Theoretical proposal for the experimental realisation of a monochromatic electromagnetic knot
HHow to make a monochromatic electromagnetic knot
How to make a monochromatic electromagnetic knot
R. P. Cameron, a) W. Löffler, and K. D. Stephan SUPA and Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK Huygens-Kamerlingh Onnes Laboratorium, Leiden University, 2333 Leiden, CA, The Netherlands Ingram School of Engineering, Texas State University, San Marcos, Texas USA (Dated: 26 January 2021)
We describe an antenna designed to generate monochromatic electromagnetic knots and other “unusual electromagneticdisturbances” in the microwave domain. Our antenna is a spherical array of radiating dipolar elements configured togenerate the desired electromagnetic field pattern near its centre. We show numerically that an "electromagnetic tangle"of frequency 2 .
45 GHz can be generated using an antenna of radius of 61 . I. INTRODUCTION
Electromagnetic knots were first described by Rañada andcollaborators in a series of works based on earlier stud-ies by Woltjer and Moffatt , making use of the Batemanconstruction . Several properties of these Rañada-Hopf elec-tromagnetic knots have been worked out in theory, includingtheir orbital angular momentum and helicity , and similarsolutions of Maxwell’s equations have been found more re-cently in the form of propagating light beams . Althoughanalogous structures have been observed experimentally inliquid crystals and in fluid dynamics , no feasible proposalhas been given to date for the realisation of a Rañada-Hopftype electromagnetic knot, the closest work being a theoreticalproposal involving plasma physics and self-organisation.A key issue is that Rañada-Hopf type electromagnetic knotsare polychromatic ; they have a broad frequency spectrum,which is difficult to realise in the laboratory.One of us recently described a collection of “unusual elec-tromagnetic disturbances” , which are essentially electro-magnetic standing waves that appear to be well localised in3D, even in complete vacuum. Included amongst these arevarious monochromatic electromagnetic knots. In this paperwe describe an antenna designed to generate (approximate)unusual electromagnetic disturbances in the microwave do-main, including monochromatic electromagnetic knots. Weshow that the electromagnetic field generated by our antennatends towards the (exact) form described in under certainlimits and that the antenna can generate linked and knottedelectric and magnetic field lines under experimentally realis-able conditions.In what follows we work in an inertial frame of referencedescribed by right-handed Cartesian coordinates x , y and z with associated unit vectors ˆ x , ˆ y and ˆ z and time t . We willalso make use of cylindrical coordinates s , φ and z with as-sociated unit vectors ˆ s = ˆ s ( φ ) , ˆ φ = ˆ φ ( φ ) and ˆ z as well asspherical coordinates r , θ and φ with associated unit vectorsˆ r = ˆ r ( θ , φ ) , ˆ θ = ˆ θ ( θ , φ ) and ˆ φ = ˆ φ ( φ ) . The position vectoris r = x ˆ x + y ˆ y + z ˆ z = s ˆ s + z ˆ z = r ˆ r . We adopt the SI system a) Electronic mail: [email protected] of units, where ε is the electric constant, µ is the magneticconstant and c = / √ ε µ is the speed of light. II. UNUSUAL ELECTROMAGNETIC DISTURBANCES
In this section we briefly summarise the unusual electro-magnetic disturbances described in . We work in vacuum.The simplest disturbances, referred to as “unusual electro-magnetic disturbances of the first kind”, can each be thoughtof as a continuous spherical superposition of plane electro-magnetic waves with common amplitude, phase at the origin( r = E = E ( r , t ) and magnetic field B = B ( r , t ) are E = ℜ (cid:40) E (cid:104) ˜ A (cid:48) ( ˆ s g + ˆ z h ) + i˜ B (cid:48) ˆ φφφ f (cid:105) e − i ω t (cid:41) (1) B = ℜ (cid:40) E c (cid:104) i˜ A (cid:48) ˆ φφφ f − ˜ B (cid:48) ( ˆ s g + ˆ z h ) (cid:105) e − i ω t (cid:41) (2)with f = (cid:90) π J ( k sin ϑ s ) cos ( k cos ϑ z ) sin ϑ d ϑ , (3) g = − (cid:90) π J ( k sin ϑ s ) sin ( k cos ϑ z ) sin ϑ cos ϑ d ϑ , (4) h = − (cid:90) π J ( k sin ϑ s ) cos ( k cos ϑ z ) sin ϑ d ϑ , (5)where E is an electric-field strength, ˜ A (cid:48) and ˜ B (cid:48) are con-stants that derive from the polarisation state of the waves and ω = ck is the angular frequency of the disturbance, k beingthe angular wavenumber. A particular unusual electromag-netic disturbance of the first kind is determined by specify-ing ˜ A (cid:48) and ˜ B (cid:48) . For example: taking ˜ A (cid:48) = i and ˜ B (cid:48) = A (cid:48) = B (cid:48) = − i corresponds to each wave being linearly polarisedalong the azimuthal direction and gives an “electric ring”; tak-ing ˜ A (cid:48) = / √ B (cid:48) = i σ / √ σ = ± a r X i v : . [ phy s i c s . c l a ss - ph ] J a n ow to make a monochromatic electromagnetic knot 2as a (monochromatic) electromagnetic knot as it has linkedand torus-knotted electric and magnetic field lines.Unusual electromagnetic disturbances of the first kind canbe superposed in various ways to create more exotic struc-tures, referred to as “unusual electromagnetic disturbances ofthe second kind”.We focus on the generation of an (approximate) electro-magnetic tangle in section IV. III. GENERAL ANTENNA DESIGN
In this section we describe in general terms an antenna de-signed to generate (approximate) unusual electromagnetic dis-turbances in the microwave domain.Our antenna is an array of 2 N driven elements, each takingthe form of a centre-fed dipole of length L driven at angularfrequency ω = π / T = ck = π c / λ , where T is the pe-riod, k is the angular wavenumber in vacuum and λ is thewavelength in vacuum. The elements are located in pairs at N points uniformly distributed on the surface of a sphere of ra-dius R centred on the origin ( r = A -type” element of the pair, isaligned with the polar unit vector ˆ θ and is so-named becauseit corresponds to the ˜ A (cid:48) contributions in (1) and (2). The otherelement of the pair, henceforth referred to as the “ B -type” el-ement of the pair, is aligned with the azimuthal unit vector ˆ φ and is so-named because it corresponds to the ˜ B (cid:48) contributionsin (1) and (2). The antenna is embedded in a medium of re-fractive index n = c √ ε µ = ck / ω , where ε is the permittivity, µ is the permeability and k is the angular wavenumber in themedium.Consider the X -type element of the n th pair ( X ∈ { A , B } , n ∈{ , . . . , N } ). The element is centred on position r n = x n ˆ x + y n ˆ y + z n ˆ z ( | r n | = R ) and is aligned with the unit vector ˆ u X n ( ˆ u A n = ˆ θ ( ˆ r n ) and ˆ u B n = ˆ φ ( ˆ r n ) ). Let the electric current I X n = I X n ( l , t ) in the element be I X n = ℜ (cid:104) I X n w e − i ( ω t − ϕ X n ) (cid:105) , (6)where I X n is the peak electric current, w = w ( l ) is a functionthat dictates the spatial distribution of the electric current (0 ≤ w ≤ l is a coordinate with origin at r n that increases inthe direction of ˆ u X n ( − L / ≤ l ≤ L /
2) and ϕ X n is a phaseconstant. Assuming that the elements are half-wave dipoles( L = λ / w = cos ( k l ) . (7)Other types of element, like short dipoles for example, canbe described using other forms for w . The electric field E X n = E X n ( r , t ) and magnetic field B X n = B X n ( r , t ) generatedby the element are obtained by decomposing the element intoa line of infintiesimal dipolar elements and integrating their contributions , giving E X n = ℜ (cid:40) (cid:90) L / − L / i µω I X n w e i ( k | r − r n − ˆ u X n l |− ω t + ϕ X n ) π | r − r n − ˆ u X n l | (cid:34) ˆ u X n (cid:32) + i k | r − r n − ˆ u X n l | − k | r − r n − ˆ u X n l | (cid:33) − ( r − r n − ˆ u X n l ) | r − r n − ˆ u X n l | ˆ u X n · ( r − r n − ˆ u X n l ) | r − r n − ˆ u X n l | (cid:32) + k | r − r n − ˆ u X n l | − k | r − r n − ˆ u X n l | (cid:33)(cid:35) d l (cid:41) (8) B X n = ℜ (cid:34) (cid:90) L / − L / i µ n ω I X n w e i ( k | r − r n − ˆ u X n l |− ω t + ϕ X n ) π c | r − r n − ˆ u X n l | ( r − r n − ˆ u X n l ) × ˆ u X n | r − r n − ˆ u X n l | (cid:32) + i k | r − r n − ˆ u X n l | (cid:33) d l (cid:35) . (9)The electric field E = E ( r , t ) and magnetic field B = B ( r , t ) generated by the antenna as a whole are superpositions of thefields generated by the individual elements: E = ∑ X = A , B N ∑ n = E X n (10) B = ∑ X = A , B N ∑ n = B X n . (11)The integrals in (8) and (9) can be evaluated numerically.We will now elucidate the conditions under which our an-tenna functions as desired, specialising to vacuum ( µ = µ and k = k ) for a direct comparison with the results presentedin and summarised in section II. The antenna is designed togenerate an approximation to an unusual electromagnetic dis-turbance internally, near the origin ( r = r max inside the antenna( r max < R ), centred on the origin (0 ≤ | r | (cid:46) r max ). Assumingthat the antenna is many wavelengths in radius ( k R (cid:29)
1) andthat we are in the far-field regime with respect to each of theelements ( k ( R − r max ) (cid:29) E ≈ ℜ (cid:40) ∑ X = A , B N ∑ n = (cid:90) L / − L / i µ ω I X n w e i ( k | r − r n − ˆ u X n l |− ω t + ϕ X n ) π | r − r n − ˆ u X n l | (cid:34) ˆ u X n − ( r − r n − ˆ u X n l ) | r − r n − ˆ u X n l | ˆ u X n · ( r − r n − ˆ u X n l ) | r − r n − ˆ u X n l | (cid:35) d l (cid:41) (12) B ≈ ℜ (cid:34) ∑ X = A , B N ∑ n = (cid:90) L / − L / i µ ω I X n w e i ( k | r − r n − ˆ u X n l |− ω t + ϕ X n ) π c | r − r n − ˆ u X n l | ( r − r n − ˆ u X n l ) × ˆ u X n | r − r n − ˆ u X n l | d l (cid:35) . (13)Assuming that the spherical region lies sufficiently well withinthe antenna ( r max (cid:28) R ) that the electromagnetic wave pro-duced by each element appears planar, we take (noting thatow to make a monochromatic electromagnetic knot 3 L = λ / (cid:28) R ) 1 | r − r n − ˆ u X n l | ≈ R , (14)e i k | r − r n − ˆ u X n l | ≈ e i k ( R − r · ˆ r n ) , (15)ˆ u X n − ( r − r n − ˆ u X n l ) | r − r n − ˆ u X n l | ˆ u X n · ( r − r n − ˆ u X n l ) | r − r n − ˆ u X n l | ≈ ˆ u X n (16) ( r − r n − ˆ u X n l ) × ˆ u X n | r − r n − ˆ u X n l | ≈ ˆ u X n × ˆ r n (17)to see (12) and (13) reduce further still to E ≈ ℜ (cid:40) ∑ X = A , B N ∑ n = i µ ω I X n W e i [ k ( R − r · ˆ r n ) − ω t + ϕ X n ] π R ˆ u X n (cid:41) (18) B ≈ ℜ (cid:40) ∑ X = A , B N ∑ n = i µ ω I X n W e i [ k ( R − r · ˆ r n ) − ω t + ϕ X n ] π cR ˆ u X n × ˆ r n (cid:41) (19)with W = (cid:90) L / − L / w d l . (20)Assuming that there are a large number of element pairs ( N (cid:38) π r / λ ) and that these are indeed uniformly distributed,we replace the discrete summation over N with a continuousintegral over 4 π sr to see (18) and (19) reduce finally to E ≈ ℜ (cid:40) (cid:90) π (cid:90) π i µ ω NW e i { k [ R − r · ˆ r ( ϑ , ϕ )] − ω t } π R (cid:104) I A ( ϑ , ϕ ) e i ϕ A ( ϑ , ϕ ) ˆ θ ( ϑ , ϕ )+ I B ( ϑ , ϕ ) e i ϕ B ( ϑ , ϕ ) ˆ φ ( ϕ ) (cid:105) sin ϑ d ϑ d ϕ (cid:41) (21) B ≈ ℜ (cid:40) (cid:90) π (cid:90) π i µ ω NW e i { k [ R − r · ˆ r ( ϑ , ϕ )] − ω t } π cR (cid:104) I A ( ϑ , ϕ ) e i ϕ A ( ϑ , ϕ ) ˆ φ ( ϕ ) − I B ( ϑ , ϕ ) e i ϕ B ( ϑ , ϕ ) ˆ θ ( ϑ , ϕ ) (cid:105) sin ϑ d ϑ d ϕ (cid:41) , (22)where we have introduced continuous peak electric currentfunctions I X = I X ( θ , φ ) as well as continuous phase constantfunctions ϕ X = ϕ X ( θ , φ ) defined such that I X n = I X ( ˆ r n ) and ϕ X n = ϕ X ( ˆ r n ) ( X ∈ { A , B } , n ∈ { , . . . , N } ). To generate an(approximate) unusual electromagnetic disturbance of the first kind we take the I X and the ϕ X to be constants, correspondingto each A -type element having the same (possibly zero) peakelectric current and phase constant and similarly for each B - type element. In this case, (21) and (22) are E ≈ ℜ (cid:40) E (cid:104) ˜ A (cid:48) ( ˆ s g + ˆ z h ) + i˜ B (cid:48) ˆ φ f (cid:105) e − i ω t (cid:41) (23) B ≈ ℜ (cid:40) E c (cid:104) i˜ A (cid:48) ˆ φφφ f − ˜ B (cid:48) ( ˆ s g + ˆ z h ) (cid:105) e − i ω t (cid:41) (24)with E ˜ A (cid:48) = i µ ω NW π R I A e i ( k R + ϕ A ) (25) E ˜ B (cid:48) = − i µ ω NW π R I B e i ( k R + ϕ B ) . (26)Comparing (23) and (24) with (1) and (2) we see that theantenna can indeed generate an (approximate) unusual elec-tromagnetic disturbance of the first kind, as claimed. A ro-tated and/or translated version of this disturbance can be gen-erated using suitably modified forms for the I X and/or the ϕ X .To generate an (approximate) unusual electromagnetic distur-bance of the second kind we need only superpose the electriccurrent distributions associated with each of its constituent un-usual electromagnetic disturbances of the first kind. IV. A SPECIFIC EMBODIMENT
In this section we describe a specific embodiment of ourantenna and its use to generate an (approximate) electromag-netic tangle of frequency f = .
45 GHz. We work in vacuumand focus on the electric field.We consider 2 N =
40 elements in the form of half-wavedipoles ( L = λ / = .
12 cm), arranged in pairs on the ver-tices of a regular dodecahedron with circumradius equal tofive wavelengths ( R = λ = . I A e i ϕ A = I e i ( − k R − π / ) (27) I B e i ϕ A = I e i ( − k R + π ) , (28)where I is a peak electric current. This corresponds to ˜ A (cid:48) ∝ / √ B (cid:48) ∝ i / √
2, as can be seen by comparing (27) and(28) with (25) and (26).We expect to find a reasonable approximation to an electro-magnetic tangle in a spherical region of radius r max = λ (cid:114) N π = . λ (29)centred on the origin ( r = r max to be divisible into N patches of area equal to a half wavelength squared ( λ / ) ;inside this sphere our array of elements is effectively quasi-continuous.The degree of localisation of the (approximate) electromag-netic tangle can be examined by considering the root-mean-square E rms = E rms ( r ) of the electric field, given by E rms = (cid:115) T (cid:90) t + T t | E | d t , (30)ow to make a monochromatic electromagnetic knot 4where t is an arbitrary initial time. Numerical calculationreveals that our (approximate) electromagnetic tangle is rea-sonably well localised, with E rms taking on a local maximumvalue of 0 . µ ω I / π at the origin ( r = E rms is shown in Fig. 2.The structure of the electric field lines at time t can be ex-amined by integrating the streamline equationd e ( τ ) d τ = E [ e ( τ ) , t ] | E [ e ( τ ) , t ] | , (31)which gives the trajectory e = e ( τ ) of an electric field line as e ( τ ) = e ( ) + (cid:90) τ E [ e ( τ (cid:48) ) , t ] | E [ e ( τ (cid:48) ) , t ] | d τ (cid:48) , (32)where e ( ) is the seed position and it is assumed that theelectric field is non-zero ( E [ e ( τ (cid:48) ) , t ] (cid:54) =
0, 0 ≤ τ (cid:48) ≤ τ ). The(exact) electromagnetic tangle described in and alluded toin section II with ˜ A (cid:48) = / √ B (cid:48) = i / √ z axis and an electric field that vanishes everywhere at time t =( / + q ) T /
2, where q ∈ { , ± , . . . } is an integer. The struc-ture of the electric field lines at other times ( t = ( / + q ) T / x axis. For the sake of brevity and without significant loss ofgenerality we focus, therefore, on the structure of the electricfield lines at t = π q with seed positions on the positive x axis( e ( ) = e x ( ) ˆ x with 0 < e x ( ) < r max ). Numerical calculationreveals a dichotomy: electric field lines seeded close to the z axis quickly exit the spherical region 0 ≤ | r | (cid:46) r max = . λ and go on to trace out chaotic paths, quite distinct from thoseseen in the exact electromagnetic tangle; electric field linesthat remain within the region 0 ≤ r (cid:46) r max are torus knots (in-cluding the trivial ‘knot’), qualitatively similar to those foundin the exact electromagnetic tangle. A train of electric fieldlines are shown in Fig. 3 for increasing integration lengthsand a selection of torus-knotted electric field lines are plottedindividually in Fig. 4.Although we have focussed on the electric field, we notefor completeness that our antenna also generates torus-knottedmagnetic field lines, with an overall structure similar that ofthe torus-knotted electric field lines only shifted in time bya quarter cycle, as is the case for the (exact) electromagnetictangle described in and alluded to in section II.In summary, our antenna can successfully generate an (ap-proximate) electromagnetic tangle, with linked and torus-knotted electric and magnetic field lines, as claimed. V. OUTLOOK
There is much still to be done, not least an experimentaldemonstration of our antenna design.Although our focus in this paper has been on the microwavedomain, we recognise that analogous ideas might be pursuedin other frequency domains. For example: a monchromaticelectromagnetic knot might be generated in the visible do-main by superposing multiple laser beams or perhaps more
FIG. 1. A specific embodiment of our antenna, complete with a 5 × magnified view of the n = E rms of the electric field in the x = y = z = ≤ | r | ≤ R ), configured to generate an (approximate)electromagnetic tangle. Included is a 2 . × magnified view centredon the origin ( r = π E rms / . µ ω I > ow to make a monochromatic electromagnetic knot 5 FIG. 3. Streamline plots of eleven electric field lines at time t = π q in an (approximate) electromagnetic tangle generated by our antenna, theelectric field lines having been seeded at 11 positions equally spaced along the positive x axis in the range ( , λ / ] . Three, increasing valuesof the integration length τ are shown. Segments of electric field lines that lie outside a sphere of radius r max = . λ centred on the origin( r =
0) are coloured red. The streamlines were calculated numerically using the fourth-order Runge-Kutta method with a step size equal to λ / t = π q in an(approximate) electromagnetric tangle generated by our antenna. The seed position e ( ) of each field line has been chosen carefully suchthat the line closes on itself within the integration length of τ = λ . The streamlines were calculated numerically using the fourth-orderRunge-Kutta method with a step size equal to λ / ow to make a monochromatic electromagnetic knot 6 TABLE I. The Cartesian coordinates of the element pairs, where g = ( + √ ) / n √ x n / R √ y n / R √ z n / R n √ x n / R √ y n / R √ z n / R − / g − g − / g − g − − g / g − g − / g − − g − / g − − − g / g − − − / g g
08 1 − − − / g g
09 0 1 / g g − / g − g
010 0 − / g g
20 1 / g − g elegantly by exciting the appropriate modes in an optical cav-ity.Our antenna could be used to locally excited plasma.One motivation for doing so is the possible connectionhypothesised in between unusual electromagnetic distur-bances and the as-yet unexplained natural phenomenon of balllightning ; a modern take, perhaps, on some old ideas . ACKNOWLEDGMENTS
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