Needles of longitudinally polarized light: guidelines for minimum spot size and tunable axial extent

Harold Dehez; Alexandre April; Michel Piché

doi:10.1364/OE.20.014891

1. Introduction

Strongly focused optical beams and smaller focal spots are subjects of considerable interest from both theoretical and practical viewpoints. In fact, a better understanding of the electromagnetic field of tightly focused beams in the focal region leads to practical applications, such as electron acceleration [1], high-resolution microscopy [2], optical trapping [3], and optical data storage [4]. As early as 1872, Lord Rayleigh showed in paraxial optics that illuminating a narrow annular aperture placed in front of a lens produces in the focal region an intensity distribution of light that is proportional to the square of a Bessel function of the first kind of order zero; the full width at half-maximum (FWHM) of this beam is 0.36 λ/NA [5]. However, in the nonparaxial regime, the vector nature of light cannot be ignored to correctly describe optical beams focused by high-NA systems.

Richards and Wolf provided a theoretical framework through which strongly focused beams can be studied, given the field distribution of the collimated input beam at the entrance pupil of the optical system [6]. Some authors studied the focusing properties of optical beams with spatially inhomogeneous polarizations (e.g. radial and azimuthal polarizations) focused by an aplanatic lens [7–10] while others analyzed the electromagnetic field of optical beams at the focus of a parabolic mirror [11–15]. Those studies showed that the choice of the focusing system as well as the state of polarization strongly affect the properties of the focused optical beams.

A focal spot smaller than the diffraction-limit can be reached using an incident radially polarized beam on a high-NA system [16–19]. In fact, when such a beam is tightly focused, the radial symmetry of the electromagnetic field yields a perfectly symmetric focal spot with a strong longitudinal electric field component. Moreover, it has been shown, both theoretically and experimentally, that a parabolic mirror can focus a radially polarized laser beam to a considerably tighter focal spot than what can be achieved with an aplanatic lens [13,20,21].

For some applications, it may be desirable to obtain a very narrow diffraction pattern with a tunable axial extent. For instance, in optical data storage, one would expect that the focal point possesses a small transverse dimension but a long focal depth [22]; this leads to a larger data density as well as a better tolerance for beam focusing. In biomedical imaging, temporal resolution is strongly enhanced using beams with a long focal depth in two-photon laser scanning microscopy [23] or plane illumination microscopy [24].

Subwavelength and nearly nondiffracting beams can be obtained when the light is longitudinally polarized. This type of polarized beam is of growing interest in applications such as fluorescent imaging [25], coherent anti-Stokes Raman scattering microscopy [26], second-harmonic generation [27], and material ablation [28]. Wang et al. focused a radially polarized Bessel–Gauss beam with a binary optical element and a lens to generate a longitudinally polarized beam that propagated without divergence over 4λ and that was highly localized in the transverse direction with a FWHM of 0.43λ [29]; the dimensions of the focal line was fixed by the shape of the binary optics. Kitamura et al. produced a longitudinally polarized focal line by focusing a finite annulus of light with a high-numerical focusing lens. The transverse FWHM was approximately 0.4λ with a depth of focus of more than 4λ [30]. More recently, Rajesh et al. demonstrated that the formation of longitudinally polarized light can be achieved by tightly focusing a double ring radially polarized beam with a diverging aberrated lens in front of a high-NA converging lens (high-NA lens axicon); the resulting longitudinally polarized beam presents a nondiffractive region with a length of 8λ and a transverse FWHM of 0.45λ [31].

If an infinitesimally narrow annulus of radially polarized light is tightly focused, the electric field in the focal region is dominated by its longitudinal component, whose amplitude profile follows a Bessel function of the first kind of order zero [19]. This focused field is in fact a longitudinally polarized Bessel (nondiffracting) beam that has infinite extent along the longitudinal axis and the smallest width in the transverse direction, i.e. 0.36λ/NA [19,22]. However this longitudinally polarized Bessel beam is known to be nonphysical: to obtain this nondiffracting beam, one has to focus an infinitesimally thin annulus of light while only annuli of light of finite width can physically be produced.

In this paper, we are interested in longitudinally polarized focal lines, called needles of longitudinally polarized light, that are physically realizable and whose width along the transverse direction can reach the theoretical limit. We consider the realistic case of a narrow annulus (of finite width) of radially polarized light, produced in the far field of an axicon, and focused by a parabolic mirror or an aplanatic lens. We provide the theoretical guidelines to show that the needle of longitudinally polarized light thus created presents the smallest transverse dimension and a tunable axial extent.

The paper is organized as follows. In Section 2, we recall the theory, based on the Richards–Wolf vector diffraction integral, of the focusing of radially polarized light. In Section 3, we present the results obtained for the focusing of a thin annulus of radially polarized light; the resulting needle of longitudinally polarized light is characterized in Section 4. In Section 5, we discuss the domain of validity of the analytical solutions given in Section 3. Finally, in Section 6, we propose an experimental setup that can be used to generate such needles of longitudinally polarized light. Concluding remarks are given in Section 7.

2. Focusing radially polarized beams beyond the paraxial approximation

In this work we use the Richards–Wolf theory, which consists in a vector diffraction integral useful to analyze tightly focused beams [6]. Consider an incident collimated beam, whose electric field has prescribed spatial amplitude distribution and state of polarization, in the entrance pupil of a given optical system of focal length f. The wave at the exit pupil of the system converges toward its focal point. We employ cylindrical coordinates (r,ϕ,z) near the focus, with the origin located at the focal point. Since the system has a high numerical aperture, a vector diffraction theory is required to calculate the electric field $E (r, ϕ, z)$ in the neighborhood of the focus. The Richards–Wolf vector field equation is an integral over the incident vector field amplitude $A (α, β) \equiv ℓ_{0} (α, β) \hat{a} (α, β)$ on a spherical aperture of focal radius f [8,12,13,17]:

E (r, ϕ, z) = \frac{E_{o}}{2 π} \iint_{Ω} q (α) A (α, β) \exp (- j k • r) d Ω,

where E_o is a constant amplitude, α and β are the polar and the azimuthal angles of the incident plane wave component, respectively, k is the wave vector oriented toward the focus, r is the position vector, and

d Ω = \sin α d α d β

is the element of solid angle. The function q(α) is the so-called apodization factor of the system, obtained from energy conservation and geometric considerations. The function

ℓ_{0} (α, β)

is the amplitude distribution of the collimated input beam at the entrance pupil and

\hat{a} (α, β)

is a unit vector representing the polarization direction of the electric field near the focus.

The integration is done over the solid angle Ω that covers the entrance pupil of the optical system. Here the time dependence exp(jωt) is omitted and we have neglected the constant phase factor due to the total path length.

The electric field defined by Eq. (1) can be seen as a superposition of plane waves $\exp (- j k • r)$ having different amplitudes. Angle α is the angle of depression between the z axis and wave vector k; specifically, it is defined such that the Cartesian components of wave vector k of the constitutive plane waves of the converging beam are respectively given by $k_{x} = - k \sin α \cos β$ , $k_{y} = - k \sin α \sin β$ and $k_{z} = k \cos α$ , where $k \equiv | k | = 2 π / λ$ is the wave number and λ is the wavelength [17]. Therefore, the dot product in the kernel of the integral is explicitly $k • r = k z \cos α - k r \sin α \cos (ϕ - β)$ . We assume the optical system exhibits a symmetry of revolution; hence the solid angle Ω subtended by that the system is covered with $0 \leq β \leq 2 π$ and $α_{\min} \leq α \leq α_{\max}$ where $α_{\min}$ and $α_{\max}$ are the minimum and the maximum values of the angle α, respectively.

With the help of the Richards–Wolf theory, we now analyze a radially polarized annular beam focused by a parabolic mirror or an aplanatic lens. The optical axis of the collimated beam coincides with the axis of revolution of the paraboloid or the lens (Figs. 1a and 1b). For an entrance pupil illuminated with a radially polarized beam, the amplitude distribution $ℓ_{0} (α)$ is azimuthally symmetric. For an incident radially polarized beam, the electric field vector is everywhere oriented radially with respect to the optical axis. We consider that the electric field in the entrance pupil of the parabolic mirror is oriented along the unit vector $- {\hat{a}}_{r}$ shown in Fig. 1a while the electric field in the entrance pupil of the aplanatic lens is oriented along the unit vector ${\hat{a}}_{r}$ shown in Fig. 1b; in both cases, the unit vector $\hat{a} (α, β)$ may be written as [8,13,17]

\hat{a} (α, β) = {\hat{a}}_{x} \cos α \cos β + {\hat{a}}_{y} \cos α \sin β + {\hat{a}}_{z} \sin α,

where

{\hat{a}}_{x}, {\hat{a}}_{y}, {\hat{a}}_{z}

are unit vectors oriented along the Cartesian axes x, y, z. Note that the orientations of the polarization vectors in Fig. 1a take into account the phase jump of π radians due to the reflection at the surface of the mirror. Equation (1) can be rewritten explicitly as

\begin{matrix} E (r, ϕ, z) = \frac{E_{o}}{2 π} \int_{0}^{2 π} \int_{α_{\min}}^{α_{\max}} q (α) ℓ_{0} (α) \hat{a} (α, β) \\ \times \exp [- j k (z \cos α - r \sin α \cos (ϕ - β))] \sin α d α d β, \end{matrix}

By integrating this relation over β [6,17], one finds that the nonzero components of the electric field near the focus are

E_{r} (r, z) = j E_{o} \int_{α_{\min}}^{α_{\max}} q (α) ℓ_{0} (α) \sin α \cos α \exp (- j k z \cos α) J_{1} (k r \sin α) d α,

E_{z} (r, z) = E_{o} \int_{α_{\min}}^{α_{\max}} q (α) ℓ_{0} (α) \sin^{2} α \exp (- j k z \cos α) J_{0} (k r \sin α) d α,

where we have written the cylindrical components of the electric field using

E_{r} = E_{x} \cos ϕ + E_{y} \sin ϕ

and

E_{ϕ} = - E_{x} \sin ϕ + E_{y} \cos ϕ \equiv 0

. Equations (4a) and (4b) show that the focused field consists of a radially polarized beam with a longitudinal component; the electric field structure has no azimuthal component.

Fig. 1 The geometry of a) the parabolic mirror and b) the aplanatic lens.In b), the incident rays are refracted by a reference sphere (dashed line) of radius equal to the focal length of the aplanatic lens.

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3. Focusing a narrow annulus of radially polarized light

A very sharp intensity profile in the focal region can be obtained when a very narrow annulus of light is incident on the focusing system. Such an annulus of light can be modeled by the Dirac delta $ℓ_{0} (α) = δ (α - α_{0})$ , where $α_{0}$ is the angular position of the incident annulus of light on the focusing system (see Fig. 1). Substituting this expression for $ℓ_{0} (α)$ in Eqs. (4a) and (4b), assuming that $α_{0}$ is comprised between $α_{\min}$ and $α_{\max}$ , one finds

E_{r} (r, z) = j E_{o} q (α_{0}) \cos α_{0} \sin α_{0} \exp (- j k z \cos α_{0}) J_{1} (k r \sin α_{0}),

E_{z} (r, z) = E_{o} q (α_{0}) \sin^{2} α_{0} \exp (- j k z \cos α_{0}) J_{0} (k r \sin α_{0}) .

For the special case

α_{0} = π / 2

, Eqs. (5a) and (5b) reduce to

E_{r} (r, z) = 0

and

E_{z} (r, z) = E_{o} q (π / 2) J_{0} (k r)

(pure longitudinally polarized light). This result is in agreement with that obtained by Grosjean and Courjon [19]: when an infinitesimally thin, radially polarized annulus of light is focused by an aplanatic lens with NA = 1 (i.e. when

α_{0} = π / 2

), there is no transverse component of the electric field at the focus and the longitudinal component of the field is proportional to the Bessel function J₀(kr). Grosjean and Courjon also demonstrated that such an amplitude profile provides the best confinement of light in the transverse direction. Since Eqs. (4a) and (4b) depend on the specific focusing system only via the constant amplitude factor

q (α_{0})

, this conclusion is valid for both the aplanatic lens and the parabolic mirror. It can be shown that the FWHM of the focal spot in the transverse direction is 0.36λ. However, this result remains theoretical, because the optical beam obtained in this case is a nondiffracting Bessel beam that carries an infinite amount of power.

Expressions for a physically realizable needle of the longitudinally polarized light in the focal region of a focusing system can be obtained if the width of the incident annulus of light is finite. We consider now the more realistic case of an incident illumination having the shape of a Gaussian annulus $ℓ_{0} (α) = {(\sqrt{π} Δ α)}^{- 1} \exp [- {(α - α_{0})}^{2} / Δ α^{2}]$ , where $Δ α$ is the angular width of the incident annulus of light (see Fig. 1) – note that this amplitude distribution reduces to the Dirac delta distribution in the limit of an infinitesimally thin annulus of light. According to Eqs. (4a) and (4b), the integrals to be solved read

E_{r} = \frac{j E_{o}}{\sqrt{π} Δ α} \int_{α_{\min}}^{α_{\max}} q (α) \sin α \cos α \exp [- {(\frac{α - α_{0}}{Δ α})}^{2} - j k z \cos α] J_{1} (k r \sin α) d α,

E_{z} = \frac{E_{o}}{\sqrt{π} Δ α} \int_{α_{\min}}^{α_{\max}} q (α) \sin^{2} α \exp [- {(\frac{α - α_{0}}{Δ α})}^{2} - j k z \cos α] J_{0} (k r \sin α) d α .

These integrals may be evaluated numerically. However, if the annulus of light is assumed to be sufficiently narrow (i.e.

Δ α < < α_{0}

), it is possible to evaluate analytically the integrals in Eqs. (6a) and (6b). In this case, the following practical expressions for the components of the electric field in the focal region of the focusing system may be found (see Appendix)

\begin{matrix} E_{r} (r, z) \approx j E_{o} q (α_{0}) \sin α_{0} \cos α_{0} \exp (- z^{2} / z_{0}^{2} - j k z \cos α_{0}) \\ \times {[1 + j Δ α U_{r} (α_{0}) (z / z_{0}) + \frac{1}{2} Δ α^{2} V_{r} (α_{0})] J_{1} (v) + \frac{1}{4} Δ α^{2} v J_{2} (v)}, \end{matrix}

\begin{matrix} E_{z} (r, z) \approx E_{o} q (α_{0}) \sin^{2} α_{0} \exp (- z^{2} / z_{0}^{2} - j k z \cos α_{0}) \\ \times {[1 + j Δ α U_{z} (α_{0}) (z / z_{0}) + \frac{1}{2} Δ α^{2} V_{z} (α_{0})] J_{0} (v) + \frac{1}{4} Δ α^{2} v J_{1} (v)}, \end{matrix}

where

v \equiv k r \sin α_{0}

and

z_{0} \equiv \frac{2}{k \sin α_{0} Δ α} = \frac{λ}{π \sin α_{0} Δ α} .

In Eqs. (7a) and (7b), we have also introduced the following functions

U_{r} (α_{0}) = \frac{q^{'} (α_{0})}{q (α_{0})} + 2 \cot (2 α_{0}),

V_{r} (α_{0}) = \frac{2 q^{'} (α_{0})}{q (α_{0})} \cot (2 α_{0}) + \frac{q^{″} (α_{0})}{2 q (α_{0})},

U_{z} (α_{0}) = \frac{q^{'} (α_{0})}{q (α_{0})} + 2 \cot α_{0},

V_{z} (α_{0}) = \frac{2 q^{'} (α_{0})}{q (α_{0})} \cot α_{0} + \frac{q^{″} (α_{0})}{2 q (α_{0})},

where

q^{'} (α_{0})

and

q^{″} (α_{0})

are the first-order and the second-order derivatives, respectively, of the apodization factor

q (α)

with respect to its argument, evaluated at

α = α_{0}

. The condition for which Eqs. (7a) and (7b) are valid is that

Δ α

is much smaller than

α_{0}

and

| q (α_{0}) / q^{'} (α_{0}) |

. In the special case of an infinitesimally thin annulus (

Δ α \to 0

), Eqs. (7a) and (7b) reduce to Eqs. (5a) and (5b), as expected.

The general results are now applied to the specific cases of the parabolic mirror and the aplanatic lens. The apodization factor of a parabolic mirror is $q (α) = 2 / (1 + \cos α)$ [12,13,31], whereas the apodization factor of an aplanatic lens is $q (α) = \cos^{1 / 2} α$ [6,17,32]. Equations (9a)–(9d) are then computed for both cases (Table 1 ).

Table 1. Expressions of Eqs. (9a)–(9d) for a parabolic mirror and an aplanatic lens.

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Equations (7a) and (7b) are valid for the parabolic mirror if $Δ α < < \min {α_{0}, \cot (\frac{1}{2} α_{0})}$ while they are valid for the aplanatic lens if $Δ α < < \min {α_{0}, 2 \cot α_{0}}$ .

To get more physical insight in these results, we can approximate each component of the electric field by its first term, which is easier to interpret physically and manipulate mathematically:

E_{r} (r, z) \approx j E_{o} q (α_{0}) \sin α_{0} \cos α_{0} \exp (- z^{2} / z_{0}^{2} - j k z \cos α_{0}) J_{1} (k r \sin α_{0}),

E_{z} (r, z) \approx E_{o} q (α_{0}) \sin^{2} α_{0} \exp (- z^{2} / z_{0}^{2} - j k z \cos α_{0}) J_{0} (k r \sin α_{0}) .

In light of Eqs. (10a) and (10b), one can say as a crude approximation that, independently of the focusing system, the radial component of the electric field of the needle of longitudinally polarized light is proportional to the Bessel function

J_{1} (k r \sin α_{0})

modulated axially by the Gaussian function

\exp (- z^{2} / z_{0}^{2})

while its longitudinal component is proportional to the Bessel function

J_{0} (k r \sin α_{0})

modulated axially by the same Gaussian function. In other words, the needle of longitudinally polarized light described by Eqs. (10a) and (10b) is a radially polarized Bessel beam whose longitudinal profile is modulated by a Gaussian envelope.

The electric energy density in the focal region is proportional to the square modulus of the electric field of the beam, herein called the intensity I:

I (r, z) \equiv {| E (r, z) |}^{2} = {| E_{r} (r, z) |}^{2} + {| E_{z} (r, z) |}^{2} .

An explicit but approximate expression for the intensity distribution of a needle of longitudinally polarized light can be obtained by substituting Eqs. (10a) and (10b) in Eq. (11):

I (r, z) \approx I_{o} \exp (- 2 z^{2} / z_{0}^{2}) [J_{0}^{2} (k r \sin α_{0}) + \cot^{2} α_{0} J_{1}^{2} (k r \sin α_{0})],

where

I_{o} \equiv I (0, 0) = {| E_{o} |}^{2} q^{2} (α_{0}) \sin^{4} α_{0}

is a constant (the total intensity at the focal point). It is seen that, to a first approximation, the electric field of the needle of longitudinally polarized light does not depend on the specific focusing system, apart from the constant factor q²(α₀) in the total intensity at the focal point. The intensity distribution of a needle of longitudinally polarized light exhibits an oscillatory behavior with sidelobes in the transverse direction while it follows a Gaussian function in the longitudinal direction (Fig. 2 ).

Fig. 2 Intensity distribution of a needle of longitudinally polarized light, normalized to its maximum value, for α₀ = 75° and Δα = 0.01 rad, as computed with Eq. (12).

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4. Characterization of the needle of longitudinally polarized light

Using the expressions of the electric field defined by Eqs. (6a) and (6b), it is possible to analyze numerically the effects of the focusing angle α₀ and the angular thickness Δα of the annular illumination on the needle of longitudinally polarized light produced by a parabolic mirror or an aplanatic lens. Equation (12) can also be exploited to obtain some simple analytical relations useful to characterize the dimensions of the needle of longitudinally polarized light. First, we evaluate the ratio of the maximum value of the radial component of the electric field over the maximum value of its longitudinal component. Second, we find the width of the needle of longitudinally polarized light in the transverse direction. Third, we evaluate the length of the focal line in the longitudinal direction.

The electric field of the needle of longitudinally polarized light is indeed dominated by its longitudinal component. For the range of angular thickness $0 \leq Δ α \leq 0.2 rad$ , only the focusing angle α₀ has a significant impact on the relative amplitude of the transverse and the longitudinal components of the electric field at focus. From Eqs. (10a) and (10b), one finds that the maximum values of the radial and the longitudinal components of the electric field in the focal plane (z = 0) are $E_{r, \max} \approx (0, 582) j E_{o} q (α_{0}) \sin α_{0} \cos α_{0}$ and $E_{z, \max} \approx E_{o} q (α_{0}) \sin^{2} α_{0}$ , respectively (note that the maximum value of the Bessel function of the first kind of order 1 is 0,582). Accordingly, the ratio of the square of the maximum values of the radial and longitudinal components of the electric field is approximately given by

\frac{{| E_{r} |}_{\max}^{2}}{{| E_{z} |}_{\max}^{2}} \approx 0, 34 \cot^{2} α_{0} .

The smaller the value of Δα is, the more accurate is Eq. (13). It is seen that the power transferred from the transverse component of the electric field of the beam to its longitudinal component increases as the value of α₀ gets closer to 90°. For focusing angles α₀ above 75°, the transverse component E_r of the electric field can be considered negligible with respect to the longitudinal component E_z: the focused beam is then longitudinally polarized. Rigorous calculations of the ratio

| E_{r} |_{\max}^{2} / | E_{z} |_{\max}^{2}

for selected focusing parameters, computed with Eqs. (6)a) and (6b), are given in Table 2 . These values confirm that the results are very slightly dependent on the specific choice of the focusing system.

Table 2. Evolution of the amplitude of the transverse component of the electric field compared to the amplitude of its longitudinal component with the focusing angle, for a fixed angular thickness of $Δ α = 0.1 rad$ .

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However, considering the geometric limitations of the focusing system used to generate the needle of longitudinally polarized light, higher focusing angles α₀ can be obtained using a parabolic mirror. A focusing angle of 90° can be achieved with a parabolic mirror, whereas the maximum value of focusing angle for an aplanatic lens is limited in practice to approximately 75°. This limitation has a direct effect on the ratio $| E_{r} |_{\max}^{2} / | E_{z} |_{\max}^{2}$ : the only way to obtain a pure longitudinal polarization is to use a parabolic mirror (as opposed to an aplanatic lens).

The width of the focal line is defined by the FWHM of its intensity profile in the transverse direction. The values of the transverse FWHM, rigorously calculated with Eqs. (6a) and (6b), are approximately independent of the choice of the focusing system, especially when the annulus of light is very narrow (Fig. 3 ). Moreover, as seen in Fig. 3, the transverse FWHM of the needle of longitudinally polarized light is almost independent of the angular thickness Δα, for any focusing angle α₀, as long as a narrow annular illumination is used. Especially with a parabolic mirror and a focusing angle of 90°, the transverse FWHM is increased by less than 1% from the theoretical focusing limit of 0.36λ when $Δ α \leq 0.2 rad$ . Hence, the theoretical case of an infinitesimally thin annulus of light is not necessary to obtain the smallest transverse FWHM. Such a range of thickness is easily feasible experimentally using the setup that will be presented in Section 6.

Fig. 3 Transverse FWHM of a needle of longitudinally polarized light as a function of the angular thickness of a radially polarized annulus of light focused by (a) a parabolic mirror and (b) an aplanatic lens. Several focusing angles α₀ between 45° and 90° are presented (note that α₀ = 90° is not achievable when an aplanatic lens is used).

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The results presented in Fig. 3 follow from the fact that the electric field in the focal plane (z = 0) is independent of the angular thickness Δα of the annular illumination [see Eqs. (10a) and (10b)]. Accordingly, for sufficiently small values of Δα, one can estimate to a relatively high accuracy the transverse FWHM of the intensity profile by finding the value of the radial coordinates that solves the following transcendental equation:

I (r, z) / I (0, z) = J_{0}^{2} (k r \sin α_{0}) + \cot^{2} α_{0} J_{1}^{2} (k r \sin α_{0}) = \frac{1}{2} .

Figure 4 shows the transverse FWHM so obtained, in units of λ, as a function of the focusing angle α₀: it is minimum (FWHM = 0.36λ) at α₀ = 90° and it grows to 0.76λ at α₀ = 45°. Considering that the maximum value of focusing angle for aplanatic lenses is limited to approximately 75°, the only way to reach the transverse focusing limit of 0.36λ is to use a parabolic mirror. A more realistic focusing limit with an aplanatic lens is 0.38λ, the value of the transverse FWHM of the focal line obtained with a focusing angle of 75°.

Fig. 4 The transverse FWHM, as a function of the focusing angle, of a needle of longitudinally polarized light produced by an arbitrary focusing system.

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The length of the focal line is defined by the FWHM of its intensity profile in the longitudinal direction. Even though the apodization factor q(α) changes with the focusing system, the values of the longitudinal FWHM as a function of the angular thickness Δα for a given focusing angle α₀, computed with Eqs. (6a) and (6b), are approximately the same for each focusing system, especially when Δα is small (Fig. 5 ). With the help of Eq. (12), an analytical expression for the longitudinal FWHM, valid for sufficiently small values Δα, can be found. It is obtained by twice the longitudinal coordinate z that is solution of the equation $I (0, z) / I_{o} = \exp (- 2 z^{2} / z_{0}^{2}) = \frac{1}{2}$ . This equation can be solved analytically and yields

Longitudinal FWHM \approx z_{0} {(2 \ln 2)}^{1 / 2} = \frac{λ {(2 \ln 2)}^{1 / 2}}{π \sin α_{0} Δ α},

where Eq. (8) has been used. Hence, the length of the needle of longitudinally polarized light in the z-direction is proportional to the wavelength λ and is inversely proportional to the angular width Δα of the annulus of incident light. Therefore, the extent of the focal spot can be tuned by changing the value of Δα. With this method, it is then possible to generate a longitudinally polarized focus line with a tunable longitudinal extent and a fixed transverse size.

Fig. 5 Longitudinal FWHM of the focal spot as a function of the angular thickness of a radially polarized annulus of light focused by (a) a parabolic mirror and (b) an aplanatic lens. Several focusing angles α₀ comprised between 45° and 90° are presented. The insets give a zoom around practical values of Δα.

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The geometry has an important impact on the relation between the angular thickness Δα of the annulus of light and the radial thickness ΔR at the entrance pupil recorded at half maximum (see Fig. 1). This relation depends on the focusing angle α₀ and the distance h between the optical axis and the conjugate ray for each focusing system:

\frac{Δ R}{R} = 2 {(\ln 2)}^{1 / 2} [\frac{h (α_{0} + Δ α) - h (α_{0} - Δ α)}{h (α_{0})}] \approx 4 {(\ln 2)}^{1 / 2} \frac{h^{'} (α_{0})}{h (α_{0})} Δ α,

where

h^{'} (α_{0})

is the first derivative of

h (α)

with respect to its argument, evaluated at

α = α_{0}

. With

h (α) = 2 f \tan (\frac{1}{2} α)

for the parabolic mirror and

h (α) = f \sin α

for the aplanatic lens, where f is the focal length of the system [32], Eq. (16) becomes explicitly

Δ R / R = 3, 33 \csc α_{0} Δ α

for the parabolic mirror and

Δ R / R = 3, 33 \cot α_{0} Δ α

for the aplanatic lens. Considering the same incident annulus of light and the same focusing angle α₀, the angular thickness is much higher using an aplanatic lens than using a parabolic mirror (because

\tan α_{0} > \sin α_{0}

, especially when α₀ approaches 90°). More precisely, for a given ratio ΔR/R and focusing angle α₀, the angular thickness Δα is secα₀ times larger using an aplanatic lens rather than a parabolic mirror. As a consequence, by virtue of Eq. (15), the needles of longitudinally polarized light in the focal region of a parabolic mirror are longer than the needles in the focal region of a lens. Using parabolic mirrors seems to be a better solution to confine light in the transverse direction and to increase the longitudinal extent of the focal spot; however it suffers from a strong coma in oblique illumination [12,14], which is not the case with aplanatic lenses. This could be a major drawback in applications for which angle scanning is used (as with most of laser scanning microscopes). Therefore, depending on the application, one may prefer to use a parabolic mirror or an aplanatic lens to generate a needle of longitudinally polarized light.

5. Domain of validity of the analytical solutions

Most of the results presented in Section 4 are based on numerical calculations with Eqs. (6a) and (6b), but the same analysis can be made using the analytical solutions given by Eqs. (7a) and (7b). It was already mentioned that these analytical expressions are valid for $Δ α < < \min {α_{0}, \cot (\frac{1}{2} α_{0})}$ with the parabolic mirror and $Δ α < < \min {α_{0}, 2 \cot α_{0}}$ with the aplanatic lens. It is possible to go further by comparing directly analytical and numerical solutions. An alternative domain of validity can be defined as the range of angular thickness for which the difference between the axial and transversal FWHMs obtained numerically and those obtained analytically is less than 1%. This domain depends essentially on the focusing angle α₀ (Table 3 ).

Table 3. Domain of validity of Eqs. (7a) and (7b) for several focusing angles between 45° and 90°. In this domain, the difference between numerical FWHMs and analytical FWHMs is less than 1%. The smaller domain considering transverse and longitudinal FWHMs is given here.

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Within the domain of validity given in Table 3, the analytical description developed here is advantageous. As no integration is required, computation time is reduced, and sampling errors are avoided. In fact, analytical solutions [Eqs. (10a) and (10b)] further give a better idea of the behavior of the electric fields at the focus and Eqs. (13)–(15) provide useful information, easier to compute, about the dimensions of the needle of longitudinally polarized light.

6. Narrow annulus of light produced by the far field of a Bessel–Gauss laser beam

In the far field, the transverse distribution of the electric field of a Bessel–Gauss beam is a thin annulus of light. Here we propose to focus this ring of light to generate a needle of longitudinally polarized light. This narrow annulus of light can be generated experimentally using an axicon and a lens (Fig. 6 ) [33].

Fig. 6 System to generate radially polarized annulus of light using a lens of focal length f₀ and an axicon.

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An axicon is a conical refractive element that focuses light into a line instead of a point and the transverse amplitude profile of the electric field of the focused beam corresponds to a Bessel–Gauss beam. To produce the thin annulus of light, an optical Fourier transform can be achieved using a simple lens. The width at half maximum ΔR and the radius of the annulus R depend on the angle of deviation γ after the axicon, the wavelength λ of the illumination, the focal length f₀ of the lens, and the size w₀ of the incident laser beam according to:

R = f_{0} γ

Δ R = \frac{λ f_{0}}{π w_{0}}

To generate a radially polarized ring of light, a TM₀₁ laser beam is incident on the axicon. Several methods to produce a TM₀₁ laser beam have been demonstrated. Axially symmetric intracavity devices were shown to force a laser to oscillate in radially polarized modes [34]. Polarization converters are also widely used and commercially available to generate TM₀₁ laser beams; the conversion is based on birefringent elements such as a quadrant of four half-wave plates [18] or a liquid crystal cell [35].

In brief, with this system, the length of the needle of light can easily be tuned by changing the size of the incident beam on the axicon. Besides, the angular width of the annulus of light depends not only on the radial width, but also on the focusing angle of the parabolic mirror or the aplanetic lens. At ΔR/R = constant, the angular thickness increases with the focusing angle [see Eq. (16)]. As both focusing systems (the parabolic mirror or the aplanatic lens) can be used at several focusing angles by simply varying the radius R of the incident annulus of light (Fig. 1), the axial extent of the needle of longitudinally polarized light can also be tuned by changing this radius, i.e. by changing the focal length f₀ of the Fourier transform lens (Fig. 6). For any radial thickness below the ratio ΔR/R = 10%, the focusing limit of 0.36λ can be obtained. Examples of focal spots are presented in Table 4 with a fixed ΔR/R ratio of 5% and different focusing angles.

Table 4. Comparison of the spot size in the focal region of a parabolic mirror and an aplanatic lens for several focusing angles between 45° and 90° and a fixed radial thickness of the incident annulus of light: $Δ R / R = 5 %$ .

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7. Conclusion

In summary, we presented an expression for the electric field of a needle of longitudinally polarized light. This beam is produced by focusing a narrow annulus of radially polarized light with a high-NA optical system, such as a parabolic mirror or an aplanatic lens. Based on analytical expressions and numerical simulations of the electric field in the focal region, we demonstrated for each focusing system that the dimensions of the focal line depends on the wavelength, the focusing angle, and the angular thickness of the annular illumination. The longitudinal extent of the focal line is then tunable by simply changing the thickness of the annular incident beam. The FWHM in the transverse direction of the needle of longitudinally polarized light can be as small as 0.36λ when the annulus is focused at an angle of 90° by a parabolic mirror, which is still smaller than the smallest experimentally demonstrated focal spot that has a FWHM of about 0.42λ, also demonstrated with a parabolic mirror [20]. We also compared the performances of the two different focusing systems. Contrary to aplanatic lenses, parabolic mirrors are not limited to focusing angles of approximately 75° or less; they are better candidates to generate the longest and the thinnest needles of longitudinally polarized light. However, using aplanatic lenses remains a more versatile solution, because such lenses are readily available and are relatively well behaved with oblique illumination.

We have also proposed a method to generate needles of longitudinally polarized light using a Fourier transform lens, an axicon, and a focusing system (a parabolic mirror or an aplanatic lens). The required thickness of the annular illumination can easily be achieved using the lens-axicon system with appropriate parameters. To our knowledge, the transverse FWHM of 0.36λ, that could be achieved with a parabolic mirror used at the focusing angle of 90°, is the smallest size that can be obtained experimentally when a continuous wave laser beam is focused.

Appendix: Evaluation of the integrals (6a)–(6b)

We present in this Appendix the steps followed to find Eqs. (7a) and (7b) by solving the integrals in Eqs. (6a) and (6b). We assume that the annulus of light is very thin, i.e. $Δ α < < α_{0}$ . With this assumption, we note that the integration domain is mainly restricted near $α = α_{0}$ ; it means that the limits of integration can be extended to infinity without altering considerably the value of the integrals. Let us shift to the dummy variable: $θ = α - α_{0}$ . Furthermore, since the integrals take a significant value only for $θ < < 1$ , we use the small-angle approximation, i.e. $\sin θ \approx θ$ , which gives

\sin α \cos α \approx \sin α_{0} \cos α_{0} [1 + 2 θ \cot (2 α_{0})],

\sin^{2} α \approx \sin^{2} α_{0} (1 + 2 θ \cot α_{0}) .

Also, we use

\cos α \approx \cos α_{0} - θ \sin α_{0}

in the exponentials as well as

\sin α \approx \sin α_{0} \cos θ

in the Bessel functions, the former being valid if

\frac{1}{2} k z Δ α^{2} \cos α_{0} < < 1

and the latter being approximately true if

| \sin θ \cos α_{0} | < < | \cos θ \sin α_{0} |

. The integrals (6a) and (6b) then become

\begin{matrix} E_{r} (r, z) = \frac{j E_{o} \sin α_{0} \cos α_{0}}{\sqrt{π} Δ α} \exp (- j k z \cos α_{0}) \int_{- \infty}^{\infty} q (α) [1 + 2 θ \cot (2 α_{0})] \\ \times \exp (- \frac{θ^{2}}{Δ α^{2}} + j k z θ \sin α_{0}) J_{1} (k r \sin α_{0} \cos θ) d θ, \end{matrix}

\begin{matrix} E_{z} (r, z) = \frac{E_{o} \sin^{2} α_{0}}{\sqrt{π} Δ α} \exp (- j k z \cos α_{0}) \int_{- \infty}^{\infty} q (α) [1 + 2 θ \cot α_{0}] \\ \times \exp (- \frac{θ^{2}}{Δ α^{2}} + j k z θ \sin α_{0}) J_{0} (k r \cos θ \sin α_{0}) d θ . \end{matrix}

In order to evaluate analytically these integrals, we employ the product theorem of the Bessel functions, which can be written as (see Eq. (8).535 in [36])

J_{ν} (x y) = y^{ν} \sum_{n = 0}^{\infty} \frac{x^{n} {(1 - y^{2})}^{n}}{(2 n)!!} J_{ν + n} (x) .

Equation (20) is used with

x = k r \sin α_{0}

,

y = \cos θ

, and

ν = 0 or 1

. Moreover, we expand the apodization factor

q (α)

in a Taylor series about

α = α_{0}

:

q (α) = \sum_{s = 0}^{\infty} \frac{q^{(s)} (α_{0})}{s!} θ^{s},

where

q^{(s)} (α_{0})

is the s^th derivative of

q (α)

with respect to its argument, evaluated at

α = α_{0}

. Substituting Eqs. (20) and (21) in Eqs. (19a) and (19b) and, using the small-angle approximation for θ, we find

\begin{matrix} E_{r} = \frac{j E_{o} \sin α_{0} \cos α_{0}}{\sqrt{π} Δ α} \exp (- j k z \cos α_{0}) \sum_{n = 0}^{\infty} \sum_{s = 0}^{\infty} \frac{q^{(s)} (α_{0})}{s!} \frac{{(k r \sin α_{0})}^{n} J_{n + 1} (k r \sin α_{0})}{(2 n)!!} \\ \times \int_{- \infty}^{\infty} θ^{2 n + s} [1 + 2 θ \cot (2 α_{0})] \exp (- \frac{θ^{2}}{Δ α^{2}} + j k z θ \sin α_{0}) d θ, \end{matrix}

\begin{matrix} E_{z} = \frac{E_{o} \sin^{2} α_{0}}{\sqrt{π} Δ α} \exp (- j k z \cos α_{0}) \sum_{n = 0}^{\infty} \sum_{s = 0}^{\infty} \frac{q^{(s)} (α_{0})}{s!} \frac{{(k r \sin α_{0})}^{n} J_{n} (k r \sin α_{0})}{(2 n)!!} \\ \times \int_{- \infty}^{\infty} θ^{2 n + s} [1 + 2 θ \cot α_{0}] \exp (- \frac{θ^{2}}{Δ α^{2}} + j k z θ \sin α_{0}) d θ . \end{matrix}

To solve the remaining integrals, we use the following result (see Eq. (3.462.4) in [36]):

\int_{- \infty}^{\infty} θ^{p} \exp (- \frac{θ^{2}}{Δ α^{2}} + j k z \sin α_{0} θ) d θ = \sqrt{π} Δ α {(j \frac{1}{2} Δ α)}^{p} \exp (- \frac{z^{2}}{z_{0}^{2}}) H_{p} (\frac{z}{z_{0}}),

where

H_{p} (\cdot)

is the Hermite polynomial of degree p and z₀ is defined by Eq. (8). Using Eq. (23) in integrals (22a) and (22b) yields the general expression for the electric field in the focal region of the focusing system:

\begin{matrix} E_{r} (r, z) = j E_{o} q (α_{0}) \sin α_{0} \cos α_{0} \exp (- \frac{z^{2}}{z_{0}^{2}} - j k z \cos α_{0}) \sum_{n = 0}^{\infty} \sum_{s = 0}^{\infty} C_{n, s} \\ \times [H_{2 n + s} (\frac{z}{z_{0}}) + j Δ α \cot (2 α_{0}) H_{2 n + s + 1} (\frac{z}{z_{0}})] J_{n + 1} (k r \sin α_{0}), \end{matrix}

\begin{matrix} E_{z} (r, z) = E_{o} q (α_{0}) \sin^{2} α_{0} \exp (- \frac{z^{2}}{z_{0}^{2}} - j k z \cos α_{0}) \sum_{n = 0}^{\infty} \sum_{s = 0}^{\infty} C_{n, s} \\ \times [H_{2 n + s} (\frac{z}{z_{0}}) + j Δ α \cot α_{0} H_{2 n + s + 1} (\frac{z}{z_{0}})] J_{n} (k r \sin α_{0}), \end{matrix}

with the coefficients

C_{n, s} \equiv \frac{q^{(s)} (α_{0})}{q (α_{0})} \frac{{(j \frac{1}{2} Δ α)}^{2 n + s} {(\frac{1}{2} k r \sin α_{0})}^{n}}{s! n!} .

The approximate and explicit expressions for the components of the electric field in the focal region of the focusing system, given by Eqs. (7a) and (7b), can be found if only the dominant terms (small n and s) are kept in Eqs. (24a) and (24b).

Acknowledgments

The authors acknowledge support from the Natural Sciences and Engineering Research Council of Canada (NSERC), the Fonds québécois de recherche sur la nature et les technologies (FQRNT), the Canadian Institutes of Health Research (CIHR), the Canadian Institute for Photonic Innovations (ICIP/CIPI), and the Centre d’optique, photonique et laser (COPL).

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	Parabolic mirror	Aplanatic lens
$U_{r} (α_{0})$	$\csc α_{0} - \tan α_{0}$	$\cot α_{0} - \frac{3}{2} \tan α_{0}$
$V_{r} (α_{0})$	$\sec α_{0} \tan^{2} (\frac{1}{2} α_{0}) + \frac{3}{2} \cot α_{0} \tan (\frac{1}{2} α_{0})$	$\frac{3}{8} \tan^{2} α_{0} - \frac{3}{4}$
$U_{z} (α_{0})$	$\cot (\frac{1}{2} α_{0})$	$2 \cot α_{0} - \frac{1}{2} \tan α_{0}$
$V_{z} (α_{0})$	$\frac{1}{2} \sec^{2} (\frac{1}{2} α_{0}) (1 + \frac{3}{2} \cos α_{0})$	$- \frac{1}{8} \tan^{2} α_{o} - \frac{5}{4}$

	Focusing angle α₀
	45°	60°	75°	90°
${\| E_{r} \|}_{\max}^{2} / {\| E_{z} \|}_{\max}^{2}$ (%)
with the parabolic mirror	32.5	10.9	2.3	0
with the aplanatic lens	33.1	11.3	2.6	—

	Focusing angles α₀
	45°	60°	75°	90°
Maximum thicknesses Δα (in rad)
with the parabolic mirror	0.06	0.10	0.14	0.12
with the aplanatic lens	0.06	0.14	0.10	—

		Focusing angles α₀
		45°	60°	75°	90°
with the parabolic mirror
	transverse FWHM	0.76λ	0.46λ	0.38λ	0.36λ
	longitudinal FWHM	49.9λ	33.3λ	26.7λ	25.0λ
with the aplanatic lens
	tranverse FWHM	0.76λ	0.46λ	0.38λ	—
	longitudinal FWHM	35.3λ	16.6λ	7.0λ	—

	Parabolic mirror	Aplanatic lens
$U_{r} (α_{0})$	$\csc α_{0} - \tan α_{0}$	$\cot α_{0} - \frac{3}{2} \tan α_{0}$
$V_{r} (α_{0})$	$\sec α_{0} \tan^{2} (\frac{1}{2} α_{0}) + \frac{3}{2} \cot α_{0} \tan (\frac{1}{2} α_{0})$	$\frac{3}{8} \tan^{2} α_{0} - \frac{3}{4}$
$U_{z} (α_{0})$	$\cot (\frac{1}{2} α_{0})$	$2 \cot α_{0} - \frac{1}{2} \tan α_{0}$
$V_{z} (α_{0})$	$\frac{1}{2} \sec^{2} (\frac{1}{2} α_{0}) (1 + \frac{3}{2} \cos α_{0})$	$- \frac{1}{8} \tan^{2} α_{o} - \frac{5}{4}$

Needles of longitudinally polarized light: guidelines for minimum spot size and tunable axial extent

Abstract

1. Introduction

2. Focusing radially polarized beams beyond the paraxial approximation

3. Focusing a narrow annulus of radially polarized light

4. Characterization of the needle of longitudinally polarized light

5. Domain of validity of the analytical solutions

6. Narrow annulus of light produced by the far field of a Bessel–Gauss laser beam

7. Conclusion

Appendix: Evaluation of the integrals (6a)–(6b)

Acknowledgments

References and links

Cited By

Figures (6)

Tables (4)

Equations (38)

Optics Express