Introduction

Transparent dielectric single microspheres support optical resonances or whispering gallery modes upon the resonant light illumination1,2,3. The quality factor of these resonances is improved significantly when the microspheres are excited with suitable geometries4,5,6,7,8. The microspheres are also found to support super-resonances for specific values of size parameters9,10. However, narrow-focused and intense electromagnetic beams are generated at the shadow side of these microspheres upon the non-resonant light illumination. These are non-evanescent beams and are named ‘photonic nanojets’ (PNJs)11,12,13,14,15,16,17. These PNJs are generated from solid/core–shell dielectric microparticles illuminated by a plane wave18,19, spherical wave20, Gaussian beam21,22, Bessel beam23,24, etc. The origin of the PNJ is explained based on the forward Mie scattering and focused near-field diffraction25,26,27.

It is to be noted that the PNJs generated by monochromatic light are found useful for enhancing Raman scattering28,29,30,31, fluorescence signals32, nanoparticles sensing33, optical data storage34, laser surgery35, optical transport36, two-photon fluorescence37, nanolithography38, biomedical research39 etc. On the other hand, several researchers have generated the PNJs using the focused polychromatic light used for enhancing the backscattering signals of nanoparticles40, photoacoustic spectroscopy41, white light nanoscopy42,43,44,45,46,47,48,49, low-coherence phase-shifting interference microscopy50, Mirau interferometry51, and super-resolution imaging52,53,54. It is worth mentioning here that several theoretical studies are reported on PNJs of single microspheres illuminated by monochromatic plane and Gaussian waves18,55,56,57,58,59. The role of key parameters such as incident beam waist (ω0), refractive index of the surrounding medium (nm), radius (Rs), and refractive index (ns) of the microspheres on the characteristic parameters of the PNJs is reported in detail. Few theoretical studies related to the PNJs of single dielectric microspheres under pulsed irradiation (or coherent broadband illumination) are also reported in the literature60,61. Under the pulsed irradiation, the PNJs appear to be non-stationary and these are found useful in micromachining beyond the diffraction limit62, microscopy63, and non-linear optics64. However, details of the characteristic parameters of the PNJs generated by focused polychromatic light (or broadband radiation) and the role of Rs, nm, and ω0 on the characteristic parameters of the PNJs generated by broadband radiation are not reported in the literature. Moreover, the differences in the PNJs generated by focused monochromatic light and polychromatic light are not reported. These details would be useful for optimizing the experimental signals obtained using PNJs generated by polychromatic light. Therefore, we have performed a theoretical investigation on the PNJ of single solid and core–shell microspheres illuminated by focused polychromatic light from different sources.

Herein, we report (1) the electric field intensity enhancement (EFIE) distribution inside and outside single dielectric microspheres illuminated by polychromatic light from different light sources such as Hg arc lamp, white LED, supercontinuum (SC) source, and Halogen lamp, (2) the role of ω0, Rs, and nm on the characteristic parameters of PNJs obtained for polychromatic illumination, (3) the characteristic parameters of the PNJs of single microspheres illuminated by monochromatic light, (4) the EFIE distribution inside and outside core–shell microspheres under monochromatic and polychromatic illumination.

Theoretical aspects

Recently, we have developed an analytical theory for estimating the electric field enhancement inside and outside single core–shell dielectric microspheres illuminated by focused monochromatic light59. Using this theory, the characteristic parameters of the PNJs generated by core–shell microspheres of different sizes and refractive indices are estimated. In the present study, the analytical theory has been extended to study the characteristic parameters of the PNJs of single solid/core–shell microspheres illuminated by focused polychromatic light (Fig. 1) from different sources.

Figure 1
figure 1

Illustration of PNJs generated by single solid microsphere (panel a) and core–shell microsphere (panel b) under focused polychromatic illumination. Here Rs, Rc, and Rsh are the radii of the solid microsphere, core, and shell, respectively. ns, nc, nsh, and nm are the refractive indices of the solid microsphere, core, shell, and surrounding medium, respectively.

Full size image

The final expressions for plotting the EFIE in the surrounding medium (ηm), core (ηc), and shell (ηsh) are given below.

$$\eta_{m} = \mathop \Sigma \limits_{p = 1}^{k} I_{m} (r,\theta ,\phi ,\lambda_{p} )$$
(1)
$$\eta_{c} = \mathop \Sigma \limits_{p = 1}^{k} I_{c} (r,\theta ,\phi ,\lambda_{p} )$$
(2)
$$\eta_{sh} = \mathop \Sigma \limits_{p = 1}^{k} I_{sh} (r,\theta ,\phi ,\lambda_{p} )$$
(3)
$$I_{m} (r,\theta ,\phi ,\lambda_{p} ) = E_{r,m}^{2} + E_{\theta ,m}^{2} + E_{\phi ,m}^{2}$$
(4)
$$I_{c} (r,\theta ,\phi ,\lambda_{p} ) = E_{r,c}^{2} + E_{\theta ,c}^{2} + E_{\phi ,c}^{2}$$
(5)
$$I_{sh} (r,\theta ,\phi ,\lambda_{p} ) = E_{r,sh}^{2} + E_{\theta ,sh}^{2} + E_{\phi ,sh}^{2}$$
(6)

here (r,θ,ϕ) are the spherical polar coordinates. λ1, λ2, λ3, …., λk represent different wavelengths present in the polychromatic light. Er,m, Er,c, and Er,sh are the enhancements in the radial component of the electric field in the medium, core, and shell, respectively. Eθ,m, Eθ,c, and Eθ,sh are the enhancements in the angular component of the electric field in the medium, core, and shell, respectively. Eϕ,m, Eϕ,c, and Eϕ,sh are the enhancements in the azimuthal component of the electric field in the medium, core, and shell, respectively. These components can be estimated using the following expressions.

$$E_{r,m} = E_{p} \left( {\lambda_{p} } \right)\cos \phi \mathop \sum \limits_{n = 1}^{\infty } C_{n} g_{n} a_{n}^{m} \left( {\lambda_{p} } \right)\left[ {\xi_{n}^{\prime \prime } \left( {k_{m} \left( {\lambda_{p} } \right)r} \right) + \xi_{n} \left( {k_{m} \left( {\lambda_{p} } \right)r} \right)} \right]P_{n}^{1} \left( {\cos \theta } \right)$$
(7)
$$E_{\theta ,m} = \frac{{E_{p} \left( {\lambda_{p} } \right)}}{{k_{m} \left( {\lambda_{p} } \right)r}}\cos \phi \mathop \sum \limits_{n = 1}^{\infty } C_{n} g_{n} \left[ {a_{n}^{m} \left( {\lambda_{p} } \right)\xi_{n}^{\prime } \left( {k_{m} \left( {\lambda_{p} } \right)r} \right)\tau_{n} \left( {\cos \theta } \right) - ib_{n}^{m} \left( {\lambda_{p} } \right)\xi_{n} \left( {k_{m} \left( {\lambda_{p} } \right)r} \right)\pi_{n} \left( {\cos \theta } \right)} \right]$$
(8)
$$E_{\phi ,m} = - \frac{{E_{p} \left( {\lambda_{p} } \right)}}{{k_{m} \left( {\lambda_{p} } \right)r}}\sin \phi \mathop \sum \limits_{n = 1}^{\infty } C_{n} g_{n} \left[ {a_{n}^{m} \left( {\lambda_{p} } \right)\xi_{n}^{\prime } \left( {k_{m} \left( {\lambda_{p} } \right)r} \right)\pi_{n} \left( {\cos \theta } \right) - ib_{n}^{m} \left( {\lambda_{p} } \right)\xi_{n} \left( {k_{m} \left( {\lambda_{p} } \right)r} \right)\tau_{n} \left( {\cos \theta } \right)} \right]$$
(9)
$$E_{r,sh} = - E_{p} \left( {\lambda_{p} } \right)\cos \phi \mathop \sum \limits_{n = 1}^{\infty } C_{n} g_{n} \left\{ \begin{gathered} a_{n}^{sh\alpha } \left( {\lambda_{p} } \right)\left[ {\psi_{n}^{\prime \prime } \left( {k_{sh} \left( {\lambda_{p} } \right)r} \right) + \psi_{n} \left( {k_{sh} \left( {\lambda_{p} } \right)r} \right)} \right] \hfill \\ - a_{n}^{sh\beta } \left( {\lambda_{p} } \right)\left[ {\xi_{n}^{\prime \prime } \left( {k_{sh} \left( {\lambda_{p} } \right)r} \right) + \xi_{n} \left( {k_{sh} \left( {\lambda_{p} } \right)r} \right)} \right] \hfill \\ \end{gathered} \right\}P_{n}^{1} \left( {\cos \theta } \right)$$
(10)
$$E_{\theta ,sh} = - \frac{{E_{p} \left( {\lambda_{p} } \right)}}{{k_{sh} \left( {\lambda_{p} } \right)r}}\cos \phi \mathop \sum \limits_{n = 1}^{\infty } C_{n} g_{n} \left\{ \begin{gathered} \left[ {a_{n}^{sh\alpha } \left( {\lambda_{p} } \right)\psi_{n}^{\prime } \left( {k_{sh} \left( {\lambda_{p} } \right)r} \right) - a_{n}^{sh\beta } \left( {\lambda_{p} } \right)\xi_{n}^{\prime } \left( {k_{sh} \left( {\lambda_{p} } \right)r} \right)} \right]\tau_{n} \left( {\cos \theta } \right) \hfill \\ - i\left[ {b_{n}^{sh\alpha } \left( {\lambda_{p} } \right)\psi_{n} \left( {k_{sh} \left( {\lambda_{p} } \right)r} \right) - b_{n}^{sh\beta } \left( {\lambda_{p} } \right)\xi_{n} \left( {k_{sh} \left( {\lambda_{p} } \right)r} \right)} \right]\pi_{n} \left( {\cos \theta } \right) \hfill \\ \end{gathered} \right\}$$
(11)
$$E_{\phi ,sh} = \frac{{E_{p} \left( {\lambda_{p} } \right)}}{{k_{sh} \left( {\lambda_{p} } \right)r}}\sin \phi \mathop \sum \limits_{n = 1}^{\infty } C_{n} g_{n} \left\{ \begin{gathered} \left[ {a_{n}^{sh\alpha } \left( {\lambda_{p} } \right)\psi_{n} \left( {k_{sh} \left( {\lambda_{p} } \right)r} \right) - a_{n}^{sh\beta } \left( {\lambda_{p} } \right)\xi_{n} \left( {k_{sh} \left( {\lambda_{p} } \right)r} \right)} \right]\pi_{n} \left( {\cos \theta } \right) \hfill \\ - i\left[ {b_{n}^{sh\alpha } \left( {\lambda_{p} } \right)\psi_{n}^{\prime } \left( {k_{sh} \left( {\lambda_{p} } \right)r} \right) - b_{n}^{sh\beta } \left( {\lambda_{p} } \right)\xi_{n}^{\prime } \left( {k_{sh} \left( {\lambda_{p} } \right)r} \right)} \right]\tau_{n} \left( {\cos \theta } \right) \hfill \\ \end{gathered} \right\}$$
(12)
$$E_{r,c} = - E_{p} \left( {\lambda_{p} } \right)\cos \phi \mathop \sum \limits_{n = 1}^{\infty } C_{n} g_{n} a_{n}^{c} \left( {\lambda_{p} } \right)\left[ {\psi_{n}^{\prime \prime } \left( {k_{c} \left( {\lambda_{p} } \right)r} \right) + \psi_{n} \left( {k_{c} \left( {\lambda_{p} } \right)r} \right)} \right]P_{n}^{1} \left( {\cos \theta } \right)$$
(13)
$$E_{\theta ,c} = - \frac{{E_{p} \left( {\lambda_{p} } \right)}}{{k_{c} \left( {\lambda_{p} } \right)r}}\cos \phi \mathop \sum \limits_{n = 1}^{\infty } C_{n} g_{n} \left[ {a_{n}^{c} \left( {\lambda_{p} } \right)\psi_{n}^{\prime } \left( {k_{c} \left( {\lambda_{p} } \right)r} \right)\tau_{n} \left( {\cos \theta } \right) - ib_{n}^{c} \left( {\lambda_{p} } \right)\psi_{n} \left( {k_{c} \left( {\lambda_{p} } \right)r} \right)\pi_{n} \left( {\cos \theta } \right)} \right]$$
(14)
$$E_{\phi ,c} = \frac{{E_{p} \left( {\lambda_{p} } \right)}}{{k_{c} \left( {\lambda_{p} } \right)r}}\sin \phi \mathop \sum \limits_{n = 1}^{\infty } C_{n} g_{n} \left[ {a_{n}^{c} \left( {\lambda_{p} } \right)\psi_{n}^{\prime } \left( {k_{c} \left( {\lambda_{p} } \right)r} \right)\pi_{n} \left( {\cos \theta } \right) - ib_{n}^{c} \left( {\lambda_{p} } \right)\psi_{n} \left( {k_{c} \left( {\lambda_{p} } \right)r} \right)\tau_{n} \left( {\cos \theta } \right)} \right]$$
(15)
$$C_{n} {\text{ = i}}^{{\left( {n + 1} \right)}} \left( { - 1} \right)^{n} \frac{2n + 1}{{n\left( {n + 1} \right)}}$$
(16)
$$a_{n}^{m} \left( {\lambda_{p} } \right) = X_{3} /X_{1} ;a_{n}^{c} \left( {\lambda_{p} } \right) = X_{4} /X_{1} ;a_{n}^{sh\alpha } \left( {\lambda_{p} } \right) = X_{5} /X_{1} ;a_{n}^{sh\beta } \left( {\lambda_{p} } \right) = X_{6} /X_{1}$$
(17)
$$b_{n}^{m} \left( {\lambda_{p} } \right) = X_{7} /X_{2} ;b_{n}^{c} \left( {\lambda_{p} } \right) = X_{4} /X_{2} ;b_{n}^{sh\alpha } \left( {\lambda_{p} } \right) = X_{8} /X_{2} ;b_{n}^{sh\beta } \left( {\lambda_{p} } \right) = X_{9} /X_{2}$$
(18)
$$\begin{aligned} X_{1} & = n_{sh}^{2} \xi^{\prime}\left( {k_{m} \left( {\lambda_{p} } \right)R_{sh} } \right)\psi^{\prime } \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)M_{1} + n_{m}^{{}} n_{c}^{{}} \xi \left( {k_{m} \left( {\lambda_{p} } \right)R_{sh} } \right)\psi \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)M_{2} \\ & \quad + n_{sh}^{{}} n_{c}^{{}} \xi^{\prime}\left( {k_{m} \left( {\lambda_{p} } \right)R_{sh} } \right)\psi \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)M_{3} + n_{m}^{{}} n_{sh}^{{}} \xi \left( {k_{m} \left( {\lambda_{p} } \right)R_{sh} } \right)\psi^{\prime } \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)M_{4} \\ \end{aligned}$$
(19)
$$\begin{aligned} X_{2} & = n_{m} n_{c} \xi^{^{\prime}} \left( {k_{m} (\lambda_{p} )R_{sh} } \right)\psi^{^{\prime}} \left( {k_{c} (\lambda_{p} )R_{c} } \right)M_{1} + n_{sh}^{2} \xi \left( {k_{m} (\lambda_{p} )R_{sh} } \right)\psi \left( {k_{c} (\lambda_{p} )R_{c} } \right)M_{2} \\ & \quad + n_{m} n_{sh} \xi^{^{\prime}} \left( {k_{m} (\lambda_{p} )R_{sh} } \right)\psi \left( {k_{c} (\lambda_{p} )R_{c} } \right)M_{3} + n_{sh} n_{c} \xi (k_{m} (\lambda_{p} )R_{sh} )\psi^{^{\prime}} \left( {k_{c} (\lambda_{p} )R_{c} } \right)M_{4} \\ \end{aligned}$$
(20)
$$\begin{aligned} X_{3} & = n_{sh}^{2} \psi^{\prime } \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)\psi^{\prime}\left( {k_{m} \left( {\lambda_{p} } \right)R_{sh} } \right)M_{1} + n_{c} n_{m} \psi \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)\psi \left( {k_{m} \left( {\lambda_{p} } \right)R_{sh} } \right)M_{2} \\ & \quad + n_{c} n_{sh} \psi \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)\psi^{\prime } \left( {k_{m} \left( {\lambda_{p} } \right)R_{sh} } \right)M_{3} + n_{m} n_{sh} \psi^{\prime } \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)\psi \left( {k_{m} \left( {\lambda_{p} } \right)R_{sh} } \right)M_{4} \\ \end{aligned}$$
(21)
$$X_{4} = n_{c}^{{}} n_{sh}^{{}} (M_{5} - M_{6} )\left[ {\xi \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)\psi^{\prime } \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right) - \xi^{\prime}\left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)\psi \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)} \right]$$
(22)
$$X_{5} = - n_{sh} \left\{ \begin{gathered} n_{sh}^{{}} \psi^{\prime } \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)\left[ {\xi \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)M_{5} - \xi \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)M_{6} } \right] \hfill \\ + n_{c}^{{}} \psi \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)\left[ {\xi^{\prime } \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)M_{6} - \xi^{\prime}\left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)M_{5} } \right] \hfill \\ \end{gathered} \right\}$$
(23)
$$X_{6} = - n_{sh} \left\{ \begin{gathered} n_{sh}^{{}} \psi^{\prime } \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)\left[ {\psi \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)M_{5} - \psi \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)M_{6} } \right] \hfill \\ + n_{c}^{{}} \psi \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)\left[ {\psi^{\prime } \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)M_{6} - \psi^{\prime}\left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)M_{5} } \right] \hfill \\ \end{gathered} \right\}$$
(24)
$$\begin{aligned} X_{7} & = n_{c}^{{}} n_{m} \psi^{\prime } \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)\psi^{\prime}\left( {k_{m} \left( {\lambda_{p} } \right)R_{sh} } \right)M_{1} + n_{sh} n_{m} \psi \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)\psi \left( {k_{m} \left( {\lambda_{p} } \right)R_{sh} } \right)M_{2} \\ & \quad + n_{sh}^{2} \psi \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)\psi^{\prime } \left( {k_{m} \left( {\lambda_{p} } \right)R_{sh} } \right)M_{3} + n_{c} n_{sh} \psi^{\prime } \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)\psi \left( {k_{m} \left( {\lambda_{p} } \right)R_{sh} } \right)M_{4} \\ \end{aligned}$$
(25)
$$X_{8} = - n_{sh} \left\{ \begin{gathered} n_{c}^{{}} \psi^{\prime } \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)\left[ {\xi \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)M_{5} - \xi \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)M_{6} } \right] \hfill \\ + n_{sh}^{{}} \psi \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)\left[ {\xi^{\prime } \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)M_{6} - \xi^{\prime}\left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)M_{5} } \right] \hfill \\ \end{gathered} \right\}$$
(26)
$$X_{9} = - n_{sh} \left\{ \begin{gathered} n_{c}^{{}} \psi^{\prime } \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)\left[ {\psi \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)M_{5} - \psi \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)M_{6} } \right] \hfill \\ + n_{sh}^{{}} \psi \left( {k_{c} \left( {\lambda_{p} } \right)R_{c} } \right)\left[ {\psi^{\prime } \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)M_{6} - \psi^{\prime}\left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)M_{5} } \right] \hfill \\ \end{gathered} \right\}$$
(27)
$$\begin{gathered} M_{1} = \xi \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)\psi \left( {k_{sh} \left( {\lambda_{p} } \right)R_{sh} } \right) - \xi \left( {k_{sh} \left( {\lambda_{p} } \right)R_{sh} } \right)\psi \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right); \hfill \\ M_{2} = \xi^{\prime}\left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)\psi^{\prime } \left( {k_{sh} \left( {\lambda_{p} } \right)R_{sh} } \right) - \xi^{\prime}\left( {k_{sh} \left( {\lambda_{p} } \right)R_{sh} } \right)\psi^{\prime } \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right) \hfill \\ \end{gathered}$$
(28)
$$\begin{gathered} M_{3} = \xi \left( {k_{sh} \left( {\lambda_{p} } \right)R_{sh} } \right)\psi^{\prime } \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right) - \xi^{\prime}\left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)\psi \left( {k_{sh} \left( {\lambda_{p} } \right)R_{sh} } \right); \hfill \\ M_{4} = \xi^{\prime}\left( {k_{sh} \left( {\lambda_{p} } \right)R_{sh} } \right)\psi \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right) - \xi \left( {k_{sh} \left( {\lambda_{p} } \right)R_{c} } \right)\psi^{\prime } \left( {k_{sh} \left( {\lambda_{p} } \right)R_{sh} } \right) \hfill \\ \end{gathered}$$
(29)
$$M_{5} = \xi \left( {k_{m} \left( {\lambda_{p} } \right)R_{sh} } \right)\psi^{\prime } \left( {k_{m} \left( {\lambda_{p} } \right)R_{sh} } \right);M_{6} = \xi^{\prime}\left( {k_{m} \left( {\lambda_{p} } \right)R_{sh} } \right)\psi \left( {k_{m} \left( {\lambda_{p} } \right)R_{sh} } \right)$$
(30)

where Rc and Rsh are the radii of the core and shell, respectively. km, ksh, and kc are the propagation constants of light in the surrounding medium, shell, and core, respectively. ξn and ψn are the Riccati-Hankel and Bessel functions, respectively65. τn and πn are the angle-dependent associated Legendre polynomials, respectively. gn is the correcting Bromwich coefficient, which is generally expressed as21

$$g_{n} = \frac{2n + 1}{{\pi n\left( {n + 1} \right)}}\frac{1}{{\left( { - 1} \right)^{n} i^{n} }}\int\limits_{0}^{\pi } {\int\limits_{0}^{\infty } {ik_{m} r\sin^{2} \theta \times f \times \exp \left( { - ik_{m} r\cos \theta } \right)\psi {}_{n}^{1} \left( {k_{m} r} \right)P_{n}^{1} \left( {\cos \theta } \right)d\theta d\left( {k_{m} r} \right)} }$$
(31)

All parameters and functions in this expression are explained in Ref.21. Further, Gouesbet et al. simplified Eq. (31) as follows66

$$g_{n} = e^{{ - {{\rho_{n}^{2} } \mathord{\left/ {\vphantom {{\rho_{n}^{2} } {\omega_{0}^{2} }}} \right. \kern-\nulldelimiterspace} {\omega_{0}^{2} }}}}$$
(32)

here ρn = (n + 0.5)(λ/2π). It is to be noted that the Eq. (32) will be valid only when the center of the microsphere is located at the center of the beam waist (as in the present study).

The expression for Ep(λp) is given below.

$$E_{p} (\lambda_{p} ) = \frac{{E_{0} (\lambda_{p} )}}{{\mathop \Sigma \limits_{p = 1}^{k} E_{0} (\lambda_{p} )}};\quad \mathop \Sigma \limits_{p = 1}^{k} E_{p} (\lambda_{p} ) = 1$$
(33)

here E0(λp) is the electric field amplitude of the radiation of wavelength λp.

It is important to note that all these equations are dependent on the refractive index (n) of the considered material. It is well known that the n value depends upon λp. In the case of polychromatic illumination, one has to consider the dispersion of the refractive index of the material. In the present study, polystyrene (PS) and fused silica have been considered as particle materials. The dispersion relations for PS67 and fused silica68 are given below in the same order.

$$n^{2} - 1 = \frac{{1.4435\lambda_{p}^{2} }}{{\lambda_{p}^{2} - 0.020216}}$$
(34)
$$n^{2} - 1 = \frac{{0.6961663\lambda_{p}^{2} }}{{\lambda_{p}^{2} - 0.0684043^{2} }} + \frac{{0.4079426\lambda_{p}^{2} }}{{\lambda_{p}^{2} - 0.1162414^{2} }} + \frac{{0.8974794\lambda_{p}^{2} }}{{\lambda_{p}^{2} - 9.896161^{2} }}$$
(35)

In addition, the dispersion curves for PS and fused silica are plotted using Eqs. (34 and 35) and shown in Fig. 2.

Figure 2
figure 2

Panels (a,b) represent the dispersion of the refractive index of PS and fused silica, respectively.

Full size image

Results and discussion

Emission spectra of different polychromatic light sources

As mentioned in the introduction, the main aim of the present work is to understand the characteristic parameters of PNJs of single solid/core–shell microspheres under focused polychromatic illumination. So, to perform the theoretical investigation on PNJs of single dielectric microspheres using equations shown in the above section, the number of wavelengths present in the incident polychromatic light and electric field amplitude at each wavelength are necessary. These details can be obtained easily by recording the spectrum of polychromatic light. Therefore, before starting the theoretical investigation, the emission spectra of different polychromatic light sources such as Halogen lamp, white LED, SC source, and Hg arc lamp available in our laboratory are recorded using the charge-coupled device (CCD) spectrometer, and obtained spectra are used in the present study.

From Panel (a) of Fig. 3, it is clear that the spectrum of the Halogen lamp is continuous and spanned from 380 to 1000 nm. The peak maximum is observed at 640 nm. In the case of white LED, two peaks are observed. The central wavelengths of sharp and broad peaks are observed at 450 nm and 565 nm, respectively. In the case of SC source operated at lower power, the spectrum is started from 500 to 1100 nm. The supercontinuum radiation is generated from a photonic crystal fiber and a sharp peak in the spectrum at 1064 nm corresponds to the pump laser (i.e., picosecond diode laser). The background in the spectrum of the Hg arc lamp is spanned from 400 to 1100 nm. As expected, several sharp peaks are observed in the range from 380 to 600 nm.

Figure 3
figure 3

Panels (a–d) represent the spectra of Halogen lamp, white LED, SC source, and Hg arc lamp, respectively. Inset in panel (d) represents a portion of the vertically zoomed spectrum.

Full size image

PNJs generated by single PS microspheres kept in water medium

Using the spectra of different polychromatic light sources (Fig. 3) and theoretical equations mentioned in Sect. 2, the EFIE distribution inside and outside single PS microspheres is plotted and shown in Fig. 4.

Figure 4
figure 4

EFIE distribution inside and outside a PS microsphere illuminated by polychromatic light from different sources. Here Rs = 3 µm, ω0 = 3 µm, and nm = 1.333. The wavelength-dependent refractive indices of PS are estimated from Eq. (34) and used for these simulations.

Full size image

In all panels, a white circle represents the microsphere boundary. The blue and red colors represent the lower and higher EFIE values. From this figure, it can be observed that the maximum EFIE (ηmax) values obtained for all the mentioned light sources are not the same due to the difference in the wavelengths and electric field amplitude of the incident radiation. The PNJ generated through the microsphere due to the SC source illumination is the least convergent among all shown in the figure because the shorter wavelengths are not present in the spectrum. In addition, the radiation from the SC source contains several larger wavelengths (beyond 1000 nm). Since the spectrum of the LED contains one intense peak at a lower wavelength region, the ηmax value observed for this source is slightly larger as compared to those obtained for other light sources.

Another characteristic parameter that plays a major role in several applications is the width of the PNJ. Therefore, the line scans along the transversal axis are taken across the PNJs generated using different light sources and shown in Fig. 5. In this figure, X = 0 indicates the center of the PNJs where the EFIE is maximum. The effective width of the PNJs (Weff) is nothing but the width of the peaks. Here the Weff is found smaller for the case of LED and it is larger for the SC source. It is worth mentioning here that the Weff values are reasonably larger due to the smaller relative refractive index of the PS microspheres in the water medium. Therefore, these PNJs are not useful for some applications such as PNJ assisted super-resolution white light nanoscopy and photoacoustic spectroscopy. However, the Weff can be lowered significantly in water medium if we use microspheres that have higher refractive indices.

Figure 5
figure 5

Variation in the EFIE along the transversal axes of the PNJs generated by a PS microsphere illuminated by polychromatic light from different sources. Here Rs = 3 µm, ω0 = 3 µm, and nm = 1.333.

Full size image

As mentioned in the introduction, several researchers have numerically/theoretically studied the PNJs generated by single dielectric microparticles under monochromatic illumination. It has been reported that the characteristic parameters of the PNJs strongly depend upon the wavelength of the incident light, size, shape, and relative refractive index of the microparticles. In addition, there is always a little deviation between the results obtained using different commercial software and theoretical approaches. Therefore, to have a proper comparison between the PNJs obtained by monochromatic and polychromatic illumination, we have performed the theoretical investigation on PNJ of single microspheres (Rs = 3 µm) illuminated by monochromatic light of different wavelengths and found that the ηmax values obtained with monochromatic light of λ = 655 nm, Halogen lamp, and Hg arc lamp are quite close. To find out whether this is the same even for microspheres of different sizes or not, we have also estimated the ηmax values for the microspheres of different sizes, illuminated by monochromatic light of λ = 655 nm, Halogen lamp, and Hg arc lamp and the obtained values are shown in panels (a) and (b) of Fig. 6, separately. In both the panels, overall, the ηmax value is found to increase with the Rs value. This could be due to the predominant forward Mie scattering in the case of larger microspheres. It is to be noted that fluctuations in the curve for λ = 655 nm indicate the partial excitation of optical resonances or whispering gallery modes of the microspheres of specific sizes. From both the panels, we can conclude that the ηmax values obtained for the Halogen light, Hg arc lamp, and monochromatic light of λ = 655 nm are nearby for all Rs values.

Figure 6
figure 6

Panels (a,b) show the comparison between the ηmax values obtained for monochromatic light of λ = 655 nm and polychromatic light from two different sources. Panels (c,d) show the comparison between the variation in the EFIE along the transversal axes of the PNJs generated by a PS microsphere illuminated by monochromatic light λ = 655 nm and polychromatic light from different sources. Panels (e,f) show the same along the longitudinal axes. In the case of the last four panels, Rs = 3 µm, and ω0 = 3 µm.

Full size image

For comparing the Weff values, the line scans along the transversal axes of the PNJs of the PS microsphere (Rs = 3 µm) illuminated by monochromatic light of λ = 655 nm, polychromatic light from Halogen lamp and Hg arc lamp are plotted and shown in panels (c) and (d) of Fig. 6. From these Panels, we can easily conclude that the Weff values are also nearly the same.

For comparing the lengths of the PNJs, the line scans along the optical axes of the PNJs of the PS microsphere (Rs = 3 µm) illuminated by monochromatic light of λ = 655 nm, polychromatic light from Halogen lamp and Hg arc lamp are plotted and shown in panels (e) and (f) of Fig. 6. It is to be noted that the FWHM of the peaks shown in these two panels represents the effective length of the PNJs (Leff). From these figures, it is clear that the Leff is slightly more in the case of both polychromatic sources.

PNJs generated by single PS microspheres kept in air medium

Figure 7 shows the EFIE distribution inside and outside single PS microspheres (in air medium) illuminated by focused polychromatic light from different sources. As in the water medium, the ηmax values obtained from the spectra of all polychromatic sources are slightly different. The ηmax observed in the air medium is significantly larger as compared to those obtained in the water medium. This is due to the strong forward Mie scattering in the air medium due to the larger relative refractive index of the microsphere. Here also, (1) the ηmax is larger in the case of white LED and it is smaller for the SC source, and (2) the width and length of the PNJs are significantly smaller in the air as compared to the water medium. Therefore, these PNJs could be recommended for PNJ assisted white light nanoscopy, photoacoustic spectroscopy, etc.

Figure 7
figure 7

EFIE distribution inside and outside a PS microsphere illuminated by polychromatic light from different sources. Here Rs = 3 µm, ω0 = 3 µm, and nm = 1.0.

Full size image

Figure 8 represents the line scans along the transversal axes of the PNJs shown in Fig. 7. In this figure also, X = 0 indicates the center of the PNJs where the EFIE is maximum. The width of the PNJs is nothing but the width of the peaks. It is apparent that only one peak is observed at X = 0 for all light sources but the side lobes are completely disappeared irrespective of the polychromatic light source. However, in the case of monochromatic illumination, along with the main peak at X = 0, several side lobes are observed in the air medium. The number and amplitude of the side lobes are significantly decreased when the λ value is increased from 400 to 1000 nm (results not shown). Here also, relatively, the width of the PNJ obtained for the SC source is slightly larger.

Figure 8
figure 8

Variation in the EFIE along the transversal axes of the PNJ generated by a PS microsphere illuminated by polychromatic light from different sources. Here Rs = 3 µm, ω0 = 3 µm, and nm = 1.0.

Full size image

In the air medium also, at first, the EFIE distribution inside and outside a PS microsphere (Rs = 3 µm) is plotted using monochromatic light of different λ values. From Panels (a) and (b) of Fig. 9, it is apparent that the ηmax values of microspheres having different Rs obtained for monochromatic light of λ = 655 nm, polychromatic light from Halogen and Hg arc lamps are nearly the same. In addition, from Panels (c–f) of Fig. 9, it can be observed that the Weff and Leff values obtained for monochromatic light of λ = 655 nm, polychromatic light from Halogen and Hg arc lamps are close. Therefore, from Figs. (6 and 9), we can safely conclude that one can use the characteristic parameters of the PNJ obtained with monochromatic light of λ = 655 nm to interpret the experimental results obtained with the PNJs generated in air and water media, using the polychromatic light from Hg arc and Halogen lamps, which are the most commonly used in different types of microscopes. A similar observation is found for the silica microspheres as well (see “Supplementary information”). This observation is very useful for the researchers because generating PNJs numerically using monochromatic light is simple but the generation of PNJs numerically using polychromatic light is a difficult task and it demands a sophisticated computing facility.

Figure 9
figure 9

Panels (a,b) show the comparison between the ηmax values obtained for monochromatic light of λ = 655 nm and polychromatic light from two different sources. Panels (c,d) show the comparison between the variation in the EFIE along the transversal axes of the PNJs generated by a PS microsphere illuminated by monochromatic light of λ = 655 nm and polychromatic light from different sources. Panels (e,f) show the same along the longitudinal axes. In the case of the last four panels, Rs = 3 µm, and ω0 = 3 µm.

Full size image

A comparison between the EFIE distribution of the PNJs obtained from the PS microsphere under the illumination of polychromatic light from a white LED and SC source with the monochromatic light of different wavelengths is also made for finding the suitable wavelength of monochromatic radiation.

It can be observed from Panels (a) and (c) of Fig. 10 that the Weff and Leff values obtained for monochromatic light of λ = 630 nm and polychromatic light from LED are very close. Similar behavior can be observed from Panels (b) and (d) of Fig. 10 for monochromatic light of λ = 800 nm and polychromatic light from SC source. Thus, the characteristic parameters of the PNJs obtained from the illumination of the monochromatic light of the mentioned wavelengths can be used to interpret the experimental results for the PNJs under the illumination by respective polychromatic light sources as mentioned earlier in this section.

Figure 10
figure 10

Panels (a,b) show the comparison between the variation in the EFIE along the transversal axes of the PNJs generated by a PS microsphere illuminated by monochromatic and polychromatic light from different sources. Panels (c,d) show the same along the longitudinal axes. For all the four panels, Rs = 3 µm and ω0 = 3 µm.

Full size image

Role of ω 0 on the PNJs of single microspheres under polychromatic illumination

Experimentally, the PNJs are generated by focusing the polychromatic light on single microspheres with the help of a microscopic objective lens with different numerical aperture (NA) values. It is well known that the NA value directly affects the ω0. Though, the effect of ω0 on the EFIE of the PNJ due to monochromatic illumination is reported in the literature69, the same due to the polychromatic illumination is again not present in the literature. Therefore, to understand the effect of ω0 on the PNJs of single microspheres under polychromatic illumination theoretically, the EFIE distribution inside and outside single microspheres is plotted by varying the ω0 value. Figure 11 shows the EFIE distribution of a single PS microsphere of Rs = 3 µm obtained for different ω0 values. Here the emission spectrum of Halogen light shown in panel (a) of Fig. 3 is used for generating all panels in Fig. 11.

Figure 11
figure 11

The role of ω0 on the EFIE inside and outside a PS microsphere illuminated by focused polychromatic light. Here Rs = 3 µm and nm = 1.0. The spectrum of the Halogen lamp is used for generating all figures.

Full size image

From Fig. 11, it can be observed that for a very small ω0 (= 0.5 µm) compared to the size of the microsphere, there is no formation of PNJ. The incident light is focused at the centre of the microsphere. When ω0 = 1 µm then the PNJ is partially visible outside the microsphere. But a reasonable amount of light is still confined inside the microsphere. Upon increasing the ω0 value from 1 to 2 µm, the PNJ is clearly visible outside the microsphere. However, the value of ηmax is low due to the weak confinement of the PNJ. When ω0 = Rs, the value of ηmax is increased further which indicates the improvement in the confinement of PNJ. These results indicate that one has to use the focused polychromatic beam with ω0 = Rs to generate the tightly confined PNJs from single dielectric microspheres.

PNJs generated by silica core-PS shell microspheres kept in air medium

The theoretical investigation is extended for studying the PNJs of core–shell microspheres illuminated by polychromatic light. Panels (a) and (b) of Fig. 12 show the EFIE distribution of single silica core-PS shell microspheres obtained using the spectra of Halogen lamp and Hg arc lamp, respectively. From these two panels, it is clear that only one PNJ is observed in all the cases due to the thin shell. In general, several short and elongated PNJs are expected in the case of a thicker shell on a large sphere59 or multi-shells on a solid sphere70 due to multiple focusing and defocusing of the incident light. The ηmax value found in the present case is lower than that obtained in the case of a solid PS microsphere of Rs = 3 µm. In the above sections, it is mentioned that the characteristic parameters of the PNJs of single solid microspheres obtained using the spectra of Halogen and Hg arc lamps are close to those obtained with the monochromatic light of λ = 655 nm. To find out the suitable wavelength of monochromatic radiation (λmono) for the present case also, the EFIE distribution is plotted inside and outside the core–shell microspheres illuminated by monochromatic radiation of different wavelengths. From Figs. 12 and 13, it is clear that the ηmax, Weff, and Leff of the PNJs generated by core–shell microsphere under illumination with polychromatic radiation and monochromatic radiation of λ = 647 nm are very close. From these results, we can also conclude that one can use the characteristic parameters of the PNJ of core–shell microspheres (having thin shells) obtained with monochromatic light of λ = 647 nm to interpret the experimental results obtained with the PNJs of core–shell microspheres generated by polychromatic light from Halogen and Hg arc lamps.

Figure 12
figure 12

(a–c) represent the EFIE distribution inside and outside a silica core-PS shell microsphere illuminated by a Halogen lamp, Hg arc lamp, and monochromatic light of λ = 647 nm, respectively. For all the panels, Rc = 2.8 µm, Rsh = 3.0 µm, ω0 = 3 µm, and nm = 1.0. The wavelength-dependent refractive indices of the PS and silica are estimated using Eqs. (34 and 35), respectively, and the obtained values are used for these simulations.

Full size image
Figure 13
figure 13

The variation in the EFIE along the transversal and longitudinal axes of the PNJs are shown in panels (a,b), respectively. For both the panels, Rc = 2.8 µm, Rsh = 3.0 µm, ω0 = 3 µm, and nm = 1.0.

Full size image

The effect of shell thickness (t) on the characteristic parameters of the PNJs is also investigated theoretically. The pink curve in panel (a) of Fig. 14 represents the variation of ηmax with the t value in the case of polychromatic radiation from the Halogen lamp. Here the t value is increased by fixing the value of Rsh at 3 µm and varying the Rc value from 2.8 to 0.5 µm. In this case, the value of ηmax is found to decrease upon increasing the t value till 1800 nm. Beyond this t value, ηmax is found to increase. For each t value, the suitable λmono has been found to generate the PNJ which has the characteristic parameters close to those obtained in the case of polychromatic radiation and mentioned in the same panel (blue color curve). In this case, the suitable λmono value is found to show an overall increase with t. When the t value is varied by fixing the Rc value at 3 µm and varying the Rsh value from 3.2 to 5.5 µm, the ηmax is observed to follow the reverse trend in the case of polychromatic illumination [panel (b) of Fig. 14] in contrast to the previous case [panel (a)]. However, the suitable λmono value is again found to increase overall with the t value.

Figure 14
figure 14

The pink curves in both the panels show the variation of ηmax with t value in the case of polychromatic light from the Halogen lamp. The blue curves in both panels represent the variation of suitable λmono with the t value. For panels (a,b), the t value is varied by fixing the Rsh and Rc value, respectively. For all cases, ω0 = Rsh and nm = 1.0.

Full size image

Conclusions

The analytical equations which are useful to understand the characteristic parameters of the PNJs of single microspheres under polychromatic illumination are given in this paper. For the first time, a theoretical investigation is carried out on the PNJs generated using the emission spectra of Halogen lamp, SC source, white LED, and Hg arc lamp. For better comparison, the PNJs are also generated theoretically using monochromatic light of different λ values. The ηmax, Weff, and Leff of the PNJ obtained for all the polychromatic sources are slightly different due to the variation in the emission spectra. Relatively, the ηmax is little larger, Weff and Leff values are slightly smaller in the case of white LED. These are found to be opposite in the case of SC source. The characteristic parameters of the PNJs of microspheres under polychromatic illumination are found sensitive with the Rs and nm as in the case of monochromatic illumination. More importantly, the characteristic parameters of the PNJs of solid microspheres obtained for all polychromatic light sources are found close to those observed for the monochromatic light of λ which is near to the central wavelength of the polychromatic light sources. This means one can use the characteristic parameters of the PNJ of the solid microspheres observed with monochromatic light with suitable λ to interpret the experimental results obtained with the PNJs generated by polychromatic light from the above-mentioned sources.

In the case of core–shell microspheres, the characteristic parameters of the PNJs obtained with polychromatic light are sensitivity to the t value as in the case of monochromatic illumination. In addition, the suitable λmono values are found to increase with t values. We believe that the observations reported here are very useful for the researchers because generating PNJs numerically using monochromatic light is simple but the generation of PNJs numerically using polychromatic light is a difficult task and it demands a sophisticated computing facility.