The angular spectrum of a vectorial laser beam is expressed in terms of an intrinsic coordinate system instead of the usual Cartesian laboratory coordinates. This switch leads to simple, elegant, and new expressions, such as for the angular spectrum of the Hertz vectors corresponding to the electromagnetic fields. As an application of this approach, we consider axially symmetric vector beams, showing nondiffracting properties of these beams, without invoking the paraxial approximation.
©2012 Optical Society of America
In many branches of physics, it is necessary to solve the vectorial wave equation in three dimensions. One of the best known and most general methods for the vectorial Helmholtz equation is the expansion in plane waves by representing the solution as a three-dimensional (3D) Fourier integral over the 3D reciprocal wavevector space. Especially in laser physics, the relevant electric and magnetic fields are in fact solutions having a directed-energy beam-character. That is, the fields propagate along a certain axis, called the optical axis and which we choose to be the axis of the Cartesian coordinate system linked with the beam. A number of mathematical methods are known to treat this case efficiently [1,2]. In this paper, we consider three methods of solution, the angular spectrum representation of the beam, the TE/TM decomposition of the beam and finally the Hertz vectors of the beam. The last method is the least well known, despite the fact that it shows a major advantage by generating particularly simple expressions for the fields. We shall indicate the interrelations between the three methods. In a first step we show how to transform the Cartesian axes to an intrinsic coordinate system linked with the propagation constants of the beam.
Physically, the new base vectors are linked with the classically known and components of polarization optics, but their expression in a Cartesian or cylindrical coordinate system is not evident.
In a second stage, we determine the angular spectrum not of the electric and magnetic fields and , but of the Hertz potential vectors and corresponding to these vectors. Then we use the results of the first phase and switch to the new coordinates. New and simple formulas are obtained. This leads to an alternative angular spectrum representation of the solution of the vectorial Helmholtz equation.
As an application of the usefulness of this representation, we will briefly discuss diffraction-free vectorial beams. We will do this without the restriction to reduce the Maxwell equations to their paraxial form. The diffraction-free solution is hence a rigorous solution and not an approximate one, satisfying in particular the zero divergence Maxwell equation for the electric field, which a paraxial solution does not satisfy.
2. INTRINSIC COORDINATE SYSTEM
Let be the wave vector of a plane wave1) we have denoted the split of the wave vector into its transverse (subscript ) and longitudinal part (subscript ). The vector forms together with the axis the meridional plane . In optical terminology, we would call it the plane of incidence (on a mirror for example, where is the normal to the mirror). The angle between the axis and the axis is the polar angle . The cylindrical coordinates are defined through their unit vectors , , and , being the azimuthal angle of the vector in the transverse plane.
The 3D locus vector is defined by
The key element in our approach is the point brought up in  and , which is that we should consider an intrinsic coordinate system. It contains three unit vectors: two of them are respectively perpendicular and parallel to the meridional plane, the third being the unit vector in the direction:
Note that does not possess a component. is perpendicular to the meridional plane, whereas is coplanar with it, just like . has only a transverse component, its azimuthal part, whereas both has a transverse component, its radial part, and a longitudinal component. Although the definition of the and vector are classically known in the literature on optical polarization, their relation [Eq. (3)] with the cylindrical base vectors is not evident. An alternative but equivalent definition of the intrinsic coordinates is
3. HERTZ POTENTIALS AND THEIR ANGULAR SPECTRUM REPRESENTATION
We will only consider free-space situations: the sources are infinitely far away from our observation region. The electric and magnetic fields can be defined in S.I. units in terms of their scalar and vector potential :6]: 6), hence eliminating the scalar and vector potential, results for the two cases resp. in the expression of the fields as a function of the Hertz vectors. For the electric Hertz vector: 7) and (8), we find the general expression for the fields in terms of the Hertz vectors:
These expressions do not look at first sight like simplifying life. The use of the Hertz potentials is advantageous, however, in case the current sources are all parallel to a common direction, say the direction [5–7]. In this situation, and far from the sources, Eq. (7) for the electric Hertz potential reduces to the set3]. The attractive aspect of the “scalar” Hertz vector representation as compared to, e.g., the mathematically equivalent modal representation is that it arrives at the same rigorous results as the modal representation, but in a much shorter and more elegant way. To give a quick example of its usefulness, consider the electric field of an electric dipole oscillating along the axis. In every direction starting from the origin, the electric dipole field has another orientation but the corresponding Hertz vector is nevertheless everywhere in space a scalar, oriented along the axis, independent of the radiation direction .
For laser applications, we are exclusively interested in directed-energy solutions of the vectorial Maxwell equations, i.e., in “beams,” rather than in “fields.” Of course we then take the direction as the propagation direction of the field energy. So, using the Hertz potentials, vectorial beams can be completely described by a scalar function. Specifically, whereas in general is a solution of the vectorial wave equation, its component in Eq. (11) is now a scalar solution of the Helmholtz wave equation with the boundary condition . The two-dimensional (2D) Fourier transform of the Hertz potential in the plane , is defined as8] 14) shows that for beam-like solutions, we do not need a 3D Fourier transform as solution of the wave equation, the dimensionality of the problem is reduced by one unit.
A. Vectorial Angular Spectrum Representation
A scalar field satisfying the scalar Helmholtz equation, can be represented as in Eq. (14) by its angular spectrum representation, i.e., the beam is linearly polarized. What happens if the beam has a vectorial character? Just by extending the argument from one dimensional (1D) to 3D, we look for a representation like
Assuming harmonic time variation, we now take the 2D Fourier transform of and at in Eq. (9) and call them the vectorial angular spectra and of the field vectors, or the spectral field vectors in short
We now take the 3D Fourier transform of the field Eq. (9) and use the theorem that the Fourier transform of the derivative of a function is equal to times the Fourier transform of the original function. In working out the expressions for , one should take into account relations such as
What we will do in this paragraph is work with the spectral field vectors and to finally obtain an angular spectrum representation of the fields and , through the intermediary of the Hertz vectors and their angular spectrum representation, according to the Fourier transforms indicated in the following diagram:
B. Transverse Spectral Hertz Vectors
Next, we proceed to decompose each angular vectorial spectrum of the Hertz potentials and in a part that is parallel to and a part that is perpendicular to :19) in Eq. (17,18), the parallel components of the spectral Hertz vectors all drop out and what rests is
C. TE/TM Decomposition
The considerations in Subsection 3.A were valid for fields in general. Because we consider beam propagation along the axis, the TE/TM decomposition theorem for the electric and magnetic field can be invoked here , using the axis as axis along which the decomposition is stated:
We now consider a source of oscillating dipoles, all oscillating in the same direction, which we choose to be the direction. The Hertz vectors have the same orientation as the currents ; hence the Hertz vectors have only a component. We assume that the dipole source is located at so that the halfspace is source free and the field differential equations become homogeneous and hence simpler. In this halfspace, the Hertz vectors keep their orientation. From here on, we assume that the Hertz vectors have only a -dependent component:
We now revert to the angular spectra of the Hertz vectors.
D. TM Case
For the moment, we only know that, by Fourier transforming the original electric Hertz vector, we obtain21), we see that . We use Eq. (20) to calculate this scalar product. This leads after some algebra to: . Next to the coordinate transformations Eq. (4,5) for the intrinsic vectors, this small equation is the second central result of this paper. It shows that the spectral magnetic Hertz vector is zero for a TM beam, leading to extremely simple equations, despite the vectorial character of the beam. This is shown by again using the intrinsic coordinate transformations, so that the angular spectra now become 23), the magnetic vector only depends on , which has no component, according to Eq. (3); so therefore the fields given by Eq. (22,23) are indeed TM. This is in complete agreement with Eqs. (10) which show that the magnetic field has no component. [Eq. (10) is not written in the Fourier domain, but for the polarizations this does not matter.]
E. TE Case
In a similar way as in the former case, this corresponds to:14) and (15) become, making use of the transformations (5):
Note that the magnitudes of the magnetic spectral vectors are just proportional copies of the magnitudes of the electric spectral vectors.
F. Spectra of the Transverse Hertz Vectors
G. Angular Spectral Representation
The electric field of the beam propagating along the axis has been decomposed in its TE and TM components:22) and (24) in 30) show the electric and magnetic fields as functions of the angular spectra of the two Hertz vectors and of the intrinsic polarization vectors and . In the next paragraph, we will express the fields directly using their angular spectra. This will lead to a relation between the angular spectra of the field vectors and of the Hertz vectors.
4. ANGULAR SPECTRUM OF THE VECTOR FIELDS
The angular spectrum representation of the vector electric field has been given in , expressed in Cartesian coordinates as:
Following another strategy to solve the same vectorial wave equation,  started from the modal representation, as commonly used in guided wave problems, combining it with the angular spectrum representation. The advantage of this approach is that it clearly identifies and separates the contributions of the transverse electric fields (TE and TM) right from the start. In laboratory Cartesian coordinates, this finally leads to1] 1] was able to separate the propagating field into two transverse contributions, which [2,4] did not indicate in this way. We will see, however, that in , essentially the same result was obtained. Expressions for the magnetic field are not given by . It is nevertheless clear that is lacking a component, as it should for the electric field of a TE mode.4], although in a slightly different format, denoted there as without explicitly pointing to the transverse character of the contributions and .
The same reasoning can be repeated for the magnetic angular spectra:22–25), and comparing these with Eq. (30,34,35), we see that
5. AXIALLY SYMMETRIC FIELDS
We now work out the general Eq. (34) for the special case of axial symmetry of the fields around the axis.1] and  have considered this case. The angular spectrum is independent of the azimuthal angle in reciprocal space and only depends on . Equation. (34) becomes  39) by just keeping the vectorial part and the differentials in place, and absorbing the other contributions in the symbol (.), we can more clearly see how the unit vectors are transformed by changing from Cartesian to cylindrical coordinates. Note that as integration variable in space is transformed into in space, which is independent of and and hence can be brought in front of the integration sign for the TE-field:
6. NONDIFFRACTING VECTORIAL BEAMS
The best known nondiffracting scalar beam in the laser literature is the Bessel beam . This beam is linearly polarized. As a short application of Eqs. (39,40), we will consider the diffraction-free propagation of an azimuthally polarized vectorial beam and extend in this way the scope of , which analyzed scalar beams.
The general solution of the scalar wave equation for a diffraction-free beam is given  as14) to arrive immediately at 41) is hence nondiffracting since the transverse part is not depending on the propagation coordinate . This result was also obtained in , but by the method of separation of the variables of the paraxial wave equation. A mathematically equivalent statement is that the Bessel function can be represented as a Whittaker integral. Its explicit form can be found in .
In summary, we have pointed out the relevance of using an intrinsic coordinate system to unify the existing angular spectrum representations. We have given the transformation formulas from intrinsic to Cartesian coordinates. The introduction of intrinsic coordinates much simplifies the angular spectral representation of vectorial beams. We have introduced the Hertz vector potential oriented along the axis and have combined it with the intrinsic coordinates to represent TE and TM beams in a particularly simple and elegant form. This leads also to the mathematical link between the angular spectra of the fields and of their Hertz vector. Next, we derived expressions for a diffraction-free vectorial beam, without invoking the paraxial approximation.
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