Polarization of electromagnetic pulses

We show the impossibility, for localized and exact solutions of the Maxwell equations, of perfect circular polarization in a fixed plane, or perfect linear polarization along a fixed direction. A measure of polarization of electromagnetic pulses is obtained by analogy with that useful in monochromatic radiation, and its limitations discussed. Using oscillatory solutions of the free-space Maxwell equations, for which all components of the electric and magnetic fields satisfy the wave equation, we construct explicit examples of TE pulses which are linearly polarized with an azimuthal electric field, TE+iTM pulses approximately linearly polarized along the propagation direction, and also approximately circularly polarized pulses. The latter have perfect circular polarization on the propagation axis.


Introduction
We remind the reader of the textbook linear and circular polarization of monochromatic plane waves, of angular frequency kc.w = In Cartesian coordinates [ ] x y z , , ,these have the complex electric and magnetic fields e E e 1, 0, 0 , 0, 1, 0 linear 1.1 We use the subscript c on the fields to denote complex.The real and imaginary parts of the electric field (assuming E 0 is real) are, for the circular polarization, In the linear case, both E c and B c oscillate in one direction (along the x and y axes, respectively).In the circular polarization case each of E c and B c have perpendicular components in phase quadrature, so the endpoint of either moves on a circle.These properties, which are special cases of elliptic polarization (the endpoint of the oscillating electric vector moves around an ellipse, once per cycle), hold everywhere in space.
The above is for plane waves, not localized transversely.Beams with finite transverse extent, but fixed wavenumber (monochromatic) also have well-defined but local polarization [1][2][3].Unlike the plane wave examples given above, the polarization now varies in space: different ellipses, in ellipticity and orientation, at different points.For monochromatic beams, there are curves C E in space of circular polarization of the electric field, and surfaces S E of linear polarization, and likewise for the magnetic field.The curves C E and surfaces S E can intersect only where E c is zero.The curves of circular polarization and surfaces of linear polarization are fixed in space [4][5][6][7][8][9].
The question arises Does polarization still make sense for pulses, which are of necessity non-monochromatic?For transverse electric (TE) pulses without azimuthal dependence, the answer is yes; such pulses have one electric field component (the azimuthal one), so as the pulse passes a given point r its electric field will oscillate (no longer sinusoidally) in one direction: it is linearly polarized.However, the direction of the electric field vector is not fixed in space, as it was in the textbook example (1.1).The lines of E of TE pulses without azimuthal dependence are circles, transverse to the net direction of propagation.Likewise for the magnetic field in transverse magnetic (TM) pulses.
In general, the polarization of a pulse makes sense only when the pulse is oscillatory enough to have a dominant wavenumber K or angular frequency cK, so that most of the pulse (at a given distance from its axis of propagation, here the z axis) has a fixed degree of polarization.For monochromatic beams the electric field at a point r may be written as a superposition of two fields in phase quadrature, ( ) The definition of polarization to be given in section 2 assumes that there is some semblance of this behaviour in the pulse.The oscillatory pulse G K to be defined in section 3 has a length a and a wavenumber K as parameters.Its focal region, centred on the space-time origin, is of size a.Near the space-time origin (on the scale of a) the dominant part is a plane wave, ( )  e , iK z ct and the fields derived from G K are of the form given above, with cK.w = We shall give examples of linearly polarized pulses in sections 3 and 4, and of circularly polarized pulses in section 5.But first we quantify polarization.

Degree of linear polarization
There are various measures of the degree of polarization of monochromatic waves [1][2][3], linear or circular at the extremes, elliptic in general, one of which will be generalized here for use with electromagnetic pulses.This is the degree of linear polarization [10,11], the generalization of which for the complex time-dependent electric field , , L is unity for linear polarization (real and imaginary parts of the complex field colinear, as for example in (1.1)).
L is zero for circular polarization, in which the real and imaginary parts of the field are perpendicular and equal in magnitude, as in (1.2) to (1.4): In between 0 L = and 1 L = lie the various degrees of elliptic polarization, with ratio of minor axis E 2 to major axis E 1 given by ( ) The plane of the polarization ellipse is that of E E , .
r i It is assumed that the pulse is oscillatory, with a dominant wavenumber, as discussed at the end of the Introduction.
Non-existence theorems (i) Localized pulses cannot be exactly linearly polarized in a fixed direction.For suppose that the complex electric field is x ¶ = and thus F is independent of x, and cannot be localized in the x direction.
(ii) Localized pulses cannot be exactly circularly polarized in a fixed plane.For suppose that the complex electric field is where F is now real.In free space x y ¶ + ¶ = or and thus F is independent of x and of y, and cannot be localized transversely to the z direction.
Note that 'in free space' features in the proofs: the theorems do not apply to pulse propagation in waveguides, for example.
The measure of linear polarization defined in (2.1) is useful only when it is exactly (as in the TE and TM cases mentioned above, and discussed in the next section), or approximately independent of time, while the pulse passes the observation point.The theorems above suggest that fields which can be locally approximated by plane waves of the form ( ) f z ct may have well-defined polarization.

TE and TM pulses
Free space electromagnetic fields which satisfy the Maxwell equations may be constructed from a vector potential ( ) A r t , and a scalar potential ( ) r V t , which satisfy the Lorenz condition • A V 0 ct  + ¶ = and the wave equation.We shall use forward-propagating solutions of the form, in cylindrical polar coordinates e dk e dq w k q e J q k , , , , , satisfies the Lorenz condition, and gives us the transverse electric pulse with The wavefunction y is complex, and so is the vector potential.Using the fact that y satisfies the wave equation, the complex fields simplify to We have assumed azimuthal variation e , imf as in equation (3.1).In the m 0 = case, the electric field has one non-zero component, E , f and is perfectly linearly polarized, everywhere.The electric field lines are circles; the direction of E is not the same everywhere, as it was in the textbook plane wave polarization of equation (1.1).Thus theorem (i) is not violated.
Likewise, TM pulses, obtained from the above by the duality transformation E B B E , ,  -have perfect linear polarization of the magnetic field if based on m 0 = wavefunctions.Reference [13] gives an example of TE and TM pulses, based on the one-parameter wavefunction Oscillatory pulses may be formed from a wavefunction parametrized by a length a and a wavenumber K: (G and G K are normalized to unity at the space-time origin).In the limit Ka 0  we regain the sub-cycle pulse G.The pulse G K will be approximately monochromatic, and characterized by dominant wavenumber K, when  Ka 1. Regarding time dependence of the G K pulse: at the spatial origin this is ( ) The pulse amplitude is maximum at time zero, with a Lorentzian decay on either side.The period T of the damped oscillation at the origin is given by KcT 2 arctan 2 .
1 the period tends to Kc 2 , / p corresponding to angular frequency Kc.
TE and TM pulses based on G and G K have been discussed in [13]; these are linearly polarized, as shown above.The self-dual TE+iTM pulses are the subject of the next section.

Self-dual TE+iTM pulses
We shall specialize to the m 0 = axially symmetric solutions of the wave equation, ( ) z t , , .y y r = Then the complex magnetic field is, in cylindrical coordinates ( ) z , , , r f The resulting energy and momentum densities are, from (4.1), = the degree of linear polarization L is the same for the electric and magnetic fields.The total energy, momentum, and angular momentum of general TE+iTM pulses are given in terms of the wavenumber weight function ( ) w k q , of (3.1) by ([12], section 4.5),  = The contours of L are ( ) 0.1 0.1 0.9; on the axis 1, L = identically.Note that the pulse is hollow in momentum: all components of p are zero on the propagation axis 0 r = .
K k a 2 = -In our case m 0, = so J 0 z = according to (4.8).But note that when m 0 = the z component of the angular momentum density j z is not zero: j p , z r = f with p f given in (4.6), which is not zero in general but always integrates to zero over all of space.For G K y = the total energy and momentum are, either from (4. Of particular interest here is the degree of linear polarization L (the same for the electric and magnetic fields for self-dual pulses).It is no longer identically unity, perfect linear polarization, as in the TE case.However, 1 L = on the pulse axis, because the r and f components of B c in (4.3) contain a derivative with respect to , so only the z component is not zero on 0. r = Since B c and E c point along the z direction on the axis of propagation, the momentum of TE+iTM pulses is zero there.Figure 1 shows L and the energy and momentum densities u p p , , z r in a longitudinal section at t 0 = for the TE+iTM pulse based on G , K when Ka 2. =

'Circularly polarized' ('CP') self-dual pulses
We shall define and discuss the properties of electromagnetic pulses which are approximately circularly polarized in a fixed plane.Self-dual pulses which are circularly polarized in the plane wave limit are formed as follows: From section 4.3 of [12], we define the complex fields 2) the wavefunction y is a solution of the wave equation, and the fields are those of a time-dependent selfdual pulse.Note that the divergence of the vector potential is zero, so the Lorenz condition is satisfied if the scalar potential is a constant.The complex magnetic field has the Cartesian components In the monochromatic plane-wave limit The corresponding real fields are, in this limit, This is the textbook circularly polarized wave.Theorem (ii) in section 2 states that pulses which are everywhere circularly polarized in a fixed plane do not exist.The textbook exception just mentioned is the non-physical limit of an infinite plane wave.This is why we have the quotes around 'CP': perfect circular polarization in a fixed plane is only possible in the plane-wave limit.We wish to find the energy and momentum densities of pulses with the complex fields B c of (5.3) and We shall consider the case m 0 = in (3.1), when the wave function is independent of the azimuthal angle .
f Since the complex vector potential and magnetic field in cylindrical coordinates are as in equation (4.16) and (4.17) of [12]: The energy and momentum densities are as in ( ) 4.18 of [12]:    Of prime interest here is the degree of linear polarization L defined in (2.1).Again, since we are dealing with self-dual fields, L is the same for the electric and magnetic fields: the complex fields are related by E B i .
zero on the axis 0 r = at t 0, = and is never unity: the polarization is perfectly circular on the axis, and nowhere linearly polarized.In the focal plane z 0 = and at t 0, =   2 illustrates L and the energy and momentum densities in longitudinal section of the 'CP' pulse based on G K at time zero, when the pulse is concentrated in its focal region.Figure 3 shows the same properties in the transverse section z 0. = In a perfectly circularly polarized field, the real and imaginary parts would be orthogonal and equal in magnitude, giving 0. L = We see from the upper part of figure 3 that in fact  0.01 L for most of the circle a r < in the focal plane.For the 'CP' pulse based on G , K we show below that, on the axis 0, r = the complex electric field has the form the z component is zero, and the x and y components differ by a factor of i.The function ( ) f t is complex, f g ih, = + and the real and imaginary parts of the electric field are , , , 0 5.14 Hence, on the propagation axis, the 'CP' pulse is always perfectly circularly polarized: The proof of perfect circular polarization on 0 r = follows from the expression (5.7) for the complex magnetic field, and the form (3.5) of G .
K We use from which the stated Cartesian form of the complex electric field follows.

Discussion
We have presented exact solutions of the free-space Maxwell equations which are localized in space-time, for which all components of the electric and magnetic fields satisfy the wave equation.We showed that, for localized and exact solutions of the Maxwell equations, perfect circular polarization in a fixed plane, or perfect linear polarization along a fixed direction are not possible.
Nevertheless, explicit examples showed that TE pulses with m 0 = are linearly polarized with an azimuthal electric field, and that approximately circularly polarized pulses may be constructed.The latter do have perfect circular polarization on the propagation axis.
In the published literature electromagnetic pulses are often modelled by plane waves ( ) f z ct - [16][17][18] or by monochromatic beams, often in the paraxial approximation [19][20][21][22][23].In many cases the analysis is more complicated than for the localized pulses presented here, for which the space-time variation is known (and thus also the polarization), together with the pulse total energy, momentum, and angular momentum.
Regarding the practical applications of electromagnetic pulse polarization: in fields such as telecommunications or medical imaging the pulses are nearly monochromatic, and the 'beam' definition of the degree of linear polarization will suffice.The theory presented here will be relevant in the scattering of very short pulses by electrons, atoms and molecules.
Self-dual electromagnetic fields are unchanged under the duality transformation[13,14] self-dual: for example, equating the real and imaginary parts of duality transformation.The energy and momentum densities for the self-dual fields are[12, section 3.3]

consider
TE+iTM pulses, formed by the superposition of a TE pulse and a TM pulse in phase quadrature.The TE pulse has the scalar potential constant, and the complex vector potential[ the TM pulse is the dual of the TE pulse, formed by E

¥
These values are twice those for TE and TM pulses based on the same wavefunction.(We omit a factor of dimension energy times length; the wavefunction is dimensionless, and normalized to unity at the space-time origin.)We shall give an example of an oscillatory TE+iTM pulse, based on the wavefunction G K of (3.5), which has the weight function[15]

Figure 1 .
Figure 1.The degree of linear polarization L (upper diagram) and the energy and momentum densities u p p , , z r (lower diagram), shown in a longitudinal section at t 0 = for the TE+iTM pulse based on G , K with Ka 2.= The contours of L are ( ) 0.1 0.1 0.9; on the axis 1, L = identically.Note that the pulse is hollow in momentum: all components of p are zero on the propagation axis 0 r = .

f¥
These values are obtained by spatial integration, or more easily from the wavenumber integration formulae (4.66) of[12],Note that the angular momentum is non-zero, even though the azimuthal winding number m is zero and the wavefunction y does not depend on .fThe self-dual 'CP' fields defined by (5.1) and (5.2) have a twist in them.

Figure 2 .
Figure 2. Longitudinal sections of the 'CP' pulse based on G , K for t Ka 0, 2. = = Upper diagram: The degree of linear polarization .L The contours of L are ( ) 0.002 0.002 0.012; on the axis 0, L = indicating perfect circular polarization.Lower diagram: The energy and momentum densities, for the same parameters.The azimuthal momentum density is shown in figure 3.

Figure 3 .
Figure 3. 'CP' pulse in transverse section at z 0, = based on the wavefunction G .K Upper diagram: the degree of linear polarization ; L contours are at ( ) 0.002 0.002 0.010; 0 L = at the origin.Lower diagram: energy and azimuthal momentum densities.The parameters are t Ka 0, 2. = = The pulse is travelling toward the reader.
8)or by integration of the densities u and p , In between the zero values at r equal to zero and infinity, the focal plane L has a maximum value, except in the