Tuning of the Gouy phase variation for radially polarized laser beams

Shanny Pelchat-Voyer; Michel Piché

doi:10.1364/OSAC.418380

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

(1)$$\ell_0(r') = (r'/w_0)^n\exp({-}r'^2/w_0^2),$$

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 from

(2)$$\begin{aligned} \begin{bmatrix} \tilde{E}_r \\ \tilde{E}_z \end{bmatrix}= \int_{0}^{\alpha_{\textrm{sa}}} q(\alpha) \ell_0(\alpha) \begin{bmatrix} i \cos \alpha\sin\alpha \ J_1(kr\sin\alpha) \\ \sin^2 \hspace{-0.7mm} \alpha \ J_0(kr\sin\alpha) \end{bmatrix} \exp \left({-}ikz \cos\alpha\right) \ d\alpha , \end{aligned}$$

where $J_n(\cdot )$ is the Bessel function of the first kind of order $n$, $\alpha$ is the angular variable comprised between $0$ and $\alpha _{\textrm {sa}}$ (the semi-aperture angle of the focusing optics), $z$ and $r$ are respectively the longitudinal and radial coordinates of the focused beam and $k$ is the wave vector. The apodization function $q(\alpha )$ ensures the energy conservation for a given focusing system; its expression is given in Table 1 for a parabolic mirror and an aplanetic lens, along with the relation between the ratio $r'/w_0$ and the angular variable $\alpha$ [29], used to properly express the illumination (1) according to the focusing geometry.

Fig. 1. (a) Normalized incident illumination described by (1) for different values of $n$. (b) On-axis Gouy phase $\phi _{G_z}$ obtained for different values of $n$ using $ka=10$ and $\alpha _{\textrm {sa}}=\pi$ for a parabolic mirror. (c) Total variation of the on-axis Gouy phase $\phi _{G_z}$ as a function of $n$. The vertical line indicates the result for the standard TM$_{01}$ beam.

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Table 1. Apodization function and geometrical factor for a parabolic mirror and an aplanetic lens

View Table

Note 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],

(3)$$\phi_{G_z} = \textrm{arg} \left[ \tilde{E}_z \right] + kz$$

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

(4)$$\Delta \phi_{G_z}(n) = \frac{(n+3)\pi}{2},$$

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.

Fig. 2. Intensity in the focal plane of the radial component (orange), the longitudinal component (blue) and the their total contribution (black) for various levels of focusing and various powers $n$: (a-c) $n=1/2$, (d-f) $n=1$, which corresponds to the TM$_{01}$ beam, (g-i) $n=5$. Each curve is normalized according to the maximum total intensity for the given values of $ka$ and $n$.

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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

(5)$$q(\alpha) \simeq 1, \qquad \ell_0(\alpha) \simeq \left( \frac{ka}{2} \right)^{n/2} \alpha^n \exp \left\{ - \frac{ka}{2} \alpha^2 \right\}$$

which means that the integral to solve for the longitudinal component is of the form

(6)$$\tilde{E}_{z}\simeq \frac{E_0 e^{{-}ikz}}{2}\int_{0}^{\infty} \alpha^{n+2} J_0(kr\alpha) \exp \left\{ - \left( \frac{1}{\Delta\alpha^2} - \frac{ikz}{2} \right) \alpha^2 \right\} d\alpha,$$

where we used $\Delta \alpha ^2 = 2/ka$ to define the incident angular width and where $\alpha _{\textrm {sa}} \rightarrow \infty$ has been used for the upper bound of the integral since the aperture does not impact the paraxial regime. To solve this integral, we use the following identity from Ref. [33],

(7)$$\int_{0}^{\infty} x^\mu e^{{-}ax^2} J_\nu (\beta x) dx = \frac{\beta^\nu \Gamma\left(\frac{\nu}{2} + \frac{\mu}{2} + \frac{1}{2}\right) }{2^{\nu+1}a^{(\mu + \nu + 1)/2} \ \Gamma(\nu+1) } \hspace{1mm} {}_1F_1 \left[ \frac{\nu+\mu+1}{2}; \nu+1; - \frac{\beta^2}{4a} \right],$$

which is valid for $\textrm {Re} \{ a\} > 0$ and $\textrm {Re} \{ \mu + \nu \} > -1$ and where $\Gamma (\cdot )$ is the gamma function and ${}_1F_1 (\cdot )$ is the Kummer confluent hypergeometric function. In the paraxial limit, the longitudinal electric field thus has the following form

(8)$$\begin{aligned} \tilde{E}_{z} &\simeq \frac{E_{0,n} e^{{-}ikz} }{ \left( 1 - i\zeta \right)^{(n+3)/2}} \ {}_1F_1 \left[ \frac{n+3}{2}; 1; - \frac{\rho^2}{ \Delta\alpha^2 \left( 1- i\zeta \right)} \right]\\ & \simeq \frac{E_{0,n} e^{{-}ikz} }{ \left( 1 - i\zeta \right)^{(n+3)/2}} \ \exp \left\{ - \frac{\rho^2}{ \Delta\alpha^2 \left( 1- i\zeta \right)} \right\} \hspace{1mm} {}_1F_1 \left[ -\frac{n+1}{2}; 1; \frac{\rho^2}{ \Delta\alpha^2 \left( 1- i\zeta \right)} \right]\end{aligned}$$

with $\zeta = kz\Delta \alpha ^2/2$, $\rho = kr \Delta \alpha ^2/2$ and $E_{0,n} = E_0\Delta \alpha ^{n+3} \ \Gamma \left ( \frac {n+3}{2} \right )/4$ and where we used the following identity of the confluent hypergeometric function

(9)$${}_1F_1 (a; b; -x) = e^{{-}x} {}_1F_1(b-a; b;x)$$

to make the Gaussian dependence explicit. The solution described by Eq. (8) is the same obtained from the fractionalization of the scalar Laguerre-Gaussian paraxial beams in Ref. [34] for an angular index of zero and a radial index corresponding to $(n+1)/2$ (keeping in mind that $ka \sim kz_R$ in the paraxial limit [30,31], where $z_R$ is the Rayleigh length). As such, the illumination specified by Eq. (1) provides a pathway to generate a radially polarized version of the fractionnalized scalar solution of Ref. [34] in a context where a strongly focused generalization can be produced. Even though it has not been made clear in the literature, this is also the same solution as the Hypergeometric-Gaussian beam of type II (HyGG-II) for integer indexes [35,36], a member of the family of hypergeometric paraxial beams [37].

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

(10)$$\begin{aligned} e^{{-}ikz}\left( 1 - i\zeta \right)^{-(n+3)/2} &= e^{{-}ikz} \left[\sqrt{1+\zeta^2} \ e^{{-}i\tan^{{-}1} \zeta} \right]^{-(n+3)/2}\\ &= \frac{\exp \left[ i \left(- kz + \frac{(n+3)}{2} \tan^{{-}1} \zeta \right) \right] }{(1+\zeta^2)^{(n+3)/4}}. \end{aligned}$$

which means that the Gouy phase, obtained using Eq. (3), and the total Gouy phase variation are respectively given by

(11)$$\phi_{G_z} = \frac{(n+3)}{2}\tan^{{-}1}\zeta, \qquad \Delta \phi_{G_z} = \frac{(n+3)\pi}{2}$$

as expected from numerical computations. As we can see from (7), the on-axis Gouy phase is tied to the power $(\mu + \nu +1)/2$. Hence the power of $\alpha$ and the order of the Bessel function are the two contributing factors to the Gouy phase variation. As mentioned by Ref. [34], these solution are smoothly connected to elegant Laguerre-Gauss (eLG) solutions. Thereby, Eq. (8) can be written as

(12)$$\tilde{E}_{z} \simeq \frac{ E_{0,n} e^{ {-}ikz}}{ \left( 1 - i\zeta \right)^{(n+3)/2}} \exp \left\{ - \frac{\rho^2 }{ \Delta\alpha^2 \left( 1- i\zeta \right)} \right\} L_{\frac{n+1}{2}}^{0} \left[ \frac{\rho^2 }{ \Delta\alpha^2 \left( 1 -i\zeta \right)} \right].$$

where the following identity [33]

(13)$$L_\gamma^\sigma (x) = \frac{(\sigma+\gamma)!}{\sigma!\gamma!} \ {}_1F_1 (-\gamma;\sigma+1;x),$$

was used. In this case, the solution of Eq. (12) is proportional to a polynomial only for $n=2m+1$, for $m$ a positive integer. For other values of $n$, only the integral representation of the Laguerre polynomial allows a mathematical description. Likewise, applying the paraxial approximation to Eq. (2) for the radial component, the following result is obtained,

(14)$$\tilde{E}_{r} \simeq \frac{ i E_{0,n} e^{ {-}ikz} kr}{ (n+1) \left( 1 - i\zeta \right)^{(n+3)/2}} \exp \left\{ - \frac{\rho^2 }{ \Delta\alpha^2 \left( 1- i\zeta \right)} \right\} L_{\frac{n-1}{2}}^{1} \left[ \frac{\rho^2 }{ \Delta\alpha^2 \left( 1 -i\zeta \right)} \right].$$

where the on-axis Gouy phase is undefined since the field is zero for $r=0$. Nonetheless, the near axis Gouy phase is described by the same factor $(1-i\zeta )^{-(n+3)/2}$, since the order of the Bessel function is increased by one while the power of $\alpha$ is decreased by the same amount in the integral for the radial component compared to the longitudinal component. For a given level of focusing, the normalization constants of $\tilde {E}_z$ and $\tilde {E}_r$ differ by a $1/(n+1)$ factor, in agreement with the observed increasing ratio $|\tilde {E}_z|^2_{\textrm {max}}/|\tilde {E}_r|^2_{\textrm {max}}$ for higher values of $n$ in Fig. 2. The consistency of the paraxial solutions given by Eqs. (12) and (14) with the numerical solution of Eq. (2) in the paraxial regime is shown in Fig. 3 for $n=1/2$ and $n=8$. A small deviation from the numerical computation is observed for $n=8$, which increases in importance for higher values of $n$. As mentioned before, this is due to the increase in beam width, which implies that the first-order paraxial solution no longer captures the entire field profile for very large values of $n$.

Fig. 3. Normalized intensity in the focal plane of the radial and longitudinal field components obtained numerically from the Richards and Wolf integrals (Eq. (2)) for $ka=200$ and $\alpha _{\textrm {sa}}=\pi$ (solid lines) and from Eqs. (12) and (14) with $\Delta \alpha = 0.1$ (dotted lines). (a) For $n=1/2$, (b) for $n=8$.

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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

(15)$$\ell_0(r') = \frac{r'}{w_0} \exp \left( - \left| \frac{r'}{w_0} \right|^{2b} \right) \ \ \textrm{with} \ \ b = 1, \ 1.5, \ 2, \ 2.5, \ \cdots$$

and to a hyperbolic secant raised to power $p$ such as

(16)$$\ell_0(r') = \frac{r'}{w_0} \textrm{sech} \left( r'/w_0 \right)^p, \ \ \textrm{with} \ \ p = 1, \ 2, \ 3, \ \cdots$$

to investigate its effect on the on-axis Gouy phase variation. The results are presented in Fig. 4 where the stability of the Gouy phase over different values of $b$ and $p$ can be observed. Consequently, even though the field envelope as well as the parameter $ka$ influence the illumination pattern, they 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.

Fig. 4. Normalized incident illumination described by Eq. (15) in (a) and by Eq. (16) in (c) for different values of $b$ and $p$ with their respective on-axis Gouy phase shift $\phi _{G_z}$ in (b) and (d) for $ka=10$ and a parabolic mirror with $\alpha _{\textrm {sa}}=\pi$. The case $b=1$ (the TM$_{01}$ beam) is also depicted in (c-d) for comparison.

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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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	Parabolic mirror	Aplanetic lens
$q (α)$	$\sec^{2} (α / 2)$	$\cos^{1 / 2} α$
$(r^{'} / w_{0})^{n}$	$(2 k a)^{n / 2} \tan^{n} (α / 2)$	$(\frac{k a}{2})^{n / 2} \sin^{n} α$

Tuning of the Gouy phase variation for radially polarized laser beams

Abstract

1. Introduction

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

3. Tuning of the Gouy phase in the paraxial limit

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

5. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Tables (1)

Equations (16)

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