## Abstract

We discuss a method to produce a radially polarized laser beam with a tunable on-axis Gouy phase total variation different from the standard value of 2*π*. It is shown that structuring the derivative of the field near the optical axis of a single-ring-shaped illumination focused by an aplanetic lens or a parabolic mirror, one could obtain any value between 3*π*/2 and large multiples of *π*. Our results might prove useful for experiments involving the field pattern of a TM_{01} beam that requires specific phase matching conditions.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Radially polarized laser beams (RPLBs), a subclass of cylindrical vector beams [1], have recently gained a significant amount of interest due to their particular focusing properties. Tightly focused RPLBs are known to develop a dominant on-axis longitudinal electric field and to exhibit a smaller spot size than their linearly polarized counterparts [2,3]. They have been used in various applications such as optical trapping [4,5] and high-resolution microscopy [6]. In some applications, such as direct electron acceleration in vacuum [7–10] and second-harmonic generation [11–14], phase matching conditions require an understanding of the phase carrier behavior of the longitudinal electric field, rendering essential the knowledge of the on-axis Gouy phase for strongly focused RPLBs.

The Gouy phase is the well-known phase shift experienced by any focused wave around focus [15] and its physical interpretation is still under discussions nowadays [16]. Analysis of the Gouy phase for the strongly focused RPLB of first order (or TM$_{01}$) has been carried out both numerically [17–20], using Richards-Wolf vector integrals [21] and experimentally [22], using terahertz microscopic imaging. It has been found that when diffraction effects with the edges of the optics are negligible and that only the on-axis behavior is considered, the total variation of the Gouy phase for the longitudinal electric field is $2\pi$ [20], in agreement with theoretical predictions [23,24]. This is true for an illumination of the form $\ell _0(r') = (r'/w_0)\exp (-r'^2/w_0^2)$ in the entrance pupil of the optical system, where $r'$ is the radial coordinate and $w_0$ is the beam waist. Obviously, since the slope of the Gouy phase around focus is more abrupt for more strongly focused beam [18], one could use the focusing level to adjust the Gouy phase variation, decreasing its value over a limited zone around focus. Also, one could manifestly use higher order TM$_{n1}$ modes to increase the Gouy phase variation according to the integer order $n$ [25,26]. There is however, as of now, no known method to adjust the *total* variation of the on-axis Gouy phase of a *single-ring-shaped* RPLB, an interesting feature for experiments that cannot be limited to a small zone around focus. In this paper, we propose to exploit the fact that the order of the axial discontinuity of the pupil function in Richards-Wolf integrals directly influences the Gouy phase [27]. By appropriately modeling the incident illumination, as shown in the next section, we obtain an on-axis Gouy phase variation that can take any value larger than $3\pi /2$, while preserving the interesting features of a strongly focused TM$_{01}$ beam.

## 2. Tuning of the Gouy phase in the strongly focused limit

We propose to use in Richards-Wolf integrals an incident illumination of the form

for $n>0$ to ensure a solution with finite energy. In general, $n$ needs not be an integer and only affects the derivative of the central hole (or the degeneracy of the zero) of the single-ring-shaped illumination, as can be seen in Fig. 1(a). Such an illumination pattern could, in principle, be created in the entrance pupil of an optical system using a combination of a semi-transparent plate with variable transmission to shape the profile and a polarization converter to obtain radial polarization. Note that since the proposed illumination is not a fundamental beam mode, imaging systems would be necessary to transport the illumination profile up to the desired entrance plane without distortion. Following the formalism of [28], the complex spatial part of the electric field of an RPLB is obtained fromNote that $ka$ is an adimensional parameter defined by the wavevector $k$ multiplied by $a = 2f^2/kw_0^2$, where $f$ is the focal length of the system [30]. It describes the focusing level, such that $ka \gg 1$ is the paraxial limit, while $ka\leq 10$ corresponds to the turning point from a radial to a longitudinal dominant electric field component for the TM$_{01}$ beam and can be considered as the beginning of the strongly focused limit [30,31]. Evaluating numerically the longitudinal electric field using (2) with the illumination of (1) for an ideal parabolic mirror ($\alpha _{\textrm {sa}} = \pi$), with $ka=10$ and extracting the on-axis Gouy phase using [32],

we obtain the results showcased in Fig. 1(b) for different values of $n$. The total variation of the Gouy phase, defined by $\Delta \phi _G = \lim _{z \rightarrow \infty } \phi _{G} - \lim _{z \rightarrow -\infty } \phi _{G}$, clearly evolves with the value of $n$. Repeating the same process for more values of $n$, including non-integers, Fig. 1(c) is obtained, where one can clearly extract the linear relation with $n=1$ giving the expected value of $2\pi$ for the total on-axis variation of the Gouy phase for the TM$_{01}$ beam. The total variation of the Gouy phase in this model is thus limited by $\Delta \phi _{G_z}(n) > 3\pi /2$, where this minimal value is excluded since $n=0$ is not consistent with a radial polarization. Consequently, changing the slope of the central hole of the illumination (directly related to the power $n$ as can be seen on Fig. 1(a)) allows to tune the total variation of the on-axis Gouy phase according to Eq. (4).It should be noted that our particular choice of focusing level does not influence the result for the total variation of the Gouy phase and that the paraxial and the more strongly focused limits are described by the same behavior. In general, a smaller value of $ka$ only implies a steeper variation around focus and vice versa. In some limiting cases, when edges effects of the optics aperture become non-negligible, oscillations might appear near the focal region but they do not influence the asymptotic behavior. Accordingly, other semi-aperture angles (or other axisymetric focusing systems) could be used as long as $ka$ is large enough to avoid important diffraction effects [20]. Even though there is a noticeable shift of the illumination function’s first moment in Fig. 1(a), it is *not* the cause for the change in the Gouy phase, since the variation of $ka$ also modifies the first moment without affecting the total Gouy phase variation. The modification of the Gouy phase variation with power $n$ can thus only be related to the degeneracy of the on-axis zero of Eq. (1). This means that the Gouy phase is determined in the *paraxial* regime, in the sense that it is determined by the near axis behavior. It explains why beams with different levels of focusing, which differ by their illumination further from the optical axis, give the same total variation of the Gouy phase.

In terms of intensity pattern, one can notice that the single-ring shape is essentially preserved in the paraxial limit (see Fig. 2(a,d,g)) for different values of $n$. In general, we observe that increasing $n$ is favoring the longitudinal electric field component versus the radial component (compare Fig. 2(b) and (h) for example). This is expected since increasing $n$ increases the width of the central hole (and thus broadens the whole illumination), creating a stronger effective focusing. Figure 2(c,f,i) shows that the strongly focused intensity profile is stable over a large range of values of $n$. Also, notice that Fig. 2(a-c) portrays the intensity for a non-integer value of $n$ and that its behavior does not differ from that obtained from integer powers.

## 3. Tuning of the Gouy phase in the paraxial limit

To get a better grasp of the numerical solution obtained in Figs. 1 and 2, we solve analytically the integrals of Eq. (2) in the paraxial limit, which is valid for $\alpha \ll 1$ and $ka \gg 1$. Using the small angle approximation (keeping only the first order in the series expansion), and the illumination defined by Eq. (1) and Table 1, we obtain for *both* the parabolic mirror and the aplanetic lens

On-axis ($\rho =0$), both the Gaussian and hypergeometric function contributions in Eq. (8) are unity, and the Gouy phase is solely determined by the following term

As a final note on the paraxial solutions (Eqs. (12) and (14)), let us mention that they also correspond to the paraxial limit of the solutions of Ref. [31] (for $n=2m+1$), obtained by solving exactly Maxwell’s equations for TM and TE beams using a generalized eLG beam as a starting point. We can verify numerically (not shown) that even in the strongly focused limit, the integral of Eq. (2) (with the illumination given by Eq. (1)) and the solutions of Ref. [31] give a very similar description of the RPLBs as already discussed in Ref. [30] for the particular case of the TM$_{01}$ beam. However, the intrinsic discrete nature of those solutions prevents us from using them to smoothly tune the Gouy phase, as it is possible in the present formalism.

## 4. Influence of the field envelope on the Gouy phase variation

Although changing the shape of the incident illumination seems a reasonable way to tune the Gouy phase, it is not a trivial concept as we show next. Using $n=1$ in the illumination function of Eq. (1), we now modify the field envelope from a Gaussian to a supergaussian such as

*do not*influence the total variation of the Gouy phase. Only the central slope of the illumination, determined by the parameter $n$, allows the tuning of the total variation of the Gouy phase in accordance to Eq. (4). This is encouraging for the experimental production of an RPLB with a tailored Gouy phase, since only the near axis derivative must be taken care of.

## 5. Conclusion

In conclusion, we have shown that shaping the central derivative of a single-ring illumination can be used to tune the total variation of the Gouy phase of an RPLB. Using this shaping, the intensity of the strongly focused limit behaves essentially as the TM$_{01}$ beam’s intensity while the total Gouy phase variation can be adjusted for any non-integer multiple of $\pi$ starting from $3/2$. This method could prove useful in applications requiring a suitable control on the phase carrier of the longitudinal electric field of a strongly focused RBLB. It also reveals that some experimental care must be taken to create the expected TM$_{01}$ beam since the Gouy phase is very sensitive to the slope of the central hole of the incident illumination. We thus foresee that methods to produce RPLBs with a sharp radial variation near the optical axis, such as combining rotated half-wavelength plates [38], might produce a different Gouy phase variation than the one expected.

## Funding

Natural Sciences and Engineering Research Council of Canada (05753-2015).

## Acknowledgments

The authors acknowledge support through scholarships from NSERC and the Fonds de recherche du Québec - Nature et technologies (FRQNT).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

No data were generated or analyzed in the presented research.

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