Linearly polarized electromagnetic pulses

We construct theoretically a set of space-time localized electromagnetic pulses, characterized by a wavenumber K and a length a. Their linear polarization becomes more perfect as Ka increases and the pulse becomes more nearly monochromatic. Two measures of the degree of linear polarization are explored: one that gives the polarization at any point in space-time, and another that integrates the electric intensity over the focal plane of the pulse as the pulse passes through. The polarization measures and the total energy, momentum, and angular momentum of these pulses are calculated for all K and a. The fields of these pulses satisfy the Maxwell equations exactly.


Introduction
Electromagnetic pulses are sometimes modelled just as a time variation f t ( ) [1][2][3], by plane waves f z ct ( ) [4,5], by pulsed beams [6], or by a solution of the wave equation in the paraxial approximation [7].A previous paper [8] deals with the polarization of space-time localized solutions of Maxwell's equation.We defined a degree of linear polarization L r t , ( )for pulses; L = 1 for perfect linear polarization, L = 0 for perfect circular polarization.In contrast to monochromatic beams, L now depends on time as well on position, in general.We showed that perfect linear polarization along a fixed direction is not possible for any space-time localized pulse.(The non-existence of perfect linear polarization statement holds in free space and does not apply to propagation in waveguides).We also showed that perfect circular polarization in a fixed plane is not possible.However, 'circularly polarized' pulses were defined which have perfect circular polarization (L = 0) on the propagation axis.Elsewhere the pulse polarization is not completely circular.We used quotes, 'CP', to designate that the circular polarization is not perfect.
This paper is concerned with generating theoretically 'linearly polarized' ('LP') pulses, and with exploring their properties.The textbook linearly polarized monochromatic plane waves, of angular frequency w = kc, have in Cartesian coordinates x y z , , [ ]the complex electric and magnetic fields The subscript c on the fields denotes complex: i In this idealized case E c and B c oscillate in one direction (along the x and y axes, respectively).The real and imaginary parts of the electric field (assuming E 0 is real) are , 0, 0 , sin 0, 0 1.2 As in [1], we define the degree of linear polarization for the electric field by L E is identically unity for the plane wave of (1.1) or (1.2); likewise for the magnetic polarization L .
B The measure ( ) | ( )| | ( )| which applies without qualification to monochromatic beams [9,10].In such beams the measures L r , E ( ) and the degree of magnetic linear polarization L = r B r B r , contrast, is a function of time at any fixed point in space, and is physically meaningful only if it is almost constant as the pulse passes a given point in space.
There are also differences in how electromagnetic pulses and beams are derived from solutions of the wave equation.For monochromatic beams the time dependence - e ikct of all the (complex) solutions reduces the wave equations to Helmholtz equations.Further, the Lorenz condition  + ¶ = A V 0 ct • on the vector potential A and scalar potential V becomes  -= A ikV 0, • so V is known once A is specified.In setting up the pulse problem, on the other hand, we usually satisfy the Lorenz condition by choosing the vector potential to be the curl of a vector made up from solutions of the wave equation, and taking the scalar potential to be a constant.This is the method by which we (theoretically) construct linearly polarized pulses in section 3.
Note that L E remains defined even in space-time regions where the field strength tends to zero, when the polarization becomes physically irrelevant.An alternative measure of linear polarization, which integrates the electric intensity over the focal plane of the pulse as the pulse passes through it, is defined below and applied in section 5.For pulse propagation along the z axis, with = z 0 being the focal plane, the degree of electric linear polarization along the x direction may be characterized by a single number, This measure is applied separately to the real or imaginary parts of the complex field E .c The P E value is the same for the real and imaginary parts for 'LP' pulses derived from the pulse wavefunction G , K as we shall see.A full description of polarization of monochromatic electromagnetic waves is in terms of the Stokes parameters ( [11], sections 1.4.2 and 10.8.3) which are proportional to the field strength squared: Three of the four Stokes parameters are independent, since = + + s s s s .For space-time localized pulses, which are necessarily not monochromatic, the polarization along for real E.This is different for pulses derived from the real and imaginary parts of the complex i An advantage of the definitions (1.3) and (1.4) is that they give one value for both pulses originating in the same complex solution y of the wave equation.While L r t , E ( )gives the local polarization, P E gives one number for a given pulse, an integrated measure.
The simplest TE and TM pulses are exactly linearly polarized, but azimuthally.The self-dual TE+iTM pulses are approximately linearly polarized, along the propagation direction.These were considered in [8].In this paper we shall discuss pulses which are linearly polarized along a fixed transverse direction, approximately.These are the pulse version of the textbook transversely polarized plane waves, with complex fields given in (1.1), and are what is usually meant by linear polarization of electromagnetic waves.In section 2 we show how localized solutions of the wave equation are constructed theoretically.Electromagnetic pulses which become polarized linearly along a fixed direction, in the limit of large wavenumber, are defined in section 3. The properties of a specific 'LP' pulse based on the oscillatory wavefunction G K are explored in section 4. In the Discussion (section 5) and the appendix we give some properties of P , E the integral of the square of the electric field over the focal plane of the pulse and over the time the pulse takes to pass through.For both measures the linearity of the polarization of the pulses to be considered increases as the pulse becomes more monochromatic.

Localized solutions of the wave equation
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 wave equation and the Lorenz condition We shall use forward-propagating solutions in cylindrical polar coordinates r f z , , , ( ) of the form [12] ò ò The simplest space-time localized wavefunction with = m 0 is [13, 14] Note that the (complex) length R is the hypotenuse of a right-angled triangle in which the orthogonal sides are the radial distance r and the length a augmented by ict.The expression for G may be generalized [15] to the solution of the wave equation containing an arbitrary differentiable function F, of which a special case is the oscillatory wavefunction G , G and G K are normalized to unity at the space-time origin.The pulse G K is characterized by the length a and the wavenumber K.The longitudinal extent of the scalar pulse is of order a 2 , the transverse extent of order - K , 1 and the number of oscillations within the wave packet of order Ka (see section V of [15], which includes an equation for the divergence angle of TE and TM pulses based on G K ).The pulse G K becomes approximately monochromatic, with dominant wavenumber K, when Ka 1.  G K becomes the sub-cycle pulse G in the limit  Ka 0. The polarization of TE and TM pulses based on G K has been discussed in [8]; those with = m 0 are linearly polarized, azimuthally, not in a fixed direction.
'Linearly polarized' ('LP') pulses based on G , K which have the electric field predominantly in one transverse direction, are discussed in section 3. The inverted commas are used to emphasize that the linear polarization is not perfect, except in the physically unrealizable plane wave limit.

'LP' pulses, theory
A transversely bounded pulse which is linearly polarized in a fixed direction does not exist (Theorem (i) of [8]).Nevertheless, as we shall show, a pulse can be formulated theoretically which, in the plane-wave limit, is polarized along a fixed direction.We start with a divergence-free vector potential (so that with the scalar potential being constant, the Lorenz condition is satisfied), The function y is a forward-propagating solution of the wave equation, in the form (2.1).The complex fields derived from this vector potential have identically zero y component of the electric field: the z component of the field will be zero in the = x 0 plane, for solutions of the wave equation of the form ) Hence, for this family of solutions, 'LP' pulses will always be perfectly linearly polarized in the = x 0 plane, in which the x component of the electric field is the only one not zero.This holds at all times and for all y and z.
The Maxwell equations are satisfied because y satisfies the wave equation.The plane-wave limit y = f z ct ( )gives fields proportional to those of (1.1), linearly polarized and mutually perpendicular.In cylindrical coordinates r f z , , ( ) the formulae for the complex fields become, when y is independent of the azimuthal angle, and on using The energy densities are different for the real and imaginary parts of the fields, However, the total energy of the pulse is the same, whether u r or u i are integrated over space, as is shown below equation (3.12).The total energy is also independent of time ( [12], section 1.2), so the spatial integrals

{ ( ) ( )} ( )
Note the limiting values of the ratio cP U z / at small and large Ka: As expected, the ratio tends to unity from below as  ¥ Ka .Note that > U cP , z which follows from > u cp z ([12], section 1.4).
The total energy, total momentum and total angular momentum of the general 'LP' pulse may be expressed as integrals over the wavenumbers: The expressions given in (3.8) and (3.9) follow from (3.11) with w k q , ( )given by, for the pulse G , The form of (3.11) shows that the real and imaginary parts y y , r i of the general solution of the wave equation (2.1) give the same total energy, momentum, and angular momentum: complex conjugation of y is equivalent to space inversion and time reversal together with  w w .* Pulses with azimuthal winding number > m 0 may be obtained from G K by repeated operation with r x

{ ( ) ( )} ( )
The limiting values of the ratio cP U z / at small and large Ka are ( ) ( ) The method of proof of (3.11) is as in [12], section 4.5; we give an outline only.When y (and thus also y*) are given by (2.1), there are seven levels of integration in the evaluation of the energy: four over the wavenumbers ¢ ¢ k q k q , , , ,and three over r f z , , .The integrand contains The azimuthal dependence in the factors f f e e , im im cancels, so the integration over f gives p 2 .The integration over z selects ¢ = q q: dz e q q 2 3 .1 7 The integration over r also gives a delta function, At this point we have integrated over the spatial coordinates; integrations remain over k q , and ¢ k .The delta function of (3.18) ) The seven-fold integration is thus reduced to just the double integration of (3.11).
Regarding the degree of linear polarization L, we see from the definition (2.1) and the field given in (3.5) that, when = m 0 and so This is unity (perfect linear polarization) when f = cos 0, that is, in the = x 0 plane.L is also unity on the propagation axis r = 0: so the y ¶ ¶ r ct terms are zero on the axis, and L  1.However, off-axis and away from the = x 0 plane, there is also the possibility of L = 0 (perfect circular polarization), which we shall explore in the next section.

'LP' pulses based on G K
Throughout this section the Figures will show aspects of the two 'LP' pulses derived from the complex scalar solution G K of the wave equation.We shall use the parameter combination = Ka 2, corresponding to a short pulse, but with some oscillation.The time variation of the complex electric field component E t 0, x ( )at the spatial origin (which is the center of the focal region), is shown in figure 1. E t 0, y ( ) and E t 0, z ( ) are zero.The quantities plotted are the real and imaginary parts of ) We see that at large Ka the decrease in the field amplitude at large ct | | is slower, as - ct ,

2
| | than at small Ka, when it decays as - ct .

4
| | Pulses with large Ka pulse last longer.Note also the functional form: the coefficients a and b of the oscillatory terms are algebraic, rather than exponential or Gaussian.
The energy and momentum densities are different for the pulses derived from real and imaginary parts of E B , .
c c (The total energy and total momentum are the same, as discussed in section 3). Figure 2 shows the densities for the real parts, figure 3 for the imaginary parts.In figures 2 and 3 the energy density contours are approximately at 0.1 to 0.9 of the maximum density, spaced evenly.
The energy and momentum densities derived from E B , r r are both maximal at the space-time origin.For the 'LP' pulse based on G K we find that the ratio of p z to u is then This ratio is in accord with the general inequality u cp z  ( [12], section 1.4).Note that equality is attained in the limit  ¥ Ka .Figure 4 shows the degree of linear polarization L E in longitudinal and transverse sections.We note the region where L  0, E corresponding to circular polarization.The very definition of the polarization measure assumes almost monochromatic oscillation, which is not the case at the parameter value = Ka 2. Nevertheless, the existence and location of L = 0 E is of interest.We give below the functional form determining the zeros of

Discussion
The experimenter may well object to the measure of linear polarization L which has been used in this paper, since this strictly applies only in the monochromatic limit, and pulses are never monochromatic.We therefore construct another measure, of perhaps more experimental value, since it applies to the maximum intensity region, the focal plane of the pulse.This is the ratio of the electric intensity of interest, E , x 2 integrated over the focal plane = z 0 and over all time, to the same integrations applied to the total electric intensity, ( ) (For the 'LP' pulses defined here E y is zero; we have given the general definition.)The calculations of P E are more complicated than those of L .E Some detail is given in the appendix.P E is the same for the real and for the imaginary parts of the electric field derived from G : The exponential integral Ei 1 is defined by At large Ka the polarization measure P E tends to unity, as expected: ( ) ( ) ( )  ).The measure P E is in terms of the squared electric field, the electric intensity.
The deviation from perfect linear polarization in the field amplitude is thus of order -Ka .
1 ( ) A reviewer has pointed out that the same has been found for circularly polarized pulses ( [16], see in particular their equation (6), and also [17]).
P E goes to zero at small Ka, logarithmically, as --Ka ln .
1 1 { ( ) } This aspect is discussed in the appendix.Like L, the polarization measure P loses physical relevance as  Ka 0. Figure 5 shows the Ka dependence.Another aspect of experimental importance is pulse shape.Theoretical treatments often impose a pulse shape, for example Gaussian.Modelling and measurement show that pulses reach high irradiance levels well before a Gaussian pulse of the same peak irradiance [18].The same is observed in modelling by ideally Fourier transform limited pulses [19].Using pulse fields that satisfy the Maxwell equations, have variable wavenumber and scale and with analytically simple waveforms, is now possible with the G K solutions of the wave equation.The integrals to be evaluated are of the form For the evaluation of the focal plane ratio of integrals in (5.1) we set z = 0 after the derivative with respect to z has been performed in y = ¶ ¶ E .
x z c t The result is (5.2), the same for fields obtained from y r and from y .

∭
The pulse derived from G, the zero Ka limit of G , K has finite total energy and total momentum, independent of time.However, for 'LP' pulses based on G, the integral

2
The degree of linear polarization is related to s 0 and s 3 by L = - Differentiation with respect to r multiplies the weight function w k q ,

Figure 1 .
Figure 1.The variation with time of the electric field + E E i r ix ( ) at the center of the focal region, 0 E t , , x () plotted for = Ka 2.

Figure 2 . 2 1 (2
Figure 2. The upper two figures show the contours of the energy density and the momentum density (arrows) in the zx and zy planes.The bottom figure shows the energy density and the electric field, seen in the = z 0 transverse section.All three diagrams show quantities based on the real parts of the fields at = t 0, with = Ka 2.

Figure 3 .
Figure 3. Contours of the energy density and the momentum density (arrows) in the zx and zy planes (upper two figures), and the energy density and the electric field, seen in the = z a 2 / transverse section (bottom figure).Based on the imaginary parts of the fields at = t 0, with = Ka 2.

Figure 4 . 2 .
Figure 4. 'LP' pulse based on the wavefunction G .K Top: Degree of linear polarization in the zx plane.Bottom: the degree of linear polarization L in the = z 0 plane.The contours of L are at 0.1(0.1)0.9.The parameters are = = t Ka 0, 2. The diamonds show the positions » x a 0.8138 where the yellow curves of L = 0, corresponding to perfect circular polarization, intersect the figure.
Thus, for the 'LP' pulses based on G , K the deviation from perfect linear polarization, P = 1, The deviation is due the longitudinal component E z in this case ( = E 0 y x measure P E goes to zero.This is due to the logarithmic divergence of the integral -¥ ¥ dxdydtE .z 2

2 1 |
diverges: after integration over r and f the time integrand has an infinite tail: it varies as - t , | leading to logarithmic divergence.
We find, for the 'LP' pulses based on G , Of particular interest is the net momentum of the pulse in the direction of propagation.For the 'LP' pulses based on G , 1 ( )is also different for the real and imaginary parts, namely for E B , r r or E B , , i i but again the total momentum is the same for the real and imaginary parts of the fields.