the method of fourier transforms applied to
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European Journal of Physics
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The method of Fourier transforms applied to electromagnetic knotsTo cite this article: M Arrayás and J L Trueba 2019 Eur. J. Phys. 40 015205
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The method of Fourier transforms applied toelectromagnetic knots
M Arrayás1 and J L Trueba
Área de Electromagnetismo, Universidad Rey Juan Carlos, Tulipán s/n, 28933Móstoles, Madrid, Spain
E-mail: [email protected] and [email protected]
Received 13 September 2018, revised 29 October 2018Accepted for publication 5 November 2018Published 7 December 2018
AbstractThe Fourier transform method can be applied to obtain electromagnetic knots,which are solutions of Maxwell equations in a vacuum with non-trivialtopology of the field lines and special properties. The program followed in thiswork allows us to present the main ideas and the explicit calculations atundergraduate level, so they are not obscured by a more involved formulation.We make use of the helicity basis for calculating the electromagnetic helicityand the photon content of the fields.
Keywords: Fourier transform, electromagnetic knot, helicity basis
(Some figures may appear in colour only in the online journal)
1. Introduction
The application of topology ideas to electromagnetism has led to the finding of new solutions ofMaxwell equations in a vacuum [1], some of them with the property of non-nullity [2] and withother interesting properties in terms of helicity exchange [3] and spin-orbital decomposition [4].
In this article we want to introduce one of the methods used in the current research ofelectromagnetic knots at undergraduate level. Only a basic knowledge of the Fourier transform isrequired by the student. The content of this article can be proposed to students as a set of advancedproblems exploring special topics such as duality [5] and gauge transforms, the formulation of theCauchy problem in electromagnetism, the helicity basis [6], and photon polarization.
We hope that it will help to clarify and review some concepts and ideas through explicitexamples. For instance, in many introductory courses, electromagnetic fields in a vacuum arepresented in terms of planar waves. The planar electromagnetic waves are null fields [7], and thisleads quite often to the misconception that all the electromagnetic fields in a vacuum must satisfy
European Journal of Physics
Eur. J. Phys. 40 (2019) 015205 (17pp) https://doi.org/10.1088/1361-6404/aaee2d
1 Author to whom any correspondence should be addressed.
0143-0807/19/015205+17$33.00 © 2018 European Physical Society Printed in the UK 1
the property = =∣ ∣ ∣ ∣ ·cE B E B, 0. Here we present explicit calculations of non-null torus fieldsusing only Fourier transforms, so we obtain a solution where electric and magnetic fields are notorthogonal. At the same time the student is calculating a rather non-trivial solution of Maxwellequations in a vacuum, with many of its properties being the subject of undergoing research [8, 9].
We make a brief summary of Maxwell theory in a vacuum, formulate the Cauchyproblem, and introduce the Fourier decomposition of the initial values of the fields. We showhow to solve the time evolution by making the inverse Fourier transform so expressing thefields as wave packages determined by the initial conditions. We then obtain the potentialassociated with the fields in the Coulomb gauge.
The helicity basis is used to express the potential as combinations of circularly polarizedwaves. The helicity basis is useful to introduce the magnetic, electric, and electromagnetic heli-cities. The helicity of a vector field is a useful quantity to characterize certain topological prop-erties of the fields, such as the linkage of the field lines, and the electromagnetic helicity can berelated to the difference in the right- and left-photon content of the field in quantum mechanics.
We will apply this program to a set of knotted electromagnetic fields, including theparticular case of the celebrated Hopt–Rañada solution, called the Hopfion.
2. Maxwell theory in vacuum
Electromagnetism in a vacuum three-dimensional space can be described in terms of two realvector fields, E and B, called the electric and magnetic fields respectively. Using the SI ofunits, these fields satisfy the following Maxwell equations in a vacuum:
= ´ +¶¶
=· ( )t
B EB
0, 0, 1
= ´ -¶¶
=· ( )c t
E BE
0,1
0, 22
where c is the speed of light, t is time, and ∇ denotes differentiation with respect to spacecoordinates = ( )x y zr , , . The first two equations (1) are identities using the following four-vectorelectromagnetic potential in the Minkowski spacetime with coordinates = = =(x ct x x x y, , ,0 1 2
= )x z3 and metric = - - -( )g diag 1, 1, 1, 1 :
=m ⎜ ⎟⎛⎝
⎞⎠ ( )A
V
cA, . 3
In equation (3), A0=V/c, where V is the scalar electric potential, and = ( )A A AA , ,1 2 3 is thevector electromagnetic potential. Greek indices such as μ take values 0, 1, 2, 3 and latinindices such as i take values 1, 2, 3. An electromagnetic field tensor mnF can be defined as
= ¶ - ¶mn m n n m ( )F A A , 4
with components related to the electric and magnetic fields through
=
=- ( )
c F
F
E
B
,1
2. 5
ii
i ijkjk
0
In three-dimensional quantities, equation (4) is
=- -¶¶
= ´ ( )
Vt
EA
B A
,
. 6
Eur. J. Phys. 40 (2019) 015205 M Arrayás and J L Trueba
2
The first pair (1) of Maxwell equations become a pair of identities by using (6). The dynamicsof electromagnetism in a vacuum are then given by the second pair (2) of Maxwell equations,which can be written in spacetime quantities as
¶ =mmn ( )F 0. 7
Maxwell equations (1)–(2) are invariant under the map -( ) ( )c cE B B E, , , as stated in[5]. This means that, in a vacuum, it is possible to define another four-potential,
= ¢m ( ) ( )C c V C, , 8
so that the dual of the electromagnetic tensor mnF , defined as
* e=mn mnabab ( )F F
1
2, 9
satisfies
* = - ¶ - ¶mn m n n m( ) ( )Fc
C C1
, 10
or, in terms of three-dimensional fields,
= ´ = ¢ +¶¶
( )Vc t
E C BC
,1
. 112
In this dual formulation, the pair of equations in (2) are simply identities when definitions (11)are imposed, and the dynamics of electromagnetism in a vacuum are given by the first pair ofMaxwell equations (1), written in terms of components of the four-potential Cμ.
Alternatively, Maxwell equations in a vacuum can be described in terms of two sets ofvector potentials, as in definitions (4) and (10), that satisfy the duality condition (9). In thisway, dynamics are encoded in the duality condition.
3. Fourier decomposition for electromagnetic fields in vacuum
The method of Fourier transform [10] can be used to get solutions of Maxwell equations in avacuum (1)–(2) from initial conditions for the electric and the magnetic fields. Here wereview some basics for the case of electromagnetic fields defined in all the three-dimensionalspaces R3 that satisfy equations (1)–(2).
Suppose that the initial values of the electric and magnetic fields,
= == =
( ) ( )( ) ( ) ( )
tt
E r E rB r B r
, 0 ,, 0 , 12
0
0
are given and defined in R3. It is necessary that they satisfy the conditions
= =
·· ( )
EB
0,0. 13
0
0
These conditions assure that the electric and the magnetic fields will be divergenceless at anytime, since Maxwell equations in a vacuum conserve them. For example, in the case of theelectric field,
¶¶
= ¶¶
= ´ =⎛⎝⎜
⎞⎠⎟( · ) · · ( ) ( )
t tcE
EB 0, 142
where equation (2) has been used. A similar computation can be done for the magnetic field.
Eur. J. Phys. 40 (2019) 015205 M Arrayás and J L Trueba
3
To solve Maxwell equations in a vacuum with Cauchy conditions given by fields (12)that satisfy (13), plane wave solutions can be proposed, such that they can be decomposed inFourier terms,
ò
òp
p
= +
= +
w w
w w
- - - +
- - - +
( )( )
( ( ) ( ) )
( )( )
( ( ) ( ) ) ( )
( · ) ( · )
( · ) ( · )
/
/
t k e e
t k e e
E r E k E k
B r B k B k
,1
2d ,
,1
2d , 15
i t i t
i t i t
k r k r
k r k r
3 23
1 2
3 23
1 2
where w = ck and = ·k k k . Taking t=0 in (15), we get
ò
òp
p
= +
= +
-
-
( )( )
( ( ) ( ))
( )( )
( ( ) ( )) ( )
·
·
/
/
k e
k e
E r E k E k
B r B k B k
1
2d ,
1
2d , 16
i
i
k r
k r
0 3 23
1 2
0 3 23
1 2
so that the vectors
= += +
( ) ( ) ( )( ) ( ) ( ) ( )
E k E k E kB k B k B k
,, 17
0 1 2
0 1 2
are inverse Fourier transforms of the initial values (12),
ò
òp
p
=
=
( )( )
( )
( )( )
( ) ( )
·
·
/
/
r e
r e
E k E r
B k B r
1
2d ,
1
2d , 18
i
i
k r
k r
0 3 23
0
0 3 23
0
and they satisfy
==
· ( )· ( ) ( )
k E kk B k
0,0, 19
0
0
because of (13).Now, in expressions (15), we impose the pair of Maxwell equations ´ =B
¶ ¶( )c tE1 2 and ´ = -¶ ¶tE B . Taking then t= 0 and using (17), we get the con-ditions
´ = -
´ =- -
( ) ( ( ) ( ))
( ) ( ( ) ( )) ( )
ckk
c
k E k B k B k
k B k E k E k
,
. 20
0 1 2
0 1 2
From (20), we obtain a solution for the vectors in k-space, namely
= - ´
= + ´
= + ´
= - ´
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
c
c
c
c
E k E k e B k
E k E k e B k
B k B k e E k
B k B k e E k
1
2 2,
1
2 2,
1
2
1
2,
1
2
1
2, 21
k
k
k
k
1 0 0
2 0 0
1 0 0
2 0 0
Eur. J. Phys. 40 (2019) 015205 M Arrayás and J L Trueba
4
where we have defined the unit vector
= ( )k
ek
. 22k
Consequently, the time-dependent fields are obtained computing first the Fourier transforms(18) of the initial values (12), and then performing the Fourier integrals
ò
òp
w w
pw w
= - ´
= + ´
-
- ⎜ ⎟⎛⎝
⎞⎠
( )( )
( ( ) ( ) )
( )( )
( ) ( ) ( )
·
·
/
/
t k e t ic t
t k e ti
ct
E r E k e B k
B r B k e E k
,1
2d cos sin ,
,1
2d cos sin . 23
ik
ik
k r
k r
3 23
0 0
3 23
0 0
By examining equation (17) one can see that the complex Fourier basis components satisfy
- =- =
( ) ¯ ( )( ) ¯ ( ) ( )
E k E kB k B k
,
, 240 0
0 0
where E0 is the complex conjugate of E0.The Fourier transforms can be written in other ways. We begin by takingw = +w w-( )t e ecos 2i t i t , w = -w w-( )t e e isin 2i t i t , in equation (23), to get
ò
ò
p
p
= - ´
+ + ´
= + ´
+ - ´
w
w
w
w
-
- -
-
- -
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎡⎣⎢
⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
⎡⎣⎢
⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
( )( )
( ) ( )
( ) ( )
( )( )
( ) ( )
( ) ( ) ( )
·
·
·
·
t d k e ec
e ec
t d k e ec
e ec
E r E k e B k
E k e B k
B r B k e E k
B k e E k
,1
2
1
2 2
1
2 2,
,1
2
1
2
1
2
1
2
1
2. 25
i i tk
i i tk
i i tk
i i tk
k r
k r
k r
k r
3 23
0 0
0 0
3 23
0 0
0 0
Next we perform the change -k k in the second terms of both fields using that
ò ò
òp p
p
= = - -
= -
-¥
¥-
¥
-¥
-¥
¥
( )( )
( )( )
( ) ( )
( )( ) ( )
· ·
·
/ /
/
f k F e k F e
k F e
r k k
k
1
2d
1
2d
1
2d . 26
i i
i
k r k r
k r
3 23
3 23
3 23
Taking into account the properties (24), we get
ò
ò
p
p
= - ´
+ - ´
= + ´
+ + ´
-
-
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎡⎣⎢
⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
⎡⎣⎢
⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
( )( )
( ) ( )
( ) ( )
( )( )
( ) ( )
( ) ( ) ( )
/
/
t k ec
ec
t k ec
ec
E r E k e B k
E k e B k
B r B k e E k
B k e E k
,1
2d
1
2 2
1
2 2,
,1
2d
1
2
1
2
1
2
1
2, 27
ikxk
ikxk
ikxk
ikxk
3 23
0 0
0 0
3 23
0 0
0 0
where we have introduced the four-dimensional notation w= - ·kx t k r.
Eur. J. Phys. 40 (2019) 015205 M Arrayás and J L Trueba
5
4. Fourier transforms of vector potentials in the Coulomb gauge
As stated in equations (6) and (11), any electromagnetic field in a vacuum can be written interms of two sets of four-vector electromagnetic potentials =m ( )A V c A, and = ¢m ( )C cV C,so that
=- -¶¶
= ´
= ´ = ¢ +¶¶
( )
Vt
Vc t
EA
C
B AC
,
1. 28
2
There are some circumstances, as computations of electromagnetic helicity or spin, in whichthe Fourier transforms of these four-vector potentials can be useful. Here we give a simplerecipe to get those Fourier transforms in a similar form to equation (27) in a special case:when the four-vector potentials Aμ and Cμ are taken to satisfy the Coulomb condition.
It is known that Aμ and Cμ are not totally defined by equation (28). This property iscalled gauge freedom and it means that, if we have solutions mA1 and mC1 that satisfyequations (28), then any other potentials mA2 and mC2 given by
= +¶¶
= -
¢ = ¢ +¶¶
= -
( ) ( )
( ) ( ) ( )
V Vtf t f t
V Vc t
g t g t
r A A r
r C C r
, , , ,
1, , , , 29
2 1 2 1
2 1 2 2 1
in which ( )f tr, and ( )g tr, are functions of space and time, are also solutions ofequation (28). Fixing the gauge is choosing a particular form for the vector potentials. A well-known possibility is the Coulomb gauge, in which the vector potentials are chosen so that
= =¢ = =
·· ( )
VV
AC
0, 0,0, 0. 30
As a consequence, they satisfy the duality equations
= ´ =¶¶
= ´ = -¶¶
( )
c t
t
B AC
E CA
1,
. 31
2
In this gauge, since equations (30) and (31) are similar to the Maxwell equations for theelectric and magnetic fields in a vacuum, and by comparison with (27), one can propose forthe potentials the following Fourier decomposition in terms of plane waves:
ò
òp
p
= +
= +
-
-
( )( )
[ ( ) ( )]
( )( )
[ ( ) ( )] ( )
/
/
t k e e
tc
k e e
A r a k a k
C r c k c k
,1
2d ,
,2
d , 32
ikx ikx
ikx ikx
3 23
3 23
where the factor c in C is taken for dimensional reasons. Taking time derivatives and using(31),
Eur. J. Phys. 40 (2019) 015205 M Arrayás and J L Trueba
6
ò
òp
p
=-¶¶
= -
=¶¶
= - +
-
-
( )[ ( ) ( ) ( ) ( )]
( )[ ( ) ( ) ( ) ( )] ( )
/
/
tk e ick e ick
c tk e ik e ik
EA
a k a k
BC
c k c k
1
2d ,
1 1
2d . 33
ikx ikx
ikx ikx
3 23
2 3 23
By comparison with equation (27), one gets
= - ´
=-
+ ´
( ) ( ( ) ( ))
( ) ( ( ) ( )) ( )
i
ckc
i
ckc
a k E k e B k
c k B k e E k
2,
2. 34
k
k
0 0
0 0
Consequently, in the Coulomb gauge, the vector potentials of the electromagnetic field in avacuum can be decomposed in the Fourier basis as
ò
ò
p
p
= - + ´
+ - ´
= + ´
+ - - ´
-
-
( )( )
[ ( ( ) ( ))
( ( ) ( ))]
( )( )
[ ( ( ) ( ))
( ( ) ( ))] ( )
/
/
ti
c
k
ke c
e c
ti k
ke c
e c
A r E k e B k
E k e B k
C r B k e E k
B k e E k
,2
d
2
,
,2
d
2
. 35
ikxk
ikxk
ikxk
ikxk
3 2
3
0 0
0 0
3 2
3
0 0
0 0
5. Helicity basis
The helicity Fourier components appear when the vector potentials A and C, in the Coulombgauge, are written as a combination of circularly polarized waves [6], as
ò
ò
m
pm
p
= + +
= - +
-
-
( )( )
[ ( ( ) ( ) ( ) ( )) ]
( )( )
[ ( ( ) ( ) ( ) ( )) ] ( )
/
/
tc k
ke a a C C
tc c k
ki e a a C C
A r k e k k e k
C r k e k k e k
,2
d
2. . ,
,2
d
2. . , 36
ikxR R L L
ikxR R L L
03 2
3
03 2
3
where ÿ is the Planck constant, μ0 is the vacuum magnetic permeability and C C. . meanscomplex conjugate. The Fourier components in the helicity basis are given by the complexunit vectors ( )e kR , ( )e kL , = ke kk , and the helicity components ( )a kR , ( )a kL that, in thequantum theory, are interpreted as annihilation operators of photon states with right- and left-handed polarization, respectively ( ¯ ( )a kR , ¯ ( )a kL are, in quantum theory, creation operators ofsuch states) [6].
The unit vectors in the helicity basis are taken to satisfy the relations
= - = - - = -= = = = =
´ = ´ = ´ =´ =- ´ = ´ = -
¯ ( ) ( ) ( ) ( )· · · · ·
( )i i i
e e e k e k e k e ke e e e e e e e e e
e e e e e ee e e e e e e e e
, , ,0, 0, 1,
0,, , . 37
R L R L L R
k R k L R R L L R L
k k R R L L
k R R k L L R L k
Eur. J. Phys. 40 (2019) 015205 M Arrayás and J L Trueba
7
These relations can be achieved by taking real vectors e1, e2 in k-space such that
= = = = =
´ = ´ = ´ =
- =- - =
· · · · ·
( ) ( ) ( ) ( ) ( )
e e e e e e e e e e
e e e e e e e e e
e k e k e k e k
1
2, 0,
1
2, , ,
, , 38
k k
k k k
1 1 2 2 1 2 1 2
1 2 1 2 2 1
1 1 2 2
and constructing the complex unit vectors
= += - ( )
ii
e e ee e e
,. 39
R
L
1 2
1 2
The relation between the helicity basis and the planar Fourier basis can be understood bycomparing equations (35) and (36). It is found that
m
m
+ = - + ´
- = + ´
( ¯ ( ) ¯ ( ))
( ¯ ( ) ¯ ( )) ( )
a ac c k
i ic
a ac c k
c B
e e E k e B k
e e k e E k
1
2,
1
2. 40
R R L L k
R R L L k
00 0
00 0
From these expressions,
m
m
= - + ´ -
=-
+ - ´ +
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎡⎣⎢⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
⎡⎣⎢⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
¯ ( ) ¯ ( ) ¯ ( ) ¯ ( )
¯ ( ) ¯ ( ) ¯ ( ) ¯ ( ) ( )
ac k
i
ci
i
c
ac k
i
ci
i
c
e B k E k e B k E k
e B k E k e B k E k
1
2 2,
1
2 2. 41
R R k
L L k
00 0 0 0
00 0 0 0
Consequently, in the helicity basis, fields in a vacuum are written as a superposition ofcircularly polarized waves. The electric and magnetic fields of an electromagnetic field in avacuum, and the vector potentials in the Coulomb gauge, can be expressed in this basis as (wedo not write the dependence on k in the following expressions for clarity):
ò
ò
ò
ò
m
pm
pm
pm
p
= + - +
= - + -
= + + +
= - - -
-
-
-
-
( )( )
[ ( ) ( )]
( )( )
[ ( ) ( )]
( )( )
[ ( ) ( )]
( )( )
[ ( ) ( )] ( )
/
/
/
/
tc c
kk
i e a a i e a a
tc
kk
e a a e a a
tc
kk
e a a e a a
tc c
kk
i e a a i e a a
E r e e e e
B r e e e e
A r e e e e
C r e e e e
,2
d2
,
,2
d2
,
,2
d1
2,
,2
d1
2. 42
ikxR R L L
ikxR L L R
ikxR R L L
ikxR L L R
ikxR R L L
ikxR L L R
ikxR R L L
ikxR L L R
03 2
3
03 2
3
03 2
3
03 2
3
The unit vectors in this basis satisfy equation (37) and the basis elements are related to theplanar Fourier basis through equations (40) or (41).
The helicity basis can be used in relation to the helicity of a vector field [11–16], which isa useful quantity in the study of topological configurations of electric and magnetic lines. Inthe case of electromagnetism in a vacuum, the magnetic helicity can be defined as the integral
Eur. J. Phys. 40 (2019) 015205 M Arrayás and J L Trueba
8
òm= · ( )h
cr A B
1
2d , 43m
0
3
and the electric helicity [17–20] is
ò òe
m= =· · ( )h
cr
crC E C E
2d
1
2d , 44e
0 33
0
3
where e m= ( )c102
0 is the vacuum electric permittivity. It can be proved [3, 4] that (i) if theintegral of ·E B is zero, both the magnetic and the electric helicities are constant during theevolution of the electromagnetic field, (ii) if the integral of ·E B is not zero, the helicities arenot constant but they satisfy = -/ /h t h td d d dm e , so we get an interchange of helicitiesbetween the magnetic and electric parts of the field, and (iii) for every value of the integral of
·E B, the electromagnetic helicity = +h h hm e is a conserved quantity.Using the helicity basis defined in (36), the electromagnetic helicity h of an electro-
magnetic field in a vacuum can be written as
ò= + = -( ( ) ( ) ( ) ( )) ( )h h h k a a a ak k k kd . 45m e R R L L3
From the usual expressions for the number of right- and left-handed photons in quantummechanics,
òò
=
=
( ) ( )
( ) ( ) ( )
N k a a
N k a a
k k
k k
d ,
d , 46
R R R
L L L
3
3
we can write
= -( ) ( )h N N . 47R L
Consequently, the electromagnetic helicity is the classical limit of the difference between thenumbers of right-handed and left-handed photons [17, 18, 21].
6. Fourier method in the case of a set of knotted electromagnetic fields
There is a set of so-called non-null torus electromagnetic knots [1, 2] that are exact solutionsof Maxwell equations in a vacuum with the property that, at a given initial time t=0, allpairs of lines of the field =( ) ( )B r B r, 00 are linked torus knots, and that the linking numberis the same for all the pairs. Similarly, all pairs of lines of the field =( ) ( )E r E r, 00 are linkedtorus knots and the linking number of all pairs of lines is the same.
The initial values ( )B r0 and ( )E r0 for the magnetic and the electric fields of these non-null torus electromagnetic knots can be obtained [2] from two complex scalar fields f ( )tr,and q ( )tr, through the equations
pf f
ff=
´ +
( )¯
( ¯ )( )a
iB r
2 1, 480 2
pq q
qq=
´ +
( )¯
( ¯)( )a c
iE r
2 1, 490 2
where a is some constant proportional to the product mc 0 to give the electromagnetic field itscorrect dimensions in SI units. For historical purposes, we note that a similar construction waspresented by Bateman in 1915 [22]. Written in this way, the magnetic field lines are given bythe level curves of the scalar field f ( )r and the electric field lines are given by the level curves
Eur. J. Phys. 40 (2019) 015205 M Arrayás and J L Trueba
9
of the scalar field q ( )r . If we take
f =+
+ -( ) ( )
( ( ))( )
( )
( )X iY
Z i Rr
2 2
2 1, 50
n
m2
where n m, are positive integers and we have used the notation
=-∣ ∣
( )( )ff
f, 51n
n
n 1
in which f is a complex function of space coordinates. In (50), we use dimensionlesscoordinates (X, Y, Z), related to the physical ones (x, y, z) in the SI of units by
=( ) ( ) ( )X Y Zx y z
L, ,
, ,, 52
0
where L0 is a constant with dimensions of length, and
= + + =+ +
= ( )R X Y Zx y z
L
r
L. 532 2 2 2
2 2 2
02
2
02
All the magnetic lines of the field defined by (48) are (n, m) torus knots. This initial magneticfield is
p=
+- - -
+ - -⎛⎝⎜
⎞⎠⎟( )
( )( )a
L RmY nXZ mX nYZ n
X Y ZB r
8
1, ,
1
2. 540
02 2 3
2 2 2
Moreover, the linking number of every pair of magnetic lines at t=0 is equal to nm.Similarly, for the initial electric field we use the complex scalar field
q =+
+ -( ) ( )
( ( ))( )
( )
( )Y iZ
X i Rr
2 2
2 1, 55
l
s2
where l, s are positive integers. Using (49), we get
p=
+- - +
- +⎛⎝⎜
⎞⎠⎟( )
( )( )c a
L RlX Y Z
lXY sZ lXZ sYE r8
1
1
2, , . 560
02 2 3
2 2 2
This initial electric field is such that all its electric lines are (l, s) torus knots and any pair ofelectric lines has a linking number equal to l s. These topological properties only happen fort=0. This is due to the fact that the topology changes during time evolution if one of theintegers (n, m, l, s) is different from any of the others (see [2]).
In figure 1 we can see the case (n, m, s, l)=(3, 2, 3, 2), which is the trefoil case. Thefield lines are trefoil knots and they are linked at the initial time. We have plotted a few fieldelectric and magnetics lines calculated from (54) and (56) starting from the same points in themiddle of the figure. Clearly they are not orthogonal.
Let us compute the magnetic and electric fields from the Cauchy conditions (54) and(56) that satisfy the required conditions = =· ·B E 00 0 . We will use the dimen-sionless time =T ct L0. Before the computations, we can use (23) to build the complexvector field
Eur. J. Phys. 40 (2019) 015205 M Arrayás and J L Trueba
10
òpw
w
+ = +
+ ´ +
- ⎜ ⎟
⎜ ⎟
⎡⎣⎢⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
( ) ( )( )
( ) ( )
( ) ( ) ( )
·/
t it
ck e i
ct
ic
t
B rE r
B kE k
e B kE k
,, 1
2d cos
sin , 57
i
k
k r3 2
30
0
00
to show that we need first to compute the complex vector field +( ) ( )i cB k E k0 0 . Using(18), we get
òp+ = +⎜ ⎟⎛
⎝⎞⎠( ) ( )
( )( ) ( ) ( )·
/i
cr e i
cB k
E kB r
E r1
2d , 58ik r
00
3 23
00
where ( )B r0 and ( )E r0 are given by (54) and (56) respectively. We are going to usedimensionless coordinates in k-space, defined by
w= = =( ) ( ) ( )K K K L k k k K L k
L
c, , , , , . 59x y z x y z0 0
0
The integrals in (58) can be done to give
p+ = - -
+ -
+ + - -
+ -
- ⎡⎣⎢( ) ( ) ( )
( )
( )
( )] ( )
ic
L a e n
KK K K K K K
i m K K
il
KK K K K K K
s K K
B kE k
2, ,
, , 0
, ,
0, , , 60
K
x z y z x y
y x
y z x y x z
z y
00 0 2 2
2 2
so the Fourier transform of the initial magnetic field ( )B r0 is
p= - -
+ -
- ⎡⎣⎢( ) ( )
( )] ( )
L a e n
KK K K K K K
i m K K
B k2
, ,
, , 0 , 61
K
x z y z x y
y x
00 2 2
and the Fourier transform of the initial magnetic field ( )E r0 is
p= + - -
+ -
- ⎡⎣⎢( ) ( )
( )] ( )
L c a e l
KK K K K K K
i s K K
E k2
, ,
0, , . 62
K
y z x y x z
z y
00 2 2
Figure 1. Electric and magnetic lines given for (n, m, s, l)=(3, 2, 3, 2), the trefoil atinitial time starting from same points. The field lines are linked with each other. Theelectric and magnetic fields are not orthogonal.
Eur. J. Phys. 40 (2019) 015205 M Arrayás and J L Trueba
11
Note that the Fourier transforms (61) and (62) satisfy the conditions - =( ) ¯ ( )B k B k0 0 and- =( ) ¯ ( )E k E k0 0 , as was established in (24). According to (57), we also need to compute,
using (60), the complex vector
p´ + = -
+ - -
+ -
+ - -
-⎜ ⎟⎛⎝
⎞⎠
⎤⎦⎥
( ) ( ) [ ( )
( )
( )
( ) ( )
ic
L a en K K
im
KK K K K K K
i l K K
s
KK K K K K K
e B kE k
2, , 0
, ,
0, ,
, , . 63
k
K
y x
x z y z x y
z y
y z x y x z
00 0
2 2
2 2
The next goal is to compute the integral (57) that, in terms of dimensionless coordinates in x-space and k-space, can be written as
òp+ = +
+ ´ +
- ⎜ ⎟
⎜ ⎟
⎡⎣⎢⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
( ) ( )( )
( ) ( )
( ) ( ) ( )
·/
t it
c LK e i
cKT
Ki
cKT
B rE r
B KE K
KB K
E K
,, 1
2d cos
sin . 64
iK R3 2
03
30
0
00
It is interesting to compute first the following integrals,
ò
ò
ò
ò
p p
p p
p p
p p
= =+
= =- +
+
= =+
= =+ -
+
--
-
--
-
( )
( )( )
·
·
·
·
Ia
LK e KT
e
K
a
L
A
A T
Ia
LK e KT
a
L
A T AT
A T
Ia
LK e KT
e
K
a
L
T
A T
Ia
LK e KT
a
L
T AT A T
A T
4d cos
2,
4d cos
2
2,
4d sin
2,
4d sin
2
2, 65
ci
K
ci
si
K
si
K R
K R
K R
K R
1 202
3
02 2 2
2 202
3
02
2 2 2
2 2 2
1 202
3
02 2 2
2 202
3
02
3 2
2 2 2
in which
=- + ( )A
R T 1
2. 66
2 2
Now we make one useful observation. If, in the integral (64), we need to compute a termcontaining Kx times one of the integrands of the expressions (65), we can do it in this way,
òp=
¶¶
=--
( )·a
LK e KT
e
KK i
I
XiX
I
A4d cos
d
d, 67i
K
xc cK R
202
3 1 1
and the same happens with I2c, I1s, and I2s. If we need to compute a term containing Kx2, we
can make
òp= - --
-( )·a
LK e KT
e
KK
I
AX
I
A4d cos
d
d
d
d, 68i
K
xc cK R
202
3 2 1 22
12
Eur. J. Phys. 40 (2019) 015205 M Arrayás and J L Trueba
12
and, if we need to compute a term containing Kx Ky,
òp= --
-( )·a
LK e KT
e
KK K XY
I
A4d cos
d
d. 69i
K
x ycK R
202
32
12
Defining the quantities
= -= -
( )( ) ( )
P T T A
Q A A T
3 ,
3 , 70
2 2
2 2
we can finally compute (64), obtaining that the set of non-null torus electromagnetic knots canbe written as
p=
++
( )( )
( )ta
L
Q P
A TB r
H H, 71
02
1 22 2 3
p=
-+
( )( )
( )ta c
L
Q P
A TE r
H H, 72
02
4 32 2 3
where
= - + + - - -- - + + +
+
=+ - - -
- - + + +
= - + + - - -- - + + +
+
=+ - - -
- - + + +
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
( )
nXZ mY sT nYZ mX lTZ nZ X Y T
lTY
sX Y Z T
mTY sXY lZ mTX sXZ lY nT
mXZ nY lT mYZ nX sTZ mZ X Y T
sTY
lX Y Z T
nTY lXY sZ nTX lXZ sY mT
H
H
H
H
, ,1
2.
1
2, , .
, ,1
2.
1
2, , .
73
1
2 2 2 2
2
2 2 2 2
3
2 2 2 2
4
2 2 2 2
7. Helicity basis for the set of electromagnetic knots
Let us consider the helicity basis for the electromagnetic fields we have computed, since it isinteresting to understand certain physical quantities of the electromagnetic field such asenergy, helicity, and linear and angular momentum. Using expressions (61) and (62), we get
p
p
= - -
- -
= + - -
- -
-
-
⎡⎣⎢
⎡⎣⎢
¯ ( ) ( )
( )]
¯ ( ) ( )
( )] ( )
L a e n
KK K K K K K
i m K K
L c a e l
KK K K K K K
i s K K
B k
E k
2, ,
, , 0 ,
2, ,
0, , , 74
K
x z y z x y
y x
K
y z x y x z
z y
00 2 2
00 2 2
Eur. J. Phys. 40 (2019) 015205 M Arrayás and J L Trueba
13
so that
p- = - -
+ -- -
- + - -
- ⎡⎣⎢
⎤⎦⎥
¯ ( ) ¯ ( ) ( )
( )( )
( ) ( )
i
c
L a e n
KK K K K K K
s K K
i m K K
il
KK K K K K K
B k E k2
, ,
0, ,
, , 0 ,
, , , 75
K
x z y z x y
z y
y x
y z x y x z
0 00 2 2
2 2
and
p´ - = - -
- + - -
- - -
- -
-⎜ ⎟⎛⎝
⎞⎠
⎤⎦⎥
¯ ( ) ¯ ( ) [ ( )
( )
( )
( )] ( )
i
c
L a en K K
s
KK K K K K K
im
KK K K K K K
i l K K
e B k E k2
, , 0
, , ,
, ,
0, , . 76
k
K
y x
y z x y x z
x z y z x y
z y
0 00
2 2
2 2
According to (41),
m p=
´+
- - + + -
-+
+ - - + + -
-
⎡⎣⎢
⎤⎦⎥
⎡⎣⎢
⎤⎦⎥
( ) ( )( )
( ) ( )( ) ( )
aa
c
L e
K
n m
KK K K K K K l s K K
il s
KK K K K K K n m K K
e4
, , 0, ,
, , , , 0 . 77
R R
K
x z y z x y z y
y z x y x z y x
0
03 2
2 2
2 2
Similarly,
p+ = - -
- -- -
+ + - -
- ⎡⎣⎢
⎤⎦⎥
¯ ( ) ¯ ( ) ( )
( )( )
( ) ( )
i
c
L a e n
KK K K K K K
s K K
i m K K
il
KK K K K K K
B k E k2
, ,
0, ,
, , 0 ,
, , , 78
K
x z y z x y
z y
y x
y z x y x z
0 00 2 2
2 2
and
p´ + = - -
+ + - -
- - -
+ -
-⎜ ⎟⎛⎝
⎞⎠
⎤⎦⎥
¯ ( ) ¯ ( ) [ ( )
( )
( )
( )] ( )
i
c
L a en K K
s
KK K K K K K
im
KK K K K K K
i l K K
e B k E k2
, , 0
, , ,
, ,
0, , , 79
k
K
y x
y z x y x z
x z y z x y
z y
0 00
2 2
2 2
Eur. J. Phys. 40 (2019) 015205 M Arrayás and J L Trueba
14
from which
m p=
´-
- - + - -
--
+ - - + - -
-
⎡⎣⎢
⎤⎦⎥
⎡⎣⎢
⎤⎦⎥
( ) ( )( )
( ) ( )( ) ( )
aa
c
L e
K
m n
KK K K K K K s l K K
il s
KK K K K K K n m K K
e4
, , 0, ,
, , , , 0 . 80
L L
K
x z y z x y z y
y z x y x z y x
0
03 2
2 2
2 2
One interesting application of these results appears by multiplying a eR R and its complexconjugate, and the same for a eL L. Using (37),
= == =
( ¯ ¯ ) · ( ) ¯ ( · ) ¯( ¯ ¯ ) · ( ) ¯ ( · ) ¯ ( )a a a a a aa a a a a a
e e e ee e e e
,. 81
R R R R R R L R R R
L L L L L L R L L L
In our case,
m p
m p
= + + + + +
+ + +
= - + + - +
+ - -
-
-
¯ (( ) ( ) ( ) ( )
( )( ) )
¯ (( ) ( ) ( ) ( )
( )( ) ) ( )
a aa
c
L e
Kn m K K l s K K
n m l s KK
a aa
c
L e
Kn m K K l s K K
n m l s KK
8
4 ,
8
4 . 82
R R
K
x y y z
y
L L
K
x y y z
y
0
03 2
2 2 2 2 2 2
0
03 2
2 2 2 2 2 2
From (46), the classical expressions corresponding to the quantum mechanical number ofright- and left-handed photons for our set of electromagnetic knots are
ò ò
ò ò
m
m
= = =+ + +
= = =- + -
( ) ( )
( ) ( ) ( )
N k a aL
K a aa
c
n m l s
N k a aL
K a aa
c
n m l s
d1
d8
,
d1
d8
, 83
R R R R R
L L L L L
3
03
3
0
2 2
3
03
3
0
2 2
where we have to remember that the constant a is proportional to the product mc 0 because ofdimensional reasons. In the particular case of the Hopfion, for which n=m=l=s=1, wehave
m=
=
( )
( ) ( )
Na
c
N
Hopfion ,
Hopfion 0. 84
R
L
0
If, moreover, we set the arbitrary constant a to have the value m=a c 0, for the set ofelectromagnetic knots we are studying we get
ò ò
ò ò
= = =+ + +
= = =- + -
( ) ( )
( ) ( ) ( )
N k a aL
K a an m l s
N k a aL
K a an m l s
d1
d8
,
d1
d8
, 85
R R R R R
L L L L L
3
03
32 2
3
03
32 2
Eur. J. Phys. 40 (2019) 015205 M Arrayás and J L Trueba
15
and for the Hopfion,
==
( )( ) ( )
NN
Hopfion 1,Hopfion 0, 86
R
L
so the classical value corresponding to the number of right-handed photons in the Hopfion is1, and the classical value corresponding to the number of left-handed photons is 0.
8. Conclusions
Solutions to Maxwell equations in a vacuum can be obtained by taking an initial electro-magnetic field configuration, and by Fourier transform, propagating it in time. This con-struction is used in the generation of non-null toroidal electromagnetic fields [2]. Here wehave presented a detailed calculation suitable for students with a basic mathematicalbackground.
The helicity is a property which characterizes certain non-trivial topological behavior ofthe field and can be related to its photon content when the field is quantized. The Fourierdecomposition can be expressed in terms of the helicity or circularly polarized basis which isconvenient in order to calculate the helicity.
We have applied the Fourier method to compute a set of solutions such that at a particulartime the field lines are torus knots. The set includes the Hopfion (which is a null field) and ingeneral many non-null fields.
We believe that the calculations involved are at undergraduate level, so any student witha minimum background can rework and check them. Maxwell equations remain a funda-mental cornerstone of our understanding of nature, but one might think that the classicalelectromagnetic theory of light has obtained its limits of serviceability in terms of funda-mental research. We hope that this work proves the contrary. Solutions with new topologicalproperties may pave the road for future discoveries.
Acknowledgments
This work is supported by the research grant from the Spanish Ministry of Economy andCompetitiveness ESP2017-86263-C4-3-R.
ORCID iDs
M Arrayás https://orcid.org/0000-0003-2225-6966
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