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# Nano-jet related to Bessel beams and to super-resolutions in microsphere optical experiments

- Yacob Ben-Aryeh
^{1}Email author

**4**:3

https://doi.org/10.1140/epjti/s40485-017-0038-5

Β© The Author(s); licensee Springer on behalf of EPJ. 2017

**Received:**12 March 2017**Accepted:**21 June 2017**Published:**29 June 2017

## Abstract

The appearance of a Nano-jet in the microsphere optical experiments is analyzed by relating this effect to non-diffracting Bessel beams. By inserting a circular aperture with a radius of subwavelength dimension in the EM waist, and sending the transmitted light into a confocal microscope, the EM oscillations by the different Bessel beams are avoided. On this constant EM field evanescent waves are superposed. While this effect improves the optical-depth of the imaging process, the object fine-structures are obtained, from the modulation of the EM fields by the evanescent waves. The use of a combination of the microsphere optical system with an interferometer for phase contrast measurements is described.

## Keywords

- Microsphere
- Nano-jet
- Bessel beams
- Super-resolution
- Evanescent waves
- Phase contrast

## Introduction

Researchers spent much effort in pushing the frontier of optical resolution in a quest to image very small objects. Unfortunate optical resolution is limited by diffraction and dispersion effects. Diffraction causes light beams to spread in transverse direction during their propagation. A temporal pulse of light analogously spreads owing to material dispersion. The quest for ultimate super resolution has been described in a short Review [1], treating incoherent super-resolution techniques, coherent super resolutions schemes, near field super-resolution techniques, etc. Quite long ago Silberberg [2] has shown that it is possible for self-focusing nonlinearity to compensate both spatial and temporal spreading of the light pulse. The resulting wave packet is a spatiotemporal soliton which is referred to as a βlight bulletβ). Minardy et al. [3], by using arrays of wave guides, demonstrated a nonlinear propagation of a 3D localized optical wave packet. Belic et al. [4] obtained exact spatiotemporal periodic travelling wave solutions to the generalized (3 + 1)βdimensional nonlinear Schrodinger equation and used these solutions to construct analytical light bullet solutions. Scattering effects have been found to be an impediment to focusing and imaging. Light scattering, however, does not lead to an irretrievable loss of information. The information might be scrambled into disordered interference patterns called laser speckles. Para-axial speckle based metrology systems have been analyzed by Kelly et al. [5]. There are various techniques for imaging inside scattering media. Methods for controlling waves in space and time for imaging and focusing in complex media have been developed by Mosk et al. [6]. A different approach to super resolution phenomena has been discussed in relation to microsphere optical experiments (see e.g. [7β10]). The microsphere optical system is different from those described in [1β6], as in this system the information on the fine structures of the object is obtained from the evanescent waves transmitted through the microsphere. Also as the microsphere is composed of dielectric materials there are not any nonlinear effects in this system. High resolutions were obtained in microsphere optical experiments (up to 50 nm in [7β9] and 25 nm in [10]). Such high resolutions might have an impact in the fields of integrated optics, microchips, photo lithography plus plasma etching, photo resists etc., in which very high resolutions are needed. Therefore the improvement of resolutions in these systems is of utmost importance.

There are various different studies about the high resolutions obtained in microsphere optical experiments. Pang et al. [11] and Sundaram et al. [12] claimed that the physical mechanism by which high resolutions are obtained in microsphere experiments is unclear. Hao et al. [13] described the microsphere as a channel that connects the near field evanescent waves and the transmission one in the far field. In some works by the present author [14β17] the enhancement of resolutions which are higher than the Abbe limit has been related to the field of scanning near-field optical microscopy (SNOM) [18]. In this field evanescent waves are produced in which one component of the optical wave vector is imaginary (e.g. in the symmetric *z* axis), leading to a decay of the wave in this direction. Other components of the wave vector increase according to the Helmholtz equation, thus decreasing the effective wavelength and correspondingly increase the resolution. In previous work [17] an estimate to the increase of resolution relative to the Abbe limit has been given, and the condition for converting the evanescent to propagating waves has been derived. It has been shown in various works [19β23] that the increase of the refractive index of microspheres can enhance the imaging resolution and quality. The quantitative analysis made in previous work [17] is in a good agreement with these results.

There are special effects which were obtained with microspheres. It has been shown that by using microsphere-chain waveguides [24, 25] focusing and resolutions can improved. Microsphere near-field nano-structures were observed and analyzed using picosecond pulses [26]. Viruses were observed by microsphere optical nanoscopy [27]. It has been shown that it is possible to control the focusing properties of the microsphere by using pupil masks [28]. Resolution phase contrast imaging by digital holography has been developed and observed [29]. There are various articles relating the microsphere high resolutions to Nano-jets where optical beams are produced near the focal point which have a very narrow waist [30β34]. Appearance of Nano-jets in the microsphere optical experiments can be analyzed by the use of the Mie theory which shows certain deviations from the simple geometric optics approach [31]. This method does not give analytical results, as it requires the summation of a large number of terms for moderate sphere size. An analytical description of the microsphere axially symmetric focusing with strong aberration has been developed by the use of the Bessoid integral [35] and the results are in agreement with calculations by Mie theory. However, such calculations do not explain the high resolutions obtained in microsphere optical experiments.

The main issue of the present paper is show how to avoid diffraction effects in microsphere optical experiments. We describe a combination of a microsphere with an orifice of subwavelength dimensions in the focal plane to affect the profile of the out coming classical electromagnetic (EM) wave-front which will improve the resolution. (Such an orifice was described and constructed in [36]). While geometric optics describes the mean propagation of light the diffraction effects are described as spherical aberration. The full description of the diffraction effects in this system is very complicated [35]. However, by introducing a subwavelength aperture in the focal plane the imaging process is described by the use of Bessel function of zero order [37β39]. While usually a Bessel function is quite broad by restricting the transmitted EM beam to the orifice, the diffraction effects are eliminated and a very good optical depth is obtained. By using the present approach we analyze also the use of a combination of the microsphere with an interferometer for phase contrast measurements.

## Microsphere optical Nano-jet related to non-diffractive Bessel beams

## A combination of the microsphere with an interferometer for phase contrast measurements

*U*

_{1}after its transmittance through the thin layer of a phase object can be described as

We inserted here the phase of the object *Ο*(*x*
_{0},β*y*
_{0}) which is a function of the object (*x*
_{0},β*y*
_{0}) coordinates. This EM field is converging by the microsphere into the focal plane and transmitted through a small circular aperture inserted in the plane barrier located on the focal plane of the microsphere.

The EM field *U*
_{3} on the circular aperture is composed of the Nano-jet and its modulation *Ο*(*x*
_{0},β*y*
_{0}) by the evanescent waves, as described in the previous analysis related to Fig. 1. While the EM field *U*
_{3} might propagate into a confocal microscope measuring fine structures light intensities, it cannot measure directly the objects phases. For the purpose of measuring phase fine structures we use the second EM field *U*
_{2} reflected from BS1 that after reflection from mirrors M1 and M2 and transmittance through a phase shifter (PS) is given by*U*
_{2}βexp(β*iΞ±*). By using the phase shifter (PS) the phase difference between the beams *U*
_{2}βexp(β*iΞ±*) and *U*
_{3} can be changed. The EM fields *U*
_{3} and *U*
_{2}βexp(β*iΞ±*) are recombined by the second beam splitter (BS2) leading to interference between the two beams with light intensity |*U*
_{4}|^{2}which can be described schematically as |*U*
_{4}(*x*
_{4},β*y*
_{4})|^{2}β=β|*CU*
_{3}|^{2}β+β|*U*
_{2}|^{2}β+β2|*CU*
_{3}||*U*
_{2}|βcos[*Ο*(*Mx*
_{0},β*My*
_{0})β+β*Ξ±*] (13)

Here the constant C represents the relative intensity between the two beams. *Ο*(*Mx*
_{0},β*My*
_{0}), represents the phases of the phase object where in a simple geometric approach they can be magnified by factor M. *Ξ±* represents the phase difference between the two beams which is controlled by the phase shifter PS. The conversion of the phases of the phase object to light intensities is demonstrated by the physical scheme described Fig. 2 and the schematic analytical relation given by eq. 13. The realization of such phase measurements is especially important for biological systems but so far such proposed system has not been exploited.

## Conclusions

The Nano-jet in microsphere optical experiments is found to be produced by propagating Bessel beams. By using a plane barrier in the focal plane with a circular aperture with a radius in the order of subwavelength dimensions the oscillations produced in the Nano-jet are eliminated, as all the bright cores of the various Bessel beam are overlapping. The information in this system is obtained from modulation of the Nano-jets by evanescent waves transmitted through the microsphere. This system is expected to give very high resolutions which might be applied in various fields of integrated optics.

A combination of the microsphere with an interferometer is described for phase contrast measurements, as analyzed in the present work. Such system might be used for imaging phase objects and is especially important for imaging biological systems.

## Declarations

### Acknowledgements

The author would like to thank Prof. S. Lipson for interesting discussions.

### Authorsβ contributions

The author approves the final original paper written by him.

### Competing interests

The author declares he has no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authorsβ Affiliations

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