## Abstract

Microsphere-based microscopy systems have garnered lots of recent interest, mainly due to their capacity in focusing light and imaging beyond the diffraction limit. In this paper, we present theoretical foundations for studying the optical performance of such systems by developing a complete theoretical model encompassing the aspects of illumination, sample interaction and imaging/collection. Using this model, we show that surface waves play a significant role in focusing and imaging with the microsphere. We also show that by designing a radially polarized convergent beam, we can focus to a spot smaller than the diffraction limit. By exploiting surface waves, we are able to resolve two dipoles spaced 98 nm apart in simulation using light at a wavelength of 402.292 nm. Using our model, we also explore the effect of beam geometry and polarization on optical resolution and focal spot size, showing that both geometry and polarization greatly affect the shape of the spot.

© 2015 Optical Society of America

## 1. Introduction

Optical imaging faces a fundamental limit because of diffraction, and thus sub-wavelength imaging has been a topic of great interest for the last several decades. Among well-established techniques, solid immersion has been known for improving spatial resolution [1] and increasing optical collection efficiency [2]. A bulk solid immersion lens (SIL) with a radius of 1.5 mm was used to resolve features with a resolution of 110 nm using 1.342 μm wavelength laser illumination [3, 4].

Instead of using the bulk SIL, a micro-SIL was predicted to focus light to a tighter spot [5, 6] and has been reported to be able to image features beyond the diffraction limit in free space [5, 7]. Furthermore, stimulated emission depletion (STED) microscopy at visible wavelengths using a micro-SIL was able to reach a resolution of 2.4 ± 0.3 nm, 1.8 times finer than without the micro-SIL [8]. This ability of the micro-SIL is simply attributed to near-field focusing and magnification effects [7].

Instead of using a micro-SIL, which is a truncated microsphere, a complete-microsphere-based visible light confocal microscope was developed, achieving a resolution of 25 nm [9]. The microsphere-based microscope is currently an active topic in both theory [10] and experiment [11]. In theory, researchers have posited many different explanations for the ability of the microsphere-based microscope to achieve resolution beyond the diffraction limit; the sub-wavelength focusing ability was explained as being similar to superoscillatory lenses in [9], while photonic nanojet formation and optical resonance were proposed as the cause in [12]. Meanwhile, [13] explained the phenomenon using virtual imaging and ray optics, while [14] provided an explanation using near-field optics and geometrical optics. However, super oscillations occur in a focusing system and not an imaging system—the high side-lobes commonly associated with super oscillations would deteriorate image quality in the latter. Hence, super oscillations alone would not be a complete explanation for super-resolved *imaging*. Likewise, photonic nanojets are also a focusing phenomenon, and may not be applicable to describing imaging, as was pointed out in [15]. Lastly, explanations based on ray optics and virtual imaging are not appropriate for a microsphere since ray optics has been known to be not applicable for near-field interactions.

Based on vectorial electromagnetic analysis, Duan et al. in [15] show that the microsphere-based microscope cannot resolve two points separated less than 100 nm apart and that the best resolution is obtained when optical resonance occurs inside the microsphere. However, since their simulations do not agree with previous results, the authors acquiesce that their theoretical treatment may be incomplete. In this paper, we propose using multipole and plane wave expansions [16] to rigorously model the entire microsphere-based microscope system, including both the focusing (illumination) and imaging sub-systems. Our theoretical model can be used for imaging a designed feature which mimicked a real object and hence provides an insightful understanding about the microsphere-based microscopy.

## 2. Theory

A complete model of a microscope requires a full understanding of three processes: illuminating the specimen with incident light, interaction of the specimen with the incident light, and collection of scattered light from the specimen [17]. In general, all three processes contribute to the resolution attainable by the microscope, and hence it is necessary to study all three as a complete system. Interaction between the incident light and the specimen can be solved numerically as was presented in [17]. Hence, we will focus on illumination and collection in this paper. First, let us rigorously derive the field at the output of a conventional imaging system; this result will serve as a foundation for the more complex derivation with regards to a microsphere-based microscopy.

#### 2.1. Imaging in a conventional microscope

Figure 1a) shows a simplified diagram of a conventional infinity-corrected imaging system. The source object *O*, fully contained inside sphere *S*, radiates electromagnetic energy in all directions. The portion of this radiation that falls within a cone with half-angle *α _{m}* <

*π*/2 is captured by the objective lens

*L*

_{1}and collimated into a beam, which is then focused by tube lens

*L*

_{2}onto a CCD detector to generate the image. For this derivation, we will assume that the source is emanating fully coherent light at a wavelength of

*λ*, and we will model this as a collection of electromagnetic multipole fields. Each of these electromagnetic multipole fields can be treated separately as shown in Fig. 1b), and we will derive its output by modeling each lens as a section of a Gaussian reference sphere (GRS).

First, we adopt terminologies from Devaney and Wolf [18] and expand the source radiation field as the following sum of electromagnetic multipole fields ${\mathbf{N}}_{\mathit{lm}}^{(1)}$ and ${\mathbf{M}}_{\mathit{lm}}^{(1)}$:

*r̂*,

*θ̂*, and

*ϕ̂*are the spherical unit vectors in spherical coordinates with the origin at the source,

*i*is the imaginary unit,

*k*= 2

*π/λ*is the wave number. Since there exist different definitions of the scalar spherical harmonics [22, 23], we adopt the definition of Arfken and Weber [22] and explicitly write it here for convenience:

*l*and order

*m*and is defined as follows:

*in the definition of the associated Legendre polynomial but include it in the definition of the scalar spherical harmonics. Consequently, our definition of the scalar spherical harmonics is the same as that in [20, 21]. For convenience, we can express Eq. (1) in the form of*

^{m}**E**(

*r̄*) =

*r̂E*+

_{r}*θ̂E*+

_{θ}*ϕ̂E*[3, 16] where

_{ϕ}*l*=

*L*.

Now the general expression in Eq. (5) can be simplified using a far-field approximation assuming we are operating in that regime. It is worth noting that Eq. (5) is valid everywhere outside the source and in general the electromagnetic wave associated with Eq. (5) does not behave as a spherical wave near the source. For this paper, we define the far-field regime to be one where the radiating field can be accurately approximated by spherical waves, i.e. when the radial coordinate *r* satisfies the following inequality:

*f*

_{1}is long enough) that it is located in the far-field regime, and thus a far-field approximation would be valid. Assuming Eq. (6) holds, for all

*l*≤

*L*we can make the following approximation: which in turn allows us to write

Since the objective lens *L*_{1} is located in the far-field, we can use
ray optics to solve for light refracting through it [24]. The vectorial coordinate system, which
we will use, is given in Fig. 1(b). At
the Gaussian reference sphere *GRS*_{1}, the incident
vectorial electric field
*E*_{α1}*α̂*_{1}
+
*E*_{β1}*β̂*_{1}
can be associated with a ray traveling from the centre of the
*GRS*_{1}. This ray is then refracted and becomes a
collimated ray, which is associated with an electric field
*E _{ρ}ρ̂* +

*E*. We can show that

_{φ}φ̂*L*

_{2}which, in this paper, is a conventional lens with a long focal length

*f*. The focused ray is associated with an electric field

*E*+

_{α}α̂*E*where

_{β}β̂*L*

_{2}can be estimated using the well-known angular spectrum representation:

*l*= 1). In this case, the formula in Eq. (16) is equivalent to the dyadic Green function of the imaging system. However, if the object is big compared to the wavelength, the radiation field will contain many multipole fields. In fact, Eq. (16) expresses the focal field as a sum of diffraction integrals applied to the constituent multipole fields from the source [25].

#### 2.2. Illumination and imaging in a microsphere-based microscope

Now let us consider a modification to the conventional microscope—inserting a microsphere between the object and the objective lens and adding a beam splitter for illumination, as shown in Fig. 2. We will refer to this new optical recipe as the microsphere-based microscope. Incident light reflects off the beam splitter and illuminates the specimen after passing through the objective lens and microsphere. This light then interacts with the specimen, which in turn generates some outgoing radiation, represented as a collection of dipole fields by the numerical method [17]. This outgoing light passes through the microsphere and is then imaged by the two lenses, similar to the conventional microscope. Since our derivation for a conventional microscope is general enough for *any* coherent source located inside some finitely large sphere *S*, a model for the imaging aspect of the microscope can be derived by constructing a multipole-based expression for the field generated by a dipole behind the microsphere. As was mentioned before, understanding the microscope requires understanding the illumination aspect as well, so we will first start with a derivation for the focusing field before presenting the results for the imaging aspect.

### 2.2.1. Illumination through a microsphere

A converging wave from a positive lens is incident on a microsphere, as shown in Fig. 3(a), where *k̄* is the wave vector, *α _{m}* is the half-angle of the incident cone of light,

*R*is the radius of the microsphere centered at point

*O*,

*ρ*is the radial axis of a cylindrical coordinate system centered at

*O*,

*z*is the longitudinal axis as well as optical axis,

*F*and

*f*are the focus and focal length of the lens, respectively, and

*d*is the distance between

*F*and

*O*.

It is a nontrivial task to determine the multipole expansion coefficients for a converging wave.
Given an incident plane wave approaching the lens (which we will assume to
be aplanatic for this derivation), we can express the electric field on the
GRS as **E*** ^{GRS}* =

*E*+

_{α}α̂*E*and the multipole coefficients of the focused beam can be estimated by matching

_{β}β̂**E**

*with Eq. (5) as shown in [3, 25, 26, 27]:*

^{GRS}*h*

^{(1)}, the spherical Hankel function of the first kind, are replaced with the spherical Hankel function of the second kind,

*h*

^{(2)}) to obtain an expression for the vectorial field of a converging wave. Note that this expression would be based on a spherical coordinate system centered on

*F*in Fig. 3(a). In practice, the focus of the objective lens and the center of the sphere are not coincident (i.e.

*d*> 0), and hence the coordinate system for this converging wave field would

*not*be centered on the microsphere. Thus, to use previous results for scattering and focusing by a microsphere, we would need to translate the coordinate system for the converging wave’s vectorial field so that the origin is at the center of the microsphere. We do this by employing the translational addition theorem [28] and obtain the following multipole expansion coefficients [3]:

*l′*is sufficiently large (i.e. ${l}^{\prime}>kd+1.8\eta \sqrt[3]{kd}$ for a desired accuracy of 10

^{−η}). The multipole expansion coefficients in Eq. (18) can be substituted into a modified version of Eq. (5) (where instances of ${h}_{l}^{(1)}$ are replaced with the spherical Bessel function

*j*) to obtain an expression for the translated vectorial field.

_{l}Using this translated field, we can now derive expressions for multipole expansion coefficients corresponding to the vectorial field both inside and outside the microsphere using the boundary condition [3]. Inside the microsphere, we obtain the following multipole expansion coefficients:

*h*

^{(1)}are replaced with the spherical Bessel function of the first kind and instances of

*k*are replaced with

*k*). In Eq. (19),

_{s}*ε*and

*ε*are the permittivity of the surrounding material and the microsphere respectively,

_{s}*k*and

*k*are the wave numbers for the surrounding material and the microsphere respectively, and the internal Mie scattering coefficients are given by

_{s}*c*and

_{l′}*d*. For the scattering field, we have the following expression for the multipole expansion coefficients

_{l′}*a*and

_{l′}*b*are the external Mie scattering coefficients. We followed the derivation procedure in [29, 30] and derived our scattering coefficients in [27, 3], we list them here for convenience:

_{l′}*Ĵ*(

_{l}*x*) =

*xj*(

_{l}*x*) is Riccati-Bessel function; ${\widehat{H}}_{l}^{(1)}(x)=x{h}_{l}^{(1)}(x)$ and ${\widehat{H}}_{l}^{2}(x)=x{h}_{l}^{(2)}(x)$ are Riccati-Hankel functions.

For completeness, we would like to include results for a plane wave (shown in Fig. 3(b)) for comparison. An incident linearly
polarized plane wave **E*** ^{inc}* =

*x̂E*

_{0}

*e*

^{ikr cosθ}can be expressed in terms of the electromagnetic multipole fields with the following multipole expansion coefficients:

### 2.2.2. Far-field imaging through a microsphere

As mentioned before, the light radiating back from an illuminated object can be modeled by a number of discrete dipoles, and thus now we will present a rigorous expression for the electromagnetic field at the CCD due to an axis-aligned dipole (along *x̂*, *ŷ* or *ẑ*) placed behind the microsphere, i.e. we derive the dyadic Green function for the collection subsystem. The output field due to an arbitrary dipole can then be calculated using the dyadic Green function with the projected length of the dipole moment along the three axes.

Consider a dipole located next to but outside of a microsphere; it will radiate what we will call an incident field, and the microsphere interacts with this field, creating a scattering field. What arrives at lens *L*_{1} is a sum of these two fields, and we can derive multipole coefficients for the total field resulting from a dipole along each of the basic directions *x̂*, *ŷ* and *ẑ*. The derivation, not explicitly shown here, is similar to the previous section—derive an expression for the incident field centered on the microsphere and apply the boundary condition [3]. The resulting coefficients are as follows:

- A Hertzian electric dipole with the current dipole moment Il pointing in the
*x̂*direction:$$\begin{array}{l}{p}_{\mathit{El}}^{m}=\frac{\omega \mu Il}{2\sqrt{6\pi}}\left({a}_{l}\left[{A}_{\mathit{lm}}^{1,1}-{A}_{\mathit{lm}}^{1,-1}\right]+\left[{C}_{\mathit{lm}}^{1,1}-{C}_{\mathit{lm}}^{1,-1}\right]\right),\\ {p}_{\mathit{Ml}}^{m}=-i\frac{\omega \mu Il}{2\sqrt{6\pi}}\left({b}_{l}\left[{B}_{\mathit{lm}}^{1,1}-{B}_{\mathit{lm}}^{1,-1}\right]+\left[{D}_{\mathit{lm}}^{1,1}-{D}_{\mathit{lm}}^{1,-1}\right]\right),\end{array}$$where the translational coefficients (*A*,*B*,*C*and*D*) are calculated via the method in [28]. - A Hertzian electric dipole with the current dipole moment Il pointing in the
*ŷ*direction:$$\begin{array}{l}{p}_{\mathit{El}}^{m}=\frac{i\omega \mu Il}{2\sqrt{6\pi}}\left({a}_{l}\left[{A}_{\mathit{lm}}^{1,1}+{A}_{\mathit{lm}}^{1,-1}\right]+\left[{C}_{\mathit{lm}}^{1,1}+{C}_{\mathit{lm}}^{1,-1}\right]\right),\\ {p}_{\mathit{Ml}}^{m}=\frac{\omega \mu Il}{2\sqrt{6\pi}}\left({b}_{l}\left[{B}_{\mathit{lm}}^{1,1}+{B}_{\mathit{lm}}^{1,-1}\right]+\left[{D}_{\mathit{lm}}^{1,1}+{D}_{\mathit{lm}}^{1,-1}\right]\right).\end{array}$$ - A Hertzian electric dipole with the current dipole moment Il pointing in the
*ẑ*direction:$${p}_{\mathit{El}}^{m}=\frac{\omega \mu Il}{2\sqrt{3\pi}}\left({a}_{l}{A}_{\mathit{lm}}^{1,0}+{C}_{\mathit{lm}}^{1,0}\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{p}_{\mathit{Ml}}^{m}=-i\frac{\omega \mu Il}{2\sqrt{3\pi}}\left({b}_{l}{B}_{\mathit{lm}}^{1,0}+{D}_{\mathit{lm}}^{1,0}\right).$$

We note that Eqs. (23), (24), and (25) are the multipole expansion
coefficients of the corresponding total fields expressed in a coordinate
system where the origin is at the center (*O′*) of
the microsphere, as shown in Fig. 4.
However, from the point of view of the far field, the dipole appears to be
at position *O* due to the presence of the microsphere. Thus,
for aberration-free imaging, the lens *L*_{1} has to
be positioned such that *O* is at the front focal plane of
the lens, not *O′*. To properly analyze this
situation, we would have to shift the coordinate system’s origin
from *O′* to *O*, i.e. substituting
Eqs. (23), (24), and (25) into Eq. (18). Alternatively, we can
approximate the multipole coefficients in the coordinate system
*O* by employing the fact that in practical cases, the
distance between *O* and *O′* is much
less than the focal length of the lens *L*_{1}
(*d* ≪ *f*_{1}). This
means that the electromagnetic wave on the GRS can be approximated as a
spherical wave in both *O′* and
*O*-coordinate systems. At a far-field position
*P* (*r*, *θ*,
*ϕ*), we can approximate
*θ* ≈
*θ′*, *ϕ*
=*ϕ′*, *r*
= *r′* +*d*
cos*α*_{1}. For wave vector
*k̄*_{1} (*k*,
*α*_{1},
*β*_{1}) at *P*, we have
*k* = *k′*,
*α*_{1} ≈
*α′*,
*β*_{1} =
*β′*, and
*e ^{ikr′}/r′* ≈

*e*

^{−ikd cosα1}

*e*. Using these approximations and Eq. (9), we can express the multipole coefficients ${\overline{p}}_{\mathit{El}}^{m}$ and ${\overline{p}}_{\mathit{Ml}}^{m}$ representing the field in the

^{ikr}/r*O*-coordinate system in terms of those in the

*O′*-coordinate system ( ${p}_{\mathit{El}}^{m}$and ${p}_{\mathit{El}}^{m}$ from Eqs. (23), (24), and (25)):

## 3. Discussion

#### 3.1. Resonance in the microsphere

When light is scattered by a sphere, the total energy lost from the incident wave can be determined by using the optical cross-section theorem [29], which relates the rate at which energy is lost from the incident field to the amplitude of the scattered field in the direction of incidence (the forward direction). The mathematical formula of the theorem expresses the extinction cross-section in terms of the forward scattering amplitude, and it was first derived by van de Hulst. The extinction cross section for scattering by a microsphere contains a series of spikes as a function of wavelength [31]. It has been shown that these spikes manifest optical resonances [31], and they have been investigated and exploited in numerous applications, such as biosensing [34] and environmental monitoring [35]. Each of the spikes corresponds to one scattering coefficient (*a _{l}* or

*b*), i.e. the optical resonance is due to one particular partial wave ( ${N}_{l}^{m}$ or ${M}_{l}^{m}$) temporarily trapped inside the microsphere. More intuitively, these resonances can be shown to be analogous to quantum-mechanical shape resonances in which the electromagnetic energy is temporarily trapped near the surface of the microsphere in a dielectric potential well [36].

_{l}Let us consider the rays associated with a so-called impact parameter *D*
traveling close to the edge of the microsphere, as shown in Fig. 3(b) [31]. These rays tunnel through the centrifugal barrier to
the surface of the microsphere and are then totally internally reflected
multiple times inside the near-surface annulus (*D/n* ≤
*r* ≤ *R*), each time only losing a
small amount of energy [31]. They are usually referred to as surface waves, which in
turn account for the optical resonances [37]. Resonance occurs when the ray associated with one of
the partial waves, after traveling around the microsphere, is in phase with
itself when it has traveled to the point where it originally entered the
microsphere—there is a standing wave along the surface of the
microsphere. Nussenzveig defines the multipole fields associated with these rays
as being in the edge domain; these fields are associated with multipole orders
*l*_{−} < *l*
< *l*_{+}, where ${l}_{\pm}\approx kR\pm {\eta}_{0}\sqrt[3]{kR}$ and *η*_{0} is a
constant of order unity (empirically, *η*_{0}
> 3) [38]. Since
optical resonance is completely due to surface waves, we will consider only edge
domain multipole fields in our analysis.

Let us consider a microsphere with radius *R* = 2.37 μm and a
refractive index *n _{s}* = 1.46 to match the
parameters in [13, 15]. With these parameters and a
choice of

*η*

_{0}= 4,

*λ*= 400 nm we obtain:

*l*= 40 and

*l*= 43. We plot the Mie scattering coefficients

*c*

_{40},

*d*

_{40},

*c*

_{43}, and

*d*

_{43}as a function of the wavelength

*λ*for 0.35 μm <

*λ*< 0.5 μm in Fig. 5(a). We observe from the plot that there are several visible peaks for each scattering coefficient (in fact, there are an infinite number of such peaks over the electromagnetic spectrum); the width of the peaks decreases as wavelength increases. The largest peaks for each of the scattering coefficients are essentially spikes. We do not consider wavelengths shorter than 0.35 μm because the peaks in the scattering coefficients start widening so much that the maxima are too low, resulting in negligible resonance. We also do not consider wavelengths longer than 0.5 μm because the peaks are so narrow that they are not applicable in practice.

In Fig. 5(b), we show the magnitude of the Mie scattering
coefficients for 1 ≤ *l* ≤ 50 at
*λ* = 401.6345 nm (the location of the middle
*d*_{43} peak in Fig.
5(a)) as well as just
|*d _{l}*| at

*λ*= 403.07 nm (the off-resonant case). We observe that the resonant scattering coefficient

*d*

_{43}= −21.3

*i*is much higher in magnitude than the other coefficients. This high value of the scattering coefficient represents the constructive interference of the partial wave

*l*= 43 inside the microsphere. It has been known that Mie scattering does not explain the effect of the multiple reflections inside the microsphere [27].

To further our understanding of the multiple reflections occurring inside the microsphere, it is
necessary to adopt the Debye series, since standard Mie theory considers the
entire process as a whole [25,
31]. The Debye series
expresses each of the Mie scattering coefficients
(*a _{l}*,

*b*,

_{l}*c*,

_{l}*d*) in terms of an infinite summation. Each term represents one particular scattering event occurring at the microsphere’s surface. In the case of resonance, we need to include several dozen terms in order for the Debye series to converge to the Mie scattering coefficient. For off-resonance, we only need to include several terms for convergence [31]. The constructive contribution of many internal reflections physically explain the high magnitude of the resonant scattering coefficient,

_{l}*d*

_{43}in this particular example.

#### 3.2. Microsphere focusing of a converging wave

Using our derivation for illumination through a microsphere, we will now show by example that a microsphere can focus light from a converging wave into a tight three-dimensional spot; such focal spots are important to many applications, e.g. [39]. We consider a converging wave, the so-called axial dipole wave [3], formed by an aplanatic lens focusing a radially polarized wave. Substituting the description of this radially polarized wave (*E _{α}* = sin

*α*,

*E*= 0) into Eq. (17) yields the following multipole expansion coefficients:

_{β}Using the derived results in Section 2.2.1 with parameters *λ* = 401.6345 nm, *R* = 2.37 μm, *d* = 3*R*, and *α _{m}* = arcsin(

*R/d*), we can compute the electric intensity of the illuminated microsphere, shown in Fig. 6. It should be noted that

*λ*= 401.6345 nm is the resonant wavelength for the TE mode ( ${d}_{l}{p}_{\mathit{Ml}}^{m}{M}_{l}^{m}$ with

*l*= 43) as indicated in Fig. 5. However, a converging beam with radial polarization is completely described by the TM modes, i.e. ${p}_{\mathit{Ml}}^{m}=0$ which explains that we do not observe any resonance in Fig. 6.

Figure 6 shows that the field is highly confined to the
surface, especially along the optical axis. The focal spot has a FWHM of 234 nm
(≈ *λ*/1.72) as indicated by the green curve in
Fig. 7(a). It has been known that
focusing a radially polarized beam with a high numerical aperture (NA) aplanatic
lens results in a tighter focal spot than that of focusing a linearly polarized
beam [32, 33]. But a radially polarized beam with a low
NA may result in a focal spot of a donut-shaped spot. In our case, the incident
beam is a low NA (= 1/3) but the focus is tight. This tight spot is
attributed to the microsphere which guides light to converge to the focus with a
semi-angle of 90° as observed in Fig.
6, i.e. the effective NA is 1. To make the focal spot even tighter,
we can increase the contribution of the surface wave to the focal spot by
blocking the center portion of the incident beam (i.e. block all of the light
within a cone with semi-angle *α*_{0}). Using
this annular focusing technique, we can reduce the FWHM to 171 nm, represented
by the blue curve in Fig. 7(a).

An alternative approach would be to move the microsphere toward the lens, i.e. increasing
*d*. At *d* = 7*R*, we
obtain a FWHM of 174 nm as represented by the pink curve in Fig. 7(a). We can further tighten the spot
transversally by tuning the wavelength so that resonance occurs. Figure 8 shows two such wavelengths: 397.44
nm and 439.44 nm. Resonance is characterized by a radial mode number
*p* (*p* peaks along the radial direction) and
an angular mode number *l*. We excite the microsphere with
(*l* = 43, *p* = 2) and
(*l* = 43, *p* = 1) in Fig. 8(a) and Fig. 8(b), respectively. The
resulting transversal cross section for the 439.44 nm case is represented by the
red curve in Fig. 7a), with a FWHM of 138
nm. The transversal cross section for the 397.44 nm is nearly identical and thus
we have omitted it from Fig. 7(a). One
important property of focusing a radially polarized beam is that the side-lobes
of the electric intensity distribution are relatively low compared to those of
linear and circular polarizations. The low side-lobes are important in imaging
applications since high side-lobes may cause distortion and poor contrast
[15].

We note that this FWHM is close to *λ*/(2*n _{s}*), where

*n*is the index of refraction for the interior of the microsphere, although it is not directly dependent on the wavelength;

_{s}*λ*/(2

*n*) is 136 nm and 150.5 nm for

_{s}*λ*= 397.44 nm and

*λ*= 439.44 nm, respectively, although the FWHM remained almost the same at 138 nm in both cases. In fact, the FWHM was smaller than

*λ*/(2

*n*) in the

_{s}*λ*= 439.44 nm case. In this resonant case, the effective NA of the incident beam is approximately equal to

*n*which is attributed to the dominant contribution of the surface wave.

_{s}#### 3.3. Convergent versus planar incident field

For scanning or confocal microsphere-based microscopy, the illumination usually involves a converging beam, although analysis of such systems have sometimes been conducted using a plane wave illumination model, e.g. in [9]. However, the beam geometry and polarization play a significant role in determining the shape of the focal spot, as we will now discuss.

For *λ* = 439.44 nm, an incident converging wave with radial
polarization results in a tight spot as observed in Fig. 8(b). However, if we switch to a
circularly-polarized plane wave at the same wavelength, the result is a wider
donut-shaped spot, as demonstrated by the double-lobed pattern in Fig. 9(a). To obtain a tight spot on the
microsphere’s surface with circular or linear polarization, we may
choose to excite a TE mode in the microsphere instead of the TM mode. For
example, if we excite the microsphere at a wavelength of 445.5127 nm, this will
induce resonance for the mode *l* = 43,
*p* = 1
(|*d*_{43}| = 280.258), as shown
in Fig. 9(b). Cross sections of the
electric intensities at the focus along the *x̂*
direction are shown in Fig. 7(b) for
incident circularly polarized plane waves as well as plane waves linearly
polarized in the *x̂* and *ŷ*
directions. The FWHM along the *x̂* direction for
circular polarization is 124.8 nm, while for linear polarization, the FWHMs are
172.2 nm and 98.3 nm for polarization along *x̂* and
*ŷ*, respectively. The asymmetry in the FWHMs for
linear polarization is similar to the asymmetry when focusing linearly polarized
light using an aplanatic lens [24].

To study the focusing property of the microsphere off-resonance, we choose the nonresonant
wavelength of *λ* = 403.07 nm. Figure 9(c) shows a photonic jet formed after the
microsphere for the case of off-resonance [40]. The electric intensity along the
transversal direction at the maximum intensity of the photonic jet is
represented by the solid pink curve in Fig.
7(b), and it has a FWHM of 268.4 nm (≈
*λ*/1.5). Using the Debye series, we can show that
the formation of the photonic jet is mainly due to light refraction at the front
and back interfaces, i.e. we can ignore the contribution of the surface wave to
the formation. In [41],
the strong focusing ability of the microsphere is explained as a consequence of
the interaction between the incident wave and the microsphere which enhances the
high spatial frequency components in the angular spectrum content of the
scattering field.

Looking at Fig. 7(b), it is obvious that even with an incident plane wave, the shape of the focal spot depends greatly on the incoming polarization. A comparison with the convergent wave results in Fig. 7(a) also shows that geometry plays an important role as well. Based on these observations, it would appear that Yan et al.’s superoscillatory explanation for their 25 nm resolution is incomplete [9], since an analysis of plane wave illumination was used to explain results from an experiment that uses convergent wave illumination. It is not clear what state of polarization the aforementioned convergent wave is in, but if it were radially polarized, then it would form a tight focal spot near the surface of the microsphere without significant side lobes, as shown in Fig. 7a).

#### 3.4. Far-field imaging

Now that we have examined the illumination aspect of the microsphere-based microscope with simulation examples, let us also do the same for the imaging/collection aspect as well. However, recall earlier that we assumed that the objective lens operates in the far-field regime, i.e. Eq. (6) holds. Thus, before we present imaging simulation results, we would first like to present some numerical examples supporting this far-field assumption.

Since any object can be modeled as a sum of dipoles, we will examine the case of a single dipole placed at a distance *d* from the center of the microsphere with specific wavelengths; any conclusions can be extended to general objects because we are in effect investigating the dyadic Green function. Let us investigate the magnitude of the multipole expansion coefficients. Since we are only interested in objects *outside* the microsphere (i.e. *d* ≥ *R*), the truncation multipole number *L* will only depend on the translational distance *d*.

Figure 10 shows the magnitudes of the first 45 multipole expansion coefficients for a *x̂*-dipole on the surface of the microsphere (*d* = *R* and *R* = 2.37 μm) at two different wavelengths—450 nm for an off-resonant case and 445.5127 nm for a resonant case. As evident from the magnitude plots, we can accurately model the total field even if we ignore all coefficients with *l* > 40 for the off-resonant case and *l* > 45 for the resonant case. This is due to the fact that *kd* ≈ 33 for both cases, and if we apply the rule of thumb from Section 2.2.1, we get that *L* = *kd* + 3(*kd*)^{(1/3)} ≈ 43.

In practice, a microscope usually uses an objective lens with a focal length in millimeters. For
example, the Olympus MDPlan 80×/0.9 objective from [13] has an effective focal length
of *f*_{1} = 2.25mm, and thus
*kf*_{1} ≈ 3 × 10^{4} for
both wavelengths. This is more than an order of magnitude greater than
*L*(*L* + 1)/2 ≈ 946, and
hence the far-field condition in Eq.
(6) is satisfied, an assumption we made for our theoretical
model.

Now that we have validated the far-field condition, we will discuss the imaging results shown in
Fig. 11 for a pair of mutually
incoherent dipoles 109 nm apart in three different situations—a) without
a microsphere, b) behind a microsphere in off-resonance, and c) behind a
microsphere in resonance. Imaging without a microsphere, as shown in Fig. 11(a), demonstrates that a
conventional microscope is unable to resolve the two dipoles due to the
diffraction limit. Although using a microsphere without resonance imparts
magnification on the sample, as shown in Fig.
11(b), we still cannot resolve the two dipoles. It is only with
microsphere in resonance (i.e. at *λ* = 445.5127
nm) that we can resolve the two, as seen in Fig.
11(c). The resonant condition requires a laser with a high
monochromaticity. We adopt Johnson’s derivation in [36] and derive an analytical
formula for estimating the laser linewidth Δ*λ*
for efficiently exciting a resonant TE mode as follows:

*n*= 1.46,

_{s}*λ*

_{0}= 0.4455127 μm,

*R*= 2.37 μm,

*k*

_{0}×

*R*= (2

*π/λ*

_{0}) × (2.37) = 33.4247, and

*l*= 43 into Eq. (29) and obtain Δ

*λ*≈ 6.5 × 10

^{−4}nm. This accuracy can be experimentally produced –for an example– by using optimized solid-state multiple-prism grating laser oscillators which is shown to produce a laser with a laser linewidth Δ

*λ*≈ 4 × 10

^{−4}

*nm*(Δ

*f*= 350

*MHz*) [42].

We also note that the microsphere in resonance transfers energy to the far-field region more efficiently. The maximum intensity in Fig. 11c) is much larger than that in Fig. 11(b). This is due to the evanescent wave on the microsphere being converted into a propagating wave more efficiently. This wave conversion has been previously exploited in microdroplet lasing, in which the microsphere’s surface plays the role of resonance cavity mirrors in a standard laser [31].

Obviously, resolution and conversion efficiency depend on the resonant wavelength, i.e. the
resonant mode. For example, at a resonant wavelength of
*λ* = 402.2920 nm, we can resolve two dipoles
placed 98 nm apart. Furthermore, the resonant scattering coefficient
|*d*_{48}| = 628.6162 is
slightly more than twice that of the resonant scattering coefficient from the
previous case, making the maximum intensity more than 5 times higher compared to
that in Fig. 11(c).

It should be noted that the microsphere induces a magnification of 2.93 in the off-resonant case and a magnification of 1.83 in the resonant case. Ray optics gives an approximate magnification of *n _{s}*/(2 −

*n*) ≈ 2.70 [13], quite close to that of the non-resonant case. This accurate approximation is due to the fact that refraction plays a dominant role in off-resonance, while that is not the case in resonance. However, even with a higher magnification of 2.93, the off-resonant case does not yield a higher resolution than that of the resonant case [15]. Given these results, it is evident that coupling from the evanescent field onto a propagating field plays a pivotal role in resolution enhancement with microsphere-based microscopy.

_{s}#### 3.5. Future directions

As mentioned before, a complete model of the microsphere-based microscope comprises three components: focusing of the incident light, interaction of the incident light with the sample, and imaging of the scattered light. As a next step, we would like to apply the rigorous focusing and imaging models developed in this paper along with the numerical model for sample-illumination interaction in [17] to properly simulate a microsphere-microscopy system imaging specific specimens in various modalities, such as wide-field, scanning and confocal. In doing so, we would be able to validate our theory and provide a physically rigorous explanation for previously obtained experimental results, such as those from [9, 13]. We would also be able to use our complete model to further explore the resolution limits of microsphere-based microscopy in a principled fashion.

## Acknowledgments

We thank Dr. Zhengyun Zhang for proofreading our manuscript. This research was supported by the National Research Foundation Singapore through the Singapore-MIT Alliance for Research and Technology’s Center for Environmental Sensing and Modeling interdisciplinary research program. This research was also supported by the National Research Foundation, Prime Ministers Office, Singapore under its Competitive Research Programme (CRP Award No. NRF-CRP10-2012-04).

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