Focusing Hemispherical Waves

Abstract

The field in the focal plane of a hemispherically focused wave in an aplanatic optical system can be expressed in an analytical form. In fact, many different cases of hemispherically focused scalar and vectorial waves can be analytically expressed. We consider focusing with linear or circular polarized illumination as well as other cases such as electric dipole, transverse electric, and radial polarizations. We also investigate 4Pi focusing and the focusing of vortex waves. These results can be applied to focusing with microscope objectives of high numerical aperture.

1. Introduction

We consider focusing hemispherical waves in both scalar and vectorial theory. Carter numerically calculated and plotted the intensity and phase in the focal region of a uniformly weighted scalar hemispherical wave [1]. Carter also gave analytic expressions for the field in the focal plane and along the optical axis. Bertilone found an analytic expression for this diffracted field in terms of the Lommel function of two variables, $U_0(u,v)$ [2,3]. Arnoldus extended this result to the electromagnetic case [4,5,6]. Richards and Wolf gave integral expressions for the focal region of a lens of high numerical aperture (NA), satisfying the sine condition, illuminated by a linearly polarized wave, based on theory developed by Ignatowsky [7,8]. They plotted the intensity in the focal region for different angular semiapertures $\alpha$, including the limiting case of hemispherical focusing. Recently, we found that the field in the focal plane for the hemispherical focusing of many different examples of scalar and vector waves can be evaluated analytically, improving both computational speed and accuracy as well as avoiding sampling and zero-padding complications in fast Fourier transformation. These results are presented here. The analytical expressions for the variation in the field along the optical axis are easy to obtain and are not discussed here.

We find that high-NA scalar theory is not an improvement over a paraxial theory. The results for vectorial hemispherical focusing can give useful approximate predictions for high-NA lenses such as microscope objectives. We investigate the transverse behavior of 4Pi focusing and the focusing of vectorial vortex beams.

2. Scalar Theory

For paraxial scalar theory (Par), the amplitude point spread function (PSF) in the focal plane is $I\left(v\right)$, where $v=k\rho$ is a dimensionless cylindrical radius, $\rho$ is true cylindrical radius and $k=2\pi /\lambda$. This gives the simple Airy disk, where the amplitude is given by the integral [9]

Paraxial (Par)

$$I={\int }_0^1{J}_0\left(vt\right)t⠀\mathrm{d}t={\displaystyle \frac{J_1\left(v\right)}{v}};I(v=0)=\frac{1}{2},$$

(1)

where t is a normalized radius in the pupil plane, and $J_{\nu }$ is a Bessel function of the first kind of order $\nu$. The amplitude is given in dimensionless units, which can be normalized using the value at $v=0$, as is applied throughout this paper. The transverse optical coordinate for a general NA is $v=k\rho sin\alpha$, where $\alpha =\pi /2$ for the hemispherical case.

Next, we consider high-NA scalar theory. Here, we must choose the pupil weighting $a\left(\theta \right)$ [10]. The diffraction integral is

$$I={\int }_0^{\pi /2}a\left(\theta \right)J_0(vsin\theta )sin\theta ⠀d\theta .$$

(2)

We consider a few different cases, the first for a uniform angular spectrum, where the amplitude pupil weighting is constant, $a\left(\theta \right)=1$, equivalent to an optical system that satisfies the Herschel condition (ScHr). For this case,

Scalar Herschel (ScHr)

$$I={\displaystyle \frac{sin\left(v\right)}{v}}=j_0\left(v\right);I(v=0)=1.$$

(3)

where $j_{\nu }$ is a spherical Bessel function of the first kind of order $\nu$.

This analytical result is well known [1,2]. The field in the focal plane is the same as for a complete spherical incoming wave. Some of the integrals for different cases appear as examples of Straubel filters [11] (Sonine integrals, 11.4.10 in Ref. [12]). ScHr maximizes the intensity at the focal point for a given input power, so it is the high-NA equivalent of the Luneburg apodization problem [13].

The second case we consider is for a scalar aplanatic system (ScApl), i.e., an aberration-free system that satisfies the Abbe sine condition ($a\left(\theta \right)={cos}^{1/2}\theta$) [14]. For this case, the analytical expression for the amplitude PSF is

Scalar aplanatic (ScApl)

$$I=\Gamma \left({\displaystyle \frac{3}{4}}\right){\displaystyle \frac{J_{3/4}\left(v\right)}{2^{1/4}{\left(v\right)}^{3/4}}};I(v=0)=\frac{2}{3},$$

(4)

where $\Gamma$ is a Gamma function.

Other important cases are the corresponding perfect cases, where $a(\theta$) is multiplied by $cos\theta$. For the perfect Herschel case (ScPHr), the amplitude in the focal plane is identical to that in the paraxial case. This is also equivalent to illuminating with a scalar dipole field, equal to the scalar Green function for the first Rayleigh–Sommerfeld diffraction integral [15]. Note that the diffracted field of an infinitesimal two-dimensional (2D) area is not the same as the field of a (3D) point source [16]. In addition to a uniform hemispherical wave, Bertilone gave an analytical expression for the field variation over all space for the scalar dipole case [2]. The perfect Herschel case maximizes the intensity at the focal point for a given integrated intensity in the focal plane.

Another interesting example is the perfect Helmholtz case (ScPHz). For the Helmholtz case, which holds for a planar optical element such as a diffractive or holographic lens, $a\left(\theta \right)={sec}^{3/2}\theta$, so for the perfect Helmholtz case, $a\left(\theta \right)={sec}^{1/2}\theta$. Then, the integral becomes

Perfect Helmholtz (ScPHz)

$$I=\Gamma \left({\displaystyle \frac{1}{4}}\right){\displaystyle \frac{J_{1/4}\left(v\right)}{2^{3/4}{\left(v\right)}^{1/4}}};I(v=0)=2,$$

(5)

The intensity in the focal plane for these different scalar cases is shown in Figure 1. It is seen that all the scalar hemispherical curves are narrower than the paraxial prediction, except for the perfect Herschel case, which is identical to it. The narrowing of the PSF is associated with an increase in the strength of the side lobes, a well-known relationship.

3. Vectorial Theory for Linearly or Circularly Polarized Illumination

Next, we consider the high-NA vectorial Richards and Wolf theory for time-averaged electric energy density for circularly polarized illumination for hemispherical focusing [8]. For an aplanatic system (Apl), the weighting factor is $a\left(\theta \right)={cos}^{1/2}\theta$, and generalizing Richards’s and Wolf’s integrals $I_0,I_1,I_2$,

$$\begin{array}{cc}\hfill I_0=& {\int }_0^{\pi /2}a\left(\theta \right)(1+cos\theta )sin\theta J_0(vsin\theta )exp(iucos\theta )⠀d\theta ,\hfill \\ \hfill I_1=& {\int }_0^{\pi /2}a\left(\theta \right){sin}^2\theta J_1(vsin\theta )exp(iucos\theta )⠀d\theta ,\hfill \\ \hfill I_2=& {\int }_0^{\pi /2}a\left(\theta \right)(1-cos\theta )sin\theta J_2(vsin\theta )exp(iucos\theta )⠀d\theta \hfill \end{array}$$

(6)

where $u=kz$ is the axial optical coordinate. These can be evaluated analytically for the focal plane $u=0$ to give

Linear aplanatic (Apl)

$$\begin{array}{cc}\hfill I_0=& \Gamma \left(\frac{3}{4}\right){\displaystyle \frac{J_{3/4}\left(v\right)}{2^{1/4}{v}^{3/4}}}+\Gamma \left(\frac{5}{4}\right){\displaystyle \frac{2^{1/4}{J}_{5/4}\left(v\right)}{v^{5/4}}};I_0(v=0)=\frac{16}{15}\hfill \\ \hfill I_1=& \Gamma \left(\frac{3}{4}\right){\displaystyle \frac{J_{7/4}\left(v\right)}{2^{1/4}{v}^{3/4}}},\hfill \\ \hfill I_2=& \Gamma \left(\frac{1}{4}\right)\left[{\displaystyle \frac{J_{1/4}\left(v\right)}{2^{3/4}{v}^{9/4}}}+{\displaystyle \frac{J_{5/4}\left(v\right)}{2^{7/4}{v}^{5/4}}}\right]-\Gamma \left(\frac{3}{4}\right)\left[{\displaystyle \frac{2^{3/4}{J}_{-1/4}\left(v\right)}{v^{7/4}}}+{\displaystyle \frac{J_{3/4}\left(v\right)}{2^{1/4}{v}^{3/4}}}\right].\hfill \end{array}$$

(7)

The normalized electric fields for the x- or y-polarized orientation are given by [8]

$$\begin{array}{cc}\hfill {\mathbf{E}}^{\left(x\right)}=& (I_0+I_{2}cos2\varphi )\mathbf{i}+I_{2}sin2\varphi ⠀\mathbf{j}-2i{I}_{1}cos\varphi ⠀\mathbf{k},\hfill \\ \hfill {\mathbf{E}}^{\left(y\right)}=& I_{2}sin2\varphi ⠀\mathbf{i}+(I_0-I_{2}cos2\varphi )\mathbf{j}+2i{I}_{1}sin\varphi ⠀\mathbf{k}.\hfill \end{array}$$

(8)

The first of these equations can be written as

$${\mathbf{E}}^{\left(x\right)}=(I_0+2I_1)\mathbf{i}+I_2(cos2\varphi ⠀\mathbf{i}+sin2\varphi ⠀\mathbf{j})-2I_1(\mathbf{i}+icos\varphi ⠀\mathbf{k}),$$

(9)

which is equal to the coherent sum of a linearly polarized component, a polarization singularity of order two, and a component elliptically polarized in an x-y meridional plane. The time-averaged electric energy density, $W_E$, in the focal plane is $W_E=\left(\right|I_0{|}^2+2|I_1{|}^2+|I_2{|}^2)+2(|I_1{|}^2+\Re \left\{I_0{I}_2^{★}\right\})cos2\varphi$, and for the y orientation, $W_E=\left(\right|I_0{|}^2+2|I_1{|}^2+|I_2{|}^2)-2(|I_1{|}^2+\Re \left\{I_0{I}_2\right\})cos2\varphi$.

For circularly polarized illumination given by the sum of two orthogonally oriented components added in phase quadrature, the electric field is

$$\mathbf{E}=(I_0+I_2{e}^{2i\varphi })\mathbf{i}+i(I_0-I_2{e}^{2i\varphi })\mathbf{j}-2i{I}_1{e}^{i\varphi }\mathbf{k},$$

(10)

the electric energy density is given by $W_E=\left(\right|I_0{|}^2+2|I_1{|}^2+|I_2{|}^2)$, and the power flow in the focal plane $S=\left(\right|I_0{|}^2-\left|I_2{|}^2\right)$ [17].

Another interesting case is the corresponding Richards and Wolf vectorial solution for a uniform angular spectrum $a\left(\theta \right)=1$, given by the Herschel condition (Hr). Then, we find that

Linear Herschel (Hr)

$$\begin{array}{cc}\hfill I_0=& {\displaystyle \frac{J_1\left(v\right)+sinv}{v}}={\displaystyle \frac{J_1\left(v\right)}{v}}+j_0\left(v\right);I_0(v=0)=\frac{3}{2}\hfill \\ \hfill I_1=& {\displaystyle \frac{sinv-vcosv}{v^2}}=j_1\left(v\right),\hfill \\ \hfill I_2=& {\displaystyle \frac{2J_0\left(v\right)+v{J}_1\left(v\right)-2cosv-vsinv}{v^2}}=g_0\left(v\right)-f_0\left(v\right).\hfill \end{array}$$

(11)

The functions $g_0\left(v\right)$ and $f_0\left(v\right)$ are introduced in Equations (41) and (42). For the perfect Herschel condition (PHr), we find that, for off-axis illumination,

Perfect linear Herschel (PHr)

$$\begin{array}{cc}\hfill I_0=& {\displaystyle \frac{v^2{J}_1\left(v\right)-vcosv+sinv}{v^3}}={\displaystyle \frac{J_1\left(v\right)+j_1\left(v\right)}{v}};I_0(v=0)=\frac{5}{6}\hfill \\ \hfill I_1=& {\displaystyle \frac{J_2\left(v\right)}{v}},\hfill \\ \hfill I_2=& {\displaystyle \frac{3sinv-vcosv-2v{J}_0\left(v\right)-v^2{J}_1\left(v\right)}{v^3}}=j_2\left(v\right)-g_0\left(v\right)+f_0\left(v\right).\hfill \end{array}$$

(12)

The optimum weighting is, however, given by the mixed dipole case (MxD), $a\left(\theta \right)=(1+cos\theta )$, which maximizes, for linear or circular polarized illumination, the electric energy density at the focus [10]. Then,

Linear mixed dipole (MxD)

$$\begin{array}{cc}\hfill I_0=& {\displaystyle \frac{2v^2{J}_1\left(v\right)+sinv-vcosv+v^{2}sinv}{v^3}}={\displaystyle \frac{2J_1\left(v\right)+j_1\left(v\right)+v{j}_0\left(v\right)}{v}};I_0(v=0)=\frac{7}{3}\hfill \\ \hfill I_1=& {\displaystyle \frac{v{J}_2\left(v\right)+sinv-vcosv}{v^2}}={\displaystyle \frac{J_2\left(v\right)+v{j}_1\left(v\right)}{v}},\hfill \\ \hfill I_2=& {\displaystyle \frac{(3-v^2)sinv-3vcosv}{v^3}}=j_2\left(v\right).\hfill \end{array}$$

(13)

An analytical expression for the electric field in the focal plane for the hemispherical focusing of a mixed dipole wave is given in Ref. [10]. Note that the mixed dipole case is equal to the sum of the Herschel and the perfect Herschel results, (Hr + PHr).

There is also a perfect mixed dipole case (PMxD) that maximizes, for linear or circular polarization, the electric energy density at the focus for a fixed integrated electric energy density in the focal plane:

Perfect linear mixed dipole (PMxD)

$$\begin{array}{cc}\hfill I_0=& {\displaystyle \frac{v^2{J}_1\left(v\right)+2v{J}_2\left(v\right)+2sinv-2vcosv}{v^3}}={\displaystyle \frac{v{J}_1\left(v\right)+2J_2\left(v\right)+2v{j}_1\left(v\right)}{v^2}};I_0(v=0)=\frac{17}{12}\hfill \\ \hfill I_1=& {\displaystyle \frac{v^3{J}_2\left(v\right)+(3-v^2)sinv-3vcosv}{v^4}}={\displaystyle \frac{J_2\left(v\right)+j_2\left(v\right)}{v}},\hfill \\ \hfill I_2=& {\displaystyle \frac{J_3\left(v\right)}{v^3}}.\hfill \end{array}$$

(14)

We believe the analytical results in Equations (4), (5), (7), (11), (12) and (14), and many of the expressions given later in this paper, are novel. The combinations of Bessel functions J and spherical Bessel functions j are interesting. Somewhat similar related expressions are given in some other papers [4,5,6,18].

Figure 2 shows the contours of constant electric energy density in the focal plane for some different examples of linear polarized (x) illumination. Apl and MxD are very similar. Hr has a more elongated central lobe and stronger side lobes. The perfect forms have a wider central lobe in the y direction and weaker side lobes.

Plots of the normalized intensity for the different cases with circular-polarized illumination are given in Figure 3. The curves for the mixed dipole weighting are not shown as the FWHM (full-width at half-maximum) for this and the vector aplanatic case agree to within $1\%$. We see that the paraxial result agrees better with the vectorial result for electric energy density than the high-NA scalar theory. The paraxial approximation predicts the FWHM for the aplanatic Richards and Wolf result with an error of $-11.9\%$, but for the scalar aplanatic theory, the error is larger, $-17.9\%$. The paraxial result also agrees better with the vectorial power flow than with the vectorial electric energy density. Note the negative power flow in Figure 4, corresponding to the vortices in the azimuthal plane.

4. Other Polarization Distributions

4.1. Electric and Magnetic Dipole Polarizations

So far, we considered the vectorial focusing of light for the illumination of a lens with linearly or circularly polarized light. However, illumination with other polarization distributions can result in transverse linearly polarized light at the focus [19,20,21,22,23,24]. In particular, the electric dipole field (ED) maximizes the electric energy density at the focus for a given input power. The magnetic dipole field (MgD) maximizes the magnetic energy density for a given power. General combinations of electric dipole, denoted by superscript ${⠀}^{\left(p\right)},$ and magnetic dipole polarizations, denoted by superscript ${⠀}^{\left(mg\right)}$, can be used as a basis for synthesizing other cases [22]. Then, the equivalent of Richards’s and Wolf’s integrals are

$$\begin{array}{cc}\hfill I_0^{\left(p\right)}=& {\int }_0^{\pi /2}{a}^{\left(p\right)}\left(\theta \right)(1+{cos}^2\theta )sin\theta ⠀J_0(vsin\theta )exp(iucos\theta )⠀d\theta ,\hfill \\ \hfill I_1^{\left(p\right)}=& {\int }_0^{\pi /2}{a}^{\left(p\right)}\left(\theta \right){sin}^2\theta cos\theta ⠀J_1(vsin\theta )exp(iucos\theta )⠀d\theta ,\hfill \\ \hfill I_2^{\left(p\right)}=& {\int }_0^{\pi /2}{a}^{\left(p\right)}\left(\theta \right){sin}^3\theta ⠀J_2(vsin\theta )exp(iucos\theta )⠀d\theta ;\hfill \\ \hfill I_0^{\left(mg\right)}=& {\int }_0^{\pi /2}{a}^{\left(m\right)}\left(\theta \right)2sin\theta cos\theta ⠀J_0(vsin\theta )exp(iucos\theta )⠀d\theta ,\hfill \\ \hfill I_1^{\left(mg\right)}=& {\int }_0^{\pi /2}{a}^{\left(m\right)}\left(\theta \right){sin}^2\theta ⠀J_1(vsin\theta )exp(iucos\theta )⠀d\theta ,\hfill \\ \hfill I_2^{\left(mg\right)}=& 0.\hfill \end{array}$$

(15)

The resulting fields are given in terms of the sums $I_n=(I_n^{\left(p\right)}+I_n^{\left(mg\right)})$.

The electric dipole case (ED) is given by $a^{\left(p\right)}\left(\theta \right)=1,a^{\left(mg\right)}\left(\theta \right)=0$, analogously for the magnetic dipole case (MgD). General mixed dipole cases, corresponding to linear or circular polarized illumination, result from taking $a^{\left(p\right)}\left(\theta \right)=a^{\left(mg\right)}\left(\theta \right)=a^{\prime }\left(\theta \right)$. Then, $a\left(\theta \right)=(1+cos\theta )a^{\prime }\left(\theta \right)$. The particular MxD case is given by $a^{\left(p\right)}\left(\theta \right)=a^{\left(mg\right)}\left(\theta \right)=1$, giving $a\left(\theta \right)=(1+cos\theta )$. So, MxD is equal to the sum ED + MgD. We have, in the focal plane $u=0$,

Electric dipole (ED)

$$\begin{array}{cc}\hfill I_0=& {\displaystyle \frac{sinv-vcosv+v^{2}sinv}{v^3}}={\displaystyle \frac{j_1\left(v\right)+v{j}_0\left(v\right)}{v}};I_0(v=0)={\displaystyle \frac{4}{3}},\hfill \\ \hfill I_1=& {\displaystyle \frac{J_2\left(v\right)}{v}},\hfill \\ \hfill I_2=& {\displaystyle \frac{3sinv-3vcosv-v^{2}sinv}{v^3}}=j_2\left(v\right).\hfill \end{array}$$

(16)

Arnoldus and Foley gave expressions for these integrals even for the defocused case, when $u\ne 0$, in terms of Lommel functions [4].

Magnetic dipole (MgD)

$$\begin{array}{cc}\hfill I_0=& {\displaystyle \frac{2J_1\left(v\right)}{v}};I_0(v=0)=1,\hfill \\ \hfill I_1=& {\displaystyle \frac{sinv-vcosv}{v^2}}=j_1\left(v\right),\hfill \\ \hfill I_2=& 0.\hfill \end{array}$$

(17)

Arnoldus gave expressions for these integrals, in terms of sums of Bessel functions, for the defocused case, $u\ne 0$ (but as the magnetic field of an electric dipole) [6]. Note that the even/odd orders of $I_n$ for ED are expressed in terms of spherical Bessel functions (j)/Bessel functions (J), and vice versa for MgD.

4.2. Transverse Electric and Transverse Magnetic Polarizations

An alternative basis uses transverse electric and transverse magnetic modes, denoted by superscripts ${⠀}^{\left(TE\right)}$ and ${⠀}^{\left(TM\right)}$. Then, the equivalent of Richards and Wolf’s integrals are [22]

$$\begin{array}{cc}\hfill I_0^{\left(TE\right)}=& {\int }_0^{\pi /2}{a}^{\left(TE\right)}\left(\theta \right)sin\theta ⠀J_0(vsin\theta )exp(iucos\theta )⠀d\theta ,\hfill \\ \hfill I_1^{\left(TE\right)}=& 0,\hfill \\ \hfill I_2^{\left(TE\right)}=& {\int }_0^{\pi /2}{a}^{\left(TE\right)}\left(\theta \right)sin\theta ⠀J_2(vsin\theta )exp(iucos\theta )⠀d\theta ;\hfill \\ \hfill I_0^{\left(TM\right)}=& {\int }_0^{\pi /2}{a}^{\left(TM\right)}\left(\theta \right)sin\theta cos\theta ⠀J_0(vsin\theta )exp(iucos\theta )⠀d\theta ,\hfill \\ \hfill I_1^{\left(TM\right)}=& {\int }_0^{\pi /2}{a}^{\left(TM\right)}\left(\theta \right){sin}^2\theta ⠀J_1(vsin\theta )exp(iucos\theta )⠀d\theta ,\hfill \\ \hfill I_2^{\left(TM\right)}=& -{\int }_0^{\pi /2}{a}^{\left(TM\right)}\left(\theta \right)sin\theta cos\theta ⠀J_2(vsin\theta )exp(iucos\theta )⠀d\theta .\hfill \end{array}$$

(18)

The particular transverse electric weighting (TE${⠀}_1$) is given by $a^{\left(TE\right)}\left(\theta \right)=1,a^{\left(TM\right)}\left(\theta \right)=0$, analogously for the transverse magnetic weighting (TM${⠀}_1$), $a^{\left(TE\right)}\left(\theta \right)=0,a^{\left(TM\right)}\left(\theta \right)=1$. General mixed dipole cases result from taking $a^{\left(TE\right)}\left(\theta \right)=a^{\left(TM\right)}\left(\theta \right)=a^{\prime }\left(\theta \right)$, giving $a\left(\theta \right)=a^{\prime }\left(\theta \right)$. The particular MxD weighting is given by $a^{\left(TE\right)}\left(\theta \right)=a^{\left(TM\right)}\left(\theta \right)=(1+cos\theta )$ giving $a\left(\theta \right)=(1+cos\theta )$. The two bases are related by the relationships $a^{\left(TE\right)}=a^{\left(p\right)}+a^{\left(mg\right)}cos\theta$, $a^{\left(TM\right)}=a^{\left(p\right)}cos\theta +a^{\left(mg\right)}$.

For an infinity tube length lens, the input electric field for TE${⠀}_1$ is circumferential, and the magnetic field is radial. For ED, the lines of electric field converge, becoming circumferential for $\theta ={90}^{\circ }$, while the lines of the magnetic field diverge, becoming radial for $\theta ={90}^{\circ }$ [25]. For TM${⠀}_1$ and MgD, the lines of the electric and magnetic fields are interchanged, as compared with TE${⠀}_1$ and ED.

We have

Transverse electric (TE${⠀}_1$)

$$\begin{array}{cc}\hfill I_0=& {\displaystyle \frac{sinv}{v}}=j_0\left(v\right);I_0(v=0)=1,\hfill \\ \hfill I_1=& 0,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill I_2=& {\displaystyle \frac{2-2cosv-vsinv}{v^2}}=g_0\left(v\right).\hfill \end{array}$$

(20)

Transverse magnetic (TM${⠀}_1$)

$$\begin{array}{cc}\hfill I_0=& {\displaystyle \frac{J_1\left(v\right)}{v}};I_0(v=0)=\frac{1}{2}\hfill \\ \hfill I_1=& {\displaystyle \frac{sinv-vcosv}{v^2}}=j_1\left(v\right),\hfill \\ \hfill I_2=& -{\displaystyle \frac{2-2J_0\left(v\right)-v{J}_1\left(v\right)}{v^2}}=-f_0\left(v\right).\hfill \end{array}$$

(21)

Be aware of the negative sign in the last line in Equation (18) for TM${⠀}_1$. Note also that $I_2$ for both TE${⠀}_1$ and TM${⠀}_1$ are zero for $v\to 0$. (TE${⠀}_1$+ TM${⠀}_1$) is equal to the linear/circular polarization Herschel case, Hr.

4.3. Perfect Weightings

There are perfect weightings corresponding to these polarizations that maximize the electric energy density at the focus for a given integrated energy density in the focal plane. We have

Perfect electric dipole (PED)

$$\begin{array}{cc}\hfill I_0=& {\displaystyle \frac{2J_2\left(v\right)+v{J}_1\left(v\right)}{v^2}};I_0(v=0)=\frac{3}{4},\hfill \\ \hfill I_1=& {\displaystyle \frac{3sinv-3vcosv-v^{2}sinv}{v^4}}={\displaystyle \frac{j_2\left(v\right)}{v}},\hfill \\ \hfill I_2=& {\displaystyle \frac{J_3\left(v\right)}{v}}.\hfill \end{array}$$

(22)

Perfect magnetic dipole (PMgD)

$$\begin{array}{cc}\hfill I_0=& {\displaystyle \frac{2(sinv-vcosv)}{v^3}}={\displaystyle \frac{2j_1\left(v\right)}{v}};I_0(v=0)=\frac{2}{3},\hfill \\ \hfill I_1=& {\displaystyle \frac{J_2\left(v\right)}{v}},\hfill \\ \hfill I_2=& 0.\hfill \end{array}$$

(23)

Perfect transverse electric (PTE${⠀}_1$)

$$\begin{array}{cc}\hfill I_0=& {\displaystyle \frac{J_1\left(v\right)}{v}};I_0(v=0)=\frac{1}{2},\hfill \\ \hfill I_1=& 0,\hfill \\ \hfill I_2=& {\displaystyle \frac{2-2J_0\left(v\right)-v{J}_1\left(v\right)}{v^2}}=f_0\left(v\right).\hfill \end{array}$$

(24)

Perfect transverse magnetic (PTM${⠀}_1$)

$$\begin{array}{cc}\hfill I_0=& {\displaystyle \frac{sinv-vcosv}{v^3}}={\displaystyle \frac{j_1\left(v\right)}{v}};I_0(v=0)=\frac{1}{3},\hfill \\ \hfill I_1=& {\displaystyle \frac{J_2\left(v\right)}{v}},\hfill \\ \hfill I_2=& {\displaystyle \frac{3sinv-vcosv-2v}{v^3}}=j_2\left(v\right)-g_0\left(v\right).\hfill \end{array}$$

(25)

ED is equal to (TE${⠀}_1$+ PTM${⠀}_1$); MgD is equal to (TM${⠀}_1$+ PTE${⠀}_1$). (PTE${⠀}_1$+ PTM${⠀}_1$) is equal to the linear/circular polarization perfect Herschel case, PHr. The mixed dipole case MxD is equal to (Hr + PHr), which is equal to (TE + TM + PTE + PTM). Contour plots of the electric energy density in the focal plane for these different polarization conditions are shown in Figure 5. TE${⠀}_1$ has a more elongated central lobe than ED and stronger side lobes. The normalized intensity along the x axis is $g_s^2\left(v\right)=j_0^4(v/2)$ (Equation (41)). MgD, TM${⠀}_1$ and PTM${⠀}_1$ all have a central spot that is divided into two peaks (heights 1.18, 3.18, and 1.37, respectively) as a result of the cross-polarization caused by $I_1$. This is an effect that has been described previously for electromagnetic focusing [26]. The perfect plots for ED and TE${⠀}_1$ exhibit an enlarged central lobe and weaker side lobes. The normalized intensity along the x axis for PTE${⠀}_1$ is $f_s^2\left(v\right)$ (Equation (42)).

4.4. Longitudinal Electric or Magnetic Dipole Polarizations

In addition, we have cases given by longitudinal electric or magnetic dipoles. Radially polarized illumination results in a longitudinal electric field at the focus [20,27,28,29,30,31]. The optimum case for maximizing the longitudinal electric field at the focus is the radial electric dipole weighting (TM${⠀}_0$) [20,31,32]. Similarly, azimuthal polarized illumination gives a longitudinal magnetic field at the focus, which is optimized for the azimuthal magnetic weighting (TM${⠀}_0$). These give:

Radial electric dipole (TM${⠀}_0$)

$$\begin{array}{cc}\hfill I_0=& {\displaystyle \frac{vcosv+v^{2}sinv-sinv}{v^3}}={\displaystyle \frac{v{j}_0\left(v\right)-j_1\left(v\right)}{v}};I_0(v=0)=\frac{2}{3}\hfill \\ \hfill I_1=& {\displaystyle \frac{J_2\left(v\right)}{v}},\hfill \end{array}$$

(26)

where $I_0$ gives the field in the longitudinal direction and $I_1$ in the radial direction.

Azimuthal magnetic dipole (TE${⠀}_0$)

$$I_1={\displaystyle \frac{sinv-vcosv}{v^2}}=j_1\left(v\right);I_0(v=0)=0.$$

(27)

$I_1$ is the field in the azimuthal direction.

Perfect radial electric dipole (PTM${⠀}_0$)

$$\begin{array}{cc}\hfill I_0=& {\displaystyle \frac{v{J}_1\left(v\right)-2J_2\left(v\right)}{v^2}};I_0(v=0)=\frac{1}{4},\hfill \\ \hfill I_1=& {\displaystyle \frac{3sinv-3vcosv-v^{2}sinv}{v^4}}={\displaystyle \frac{j_2\left(v\right)}{v}}.\hfill \end{array}$$

(28)

Perfect azimuthal electric dipole (PTE${⠀}_0$)

$$I_1={\displaystyle \frac{J_2\left(v\right)}{v}};⠀I_0(v=0)=0.$$

(29)

4.5. Cylindrically Symmetric Solutions

Other than the radial and azimuthal cases, the different polarization distributions produce a focal spot that is not cylindrically symmetric. However, just like circular polarized light can be produced by adding two orthogonally polarized plane waves in phase quadrature, cylindrically symmetric ED, MgD, TE${⠀}_1$, and TM${⠀}_1$ forms can be generated [24]. The polarization for ED is equivalent to a rotating electric dipole, which we call rot.-ED. This corresponds to an illumination that is circular polarized on the optical axis, which, for off-axis illumination, is elliptically polarized with the major axis in the radial direction, becoming azimuthally polarized with a phase singularity for $\theta \to {90}^{\circ }$. The analogous forms hold for the other cases, rot.-MgD, and rot.-TE${⠀}_1$ and rot.-TM${⠀}_1$. As (ED + MgD) gives MxD, (rot.-ED +rot.-MgD) gives circular polarized c-MxD. The case of rot.-TE${⠀}_1$ is the same as azimuthal polarization with a phase vortex of topological charge unity, and rot.-TM${⠀}_1$ is the same as radial polarization with a phase vortex of topological charge unity [22,23,24]. As the sum (TE${⠀}_1$+ TM${⠀}_1$) is equivalent to linear polarized illumination with Herschel weighting l-Hr, the sum (rot.-TE${⠀}_1$+rot.-TM${⠀}_1$) is equivalent to circular polarized Herschel weighting, c-Hr.

The variations in electric energy density in the focal plane, normalized to unity at the focus, for some of these cases, and their perfect forms, are shown in Figure 6. The sharpest central lobe is for the radial case TM${⠀}_0$, followed by TE${⠀}_1$ and then ED. The curves for Hr and MxD (in blue and magenta, respectively) are very close to each other.

The full-width at half-maximum (FWHM) of these PSFs are compared in

Table 1. The full-width at half-maximum (FWHM) of the intensity in the focal plane, measured as a value of the optical coordinate v for the different circularly symmetric cases.

Case	Abbreviation	FWHM (v Value)
Paraxial	Par	3.23
Scalar Herschel	ScHr	2.78
Scalar aplanatic	ScApl	3.02
Scalar perfect Herschel	ScPHr	3.23
Scalar perfect Helmholtz	ScPHz	2.53
Circular polarized aplanatic	c-Apl	3.67
Circular polarized Herschel	c-Hr	3.61
Circular polarized mixed dipole	c-MxD	3.64
Circular polarized perfect Herschel	c-PHr	3.79
Circular polarized perfect mixed dipole	c-PMxD	3.90
Rotating electric dipole	rot.-ED	3.10
Rotating magnetic dipole	rot.-MgD	5.02
Transverse electric (azimuthal vortex)	rot.-TE${⠀}_1$	2.87
Transverse magnetic (radial vortex)	rot.-TM${⠀}_1$	7.22
Radial electric dipole	TM${⠀}_0$	2.68
Perfect rotating electric dipole	rot.-PED	3.62
Perfect rotating magnetic dipole	rot.-MgD	4.38
Perfect transverse electric (perfect azimuthal vortex)	rot.-PTE${⠀}_1$	3.33
Perfect transverse magnetic (perfect radial vortex)	rot.-PTM${⠀}_1$	6.82
Perfect radial electric dipole	PTM${⠀}_0$	3.26
Circular polarized aplanatic 4PI	c-Apl 4Pi	3.08
Circular polarized Herschel 4Pi	c-Hr 4Pi	2.93
Circular polarized mixed dipole 4Pi	c-MxD 4Pi	3.08
Rotating electric dipole 4Pi	rot.-ED 4Pi	2.98
Rotating magnetic dipole 4Pi	rot.-MgD 4Pi	3.24
Transverse electric (azimuthal vortex) 4Pi	rot.-TE${⠀}_1$ 4Pi	2.87
Transverse magnetic (radial vortex) 4Pi	rot.-TM${⠀}_1$ 4Pi	3.33
Radial electric dipole 4Pi	TM${⠀}_0$ 4Pi	2.53

.

5. 4Pi Focusing

In a 4Pi system, two opposing objective lenses are used to approximate a full spherically focused wave, with a solid angle of $4\pi$ steradians [33]. In this section, we discuss the application of the results to such systems. The main advantage of 4Pi focusing is an increase in axial resolution, as the two counter-propagating beams interfere to produce a standing wave pattern.

A second advantage is that the electric energy density $W_E$ at the focus for a given input power is increased. For a scalar system, the intensity is increased by a factor of four and the power doubled, so the ratio is also doubled. However, for a vectorial system, the result depends strongly on the polarization distribution of the illumination. The doubling is achieved if the electric field at the focus reinforces for the two beams. For ED or TM${⠀}_0$, the two hemispheres together form a complete electric dipole wave, either oriented in a transverse or longitudinal direction, respectively. We define the normalized ratio F as 1 for these cases [10,22], so a single hemispherical beam for ED or TM${⠀}_0$ gives $F=1/2$. For TE${⠀}_1$, $F=3/4$ for 4Pi and $F=3/8$ for a single lens. For MgD, if the two beams together form a complete magnetic dipole wave, the electric field at the focus cancels, so $F=0$. The doubling is then only achieved if a phase difference of $\pi$ is introduced, which then gives $F=9/16$ rather than $F=9/32$ for a single lens. Mixed dipole polarization (as in Apl or MxD) is directive. For the complete spherical focusing of MxD, $F=1/2$, as the total energy density is divided equally into electric and magnetic forms, but, for a single hemispherical wave, $F=32/75$ or $F=7/16$ (respectively, for Apl or MxD), and $F=64/75$ or $F=7/8$ for a 4Pi system [10,34].

The third advantage of 4Pi focusing is that the transverse resolution is also improved, because some cross-polarization terms cancel [34]. For ED or TE${⠀}_1$, the hemispherical 4Pi case is exactly the same as for complete spherical focusing. We find that $I_0$ and $I_2$ are doubled as compared with a single lens, whereas $I_1$ becomes zero. For MgD or TM${⠀}_1$, for a complete sphere, $I_1$ is doubled, and $I_0,I_2$ are zero. The result is that for complete spherical focusing, in the expressions given previously for $I_n$, all the Bessel functions J (and $f_0$) cancel, leaving just the spherical Bessel functions j (and $g_0$). However, for MgD or TM${⠀}_1$ in 4Pi, the direction of the electric field vectors in the second lens are reversed to again double $I_0$ and $I_2$ and make $I_1=0$. The result is that the PSF is slightly improved for 4Pi in ED but greatly improved for MgD and TM${⠀}_1$. For TE${⠀}_1$, there is no improvement in the shape of the PSF because $I_1=0$ even for a single lens, but the PSF is still doubled in strength. For 4Pi illumination with linear or circular polarization, as in MxD, Hr, or Apl, $I_0$ and $I_2$ are doubled, and $I_1=0$ for both the electric/magnetic dipole components or the TE/TM components. Here, we considered just the focal plane. If we introduce defocus, the complex exponential term $exp(iucos\theta )$ must be replaced with $cos(ucos\theta )$ or $isin(ucos\theta )$ according to symmetry, and $I_1$ does not cancel [33].

Figure 7 shows the contours of the electric energy density in the focal plane for different 4Pi systems. For linear polarized illumination, Apl, Hr, and MxD, the width of the central lobe along the x axis is reduced. A similar but smaller change occurs for ED, but TE${⠀}_1$ (not shown) is unchanged. For MgD (also not shown), the focal spot is circularly symmetric. A single diffraction integral $I_0$ remains, which gives a focus equal to the paraxial Airy disk. For TM${⠀}_1$, the focal spot is enlarged in the y direction. The normalized intensity along the y axis is $W_E=f_s^2\left(v\right)$, where $f_s\left(v\right)$ is defined in Equation (42) and the preceding line.

So, for MgD or TM${⠀}_1$, 4Pi is not the same complete spherical focusing. For MgD, complete spherical focusing results in zero electric field at the focus.

Figure 6c shows cross-sections through the electric energy density $W_E$ in the focal plane for a variety of cylindrically symmetric 4Pi cases. The FWHMs are given in Table 1. The smallest FWHMs are TM${⠀}_0$ 4Pi, followed by $rot.$-TE${⠀}_1$ 4Pi, c-H 4Pi, and $rot.$-ED 4Pi.

6. Vortex Beams

Fields with phase vortices are currently of great interest, with applications in optical trapping, optical communications, and other areas. We distinguish between vortices $e^{\pm im\varphi }$ and ${\rho }^m{e}^{\pm im\varphi }$. We call these Riesz vortices and Laguerre vortices, respectively. A Riesz vortex is produced from a plane wave by a spiral phase plate. A Laguerre vortex appears in the core region of a Laguerre–Gauss beam.

If a Laguerre vortex is focused by a paraxial system with a circular aperture, the focused field can be expressed in terms of Lommel functions as a natural extension of the case for a uniform beam, $m=0$ [35]. Other papers have investigated the focusing of Riesz vortices [36,37]. Often, the integrals involved result in Struve functions, ${\mathbf{H}}_0,{\mathbf{H}}_1$. As high-NA focusing produces cross-polarization components, the interaction between phase and polarization vortices becomes important for tight focusing [38,39].

An important particular special case is the generation of a doughnut beam, with a dark center, as used in STED (stimulated emission depletion) microscopy [23,40,41]. An axial zero is exhibited when focusing azimuthally polarized light, TE${⠀}_0$, but, with a phase singularity of topological charge unity, rot.-TE${⠀}_1$, there is a transverse circular polarized electric field on-axis.

Linear polarized illumination (in the x direction, unit vector $\mathbf{i}$) can be written as $\mathbf{i}=-sin\varphi {\mathbf{a}}_{\varphi }+cos\varphi {\mathbf{a}}_{\rho }$. For a vortex beam of topological charge m, to evaluate Richards’s and Wolf’s integrals for the locally linear polarized case, the integrals $I_0,I_1,I_2$ are no longer sufficient, and we need azimuthal integrals of the form [38]

$$\begin{array}{cc}\hfill i_{nc}^{\left(m\right)}={\int }_0^{2\pi }cosn{\varphi }^{\prime }{e}^{im{\varphi }^{\prime }}{e}^{i\rho cos({\varphi }^{\prime }-\varphi )}d{\varphi }^{\prime }& =\pi i^m{e}^{im\varphi }\left[i^n{J}_{n+m}\left(\rho \right)e^{in\varphi }+{(-i)}^n{J}_{m-n}\left(\rho \right)e^{-in\varphi }\right],\hfill \\ \hfill i_{ns}^{\left(m\right)}={\int }_0^{2\pi }sinn{\varphi }^{\prime }{e}^{im{\varphi }^{\prime }}{e}^{i\rho cos({\varphi }^{\prime }-\varphi )}d{\varphi }^{\prime }& =\pi i^{m-1}{e}^{im\varphi }\left[i^n{J}_{n+m}\left(\rho \right)e^{in\varphi }-{(-i)}^n{J}_{m-n}\left(\rho \right)e^{-in\varphi }\right].\hfill \end{array}$$

(30)

for $n=0,1,2$. These integrals then need to be integrated over $\rho =vsin\theta$ with appropriate weighting to give $I_{nc}^{\left(m\right)},I_{ns}^{\left(m\right)}$. For the integral $i_{0c}^{\left(m\right)}$ (for $n=0$), there is still the integral over $\rho$ of a single Bessel function to give $I_0^{\left(m\right)}$, but now of $J_m\left(\rho \right)$ instead of $J_0\left(\rho \right)$. Azimuthal integrals $n=1,2$ result in the integrals of two Bessel functions rather than one. Instead of $I_{1}cos\varphi$ and $I_{1}sin\varphi$, we now need the integrals $I_{1c}^{\left(m\right)},I_{1s}^{\left(m\right)}$, and instead of $I_2$ we need $I_{2c}^{\left(m\right)}$ and $I_{2s}^{\left(m\right)}$. The result is that for the locally-linear polarized cases ED, MgD, MxD, TE${⠀}_1$, TM${⠀}_1$, Hr, for a topological charge $m=1$, the axial field at the origin usually does not exhibit a zero, and for topological charge $m=2$ the transverse field at the origin is not usually zero [38,42]. But for the special case of TE${⠀}_1$, the integrals $I_{1c}^{\left(m\right)},I_{1s}^{\left(m\right)}$ (for $n=1$) are zero, so that there is no $E_z$ for any value of m, including $m=1$. For MgD, the integrals $I_{2c}^{\left(m\right)},I_{2s}^{\left(m\right)}$ (for $n=2$) are zero, and $E_y=0$ for any m, including $m=2$.

Figure 8 shows contours of constant electric energy density in the focal plane for vortices $m=1$ and $m=2$ for the cases of ED, MgD, and MxD. For MgD, $m=2$, there is a zero at the focal point for $m=1$. For the remaining plots, there is a field at the focus, either transverse or axial, resulting from a zero value of $m-n$, so the contours are normalized to unity at the focus. The contours can take values larger than one, as specified in the figure caption.

Figure 9 shows the contours of the constant electric energy density in the focal plane for vortices $m=1$ and $m=2$ for the cases of TE${⠀}_1$, TM${⠀}_1$, and Hr (=TE${⠀}_1$ + TM${⠀}_1$). For TE${⠀}_1$, there is a zero at the focal point for $m=1$. For the remaining plots, there is a field at the focus, either transverse or axial, resulting from a zero value of $m-n$. If the illumination is locally circular polarized to give rot.-ED, rot.-TE${⠀}_1$, c-MxD, etc., the two integrals in Equation (30) are added in phase quadrature, the Bessel function $J_{m-n}$ cancels, and we are left with just $J_{m+n}$:

$$i_n^{\left(m\right)}={\int }_0^{2\pi }{e}^{i(m+n){\varphi }^{\prime }}{e}^{i\rho cos({\varphi }^{\prime }-\varphi )}d{\varphi }^{\prime }=2\pi i^{(m+n)}{J}_{n+m}\left(\rho \right)e^{i(m+n)\varphi }.$$

(31)

Then, in the integrals $I_0^{\left(m\right)},I_1^{\left(m\right)},I_2^{\left(m\right)}$, a zero in intensity occurs for any value of m except for $m=-2,-1,0$. So, for circular polarized illumination, there is a zero for $m=1$, which is a commonly used strategy in STED microscopy, or for $m=2$.

For the case of locally linear polarized illumination with a vortex, $I_{nc}^{\left(m\right)}+i{I}_{ns}^{\left(m\right)}=I_n^m{e}^{in\varphi }$, so the integrals can be written in the form $I_{nc}^{\left(m\right)}=[I_n^m{e}^{in\varphi }+{\Delta }_n^{\left(m\right)}{e}^{-in\varphi }]/2$, $I_{ns}^{\left(m\right)}=-i[I_n^m{e}^{in\varphi }-{\Delta }_n^{\left(m\right)}{e}^{-in\varphi }]/2$. Then, we have

Electric dipole vortex (ED), $m=1$

$$\begin{array}{cc}\hfill I_0^{\left(1\right)}=& i{e}^{i\varphi }\left[{\displaystyle \frac{j_2\left(v\right)+v{j}_1\left(v\right)}{v}}\right],\hfill \\ \hfill I_1^{\left(1\right)}=& i{e}^{i\varphi }{\displaystyle \frac{J_3\left(v\right)}{v}},{\Delta }_1^{\left(1\right)}=i{e}^{i\varphi }\left[{\displaystyle \frac{2J_2\left(v\right)-v{J}_1\left(v\right)}{v^2}}\right],\hfill \\ \hfill I_2^{\left(1\right)}=& i{e}^{i\varphi }{j}_3\left(v\right),\hfill \\ \hfill {\Delta }_2^{\left(1\right)}=& i{e}^{i\varphi }\left[{\displaystyle \frac{j_2\left(v\right)-v{j}_1\left(v\right)}{v}}\right]=i{e}^{i\varphi }{h}_1\left(v\right).\hfill \end{array}$$

(32)

Electric dipole vortex (ED), $m=2$

$$\begin{array}{cc}\hfill I_0^2=& -e^{2i\varphi }\left[{\displaystyle \frac{j_3\left(v\right)+v{j}_2\left(v\right)}{v}}\right],\hfill \\ \hfill I_1^2=& -e^{2i\varphi }{\displaystyle \frac{J_4\left(v\right)}{v}},{\Delta }_1^2=-e^{2i\varphi }\left[{\displaystyle \frac{2J_3\left(v\right)-v{J}_2\left(v\right)}{v^2}}\right],\hfill \\ \hfill I_2^2=& -e^{2i\varphi }{j}_4\left(v\right),\hfill \\ \hfill {\Delta }_2^2=& -e^{2i\varphi }\left[{\displaystyle \frac{8j_2\left(v\right)-v{j}_3\left(v\right)-v^2{j}_2\left(v\right)}{v^2}}\right]=-e^{2i\varphi }{h}_2\left(v\right).\hfill \end{array}$$

(33)

Magnetic dipole vortex (MgD), $m=1$

$$\begin{array}{cc}\hfill I_0^{\left(1\right)}=& 2i{e}^{i\varphi }{\displaystyle \frac{J_2\left(v\right)}{v}},\hfill \\ \hfill I_1^{\left(1\right)}=& i{e}^{i\varphi }{j}_2\left(v\right),{\Delta }_1^{\left(1\right)}=i{e}^{i\varphi }\left[{\displaystyle \frac{j_1\left(v\right)-v{j}_0\left(v\right)}{v}}\right],\hfill \\ \hfill I_2^{\left(1\right)}=& 0,{\Delta }_2^{\left(1\right)}=0.\hfill \end{array}$$

(34)

Magnetic dipole vortex (MgD), $m=2$

$$\begin{array}{cc}\hfill I_0^{\left(2\right)}=& -e^{2i\varphi }{\displaystyle \frac{J_3\left(v\right)}{v}},\hfill \\ \hfill I_1^{\left(2\right)}=& -e^{2i\varphi }{j}_3\left(v\right),{\Delta }_1^{\left(2\right)}=-e^{2i\varphi }\left[{\displaystyle \frac{j_2\left(v\right)-v{j}_1\left(v\right)}{v}}\right],\hfill \\ \hfill I_2^{\left(2\right)}=& 0,{\Delta }_2^{\left(2\right)}=0.\hfill \end{array}$$

(35)

Transverse electric vortex (TE${⠀}_1$), $m=1$

$$\begin{array}{cc}\hfill I_0^{\left(1\right)}=& i{e}^{i\varphi }{j}_1\left(v\right),\hfill \\ \hfill I_1^{\left(1\right)}=& 0,{\Delta }_1^{\left(1\right)}=0,\hfill \\ \hfill I_2^{\left(1\right)}=& i{e}^{i\varphi }\left({\displaystyle \frac{8-8cosv-5vsinv+v^{2}cosv}{v^3}}\right)=i{e}^{i\varphi }{g}_1\left(v\right),\hfill \\ \hfill {\Delta }_2^{\left(1\right)}=& -i{e}^{i\varphi }{j}_1\left(v\right).\hfill \end{array}$$

(36)

Transverse electric vortex (TE${⠀}_1$), $m=2$

$$\begin{array}{cc}\hfill I_0^{\left(2\right)}=& -e^{2i\varphi }{j}_2\left(v\right),\hfill \\ \hfill I_1^{\left(2\right)}=& 0,{\Delta }_1^{\left(2\right)}=0,\hfill \\ \hfill I_2^{\left(2\right)}=& -e^{2i\varphi }\left({\displaystyle \frac{48-48cosv-33vsinv+9v^{2}cosv+v^{3}sinv}{v^3}}\right)=-e^{2i\varphi }{g}_2\left(v\right),\hfill \\ \hfill {\Delta }_2^{\left(2\right)}=& -e^{2i\varphi }\left[{\displaystyle \frac{j_1\left(v\right)-v{j}_0\left(v\right)}{v}}\right].\hfill \end{array}$$

(37)

Transverse magnetic vortex (TM${⠀}_1$), $m=1$

$$\begin{array}{cc}\hfill I_0^{\left(1\right)}=& i{e}^{i\varphi }{\displaystyle \frac{J_2\left(v\right)}{v}},\hfill \\ \hfill I_1^{\left(1\right)}=& i{e}^{i\varphi }{j}_2\left(v\right),{\Delta }_1^{\left(1\right)}=i{e}^{i\varphi }\left[{\displaystyle \frac{j_1\left(v\right)-v{j}_0\left(v\right)}{v}}\right],\hfill \\ \hfill I_2^{\left(1\right)}=& -i{e}^{i\varphi }\left[{\displaystyle \frac{8-8J_0\left(v\right)-6v{J}_1\left(v\right)+v^2{J}_0\left(v\right)}{v^3}}\right]=-i{e}^{i\varphi }{f}_1\left(v\right),\hfill \\ \hfill {\Delta }_2^{\left(1\right)}=& i{e}^{i\varphi }\left[{\displaystyle \frac{2J_1\left(v\right)-v{J}_0\left(v\right)}{v^2}}\right]=i{e}^{i\varphi }{\displaystyle \frac{J_2\left(v\right)}{v}}.\hfill \end{array}$$

(38)

Transverse magnetic vortex (TM${⠀}_1$), $m=2$

$$\begin{array}{cc}\hfill I_0^{\left(2\right)}=& -e^{2i\varphi }{\displaystyle \frac{J_3\left(v\right)}{v}},\hfill \\ \hfill I_1^{\left(2\right)}=& -e^{2i\varphi }{j}_3\left(v\right),{\Delta }_1^{\left(2\right)}=-e^{2i\varphi }\left[{\displaystyle \frac{j_2\left(v\right)-v{j}_1\left(v\right)}{v}}\right],\hfill \\ \hfill I_2^{\left(2\right)}=& e^{2i\varphi }\left[{\displaystyle \frac{48-48J_0\left(v\right)-44v{J}_1\left(v\right)+10v^2{J}_0\left(v\right)+v^3{J}_1\left(v\right)}{v^4}}\right]=e^{2i\varphi }{f}_2\left(v\right),\hfill \\ \hfill {\Delta }_2^{\left(2\right)}=& -e^{2i\varphi }\left[{\displaystyle \frac{2J_2\left(v\right)-v{J}_1\left(v\right)}{v^2}}\right].\hfill \end{array}$$

(39)

The General Behavior with Vortices

We can see that many of the expressions for $I_0,I_1,I_2$ for the different cases, given earlier, still hold in Equation (10), but with $J_k\left(v\right)$ or $j_k\left(v\right)$ replaced with $i^{m}exp\left(im\varphi \right)J_{k+m}\left(v\right)$ or with $i^{m}exp\left(im\varphi \right)j_{k+m}\left(v\right)$. Exceptions include the cases of $I_2^{\left(m\right)}$ for TE${⠀}_1$ and TM${⠀}_1$, and ${\Delta }_2^{\left(m\right)}$ for ED. For the former, we find that for the general $p=m+2$,

$$\begin{array}{cc}\hfill {\int }_0^{\pi /2}{J}_p(vsin\theta ){sin}^{p-1}\theta d\theta =& {\displaystyle \frac{\Gamma \left(p\right)v^p}{2^{p+1}\Gamma (2p+1)}}{⠀}_1{F}_2\left(p;p+\frac{1}{2},p+1;-{\displaystyle \frac{v^2}{4}}\right),\hfill \\ \hfill {\int }_0^{\pi /2}{J}_p(vsin\theta ){sin}^{p-1}\theta cos\theta d\theta =& {\displaystyle \frac{v^p}{2^{p+1}{p}^2\Gamma \left(p\right)}}{⠀}_1{F}_2\left(p;p+\frac{1}{2},p+1;-{\displaystyle \frac{v^2}{4}}\right),\hfill \end{array}$$

(40)

where ${⠀}_1{F}_2$ is a hypergeometric function. We define the function $g_s\left(v\right)=2(1-cosv)/v^2={[2sin(v/2)/v]}^2=j_0^2(v/2)$, so it is a type of sinc-squared function. Then, we can generate a sequence of functions $g_m\left(v\right)$:

$$\begin{array}{cc}\hfill g_0\left(v\right)=g_s-j_0\left(v\right)=& {\displaystyle \frac{2-2cosv-vsinv}{v^2}},\hfill \\ \hfill g_1\left(v\right)={\displaystyle \frac{4g_0}{v}}-j_1\left(v\right)=& {\displaystyle \frac{8-8cosv-5vsinv+v^{2}cosv}{v^3}},\hfill \\ \hfill g_2\left(v\right)={\displaystyle \frac{6g_1}{v}}-j_2\left(v\right)=& {\displaystyle \frac{48-48cosv-33vsinv+9v^{2}cosv+v^{3}sinv}{v^4}}.\hfill \end{array}$$

(41)

The coefficients were chosen to make $g_0,g_1,g_2$ all zero at the origin.

We now perform a similar operation with Bessel functions. We define the function $f_s\left(v\right)=4[1-J_0\left(v\right)]/v^2$, which behaves similarly to a sinc-squared function. Then, we have

$$\begin{array}{cc}\hfill f_0\left(v\right)={\displaystyle \frac{f_s}{2}}-{\displaystyle \frac{J_1\left(v\right)}{v}}=& {\displaystyle \frac{2-2J_0\left(v\right)-v{J}_1\left(v\right)}{v^2}},\hfill \\ \hfill f_1\left(v\right)={\displaystyle \frac{4f_0}{v}}-{\displaystyle \frac{J_2\left(v\right)}{v}}=& {\displaystyle \frac{8-8J_0\left(v\right)-6v{J}_1\left(v\right)+v^2{J}_0\left(v\right)}{v^3}},\hfill \\ \hfill f_2\left(v\right)={\displaystyle \frac{6f_1}{v}}-{\displaystyle \frac{J_3\left(v\right)}{v}}=& {\displaystyle \frac{48-48J_0\left(v\right)-44v{J}_1\left(v\right)+10v^2{J}_0\left(v\right)+v^3{J}_1\left(v\right)}{v^4}}.\hfill \end{array}$$

(42)

The functions $f_m$ and $g_m$ are illustrated in Figure 10a,b.

For ED, the corresponding integral for ${\Delta }_2^{\left(m\right)}$ is

$$h_m\left(v\right)={\int }_0^{\pi /2}{J}_{m-2}(vsin\theta ){sin}^{m+3}\theta d\theta ={\displaystyle \frac{(4m^2-4m-v^2)j_m\left(v\right)+(3-2m)v{j}_{m+1}\left(v\right)}{v^2}},$$

(43)

giving the functions $h_m\left(v\right)$,

$$\begin{array}{cc}\hfill h_0\left(v\right)=& j_2\left(v\right),\hfill \\ \hfill h_1\left(v\right)=& {\displaystyle \frac{j_2\left(v\right)-v{j}_1\left(v\right)}{v}},\hfill \\ \hfill h_2\left(v\right)=& {\displaystyle \frac{8j_2\left(v\right)-v{j}_3\left(v\right)-v^2{j}_2\left(v\right)}{v^2}}.\hfill \end{array}$$

(44)

We note that for $m=0$, ${\Delta }_2^{\left(0\right)}=I_2^{\left(0\right)}$, giving $I_{2c}^{\left(0\right)}=I_2^{\left(0\right)}cos2\varphi$, $I_{2s}^{\left(0\right)}=I_2^{\left(0\right)}sin2\varphi$. Generally, for $m=0$, ${\Delta }_n^{\left(0\right)}=I_n^{\left(0\right)}$. The functions $h_m$ are shown in Figure 10c.

7. Discussion

It was found that the field in the focal plane of a hemispherical focused wave can be written in an analytical form. This holds for scalar waves and for focused linear or circular polarized waves or some other polarization distributions. In particular, the (transverse) electric dipole form maximizes the electric energy density at the focus. The TE${⠀}_1$ form results in a narrower central lobe. The same maximum electric energy density for a transverse electric dipole field also occurs for an axial electric dipole field, which can be achieved by focusing a radially polarized wave.

An important observation from the results is that using high-numerical-aperture scalar theory does not result in a useful approximation to the vectorial theory for electric energy density in the optical case. The high-NA scalar theory results in a narrower central lobe than predicted with paraxial theory, whereas for the electric energy density, the vectorial theory results in a broader central lobe than predicted with paraxial theory.

While the analytical results described here are calculated for a hemispherical focusing system, they give approximate results for practical systems of high NA such as 0.95. Although, according to scalar theory, the width of the PSF is 5% wider for 0.95 than for the hemispherical case; for the focusing of linearly polarized light for the mixed dipole case (or by an aplanatic system), the PSF is only 2.6% wider, as a result of the cross-polarization terms. A better approximation for such a case can be developed by noting that the focused field for NA = 0.95 is the unnormalized hemispherical field minus the field for an annular aperture, the latter being approximated by a vectorial Bessel beam. This approach gives a good approximation of the shape of the curves in the focal region.

In addition to the expected spherical Bessel functions j that appear in complete spherical focused fields, the Bessel functions J also appear in the diffraction integrals for the hemispherical case. We also introduced some other functions $f_m$, $g_m$, and $h_m$ that occurred in the analytical expressions.

Two other sections of the paper considered 4Pi focusing and the focusing of vortex fields. The treatment of 4Pi focusing, in particular regarding the improvement in lateral resolution, has received surprisingly little attention in the literature.