1. Introduction

Several far-field and near-field optical microscopy techniques overcome the Abbe diffraction limit [1] and achieve super-resolution imaging of particles and features. In a recent review article [1], Huszka and Gijs give a good introduction to the most common techniques, one of which is the use of photonic jets (also called photonic nanojets or PNJs [2,3]; see Fig. 1(b) for definitions). Here, laser or white light illumination of a dielectric micro-lens (cylindrical, hemispherical, spherical) produces a highly focused beam, a PNJ, in the shadow region of the near field exterior to the lens.

 

Fig. 1. Photonic nanojet (PNJ) (a) design by amplitude- and phase-modulated illumination of a dielectric micro-lens with general shape and (b) features of a PNJ, its radial distance, waist width, and decay length, as in [5]. The full width at half maximum (FWHM) amplitude contour is with respect to the amplitude $|E^{\rm tot}|$ of the total electric field.

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PNJ formation was observed experimentally in 2006 [4], and it was since shown to depend on the micro-lens radius, shape, and refractive index relative to the surrounding medium, as well as on the illumination wavelength. [2,5] Several works exploit this to create PNJs of various shapes and sizes using lens shaping and plane wave or gaussian illumination [6,7]. The high field intensity and the small width of the PNJ focus, relative to the illuminating laser wavelength, allows highly local measurements and applications such as detection, manipulation and trapping of nanoparticles [8–10], nanopatterning and nanolithography [11,12] as well as optical imaging methods [13–15] and other optical metrology techniques like fluorescence [16] and Raman [17] microscopy, in addition to scatterometric measurements [18,19]. However, the highly localized nature of PNJs also makes it difficult to find the area of interest. It is therefore desirable to realize a steerable PNJ that can quickly scan the local region without any need for lens adjustment or sample movement. Advances in this direction have been obtained using optically trapped levitating nanoparticles [12], shaped optical fiber tips [20] and metalic masks [21,22]. Precise and quick steering of PNJs could potentially allow rapid selective photo-switching of closely spaced sites along a biomolecule [16,17], or unlock the full potential of the emerging label-free microsphere-assisted optical super-resolution microscopy; where a sub-classical sample is placed near or inside a PNJ, enabling the localization, characterization (e.g., sizing) [3,4,18,19] and even magnified optical image creation [13–15,23,24] of the sample.

We propose a method for fast and precise PNJ steering using a computed spatially inhomogeneous incident wave and a simple fixed micro-lens. The method enables rapid focus point movement without any adjustment or translation of the lens; it is not confined to the traditional spherical (3D) or circular (2D) micro-lenses; and it results in a consistently narrow PNJ and a large range of achievable PNJ positions. The results presented in this work have direct applications in particle manipulation and improvement in optical nanopattering and nanolithography. Furthermore, due to the robustness of the method, this technique combined with an adequate measuring strategy is of direct relevance for super-resolution microscopy.

The mathematical foundation of the method is described in Section 2, and a practical way to realize the computed incident fields is proposed in Section 3. We estimate theoretically the achievable PNJ waist width as a function of the radial distance in Section 4, and present numerical results for 2D circular and square cross section dielectric lenses in Section 5.

2. Tailoring the incident field

The procedure for tailoring the incident field that produces the desired PNJ is based on the solution of a Lippmann-Schwinger equation, such as [25, Eq. (8.13), p. 309], which we here derive explicitly for completeness (Eq. (8) below). Figure 1(a) shows the setup of the problem.

We consider the time-harmonic, two-dimensional transverse electric (2D TE) case with the electric field pointing out of the plane, and with the time-dependence factor $\exp (j\omega t)$ implied and suppressed; here $j$ is the imaginary unit. The operating free-space wavelength and wavenumber are $\lambda _0$ and $k_0=2\pi /\lambda _0$, respectively. A cylindrical dielectric lens with a constant refractive index $n_L>1$ and a cross-section profile $L$ is immersed in vacuum and illuminated with an incident field $E^{\rm inc}(x,y)=A(x,y)\exp (jP(x,y))$. Here a Cartesian coordinate system $(x,y,z)$ is introduced with the $z$-axis aligned with the lens axis, as in Fig. 1. Note that the problem geometry, the material parameters, and all the fields are invariant along the $z$-axis. The real-valued functions $A(x,y)$ and $P(x,y)$ give the amplitude and the phase profile of the incident field, respectively. Define the ’lens contrast’ $\alpha =k_0^{2}(n_L^{2}-1)$, the characteristic function

Here $\Phi _0(\mathbf {x})=(j/4)H_0^{(2)}(k_0|\mathbf {x}|)$ is the outgoing fundamental solution of the Helmholtz operator in the plane, and $H_0^{(2)}$ is the Hankel function of order zero and of the second kind. This suggests the following procedure for producing a PNJ at a desired location in the near field:

If the realized lens illumination produces an unsatisfactory approximation to the desired PNJ then one may repeat step 4 above using a better approximation to ${E^\textrm {inc}}$ at $\Gamma$, or one may start over and modify the curve $C$ and function $\xi$. For the numerical results presented in this paper, we always choose the curve $C$ to be a short straight linear piece located at the desired PNJ position and pointing radially away from the origin. Our numerical experimentation has shown that more complicated choices of the geometry of $C$, such as multi-component curves, closed curves, or non-radial curves, can result in involved interference patterns in the near field. While this should certainly be studied systematically and exploited for advanced near-field design, we feel it falls outside of the scope of this paper.

The curve $\Gamma$ at which the incident field profile is tailored need not be a linear piece. However, for simplicity, in this paper we always choose $\Gamma$ as illustrated in Fig. 1(a), that is, a linear piece parallel with the $x$-axis and placed just above the micro-lens. We did not investigate the practical capabilities and performance of our PNJ design method for more complicated choices of $\Gamma$.

3. Realizing the incident field profiles using spatial light modulators

The spatial profile of an optical beam as computed in Section 2 could be achieved in practice using diffractive optical elements such as phase filters or computer generated holograms (CGHs) implemented with digital micromirror devices (DMDs) or liquid crystal spatial light modulators (SLMs). These mechanisms allow to generate complex fields by effectively varying the diffraction efficiency with phase modulations [26,27] or by transferring phase modulations onto amplitude modulations using polarizing optics [28]. We here propose a mechanism, illustrated in Fig. 2, to generate the complex field required to probe the micro-lens, which leads to the generation of the nanojet based on the latter technique. We consider a linearly polarized light beam of spatial profile $u_0({ \bf x})$ propagating along the $z$ direction.

 

Fig. 2. Probe beam preparation through amplitude and phase modulation of a beam $u_0(\textbf {x})$. Using two spatial light modulators (SLMs) and polarization optics we generate an arbitrary complex field $u(\textbf {x}) = \Psi (\textbf {x}) \exp {i\Phi (\textbf {x})}$ by imprinting the phases $\varphi _1(\textbf {x})$ and $\varphi _2(\textbf {x})$, respectively with the SLMs. The dimensions of the beam are adjusted with a demagnifying telescopic system.

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Without losing generality, let us consider a Gaussian beam $u_0(\textbf {x}) = A_0 \exp (-|\textbf {x}|^{2}/w_0^{2})$ with horizontal polarization along the horizontal direction. With the use of a half-wave plate (HWP) the polarization of the beam is rotated 45${\circ }$ so that it has equal components of magnitude $A_0/\sqrt {2}$ in the horizontal ($H$) and vertical ($V$) direction. The beam is transmitted (or reflected) by a first spatial light modulator (SLM$_1$) imprinting a phase $\varphi _1(\textbf {x})$ to the horizontal component of the beam in the following way:

We next address the achievable resolution of the SLM-produced incident field profiles. In general, SLMs can generate pixelated 8-bit and 10-bit phase-maps (such as $\varphi _1(\textbf {x})$ and $\varphi _2(\textbf {x})$ above) with an arbitrary phase distribution of values discretized in steps of 2$\pi$ rad$/255\approx 24.6$ mrad ($\approx$1.4$^{\circ }$) and $2\pi$ rad$/1023\approx 6.1$ mrad ($\approx 0.35^{\circ }$) respectively. The actual performance depends on the model of the SLM and on the operating wavelength.

For the "SLM-generated" part of the numerical results in Section 5, we select the phase map resolution of 24.6 mrad, and Gaussian illumination sufficiently large to cover the area of interest of the SLM, but just enough not to generate diffraction effects due to the borders of the SLM. Since we choose to modulate the theoretical incident field profiles in Section 5 in 902 steps of 26.6 nm over a domain (the curve $\Gamma$ in Fig. 1(a)) of length 24 $\mu$m, we here start with an illumination beam of diameter (FWHM) equal to 902 pixels of the SLM display. This way, each pixel of the SLM corresponds to a pixel in the domain $\Gamma$. For an SLM with pixel size of 8 $\mu$m (such as Thorlabs EXULUS HD2), the required beam diameter is 7.2 mm and the beam waist 5.2 mm. After passing through the two SLMs, the beam will have the desired amplitude and phase profiles, however at a larger scale.

To reduce the size of the beam to 24 $\mu$m without modifying its spatial profile, we require an imager with magnification $M=300$. This can be realized with consecutive telescopic systems such that the product of their magnifications $M_i$ equals $M$, for instance two telescopes: A) & B) $M_{a,b}=10$ $\rightarrow$ $f_1$ = 1000 mm, $f_2$ = 100 mm, and a third one C) $M_{c}=3$ $\rightarrow$ $f_5$ = 300 mm, $f_6 = f_2$. With these three consecutive systems (A, B and C), the desired field is obtained at the final focal plane of the last system, which corresponds to the reference space domain $\Gamma$ above the micro-lens. The imager requires ultra-precise alignment and position placement of the demagnifying lenses and of the micro-lens, which can be found by rigorous computation. We can compensate for small diffraction effects caused by misalignment, cross-talk and aberrations by introducing some feedback to the SLM phases $\varphi _1(\textbf {x})$ and $\varphi _2(\textbf {x})$.

4. Waist-width analysis of designed photonic jets

In this section we construct a simplified theoretical model of PNJ formation, and then estimate the qualitative behavior of PNJ waist width as a function of the radial distance $\varrho \ge 0$ of the PNJ. We consider the 2D TE case (see Section 2 and Fig. 1(b)) with a circular micro-lens of cross-section radius $R_L$ (see Fig. 3.1). This analysis is based on the stable bandwidth estimates for the singular spectrum of the forward operator $F$ for the Helmholtz equation in the plane, established in [29]. The reader interested only in the performance of the PNJ design algorithm proposed in sections 2 and 3 can skip the current section entirely.

 

Fig. 3. A schematic of the 5-step theoretical procedure for estimating the full width at half maximum (FWHM) of a PNJ with a desired profile.

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We shall write $F$ for the linear operator that maps a source $s$ radiating in free space to the radiated field $F(s)$ measured at a circle of radius $R_L+\varrho$ and centered at the origin. More precisely, the forward operator $F:L^{2}(B_{R_L})\mapsto L^{2}(\partial B_{R_L+\varrho }),$

Given a desired radial distance $\varrho$, we

 

Fig. 4. Desired PNJ profiles $|E^\textrm {tot}_\textrm {PNJ}(\theta )|$ at $\partial B_{R_L+\varrho }$ ($\times 10^{5}$ V/m) for different lens radii $R_L$.

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We now fix the operating wavelength at $\lambda _0=532$ nm (common green laser). Figure 5 shows the physically viable field profiles produced using the 5-step procedure outlined above, taking $\mathcal {B}=\mathcal {B}_-$, and starting with the desired profiles from Fig. 4. We see that the achieved FWHM of the PNJ profile gets slightly closer to the desired FWHM of $0.5\lambda _0$ as we increase the lens radius $R_L$ and thereby the bandwidth $\mathcal {B}$. Figure 6 shows more desired and achieved FWHM for the four lens radii $R_L$. We again use $\mathcal {B}=\mathcal {B}_-$, in order to arrive at an upper bound for the achievable FWHM. The curves representing larger lens radii, and therefore larger bandwidths, are closer to the desired FWHM. The plot further suggests that the bandwidth of the forward operator $F$ is indeed the limiting factor for how narrow a PNJ can get. Figure 7 shows the lower and upper bounds on the achievable FWHM (based on $\mathcal {B}=\mathcal {B}_+$ and $\mathcal {B}=\mathcal {B}_-$ respectively) for the four examined lens radii. These bounds are the asymptotic values of the achieved FWHM as the desired FWHM approaches zero. We finally examine the achievable FWHM as function of the desired radial distance, again using the 5-step procedure outlined above. Figure 8 shows our numerical results; here we try to produce a PNJ with a FWHM waist-width of $0.5\lambda _0$ (as in Fig. 5). The achieved FWHM waist-width of the PNJ seems to be approximately linearly increasing with increasing radial distance. The rate of increase is lower for larger lenses, which implies that, theoretically, we can achieve higher angular resolution at large distances from the lens by increasing the lens radius.

 

Fig. 5. Resulting physically viable PNJ profiles $|E^\textrm {sca}_s(\theta )|$ at $\partial B_{R_L+\varrho }$ ($\times 10^{5}$ V/m) for different lens radii $R_L$.

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Fig. 6. Waist-width analysis of PNJ profile for four different lens radii with a radial distance of $2\lambda _0$. The black dots are the results from Fig. 5.

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Fig. 7. Angular PNJ resolution analysis using the projection from 16 with both bandwidth estimates $\mathcal {B}_{\pm }$ from [29].

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Fig. 8. Waist-width analysis with varying radial distance of a single PNJ with desired FWHM of $0.5\lambda _0$, for four different lens radii $R_L$. The black dots are the results from Figs. 4 and 5.

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5. Achieved PNJ design

We demonstrate our method of PNJ design numerically using a 2D circular cross-section lens of radius 4 $\mu$m and a 2D square cross-section lens of side length 8 $\mu$m. Both lenses have refractive index $n_L=1.4607$ (SiO$_2$ at 532 nm), and the operating wavelength is $\lambda _0=532$ nm (common green laser). The illumination is time-harmonic and 2D TE, that is, the electric field vector points out of the lens cross-section plane, and the coordinate system origin is at the center of the lens cross-section.

The curve $C$, defining the desired location of the PNJ in 3, is a straight, 10 nm-linear piece pointing radially away from the origin, and we choose $\xi (\textbf {x})=1$ for $\textbf {x}\in C$. The curve $\Gamma$ where the incident field is tailored is the straight linear piece defined by $x\in [-12\ \mu$m$,12\ \mu$m$]$, $y=4\ \mu$m. Thus $\Gamma$ is always ’just above’ the micro-lens, and parallel with the $x$-axis. We solve 3 for ${E^\textrm {tot}}$ using the Finite Element method implemented by COMSOL, although other near-field numerical methods are also applicable [31]. Furthermore, we solve 8 for ${E^\textrm {inc}}$ at $\Gamma$ using numerical quadrature and interpolation in a Matlab script. Finally, we input the computed ${E^\textrm {inc}}$ at $\Gamma$ to a second COMSOL model to find the resulting total field.

The left-most and right-most columns in Fig. 9 demonstrate the realized PNJ design using the theoretical (Section 2) and the simulated SLM-produced (Section 3) incident fields, respectively. For practical purposes, there seems little difference between the theoretical and the SLM-produced near fields. The SLM angles $\varphi _1(\textbf {x})$ and $\varphi _2(\textbf {x})$ are here discretized with the resolution of 24.6 mrad. We conjecture that the lack of symmetry in some of the profiles of the tailored incident field in the first four cases in Fig. 9 is a numerical artifact due to the finite field sampling in the steps 2 and 3 described in Section 2. Evidently, the solution compensates for the asymmetry, and correct symmetric PNJs are ultimately achieved as desired in these cases.

 

Fig. 9. Left column: PNJ scanning achieved at the single optical wavelength $\lambda _0=532$ nm (common green laser). A 2D SiO$_2$ micro-lens with a circular cross-section of radius 4 $\mu$m, or a square cross-section of side length $8\ \mu$m, is illuminated along the negative $y$-axis by a computed structured incident field (see Section 2). The plots show the amplitude (in V/m) of the resulting total near field, normalized to maximum intensity of 1. Middle column: The computed amplitude and phase profiles of the incident field that produce the desired total near field. The desired PNJ locations in $\mu$m are, from top to bottom: $(x,y)=(0,-4.532)$, $(x,y)=(0,-9.32)$, $(x,y)=(0,-4.532)$, $(x,y)=(0,-9.32)$ (radial distance $1\lambda _0$ or $10\lambda _0$), and $(x,y)=(3,-9)$. Right column: PNJ scanning achieved with the incident field profiles simulated as being produced by an SLM with phase steps of 24.6 mrad in the angles $\varphi _1(\textbf {x})$ and $\varphi _2(\textbf {x})$. See Section 3 for details.

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The numerical results indicate that our method can design the PNJ position well and over a large domain in the near field. Specifically, Fig. 10 shows the achieved radial distances for the two lens geometries over large ranges of desired radial distances. The relative error in the achieved radial distances in that figure is on average $9.9\%$ in the circular lens case (for both the theoretical and the SLM-produced incident fields), and on average $1.5\%$ (theoretical) and $2.7\%$ (SLM) in the square lens case.

 

Fig. 10. Desired vs. achieved radial distance. a) circular lens, b) square lens.

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In Fig. 11 we observe that the waist width and the decay length of the realized PNJs are increasing with increasing radial distance, and for both lens geometries. The linear nature of the waist width increase was predicted in Section 4, but overestimated, especially for large radial distances. We stress that the analysis in Section 4 uses a simple Gaussian profile to model the field traces. In conclusion, both the radial and the lateral dimension of our near-field PNJ design increase with increasing distance from the lens, with the lateral dimension growing much slower and to the point of being practically constant over large intervals of radial distances.

 

Fig. 11. Achieved waist width and decay length vs. achieved radial distance. a) circular lens, b) square lens.

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Figure 12 shows, for the theoretical incident field profiles, the maximal PNJ electric field intensity relative to the maximal tailored incident electric field intensity, as function of the achieved radial distance in the PNJ. For the circular lens, this quantity fluctuates about 30$\%$ to 35$\%$, while for the square lens it decays significantly with increasing radial distance, and seems to stabilize at about 40$\%$. The fact that we achieve PNJs with maximum amplitudes comparable to those of the tailored incident fields suggests that the phenomenon underlying our PNJ formation is not primarily focusing, but rather an interference effect.

 

Fig. 12. Achieved PNJ field amplitude maximum vs. radial distance. a) circular lens, b) square lens.

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Table 1 shows mean values of achieved PNJ parameters, for the realizations of the theoretical incident field profiles treated in Figs. 10–12. For comparison, the simulated SLM-produced incident field profiles yield a mean waist width of $0.88\lambda _0$ (circular lens) and $0.70\lambda _0$ (square lens), as well as a mean decay length of $2.24\lambda _0$ (circular lens) and $1.60\lambda _0$ (square lens). For the theoretical incident field profiles, the predicted FWHM of the achieved PNJs in the example in Fig. 8 is between approximately $0.75\lambda _0$ and $1.00\lambda _0$ for $R_L=4\ \mu$m and for radial distances between approximately zero and $3\lambda _0$. This corresponds well with the numerically obtained waist widths in Fig. 11 for the circular lens and for achieved radial distances from approximately zero to approximately $3\lambda _0$. (We stress that our PNJ design procedure from Section 2 does not include specifying a desired PNJ waist width.) We next note that the PNJs produced using a square lens seem not only more precisely steerable, but also smaller than those produced using a circular lens. Their peak intensity relative to the incident field maximum intensity is also on average greater than in the circular-lens case, albeit this intensity ratio seems to drop quickly with the radial distance of the PNJ.

Table 1. Mean values of the achieved PNJ parameters in Figs. 10–12, for theoretical incident field profiles (Section 2).

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Finally, for reference, we investigate the near-field localization occurring when a simple $-\widehat {y}$-directed uniform plane wave at wavelength $\lambda _0=532$ nm illuminates the micro-lenses considered in this section. Figure 13 shows the resulting near fields. The circular cross-section lens produces a fixed PNJ with radial distance $2.22\lambda _0$, waist width $0.84\lambda _0$, decay length $3.21\lambda _0$, and maximum PNJ field intensity (relative to maximum incident electric field intensity) $18.92$. The waist width is $0.04\lambda _0\approx 21$ nm smaller than its mean counterpart in our steerable PNJs in the circular-lens case (but larger than that for our steerable PNJs in the square-lens case), while the decay length is almost one full wavelength greater than the mean decay length in the corresponding steerable PNJs. The main difference is the fact that the steerable PNJs exhibit 60 times smaller peak intensity relative to the maximum incident field intensity, and hence probably need a more powerful illumination in order to achieve comparable maximum field strengths. Alternatively, the relatively weak PNJs are well-suited as steerable optical probes in the investigation of bio-molecules and other sensitive samples.

 

Fig. 13. Near fields, normalized to maximum intensity of 1, obtained using uniform plane wave illumination. a) circular lens, b) square lens.

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The near-field structure in Fig. 13 of the square lens illuminated by a uniform plane wave seems too far off an admissible PNJ structure for analysis.

6. Conclusion

We give a theoretical foundation for a method of spatial steering of photonic jets, for a wide range of micro-scale dielectric lens shapes used together with computed illumination profiles. We furthermore provide numerical validation of the method in the case of circular and square cross-section 2D micro-lenses. Our method produces photonic jets at the desired locations in the near field of the dielectric lens. The novelty is that we take an inverse problem approach: starting with a desired total near field ${E^\textrm {tot}}$ for a fixed micro-lens geometry, we compute an incident field ${E^\textrm {inc}}$ that produces an approximation of ${E^\textrm {tot}}$, and that specifically produces a PNJ at the desired position. We also describe a practical way of obtaining the incident field profiles using spatial light modulators (SLMs). Our work paves the way towards fast SLM-based imaging that requires no mechanical movement of the sample or of the micro-lens. We see many applications, one of which is improving the resolution and speed of confocal microscopy by replacing the scanning mirrors with SLMs and doing $x,y$, and $z$ scan with our steerable PNJ. Other applications could be nanoparticle manipulation, direct lithography writing, scatterometry, fluorescence microscopy and Raman microscopy, which all benefit from the fact that we have a highly localized steerable optical probe with a subwavelength waist width. Since we see nearly zero defocusing of the PNJ over radial distances of at least up to 15$\lambda _0$ (almost 8 $\mu$m), the technique is ideal for the abovementioned and other applications. We thus expect our steerable optical photonic jet probe to drive advances within several fields.

The observed small relative peak field strength makes our steerable PNJ probe unfavorable in certain applications. However, in other cases a steerable PNJ may be of more interest than one that is narrower but spatially fixed. We plan to explore in future work how the relative field strength can be increased. Our analysis in Section 4 leads us to expect that using larger micro-lenses will decrease the waist width of the steerable PNJ. Further, the relative peak field strength may increase with increasing size of the micro-lens.

We have not yet explored the general validity of the results in Figs. 6 and 7, that is, for other lens geometries than the circular case. Our analytical method of Section 4 is limited to circular-shaped lenses, and can therefore only make (qualitative) predictions for that case. We leave a more general analytic and numerical treatment to future work.

Finally, we did not investigate the sensitivity of our method to local minima in the optimization problem of finding the incident field profiles. It would certainly be desirable to be able to obtain a similar level of PNJ steerability using simpler incident fields. However, it is currently unclear whether such simplification is even possible.