1. Introduction

Over the past decades, considerable attention has been paid to the phenomenon of focal shift, in which the maximum on-axis intensity for a monochromatic converging wave does not occur exactly at the geometric focus, but it is slightly displaced towards the aperture plane [1, 2, 3, 4, 5, 6, 7, 8, 9]. It has been shown by different authors that this shift depends on the Fresnel number of the Gaussian apodization, for unapertured beams, and on the effective Fresnel number associated with the aperture, for apertured systems [1, 5, 7].

In imaging applications where the Fresnel number is usually large, the focal shift is negligible, however in other physical systems with Fresnel numbers closer to unity, or even smaller, the effect cannot be neglected. This phenomenon has been extensively studied in the context of coherent scalar beams, for instance it has been shown to exist in systems with an obscured pupil [6, 10], in Laguerre-Gaussian beams [11], in unapertured Bessel-Gauss beams [12] and apertured Bessel beams [13]. More recently, Hricha et al. [14] and Dong and Pu [15] have studied this effect in partially coherent beams. However, to our knowledge very few references have addressed this effect for vector beams [16, 17].

On the other hand, Mathieu beams, which constitute a family of nondiffracting optical beams [18], have been employed in different areas of optics, for instance in photonic lattices to study soliton behavior in periodic media [19], in optical tweezers for the transfer of orbital angular momentum [20], and in the study of localized X waves [21, 22]. In this paper, we study the focal shift in vector Mathieu-Gauss beams (vMG), which constitute a particular family of the more general vector Helmholtz-Gauss beams (vHzG) [23]. We start our discussion by introducing the vMG beams and setting up the mathematical formulae. Next, we discuss about the criteria for defining the focal distance of a vMG beam, and confirm the existence of a focal shift for these beams based on this criteria. The numerical results are presented and the dependence of the focal shift on the beam parameters is discussed in detail.

2. Vector Mathieu-Gauss beams

Vector Mathieu-Gauss (vMG) beams constitute a particular family of the more general vector Helmholtz-Gauss beams, introduced by Bandres and Gutiérrez-Vega [23], and correspond to localized solutions of the paraxial wave equation in elliptical coordinates. These solutions represent a monochromatic field propagating along the positive z axis, and have a transverse distribution whose amplitude is modulated by a Gaussian factor. Recently we have explored the propagation dynamics of focused vector Mathieu-Gauss beams in free space [24], and through paraxial ABCD systems [25], giving closed form expressions for their evaluation. Chafiq et al. [26] have also studied the propagation properties of vMG beams.

2.1. General description of a vector Mathieu-Gauss beam

Consider a monochromatic electromagnetic field propagating along the positive z axis. The electric and magnetic components of this field can be expressed in terms of their transverse and longitudinal components as E=(E _t+ẑ E_z)exp(ikz) and H=(H _t+ẑ H_z)exp(ikz) respectively, where $k=\omega {\left({\mu }_0{\epsilon }_0\right)}^{\frac{1}{2}}=\sqrt{k_t^2+k_z^2}$ is the wave number, and the subscripts t and z refer to transverse and longitudinal components. Throughout the rest of the paper a time dependence factor exp(-iωt) will be assumed, and it will be omitted from the equations for the sake of simplicity. For a vHzG beam the transverse component of the electric field E _t is of the form [23]

where $r=\sqrt{x^2+y^2}$ is the usual radial variable in polar coordinates, (X,Y)=(x/ζ,y/ζ) are scaled Cartesian coordinates, and ζ(z)=1+iz/z_R, with z_R being the Rayleigh distance. The function Z(ζ)=exp[k ² _t w ² ₀(ζ ^-1-1)/4] depends only on the propagation coordinate, with w ₀ being the Gaussian beam waist at plane z=0. The factor G(r,ζ) represents a Gaussian amplitude modulation given by

Finally, the function U is known to be a solution to the vector Helmholtz equation in 2D, and following Stratton [23, 27], it represents two independent solutions corresponding to the TM and TE polarizations. These solutions are of the form:

It is important to mention that the gradient ∇_T is taken over the transverse scaled coordinates. Once the electric field in Eq. (1) is determined, its corresponding magnetic field can be computed as H _t=(ε ₀/µ ₀)^1/2 ẑ×E _t, which is a direct consequence of the transversality of the fields from Maxwell’s equations.

Inserting the solution U ⁽¹⁾ from Eq. (3) into Eq. (1) we obtain the following expressions for the electric and magnetic fields of the TM polarization

A similar procedure but now employing U ⁽²⁾ will give the expressions corresponding to TE polarization. Based on the Lax series expansion of Maxwell’s equations [28], E _t and H _t would correspond to the zeroth order terms, and for the next-order of correction the presence of small longitudinal field components is required. These longitudinal terms can be computed from the previous knowledge of E _t and H _t as follows

It is important to remark that these solutions hold for the paraxial approximation, where the z component of the field is small compared to the transverse component. The longitudinal components of vMG beams include a term of the order (k_t/k), and therefore to satisfy the paraxial condition it is required that k≫k_t, i.e. the transverse spatial oscillations are many wavelengths wide.

For vMG beams, the function W(X,Y) in Eq. (3) is a scalar solution to the 2D Helmholtz equation in elliptical coordinates, we shall refer to it as the seed function, and may be expressed in terms of the m-th order radial and angular Mathieu functions [18, 29]

where Je_m(·),Jo_m(·) represent even and odd radial Mathieu functions, and ce_m(·), se_m(·) correspond to the even and odd angular Mathieu functions respectively. The (e,o) superscripts stand for even an odd solutions, m is a non-negative integer indicating the order of the functions, and q is a free parameter called the ellipticity parameter. The scaled variables (ξ,η) in the elliptical coordinates system are related to the scaled Cartesian coordinates by the relation (X,Y)=(h coshξ cos η,h sinh ξ sin η), where h is the semifocal distance of the elliptical coordinates at the plane z=0, and is also related to the transverse component of the wave vector k_t, and to the ellipticity parameter q by h=2√q/k_t.

Since the transverse structure of vMG beams is primarily determined by the gradient of the function W, as can be seen from Eq. (3), we provide an expression for the gradient in the scaled elliptic coordinates

where the scaling factors are given by $h_{\xi }=h_{\eta }=h\sqrt{{\sinh }^2\xi +{\sin }^2\eta }$ . This gradient can also be expressed in Cartesian coordinates by an appropriate transformation

doing so is helpful for numerical purposes, since it directly provides expressions for the x̂ and ŷ components.

2.2. Focusing of vector Mathieu-Gauss beams

We now consider the propagation of a vMG beam through an unapertured thin lens of focal distance f. Figure 1 shows the system under consideration, with the lens located at plane z=0, which implies ζ(0)=1, and therefore the scaled coordinates (X,Y)=(x,y). Without loss of generality we can take a TM type vMG beam as the field incident on the lens, and applying the Huygens-Fresnel diffraction integral we obtain a closed form expression for the field amplitude, this analysis has been presented in a previous work [24], and it was shown that the propagating field is described by

 

Fig. 1. Focusing system under consideration. The lens is assumed to be thin and unapertured, f defines the lens focal distance, i.e. geometrical focus, z ₀ the actual focus, and Δz the focal shift.

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where the transverse gradient ∇̃_t is taken over the complex elliptic variables (ξ̃,η̃). The factor F is defined as

and the complex elliptic variables (ξ̃,η̃) are determined by the relations

The corresponding amplitude for the magnetic field is obtained by applying the relation H _t=(ε ₀/µ ₀)^1/2 ẑ×E _t. It should be mentioned that although the expressions for the electric and magnetic fields have been derived for a TM type vMG beam, the scalar factors to the left of the gradient operator in Eq. (13) remain unaltered for TE or any other state of polarization. We point out that the following analysis is done under the paraxial approximation, however it should be mentioned that for strongly focused beams, i.e. highly non-paraxial beams, the existence of large longitudinal components has been demonstrated [31, 32], and in that case they cannot be neglected.

2.3. Focal shift in vector Mathieu-Gauss beams

A common criteria employed to define the actual focus of a beam is to consider the point where the maximum on axis intensity occurs, and the difference between this point and the geometric focus corresponds to the focal shift. However, for vMG beams, and in general for beams with an on-axis null in their intensity profile, this criterion is not applicable. To deal with this situation we adopt two different approaches, the first one was also employed by Greene and Hall [16], and Lü and Peng [30], which defines the beam’s width as the radius r_w within which 80% of the beam’s power is encircled, defining the focal plane z=z ₀ as the plane where the width is minimum. The second approach is to compute the beam’s mean square radius r_m of the diffracted beam [9], and define the focal distance as the value z=z ₀ at which r_m achieves its minimum.

2.3.1. Encircled energy criterion

The power carried by the beam is calculated by means of the Poynting vector defined as 〈S〉=Re(E×H*)/2, which can be decomposed into transverse and longitudinal components as 〈S〉=〈s〉+〈s_z〉ẑ. The time average of the longitudinal component 〈s_z〉 gives the intensity distribution, whereas the total power carried by the beam along the propagation coordinate is calculated by integrating the transverse intensity over the whole plane (x,y), namely

For a vMG beam, the s_z component of the Poynting vector is proportional to the squared amplitude of the transverse electric field, this is

where E _t is defined in Eq. (13). We look for the smallest value of r_w satisfying the condition

the actual focus of the beam is the propagation distance z ₀ at which this value occurs, to find this value Eq. (18) has to be solved numerically. Once z ₀ is found, a straightforward calculation allows the determination of the focal shift Δz=f-z ₀ using this criterion.

2.3.2. Beam’s mean squared radius (MSR)

The computation of the beam’s mean squared radius, or which is equivalent its waist with respect to the radial coordinate, is accomplished through the calculation of the second order moment for the intensity distribution [9, 33],

where r is the usual radial coordinate, r_m is the mean squared radius of the beam, and m_r=∬r〈s_z〉dA its centroid, also known as the beam’s “center of mass”. The value of the centroid is always zero for a non-decentered distribution, which means that it is always located along the propagation axis. Similarly to the encircled energy criterion, the value z ₀ where r_m achieves its minimum will be defined as the focal distance, and the focal shift is thus calculated as the difference between the geometrical focus and this value, namely Δz=f-z ₀.

3. Numerical results

An important parameter of vMG beams is the ellipticity parameter q, when this parameter is equal to zero, vMG beams reduce to the special case of vector Bessel-Gauss beams studied in [16], however as q increases the elliptic features of the beam become more evident. The transverse intensity patterns and evolution of focusedvMG beams for different orders are shown in Fig. 2.

The top row images in Fig. 2 correspond to the initial profiles at plane z=0, whereas the bottom row consists of the initial snapshots from animations showing the evolution of the beam near the focal region. Note that we have used normalized units of length for all images. The videos show the three dimensional field structure in the focal region, a relevant feature is the on-axis null of all beams close to the geometrical focus. Unless stated otherwise, all following values for the simulation parameters remain unchanged for the rest of the paper, λ=632.8 nm, lens focal distance f=80 mm, waist of Gaussian envelope w ₀=1 mm, transverse wave number k_t=0.006 µm^-1, and ellipticity parameter q=5. Three different cases are presented, for instance all distributions in Fig. 2 were obtained using a TM polarized vMG beam, whose seed function is given by Eq. (7). Plots in Fig. 3(a) and (b) also correspond to a TM type solution, however the seed function in this case is known as an helical mode [29], and is given by a linear combination between Eqs. (7) and (8), namely W(ξ,η;q)=W^e_m(ξ,η;q)+iW^o_m(ξ,η;q), a distinctive feature of helical modes is that they carry orbital angular momentum (OAM) as they propagate [20], a behavior clearly observed in the corresponding videos of Fig. 3(a) and (b). Finally, in Fig. 3(c) and (d) we present distributions for a right circularly polarized beam, constructed from the superposition of TM and TE type solutions given in Eq. (3). It is a straightforward calculation to show that right (+) and left (-) circularly polarized solutions can be constructed as E^± _t=Z(ζ)G(r,ζ)[U ⁽¹⁾±i U ⁽²⁾].

 

Fig. 2. Transverse intensity profiles of a TM type focused vMG beam with seed function W=W^e_m(ξ,η;q), q=5 and beam orders (a)m=0, (b) m=1, (c) m=2, and (d) m=7. Plots in the top row correspond to initial profiles at plane z=0, bottom row images are snapshots for animations showing the three dimensional evolution in the focal region from z/f=0.95 to 1.05. (Movie files: (a) 1.68 MB, (b) 2.07 MB, (c) 1.94 MB, and (d) 2.31 MB). [Media 1][Media 2][Media 3][Media 4]

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Fig. 3. Transverse intensity profiles of a TM type focused vMG beam with seed function W=W^e_m(ξ,η;q)+iW^o_m(ξ,η;q), q=5 and beam orders (a) m=4, and (b) m=7. Plots in (c) and (d) correspond to a right-circularly polarized vMG beam with seed function W=W^e_m(ξ,η;q), q=5, and beam orders m=0 and m=7 respectively. Plots in the top row show initial profiles at plane z=0, bottom row images are snapshots for animations showing the three dimensional evolution in the focal region from z/f=0.95 to 1.05. (Movie files: (a) 2.12 MB, (b) 2.33 MB, (c) 1.65 MB, and (d).2.25 MB) [Media 5][Media 6][Media 7][Media 8]

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Propagation in the vicinity of the geometric focus goes from z/f=0.95 to 1.05. Apart from the change in the shape of the beam, and the appearance of an on-axis null, one can appreciate the existence of a plane at which the beam is narrowest, and a plane where it achieves its maximum intensity.

The existence of a focal shift in vMG beams was confirmed by means of the two approaches previously presented in Section 2.3. For instance, Fig. 4(a) shows the plots of the beam width according to the encircled energy criterion for different values of q, and mode orders m=0 through 7. It can be readily seen from these curves that the minimum width of the beam is achieved at a plane z<f, therefore a focal shift Δz towards the lens exists. For the special case when q=0, our findings are in agreement to those reported in [16], and we observe a decreasing value for Δz as the mode order is increased, however for larger values of q this behavior is lost, and some higher order modes may present larger shifts than lower order modes, but our results suggest that no specific dependence exist between the magnitude of the focal shift and q.

It is also observed that, in general, as the value of q increases the beam width decreases, i.e. the energy is confined to a smaller radius and the beam spreading is also reduced. Similar results were obtained using the second approach, which calculates the beam waist through the second order moment of the intensity with respect to the radial coordinate. These results are presented in Fig. 4(b), and again only for the special case q=0 the focal shift decreases with an increasing mode order. It is noteworthy that the magnitude of Δz calculated in Fig. 4(b) is larger than the one for Fig. 4(a), therefore depending on the specific feature of the beam in which we are interested, we may adopt either of the two criteria to calculate the focal shift.

The dependence of the shift on other beam parameters such as its polarization state and on the Gaussian-Fresnel number is of importance. Our simulations reveal the fact that the focal shift of a vMG beam is independent of its polarization state. This can be understood from the fact that the two criteria employed to define the actual focus of the beam are based on intensity. It is seen from Eqs. (1)–(4) that the intensities for a TM and TE vMG beams are equal in magnitude, thus the magnitude of a circularly polarizedvMG beam can be computed as |E ^± _t|²=|E ^TM _t|²+|E ^TE _t|², which indeed is equivalent to two times the magnitude of either polarization. The latter result is a consequence of the orthogonality between TM and TE polarized vMG beams.

The Gaussian Fresnel number of the beam is defined as

where f is the lens focal distance. Figure 5 shows the variation of the focal shift with respect to N_w, the simulation parameters are q=5, m=0, 1, and 7. The rest of the parameters are the same as in Fig. 2. A change in N_w was achieved by varying the waist of the Gaussian envelope w ₀ from 600 µm to 2000 µm. As expected, the focal shift of the beam for a given mode order approaches zero as N_w increases, in other words, for large values of w ₀ the Gaussian amplitude modulation tends to unity, it means that in free space we would have wider beams that do not spread significantly, however smaller values of w ₀ imply narrower beams, which spread rapidly on propagation. It should also be noted from these curves that higher order modes present smaller values of Δz, as we have mentioned in previous paragraphs, but the difference between the focal shifts for different orders is significantly reduced for large values of N_w.

 

Fig. 4. Beam waist plotted against the normalized propagation distance z/f, for q=[0,5,25] and mode orders m=0 through 7. (a) Calculation of beam waist using the encircled energy criteria and (b) using the mean squared radius of the intensity distribution. The minimum beam waist is achieved just before the geometric focus, i.e. z/f=1, indicating a focal shift Δz towards the lens.

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Fig. 5. Relative focal shift versus Gaussian-Fresnel number N_w of the beam, for q=5 and beam orders m=[0,1,7]. A change in N_w was achieved by changing the Gaussian waist w ₀ from 600 µm to 2000 µm. All simulation parameters are the same as in Fig. 2. Notice how narrower beams present larger shifts.

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4. Conclusions

In this paper we have studied the behavior of vMG beams when they are focused by an unapertured thin lens. The existence of a focal shift for vMG beams was confirmed by means of two different approaches to define the actual focus of a beam, the two are based on intensity measurements, one computes the encircled energy of the beam within a radius of interest, and the other is based on the calculation of the beam waist through the second order moment of the intensity distribution. The focal shift was found to be closer to the lens, slightly displaced from the geometric focus. The magnitude of this shift is highly dependent on the beam parameters and criterion used, for the encircled energy criterion typical values are a small percentage of the lens focal distance, in the range of 0.1% to 0.25%.

We also explored the dependence of this shift on other beam parameters such as its Gaussian Fresnel number N_w and polarization. Small values of N_w correspond to narrower beams which present larger focal shifts, in contrast to wider beams with large N_w that possess smaller focal shifts. Our findings also revealed that for vMG beams the focal shift is independent of its polarization state. The latter is a direct consequence of the orthogonality between the different solutions of the vector paraxial wave equation given by Eqs. (1) and (3).