Fast calculation of tightly focused random electromagnetic beams: controlling the focal field by spatial coherence

Ruihuan Tong; Zhen Dong; Yahong Chen; Yahong Chen; Fei Wang; Fei Wang; Yangjian Cai; Yangjian Cai; Yangjian Cai; Tero Setälä; Tero Setälä

doi:10.1364/OE.386187

1. Introduction

The properties of tightly focused vector light fields have been extensively studied over the past two decades, leading to the discovery of many intriguing physical features of optical fields and a wide array of applications [1]. When a vector light beam is focused by a high numerical aperture objective, the wave vector of the beam may undergo bending and the electric field acquire a component along the optical axis, i.e., the longitudinal component is generated [2]. The electric field of a light beam can be decomposed into the radially and azimuthally polarized constituents [3], where the former will create a strong longitudinal field on tight focusing, while the latter shows purely transverse electric field distribution [4]. This implies that a tightly focused vector field, composed of tunable radially and azimuthally polarized parts, may exhibit complex and versatile polarization topologies [5], such as the unusual Möbius polarization structures [6], revealing new optical effects as evidenced by the novel light spin-orbit interaction [7,8] and extraordinary transverse spin angular momentum [9–11], as well as diverse applications, e.g. in particle manipulation [12,13] and nanoscopic sensing [14–16].

To date, the studies on tightly focused light have concentrated mainly on theoretical aspects, although experimental techniques have been gradually established [4,17–19]. Theoretical analysis of the tight-focusing properties of vector light can be performed in terms of the Richards–Wolf vectorial diffraction integral [20], which is usually a double integral for a spatially fully coherent field. Such an integral can be converted into the form of a Fourier transformation which can be computed effectively with the aid of the fast Fourier transform or chirp z-transform algorithms [21,22]. The fast focus-field calculations have been viewed as virtual tools in several practical applications including reverse shaping of the focal fields [23,24] and the analysis of microscopy performance [25]. The propagation and light-matter interaction studies with vector light of decreased spatial coherence, on the other hand, necessitate the use of electromagnetic coherence theory [26,27], such that the second-order statistical properties of the fields are involved in the two-point correlation matrix. Thus, in the context of partially coherent light the Richards–Wolf formalism leads to a four-dimensional integral [28], increasing dramatically the computation time. In our previous studies [29–32], the calculation of a tightly focused partially coherent vector beam with $512 \times 512$ spatial points takes more than hundred hours. This limits the flexible studies concerning the effect of spatial coherence of the incident beam on the physical properties of focal fields [33–37].

In this paper, we introduce a fast calculation protocol for investigating the tight focusing properties of random vectorial light fields. The method is as follows: we first reduce the four-dimensional Richards–Wolf integral to the summation of the two-dimensional integrals by employing a pseudo-mode representation of the two-point correlation matrix [38]. We then convert the two-dimensional integrals into the form of Fourier transform in the wave vector ($\mathbf {k}$) space, which can be calculated rapidly by using the chirp z-transform algorithm [21,22]. Finally, the correlation matrix of the focal field can be obtained effectively by performing a set of chirp z-transforms.

This paper is organized as follows. In Sec. 2, we describe the protocol for the fast focal field calculation. In Sec. 3, we demonstrate the method for several types of radially polarized, spatially partially coherent Schell-model sources. The role of spatial coherence in modulating the tightly focused field is also discussed in Sec. 3. We summarize the results in Sec. 4.

2. General theory

We assume that light incident on a focusing system is a statistically stationary, partially coherent and partially polarized beam-like vector field propagating along, say, the $z$ axis. The (second-order) statistical properties of such a beam in the space-frequency domain are involved in the $2 \times 2$ cross-spectral density matrix [27]

(1)$$\mathbf{W}(\mathbf{r}_1,\mathbf{r}_2) = \langle \mathbf{E}^{{\ast}}(\mathbf{r}_1)\mathbf{E}^{\mathrm{T}}(\mathbf{r}_2) \rangle.$$

Above $\mathbf {r}_1$ and $\mathbf {r}_2$ are two arbitrary spatial points in the transverse plane of the beam, $\mathbf {E}(\mathbf {r})$ is a field realization containing the components transverse to the propagation direction ($x$ and $y$ Cartesian components), whereas the asterisk, superscript T, and the angle brackets denote the complex conjugate, matrix transpose, and ensemble average, respectively. Explicit dependency on frequency has been omitted for brevity. The two-point characterization in Eq. (1) can factor in spatial variables only in the limit of full coherence [39], where the coherence matrix in Eq. (1) can be written as $\mathbf {W}(\mathbf {r}_1,\mathbf {r}_2) = \mathbf {E}^{\ast }(\mathbf {r}_1)\mathbf {E}^{\mathrm {T}}(\mathbf {r}_2)$. This fact doubles the dimension of the integrals for calculating the propagation and light-matter interaction in the case of partially coherent light.

A pseudo-mode or the coherent-mode representation can be adopted to solve the above problem. The essence of a mode representation is that a (spectrally) spatially partially coherent field can be decomposed into a sum of elementary modes that are fully coherent but mutually uncorrelated. Therefore, the propagation and light-matter interaction problems with partially coherent fields can be analyzed by means of coherent optics. Under such a representation, the coherence matrix in Eq. (1) can be represented in the series form as

(2)$$\mathbf{W}(\mathbf{r}_1,\mathbf{r}_2) = \sum_{n} \alpha_n \boldsymbol{\Phi}_n^\ast(\mathbf{r}_1) \boldsymbol{\Phi}_n^{\mathrm{T}}(\mathbf{r}_2) ,$$

where $\boldsymbol {\Phi }_n(\mathbf {r})$ and $\alpha _n$ are the mode vectors and their corresponding modal weights, respectively. The above expression is called the pseudo-mode representation if the mode vectors are not orthonormal. Such representations have attracted significant interest recently [38], and they have been adopted in many circumstances of partially coherent fields [40–44]. The traditional coherent-mode representation [26,45] has also the form of Eq. (2) but the modes are orthonormal (in a finite volume $D$) and, together with the weights, can be obtained, at least numerically, by solving the Fredholm integral equations of the form

(3)$$\int_{D} \boldsymbol{\Phi}_n^{\mathrm{T}}(\mathbf{r}_1) \cdot \mathbf{W}(\mathbf{r}_1,\mathbf{r}_2) \mathrm{d}^2 \mathbf{r}_1 = \alpha_n \boldsymbol{\Phi}_n^{\mathrm{T}}(\mathbf{r}_2).$$

By using a vectorial mode representation, the analysis of tight focusing of a partially coherent vector beam reduces to that of investigating the focusing of a set of mode vectors.

An arbitrary mode vector $\boldsymbol {\Phi }_n(\mathbf {r})$ of the incident beam contains only the transverse components. Thus, it can be decomposed into the radially and azimuthally polarized constituents, i.e.,

(4)$$\boldsymbol{\Phi}_n (\mathbf{r}) = \Phi_n^{\textrm{(rad)}}(\mathbf{r}) \hat{\mathbf{e}}_r + \Phi_n^{\textrm{(azi)}}(\mathbf{r}) \hat{\mathbf{e}}_\phi,$$

where $\Phi _n^{\textrm {(rad)}}(\mathbf {r})$ and $\Phi _n^{\textrm {(azi)}}(\mathbf {r})$ are the amplitudes of the radial and azimuthal components, respectively, and

(5)\begin{align}\hat{\mathbf{e}}_r &= \cos \phi \hat{\mathbf{e}}_x + \sin \phi \hat{\mathbf{e}}_y,\end{align}

(6)\begin{align}\hat{\mathbf{e}}_\phi &={-}\sin \phi \hat{\mathbf{e}}_x + \cos \phi \hat{\mathbf{e}}_y,\end{align}

are the unit vectors along these directions. In Eqs. (5) and (6), $\hat {\mathbf {e}}_x$ and $\hat {\mathbf {e}}_y$ are the $x$ and $y$ Cartesian unit vectors, and $\phi \in (0, 2\pi ]$ is the azimuthal angle with respect to the $x$ axis (see Fig. 1). When the mode vector passes through the objective, the radial polarization unit vector tilts and acquires a $z$ component [2]. Instead, the azimuthal polarization unit vector remains in the original form. The transmitted unit vectors thus can be written as

(7)\begin{align}\hat{\mathbf{e}}_r^{\textrm{(t)}} &= \cos \theta \hat{\mathbf{e}}_r + \sin \theta \hat{\mathbf{e}}_z,\end{align}

(8)\begin{align}\hat{\mathbf{e}}_\phi ^{\textrm{(t)}} &= \hat{\mathbf{e}}_\phi.\end{align}

In Eq. (7), $\theta$ (shown in Fig. 1) is bounded between 0 and $\theta _{\textrm {max}}= \arcsin (\mathrm {NA}/n_{\textrm {t}})$, where NA is the numerical aperture of the lens and $n_{\textrm {t}}$ is the index of refraction of its surrounding. Further, $\hat {\mathbf {e}}_z$ is the unit vector in the $z$ direction. The transmitted mode vector therefore becomes

(9)$$\boldsymbol{\Phi}_n ^{\textrm{(t)}} (\mathbf{r}, \theta) = t_r \Phi_n^{\textrm{(rad)}}(\mathbf{r}) \hat{\mathbf{e}}_r ^{\textrm{(t)}}+ t_\phi \Phi_n^{\textrm{(azi)}}(\mathbf{r}) \hat{\mathbf{e}}_\phi ^{\textrm{(t)}} ,$$

where $t_r$ and $t_\phi$ are the transmission coefficients for the radial and azimuthal polarizations, respectively. Due to the power conservation requirement and the sine condition for typical objective lenses, $t_r = t_\phi = \sqrt {\cos \theta }$ holds [20].

Fig. 1. Geometry and notation related to tight focusing of a partially coherent vector beam by a high numerical aperture objective lens with focal length $f$. In transmission through the objective the mode vector ${\Phi }_n (\mathbf {r})$ transforms into ${\Phi }^{\textrm {(t)}}_n (\mathbf {r}, \theta )$ and the related wave vector $\mathbf {k}$ points towards the focal point and forms an angle $\theta$ with respect to the $z$ axis. $\phi$ is the azimuthal angle of position vector $\mathbf {r}$. The focal plane is located at $z=0$.

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According to the Richards–Wolf method, the electric field near a focus is given by the vectorial diffraction integral of the field over the spherical surface of radius equal to the focal length $f$ of the lens. Thus, the mode vector close to focus is obtained as [2]

(10)$$\boldsymbol{\Psi}_n (\mathbf{r}, z) ={-}\frac{\mathrm{i} f}{\lambda} \iint\limits _{\Omega} \boldsymbol{\Phi}_n ^{\textrm{(t)}} (\mathbf{r}^\prime, \theta) \mathrm{e}^{\mathrm{i} (k_z z- k_x x-k_y y)} \mathrm{d} \Omega,$$

where $\lambda$ is the wavelength of the incident beam, $\mathbf {r} = (x,y)$ and $\mathbf {r}^ \prime = (x^ \prime ,y^ \prime )$ are the transverse coordinates in the output and input planes, respectively, and $z$ is the longitudinal distance from the focal point $(x,y,z) = (0,0,0)$. The two planes are defined such that the former is a plane in the focal region while the latter is located immediately after the lens. In addition,

(11)\begin{align}k_x &={-}k \sin \theta \cos \phi,\end{align}

(12)\begin{align}k_y &={-}k \sin \theta \sin \phi,\end{align}

(13)\begin{align}k_z &= k \cos \theta,\end{align}

are the components of the wave vector $\mathbf {k}$ in the Cartesian coordinates along the $x$, $y$, and $z$ directions, with $k = |\mathbf {k}| = 2\pi n_{\textrm {t}} / \lambda$. In addition, the integration in Eq. (10) extends over the solid angle $\Omega$ with

(14)$$\mathrm{d} \Omega = \sin \theta \mathrm{d} \theta \mathrm{d} \phi.$$

By using the relations of $\theta$, $\phi$ and $k_x$, $k_y$ in Eqs. (11) and (12) as well as the Jacobian determinant, we obtain

(15)$$\mathrm{d} \Omega = \frac{1}{k^2 \cos \theta} \mathrm{d} k_x \mathrm{d} k_y.$$

From the geometry of the focusing system and by using the sine condition for the objective lens, we find that the radial distance $r^\prime$ from the $z$ axis in the input plane can be written as

(16)$$r^\prime= f \sin \theta.$$

Comparing Eq. (16) with Eqs. (11) and (12), we find the relation between $\mathbf {r}^\prime$ and $k_x$, $k_y$ as follows

(17)$$x^\prime ={-}f k_x / k,$$

(18)$$y^\prime ={-}f k_y / k.$$

From Eq. (13), we obtain $\theta = \arccos (k_z/k)$, where $k_z = (k^2 - k_x^2 -k_y^2)^{1/2}$. Thus, the mode vector $\boldsymbol {\Phi }_n ^{\textrm {(t)}} (\mathbf {r}^\prime , \theta )$ in Eq. (10) can be written as $\boldsymbol {\Phi }_n ^{\textrm {(t)}} (k_x, k_y)$. Taking the result obtained in Eq. (15) into Eq. (10), we obtain

(19)$$\boldsymbol{\Psi}_n (\mathbf{r}, z) ={-}\frac{\mathrm{i} f}{\lambda k} \iint\limits _{r^\prime < R} \boldsymbol{\Phi}_n ^{\textrm{(t)}} (k_x, k_y) \mathrm{e}^{\mathrm{i}k_z z} k_z^{{-}1} \mathrm{e}^{-\mathrm{i}(k_x x +k_y y)} \mathrm{d}k_x \mathrm{d}k_y,$$

where $r^\prime < R$, with $R$ being the radius of the aperture, denotes that the integration is limited to the region of the lens (we assume that the beam fills the whole objective). By setting $\boldsymbol {\Phi }_n ^{\textrm {(t)}} (k_x, k_y) =0$ for $r^\prime > R$, the integration in Eq. (19) can be formally extended over the whole aperture plane. The integration, therefore, takes on the form of two-dimensional Fourier transform, i.e.,

(20)$$\boldsymbol{\Psi}_n (\mathbf{r}, z) ={-}\frac{\mathrm{i} f}{\lambda k} \mathcal{F} \left[ \boldsymbol{\Phi}_n ^{\textrm{(t)}} (k_x, k_y) \mathrm{e}^{\mathrm{i}k_z z} k_z^{{-}1} \right]_{k_x,k_y},$$

where $\mathcal {F} [\cdot ]_{k_x,k_y}$ denotes the transform over the variables $k_x$ and $k_y$. We remark that $k_z = (k^2 - k_x^2 -k_y^2)^{1/2}$, as a function of $k_x$ and $k_y$, must be taken into account in the Fourier transform. The transform in Eq. (20) can be solved fast with the help of Matlab algorithms, e.g., the two-dimensional fast Fourier transform algorithm and the chirp z-transform algorithm [21,22]. In our numerical examples in Sec. 3, we use the chirp z-transform algorithm to perform the calculations of the tightly focused field distributions rapidly with high accuracy.

After the mode vectors near the focal region have been obtained, the cross-spectral density matrix of a tightly focused partially coherent vector field can be evaluated as

(21)$$\boldsymbol{\mathcal{W}}(\mathbf{r}_1,z_1; \mathbf{r}_2,z_2) = \sum_{n} \alpha_n \boldsymbol{\Psi}_n^\ast(\mathbf{r}_1, z_1) \boldsymbol{\Psi}_n^{\mathrm{T}}(\mathbf{r}_2, z_2),$$

which is a $3 \times 3$ matrix. We have hence introduced a general fast calculation protocol for numerically calculating the statistical properties of a tightly focused spatially partially coherent vector beam with arbitrary polarization and coherence states. Its performance is superior compared with the conventional quadruple Richards–Wolf integral method.

3. Numerical calculation of focused radially polarized Schell-model beams

In this section, we demonstrate the speed and feasibility of our protocol by numerically calculating the tight focusing properties of several kinds of partially coherent, radially (fully) polarized incident Schell-model beams.

3.1 Schell-model beams and the complex random screens decomposition

Schell-model type optical beams, for which the degree of spatial coherence depends on the point separation only, are the most widely studied partially coherent sources and they have found use in various applications [46]. Among them, the classical Gaussian Schell-model (GSM) beam [26], whose spectral density and degree of coherence both are of Gaussian form, has been used, e.g., to realize classical ghost interference and ghost imaging [47,48] and applied in free-space optical communication to overcome the turbulence-induced degradation, beam wander, and scintillation [49]. Recently, a wide class of Schell-model beams with non-Gaussian degree of coherence has been proposed [50]. These unconventional Schell-model beams have extraordinary propagation properties. For example, a multi-Gaussian Schell-model (MGSM) beam will exhibit a flat topped beam profile in the far field or in the focal plane [51], a Laguerre-Gaussian Schell-model (LGSM) beam can show a dark hollow beam profile [52,53], and a Hermite-Gaussian Schell-model (HGSM) beam an array profile [54], although these beams have a Gaussian beam profile in the source plane. Such self-shaping effects of unconventional Schell-model beams have potential applications in optical manipulations.

For a radially polarized, spatially partially coherent Schell-model beam, the cross-spectral density matrix can be written as

(22)$$\mathbf{W}(\mathbf{r}_1, \mathbf{r}_2) =\sqrt{ \mathcal{S} (\mathbf{r}_1) \mathcal{S}(\mathbf{r}_2)} g(\Delta \mathbf{r}) \hat{\mathbf{e}}_r(\mathbf{r}_1) \hat{\mathbf{e}}_r^{\mathrm{T}}(\mathbf{r}_2),$$

where $\mathcal {S} (\mathbf {r})$ and $\hat {\mathbf {e}}_r(\mathbf {r})$ [given in Eq. (5)] denote the spectral density and the unit polarization vector of the beam at point $\mathbf {r}$, respectively. We assume that the spectral density is of the form

(23)$$\mathcal{S}(\mathbf{r}) = \frac{2 r^2}{w_0^2} e^{{-}2 r^2/w_0^2},$$

where $w_0$ characterizes the beam width and $r=\vert \mathbf {r}\vert$. In addition, $g(\Delta \mathbf {r})$, with $\Delta \mathbf {r} = \mathbf {r}_1 - \mathbf {r}_2$, represents the degree of coherence of the beam. We remark that the absolute value of $g(\Delta \mathbf {r})$ equals the electromagnetic degree of coherence defined in [39]. The coherence matrix in Eq. (22) could be decomposed into the uncorrelated coherent modes by solving the complicated Fredholm integral equations in Eq. (3). However, to increase the efficiency of our protocol, we use the recently advanced complex random screens decomposition method for the Schell-model beams [55–62], which expresses the degree of coherence as

(24)$$g(\Delta \mathbf{r}) = \frac{1}{N} \sum_{n} \mathcal{T}_n^\ast (\mathbf{r}_1) \mathcal{T}_n (\mathbf{r}_2),$$

where $N$ is the total number of the complex random screens and $\mathcal {T}_n (\mathbf {r})$ is the (complex-valued) transmittance of the screen with label $n$. We stress that the modes in the above decomposition are uncorrelated, although they are not orthogonal. Thus, the complex random screens decomposition leads to a pseudo-mode representation of the cross-spectral density matrix.

The transmittance of the screen can be obtained by [57]

(25)$$\mathcal{T}_n (\mathbf{r}) = \mathcal{F}^{{-}1}\left[ \tau_n (f_x,f_y) \right]_{f_x,f_y},$$

where $\mathcal {F}^{-1} [\cdot ]_{f_x,f_y}$ denotes the two-dimensional inverse Fourier transform over the variables $f_x$ and $f_y$, and $\tau _n (f_x,f_y)$ is a complex random function, written as

(26)$$\tau_n (f_x,f_y) = \frac{a_n(f_x,f_y) + \mathrm{i} b_n(f_x,f_y)}{\sqrt{2}} \sqrt{p(f_x,f_y)}.$$

Here $\{a_n(f_x,f_y)\}$ and $\{b_n(f_x,f_y)\}$ are independent, unit variance, Gaussian random processes, and their combination defines a complex circular Gaussian process. The random processes can be obtained by two independent randn functions in Matlab. In Eq. (26), $p(f_x,f_y)$ is the power spectrum associated with the complex random screens, given by

(27)$$p(f_x,f_y) = \int g(\Delta \mathbf{r}) \mathrm{e}^{-\mathrm{i} 2 \pi (\Delta x f_x + \Delta y f_y )} \mathrm{d}^2 \Delta \mathbf{r},$$

where $\Delta \mathbf {r} = (\Delta x,\Delta y) = (x_1-x_2, y_1-y_2)$.

For a GSM beam, we have

(28)$$g(\Delta \mathbf{r}) = \exp[- \Delta \mathbf{r}^2 / (2 \delta_0^2)],$$

where $\delta _0$ stands for the transverse coherence width of the beam. Inserting Eq. (28) into Eq. (27), the power spectrum assumes the form

(29)$$p(f_x,f_y) = 2 \pi \delta_0^2 \exp [{-}2 \pi ^2 \delta_0^2 (f_x^2 + f_y^2)].$$

For an MGSM beam,

(30)$$g(\Delta \mathbf{r}) = \frac{1}{C_0} \sum_{m=1}^{M} \left({\begin{array}{c} M \\ m\end{array}}\right) \frac{(-1)^{m-1}}{m} \exp \left(- \frac{\Delta \mathbf{r}^2}{2 m \delta_0^2} \right),$$

where $C_0 = \sum\nolimits_{m=1}^{M} \left(\begin{array}{c} M \\ m\end{array}\right) \frac {(-1)^{m-1}}{m}$ is a normalization factor, $\left(\begin{array}{c} M \\ m\end{array}\right)$ stands for the binomial coefficient, and the integer $M \geq 1$ is the beam order. In this case, the power spectrum can be expressed as

(31)$$p(f_x,f_y) = \frac{2 \pi \delta_0^2}{C_0} \sum\limits_{m=1}^{M} \left(\begin{array}{c} M \\ m\end{array}\right) ({-}1)^{m-1} \exp [{-}2 \pi ^2 m \delta_0^2 (f_x^2 + f_y^2)].$$

For an LGSM beam,

(32)$$g(\Delta \mathbf{r}) =L_l^0 \left( \frac{\Delta \mathbf{r}^2}{2 \delta_0^2} \right) \exp \left(- \frac{\Delta \mathbf{r}^2}{2 \delta_0^2} \right),$$

where $L_l^0$ is a Laguerre polynomial with mode orders $l$ and 0. The power spectrum for the LGSM beam thus is

(33)$$p(f_x,f_y) = (\pi^{2l+1} \delta_0^{2l+2} 2^{l+1}/l!) (f_x^2 + f_y^2)^l \exp [{-}2 \pi ^2 \delta_0^2 (f_x^2 + f_y^2)].$$

For an HGSM beam,

(34)$$g(\Delta \mathbf{r}) = \frac{H_{2m_x} [\Delta x / (\sqrt{2} \delta_0)] H_{2m_y} [\Delta y / (\sqrt{2} \delta_0)]}{H_{2m_x}(0) H_{2m_y}(0)} \exp \left(- \frac{\Delta \mathbf{r}^2}{2 \delta_0^2} \right),$$

where $H_{2m_x}$ and $H_{2m_y}$ denote the Hermite polynomials of orders $2m_x$ and $2m_y$, respectively. The related power spectrum is given by

(35)$$p(f_x,f_y) = \frac{ 2^{3(m_x+m_y) +1} \pi^{2(m_x+m_y)} \delta_0^{2(m_x +m_y +1)} } {H_{2m_x}(0) H_{2m_y}(0)} f_x^{2m_x} f_y^{2m_y} \exp [{-}2 \pi ^2 \delta_0^2 (f_x^2 + f_y^2)].$$

Substituting the power spectra in Eqs. (29), (31), (33), and (35) into Eq. (26), and then into Eq. (25), the complex random screens for representing the GSM, MGSM, LGSM, and HGSM beams can be obtained numerically. The mode vectors for the various radially polarized, spatially partially coherent Schell-model beams can thus be obtained by

(36)$$\boldsymbol{\Phi}_n (\mathbf{r}) = \sqrt{ \mathcal{S} (\mathbf{r}) /N } \mathcal{T}_n (\mathbf{r}) \hat{\mathbf{e}}_r,$$

and allow to express the cross-spectral density matrix in the form of Eq. (2) with $\alpha _n = 1$. According to Eq. (9), the mode vector transmitted through the numerical aperture can be written as

(37)$$\boldsymbol{\Phi}_n ^{\textrm{(t)}} (\mathbf{r}, \theta) = \sqrt{\cos \theta \mathcal{S} (\mathbf{r}) /N } \mathcal{T}_n (\mathbf{r}) \hat{\mathbf{e}}_r ^{\textrm{(t)}}.$$

Using $\hat {\mathbf {e}}_r ^{\textrm {(t)}}$ of Eq. (7) in Eq. (37), $\boldsymbol {\Phi }_n ^{\textrm {(t)}} (\mathbf {r},\theta )$ can be expressed, in the Cartesian coordinates, as

(38)$$\boldsymbol{\Phi}_n ^{\textrm{(t)}} (\mathbf{r}, \theta) = \sqrt{\cos \theta \mathcal{S} (\mathbf{r}) /N } \mathcal{T}_n (\mathbf{r}) \left\{ {\begin{array}{c} \cos\theta \cos \phi\\ \cos \theta \sin \phi \\ \sin \theta \end{array}} \right\}.$$

Here $\mathbf {r}$ can be replaced with $k_x$ and $k_y$ by using Eqs. (17) and (18), while Eqs. (11)–(13) imply

(39)$$\theta = \arccos (k_z/k),$$

(40)$$\phi = \arg ({-}k_x -\mathrm{i} k_y),$$

where $\arg$ denotes the phase of a complex function and $k_z = (k^2 - k_x^2 -k_y^2)^{1/2}$ as earlier. Finally, we put $\boldsymbol {\Phi }_n ^{\textrm {(t)}} (k_x,k_y)$ into Eq. (20) and use the chirp z-transform algorithm to perform the two-dimensional Fourier transform. By adding up the obtained $\boldsymbol {\Psi }_n (\mathbf {r}, z)$ functions through Eq. (21), the two-point coherence matrix $\boldsymbol {\mathcal {W}}(\mathbf {r}_1,z_1; \mathbf {r}_2,z_2)$ of a tightly focused radially polarized Schell-model beam in the focal area can be obtained. In this paper, we focus only on the spectral density of the focused fields. This quantity is obtained by [27]

(41)$$S(\mathbf{r},z) = S_x(\mathbf{r},z) + S_y(\mathbf{r},z) + S_z(\mathbf{r},z),$$

where $S_i(\mathbf {r},z)$, with $i=x,y,z$, is the spectral density of the $i$ polarized component, i.e., a diagonal element of $\boldsymbol {\mathcal {W}}(\mathbf {r},z; \mathbf {r},z)$.

3.2 Numerical results and discussion

To demonstrate the feasibility of the complex random screens decomposition of the various types of Schell-model beams, we first construct for each of them 2000 screens $\mathcal {T}_n (\mathbf {r})$ with $512 \times 512$ spatial resolution by using the chirp z-transform of $\tau _n(f_x,f_y)$ in Eq. (26). After that, we add up $\mathcal {T}_n (\mathbf {r})$ by using Eq. (24) to obtain an estimate for the degree of coherence $g(\Delta \mathbf {r})$ in the cases of GSM, MGSM, LGSM, and HGSM beams. Throughout the examples we assume that the spatial coherence width for all beam types is $\delta _0=0.2$ mm. The analytical expressions for the different $g(\Delta \mathbf {r})$ functions, obtained by Eqs. (28), (30), (32), and (34), are shown in the left column of Fig. 2. The related numerical constructions are displayed in the middle column while the right column shows the cross-line ($\Delta y = 0$) for both numerical and analytical expressions. Figure 2 demonstrates that the agreement between the numerical and analytical representations is excellent when 2000 screens are used. The complex random screens decomposition method, thus, is a feasible way to decompose the cross-spectral density matrix of a Schell-model beam into spatially fully coherent but mutually uncorrelated modes.

Fig. 2. Analytical (left) and numerical (middle) representations of the degree of coherence $g(\Delta \mathbf {r})$ for a GSM beam (top row), an MGSM beam of beam order $M = 10$ (second row), an LGSM beam of beam order $l = 5$ (third row), and an HGSM beam of beam order $m_x = 1, m_y = 0$ (bottom row). The spatial coherence width is $\delta _0 = 0.2$ mm in all cases. (Right) Cross-line ($\Delta y=0$) of the analytical (gray solid curves) and numerical expressions (black dashed curves).

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Next we investigate the influence of the number $N$ of the complex random screens on the numerical results. We display in Fig. 3 the similarity between the numerical and analytical expressions of $g(\Delta \mathbf {r})$ for the LGSM beam as a function of $N$. The similarity is defined as

(42)$$s = \frac{ [\iint g^{(\textrm{A})}(\Delta \mathbf{r}) g^{(\textrm{N})}(\Delta \mathbf{r}) \mathrm{d}^2 \Delta \mathbf{r} ]^2 }{\iint [g^{(\textrm{A})}(\Delta \mathbf{r})]^2 \mathrm{d}^2 \Delta \mathbf{r} \iint [g^{(\textrm{N})}(\Delta \mathbf{r})]^2 \mathrm{d}^2 \Delta \mathbf{r} },$$

where $g^{(\textrm {A})}(\Delta \mathbf {r})$ and $g^{(\textrm {N})}(\Delta \mathbf {r})$ refer to the analytical and numerical representations, respectively. According to the Cauchy–Buniakowsky–Schwarz inequality, the similarity $s$ is bounded between 0 and 1 with the notions that the larger value the higher similarity and vice versa. The shaded region and the solid curve in Fig. 3 illustrate, respectively, the standard deviation and the mean value of the similarities obtained by 10 independent simulations. It is found that the similarity increases and the deviation between the different independent simulations decreases with the increase of the screens number $N$. When $N \geq 2000$, the similarity $s$ is very close to 1, implying that the summation in Eq. (24) has converged and the numerical expression is in good agreement with the analytical one, as was noted above. We remark that, in general, the shorter is the coherence width the more terms may be needed in the mode decomposition. The possible large number of modes is an essential limiting factor of the method.

Fig. 3. Similarity $s$ between the numerical and analytical expressions of $g(\Delta \mathbf {r})$ versus the number $N$ of the complex random screens for the LGSM beam with $l=5$ and $\delta _0 = 0.2$ mm. The shaded region and the solid curve illustrate, respectively, the standard deviation and the mean value of the similarities for ten independent simulations.

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Next, we take the $2000$ complex random screens into the algorithm of fast focus-field calculation. The computation time for obtaining a tightly focused field distribution (and its spectral density) with $512 \times 512$ spatial points in the case of a radially polarized Schell-model beam is about $100$ seconds. Figure 4 shows the numerically obtained focal-plane distributions of the total spectral density, $S(\mathbf {r}, 0 )$, the transverse spectral density, $S_x(\mathbf {r}, 0 )+S_y(\mathbf {r}, 0 )$, and the longitudinal spectral density, $S_z(\mathbf {r}, 0 )$, for the radially polarized GSM, MGSM, LGSM, and HGSM beams when the numerical aperture is $\mathrm {NA}=0.95$. The beams are spatially weakly coherent and for all Schell-model types the spatial coherence width and the beam width are taken to be $\delta _0 = 0.2$ mm and $w_0 = 1$ mm, respectively. Other parameters used in the simulations are listed in the caption of Fig. 4. It is found from the figure that the total, transverse, and the longitudinal-field spectral densities are closely related to the scalar degree of coherence $g(\Delta \mathbf {r})$ of the incident beam. When the degree of coherence is a Gaussian function, the three spectral densities in the focal plane have Gaussian distributions, while for the MGSM, LGSM, and HGSM beams they display a flat-topped, dark-hollow, and two-petal profiles, respectively. These are similar to the spectral density/intensity profiles generated by these beams in the far zone or in the focal plane of a thin lens [51,52,54]. The results shown in Fig. 4 indicate that spatial coherence of a vector beam can be viewed as a degree of freedom to manipulate (or shape) both the transverse and longitudinal-field spectral densities of a highly focused field. The optical forces (e.g., the optical gradient force) at a focus, therefore, can be inversely manipulated by controlling the incident beam’s spatial coherence distribution.

Fig. 4. Numerical focal-plane distributions of the normalized total spectral density $S(\mathbf {r})$ (left), transverse spectral density $S_x(\mathbf {r})+S_y(\mathbf {r})$ (middle), and the longitudinal spectral density $S_z(\mathbf {r})$ (right) of a tightly focused radially polarized GSM beam (top row), MGSM beam of order $M = 10$ (second row), LGSM beam of order $l=5$ (third row), and HGSM beam of order $m_x = 1, m_y = 0$ (bottom row). The parameters are: $\lambda = 632.8$ nm, $\delta _0 = 0.2$ mm, $w_0 = 1$ mm, $\textrm {NA} = 0.95$, $f = 3$ mm, and $n_{\textrm {t}} = 1$.

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To examine the effect of the incident-beam coherence width $\delta _0$ on the spectral density in the focal plane, we display in Fig. 5 the numerically obtained distributions of the total spectral density, $S(\mathbf {r}, 0)$, transverse spectral density, $S_x(\mathbf {r}, 0) + S_y(\mathbf {r}, 0)$, and the longitudinal spectral density, $S_z(\mathbf {r}, 0)$, for a radially polarized LGSM beam with different $\delta _0$ values which can be achieved conveniently in the experiments [50]. We see that the spectral densities change significantly with $\delta _0$. This is because when the coherence width increases, the effect of the shape of the spatial coherence distribution decreases while the role of the polarization structure increases. Therefore, we can observe from Fig. 5 that with increasing $\delta _0$, the longitudinal-field spectral density $S_z(\mathbf {r}, 0)$ changes gradually from a dark-hollow beam profile (induced by the degree of coherence distribution) to a subwavelength spot (induced by the radial polarization distribution of the incident beam). For the transverse and total spectral densities, the dark-hollow profiles in the case of low spatial coherence disappear gradually as the coherence width increases and they reappear when $\delta _0$ becomes large. Further, we find from Fig. 5 that with the increase of $\delta _0$, the ratio of the energy carried by the longitudinal field to the total energy increases, which is consistent with the results in Ref. [32]. In addition, we note here that the number of complex random screens required for the simulations decreases with increasing coherence width. For example, $N=2000$ screens are needed for $\delta _0 = 0.2$ mm, but only $N=100$ for $\delta _0 = 3$ mm, and $N=10$ for $\delta _0 = 10$ mm. Thus, with the increase of the spatial coherence width, the computation time of our calculation algorithm decreases.

Fig. 5. Numerical focal-plane distributions of the normalized total spectral density $S(\mathbf {r})$ (left), transverse spectral density $S_x(\mathbf {r})+S_y(\mathbf {r})$ (middle), and the longitudinal spectral density $S_z(\mathbf {r})$ (right) of a tightly focused radially polarized LGSM beam of order $l=5$, for different values of the incident-field spatial coherence width $\delta _0$. The parameters are: $\lambda = 632.8$ nm, $w_0 = 1$ mm, $\textrm {NA} = 0.95$, $f = 3$ mm, and $n_{\textrm {t}} = 1$.

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Finally, we study the effect of the spatial coherence distribution, as an effective degree of freedom, on the spectral density of the longitudinal field component at the (longitudinal) $xz$ and $yz$ planes in the focal region. In Fig. 6 we display the numerically obtained $S_z(\mathbf {r}, z)$ in the $xz$ plane ($y=0$) for the radially polarized Schell-model beams with different spatial coherence distributions. Here we only consider a low coherence case with $\delta _0 = 0.2$ mm. It is observed in Fig. 6 that the form of the degree of coherence plays an important role in tailoring $S_z(\mathbf {r}, z)$ in a longitudinal plane. Specifically, the GSM-form degree of coherence creates a Gaussian beam profile in the $z$ direction, while the MGSM, LGSM, and HGSM distributions produce a flat-topped, dark-hollow, and two-petal profiles, respectively. Further, considering the rotational symmetry of the GSM, MGSM, and LGSM distributions and the rectangular symmetry of the HGSM distribution, as illustrated in Fig. 4, the three-dimensional spatial spectral density distributions near the focus display a Gaussian spot, a cylindrical flat-topped volume, an optical cage, and a two-beam-channel, respectively. The shapes can find use in particle manipulation, e.g., the cylindrical flat-topped volume is useful for increasing the trapping area, the optical cage can be used for trapping the particles with the refractive index smaller than that of the surrounding, and the beam channels can guide atoms.

Fig. 6. Numerically obtained distributions of the normalized longitudinal spectral density, $S_z(\mathbf {r}, z)$, in the $xz$ plane ($y=0$) of the focused radially polarized (a) GSM beam, (b) MGSM beam of beam order $M=10$, (c) LGSM beam of beam order $l=5$, and (d) HGSM beam of beam order $m_x = 1$, $m_y = 0$. The parameters are: $\lambda = 632.8$ nm, $\delta _0 = 0.2$ mm, $w_0 = 1$ mm, $\textrm {NA} = 0.95$, $f = 3$ mm, and $n_{\textrm {t}} = 1$.

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4. Conclusions

In summary, we introduce an effective protocol for the fast calculation of tightly focused partially coherent vector beams with arbitrary spatial coherence and polarization states. The technique is based on the vectorial pseudo-mode decomposition of the correlation matrix and the chirp z-transform algorithm for the Fourier transform in the wave vector space. We demonstrate the method by calculating the focal spectral densities of tightly focused spatially partially coherent, radially (fully) polarized Schell-model beams whose degree of coherence is developed in terms of the complex random screens representation. The computation time for obtaining the focal field distribution with $512 \times 512$ spatial points for a beam with low spatial coherence is less than 100 seconds, which is superior compared to the time required (more than 100 hours) by the conventional four-dimensional Richards–Wolf integral method.

Ultimately, our results demonstrate that the shape of the spatial degree of coherence of the incident beam can be regarded as an important resource for manipulating both the transverse and longitudinal field components in a tight focus. The underlying physical mechanism is that the field in a single point of focal region depends on the field in every point at the input plane. Hence, the two-point coherence properties of the incident beam are reflected to the single-point coherence properties in the focus. The proposed fast calculation protocol enables effective engineering of the physical properties (spectral density, detailed polarimetric features, etc.) of tightly focused fields which can be useful, e.g., in optical trapping control.

Funding

National Key Research and Development Program of China (2019YFA0705000); National Natural Science Foundation of China (11525418, 11874046, 11904247, 11974218, 91750201); Academy of Finland (308393, 320166); Innovation Group of Jinan (2018GXRC010); Priority Academic Program Development of Jiangsu Higher Education Institutions; Natural Science Research of Jiangsu Higher Education Institutions of China (19KJB140017); Natural Science Foundation of Shandong Province (ZR2019QA004); China Postdoctoral Science Foundation (2019M661915); Qinglan Project of Jiangsu Province of China.

Disclosures

The authors declare no conflicts of interest.

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Fast calculation of tightly focused random electromagnetic beams: controlling the focal field by spatial coherence

Abstract

1. Introduction

2. General theory

3. Numerical calculation of focused radially polarized Schell-model beams

3.1 Schell-model beams and the complex random screens decomposition

3.2 Numerical results and discussion

4. Conclusions

Funding

Disclosures

References

Cited By

Figures (6)

Equations (42)

Optics Express