J. Cent. South Univ. (2012) 19: 2167-2172
DOI: 10.1007/s11771-012-1260-6
Almost pure TM or TE component far field in vector cosine-Gaussian beams
ZHU Kai-cheng(朱开成), YU Yan(余燕), TANG Hui-qin(唐慧琴)
School of Physical Science and Technology, Central South University, Changsha 410083, China
? Central South University Press and Springer-Verlag Berlin Heidelberg 2012
Abstract: The vectorial structure of cosine-Gaussian beams (cGBs) is investigated in the far field regime based on the vector plane wave spectrum and the method of stationary phase. The energy flux densities of TE or TM term and the ratio of the energy flux of TE or TM term in the whole beam are demonstrated. It is found that the spot configurations of the energy flux densities associated with the TE and TM terms depend on the polarization angle and the beam parameter of the incident cGB. And the far field may be entirely transverse magnetic or transverse electric under appropriate polarization angle and beam parameter.
Key words: cosine-Gaussian beam; vectorial structure; polarization; TM component; TE component
1 Introduction
The beams passing through a paraxial optical system had been extensively studied. However, the paraxial solution is incompatible with the exact Maxwell’s equations and the paraxial approximation is no longer valid for the beams with a large divergence angle or a small spot size that is comparable with the light wavelength, therefore, rigorous nonparaxial and vectorial treats have been desired and a variety of approaches have been developed to describe the beam propagation for the vectorial nonparaxial regime from both theoretical and practical aspects [1-3]. Among those, the vector angular spectrum method, as an effective method to resolve the Maxwell’s equations, has received enormous theoretical attention and has been applied extensively during the past decades [4-17]. Within the frame of the vectorial theory, the general solution of the Maxwell’s equations can be written as a sum of two terms, i. e., TE term and TM term, in terms of the vector angular spectrum of electromagnetic field. In the far field, the TE and TM terms are orthogonal to each other and can be detached. The isolated TE term may be used to improve the density of optical storage [11]. Based on the vector plane wave spectrum, the vectorial structure of a large number of light beams including Gaussian beams, Hermite- Gaussian beams, Bessel-Gaussian beams and so on are widely investigated. For example, WU et al [10] analytically studied the vectorial structure of a hollow Gaussian beam in the far-field by means of the vector plane wave spectrum and the method of stationary phase. On obtaining the analytical vectorial structure of a hollow Gaussian beam, they derived the energy flux distributions of the hollow Gaussian beam [10]. Besides, ZHU et al [14-15] and TANG et al [16] investigated the far field vectorial structure of a helical hollow Gaussian beam and apertured Gaussian beam making use of the vectorial structure theory. They found another potential application of the far field vectorial structure in information encoding and transmission for free-space communications in addition to re-focusing to enhance the optical storage density based on the obtained results. Moreover, this encoding scheme has the benefit of easy implementation without modulating any properties of the light source.
Hermite–sinusoidal-Gaussian beams are exact solutions of the paraxial wave equation for propagation in free space and complex optical systems. Hermite– sinusoidal-Gaussian beams represent more general beams, and the cosine-Gaussian beams (cGB), cosh-Gaussian beams, Gaussian beams and Hermite– Gaussian beams could be regarded as their special cases. These beams are used in applications for which efficient extraction of energy is required [17-19]. The propagation properties of Hermite–cosine-Gaussian beams through a paraxial optical ABCD system with a hard-edge aperture have been examined. Also, the propagation characteristics of a cGB in turbulent atmosphere have been studied. The analytical expression characterizing the propagation of non-paraxial truncated cGB beams in free space has been derived, and the corresponding beam quality in the far field has also been investigated [20]. In this work, with the help of the vectorial angular spectrum representation, the vectorial structure of a cGB is investigated in detail. Although ZHOU et al investigated the similar problem associated with cosh-Gaussian beam energy flux densities in detail, they did not demonstrate the energy fluxes associated with the TE, TM term and the whole far field never revealed the almost pure TM or TE component character exhibiting in the cosh-Gaussian beam in the far field [12]. By means of the method of stationary phase, the analytical expressions of the TE and TM terms are derived for a linearly polarized cGB in the far field. Numerical results of the energy flux densities of the field components and the whole field are analyzed. To the best of our knowledge, it is firstly found that the far field may be entirely transverse magnetic or electric under appropriate polarization angle and beam parameter.
2 Analytical expression for linearly polarized cGB
Considering a Cartesian coordinate system (x,y,z) and assuming a linearly polarized cGB propagating toward the half space z≥0, the initial transverse electric fields of the cGB at the boundary plane z=0 read as
(1)
where w0 is the beam waist width, Jones vector 
represents the linearly polarized state, α is the polarized angle with respect to the x-axis, and β is a beam parameter related to the cosine part. If β=0, the model beam reduces to a linearly polarized Gaussian beam. Moreover, the beam can be also generated through superposition of two decentered Gaussian beams or a Gaussian beam through a cosine amplitude grating.
Although the similar treatments have been done in Ref. [12], here, we still give main steps to be complete. By using the vector plane wave spectrum representation, the electric field of vectorial cGB propagating toward half free space z≥0 can be obtained as [10, 12]
(2)
where 
γ=(1-p2-q2)1/2, k=2π/λ, λ is the wavelength in medium, Ax=Ax(p,q) and Ay=Ay(p,q) are the transverse angular spectra.
According to the boundary condition of initial transverse electric field, the transverse angular spectra read as
(3)
where
(4)
Based on the vectorial structure theory of non-paraxial electromagnetic beam, the propagating electric field can be written as a sum of two terms, that is
(5)
where
(6)
(7)
Similarly, the propagating magnetic field of vectorial beams can also be divided into the corresponding TE and TM terms:
(8)
where
(9)
(10)
In the far field regime, the method of stationary phase is applicable [10]. Therefore, we can obtain the analytical far field as
(11)
(12)
(13)
(14)
where 
and the radial and azimuthal unit vectors are defined as 
and 
respectively.
3 Energy flux distributions in far field
The energy flux distributions at the constant z plane are given by the z component of their time average Poynting vector:
(15)
(16)
where the angle brackets indicate a time average and the asterisk * denotes complex conjugation.
Making use of Eqs. (15) and (16), the numerical evaluations are done and it is found that, for linearly polarized cGBs, the far field patterns of STE, z and STM, z may be two-lobe, two- or four-spot structures depending on the polarization angle α and the beam parameter β. And the total energy flux density Sz=STE, z+STM, z is always a one- or two-spot configuration determined by the parameter β, which can be seen from Fig. 1. For a sufficiently small β, these energy flux density patterns of TE- and TM-term are a two-lobe or eight-figure structure whose originations are mutual perpendicular, and the pattern of Sz is approximately a spot which resemblances to that of a Gaussian beam. But for an appropriately large β, energy flux density patterns of TE- and TM-term are a two- or four-spot structure and the pattern of Sz is always a two-spot configuration. The larger the β parameter is, the farther the spots separate. However, the two- or four-spot occurrence of TE- and TM-term also depends on α for given large β. For example, for α=0, the pattern of STE, z consists of four spots and that of STM, z consists of two spots but the spot configurations just completely exchange for α=π/2, as shown in Fig. 2. More discussion of the dependence of these patterns on α is given out in Fig. 3. From these plots, we can see that, for the other polarization direction, the patterns of STE, z and STM, z are twisted and the spot sizes change with α variation. In special, when α=π/4, the twisted spot sizes associated TE and TM terms are alike for an incident cGB.
In addition, because all patterns shown in Figs. 1 and 2 are uniformly normalized to the maximum value of Sz, we can see that the energy flux density of STM, z in the far field regime is much larger than that of STE, z when β is sufficiently large. However, as a Gaussian beam behaves, for a cGB, the maximum values of the far field patterns of STM, z and STE,z are roughly the same for α=π/4.
4 Energy fluxes in far field
To accurately measure the energy fluxes associated TE and TM terms, the quantities are calculated:
(X=TE or TM) (17)
It represents the ratios of the energy fluxes associated X (X=TE or TM) terms in the whole beam. After some mathematical manipulations, we obtain
(18)
(19)
where the cofactor 
has been omitted.
Fig. 1 Energy flux densities of STE, z, STM, z and Sz of vectorial cGBs with z=15 000λ, α=0, w0=15λ and β=1 (a) and β=8 (b)
Fig. 2 Energy flux densities of STE, z, STM, z and Sz of vectorial cGBs with z=15 000λ, w0=15λ, β=8, α=π/4 (a) and α=π/2 (b)
Fig. 3 Energy flux densities STE, z (a) and STM, z (b) of cGB at z=5 000λ, w0=15λ, β=3 and α=0, π/9, π/4, π/2 (from left to right)
The calculation results indicate that the deviation of PTM from 0.5 only depends on β almost irrespective of the beam waist width w0. For example, for α=0 and w0 being fixed, PTM (>0.5) increases with an increase of β. When β is sufficiently large, PTM approaches unity, as shown in Fig. 4(a). However, the variation of PTM is just contrary when α=π/2 and now PTE=1-PTM is close to unity. In other words, in far field, an almost pure TM or TE term can be achieved depending on the linear polarization direction of the incident cGB for sufficiently large β. In fact, it was reported that the far field is entirely transverse magnetic for a radially polarized Laguerre-Gaussian beam [13]. Therefore, the result obtained in this work is very interesting and, to the best of our knowledge, has not been reported elsewhere. This scheme has the benefit of easy implementation because an almost pure TM or TE mode field can be obtained accordingly by properly adjusting the polarization direction of the incident cGB for a sufficiently large β.
From Eqs. (18) and (19), we can see that the dependence of the energy fluxes of TE and TM terms on the polarization angle α is periodical. The numerical results shown in Fig. 4(b) demonstrate that, like a circular GB (β=0) propagation in free space, PTE=PTM= 0.5 is always true for a general cGB when α=π/4 or α=3π/4 (omitted), at which the corresponding density patterns shown in Figs. 2 and 3 also seem alike. Hence, the fact of PTE=PTM=0.5 at these angles also implies that, from Eqs. (18) and (19), z2/r5≈1/r3 (namely r~z) holds in the far field as well as the total energy flux is practically independent of α. In addition, for other angle α, PTM always approaches a specified value rather than 1 or 0 for sufficiently large β, which can be also seen from Fig. 4(b).
Fig. 4 Variations of PTM with β for z=5 000 λ: (a) α=0; (b) w0/λ=10
5 Conclusions
1) The energy flux expressions associated the TE, TM component and the whole field are analytically derived, and their dependences on the polarization direction are investigated.
2) With the help of the numerical calculations, it is found that, in the far field regime, the TM and TE component fields always hold equal shares for a general cGB with the polarization angles of α=π/4 and α=3π/4. And the far field may quite well approach an almost pure TM or TE field depending on the polarized direction and the beam parameter of the incident cGB.
3) In summary, the discussion is helpful to understand the theoretical aspects of cGBs and reveals that the far field may be the almost entirely transverse magnetic or transverse electric field, which further enriches the cognition of the cGB. And based on the fact that the energy fluxes of TE and TM components can be controlled by varying the beam parameter, it is believed that such beam characters can be applied to information encoding and transmission in free space, which deserve to be investigated further in future.
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(Edited by HE Yun-bin)
Received date: 2011-12-24; Accepted date: 2012-04-24
Corresponding author: ZHU Kai-cheng, Professor; Tel: +86-731-88876924; E-mail: kczhu058@csu.edu.cn
