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Abstract
We demonstrate numerically the feasibility of axial and angular control of the position of a photonic nanojet (PNJ) by lossless phase-only modulation of a fixed Gaussian beam illuminating a fixed 2D circular homogeneous dielectric micro-lens. We furthermore demonstrate that our phase-only modality can be used to calibrate and improve the confinement of PNJ generation.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Photonic nanojets (PNJs) [1–3] are highly localized beams of light that can occur in the shadow near-field region of a laser-illuminated dielectric micro-lens. PNJs show promise in the context of optical microscopy with or without photosensitive labels [4–14], in nanolithography [15,16], and could potentially also be used in particle trapping and detection [17–20], and as optical tweezers [21–23]. It is in particular the possibility of the use of PNJs for super-resolution optical microscopy [4,5,8,10–12,14,25–27] that motivates our work.
There is a considerable literature describing the position and shape control of PNJs [24–62]. The methods roughly fall into one of two categories: lens shaping and illumination control. Lens shaping optimizes the geometry and the material composition of the lens, potentially employing additional structures such as supporting substrates, planar substrates with gaps [35], and combinations of two or more lenses. Illumination control entails the design of the amplitude and phase distribution, polarization and wavelength of the incident light. So far, for example, point-, line-, hollow-focus and multiple-foci PNJ were produced by structured illumination of dielectric microspheres in [36,37], and PNJ position was designed by illuminating micro-scale dielectric spheres [38–40], hemispheres and spherical caps [31,41–43], dielectric elliptical particles [44,45], cylinders/disks [46–48], cuboids [49], core-shell spheres, core-shell cylinders and core-shell cuboids [50–56], spiral axicons [57], chains of metal-dielectric spheres [58], chains of core-shell cylinders [52,59] and low-dimensional parameterized dielectric particles [33,60–62]. It was demonstrated numerically [31] that ultra-elongated PNJ with waist width down to 27% of the operating wavelength can be created robustly to fabrication errors by controlling the dimensions of an asymmetrical, spherical cap-based micro-lens. Also, in [33], 2D lens optimization was done numerically to maximize the PNJ power in a specified rectangular area behind the lens [41]. More recently, a degree of PNJ steering by controlled illumination was achieved numerically by, e.g., J. Wang and collaborators [29,30,34] for a dielectric ball micro-lens using focus, beam waist radius and axis control of an incident Gaussian beam. Moreover, in [26] we demonstrated wide-ranging numerical PNJ control in 2D using computed amplitude and phase profiles of illumination, and in [25,27] we studied numerically this scanning PNJ as a super-resolution optical microscopy tool. Our approach, however, required the use of two spatial light modulators (SLMs).
The purpose of this work is to demonstrate numerically the feasibility of PNJ steering using lossless phase-only modulation of a fixed Gaussian beam illuminating a simple fixed homogeneous micro-lens. Specifically, we fix the axis, the beam width and the focal length of our Gaussian beam illumination, and achieve spatial PNJ control by imposing a combined linear and quadratic transverse phase factor of the form $\exp j(\ell y+qy^2)$. This reduces the hardware requirements to essentially just a single SLM, as already explored in [20] in the context of optical trapping and in [63] for optical encryption. In addition, we expect our work to make practical realizations of steerable PNJs much more feasible, due to the built-in phase-corrective dynamic PNJ self-calibration that compensates for imperfections in beam/lens alignment etc.
In Section 2. we describe the scattering setup and propose a simple algorithm for PNJ steering. We present our numerical findings in Section 3, and offer our conclusions and suggestions for further work in Section 4.
2. Phase-modulated illumination for dynamic PNJ steering
Figure 1 shows the problem setup. We work with 2D TE (transverse electric) time-harmonic fields at free-space wavelength $\lambda _0=532$ nm with the time-dependence factor $e^{j\omega t}$ assumed and suppressed in the following. Specifically, the electric field vector points out of the plane and can be treated as a complex scalar. The lens is circular cylindrical, centered at the origin and of radius $R_{\ell }=2.5$ $\mu$m and refractive index $n_{\ell }=1.49$. We choose the values of $\lambda _0$, $R_{\ell }$, $n_{\ell }$ and the beam waist radius $w_0$ (see below) as Huang et al. [29], to enable comparison of results. Our incident field propagates in the $x$-direction, and will be a special case of the general form
where $\varphi (x,y)$ is a phase-only modulation of a field $E^{\rm laser}(x,y)$ originating from a laser. The phase-only modulation implies that there is no absorption, and hence no loss of light involved. To choose a particular form of the phase function $\varphi (x,y)$, we look to scalar diffraction theory [64,65]. Here, a linear and a quadratic phase factor are applied to the input field of an optical system in the transverse direction (the $y$-direction in our case), in order to systematically manipulate the output field. We therefore choose where $\ell$ and $q$ are real-valued parameters. Next, since the standard Gaussian beam is by far the easiest to produce and most widely used in laser illumination, we choose $E^{\rm laser}(x,y)$ to be a Gaussian beam. Moreover, our micro-lens will be comparable in size with the beam width of the laser illumination, and it will be centered at the beam axis. This means that the interaction of the lens with the incident beam will occur within a small angle from the beam axis (the $x$-axis), and it thus makes sense to employ the standard paraxial approximation for the Gaussian beam illumination in the following.
Fig. 1. The problem setup. The PNJ coordinates $(r_{\rm PNJ},\theta _{\rm PNJ})$ are at the maximum total field amplitude.
Finally, in this work we choose to use the Finite Element package COMSOL Multiphysics [66], which employs the following formula for the paraxial approximation (see [67] as well as the information provided by the COMSOL package):
Here $k(x,y)$ is the wavenumber satisfying
A simple empirical PNJ steering algorithm would be to interpolate within a table of pre-computed corresponding parameter pairs $(\ell,q)\leftrightarrow (r_{\rm PNJ},\theta _{\rm PNJ})$. Specifically, we can compute the total near fields resulting from the illumination of the micro-lens by our phase-modulated Gaussian beam with $(\ell,q)$ taking values over some chosen grid; then, for each pair $(\ell,q)$, we find the resulting PNJ coordinates $(r_{\rm PNJ},\theta _{\rm PNJ})$; next, we find a fit vector function $F(r_{\rm PNJ},\theta _{\rm PNJ})$ for the mapping $(r_{\rm PNJ},\theta _{\rm PNJ})\mapsto (\ell,q)$ over an appropriate dynamical range; finally, given desired PNJ coordinates $(r_{\rm PNJ},\theta _{\rm PNJ})$, we estimate the corresponding values $(\ell,q)\approx F(r_{\rm PNJ},\theta _{\rm PNJ})$ and apply the transverse phase modulation $\exp j(\ell y+qy^2)$ to the incident Gaussian beam.
3. Numerical results
For illustration, Figs. 2 and 3 show the normalized total near fields resulting from the Gaussian beam illumination for various values of the phase modulation parameters $(\ell,q)$. Our near-field solver here is COMSOL Multiphysics, which we find sufficiently fast and accurate for the present work when the maximum mesh element size is 100 nm. Refining the mesh indicates numerical convergence of the results. A generalized Lorenz-Mie theory (GLMT) solver [68] is an obvious alternative and will be considered in a future development.
Fig. 2. PNJ steering by phase-only modulation of a fixed incident Gaussian beam, with $[\ell ]=\mu$m$^{-1}$ and $[q]=\mu$m$^{-2}$. The total electric fields shown are normalized to maximum amplitude of 1 V/m. The bottom right image ($\ell =0$ $\mu$m$^{-1}$, $q=0$ $\mu$m$^{-2}$) results from illumination by the unmodulated Gaussian beam, and is not confined to a single high-intensity area.
Fig. 3. The PNJs of Fig. 2 with internal lens fields included. All fields are normalized to maximum of 1 V/m. The electric field intensity inside the lens can exceed that of the PNJ and make the latter less visible in the normalized images.
Consistent with our observations in general, increasing the numerical value of $\ell$ increases the angular deflection of the PNJ from the incident beam axis, while increasing the numerical value of $q$ decreases the axial distance of the PNJ from the lens. Moreover, we see that the phase-only modulated illumination results in significantly increased field confinement of the generated PNJs (compare the field images in Fig. 2 with the bottom right image there). Note that all the field images in Fig. 2 are normalized such that the maximum field amplitude is 1 V/m. It is evident from the figure that the near field is confined to a single small feature (PNJ) when modulated illumination is used, whereas its intensity is spread out over several (at least three) relatively large regions when unmodulated illumination is used.
We would ideally like to obtain a two-dimensional interpolating surface allowing the estimation of the parameter values $\ell$ and $q$ given the desired coordinates $(r,\theta )$ of the PNJ. However, our numerical experimentation has shown some instability of the proposed interpolation method if both $\ell$ and $q$ are allowed to vary over a wide range of values, partly due to an unsystematic occurrence of high-intensity near-field side lobes for certain value pairs $(\ell,q)$. While this could potentially be addressed by (tedious) manual inspection of each computed near field, we shall in the remainder of this section, as a first step of the analysis, restrict ourselves to the automated interpolation approach of Section 2. for the control of the PNJ radial coordinate $r_{\rm PNJ}$ with $\theta _{\rm PNJ}=0$. Following the procedure outlined in Section 2, we compute the PNJ positions along the $x$-axis produced by our phase-modulated Gaussian beam illumination with $\ell =0$ $\mu$m$^{-1}$ and for $q=1.7,1.8,\dots,4$ $\mu$m$^{-2}$. Our choice of the range of $q$ is based on numerical experimentation and it is made to exclude significant outliers and instances of identical $r_{\rm PNJ}$ coordinates occurring for multiple values of $q$. Figure 4 shows the resulting PNJ positions and peak electric field amplitudes. Next, we find the least-squares quadratic fit for the function $q(r_{\rm PNJ})$,
This fit, shown in Fig. 5, has root-mean-square error (RMSE) of 0.2439 and covers the range of $r_{\rm PNJ}$ from 2.50 $\mu$m to 3.64 $\mu$m. Finally, we choose a set of 10 desired $r_{\rm PNJ}$ coordinates distributed evenly over the range from 2.5 $\mu$m to 4 $\mu$m, and use interpolation to find the corresponding values of the parameter $q$. Figure 6 shows the resulting desired vs. achieved PNJ radial coordinates $r_{\rm PNJ}$, as well as the relative error
Fig. 4. Left: PNJ positions with phase modulation factors $(\ell,q)$ for $\ell =0$ $\mu$m$^{-1}$, $q=1.7,1.8,\dots,4$ $\mu$m$^{-2}$. Right: The PNJ peak electric field amplitudes.
Fig. 6. Radial (axial) control of PNJ position. Left: desired vs. achieved PNJ radial coordinates $r_{\rm PNJ}$. The straight linear piece is the ideal. Right: the relative error in the achieved $r_{\rm PNJ}$. The mean relative error here is 2.54% and the maximum relative error is 5.02%.
Fig. 7. Longitudinal profiles (left) and transverse profiles (right) of some of the achieved PNJs. Decay length is measured from the location of the field maximum to the location of the first half-maximum (black dot). The waist width is the distance between the two half-maxima (black dots).
Fig. 8. Decay lengths and waist widths of the achieved PNJs, relative to the operating wavelength $\lambda _0=532$ nm.
4. Conclusion and further work
As a proof-of-principle, we have demonstrated numerically the feasibility of axial and angular position control of photonic nanojets using lossless phase-only modulation of a fixed Gaussian beam illuminating a fixed homogeneous dielectric micro-lens. A simple numerical scheme based on interpolation of a phase modulation parameter within a pre-computed library allowed the axial control of the PNJ position with a mean relative error of less than $3\%$. Furthermore, our phase-only modulated illumination resulted in significantly increased field confinement of the generated PNJs. We anticipate that the combined ability to steer and calibrate PNJs quickly, precisely, and without opto-mechanical intervention will be substantial in a laboratory setting. An elegant feature of our proposed approach is that it requires only a single spatial light modulator (SLM) or a fixed phase mask. This carries all the benefits of a lossless, self-aligned, compact, "optical impedance matching" system configuration. We expect our approach to generalize directly to the three-dimensional case, as long as the PNJ coordinates are close enough to the axis of the incident beam for the paraxial approximation to hold. Obviously a suitable theory for the demonstrated PNJ steering needs to be developed, as well as a computational framework that is more robust to the intricate near-field fluctuations as a function of the phase modulation parameters $\ell$ and $q$. Finally, we hope that our work will spark a practical development of adaptive optics instruments to further enhance and study the engineering of PNJs.
Funding
Novo Nordisk Fonden (NNF16OC0021948); European Metrology Programme for Innovation and Research (20FUN02/f10 POLight); Villum Fonden (25893).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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