Clarification for the fields of different radially polarized Laguerre–Gaussian light beams

Spencer W. Jolly; Miguel A. Porras; Miguel A. Porras

doi:10.1364/OL.464118

Introduction. The lowest order of the cylindrically symmetric vector beams [1] is the radially polarized (RP) beam. RP beams have important applications such as microscopy due to their sharper focus [2], manipulating and trapping macroscopic particles [3], and accelerating charged particles due to their large longitudinal field when sharply focused [4]. Just as with the Laguerre–Gaussian (LG) and elegant Laguerre–Gaussian (eLG) beams with linear polarization, RP beams can be extended to higher orders. In this work, we will clarify different solutions to the paraxial wave equation assuming radial polarization and involving Laguerre polynomials. These three solutions are radially polarized elegant Laguerre–Gaussian (RPeLG) beams, radially polarized standard Laguerre–Gaussian (RPsLG) beams, and radially polarized vortex-based Laguerre–Gaussian (RPvLG) beams.

We outline the framework used to derive the different RP LG beams, and follow that with a separate derivation for each beam. Then we compare their characteristics for both the transverse and longitudinal fields in the near-field and far-field.

Framework. Electromagnetic beams can be constructed using a vector potential $\mathbf {A}$ that has only one non-zero component along a certain Cartesian direction [5]. For $\mathbf {A}=A_0\psi \exp (i\eta )\mathbf {e}_i$ (with $\eta =\omega t-kz$, $\omega$ the frequency of the wave, $k=\omega /c$, and $\mathbf {e}_i$ a unit vector along a Cartesian direction), the wave equation for $\mathbf {A}$ produces, in the paraxial approximation ($|\partial \psi /\partial z|\ll k|\psi |$) and assuming cylindrical symmetry, the constraint

(1)$$\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial\psi}{\partial\rho}\right)-4i\frac{\partial\psi}{\partial\zeta}=0.$$

Here, the normalized radius and axial coordinate are $\rho =r/w_0$ and $\zeta =z/z_R$, where $w_0$ and $z_R=kw_0^2/2$ characterize the beam waist and the associated Rayleigh range, respectively. In the Lorenz gauge, the electric and magnetic fields can be evaluated from [5]

(2)$$\mathbf{E}={-}i\omega\mathbf{A}-\frac{i\omega}{k^2}\nabla(\nabla\cdot\mathbf{A}), \quad \mathbf{B}=\nabla\times\mathbf{A}.$$

The choice of $\mathbf {A}$ along a transversal Cartesian direction leads to a linearly polarized beam along that direction. Instead, for RP beams, $\mathbf {A}$ points in the axial direction $\mathbf {e}_z$ [6,7]. Equations (2) in cylindrical coordinates $(r,\theta,z)$ then yield the non-vanishing components of the electric and magnetic fields as

(3)$$\begin{aligned} E_r & ={-}A_0\frac{\omega}{kw_0}\frac{\partial\psi}{\partial\rho}e^{i\eta}, \end{aligned}$$

(4)$$\begin{aligned} E_{z} & ={-}A_0\frac{2\omega }{k z_R}\frac{\partial\psi}{\partial\zeta}e^{i\eta}, \end{aligned}$$

and $B_\theta =E_r/c$, where we have again performed the paraxial approximation ($|\partial \psi /\partial z|\ll k|\psi |$). We note that the above radial and axial fields are consistent with those obtainable from the Lax perturbation method [8]. From Eq. (1), the slowly varying parts, $\psi _r$ and $\psi _z$, of the radial and axial electric fields [Eqs. (3) and (4) without the $e^{i\eta }$ factor] are related as the lowest-order, paraxial fields in [8], namely,

(5)$$\psi_z={-}\frac{i}{k}\left(\frac{\partial \psi_r}{\partial r} + \frac{1}{r}\psi_r\right).$$

With the lowest-order, Gaussian solution to Eq. (1),

(6)$$\psi_0=f e^{{-}f\rho^2} ,\quad f=i/(i+\zeta),$$

as the potential, Eqs. (3) and (4) produce the fundamental RP beam

(7)$$E_r= E_0 f^2\rho e^{{-}f\rho^2}e^{i\eta},$$

(8)$$E_z={-}E_0 i\epsilon f^{2}\left[1-\rho^2{f}\right]e^{{-}f\rho^2}e^{i\eta},$$

with $\epsilon =w_0/z_R$ and $E_0=\epsilon \omega {A_0}$. Nonparaxial corrections have been calculated in the past [9]. The on-axis ($\rho =0$) axial field can be expressed in the more familiar form:

(9)$$E_z=\frac{-E_0 i\epsilon}{(1+\zeta^2)}e^{i2\phi_G}e^{i\eta},$$

where $\phi _G=\tan ^{-1} \zeta$ is Gouy’s phase.

High-order radially polarized beams. For decades, solutions to the wave equation have been extended to higher orders, and this is true as well for RP solutions which have been described with equations [10–12], calculated via vector-based diffraction integrals [13,14], and even produced experimentally [15,16]. We will derive analytical equations for the three cases.

1. Radially polarized elegant Laguerre-Gaussian beams. Reference [11] constructs an RP LG beam from the potential $\mathbf {A}=A_0\psi _\textrm {e} exp(i\eta )\mathbf {e}_z$ with

(10)$$\psi_\textrm{e} = f^n L_n(f\rho^2)\psi_0,$$

where $\psi _0$ and $f$ are as in Eq. (6) and $L_n(\cdot )$ is the Laguerre polynomial of order $n$. The potential in Eq. (10) actually corresponds to the so-called elegant Laguerre–Gaussian (eLG) beam [17,18], although that was not specified at the time, and is the simplest extension of Eq. (6) to higher orders. The eLG is named as such due to its elegant mathematical form characterized by the complex argument of the Laguerre polynomial in $\psi _\textrm {e}$ matching that within the exponential. In Ref. [11], the authors calculate nonparaxial corrections to the RPeLG beam as well, but here we are more interested in the diverse paraxial high-order RP fields.

The fields of RPeLG beams of order $n$ are calculated from Eqs. (10), (3), and (4) to be

(11)$$E_{r}= E_0 f^{n+2} \rho L_{n}^{(1)}(f\rho^2) e^{{-}f\rho^2}e^{i\eta},$$

(12)$$E_{z}={-}E_0 i\epsilon f^{n+2}(n+1)L_{n+1}(f\rho^2)e^{{-}f\rho^2}e^{i\eta},$$

with all notation as before, and where $L_{n}^{(1)}(\cdot )$ is the generalized Laguerre polynomial of radial order $n$ and azimuthal order 1. If $\rho =0$, the only non-zero field is along $\mathbf {e}_z$,

(13)$$E_z={-}E_0 i \epsilon f^{n+2}(n+1)e^{i\eta} ,$$

which can be expressed in the more recognizable format

(14)$$E_z=\frac{-E_0 i\epsilon(n+1)}{(1+\zeta^2)^{(n+2)/2}}e^{i(n+2)\phi_G}e^{i\eta}.$$

If $n=0$, then this clearly simplifies to the Gaussian case, but with non-zero $n$, the amplitude decreases faster with $\zeta$ along with an increased Gouy’s phase. Note that with increasing $n$, the RPeLG should approach an RP Bessel–Gauss beam, as is the case with linear polarization [19].

2. Radially polarized standard Laguerre–Gaussian beams. Because the fields of the RPeLG beam correspond to an eLG beam, but not to an sLG beam, which first appeared in Ref. [20], a clarification is necessary. The electric field of the linearly polarized sLG beam is (with azimuthal order $l=0$)

(15)$$\begin{aligned} E= & \frac{E_0}{\sqrt{1+\zeta^2}}L_n\left(\frac{2\rho^2}{1+\zeta^2}\right) \end{aligned}$$

(16)$$\begin{aligned}\times \exp\left[\frac{-i\rho^2}{(i+\zeta)} +i(2n+1)\phi_G\right] e^{i\eta}, \end{aligned}$$

which is a well-known high-order solution to the paraxial wave equation. A longitudinal potential $\mathbf {A}=A_0\psi _\textrm {s}\exp (i\eta )\mathbf {e}_z$ can be constructed using the field of the sLG beam in Eq. (15), and will produce, according to the standard “Laguerre–Gaussian” terminology, an RPsLG beam. The prospective of finding this form of RP LG was mentioned in Ref. [6], and acknowledged as well in Ref. [7], but not solved to our knowledge. It is clear that the case with the eLG is simpler, hence again why it is referred to as “elegant,” but the sLG retains a Laguerre polynomial of a real argument.

From Eq. (15), and noting that $f=e^{i\phi _G}/\sqrt {1+\zeta ^2}$ and $ff^\star =1/(1+\zeta ^2)$, the potential for the RPsLG beam can be formulated as

(17)$$\psi_\textrm{s} = L_n(2ff^*\rho^2)\left(\frac{f}{f^*}\right)^{n}\psi_0.$$

The fields for the RPsLG beam of order $n$ are then

(18)$$\begin{aligned} E_r & = E_0 \frac{f^{n+2}}{(f^\star)^n}\rho\Big[ L_{n}(2ff^\star\rho^2) +2f^\star L_{n-1}^{(1)}(2ff^\star\rho^2)\Big]\\ & \times e^{{-}f\rho^2}e^{i\eta},\end{aligned}$$

(19)$$\begin{aligned}E_z & ={-}E_0 i\epsilon \frac{f^{n+1}}{(f^*)^n} \bigg\{\left[(n+1)f+n f^\star\right]L_{n}(2ff^\star\rho^2)\\ & - \rho^2\Big[f^2L_{n}(2ff^\star\rho^2) +2ff^\star(f-f^\star) L_{n-1}^{(1)}(2ff^\star\rho^2)\Big] \bigg\}\\& \times e^{{-}f\rho^2}e^{i\eta}. \end{aligned}$$

If $\rho =0$, then we have the only non-zero fields along $\mathbf {e}_z$,

(20)$$E_z ={-}E_0 i\epsilon \frac{f^{n+1}}{(f^\star)^n}\Big[(n+1)f+n f^\star\Big]e^{i\eta} ,$$

which can be expressed once again in more recognizable form as

(21)$$E_z=\frac{-E_0 i\epsilon}{(1+\zeta^2)}e^{i(2n+1)\phi_G} \Big[(n+1)e^{i\phi_G} + ne^{{-}i\phi_G}\Big]e^{i\eta}.$$

These fields differ with the fields in Eqs. (11,12) as the amplitude term does not depend on a power of $n$, which matches the conventional linearly polarized Laguerre–Gaussian beam. To our knowledge the expression of RPsLG beams have not been derived before. If $n=0$, the RPsLG simplifies to the fundamental RP beam, supporting the validity of the solution.

3. Radially polarized vortex-based Laguerre–Gaussian beams. There is a third RP solution to the paraxial wave equation that involves Laguerre polynomials and that is not directly derived from a vector potential, but from the Lax method [8]. This solution can be constructed via a sum of LG beams of opposite vorticities $l=\pm 1$ [21], and hence we will therefore refer to as a radially polarized vector-based Laguerre–Gaussian (RPvLG) beam.

If $E=\psi (\rho,\zeta,\theta )e^{i\eta }=\psi _r(\rho,\zeta )e^{\pm i\theta }e^{i\eta }$ are vortex solutions to the paraxial wave equation without cylindrical symmetry,

(22)$$\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial\psi}{\partial\rho}\right)+ \frac{1}{\rho^2}\frac{\partial^2\psi}{\partial\theta^2}-4i\frac{\partial\psi}{\partial\zeta}=0,$$

then we can construct an RP solution via the superposition of right-handed and left-handed circularly polarized vortex beams:

(23)$$\begin{aligned}& \frac{\psi_r(\rho,\zeta)}{\sqrt{2}}\left[e^{{-}i\theta}\left(\frac{\mathbf{e}_x\!+\!i\mathbf{e}_y}{\sqrt{2}}\right)+e^{i\theta}\left(\frac{\mathbf{e}_x\!-\!i\mathbf{e}_y}{\sqrt{2}}\right)\right]e^{i\eta}\\ & = \psi_r(\rho,\zeta)e^{i\eta}\mathbf{e}_r=E_r\mathbf{e}_r, \end{aligned}$$

where $\mathbf {e}_r$ is a unit radial vector. Then the axial electric field $E_z=\psi _ze^{i\eta }$ is given by Eq. (5). Selecting $E$ as eLG beams of azimuthal order $l=\pm 1$ and radial order $n$, one arrives at the same RPeLG beam obtained from the eLG as with the vector potential. However, selecting $E$ as sLG beams of the same order, the resulting radial and axial electric fields are

(24)$$E_{r}= E_0\frac{f^{n+2}}{(f^\star)^n}\rho L_{n}^{(1)}(2ff^\star\rho^2)e^{{-}f\rho^2}e^{i\eta},$$

(25)$$\begin{aligned} E_{z}&={-}E_0 i\epsilon\frac{f^{n+2}}{(f^\star)^n}\left[(n+1)L_{n}(2ff^\star\rho^2)-\rho^2f L_{n}^{(1)}(2ff^\star\rho^2)\right]\\ &\times e^{{-}f\rho^2} e^{i\eta}. \end{aligned}$$

If $\rho =0$, the only non-zero field is along $\mathbf {e}_z$,

(26)$$E_z ={-}E_0 i\epsilon \frac{f^{n+2}}{(f^*)^n}(n+1)e^{i\eta},$$

which can finally be expressed in the more recognizable form as

(27)$$E_z= \frac{-E_0 i\epsilon }{(1+\zeta^2)}e^{i(2n+2)\phi_G}(n+1)e^{i\eta}.$$

Note that with $n=0$, the RPvLG beam also reduces to the fundamental RP beam, supporting as well the validity of the solution. The radial field of the RPvLG beam corresponds to the purely radial field derived in Ref. [10], which is accompanied here with the associated longitudinal component.

Comparison of the three solutions and discussion. All three solutions differ in their transversal and axial structure. As seen in Fig. 1, the transversal amplitude profiles of the radial and axial components in the near-field ($\zeta =0$) feature brighter $n+1$ rings and decreasing position of the zeros from RPeLG to RPsLG to RPvLG beams. The radial and axial components of RPsLG and RPvLG beams continue to feature $n+1$ rings at far-field ($\zeta \gg 1$), while the RPeLG beam becomes a single annulus. The rings at far-field continue to be more pronounced for RPvLG beams than for RPsLG beams, but the position of the zeros is reversed compared to the near-field. A relevant feature of the RPvLG beams is their propagation-invariant radial component, a property shared with the Cartesian polarized sLG beam. Overall, RPsLG beams represent an intermediate situation between RPeLG and RPvLG beams. With regards to the focusing capability of these RP LG beams, it is clear from the above description that the radial and axial components of RPvLG beams have tighter first ring and central lobe at their focus $\zeta =0$. If, however, the outer rings are taken into account, or are detrimental for some application, RPsLG beams are more convenient with regards to focusing capability.

Fig. 1. Comparison of (a),(b) the near-field ($\zeta =0$) and (c),(d) the far-field ($\zeta =100$) profiles of both $E_{r}$ and $E_{z}$, for RPeLG, RPsLG, and RPvLG beams (blue solid, orange dashed, and green dash–dotted lines, respectively) with $n=1$ to 5 (top row to bottom row). Here, ${E_{r}}$ for the RPsLG is normalized by $\sqrt {2}$, ${E_{z}}$ for the RPeLG and RPvLG is normalized by $(n+1)$, and ${E_{z}}$ for the RPsLG is normalized by $(2n+1)$.

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Interestingly, the axial component of the RPsLG and RPvLG solutions decay with $\zeta$ as $1/(1+\zeta ^2)$ irrespective of $n$, much slower than the faster decay of RPeLG beams, $1/(1+\zeta ^2)^{(n+2)/2}$ with increasing $n$, as observed in Fig. 2 at $\rho =0$.

Fig. 2. Comparing the normalized on-axis (a),(c) amplitude and (b),(d) phase for the RPeLG, RPsLG, and RPvLG (blue solid, orange dashed, and green dash–dotted lines, respectively) for $E_{z}$ with (a),(b) $n=1$ and (c),(d) $n=5$.

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RPeLG and RPvLG beams have an on-axis phase evolution that is a multiple of $\phi _G$. However, RPsLG beams in Eqs. (18,19) have the unique property of being described as a superposition of two modes that have phases differing by $2\phi _G$. Looking closely at Eq. (21), with small $n$, the on-axis phase through the focus deviates more pronounced from a multiple of $\phi _G$, but with large $n$, the phase of the RPsLG beam approaches that of the RPvLG beam, as seen in Fig. 2. It is interesting to note that the RPeLG beam has the fastest changing on-axis amplitude, but the slowest changing phase.

The beams presented here could be generated similarly to Ref. [16] when a Gaussian beam passes through a binary phase mask with $n$ concentric circular rings imprinting alternate 0 and $\pi$ phase, and then is transmitted through a conventional linear-to-radial polarization converter. If the radii of each 0–$\pi$ step is set properly, it results in a good approximation to the RP LG beams described here. The RPvLG beams can also be generated by coherently superposing two LG beams of topological charges $\pm 1$, radial order $n$, and opposite circular polarizations, identical to the theoretical procedure.

In addition to the intrinsic interest of these three RP LG involving LG polynomials, they provide bases for building rather arbitrary RP beams. For example, $E_r =\sum _{n=0}^\infty g^{(n)} E_r^{(n)}$ and $E_z =\sum _{n=0}^\infty g^{(n)} E_z^{(n)}$, where $(E_r^{(n)}, E_z^{(n)})$ are given by Eqs. (11,12), (18,19), or (24,25), and $g^{(n)}$ are arbitrary coefficients, are new RP beams. Using the orthogonality properties of LG polynomials, it can be verified that the choices

(28)$$g^{(n)}= \frac{2}{n+1}\int_0^\infty d\rho \rho^2 L_n^{(1)}(\rho^2)\psi_r(\rho)$$

or

(29)$$g^{(n)}= \frac{8}{n+1}\int_0^\infty d\rho \rho^2 \exp(-\rho^2)L_n^{(1)}(2\rho^2)\psi_r(\rho)$$

in the above series with RPeLG beams or RPvLG beams, respectively, provide the propagated field of the RP beam with prescribed radial profile $E_r(\rho,0)=E_0\psi _r(\rho )e^{i\eta }$ at the waist $\zeta =0$. The basis of RPsLG appears to be less useful for this purpose because of its lack of evident orthogonality properties. A detailed study of the efficiency of these bases is deferred to future work.

Conclusion. We have described basic methods for deriving radially polarized solutions to the paraxial wave equation, and used them to derive and compare three solutions all involving Laguerre polynomials. Using the vector potential method, we found the radially polarized elegant Laguerre–Gaussian beam and the radially polarized standard Laguerre–Gaussian beam. Using a sum of vortex beams and the Lax method, we derived a third solution named the radially polarized vortex-based Laguerre–Gaussian beam. The three light beams differ in their transverse and longitudinal field profiles, both in the near-field and far-field, and in both the phase and amplitude evolution.

Our study establishes a clear distinction between three light beams, all of which could be qualified as high-order, radially polarized beams. This clarification will be useful for future work involving high-order cylindrical vector beams and their applications.

Funding

Ministerio de Ciencia, Innovación y Universidades (FIS2017-87360-P, PGC2018-093854-B-I00); Horizon 2020 Framework Programme (801505).

Acknowledgments

S.W.J. has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 801505. M.A.P. acknowledges support from Projects No. PGC2018-093854-B-I00 and FIS2017-87360-P of the Spanish Ministerio de Ciencia, Innovación y Universidades.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

REFERENCES

1. Q. Zhan, Adv. Opt. Photonics 1, 1 (2009). [CrossRef]

2. R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003). [CrossRef]

3. Q. Zhan, Opt. Express 12, 3377 (2004). [CrossRef]

4. C. Varin, M. Piché, and M. A. Porras, Phys. Rev. E 71, 026603 (2005). [CrossRef]

5. L. W. Davis, Phys. Rev. A 19, 1177 (1979). [CrossRef]

6. L. W. Davis and G. Patsakos, Opt. Lett. 6, 22 (1981). [CrossRef]

7. K. T. McDonald, “Gaussian Laser Beams with Radial Polarization,” https://puhep1.princeton.edu/mcdonald/examples/axicon.pdf.

8. M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975). [CrossRef]

9. Y. I. Salamin, Opt. Lett. 31, 2619 (2006). [CrossRef]

10. A. A. Tovar, J. Opt. Soc. Am. A 15, 2705 (1998). [CrossRef]

11. S. Yan and B. Yao, Opt. Lett. 32, 3367 (2007). [CrossRef]

12. A. April, Opt. Lett. 33, 1563 (2008). [CrossRef]

13. Y. Kozawa and S. Sato, Opt. Lett. 31, 820 (2006). [CrossRef]

14. Y. Kozawa and S. Sato, J. Opt. Soc. Am. A 24, 1793 (2007). [CrossRef]

15. Y. Kozawa and S. Sato, J. Opt. Soc. Am. A 27, 399 (2010). [CrossRef]

16. Y. Kozawa, T. Hibi, A. Sato, H. Horanai, M. Kurihara, N. Hashimoto, H. Yokoyama, T. Nemoto, and S. Sato, Opt. Express 19, 15947 (2011). [CrossRef]

17. A. E. Siegman, J. Opt. Soc. Am. 63, 1093 (1973). [CrossRef]

18. S. Saghafi and C. J. R. Sheppard, J. Mod. Opt. 45, 1999 (1998). [CrossRef]

19. M. A. Porras, R. Borghi, and M. Santarsiero, J. Opt. Soc. Am. A 18, 177 (2001). [CrossRef]

20. H. Kogelnik and T. Li, Appl. Opt. 5, 1550 (1966). [CrossRef]

21. M. A. Porras, Phys. Rev. A 103, 033506 (2021). [CrossRef]

Clarification for the fields of different radially polarized Laguerre–Gaussian light beams

Abstract

Funding

Acknowledgments

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Data availability

REFERENCES

Data availability

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Optics Letters