Confocal microscopy with a radially polarized focused beam

Peiwen Meng; Silvania Pereira; Paul Urbach

doi:10.1364/OE.26.029600

1. Introduction

Many super-resolution microscopes such as STED [1], PALM [2], and STORM [3] have become very important in science and applications because of their high quality images. While the applications of these super-resolution microscopy are attractive, there are some disadvantages like extremely expensive systems, time-consuming data acquisition and choices of fluorescent dyes, which limit the techniques in some cases. Although the lateral resolution of conventional confocal imaging systems is limited because of the diffraction limit, confocal microscopy has a wide range of applications in the biological and medical sciences [4–6], as well as nano-crystal imaging and spectroscopy [7, 8]. Thus, the development of a cheap and simplified confocal system combined with super-resolution imaging is very attractive.

The total point spread function (PSF) of the confocal system is determined by both the excitation PSF and the detection PSF, where the former is related to the focused illumination and the latter depends on the small pinhole or a fiber in the detector plane [9]. A very small pinhole which can remove the out of focus information plays an important role in improving both the axial and lateral resolutions in confocal microscopy. Wilson [10] compares the images theoretically and experimentally of scatterers using conventional and confocal microscopes. However, only linearly and circularly polarized beams are considered there. Cylindrical vector beams are well-established tools in optics because of the applicability of radially polarized beams in several areas. For example, using radially polarized light, the longitudinal electric component of the illuminating focused spot can be made small [11–13]. Provided the sample interacts only with the longitudinal component, higher resolution can be obtained. For a high numerical aperture (NA) system, and a radially polarized pupil field, the longitudinal component can be enhanced compared to the transverse component by using an annular aperture in the pupil of the focusing objective. The latter also results in a tighter focusing spot size of the longitudinal component in the focal plane [14–16], although this is at the expense of stronger sidelobes which will reduce the imaging quality. However, annular apertures do not give the largest longitudinal component in focus. Other pupil filter functions, for example, a BOE [17–20], a parabolic mirror and a flat diffractive lens [21] are proposed to achieve sharper focal spot sizes and as a consequence, higher resolution. By shaping the radially polarized pupil field such that the amplitude increases monotonically in a specific way as function of the pupil radius, the full-width-at-half-maximum (FWHM) of the intensity of longitudinal component reaches a value that is 15% to 30% lower than that of the spot of a focused linear polarized pupil field [22]. Spot-size reduction by means of focusing the optimized radially polarized light is shown experimentally according to theoretical productions [23]. However, in a confocal configuration, when strong longitudinal field excites a longitudinal sensitive sample, for example, a dipole is oriented parallel to the optical axis, the consequence is that its image with low NA that is formed at the pinhole side has a zero at the center [24]. In order to make the signal detectable, a polarization convertor placed in the path of detection plane is proposed to engineer the detection PSF from a doughnut shape to one with maximum in the center [25]. In the reference [26], good imaging results are achieved with the confocal microscope together with a polarization convertor experimentally. However, the theoretical analysis of the entire system is not complete.

In this paper, we present a fully rigorous vectorial theory to describe the whole confocal imaging process in a high NA system with spatially shaped radially polarized illumination. The optimized pupil field which maximizes the longitudinal electric field component in the focal point [22] is compared with the linearly polarized and traditional radially polarized pupil field to validate that the former one gives higher lateral resolution. The annular pupil field with radially polarized illumination is also considered. We use two longitudinally oriented electric dipoles at variable distances as test object. Other orientations can be easily considered. A suitable pinhole size is chosen before the detector plane to enhance the system performance, and a polarization convertor is inserted in the collimated optical path before the pinhole to transform the radially polarized light scattered by the object back to linearly polarized light to optimize the power of the light after the pinhole.

2. Theoretical analysis

In scalar diffraction theory the focal field of a lens is considered without taking into account the vectorial nature of the light field. However, the scalar method is not valid for optical systems of high NA that we study in this paper and vectorial theory is needed to describe the field near the focal plane. Originating from Ignatovsky’s diffraction theory [27], later studied in detail in [28, 29], the solution is referred as the vectorial Richard-Wolf integral.

Rigorous analysis of vectorial image formation in the confocal system can be found in [9]. In this paper we consider different polarizations and optimized pupil field for the illumination. Fig. 1(a) shows the configuration of the confocal imaging system. In order to make the whole analytical process clear, we consider first the focusing spot which illuminates the sample. This focused spot excites electric dipoles in the sample which are then imaged on the detector. The imaging of the excited dipoles is studied in subsequent sections. All optical fields that we consider in this paper are time harmonic with time dependence given by the factor exp(−iωt), where ω > 0 is the frequency. This factor is omitted from all formulas below.

Fig. 1 Schematic of the confocal microscope. A beam splitter divides the excitation path and detection path into two arms. A laser beam is focused onto the sample by a high NA objective lens L₁. The light scattered by the sample is collected by the same objective lens and focused by a small NA lens onto a pinhole in front of a detector. a) The complete confocal microscopy system, b) Focusing and dipole excitation, c) Imaging and polarization conversion.

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2.1. Dipole excitation using a focused spot

Figure 1(b) describes the process of focusing the illumination beam by lens L₁. This focused field subsequently excites electric dipoles in the sample. We choose a coordinate system (x, y, z) with z-axis coinciding with the optical axis and origin at the Gaussian focal point of lens L₁ and such that the illuminating beam propagates in the positive z-direction. When focused in a homogeneous medium with real refractive index n, the electric and magnetic fields in the focal region of lens L₁ can be expanded into plane waves:

\begin{matrix} E^{e} (r) = \frac{1}{4 π^{2}} \iint_{k_{⊥} \leq k_{0} {NA}_{1}} A (k_{⊥}) \exp (i k \cdot r) d^{2} k_{⊥}, \\ H^{e} (r) = \frac{1}{4 π^{2}} \frac{1}{ω μ_{0}} \iint_{k_{⊥} \leq k_{0} {NA}_{1}} A \times A (k_{⊥}) \exp (i k \cdot r) d^{2} k_{⊥}, \end{matrix}

where

\vec{A} (\vec{k})

is the vector amplitude of the plane wave with the wave vector

\vec{k} = (k_{x}, k_{y}, k_{z})

, k_⊥ = (k_x, k_y) is the part of the wave vector perpendicular to the optical axis, with

k_{⊥} = \sqrt{k_{x}^{2} + k_{y}^{2}}

its length,

k_{z} = \sqrt{k^{2} - k_{⊥}^{2}}

, where k = k₀n with k₀ = 2π/λ₀ with λ₀ the wavelength in vacuum. Note that the transverse wavenumber k_⊥ satisfies: 0 ≤ k_⊥ ≤ k₀NA₁, where

{NA}_{1} = n \sin θ_{\max}^{(1)}

with

θ_{\max}^{(1)}

the maximum angle between the wave vectors and the positive z direction.

To define the polarizations of the plane waves, we define a positively oriented orthonormal basis $\hat{k}$ , $\hat{p}$ , $\hat{s}$ in reciprocal k−space by:

\begin{matrix} \hat{k} = \frac{k_{x}}{k} \hat{x} + \frac{k_{y}}{k} \hat{y} + \frac{k_{z}}{k} \hat{z} \\ \hat{p} (k_{⊥}) = \frac{k_{x} k_{z} \hat{x} + k_{y} k_{z} \hat{y} - (k_{x}^{2} + k_{y}^{2}) \hat{z}}{k \sqrt{k_{x}^{2} + k_{y}^{2}}}, \\ \hat{s} (k_{⊥}) = \frac{- k_{y} \hat{x} + k_{x} \hat{y}}{\sqrt{k_{x}^{2} + k_{y}^{2}}}, \end{matrix}

Since the electric field is free of divergence, the vector amplitude A can be written as:

A (k_{⊥}) = A_{p} (k_{⊥}) \hat{p} (k_{⊥}) + A_{s} (k_{⊥}) \hat{s} (k_{⊥}),

where A_p and A_s are the components of A on the basis of

\hat{p}

,

\hat{s}

.

The electromagnetic field in focus can be therefore rewritten as:

\begin{matrix} E^{e} (r) = \frac{1}{4 π^{2}} \iint_{k_{⊥} \leq k_{0} {NA}_{1}} (A_{p} \hat{p} + A_{s} \hat{s}) e^{i k \cdot r} d^{2} k_{⊥}, \\ H^{e} (r) = \frac{1}{4 π^{2}} \frac{1}{ω μ_{0}} \iint_{k_{⊥} \leq k_{0} {NA}_{1}} (- A_{s} \hat{p} + A_{p} \hat{s}) e^{i k \cdot r} d^{2} k_{⊥} . \end{matrix}

According to the vectorial diffraction theory [27–29], the plane wave amplitudes A_p, A_s are linked to the radial and azimuthal components of the pupil field by:

A_{p} (k_{⊥}) = \frac{2 π i f_{1}}{\sqrt{k k_{z}}} E_{ρ}^{e, p} (ρ_{p}, φ_{p}), A_{s} (k_{⊥}) = \frac{2 π i f_{1}}{\sqrt{k k_{z}}} E_{φ}^{e, p} (ρ_{p}, φ_{p}),

where, f₁ is the focal distance of the objective lens L₁ and ρ_p, φ_p are polar pupil coordinates defined by:

ρ_{p} = f_{1} \frac{k_{⊥}}{k_{0} n}, ρ_{p} \cos φ_{p} = - f_{1} \frac{k_{x}}{k_{0} n}, ρ_{p} \sin φ_{p} = - f_{1} \frac{k_{y}}{k_{0} n},

Note that

\cos φ_{p} = - k_{x} / k_{⊥}, \sin φ_{p} = - k_{y} / k_{⊥} .

Next we consider the focusing of two pupil fields in more details.

2.1.1. Focusing of a linear polarized pupil field

When the pupil field is in all points of the pupil linearly polarized parallel to the x-direction, we have:

E_{ρ}^{e, p} (ρ_{p}, φ_{p}) = g (ρ_{p}) \cos φ_{p}, E_{φ}^{e, p} (ρ_{p}, φ_{p}) = g (ρ_{p}) \sin φ_{p},

where g(ρ_p) is the amplitude which may be a function of the radius. Using Eq. (7), we get:

A_{p} (k_{⊥}) = \frac{- 2 π i f_{1} k_{x}}{\sqrt{k k_{z}} k_{⊥}} g (k_{⊥}), A_{s} (k_{⊥}) = \frac{- 2 π i f_{1} k_{y}}{\sqrt{k k_{z}} k_{⊥}} g (k_{⊥}),

where for brevity, we have written g(k_⊥) instead of g(f₁k_⊥/k).

We substitute Eq. (9) into Eq. (4) and use the following integrals [30]:

\begin{matrix} \int_{0}^{2 π} \cos n ϕ e^{i x \cos (ϕ - φ)} d ϕ = 2 π i^{n} J_{n} (x) \cos n φ, \\ \int_{0}^{2 π} \sin n ϕ e^{i x \cos (ϕ - φ)} d ϕ = 2 π i^{n} J_{n} (x) \sin n φ, \end{matrix}

where J_n(x) is the Bessel function of nth order. By using cylindrical coordinates in the focal region, we then obtain for the electric field in the focal region:

E^{e} (ρ, φ, z) = \frac{- i f_{1}}{2 k^{3 / 2}} (\begin{matrix} - I_{00} (ρ, z) - I_{02} (ρ, z) \cos 2 φ \\ - I_{02} (ρ, z) \sin 2 φ \\ 2 i I_{01} (ρ, z) \cos φ \end{matrix}),

where

I_{00} (ρ, z) = \int_{0}^{k_{0} {NA}_{1}} g (k_{⊥}) (k + k_{z}) J_{0} (k_{⊥} ρ) \frac{k_{⊥}}{\sqrt{k_{z}}} e^{i k_{z} z} d k_{⊥},

I_{01} (ρ, z) = \int_{0}^{k_{0} {NA}_{1}} g (k_{⊥}) k_{⊥} J_{1} (k_{⊥} ρ) \frac{k_{⊥}}{\sqrt{k_{z}}} e^{i k_{z} z} d k_{⊥},

I_{02} (ρ, z) = \int_{0}^{k_{0} {NA}_{1}} g (k_{⊥}) (k - k_{z}) J_{2} (k_{⊥} ρ) \frac{k_{⊥}}{\sqrt{k_{z}}} e^{i k_{z} z} d k_{⊥} .

2.1.2. Focusing of a radially polarized pupil field

For a pupil field which is in all points of the pupil polarized in the radial direction, the radial and azimuthal components of the electric pupil field are given by:

E_{ρ}^{e, p} (ρ_{p}, φ_{p}) = g (ρ_{p}), E_{φ}^{e, p} (ρ_{p}, φ_{p}) = 0,

where the amplitude g is a function of the pupil radius. By using Eq.(5), we get for the p- and s-components of the electric field of the plane waves in the focal region:

A_{p} (k_{⊥}) = \frac{2 π i f_{1}}{\sqrt{k k_{z}}} g (k_{⊥}), A_{s} (k_{⊥}) = 0,

where (as above) we have written g(k_⊥) instead of the formally more correct g(f₁k_⊥/k). Then with Eq. (4) and the integrals of Eq. (10), the electric field in the focal region becomes in terms of cylindrical coordinates:

E^{e} (ρ, φ, z) = \frac{- f_{1}}{k^{3 / 2}} (\begin{matrix} I_{11} (ρ, z) \cos φ \\ I_{11} (ρ, z) \sin φ \\ i I_{10} (ρ, z) \end{matrix}),

where

I_{10} (ρ, z) = \int_{0}^{k_{0} {NA}_{1}} g (k_{⊥}) k_{⊥} J_{0} (k_{⊥} ρ) \frac{k_{⊥}}{\sqrt{k_{z}}} e^{i k_{z} z} d k_{⊥},

I_{11} (ρ, z) = \int_{0}^{k_{0} {NA}_{1}} g (k_{⊥}) k_{z} J_{1} (k_{⊥} ρ) \frac{k_{⊥}}{\sqrt{k_{z}}} e^{i k_{z} z} d k_{⊥} .

We discuss now two special radially polarized pupil fields. The first is the pupil field derived in [22] which gives the largest possible longitudinal (i.e. z-component) of the electric field component in the focal point, for a given power P₀ in the pupil of the lens. This solution makes the longitudinal component of the focused electric field quite narrow. It was shown in [22] that the amplitude of this optimum pupil field is given by:

g (k_{⊥}) = - \frac{k_{⊥}^{3 / 2} k^{1 / 2}}{2 π i f_{1} k_{z} Λ},

where

Λ = {(\frac{π}{P_{0}})}^{1 / 2} \frac{n^{1 / 2}}{λ_{0}} {(\frac{ϵ_{0}}{μ_{0}})}^{1 / 4} {(\frac{2}{3} - \sqrt{1 - {({NA}_{1} / n)}^{2}} + \frac{1}{3} \sqrt{{(1 - {({NA}_{1} / n)}^{2})}^{3}})}^{1 / 2},

By substituting Eq. (20), Eqs. (18) and (19) become:

I_{10} (ρ, z) = - \frac{k^{1 / 2}}{2 π i f_{1} Λ} \int_{0}^{k_{0} {NA}_{1}} J_{0} (k_{⊥} ρ) \frac{k_{⊥}^{7 / 2}}{k_{z}^{3 / 2}} e^{i k_{z} z} d k_{⊥},

I_{11} (ρ, z) = - \frac{k^{1 / 2}}{2 π i f_{1} Λ} \int_{0}^{k_{0} {NA}_{1}} J_{1} (k_{⊥} ρ) \frac{k_{⊥}^{5 / 2}}{k_{z}^{3 / 2}} e^{i k_{z} z} d k_{⊥} .

The second choice for the amplitude g of the radially polarized pupil field which gives a narrow longitudinal component in the focal plane corresponds to an annular pupil:

g (ρ_{p}) = {\begin{array}{l} 1 & a - Δ ρ_{p} < ρ_{p} \leq a \\ 0 & otherwise \end{array},

where a is the radius of the pupil of the lens L₁ and Δρ_p is the width of the annular ring. Then the focal field is obtained by substituting Eq. (24) for g in Eqs. (17)–(19). As we will show in Section 3, the longitudinal component in the focal plane of the annular pupil is narrower than the optimized pupil amplitude in Eq. (20) and becomes more narrower for smaller width Δρ_p of the annular pupil aperture. However, the annular pupil field has the disadvantage that for a very small annular aperture, there is only little energy in the focal region.

2.2. Imaging without a polarization convertor

Figure 1(c) shows the imaging part of the optical system. In the Born approximation, the focused field Eⁱ excites a dipole density P^d(r_d) at the position r_d = (x_d, y_d, z = 0), where the z = 0 plane is, as before, assumed to coincide with the focal plane of lens L₁. The dipole vector is given by:

P^{d} = \overset{\leftrightarrow}{α} E^{e} (r_{d}),

where

\overset{\leftrightarrow}{α}

is the electric polarizability. It in general is a tensor, and we assume that the tensor has principal axis that are parallel to the x, y, z–axis, i.e

\overset{\leftrightarrow}{α} = (\begin{matrix} α_{x x} & 0 & 0 \\ 0 & α_{y y} & 0 \\ 0 & 0 & α_{z z} \end{matrix}) .

Let E^d be the electric field radiated by this dipole. Its plane wave amplitudes in the entrance pupil of lens L₁, i.e. for z = f₁, are given by [31]:

A^{d} (k_{⊥}) = - \frac{e^{i k_{z} f_{1}}}{2 i ϵ_{0} n^{2} k_{z}} k \times (k \times P^{d}),

Hence,

A_{p}^{d} (k_{⊥}) = \frac{e^{i k_{z} f_{1}}}{2 i ϵ_{0} n^{2} k_{z}} k^{2} P^{d} \cdot \hat{p}, A_{s}^{d} (k_{⊥}) = \frac{e^{i k_{z} f_{1}}}{2 i ϵ_{0} n^{2} k_{z}} k^{2} P^{d} \cdot \hat{s},

Using Eq. (5) and (6), the radial and azimuthal components of the pupil field are:

E_{ρ}^{d, p} (ρ_{p}, φ_{p}) = \frac{\sqrt{k k_{z}}}{2 π i f_{1}} A_{p}^{d} (k_{⊥}), E_{φ}^{d, p} (ρ_{p}, φ_{p}) = \frac{\sqrt{k k_{z}}}{2 π i f_{1}} A_{s}^{d} (k_{⊥}) .

The refocusing by the second lens L₂ with focal length f₂ yields plane wave amplitudes in image space which are given by

\begin{matrix} {\tilde{A}}_{p}^{d} ({\tilde{k}}_{⊥}) = \frac{2 π i f_{2}}{\sqrt{k {\tilde{k}}_{z}}} E_{ρ}^{d, p} (ρ_{p}, φ_{p}) = \frac{f_{2}}{f_{1}} \sqrt{\frac{k_{z}}{{\tilde{k}}_{z}}} \frac{e^{i k_{z} f_{1}}}{2 i ϵ_{0} n^{2} k_{z}} k^{2} P^{d} \cdot \hat{p}, \\ {\tilde{A}}_{s}^{d} ({\tilde{k}}_{⊥}) = \frac{2 π i f_{2}}{\sqrt{k {\tilde{k}}_{z}}} E_{φ}^{d, p} (ρ_{p}, φ_{p}) = \frac{f_{2}}{f_{1}} \sqrt{\frac{k_{z}}{{\tilde{k}}_{z}}} \frac{e^{i k_{z} f_{1}}}{2 i ϵ_{0} n^{2} k_{z}} k^{2} P^{d} \cdot \hat{s}, \end{matrix}

with

ρ_{p} = f_{2} \frac{{\tilde{k}}_{⊥}}{k}, ρ_{p} \cos φ_{p} = - f_{2} \frac{{\tilde{k}}_{x}}{k}, ρ_{p} \sin φ_{p} = - f_{2} \frac{{\tilde{k}}_{y}}{k},

where the wavenumber

k = \tilde{k}

, as the two lenses are in the same medium.

The numerical aperture NA₂ of the lens L₂ before the detector is smaller than that of the objective lens L₁, but the pupils of both lenses are identical. Hence the focal lengths of the two lenses are different. This implies the following relationship between the wave vectors of corresponding plane waves on the object side of lens L₁ and the image side of lens L₂:

k_{⊥} = \frac{f_{2}}{f_{1}} {\tilde{k}}_{⊥}, k_{z} = \sqrt{k^{2} - {(f_{2} / f_{1})}^{2} {\tilde{k}}_{⊥}^{2}} .

By substituting Eq. (30) into Eq. (4), the electric field in image space becomes:

\begin{matrix} E^{(i)} (r) = \frac{1}{4 π^{2}} [\iint_{{\tilde{k}}_{⊥} \leq k_{0} {NA}_{2}} ({\tilde{A}}_{p}^{d} ({\tilde{k}}_{⊥}) \hat{p} ({\tilde{k}}_{⊥}) + {\tilde{A}}_{s}^{d} ({\tilde{k}}_{⊥}) \hat{s} ({\tilde{k}}_{⊥})) e^{i \tilde{k} \cdot r} d^{2} {\tilde{k}}_{⊥}] \cdot P^{d} \\ = \frac{k^{2} f_{2}}{4 π i ϵ_{0} n^{2} f_{1}} [\iint_{{\tilde{k}}_{⊥} \leq k_{0} {NA}_{2}} \frac{1}{\sqrt{k_{z} {\tilde{k}}_{z}}} (\hat{p} ({\tilde{k}}_{⊥}) \otimes \hat{p} (\frac{f_{2}}{f_{1}} {\tilde{k}}_{⊥}) + \hat{s} ({\tilde{k}}_{⊥}) \otimes \hat{s} (\frac{f_{2}}{f_{1}} {\tilde{k}}_{⊥})) \cdot e^{i k_{z} f_{1}} e^{i \tilde{k} \cdot r} d^{2} {\tilde{k}}_{⊥}] \cdot P^{d}, \end{matrix}

where

\begin{matrix} \hat{p} ({\tilde{k}}_{⊥}) = \frac{1}{k^{2} {\tilde{k}}_{⊥}} (\begin{matrix} {\tilde{k}}_{x} {\tilde{k}}_{z} \\ {\tilde{k}}_{y} {\tilde{k}}_{z} \\ - {\tilde{k}}_{⊥}^{2} \end{matrix}), \hat{p} (\frac{f_{2}}{f_{1}} {\tilde{k}}_{⊥}) = \frac{1}{k^{2} {\tilde{k}}_{⊥}} (\begin{matrix} {\tilde{k}}_{x} k_{z} \\ {\tilde{k}}_{y} k_{z} \\ - \frac{f_{2}}{f_{1}} {\tilde{k}}_{⊥}^{2} \end{matrix}), \\ \hat{s} ({\tilde{k}}_{⊥}) = \hat{s} (\frac{f_{2}}{f_{1}} {\tilde{k}}_{⊥}) = \frac{1}{k^{2} {\tilde{k}}_{⊥}} (\begin{matrix} - {\tilde{k}}_{y} \\ - {\tilde{k}}_{x} \\ 0 \end{matrix}), \end{matrix}

Using polar coordinates, the field in the image space can be written as:

\begin{matrix} E_{x}^{(i)} (ρ, φ, z) = \frac{- f_{2}}{2 ϵ_{0} n^{2} f_{1}} [(K_{00}^{0} + K_{z z}^{0}) p_{x}^{d i p} - 2 i K_{⊥ z}^{1} \cos φ p_{z}^{d i p} + (K_{00}^{2} - K_{z z}^{2}) (\cos 2 φ p_{x}^{d i p} + \sin 2 φ p_{y}^{d i p})], \\ E_{y}^{(i)} (ρ, φ, z) = \frac{- f_{2}}{2 ϵ_{0} n^{2} f_{1}} [(K_{00}^{0} + K_{z z}^{0}) p_{y}^{d i p} - 2 i K_{⊥ z}^{1} \sin φ p_{z}^{d i p} + (K_{00}^{2} - K_{z z}^{2}) (\sin 2 φ p_{x}^{d i p} - \cos 2 φ p_{y}^{d i p})], \\ E_{z}^{(i)} (ρ, φ, z) = \frac{- f_{2}}{2 ϵ_{0} n^{2} f_{1}} [2 K_{⊥ ⊥}^{0} p_{z}^{d i p} - 2 i K_{z z}^{1} (\cos φ p_{x}^{d i p} + \sin φ p_{y}^{d i p})], \end{matrix}

where

\begin{matrix} K_{00}^{n} (ρ, z) = \int_{0}^{k_{0} {NA}_{2}} {(\frac{{\tilde{k}}_{z}}{k})}^{1 / 2} \frac{{\tilde{k}}_{⊥}}{k \sqrt{k^{2} - {(f_{2} / f_{1})}^{2} {\tilde{k}}_{⊥}^{2}}} J_{n} ({\tilde{k}}_{⊥} ρ) e^{i {\tilde{k}}_{z} z} \cdot e^{i z \sqrt{k^{2} - {(f_{2} / f_{1})}^{2} {\tilde{k}}_{⊥}^{2}}} d {\tilde{k}}_{⊥}, \\ K_{⊥ ⊥}^{n} (ρ, z) = \int_{0}^{k_{0} {NA}_{2}} {(\frac{{\tilde{k}}_{z}}{k})}^{1 / 2} \frac{{\tilde{k}}_{⊥}}{k \sqrt{k^{2} - {(f_{2} / f_{1})}^{2} {\tilde{k}}_{⊥}^{2}}} q_{⊥ ⊥} ({\tilde{k}}_{⊥}) J_{n} ({\tilde{k}}_{⊥} ρ) e^{i {\tilde{k}}_{z} z} \cdot e^{i z \sqrt{k^{2} - {(f_{2} / f_{1})}^{2} {\tilde{k}}_{⊥}^{2}}} d {\tilde{k}}_{⊥}, \\ K_{⊥ z}^{n} (ρ, z) = \int_{0}^{k_{0} {NA}_{2}} {(\frac{{\tilde{k}}_{z}}{k})}^{1 / 2} \frac{{\tilde{k}}_{⊥}}{k \sqrt{k^{2} - {(f_{2} / f_{1})}^{2} {\tilde{k}}_{⊥}^{2}}} q_{⊥ z} ({\tilde{k}}_{⊥}) J_{n} ({\tilde{k}}_{⊥} ρ) e^{i {\tilde{k}}_{z} z} \cdot e^{i z \sqrt{k^{2} - {(f_{2} / f_{1})}^{2} {\tilde{k}}_{⊥}^{2}}} d {\tilde{k}}_{⊥}, \\ K_{z z}^{n} (ρ, z) = \int_{0}^{k_{0} {NA}_{2}} {(\frac{{\tilde{k}}_{z}}{k})}^{1 / 2} \frac{{\tilde{k}}_{⊥}}{k \sqrt{k^{2} - {(f_{2} / f_{1})}^{2} {\tilde{k}}_{⊥}^{2}}} q_{z z} ({\tilde{k}}_{⊥}) J_{n} ({\tilde{k}}_{⊥} ρ) e^{i {\tilde{k}}_{z} z} \cdot e^{i z \sqrt{k^{2} - {(f_{2} / f_{1})}^{2} {\tilde{k}}_{⊥}^{2}}} d {\tilde{k}}_{⊥}, \end{matrix}

with

{\tilde{k}}_{⊥} = \sqrt{{\tilde{k}}_{x}^{2} + {\tilde{k}}_{y}^{2}}

,

{\tilde{k}}_{z} = \sqrt{k^{2} + {\tilde{k}}_{⊥}^{2}}

, and

q_{⊥ ⊥} ({\tilde{k}}_{⊥}) = \frac{k_{⊥} {\tilde{k}}_{⊥}}{k^{2}}, q_{⊥ z} ({\tilde{k}}_{⊥}) = \frac{k_{⊥} {\tilde{k}}_{z}}{k^{2}}, q_{z z} ({\tilde{k}}_{⊥}) = \frac{k_{z} {\tilde{k}}_{z}}{k^{2}} .

We introduced the term

{(\frac{{\tilde{k}}_{z}}{k})}^{1 / 2}

to account for an aplanatic lens.

2.3. Imaging with a polarization convertor

The longitudinal component cannot be easily measured by the detector. Furthermore, at the center of the detector the radial component is weaker than the longitudinal component. To detect the longitudinal component, we add a polarization convertor [26] between the collimator lens and the detector lens, which transforms the radially polarized light into x-polarized light before being refocused by the detector lens. As only x-polarized light needs to be measured by the detector, we put a polarizer in front of the detector as seen in Fig. 1(c). Then, the field in image space after the convertor and the polarizer becomes

E^{(i), p} (ρ_{p}, φ_{p}) = E_{ρ}^{d, p} (ρ_{p}, φ_{p}) \hat{x},

Due to the convertor, the radial and azimuthal components of the field in the entrance pupil of the second lens become:

E_{ρ}^{(i), p} (ρ_{p}, φ_{p}) = E_{ρ}^{(i), p} (ρ_{p}, φ_{p}) \cos ϕ, E_{φ}^{(i), p} (ρ_{p}, φ_{p}) = - E_{ρ}^{d, p} (ρ_{p}, φ_{p}) \sin ϕ .

The vector amplitudes of the plane waves in image space corresponding to the pupil field have p- and s- components are given by:

{\tilde{A}}_{p} ({\tilde{k}}_{⊥}) = \frac{2 π i f_{2}}{\sqrt{\tilde{k} {\tilde{k}}_{z}}} E_{ρ}^{(i), p} (ρ_{p}, φ_{p}), {\tilde{A}}_{s} ({\tilde{k}}_{⊥}) = \frac{2 π i f_{2}}{\sqrt{\tilde{k} {\tilde{k}}_{z}}} E_{φ}^{(i), p} (ρ_{p}, φ_{p}),

Substituting Eq. (40) into Eq. (4) and using polar coordinates, finally, gives for the field in image space can be deduced as:

\begin{matrix} E_{x}^{(i)} (ρ, φ, z) = \frac{- f_{2}}{2 ϵ_{0} n^{2} f_{1}} [\frac{i}{2} (K_{z z}^{3} - K_{z}^{3}) (\cos 3 φ p_{x}^{d i p} + \sin 3 φ p_{y}^{d i p}) - \frac{i}{2} K_{z z}^{1} (3 \cos φ p_{x}^{d i p} + \sin φ p_{y}^{d i p}) \\ - \frac{1}{2} K_{z}^{1} (\cos φ p_{x}^{d i p} + 3 \sin φ p_{y}^{d i p}) - (K_{⊥ z}^{2} - K_{⊥}^{2}) \cos 2 φ p_{z}^{d i p} + (K_{⊥ z}^{0} + K_{⊥}^{0}) p_{z}^{d i p}], \\ E_{y}^{(i)} (ρ, φ, z) = \frac{- f_{2}}{2 ϵ_{0} n^{2} f_{1}} [\frac{i}{2} (K_{z z}^{3} - K_{z}^{3}) (\sin 3 φ p_{x}^{d i p} - \cos 3 φ p_{y}^{d i p}) - \frac{i}{2} (K_{z z}^{1} - K_{z}^{1}) (\sin φ p_{x}^{d i p} + \cos φ p_{y}^{d i p}) \\ - (K_{⊥ z}^{2} - K_{⊥}^{2}) \sin 2 φ p_{z}^{d i p}], \\ E_{z}^{(i)} (ρ, φ, z) = \frac{- f_{2}}{2 ϵ_{0} n^{2} f_{1}} [- K_{z ⊥}^{1} p_{x}^{d i p} + K_{z ⊥}^{2} (\cos 2 φ p_{x}^{d i p} + \sin 2 φ p_{y}^{d i p}) + 2 i K_{⊥ ⊥}^{1} \cos φ p_{z}^{d i p}], \end{matrix}

where

\begin{matrix} K_{z ⊥}^{n} (ρ, z) = \int_{0}^{k_{0} {NA}_{2}} {(\frac{{\tilde{k}}_{z}}{k})}^{1 / 2} \frac{{\tilde{k}}_{⊥}}{k \sqrt{k^{2} - {(f_{2} / f_{1})}^{2} {\tilde{k}}_{⊥}^{2}}} q_{z ⊥} ({\tilde{k}}_{⊥}) J_{n} ({\tilde{k}}_{⊥} ρ) e^{i {\tilde{k}}_{z} z} \cdot e^{i z \sqrt{k^{2} - {(f_{2} / f_{1})}^{2} {\tilde{k}}_{⊥}^{2}}} d {\tilde{k}}_{⊥}, \\ K_{z}^{n} (ρ, z) = \int_{0}^{k_{0} {NA}_{2}} {(\frac{{\tilde{k}}_{z}}{k})}^{1 / 2} \frac{{\tilde{k}}_{⊥}}{k \sqrt{k^{2} - {(f_{2} / f_{1})}^{2} {\tilde{k}}_{⊥}^{2}}} q_{z} ({\tilde{k}}_{⊥}) J_{n} ({\tilde{k}}_{⊥} ρ) e^{i {\tilde{k}}_{z} z} \cdot e^{i z \sqrt{k^{2} - {(f_{2} / f_{1})}^{2} {\tilde{k}}_{⊥}^{2}}} d {\tilde{k}}_{⊥}, \\ K_{⊥}^{n} (ρ, z) = \int_{0}^{k_{0} {NA}_{2}} {(\frac{{\tilde{k}}_{z}}{k})}^{1 / 2} \frac{{\tilde{k}}_{⊥}}{k \sqrt{k^{2} - {(f_{2} / f_{1})}^{2} {\tilde{k}}_{⊥}^{2}}} q_{⊥} ({\tilde{k}}_{⊥}) J_{n} ({\tilde{k}}_{⊥} ρ) e^{i {\tilde{k}}_{z} z} \cdot e^{i z \sqrt{k^{2} - {(f_{2} / f_{1})}^{2} {\tilde{k}}_{⊥}^{2}}} d {\tilde{k}}_{⊥}, \end{matrix}

where

q_{z ⊥} ({\tilde{k}}_{⊥}) = \frac{k_{z} {\tilde{k}}_{⊥}}{k^{2}}, q_{⊥} ({\tilde{k}}_{⊥}) = \frac{k_{⊥}}{k}, q_{z} ({\tilde{k}}_{⊥}) = \frac{k_{z}}{k} .

and k_⊥ and k_z are functions of

{\tilde{k}}_{⊥}

given by Eq. (32). Other symbols are shown in Eq. (37). Again, we introduced the term

{(\frac{{\tilde{k}}_{z}}{k})}^{1 / 2}

to account for an aplanatic lens.

3. Numerical results

Through this section, we assume the light with wavelength λ = 500nm. With Eq. (11) and (17), the focal fields of linearly and radially polarized illumination can be obtained. It follows for Fig. 2 that the FWHM of the electric energy density $| \vec{E} |^{2}$ in the focal plane of the radially polarized pupil field, is slightly smaller than that of a linearly pupil field ( ${FWHM}_{t o t a l}^{r a d} \approx 0.72 λ$ , ${FWHM}_{t o t a l}^{l i n} \approx 0.76 λ$ ). In contrast, the FWHM of the squared modulus of the longitudinal component corresponding to the radially polarized pupil field $({FWHM}_{l o n g}^{r a d} = 0.48 λ)$ , is much smaller than the FWHM of $| \vec{E} |$ in the case of a linearly polarized pupil field. Thus, the polarization state of the light strongly influences the size of the focused spot in the case of high NA. If the sample is sensitive only to the longitudinal component of the focal spot, then the use of radially polarized light can result in a substantial improvement in resolution.

Fig. 2 Profiles of the squared amplitude of the longitudinal and transverse components and total energy density $| \vec{E} |^{2}$ in the focal plane in the case of a pupil field that is linearly polarized in the x–direction (a) and a pupil field that is radially polarized (b). The focusing lens has NA=0.9. The plots are normalized to the on-axis peak values.

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Figure 3 shows the comparison of the squared amplitude of the longitudinal component of the focal spots obtained in the case of radially polarized illumination with full aperture, annular aperture and optimized radially polarized illumination, as described in Eqs. (17)–(24). The optimized pupil field has FWHM which is 5.4% smaller than that of the full aperture constant pupil field. The FWHM of the squared amplitude of the longitudinal component obtained by focusing a radially polarized beam using a ring mask function (with radius 90% of the total pupil) is even smaller than the above two cases. However, the expanding side lobes in the annular case is larger than for the full aperture and the optimized case. But the energy in the focal region is of course much weaker in the annular case.

Fig. 3 Profiles of the squared amplitudes of the longitudinal components of the exciting spot in the focal plane in the case of radially polarized illumination with full aperture, annular aperture and optimized radially polarized pupil field. The plots are normalized to their on-axis maxima.

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We use expressions Eq. (35) and (41) (with/without a polarization convertor) to compute the normalized detector signal. In order to make full use of the longitudinal component, the electric dipole is set along the z-axis. A small pinhole is taken into consideration in the confocal system [32]. The criterion for choosing a suitable pinhole size is to make an aperture giving 50% of the maximum intensity [33].

As is indicated in Fig. 1, we detect the intensity with a single-pixel camera at the focal point of the detector lens, which is located at r = (0, 0, 0). However, if we assume purely longitudinal dipole excitation, then without polarization conversion, the intensity at the focal point is a local minimum, meaning that very little intensity is detected. To solve this problem, we propose to use a polarization convertor in the pupil of the lens L₂ to change the radially polarized light to linearly polarized light, which in the focal plane will result in an intensity distribution with its maximum at the focal point. The intensity at the detector plane can be obtained by Eq. (41). Figure 4 shows the final signal along the x axis when the pinhole is set before the detector and the dipole is scanned in the x-y plane at the object plane. Note that because of the use of the polarization convertor, the original radial polarization is turned into linear polarization and thus the “transversal” component in Fig. 4 corresponds to the contribution of the radially polarized light emitted by the dipole.

Fig. 4 Profiles of the intensities at the detector plane with a polarization convertor and a small pinhole with radius r = 0.18µm which is 0.36λ in terms of wavelength. The system is composed of: a high NA₁ = 0.9 focusing objective lens and a low NA₂ = 0.3 lens for detection. For illumination, radially polarized light of wavelength λ = 500nm is used. The focal field can be seen in Eq. (17). The dipole is set rigidly along the z α–axis direction (only α_zz is relevant). The plots are normalized to on-axis maxima.

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To validate that the system has the advantage of higher resolution, two dipoles close to each other need to be analyzed with the above theory. Figure 5 shows the cross sections of the detected intensity when two dipoles that are parallel are scanned by a focused spot. The polarization of the illumination is taken as either linear, radial or optimized radial as shown in Eq. (11) and (17)–(24). For linearly x-polarized illumination (diamond dot line), the dipoles are set along the x-axis (only α_xx is relevant). If we want to roughly distinguish the two dipoles, the distance between them should be larger than 0.6λ according to the Rayleigh criterion. This value is approaching the diffraction limit $0.61 \frac{λ}{NA}$ . However, for normal radial illumination (solid line), radial illumination with a ring mask (dashed line) and optimized radial illumination (dot line), with the dipoles set along the z-axis (only α_zz is relevant), the smallest distance at which they can be distinguished is reduced to 0.4λ. Meanwhile, the optimized one gives more obvious contrast than the normal radial case and the radial annular case gives the highest contrast. Even when the distance between the two dipoles decreases to 0.36λ, it can be seen that for the radial excitation with a ring mask and optimized radially polarized excitation, the two dipoles can be distinguished better than for the other two focused spots. Moreover, at this distance, note that in the case of linear polarization, the dipoles cannot be resolved.

Fig. 5 Profiles of the detected intensities as function of scanning distance of two separated dipoles for different kinds of illumination and for several distances between the dipoles. The system is composed of two lenses with NA₁ = 0.9, and NA₂ = 0.3 and is illuminated by light with a wavelength of λ = 500nm. The illumination is either linearly, radially or optimized radially polarized. The two dipoles are set along the x axis for the linear polarization and along the z axis for the other radially polarized cases. A polarization convertor is added in the case of radially polarized excitation. Four different distances between the dipoles are chosen: d=0.8λ (red line), d=0.6λ (black line), d=0.4λ (yellow line), and d=0.36λ (blue line). The plots are normalized to the on-axis peak intensity.

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The visibility as a function of d/λ can be seen in Fig. 6. Here the visibility is defined as:

visibility = \frac{I_{m a x} - I_{m i n}}{I_{m a x} + I_{m i n}} .

Here I_max is the normalized intensity which equals 1 and I_min is the value of the intensity at r = (0, 0, 0) as shown in Fig. 5. When d/λ > 1.2, for the four cases, the two dipoles can be resolved very well. When 0.2 < d/λ < 1.2, it is clearly seen that the visibility value of linear excitation is smaller than the other three cases, which means the worst resolution. While the visibility value of radial excitation with the annular objective is the highest in this range. When 0.2 < d/λ < 0.3, the resolution of the system is almost the same for the three radially polarized light excitation cases. When d/λ < 0.2, the dipoles can not be resolved any more in all cases because of the diffraction limit. This property of visibility agrees well with Fig. 5.

Fig. 6 Profiles of the visibility as a function of the d/λ for the four cases of pupil fields. The system is the same as shown in Fig. 5.

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4. Conclusion

To conclude, we present a complete vectorial theoretical analysis to describe the whole imaging process of the high NA confocal system from the illumination point to the imaging plane. Strong longitudinal component as well as shaper focal spots are achieved by comparing the linear $({FWHM}_{t o t a l}^{l i n} \approx 0.76 λ)$ and radial $({FWHM}_{t o t a l}^{r a d} \approx 0.72 λ)$ polarization pupil fields. In order to obtain the higher resolution, we apply an optimized pupil field with an amplitude distribution that varies monotonically in the radial direction, which provides 5.4% tighter focused spot for NA = 0:9 than the full aperture constant pupil field. Additionally, the condition of radially polarized illumination with an annular objective lens is also considered to provide the smallest focal spot size, although at the expense of expanding side lobes.

Four kinds of focused spots are used to scan two parallel longitudinal dipoles with variable distances in the objective plane. In order to efficiently detect the field scattered by the longitudinally oriented dipoles, we insert a radial to linear polarization convertor before the pinhole. From our analysis, we show that for a pinhole in front of the detector with radius 0.36λ, by using a radial pupil field with the an annular lens to excite the dipoles, a distance of 0.36λ can be resolved, i.e., beyond the diffraction limit. The proposed method is easy to apply to other pupil fields and helpful to analyze confocal systems. The experiment can be conducted on the basis of the theory and simulated results with different samples to validate the superiority of the optimized radially polarized illumination combined with the confocal system.

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Confocal microscopy with a radially polarized focused beam

Abstract

1. Introduction

2. Theoretical analysis

2.1. Dipole excitation using a focused spot

2.1.1. Focusing of a linear polarized pupil field

2.1.2. Focusing of a radially polarized pupil field

2.2. Imaging without a polarization convertor

2.3. Imaging with a polarization convertor

3. Numerical results

4. Conclusion

References

Cited By

Figures (6)

Equations (44)

Optics Express