On the basis of the irradiance-moments formalism, four matrices are proposed whose elements, defined for any partially coherent field, are closely related with the second-order measurable parameters handled in the ISO standard 11146. These matrices are shown to exhibit a number of properties concerning the orientation of the transverse profile of a partially coherent beam. This behavior is described by the rotation of the principal axes of the field around its propagation axis. In addition, these matrices provide information about the spatial structure of the field (beam spread product and orbital part of the angular momentum). A new overall parameter is introduced in terms of the above matrices, which remains invariant under rotation of the Cartesian coordinate axes and upon propagation through rotationally-symmetric first-order optical systems.
© 2006 Optical Society of America
The overall spatial structure of light fields has been described by means of their irradiance-moments.[1–6] Based on this formalism, a number of measurable parameters (beam width, far-field divergence, beam propagation factor, etc) have been adopted as current ISO standards. 
In this characterization, the spatial symmetry of the transverse beam profile has revealed to be another interesting property. For instance, the light fields can be classified as stigmatic (ST), simple astigmatic (SA) and general astigmatic (GA) fields (associated to rotational symmetry, orthogonal symmetry and other kind of symmetry, respectively). [8–10]
In the most generalized case, the irradiance distribution across the beam profile rotates as the field freely propagates. [8–15] This happens, for example, when a He-Ne TEM00 laser beam travels through two cylindrical non-parallel and non-orthogonal lenses separated by a free space. The output field would then show a rotating elliptical spot upon propagation. In this sense, it should be noted that there are non-rotating GA beams which behave similar to ST and SA fields: the internal features of these GA beams remains hidden if only rotationally-symmetric optics is used, and can only be revealed after propagation through cylindrical lenses. These fields are usually called pseudo-ST (PST) and pseudo-SA (PSA) beams (also called pseudo-type fields).[12,13,16] SA- and PSA-beams look similar in free space, but PSA fields carry orbital angular momentum, a characteristic that the SA beam does not have.
On the basis of the irradiance-moments formalism, the spatial orientation of a general partially coherent beam at each transverse plane has been described in the literature by means of a pair of orthogonal axes (principal axes) defined in terms of the second-order coherence features of the field.[5, 17] Within this framework, in the present work four matrices will be introduced, which are closely connected with the orientation of the beam profile. Moreover, the elements of these matrices allow describing the overall spatial structure of the beam. In addition, in terms of such matrices, a new parameter is also proposed which remains invariant under propagation through stigmatic optical systems.
The paper is then arranged as follows: In the next section the basic formalism is shown. Section 3 defines the proposed matrices, whose properties are also included in the same section. The new parameter is introduced in Section 4, and some concluding remarks are summarized in Section 5. Finally, some demonstrations are given in the Appendices.
2. Basic formalism
Let us consider a quasimonochromatic partially coherent beam. Here we are interested on the overall behaviour of the field, which is described by means of the so-called beam irradiance moments. As is well known, these parameters (denoted by sharp brackets) can be defined in terms of the Wigner distribution function, h(r,η,z), associated to the field, in the form
where m, n, p and q are integer numbers, r = (x, y) denotes the two-dimensional position vector transverse to the propagation direction z, kη = (ku, kv) = (kx, ky) provides the wavevector components along the x- and y-axes (accordingly, u and v represent angles of propagation, without taking the evanescent waves into account) and Io = ∫ h dr dη is proportional to the total beam power. The four first-order moments, <x>, <y>, <u> and <v>, characterize the centre of the beam and its mean direction. For simplicity, in what follows we assume that these moments vanish (this is not a restriction, since it is equivalent to a shift of the Cartesian coordinate system). On the other hand, the (squared) beam width at a plane z = constant and the (squared) far-field divergence are represented by < x 2 + y 2 > and < u 2 + v 2 >, respectively. In addition, the crossed moment <xu + yv> gives the position of the beam waist through the condition <xu + yv> = 0.
The spatial orientation of the irradiance profile can be characterized[5,17] by the orientation of two orthogonal axes (the so-called principal axes) for which the crossed x-y moment vanishes, i.e., < xy > = 0 . In this case, we know that the beam widths < x 2 >1/2 and < y 2 >1/2 reach their extreme values along such axes. Since, in general, the spatial profile rotates as the field propagates in free space, the principal axes are used to describe this rotation: it would suffice to determine the angle that the principal axes make with some fixed laboratory coordinate system.
3. Matrices and properties
In terms of the irradiance moments, let us now introduce the following set of four 2×2 matrices:
As is apparent from the definition, the near and far field behavior can be inferred from the diagonal elements of M 0. Furthermore, the non-diagonal elements provide the position of the waist plane and its determinant gives the beam quality parameter. 
On the other hand, the matrix M 3 contains information about the extrinsic (also called orbital) part of the angular momentum of the beam. Finally, the matrices M 1 and M 2 include all the second-order parameters required to know the orientation of the beam profile freely propagating. To show this, let θ be the angle between the principal axes of the beam and the laboratory reference axes. It can be shown that the evolution of θ under free propagation is given by the expression
where the moments are calculated (or measured) at a certain plane, say, z = 0. It is immediate to see that the coefficients of the z-polynomial in the numerator of Eq. (3) are the elements of M 2, and the coefficients in the denominator correspond to the matrix M 1. Note that a rotation by the angle θ would diagonalize the beam submatrix This does not occur, however, for the submatrices and when a general field is considered. 
It can be shown that the above four matrices have the following general properties:
- A beam does not rotate when freely propagates if and only if
a) at least one of the matrices, M 1 or M 2, are zero,
a) M 1 and M 2 are proportional.
where c is a constant.
- A beam does not rotate under free propagation if and only if there exists a coordinates system with respect to which M 1 and/or M 2 are zero.
The demonstration follows at once from property (i) when either M 1 or M 2 are zero. Let us then consider the third possibility considered in that property, namely, M 2 = cM 1. Under rotation of the transverse Cartesian axes around the propagation direction z, the matrices M 1 and M 2 become
where α is the rotation angle, and the primes refer to the values after rotation. After substitution of the condition M 2 = cM 1 into Eqs. (5), we get
It is now easy to see that M′2 = 0 when α = α 0, where
In addition, M′1 equals zero when the rotation angle take the value α= α 0 + π/4. We thus see that, after a certain rotation of the coordinates axes, we could make M 1 or M 2 equal to zero, and property (i) would apply again.
- The matrices M i, i = 0, 1, 2, 3, propagate through stigmatic optical systems according with the law
where denotes the 2×2 ABCD matrix representing the stigmatic optical system, and t means transposition.
Let us first recall the paraxial law for the second-order moments of a partially coherent beam propagating through a general first-order optical system: 
where P represents the 4×4 ABCD matrix associated to the general optical system and B is the beam moments matrix
The propagation law (8) for the matrices M i, i = 0, 1, 2, 3, would then follow by taking into account that the 4×4 ABCD matrix of a stigmatic optical system reduces to where I denotes the 2×2 identity matrix and a, b, c and d are constants. To illustrate the demonstration procedure we will next show the propagation law (8) for the matriz M 3.
Since the non-zero elements of the matrix M 3 are ±(<xv> - <yu>), it is easy to see that the propagation law for this matrix can be written in the form
- The matrices M 0 and M 3 are invariant under rotation of the Cartesian axes around the propagation axis z.
Note first that a transverse axes rotation is equivalent to a first-order optical system characterized by the 4×4 ABCD matrix where α is the rotation angle. From the application of Eqs. (9) and (10), we get, for example,
where the subscript rot denotes the values after rotation. We then have
Accordingly, the matrix M 3 remains invariant under rotation. In a similar way, it can be shown that the elements of M 0 does not change after rotation of the Cartesian axes.
4. Overall parameter
In terms of the above matrices, let us now introduce the following parameter:
where “tr” stands for the trace of the matrices. It can be shown that this parameter exhibits several interesting properties, namely,
- The parameter g does not change under propagation through stigmatic optical systems.
Let us consider a partially coherent beam propagating through a stigmatic optical system characterized by its ABCD matrix S. Taking into account the propagation law (9), we have
Accordingly, after using the cyclic property of the trace, we get
It then follows at once from Eq. (11) that g remains invariant through this kind of optical systems.
- The parameter g remains constant under rotation of the Cartesian axes around the propagation axis.
The proof is given in Appendix A.
- g ≥ 0. In particular, g = 0 if and only if the orientation of the transverse profile does not change when the beam freely propagates.
The proof is given in Appendix B.
We thus see that the parameter g can be useful to characterize the orientation of the beam profile of any partially coherent field traveling through rotationally-symmetric optical systems (note that free propagation is a particular case).
5. Concluding remarks
Note first that, in terms of the matrices defined in Section 3, one can write three overall parameters, which remain invariant through stigmatic optical systems and also under rotation of the Cartesian reference axes:
In these expressions, Q 3D is the beam quality parameter[5, 18, 19] (also called beam spread product), and it measures the joint capability of the beam for simultaneous near and far-field focusing. The parameter L 2 provides information about the orbital part of the angular momentum of the beam, and we have shown that g characterizes the rotation capability of a freely propagating field. In addition, it should be remarked that all these parameters can be obtained from measurements at a transverse plane z, in terms of the second-order irradiance moments of the beam at such plane. Note also that both the parameter g and the matrices M i, i = 0, 1, 2, 3, also apply to the description of coherent light fields.
Finally, it should be mentioned that the matrices introduced in this work are closely connected with the parameter J (see, for example, Ref. 5) through the relation.
It is interesting to recall that this parameter J is invariant upon propagation through general first-order optical systems.
In this Appendix we will show that the parameter g remains constant under rotation of the Cartesian axes around the propagation axis z.
where again α is the rotation angle and the prime refers to the values after rotation. Taking this into account, it is not difficult to see that
Let us now write
which is the result we are looking for.
We will first show that g ≥ 0 . For simplicity, from now on let us consider that the matrices are evaluated at the waist plane of the beam. In addition, the coordinates axes are chosen to fulfil < uv > = 0 (this is not a true restriction, because the parameter g does not change under rotation of the Cartesian axes). Taking this into account, the matrices M 0 and M 2 reduce to
with < r 2 > = < x 2 + y 2 > ; < η 2 > = < u 2 + v 2 >, and
where the irradiance moments should be evaluated at the waist plane. It then follows
Therefore we have
After some straightforward calculations we finally get
Let us now show that g = 0 if and only if the orientation of the beam profile remains invariant upon free propagation.
We first assume that the transverse profile does not rotate. In this case, from the property (i) of the matrices M i (see Section 3), along with the definition of g [cf. Eq. (11)], it follows at once that this parameter should be equal to zero.
To complete the demonstration, we will next show that the condition g = 0 would involve a non-rotating beam profile.
Expression (B.8) and the equality g = 0 imply the simultaneous fulfilment of the following three equations:
The trivial solutions of these equations are (a) M 1 = M 2 = 0; and (b) M 1 or M 2 equal to zero. In both cases, property (i) of Section 3 would again conclude that the beam does not rotate. Let us then consider that M 1 and M 2 differ from zero. We have two possibilities, namely, < u 2 - v 2 > = 0 ; and < u 2 - v 2 > ≠ 0 . This second possibility would imply (see Eqs. (B.10), (B-11))
so that M 2 = 0, which, in turns, implies that the orientation of the beam profile will not change.
Let us finally assume the other possibility, i.e., < u 2 - v 2 > = 0 . Since M 1 and M 2 are not zero, it is not difficult to see that M 1 and M 2 should be proportional. Again, property (i) of Section 3 enables us to conclude that the beam profile does not rotate upon free propagation, Q. E. D.
This work has been supported by the Ministerio de Educación y Ciencia of Spain, project FIS2004-1900, and by the Universidad Complutense-Comunidad de Madrid, within the framework of the Research Groups Program 2005-06, 910335.
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