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Characterization and Application of Nanomaterials 2024, 7(2), 6031. 
https://doi.org/10.24294/can.v7i2.6031 

1 

Article 

Super-resolution by converting evanescent waves in microsphere to 
propagating waves and light transmitted from its surface to nano-jet 

Y. Ben-Aryeh 

Physics Department, Technion-Israel-Institute of Technology, Haifa 32000, Israel; phr65yb@physics.technion.ac.il 

Abstract: The electro-magnetic (EM) waves transmitted through a thin object with fine 

structures are observed by a microsphere located above the thin object. The EM radiation 

transmitted through the object produces both evanescent waves, which include information on 

the fine structures of the object (smaller than a wavelength), and propagating waves, which 

include the large image of the object (with dimensions larger than a wavelength). The super-

resolutions are calculated by using the Helmholtz equation. According to this equation, 

evanescent waves have an imaginary component of the wavevector in the z direction, leading 

the components of the wavevector in the transversal directions to become very large so that the 

fine structures of the object can be observed. Due to the decay of the evanescent waves, only a 

small region near the contact point between the thin object and the microsphere is effective for 

producing the super resolution effects. The image with super-resolution can be increased by a 

movement of the microsphere over the object or by using arrays of microspheres. Both 

propagating and evanescent waves arrive at the inner surface of the microsphere. A coupling 

between the transmitted EM waves and resonances produced in the dielectric sphere, possibly 

obtained by the Mie method, leads to a product of the EM distribution function with the transfer 

function. While this transfer function might be calculated by the Mie method, it is also possible 

to use it as an experimental function. By Fourier transform of the above product, we get 

convolution between the EM spatial modes and those of the transfer function arriving at the 

nano-jet, which leads the evanescent waves to become propagating waves with effective very 

small wavelengths and thus increase the resolution. 

Keywords: microsphere; super-resolution; evanescent waves; nano-jet; transfer function; Mie 

method 

1. Introduction 

Any microscopic image can be magnified by using a microscope. But observing 
sub-wavelength structures is difficult because of the Abbe diffraction limit [1], by 

which light with wavelength 𝜆  travelling in medium with refractive index 𝑛  and 

aperture angle 𝜃 will make a spot with radius. 

𝑑 =
2(𝑛 𝑠𝑖𝑛 𝜃)

𝜆
 (1)

The term 𝑛 𝑠𝑖𝑛 𝜃is called the numerical aperture (𝑁𝐴)and the Abbe limit is of 

order 𝜆/2. 
I analyze in the present article the mechanism by which evanescent waves 

incident on a microsphere are converted into propagating waves, and by such 
conversion high resolution is obtained in the image by the microsphere, which is much 
beyond the Abbe limit. There is much interest in the optical properties of the 
microsphere system. Many studies on the super-resolution obtained by the 
microsphere system were described in recent articles [2–10]. The mechanisms by 

CITATION 

Ben-Aryeh Y. Super-resolution by 
converting evanescent waves in 
microsphere to propagating waves 
and light transmitted from its surface 
to nano-jet. Characterization and 
Application of Nanomaterials. 2024; 
7(2): 6031. 
https://doi.org/10.24294/can.v7i2.6031 

ARTICLE INFO 

Received: 26 April 2024 
Accepted: 15 May 2024 
Available online: 15 July 2024 

COPYRIGHT 

 
Copyright © 2024 by author(s). 
Characterization and Application of 
Nanomaterials is published by 
EnPress Publisher, LLC. This work is 
licensed under the Creative 
Commons Attribution (CC BY) 
license. 
https://creativecommons.org/licenses/
by/4.0/ 



Characterization and Application of Nanomaterials 2024, 7(2), 6031.  

2 

which nan-jets are produced in the microsphere system were analyzed [11–16]. These 
works and many previous ones led to controversy about the origin of the observed 
super-resolution effects. In certain works, the super-resolution was related to 
evanescent waves emitted from the object, which include the fine structures of the 
object. In these works, the production of the nan-jet was considered a separate effect, 
not related to super resolution. On the other hand, it was claimed in other works that 
the super-resolution effects are produced by the same mechanisms that produce the 
nano-jet, while the evanescent waves might have only minor effect. In the present 
work, I show that the super-resolution effects are produced in two steps, where in the 
first step the information on the object's fine structures is obtained by the evanescent 
waves. In the second step, there is a convolution between the spatial modes of the 
evanescent waves and the spatial modes of the microsphere, which transforms the 
evanescent waves into propagating waves with very small effective wavelengths [17]. 

The super-resolution obtained by the microsphere is like the field of scanning 
near-field optical microscopy (SNOM), where the resonant effect with the tip detector 
enhances the super-resolution. We follow in the present analysis the idea that the 
evanescent wavevectors are coupled to the microsphere by resonances produced by 
Mie theory analysis [18,19]. Such Mie theory is also used for a good description of the 
nano-jet [19]. The conversion of evanescent waves to propagating waves, after the 
transmission through the microsphere surface, is obtained by convolution between the 
evanescent spatial modes and the transfer function of the microsphere modes [17]. The 
idea of using such convolution was suggested already in a previous article [20]. My 
approach to microsphere high resolution will be developed in the next sections by 
using two steps: 1) In the first step, we use the Helmholtz equation, by which large 
wave vectors are produced above the object, producing high resolutions. 2) The EM 
fields of both evanescent waves and propagating waves arriving parallel to the 
microsphere surface are preserved during transmission through the microsphere 
surface due to boundary conditions. The enhancement of the conversion of evanescent 
waves to propagating waves is due to coupling between evanescent waves and 
resonances, produced in the microsphere, for example, by Mie theory [19]. In the 
second stage we do not use the Helmholtz equation, and the above coupling is 
described as a convolution between the spatial modes of the evanescent waves and 
those of the microsphere modes, described by the transfer function [17]. Although the 
propagating waves are stronger than those of the evanescent waves, the modulation of 
the total intensity by the evanescent waves is the source of the high resolutions 
obtained by the microsphere system. 

2. Methods 

The use of evanescent waves to increase the resolution beyond the Abbe limit can 
be related to Helmholtz equation [17]. In homogenous medium this equation is given 
by 

(𝑛𝑘 ) = 𝑘 + 𝑘 + 𝑘  (2)

where 𝑘 = 2𝜋/𝜆 , 𝜆  is the wavelength in vacuum, 𝑘 , 𝑘 , 𝑘 are the wavevector 

components. The evanescent waves satisfying the relation: 



Characterization and Application of Nanomaterials 2024, 7(2), 6031.  

3 

𝑘 + 𝑘 > (𝑛𝑘 )  (3)

are arriving at the microsphere with imaginary 𝑘 . The increase of the components of 

the wavevector 𝑘 in the plane 𝑥, 𝑦 decreases the “effective” wavelength in this plane, 
and thus increases the resolution. 

As described in Figure 1, a dielectric microsphere with radius R and a refractive 

index 𝑛  is located above a thin object at a contact point O. The medium between the 

object and the microsphere has a refractive index 𝑛 . Parallel EM waves are 
transmitted in a direction perpendicular to a thin planar object, which may be 
transmitted as “propagation” waves and as “evanescent” waves. But the increase of 
resolution by the microsphere is related to the conversion of evanescent waves to 
propagating waves. 

We analyze the conversion of evanescent waves to propagating waves at a point 
P, which is on the microsphere surface. The incident and transmitted angles for the 

EM waves transmitted into the microsphere at point P are given as 𝛩 and 𝛩 , 

respectively. At this point, EM waves with wavevectors 𝑘 . 𝑘  are arriving at the 

microsphere, where 𝑘 is imaginary. The increase in the component of the wavevector 

𝑘  in the plane 𝑥, 𝑦  decreases the “effective” wavelength in this plane, and thus 
increases the resolution. But the evanescent waves decay in the perpendicular z 
direction, so that to “capture” the fine structure that is available in the evanescent 
waves, we need that the point P will be near the contact point O, so that its 
perpendicular distance to the object will be of a wavelength order. According to 
geometric optics, the microsphere has a spherical symmetry under rotation around the 
z axis, which connects the center of the microsphere at point C with the contact point 

O. The line connecting the point C with point P has an angle 𝛩 with the symmetric z 
axis. 

As shown in Figure 1, the EM waves transmitted through the microsphere are 
converging into photonic-jet (PJ), where its role in producing the high resolution is 
controversial. The original work in obtaining high resolution in microsphere imaging 
was made by Zengo Wang et al. [21]. That analysis was made with a virtual image as 
follows from the geometric optics description. Since this time, a very large number of 
papers were published on various effects in the microsphere system. In Figure 1, we 
describe a real image that is produced by using a high-refractive index microsphere 
(e.g., [22,23]). 

The focusing of light in the microsphere system is concentrated in the nano-jet 
sub-diffraction region, which does not obey the classical laws of geometrical optics. 
but might be explained by diffraction effects, known as photonic-jet (PJ) (e.g., [24,25]). 
Exact solutions for non-diffraction beams might be related to the central part of the 
photonic nano-jet [26,27]. 



Characterization and Application of Nanomaterials 2024, 7(2), 6031.  

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Figure 1. Microsphere and nano-jet. 

Propagating EM waves are transmitted through a thin object with a certain 
structure, whose image is produced by evanescent waves, arriving at the microsphere 
surface at point P, for example. The EM transmitted through the microsphere is 
converging into a non-diffraction region known as a photonic jet (PJ). 

Rigorous Mie theory predicts the interaction of light with spherical particles, and 
this theory was used to describe various properties of the PJ’s produced by the 
microsphere system [19]. The exact use of the Mie theory is usually done by numerical 
calculations, as it is obtained by the sum of many terms that do not give an analytical 
result. Using Mie theory, optical resonances in microsphere photonic nano-jets were 
observed [18]. We analyzed the properties of evanescent waves, produced by plane 
EM waves transmitted through nano-corrugated-metallic thin film, which includes 
information on its fine structures [28]. A microsphere located above the metallic 
surface collects the evanescent waves, which are converted to propagating waves. The 
magnification of the nano-structure images is explained by a geometric optics 
description, but the high resolution is related to the evanescent wave analysis. Such an 
approach for explaining the high resolutions obtained by microspheres was developed 
by using complex Snell’s law [29]. Very high resolutions by microspheres were 
reported also in other works (e.g., [30,31]). 

Maslov and Astratov [32,33] studied the origin of super-resolution in 
microsphere-assisted imaging. The imaging of the nano-jet cannot give a good 
explanation for the high resolutions obtained in the microsphere system. The 



Characterization and Application of Nanomaterials 2024, 7(2), 6031.  

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evanescent wave source cannot explain the high resolutions obtained by the 
microsphere system since its effect is too weak. They suggested using a direct 
approach to Maxwell equations, including optical principles such as point-spread 
functions, in microspherical imaging. I claim, however, that there are strong arguments 
that evanescent waves should be included in the theories about high resolutions: a) 
The propagating distance between the object and the microscope must be small enough 
to reveal the nanometric features. To generate super-resolution over large areas of the 
sample, we need to attach the microsphere to a frame, which is scanned on the sample 
in step-by-step fashion [34] or by using microspheres arrays [6]. b) There is a 
dependence of the super-resolution on the radius of the microsphere and its index of 
refraction, which is in good correspondence with the evanescent wave properties [29]. 
c) The super-resolution is related to information theory, where the information on the 
fine structures of the object area is available in the evanescent waves. Magnification 
of the image and optical transforms cannot introduce high resolution if the information 
on the object's fine structures is not available in such transforms. I find, however, that 
there is an important mechanism that enhances the transformation of evanescent waves 
into propagating waves after the microsphere surface, which is beyond classical 
geometric optics (beyond the use of complex Snell’s law [29]. To explain this new 
mechanism, it will be helpful to compare the high resolution obtained by the 
microsphere with the high resolutions obtained in metallic grating [35,36]. It was 
shown in these works that a thin film of metallic grating with arrays of subwavelength 
holes can transmit light at certain frequencies, which is order of magnitude larger than 
the light intensity incident on the area of these holes. Most investigators agree that 
these experiments, conducted originally by Ebessen et al. [35,36], are related to 
coupling the light with surface plasmons. I explained these phenomena as transmission 
enhancement by converting evanescent waves, entering the small holes to propagating 
waves, due to convolution of the evanescent spatial modes with the plasmon spatial 
modes, producing high spatial wavevectors with very small “effective” wavelengths 
[37]. Such effects were described in a similar way by relating the ‘tunelling’ of 
evanescent waves to propagating waves due to the convolution of the high spatial 
frequencies of the source with those of the detector [38]. One should consider that in 
the dielectric sphere we don’t have plasmons, but we have other coupling mechanisms 
between the evanescent waves and the microsphere EM modes. 

3. An analysis for the super resolution, in the microsphere system, 
obtained in many experiments 

In experimental studies on microspheres by other authors [2–23,30,31,34], super 
resolution effects were observed. These works raised the question: Is the super-
resolution related to evanescent waves? or is it related to the existence of the nano-jet? 
The question led to controversy between the various works. I show in the present 
theoretical analysis that a combination of two effects produces the high resolution: 1) 
The evanescent waves produced by the object are incident on the microsphere surface, 
preserving high resolutions. 2) The same mechanism which produces the nano-jet 
leads to conversion of the evanescent waves to propagating waves, but with very small 
wave lengths. The reduction of the wavelength was explained also as a quantum effect 



Characterization and Application of Nanomaterials 2024, 7(2), 6031.  

6 

[39,40] where n entangled photons lead to effective wavelength /𝑛, and there is a 
distribution of such effective wavelengths. 

We develop the analysis for the microsphere system into two parts: a) In the first 
part, we describe the propagation of evanescent waves, produced on thin planar objects 
with fine structures, to the microsphere surface. In this stage, high resolutions of the 
image are obtained related to the use of the Helmholtz equation. b) In the second part, 
we consider the propagation of both evanescent and propagating waves produced on 
the inner surface of the microsphere to the nano-jet. This propagation is described by 
the convolution of these EM fields with the microsphere modes described by a transfer 
function [17], which is related to Mie theory, but it is more convenient to use it as an 
experimental function. 

3.1. The use of Helmholtz equation for getting high resolutions by 
evanescent waves 

Let us assume that the planar surface of an object is given by z = 0, and the EM 
field in this plane is given by the Fourier transform 

𝑈(𝑥, 𝑦) = 𝑢 𝑘 , 𝑘
∞

∞

∞

∞
𝑒𝑥𝑝 −𝑖 𝑘 𝑥 + 𝑘 𝑦 𝑑𝑘 𝑑𝑘  (4)

where 𝑢 𝑘 , 𝑘  is the distribution of the EM spatial modes in the 𝑥, 𝑦plane. Then, the 

EM waves propagating from the planar surface of the object into homogenous medium 

in the space 𝑧 > 0 with a refractive index 𝑛  is given by: 

𝑈(𝑥, 𝑦, 𝑧 > 0) = 𝑢 𝑘 , 𝑘
∞

∞

∞

∞
𝑒𝑥𝑝 −𝑖 𝑘 𝑥 + 𝑘 𝑦 + 𝑘 𝑧 𝑑𝑘 𝑑𝑘  (5)

Substituting Eq. (5) into the Helmholtz equation we get: 

(𝛥 + 𝑘 )𝑈(𝑟) = 0 (6)

obtaining the result: 

𝑘 − 𝑘 + 𝑘 + 𝑘 𝑈(𝑥, 𝑦, 𝑧 > 0) = 0 (7)

Under the condition 𝑘 < 𝑘 + 𝑘 , 𝑘 is imaginary, and for such case we get 

the evanescent wave solution: 

𝑈(𝑥, 𝑦, 𝑧 > 0) = 𝑈(𝑥, 𝑦, 𝑧 = 0) 𝑒𝑥𝑝(−𝛾𝑧)  ; 𝛾 = 𝑘 + 𝑘 − 𝑘  (8)

A dielectric microsphere with refractive index 𝑛  and radius 𝑅 is located above 

a thin film of the object at a contact point 𝑂,where the medium between the object and 

the microsphere has a refraction index 𝑛 . A plane EM wave is transmitted through 
the thin film of the object in the perpendicular direction and incident on the 
microsphere at point P. The evanescent waves are incident on the microsphere surface 

near the contact point 𝑂. 
For evanescent waves, there is a decay of the wave in the z direction. The 

resolution obtained by the evanescent waves is limited by the lateral component 𝑘 of 
the wavelength given by: 

𝜆 =
2𝜋

𝑘
=

2𝜋

𝑘 + 𝑘

 
(9)



Characterization and Application of Nanomaterials 2024, 7(2), 6031.  

7 

As the evanescent waves satisfy the equation 𝑘 = 𝑘 + 𝑘 − 𝛾 , then Eq. (9) 

can be written as  

𝜆 (𝑒𝑣𝑎𝑛. ) =
2𝜋

𝑘 + 𝛾
 (10)

The minimal value of 𝜆  for propagating waves is given by: 

𝜆(𝑝𝑟𝑜𝑝. )
2𝜋

𝑘 ,
 (11)

since the minimum is obtained when 𝑘 is in the x, y plane. 

As by the Abbe limit the resolution is of order 𝜆/2, the increase of resolution 𝐹 
by using evanescent waves is given by: 

𝐹 = 𝜆 (𝑝𝑟𝑜𝑝. )/𝜆 (𝑒𝑣𝑎𝑛. ) =
𝑘 + 𝛾

𝑘
= 1 +

𝛾

𝑛 𝑘
 (12)

 
Figure 2. Evanescent waves on microsphere surface. 

The distance of the point P from the planar object is given by ℎ = 𝑅(1 − 𝑐𝑜𝑠 𝜙) 

(see Figure 2). For decay constant 𝛾 of the evanescent waves, represented in unit 

𝛾/𝑛 𝑘  , the decay of the evanescent wave at point at point P, after transversing the 

distance ℎ, is given by 

𝑒𝑥𝑝(−𝛾ℎ) = 𝑒𝑥𝑝 −
𝛾

𝑛 𝑘
𝑛 𝑘 𝑅(1 − 𝑐𝑜𝑠 𝜙 = 𝑒𝑥𝑝 −

𝛾

𝑛 𝑘
2𝜋(1 − 𝑐𝑜𝑠 𝜙)

𝑅

𝜆
 (13)

We find that this decay increases very much by increasing 𝜙, so that only a small 
region around the contact point O is efficient in obtaining the high resolution by 
evanescent waves. 



Characterization and Application of Nanomaterials 2024, 7(2), 6031.  

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3.2. Microsphere imaging by a transfer function from the microsphere 
surface to the nano-jet 

Let us assume that the EM field 𝐸(𝑥, 𝑦) , which is tangent to the microsphere 

surface at the point 𝑥, 𝑦 (of evanescent wave in a small region around the contact point 
O, plus a propagating wave in a larger region), is given by: 

𝐸 𝑐𝑜𝑠 𝜙  (14)

where 𝐸(𝑥, 𝑦) is the EM field before the microsphere surface (see Figure 2, and [29]). 

The EM field 𝐸  is preserved during transmission through the microsphere surface 
due to boundary conditions, and it includes both the evanescent waves and the 
propagating waves. The EM field propagating after the microsphere surface is given 
as 

tan'( , ) ( , ) ( , )E x y E x y G x y  (15)

where the ‘transfer function’ 𝐺(𝑥, 𝑦) can be related by Fourier transform [17] to 

𝐺(𝑘 , 𝑘 ): 

𝐺(𝑘 , 𝑘 ) = 𝐺(𝑥, 𝑦)
∞

∞

∞

∞
𝑒𝑥𝑝 −𝑖 𝑘 𝑥 + 𝑘 𝑦 𝑑𝑥𝑑𝑦 (16)

One should notice that 𝐺(𝑘 , 𝑘 )  should include both evanescent waves for 

which 𝑘 + 𝑘 > 𝑘 , and propagating waves for which 𝑘 + 𝑘 < 𝑘 . 

Due to the small distance between the points x, y and the contact point O, the 

evanescent field 𝐸 , (𝑥, 𝑦) is approximately parallel to the x, y plane. This EM 

field is smaller from the evanescent EM field on the thin object, by the factor given 
approximately by Eq. (13). 

The Fourier amplitude 𝐴(𝑘 , 𝑘 ) is described by the Fourier inverse of 𝐸 , 

which is given by:  

𝐴(𝑘 , 𝑘 ) = 𝐸(𝑥, 𝑦)
∞∫

∞

∞∫

∞
 (17)

where 𝑘 . 𝑘  are the wavevectors in the x and y directions. The Fourier transform of 

𝐸′(𝑥, 𝑦), (given by the product of Eq. (14)) is obtained by the convolution of the spatial 

wavevectors 𝐺(𝑘 , 𝑘 ) and 𝐴(𝑘 , 𝑘 ) [17]: 

𝐸′(𝑘 , 𝑘 ) = 𝐴 𝑘′ , 𝑘′
∞

∞

∞

∞
𝐺 𝑘 − 𝑘′ , 𝑘 − 𝑘; 𝑑𝑘′ 𝑑𝑘′  (18)

The convolution given by Equation (18) includes spatial wavevectors 𝑘 − 𝑘′  

and 𝑘 − 𝑘′ which might be very large, so that the corresponding wavelengths can be 

reduced to very low values. For example, if approximately, 𝑘 − 𝑘′ = 𝑘𝑛 where n is 

a large integer, then the effective wavelength will be reduced to 𝜆/𝑛 (see the analysis 
in [39,40]). We use the convolution effect described by Equation (18), to obtain 
enhancement of the conversion of evanescent waves to propagating waves by reducing 
the effective wavelength. Although the modulation of the EM waves by evanescent 
waves is small relative to the total intensity of the EM field, such modulation is 
effective in producing the high resolution. The spreading of the spatial modes by 
convolution is like the point spread function used in [32,33], but one should not ignore 



Characterization and Application of Nanomaterials 2024, 7(2), 6031.  

9 

the super resolution obtained for the object in the first stage of the imaging process, 
which is related to the use of Helmholtz equation. 

The above reduction of the effective wavelength by convolution, was also 
described as a quantum effect where entanglement between the n photons reduces the 

effective wavelength to 𝜆/𝑛 [39,40]. Such entanglement remains true also after the 
transformation to propagating waves. 

4. Discussions and results 

The high resolution obtained in microspheres is due to the following two factors: 
a) The EM radiation transmitted through the object produces both evanescent waves, 
which include information on the fine structures of the object (smaller than a 
wavelength), and propagating waves, which include the large image of the object (with 
dimensions larger than a wavelength). Evanescent waves lead to a super-resolution, 

which is better than the Abbe limit, by the factor 𝐹 = 𝑘 + 𝑘 /𝑘 =

(𝑘 + 𝛾 )/𝑘  where 𝑘 = 𝑘 + 𝑘 − 𝑘  is the wavevector of the evanescent 

EM field, 𝑘 = −𝑖𝛾  is imaginary, and 𝑘 , 𝑘 are the transversal components. The 

evanescent waves arrive at the microsphere surface after a decay in the 𝑧 direction by 

the factor 𝑒𝑥𝑝(−𝛾ℎ). where ℎ  is the distance from the object to the microsphere 

surface, and 𝛾 is a certain decay constant., so that only the evanescent waves which 

are near the contact point 𝑂 of the object are efficient in conserving the fine structure 
imaging. But one can attach the microsphere to a frame which moves on the object 
and scans a large image, or to use arrays of microspheres. b) By using boundary 
conditions, we get on the inner microsphere surface both evanescent waves and 
propagating waves. The convolution between the EM waves, on the inner surface of 
the microsphere, and the transfer function, representing the coupling with microsphere 
EM modes, enables propagation with small effective wavelengths (smaller than a 
wavelength and correspondingly large wave vectors). This effect is due to the spread 
of the wavevectors due to the above convolution. The transfer function also includes 
the wavevectors of the evanescent waves and thus enables them to be transferred to 
the image without evanescent wave decay. The analytical description of this 
convolution is given in the article by using Equations (14–18). It is possible to use Mie 
theory for calculating the transfer function, but such calculations are very complicated 
and usually give only numerical results. We suggest, therefore, to use the transfer 
function as an empirical function, which can also be used for a description of the nano-
jet. 

Figure 1 gives only a geometric optics picture. The incident and transmitted 

angles 𝜃  and 𝜃  are derived by Snell’s law, respectively. There are different features 
in the microsphere system which can be explained by the geometric optics picture. For 
example, for a small microsphere index of refraction, we get a diverging beam with a 
virtual image, while if this index of refraction is large, then the beam is converging 

with a real image, like that in Figure 1. The angle 𝛽 between the beam converging to 

the nano-jet and the symmetric axis, and the distance 𝑟  from the point P to the 
symmetric axis, can be obtained by simple geometric calculations. But the 
microsphere super-resolutions can be calculated, only by using an analysis, similar to 



Characterization and Application of Nanomaterials 2024, 7(2), 6031.  

10 

that presented in the present article. Some features of the nano-jet may be described 
by using Bessel beams [26,27]. Figure 2 describes the transmittance of both 
evanescent waves and propagating waves from the object to the inner surface of the 
microsphere, where the fine structures of the object are included in the evanescent 
waves. 

5. Conclusion 

By using the Helmholtz equation, it was shown how the fine structures of the 
object are transferred by the evanescent EM radiation to the inner surface of the 
microsphere. The information on the fine structures of the object is included in the 
evanescent waves, and this information is transferred from the inner microsphere 
surface to the nano-jet by using the coupling between the EM fields and the EM modes 
of the microsphere, which might be calculated by Mie theory (e.g., [19]). This coupling 
leads to transmittance of both the evanescent waves and propagating waves to the 
nano-jet by using ‘transfer functions’ (see Equations (14–17) and reference [17]), 
which lead to a very small ‘effective wavelength’ for the evanescent waves, although 
they are not decaying any more. While the propagating waves are stronger than the 
evanescent waves, the modulation of the total EM radiation intensity in the nano-jet 
by the evanescent waves is the source of the super-resolutions of the microsphere 
system. The present analysis solves the controversy about whether the nano-jet or the 
evanescent waves are the source of the super-resolution effects, as the source of the 
super-resolution is composed of these two different mechanisms in two different parts 
of the microsphere system. 

Funding: The present study was supported by Technion-Mossad under grant No. 
2007256. 

Conflict of interest: The author declares no conflict of interest. 

References 

1. Lipson A, Lipson SG, Lipson H. Optical Physics. Cambridge University Press; 2010. doi: 10.1017/cbo9780511763120 

2. Zhang T, Yu H, Li P, et al. Microsphere-Based Super-Resolution Imaging for Visualized Nanomanipulation. ACS Applied 

Materials & Interfaces. 2020; 12(42): 48093-48100. doi: 10.1021/acsami.0c12126 

3. Xie Y, Cai D, Pan J, et al. Chalcogenide Microsphere‐Assisted Optical Super‐Resolution Imaging. Advanced Optical 

Materials. 2022; 10(6). doi: 10.1002/adom.202102269 

4. Li Y, Qiu C, Ji H, et al. Microsphere‐Aided Super‐Resolution Scanning Spectral and Photocurrent Microscopy for 

Optoelectronic Devices. Advanced Optical Materials. 2023; 11(16). doi: 10.1002/adom.202300172 

5. Wu G, Hong M. Optical Microsphere Nano-Imaging: Progress and Challenges. Engineering. 2024; 36: 102-123. doi: 

10.1016/j.eng.2023.10.019 

6. Zhou J, Lian Z, Zhou C, et al. Scanning microsphere array optical microscope for efficient and large area super-resolution 

imaging. Journal of Optics. 2020; 22(10): 105602. doi: 10.1088/2040-8986/abb17b 

7. Upreti N, Jin G, Rich J, et al. Advances in Microsphere-based Super-resolution Imaging. IEEE Reviews in Biomedical 

Engineering. 2024; 1-16. doi: 10.1109/rbme.2024.3355875 

8. Liu C, Ye A. Microsphere assisted optical super-resolution imaging with narrowband illumination. Optics Communications. 

2021; 485: 126658. doi: 10.1016/j.optcom.2020.126658 

9. Shang Q, Tang F, Yu L, et al. Super-Resolution Imaging with Patchy Microspheres. Photonics. 2021; 8(11): 513. doi: 

10.3390/photonics8110513 



Characterization and Application of Nanomaterials 2024, 7(2), 6031.  

11 

10. Jiang W, Wang J, Yang Y, et al. A Review of Microsphere Super-Resolution Imaging Techniques. Sensors. 2024; 24(8): 

2511. doi: 10.3390/s24082511 

11. Geints YE, E.K. Panina. Surface roughness influence on photonic nanojet parameters of dielectric microspheres. Computer 

Optics. 2023; 47(4): 559-566. doi: 10.18287/2412-6179-co-1280 

12. Gasparic V, Mayerhofer TG, Zopf D, et al. To generate a photonic nanojet outside a high refractive index microsphere 

illuminated by a Gaussian beam. Optics Letters. 2022; 47(10): 2534. doi: 10.1364/ol.459001 

13. Gasparic V, Ristic D, Mayerhofer TG, et al. Photonic nanojet of a Gaussian beam illuminated low refractive index 

microsphere in air: A comprehensive variation of parameters. Journal of Quantitative Spectroscopy and Radiative Transfer. 

2022; 282: 108121. doi: 10.1016/j.jqsrt.2022.108121 

14. Darafsheh A. Photonic nanojets and their applications. Journal of Physics: Photonics. 2021; 3(2): 022001. doi: 

10.1088/2515-7647/abdb05 

15. Mandal A, Tiwari P, Upputuri PK, et al. Characteristic parameters of photonic nanojets of single dielectric microspheres 

illuminated by focused broadband radiation. Scientific Reports. 2022; 12(1). doi: 10.1038/s41598-021-03610-3 

16. Kwon S, Park J, Kim K, et al. Microsphere-assisted, nanospot, non-destructive metrology for semiconductor devices. Light: 

Science & Applications. 2022; 11(1). doi: 10.1038/s41377-022-00720-z 

17. Goodman JW. Introduction to Fourier Optics. Roberts and Company Publishers; 2017. 

18. Lee S, Li L, Wang Z. Optical resonances in microsphere photonic nanojets. Journal of Optics. 2013; 16(1): 015704. doi: 
10.1088/2040-8978/16/1/015704 

19. Lecler S, Perrin S, Leong-Hoi A, et al. Photonic jet lens. Scientific Reports. 2019; 9(1). doi: 10.1038/s41598-019-41193-2 

20. Lee S, Li L, Ben-Aryeh Y, et al. Overcoming the diffraction limit induced by microsphere optical nanoscopy. Journal of 

Optics. 2013; 15(12): 125710. doi: 10.1088/2040-8978/15/12/125710 

21. Wang Z, Guo W, Li L, et al. Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope. Nature 

Communications. 2011; 2(1). doi: 10.1038/ncomms1211 

22. Darafsheh A, Walsh GF, Dal Negro L, et al. Optical super-resolution by high-index liquid-immersed microspheres. Applied 

Physics Letters. 2012; 101(14). doi: 10.1063/1.4757600 

23. Lai HSS, Wang F, Li Y, et al. Super-Resolution Real Imaging in Microsphere-Assisted Microscopy. PLOS ONE. 2016; 

11(10): e0165194. doi: 10.1371/journal.pone.0165194 

24. Lecler S, Takakura Y, Meyrueis P. Properties of a three-dimensional photonic jet. Optics Letters. 2005; 30(19): 2641. doi: 

10.1364/ol.30.002641 

25. Horiuchi N. Photonic nanojets. Nature Photonics. 2012; 6(3): 138-139. doi: 10.1038/nphoton.2012.43 

26. Durnin J. Exact solutions for nondiffracting beams I The scalar theory. Journal of the Optical Society of America A. 1987; 

4(4): 651. doi: 10.1364/josaa.4.000651 

27. Ben-Aryeh Y. Nano-jet related to Bessel beams and to super-resolutions in microsphere optical experiments. EPJ Techniques 

and Instrumentation. 2017; 4(1). doi: 10.1140/epjti/s40485-017-0038-5 

28. Ben-Aryeh Y. Superresolution observed from evanescent waves transmitted through nano-corrugated metallic films. Applied 

Physics B. 2012; 109(1): 165-170. doi: 10.1007/s00340-012-5193-4 

29. Ben-Aryeh Y. Increase of resolution by use of microspheres related to complex Snell’s law. Journal of the Optical Society of 

America A. 2016; 33(12): 2284. doi: 10.1364/josaa.33.002284 

30. Yan Y, Li L, Feng C, et al. Microsphere-Coupled Scanning Laser Confocal Nanoscope for Sub-Diffraction-Limited Imaging 

at 25 nm Lateral Resolution in the Visible Spectrum. ACS Nano. 2014; 8(2): 1809-1816. doi: 10.1021/nn406201q 

31. Wu G, Hong M. Optical nano-imaging via microsphere compound lenses working in non-contact mode. Optics Express. 

2021; 29(15): 23073. doi: 10.1364/oe.426231 

32. Maslov AV, Astratov VN. Origin of the super-resolution of microsphere-assisted imaging. Applied Physics Letters. 2024; 

124(6). doi: 10.1063/5.0188450 

33. Maslov AV, Astratov VN. Resolution and Reciprocity in Microspherical Nanoscopy: Point-Spread Function Versus 

Photonic Nanojets. Physical Review Applied. 2019; 11(6). doi: 10.1103/physrevapplied.11.064004 

34. Huszka G, Yang H, Gijs MAM. Microsphere-based super-resolution scanning optical microscope. Optics Express. 2017; 

25(13): 15079. doi: 10.1364/oe.25.015079 

35. Ebbesen TW, Lezec HJ, Ghaemi HF, et al. Extraordinary optical transmission through sub-wavelength hole arrays. Nature. 

1998; 391(6668): 667-669. doi: 10.1038/35570 



Characterization and Application of Nanomaterials 2024, 7(2), 6031.  

12 

36. Ghaemi HF, Thio T, Grupp DE, et al. Surface plasmons enhance optical transmission through subwavelength holes. Physical 

Review B. 1998; 58(11): 6779-6782. doi: 10.1103/physrevb.58.6779 

37. Ben-Aryeh Y. Transmission enhancement by conversion of evanescent waves into propagating waves. Applied Physics B. 

2008; 91(1): 157-165. doi: 10.1007/s00340-008-2945-2 

38. Ben-Aryeh Y. Tunneling of evanescent waves into propagating waves. Applied Physics B. 2006; 84(1-2): 121-124. doi: 

10.1007/s00340-006-2220-3 

39. Ben-Aryeh Y. Nonclassical high resolution optical effects produced by evanescent waves. Journal of Optics B: Quantum and 

Semiclassical Optics. 2003; 5(6): S553-S556. doi: 10.1088/1464-4266/5/6/002 

40. Ben-Aryeh Y. Super resolution of nanomaterials and quantum effects obtained by microspheres. Progress in Materials 

Science. 2019; 1(3): 1-21. doi:10.21926/rpm.1903003