Nano-jet Related to Bessel Beams and to Super-Resolutions in Micro-sphere Optical Experiments

The appearance of a Nano-jet in the micro-sphere optical experiments is analyzed by relating this effect to non-diffracting Bessel beams. By inserting a circular aperture with a radius which is in the order of subwavelength in the EM waist, and sending the transmitted light into a confocal microscope, EM fluctuations 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 micro-sphere optical system with an interferometer for phase contrast measurements is described.


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 "light bullet"). 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)-dimensional 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. [7][8][9][10]). The microsphere optical system is different from those described in [1][2][3][4][5][6], 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 [7][8][9] 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 [14][15][16][17] 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 [19][20][21][22][23] 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 [30][31][32][33][34]. 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 [37][38][39]. 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
Production of the Nano-jet in microsphere experiments is analyzed by superposing the spherical aberration on the geometric optics trajectories, as described in Fig  Plane EM waves are transmitted through the object, converging by the microsphere and exiting through a small circular aperture at point P located on a plane barrier at the focal plane. Evanescent waves which include the information on the fine structures of the object are transmitted into the microsphere near the contact point O and are transformed by the microsphere into propagating waves. The geometric optics trajectory for a propagating ray, which is not evanescent, and which enters the microsphere far from the contact point O is described with the locations of all relevant angles: in θ , T θ , α and β . Bessel beams producing the Nano-jet are obtained by the superposition of the spherical aberration on the geometric optics trajectory. The Nano-jet is modulated by the evanescent waves and the fine structures of the object are obtained by sending the radiation transmitted around point P into a confocal microscope.
According to Snell's law the incidence angle in θ and the transmitted angle , , , exp cos The exponential function describes the propagation of the wave vector in the z direction where ρ is the distance from the symmetric z axis, β is defined by eq. 5, and z is the distance from the focal plane. We substitute eq. 7 into the wave equation Then we get [37][38][39] ( ) We used here the azimuthal symmetry of the wave function including the relations through what is known as phase contrast microscopy [29]. Imaging of phase objects is especially important for biological systems [27,41]. The following optical system described in Fig. 2, which is based on a combination of the microsphere with an interferometer is suggested for phase contrast measurements. Phase contrast measurements are described by the use of a combination of the microsphere with an interferometer. Plane EM waves are incident on the first beam splitter (BS1) and are divided into a transmitted field 1 U , and reflected field 2 U . The EM field 1 U is transmitted through a phase object, converging by the microsphere and producing a Nano-jet at the focal plane modulated by the evanescent waves, denoted in Fig. 2, as 3 U . The EM field 2 U reflected from BS1 is reflected also from mirrors 1 M and 2 M , and is recombined with the EM field 3 U at the second beam splitter (BS2). A phase shifter (PS) produces a phase difference between the two beams. The recombined beam is going into confocal microscope.
As shown in Fig. 2 plane EM wave is incident perpendicular to the first beam-splitter (BS1) where part of the light is continuing perpendicular to a thin phase object like that of a biological tissue, and a part of it is reflected into horizontal direction and reflected from mirrors M1 and M2. The EM field with constant amplitude 1 U after its transmittance through the thin layer of a phase object can be described as We inserted here the phase of the object ( ) The EM field 3 U on the circular aperture is composed of the Nano-jet and its modulation ( ) Here the constant C represents the relative intensity between the two beams. , represents the phases of the phase object where in a simple geometric approach they can be magnified by factor M. α 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.