − r (cid:48) ) a ( r − r (cid:48) ) b | r − r (cid:48) | (cid:21) ˆ ϕ b ≈ ˆ ϕ a , (79)this reduces further to˜ E LF ≈ ˜ E (cid:48) ˆ φφφf (cid:48) e − i ω t (80)with ˜ E (cid:48) = µ ω ˜ IM ei k n p R π (81) f (cid:48) = (cid:90) π J ( k sin ϑn p s ) cos( k cos ϑn p z ) sin ϑ d ϑ. (82)Thus E LF ≈ | ˜ E (cid:48) | ˆ φφφf (cid:48) cos( ω t − arg ˜ E (cid:48) ) , (83)which is essentially the electric field E A of our electric ring: the two coincide precisely inform for n p ≈ f (cid:48) ≈ f , together with a choice of phase for I such that ˜ E (cid:48) → E is real. Our electric-ring antenna is depicted schematically in operation in Fig. 25.The geometrical requirements above are reasonably well satisfied by M = 100, R = 1 . × − m and ω / π = 1 . × s − , for example.Simple elaborations upon our design permit the generation of other unusualelectromagnetic disturbances in the radiowave or microwave domain. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D Figure 24.
The basic ingredients of our electric-ring antenna (section 5.1). onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D
Figure 25.
A schematic depiction of our electric-ring antenna (section 5.1) inoperation. Part of the antenna has been cut away to reveal the electric ring(section 3.1.3) generated by the antenna. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D5.2. Visible domain
The following observations were borne out of discussions with Sonja Franke-Arnold,Neal Radwell and Fiona C Speirits, to whom the author is grateful.The electric field E A of our electric ring can be recast as E A = 1 k (cid:90) k [ E + ( | k z | ) + E − ( | k z | )] d | k z | (84)with E ± ( | k z | ) = (cid:60) (cid:104) E ˆ φφφ J ( κs )ei ( ±| k z | z − ω t ) (cid:105) (85)an azimuthally polarised vector beam [72, 73, 74, 75, 76, 77] of transverse wavenumber κ = (cid:112) k − | k z | propagating in the ± z direction. This leads us to suggest, tentatively,that a good approximation to our electric ring in the visible domain might be generatedin the laboratory by superposing counter-propagating azimuthally polarised vectorbeams of light, each with an appropriate spread of transverse wave numbers. Ananalogous arrangement with radially polarised [37, 72, 73, 74, 75, 77] beams mightbe used to generate a good approximation to our electric globule. Electric rings andelectric globules created in this way could be used in turn as building blocks to generatemore elaborate unusual electromagnetic disturbances in the visible domain, followingthe recipes given in section 3.2, for example.The tight focussing of radially polarised vector beams has been explored inconsiderable detail elsewhere, owing to the fact that such beams can be focussed moretightly than usual [77, 78, 79]. It is interesting to note that our electric globule, whichcan be regarded as a superposition of radially polarised vector beams as describedabove, seems to reside within a sphere of sub-wavelength diameter, in accord with theaforementioned focussing properties of the beams individually. The tight focussing ofan azimuthally polarised vector beam has recently been explored [80], with interestingresults. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D
6. Outlook
We recognise many possible directions for future research, some of which are highlightedbelow.With regards to basic theory: • three spatial dimensions, not just two. • The generation of our unusual electromagnetic disturbances in the laboratory re-quires further consideration. • Exotic media might permit new types of unusual electromagnetic disturbance [82].With regards to potential applications: • Our unusual electromagnetic disturbances might enable new and / or improvedforms of display. In particular, unusual electromagnetic disturbances in the visibledomain might act as three-dimensional pixels or voxels for a new form of volu-metric 3D display, with each voxel rendered visible in air via Rayleigh scattering[83, 84, 85]. Such voxels would be superposable and non-destructive, the small sizeof each voxel would permit high-resolution volumetric images, different colours ofvoxel and therefore images could be realised, the non-isotropic radiation profiles ofthe voxels would permit emulation of occlusion and opacity and there would be noneed for mechanical parts within the image volume. • The setups that we envisage for the generation of our unusual electromagneticdisturbances in the visible domain are not entirely unlike that found in a ‘4Pi’microscope [86], suggesting possible applications for our unusual electromagneticdisturbances in imaging. • The use of electromagnetic forces to manipulate various forms of matter, includingatoms [87, 88, 89] and fusion plasmas [90], is well established. The possibilitiesoffered in such contexts by our unusual electromagnetic disturbances might beworthy of pursuit. A preliminary examination of the forces exerted upon chargedparticles by an electromagnetic knot is presented in [14]. It is particularlyinteresting to note that the helical magnetic field lines found at appropriatetimes within each torus of our electromagnetic tangle (section 3.1.5) resemble themagnetic field lines engineered in certain tokamaks to confine hot plasma: perhaps onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D our electromagnetic tangle could assist in the pursuit of efficient energy productionvia controlled thermonuclear fusion.With regards to natural occurrence: • Suggestions have already been made that novel electromagnetic disturbances[54, 55], in particular magnetic knots [29, 91, 92, 93], might underpin the phe-nomenon of ball lightning. Perhaps it is worth revisiting these hypotheses giventhe new ideas introduced in the present paper. Variants of our electric-ring antenna(5) might be employed to ionise the air in unusual geometries and thus help exploresuch hypotheses in the laboratory [54, 94, 95], for example. • An unusual electromagnetic disturbance in vacuum might appear essentiallyinvisible when viewed from a sufficient distance, at least as far as its electromagneticprofile is concerned. The disturbance should nevertheless be capable of making itspresence known gravitationally, due to its energy-momentum content [96]. It isinteresting to imagine an electromagnetic cloud in interstellar space acting as agravitational lens, for example. Could some component of dark matter [97] beattributed to unusual electromagnetic disturbances and / or analogous features inthe gravitational field? The author has found it possible to construct gravitationalanalogues [98, 99, 100] of the unusual electromagnetic disturbances introduced inthe present paper, including gravitational clouds.We will return to these and related ideas elsewhere.
7. Acknowledgments
The present work was supported by the Engineering and Physical Sciences ResearchCouncil (EPSRC) (EP/M004694/1). The author thanks Stephen M Barnett, J¨org BG¨otte, Sonja Franke-Arnold, Neal Radwell, Gergely Ferenczi, Alison M Yao, Aidan SArnold, Sophie Viaene, Marco Ornigotti, Megan R Paterson and an anonymous refereefor their advice and encouragement. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D
Appendix A. Multipolar electromagnetic waves
The following treatment echoes that given in [2].Any electromagnetic disturbance can be regarded as a superposition of multipolarelectromagnetic waves: the general solution to Maxwell’s equations (1)-(4) can be castas E = (cid:60) (cid:26) (cid:90) ∞ ∞ (cid:88) J =1 J (cid:88) M J = − J i (cid:114) (cid:126) ck(cid:15) (cid:104) ˜ α kJM J X ˜ I kJM J X (A.1)+ ˜ α kJM J Z ˜ I kJM J Z (cid:105) e − i ωt d k (cid:27) B = (cid:60) (cid:26) (cid:90) ∞ ∞ (cid:88) J =1 J (cid:88) M J = − J i c (cid:114) (cid:126) ck(cid:15) (cid:104) − ˜ α kJM J X ˜ I kJM J Z (A.2)+ ˜ α kJM J Z ˜ I kJM J X (cid:105) e − i ωt d k (cid:27) with ˜ I kJM J X ( φ, θ, r ) = (cid:114) π k i J ˜ X JM J ( φ, θ )j J ( kr ) (A.3)˜ I kJM J Z ( φ, θ, r ) = (cid:114) π i J − r (cid:26)(cid:112) J ( J + 1) ˜ N JM J ( φ, θ )j J ( kr ) (A.4)+ ˜ Z JM J ( φ, θ ) d [ kr j J ( kr )]d( kr ) (cid:27) , as well as ˜ X JM J ( φ, θ ) = r × ∇∇∇ ˜ Y JM J ( φ, θ ) (cid:112) J ( J + 1) , (A.5)˜ N JM J ( φ, θ ) = ˆ r ˜ Y JM J ( φ, θ ) (A.6)˜ Z JM J ( φ, θ ) = − ˆ r × ˜ X JM J ( φ, θ ) , (A.7)where ˜ α kJM J X and ˜ α kJM J Z are complex functions of k , J ∈ { , . . . } and M J ∈{ , . . . , ± J } ; φ , θ and r are spherical coordinates;ˆ φφφ = − ˆ xxx sin φ + ˆ yyy cos φ, (A.8)ˆ θθθ = ˆ xxx cos φ cos θ + ˆ yyy sin φ cos θ − ˆ zzz sin θ (A.9)ˆ r = ˆ xxx cos φ sin θ + ˆ yyy sin φ sin θ + ˆ zzz cos θ (A.10)are associated unit vectors; j J ( X ) is the spherical Bessel function of order J and˜ Y JM J ( X, X (cid:48) ) is the spherical harmonic function of order J , M J . A particular disturbanceis determined by specifying ˜ α kJM J X and ˜ α kJM J Z , which is equivalent to specifying thenormal variables: ‘˜ ααα ( ϕ, ϑ, k, t ) = (cid:82) ∞ (cid:80) ∞ J =0 (cid:80) JM J = − J δ ( k (cid:48) − k )( ˜ α k (cid:48) JM J X ˜ X JM J ( ϕ, ϑ ) +˜ α k (cid:48) JM J Z ˜ Z JM J ( ϕ, ϑ ))e − i ωt d k (cid:48) /k (cid:48) ’. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D The relationship between the multipolar description outlined above and the plane-wave description used in the main text is embodied by the following relationships:˜ α kJM J X = − (cid:114) π (cid:15) (cid:126) c (cid:90) π (cid:90) π ˜ X ∗ JM J ( ϕ, ϑ ) · (ˆ ϕϕϕ ˜ B + ˆ ϑϑϑ A ) √ k sin ϑ d ϑ d ϕ (A.11)˜ α kJM J Z = − (cid:114) π (cid:15) (cid:126) c (cid:90) π (cid:90) π ˜ Z ∗ JM J ( ϕ, ϑ ) · (ˆ ϕϕϕ ˜ B + ˆ ϑϑϑ A ) √ k sin ϑ d ϑ d ϕ. (A.12)For our unusual electromagnetic disturbances of the first kind (section 3.1), we find that˜ α kJM J X ∝ δ ( k − k ) δ M J and ˜ α kJM J Z ∝ δ ( k − k ) δ M J with no obvious closed form forthe J dependencies. References [1] J C Maxwell 1873
A Treatise on Electricity and Magnetism (Clarendon Press)[2] C Cohen-Tannoudji, J Dupont-Roc and G Grynberg 1989
Photons and Atoms: Introduction toQuantum Electrodynamics (Wiley)[3] D J Griffiths 1999
Introduction to Electrodynamics (Pearson Education)[4] J D Jackson 1999
Classical Electrodynamics (Wiley)[5] A Trautman 1977 Solutions of the Maxwell and Yang-Mills equations associated with hopf fibrings
Int. J. Theor, Phys. Lett. Math. Phys. J. Phys. A: Math. Gen. L815–L820[8] A F Ra˜nada 1992 Topological electromagnetism
J. Phys. A: Math. Gen. Phys. Lett. A
Phys. Lett. A
J. Opt. A: Pure Appl. Opt. S181–S183[12] W T M Irvine and D Bouwmeester 2008 Linked and knotted beams of light
Nat. Phys. Opt. Lett. J.Phys. A: Math. Theor. J. Phys. A: Math. Theor. arXiv:1106.1122 [17] A F Ra˜nada 2012 On topology and electromagnetism Ann. Phys.
A35–A37[18] M Array´as and J L Trueba 2012 Exchange of helicity in a knotted electromagnetic field
Ann. Phys.
J. Phys. A: Math.Theor. J. Phys. A: Math. Theor. Phys. Rev. Lett. arXiv:1302.1431 onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D [23] M Array´as and J L Trueba 2015 A class of non-null toroidal electromagnetic fields and its relationto the model of electromagnetic knots
J. Phys. A: Math. Theor. J. Phys. A: Math. Theor. J. Phys.A: Math. Theor. Annals Phys.
Anal. Math. Phys. Phys. Rev. Lett.
Phys. Rep.
J. Phys. A: Math. Theor. Proc.SPIE arXiv:1705.06750 [33] J F Nye and M V Berry 1974 Dislocations in wave trains
Proc. Roy. Soc. Lond. A
Proc. Roy. Soc. Lond. A
Nature
Nat. Phys. J. Opt. Proc.Roy. Soc. Lond. A
Proc. Roy. Soc. Lond. A
Proc.Roy. Soc. Lond. A
Science
Math. Mag. Mathematische Annalen Phil. Trans. Roy. Soc. Lond.
Comptes rendus hebdomadaires des s´eances de l’Acad´emie des sciences
Comptes rendus hebdomadaires des s´eances del’Acad´emie des sciences
Phil. Mag. The Optics of Rays, Wavefronts, and Caustics (Academic Press)[49] MoEDAL Collaboration 2016 Search for magnetic monopoles with the MoEDAL prototype onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D trapping detector in 8 TeV proton-proton collisions at the LHC
J. High Energy Phys.
Phil.Trans. Roy. Soc. Lond. A
Phil. Trans. Roy. Soc. Lond. A
New J.Phys. New J. Phys. J. Phys. A: Math. Gen. Eur. Trans. Electr. Power Eng. Mem. Acad.Sci. Inst. France J. Oxford Univ. Junior Scientific Club
Knot Theory (Mathematical Association of America)[59] R Takeda, N Kida, M Sotome, Y Matsui and H Okamato 2014 Circularly polarized narrowbandterahertz radiation from a eulytite oxide by a pair of femtosecond laser pulses
Phys. Rev. A Sitzungsberichten der Physikalisch-Medicinsichen Gesellschaft zu W¨urzburg Phil. Trans. Roy. Soc. Lond. Phil. Trans.Roy. Soc. Lond. Phil. Trans. Roy. Soc.Lond. J. Opt. Phys. Rev. Lett. J. Opt. Soc. Am. A Phys. Rev. Lett. Annalen der Physik Stud. Hist. Philos M. P. A Science ¨Uber das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels (Borrosch und Andr´e)[72] N Radwell, R D Hawley, J B G¨otte and S Franke-Arnold 2016 Achromatic vector vortex beamsfrom a glass cone
Nat. Comm. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D [73] M Ornigotti, C Conti and A Szameit 2016 Cylindrically polarized nondiffracting optical pulses J.Opt. Opt. Express Frontiers in Optics 2016, FTh5B.3 [76] D Pohl 1972 Operation of a Ruby laser in the purely transverse electric mode TE Appl. Phys.Lett. Adv. Opt.Phot. Opt.Comm.
Phys. Rev.Lett. Sci. Rep. Physics-Uspekhi Phys. Rev. Lett.
Phil. Mag. Phil. Mag. Phil. Mag. Opt. Lett. Phys. Rev. Lett. cm − in aholographically shaped dark spontaneous-force optical trap Phys. Rev. A Phys. Rev. A Plasma Phys. Control. Fusion Nature
J. Geophys. Res.
Phys. Rev.E Nature
Fusion Sci. Technol. Annalen der Physik Astrophys. J. Sci.Am. onochromatic knots and other unusual electromagnetic disturbances: light localised in 3D [99] S M Barnett 2014 Maxwellian theory of gravitational waves and their mechanical properties
NewJ. Phys. Phys. Rev. Lett.116