Focusing and imaging in microsphere-based microscopy.

Focusing and imaging in microsphere-based microscopy ∗

Thanh Xuan Hoang,1, Yubo Duan,1 Xudong Chen,2 and George Barbastathis1,3 1 Singapore–MIT

Alliance for Research and Technology (SMART) Centre, Singapore 138602, Singapore 2 Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Singapore 3 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA ∗ [email protected]

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 OCIS codes: (180.0180) Microscopy; (180.4243) Near-field microscopy; (220.3630) Lenses; (350.3950) Micro-optics; (350.5730) Resolution.

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Received 22 Jan 2015; revised 5 Apr 2015; accepted 16 Apr 2015; published 1 May 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.012337 | OPTICS EXPRESS 12337

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6347–6349 (1999).

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-microspherebased 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 subwavelength 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 microspherebased 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.

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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 αm < π /2 is captured by the objective lens L1 and collimated into a beam, which is then focused by tube lens L2 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 (1) (1) field as the following sum of electromagnetic multipole fields Nlm and Mlm : ∞

l

∑ ∑

E(¯r) =

(1)

(1)

[pm r ) + pm r)], El Nlm (¯ Ml Mlm (¯

(1)

l=1 m=−l m where pm El and pMl are multipole expansion coefficients [18, 19] (are also called as multipole moments in [18, 20] and are related to beam shape coefficients in [21]), These coefficients are determined from the source distribution or boundary conditions. The electromagnetic multipole (1) fields are defined in terms of the conventional spherical Hankel function of the first kind hl (kr) and the scalar spherical harmonics Ylm (θ , φ ) as follows: (1)

(1)

Nlm (¯r) = ∇ × ∇ × [¯rhl (kr)Ylm (θ , φ )]

∂ 1 d l(l + 1) (1) (1) hl (kr)Ylm (θ , φ ) + θˆ (rhl (kr)) Ylm (θ , φ ) r r dr ∂θ im 1 d (1) (rh (kr))Ylm (θ , φ ), + φˆ sin θ r dr l

= rˆ

(1)

(1)

Mlm (¯r) = ik∇ × [¯rhl (kr)Ylm (θ , φ )] ∂ m m m (1) ˆ ˆ Y (θ , φ ) + i φ Y (θ , φ ) , = −khl (kr) θ sin θ l ∂θ l

(2)

where rˆ, θˆ , and φˆ are the spherical unit vectors in spherical coordinates with the origin at the source, i is the imaginary unit, k = 2π /λ is the wave number. Since there exist different

Fig. 1. Imaging with Conventional Lenses: a) Physical Setup and b) System Modeling.

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definitions of the scalar spherical harmonics [22, 23], we adopt the definition of Arfken and Weber [22] and explicitly write it here for convenience: 1 2l + 1 (l − m)! 2 m = (−1) Pl (cos θ ) exp(imφ ) 4π (l + m)! = clm Plm (cos θ ) exp(imφ ),

Ylm (θ , φ )

m

(3)

where Plm (cos θ ) is the associated Legendre polynomial of degree l and order m and is defined as follows: 1 d l+m (4) Plm (x) = l (1 − x2 )m/2 l+m (x2 − 1)l . 2 l! dx One can notice that we omit the Condon-Shortley phase (−1)m 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 [21, 20]. For convenience, we can express Eq. (1) in the form of E(¯r) = rˆEr + θˆ Eθ + φˆ Eφ [16, 3] where l(l + 1) (1) (1) m clm pm El [hl−1 (kr) + hl+1 (kr)]Pl (cos θ ) exp(imφ ), l=1 m=−l 2l + 1 (1) (1) ∞ l hl+1 (kr) dPlm (cos θ ) l(l + 1) m hl−1 (kr) clm pEl − Eθ = k ∑ ∑ l l +1 dθ l=1 m=−l 2l + 1 Plm (cos θ ) 2l + 1 m (1) p h (kr) exp(imφ ), −m l(l + 1) Ml l sin θ (1) (1) ∞ l hl+1 (kr) Plm (cos θ ) l(l + 1) m hl−1 (kr) clm mpEl − Eφ = ik ∑ ∑ l l +1 sin θ l=1 m=−l 2l + 1 m dPl (cos θ ) 2l + 1 m (1) p h (kr) exp(imφ ). − l(l + 1) Ml l dθ ∞

Er = k ∑

l

∑

(5)

It is reasonable to assume in practice that the source radiates a limited number of electromagnetic multipole fields, and hence the infinite summations in Eq. (5) can be terminated at 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: kr L(L + 1)/2

(6)

Note that in most conventional microscopes, the objective is placed far enough from the object (i.e. the focal length f1 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: eikr (1) (7) hl (kr) ≈ (−i)l+1 kr which in turn allows us to write (1) (1) ikr hl+1 (kr) (1) (1) l(l+1) hl−1 (kr) (8) hl−1 (kr) + hl+1 (kr) ≈ 0 and 2l+1 ≈ (−i)l ekr . − l+1 l #233013 - $15.00 USD © 2015 OSA


Substituting Eqs. (7) and (8) into Eq. (5), we obtain Er ≈ 0,

Eθ ≈

eikr L l ∑ ∑ Klm (θ ) exp(imφ ), r l=1 m=−l

where

Klm (θ ) = (−i)l clm

Eφ ≈

dPlm (cos θ ) pm El

eikr L l ∑ ∑ Gml (θ ) exp(imφ ), r l=1 m=−l

(9)

m m Pl (cos θ ) , + impMl

dθ sin θ m (cos θ ) m P dP m l m l (cos θ ) mpEl . + ipMl sin θ dθ

l−1 Gm clm l (θ ) = (−i)

Equation (9) represents the fact that an electromagnetic wave behaves as a spherical wave at a position far away from the source. Since the objective lens L1 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 GRS1 , the incident vectorial electric field Eα1 αˆ 1 + Eβ1 βˆ1 can be associated with a ray traveling from the centre of the GRS1 . This ray is then refracted and becomes a collimated ray, which is associated with an electric field Eρ ρˆ + Eϕ ϕˆ . We can show that Eα1 = Eθ ≈ Eβ1 = Eφ ≈

eik f1 f1 eik f1 f1

L

l

∑ ∑

Klm (α1 ) exp(imβ1 ),

(10)

Gm l (α1 ) exp(imβ1 ),

(11)

l=1 m=−l L

l

∑ ∑

l=1 m=−l

Eα Eρ = √ 1 , cos α1

Eβ Eϕ = √ 1 . cos α1

(12)

The collimated beam is then focused by the tube lens L2 which, in this paper, is a conventional lens with a long focal length f . The focused ray is associated with an electric field Eα αˆ + Eβ βˆ where f1 sin α1 , β = β1 , f

√ cos α Eα = Eρ cos α = Eα1 , cos α1 sin α =

(13) √ Eβ = Eϕ cos α = Eβ1

cos α . cos α1

(14)

The focal field at the focus of lens L2 can be estimated using the well-known angular spectrum respresentation: ⎡ ⎛ ⎞ αm 2π cos β cos α ik f ik f e sin α d α d β ⎣Eα ⎝ sin β cos α ⎠ E(x, y, z) = − 2π 0 0 − sin α ⎞⎤ ⎛ − sin β (15) +Eβ ⎝ cos β ⎠⎦ eik[ρ sin α cos(β −φ )+z cos α ] . 0 Substituting Eq. (14) into Eq. (15), we can write the focal field as ⎛ m−1 ⎞ Il (¯r) + Ilm+1 (¯r) l L E(x, y, z) = ∑ ∑ ⎝ i(Ilm−1 (¯r) − Ilm+1 (¯r)) ⎠ , l=1 m=−l −2Ilm (¯r) #233013 - $15.00 USD © 2015 OSA

(16)


where

ik f eik f m+1 i(m+1)φ αm i e Jm+1 (kρ sin α )[Ulm (α ) cos α + iVlm (α )]eikz cos α sin α dα , 2 0 ik f eik f m−1 i(m−1)φ αm i Ilm−1 (¯r) = − e Jm−1 (kρ sin α )[Ulm (α ) cos α − iVlm (α )]eikz cos α sin α dα , 2 0 ik f eik f m imφ αm i e Ilm (¯r) = − Jm (kρ sin α )Ulm (α ) sin α eikz cos α sin α dα , 2 0

eik f1 m eik f1 m cos α cos α m m Ul (α ) = K (α1 ) , Vl (α ) = G (α1 ) . f1 l cos α1 f1 l cos α1 Ilm+1 (¯r) = −

Equation (16) gives the vector field impinging on the CCD. When the object is small compared to the wavelength, the source can be modeled as a single dipole (i.e. 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

Fig. 2. Microsphere based Microscopy.

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 multipolebased 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 k¯ is the wave vector, αm is the half-angle of the incident cone of light, R is the radius #233013 - $15.00 USD © 2015 OSA


Fig. 3. Illumination with a) An incident converging wave and b) An incident plane wave.

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 EGRS = Eα αˆ + Eβ βˆ and the multipole coefficients of the focused beam can be estimated by matching EGRS with Eq. (5) as shown in [3, 25, 26, 27]: Plm (cos α ) il f eik f clm 2π α1m dPlm (cos α ) −imβ = − E − im E sin α d α d β , pm α β e El l(l + 1) 0 0 dα sin α Plm (cos α ) il f eik f clm 2π α1m dPlm (cos α ) m Eβ + im Eα e−imβ sin α d α d β . (17) pMl = l(l + 1) 0 0 dα sin α We can then substitute these coefficients into a modified version of Eq. (5) (where instances of 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]:

pm El =

∞

l

∑ ∑

l=1 m=−l

m lm m [Alm l m pEl + iBl m pMl ],

pm Ml =

∞

l

∑ ∑

m lm m [Alm l m pMl − iBl m pEl ]

(18)

l=1 m=−l

lm where Alm l m and Bl m are the translational coefficients derived in [28]. One can find details for deriving Eq. (18) in Appendix B of [3]. These translational coefficients can be ignored when √ l is sufficiently large (i.e. l > kd + 1.8η 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 (1) instances of hl are replaced with the spherical Bessel function jl ) to obtain an expression for the translated vectorial field. 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

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boundary condition [3]. Inside the microsphere, we obtain the following multipole expansion coefficients:

m p¯m El = (ε /εs )cl pEl

m p¯m Ml = (k/ks )dl pMl

(19)

which can then be substituted into a modified version of Eq. (5) (where instances of h(1) are replaced with the spherical Bessel function of the first kind and instances of k are replaced with ks ). In Eq. (19), ε and εs are the permittivity of the surrounding material and the microsphere respectively, k and ks are the wave numbers for the surrounding material and the microsphere respectively, and the internal Mie scattering coefficients are given by cl and dl . For the scattering field, we have the following expression for the multipole expansion coefficients

m p˜m El = al pEl

m p˜m Ml = bl pMl

(20)

which can then be substituted into the unmodified Eq. (5). al and bl are the external Mie scattering coefficients. We followed the derivation procedure in [29, 30] and derived our scattering coefficients in [3, 27], we list them here for convenience: al = cl =

ε ks ˆ ˆ ˆ ˆ εs k Jl (kR)Jl (ks R) − Jl (kR)Jl (ks R) , (1) (1) Hˆ l (kR)Jˆl (ks R) − εεs kks Hˆ l (kR)Jˆl (ks R) i kks , (1) (1) Hˆ l (kR)Jˆl (ks R) − εεs kks Hˆ l (kR)Jˆl (ks R)

bl = dl =

μ ks ˆ ˆ ˆ ˆ μs k Jl (kR)Jl (ks R) − Jl (kR)Jl (ks R) , (1) (1) Hˆ l (kR)Jˆl (ks R) − μμs kks Hˆ l (kR)Jˆl (ks R) i kks , (1) (1) Hˆ l (kR)Jˆl (ks R) − μμs kks Hˆ l (kR)Jˆl (ks R)

(1) (1) (2) (2) where Jˆl (x) = x jl (x) is Riccati-Bessel function; Hˆ l (x) = xhl (x) and Hˆ l (x) = xhl (x) are Riccati-Hankel functions. For completeness, we would like to include results for a plane wave (shown in Fig. 3(b)) ˆ 0 eikr cos θ can be expressed for comparison. An incident linearly polarized plane wave Einc = xE in terms of the electromagnetic multipole fields with the following multipole expansion coefficients: E E π (2l + 1) π (2l + 1) 1 0 l+1 0 l i (δm1 − δm−1 ), i (δ + δm−1 ). pm (21) pm El = Ml = k l(l + 1) k l(l + 1) m

where δab is the Kronecker delta function (i.e. δab = 1 ⇐⇒ a = b). These are analogous to the coefficients in Eq. (18), and the internal and scattering fields can be obtained in a fashion similar to the converging wave case. Likewise, for a circularly polarized plane wave Einc = E0 ikr cos θ e , the multipole expansion coefficients are: (xˆ + iy) ˆ √ 2 √ pm El 2.2.2.

=

2E0 l+1 i k

π (2l + 1) 1 δ , l(l + 1) m

√ pm Ml

=

2E0 l i k

π (2l + 1) 1 δ . l(l + 1) m

(22)

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 x, ˆ yˆ or zˆ) 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. #233013 - $15.00 USD © 2015 OSA


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 L1 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 x, ˆ yˆ and zˆ. 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: 1. A Hertzian electric dipole with the current dipole moment Il pointing in the xˆ direction: ω μ Il 1,1 1,−1 1,1 1,−1 √ a = [A − A ] + [C −C ] , pm l El lm lm lm lm 2 6π ω μ Il 1,−1 1,1 1,−1 bl [B1,1 pm Ml = −i √ lm − Blm ] + [Dlm − Dlm ] , 2 6π

(23)

where the translational coefficients (A, B, C and D) are calculated via the method in [28]. 2. A Hertzian electric dipole with the current dipole moment Il pointing in the yˆ direction: iω μ Il 1,−1 1,1 1,−1 al [A1,1 + A ] + [C +C ] , pm El = √ lm lm lm lm 2 6π ω μ Il 1,−1 1,1 1,−1 bl [B1,1 pm Ml = √ lm + Blm ] + [Dlm + Dlm ] . 2 6π

(24)

3. A Hertzian electric dipole with the current dipole moment Il pointing in the zˆ direction: ω μ Il 1,0 1,0 al Alm +Clm , pm El = √ 2 3π

ω μ Il 1,0 pm bl Blm + D1,0 Ml = −i √ lm . 2 3π

(25)

Fig. 4. Far field approximation for expressing the field in O coordinate system.

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 L1 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 #233013 - $15.00 USD © 2015 OSA


from O to O, i.e. substituting Eqs. (23) , (24), and (25) into Eq. (18). Alternatively, we can approximate the multipole coeficients 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 L1 (d f1 ). 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 α1 . For wave vector k¯ 1 (k, α1 , β1 ) at P, we have k = k , α1 ≈ α , β1 = β , and eikr /r ≈ e−ikd cos α1 eikr /r. Using these approximations and Eq. (9), we m can express the multipole coefficients p¯m El and p¯Ml representing the field in the O-coordinate m system in terms of those in the O -coordinate system (pm El and pEl from Eqs. (23) , (24), and (25)): −ikd cos α1 m pEl p¯m El = e

−ikd cos α1 m p¯m pMl Ml = e

(26)

We can then substitute into Eq. (16) to obtain the desired dyadic Green function. 3. 3.1.

Discussion 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 (al or bl ), i.e. the optical resonance is due to one particular partial wave (Nlm or Mlm ) 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]. 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 refered 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± ≈ kR ± η0 3 kR and η0 is a constant of order unity (empirically, η0 > 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 ns = 1.46 to match the parameters in [13, 15]. With these parameters and a choice of η0 = 4, λ = 400 nm we obtain: kR ≈ 37, l− ≈ 25, l+ ≈ 49. (27) There are many multipole orders in the edge domain for this case, and we will study in particular orders l = 40 and l = 43. We plot the Mie scattering coefficients c40 , d40 , c43 , and d43 as a #233013 - $15.00 USD © 2015 OSA


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 d43 peak in Fig. 5(a)) as well as just |dl | at λ = 403.07 nm (the off-resonant case). We observe that the resonant scattering coefficient d43 = −21.3i 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 (al , bl , cl , dl ) 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, d43 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 (Eα = sin α , Eβ = 0) into Eq. (17) yields the following multipole expansion coefficients: 1

pm El = −

300

il f eik f [π (2l + 1)] 2 0 δm l(l + 1)

αm dPl (cos α )

dα

0

25

c43

200 100

Magnitude

|cl|,|dl|

d43 c40 d40

0.35

0.4 λ (μm)

0.45

0.5

|bl| for resonance

15

|c | for resonance

10

|d | for resonance

a) Mie coefficients with l = 40 and l = 43.

0

pm Ml = 0.

(28)

|al| for resonance

20

5 0 0.3

sin2 α d α ,

l

Resonant mode: l=43

l

|dl| for off−resonance 10

20 30 Multipole order l

40

50

b) The first 50 Mie coefficients.

Fig. 5. a) Scattering coefficients as a function of the wavelength λ : the peaks correspond to the resonant wavelengths and b) Scattering coefficients with a resonant wavelength of λ = 401.6345 nm and an off-resonant wavelength of λ = 403.07 nm.

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Electric Intensity

4

Electric Intensity

x 10

3

3

16

2

5

2

14

4 12

1

10

0

8

−1

6

3

0 2

−1

4

−2 −3

x(μm)

x(μm)

1

−2

0

2 z(μm)

4

1

−2

2

−3

6

−2

0

2 z(μm)

4

6

Fig. 6. Electric intensity in linear (left) and logarithmic (right) scales with λ = 401.6345 nm. Light propagates from the left to the right sides.

Using the derived results in Section 2.2.1 with parameters λ = 401.6345 nm, R = 2.37 µm, d = 3R, and αm = 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 m the TE mode (dl pm Ml Ml with l = 43) as indicated in Fig.5. However, a converging beam with radial polarization is completely described by the TM modes, i.e. pm Ml = 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 90o 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 α0 ). Using this annular focusing technique, we can reduce the FWHM to 171 nm, represented by the blue curve in Fig. 7(a). 1

d=3R, αm=0.34,αo=0 d=3R,αm=0.34,αo=0.3 d=7R, Resonance, αo=0

0.6

d=7R, Off−resonance, αo=0

0.4 0.2 0 −1

−0.5

0 x(μm)

0.5

a) A converging wave.

1

Normalized Intensity

Normalized Intensity

1 0.8

0.8

CP: Resonance CP: Off−resonance LPx: Resonance

0.6

LPy: Resonance

0.4 0.2 0 −1

−0.5

0 x(μm)

0.5

1

b) A plane wave.

Fig. 7. Electric intensity at focus along the transversal direction with a) λ = 401.6345 nm (off-resonance) and λ = 439.44 nm (resonance). b) λ = 403.07 nm (off-resonance) and λ = 445.5127 nm (resonance). CP and LP refer to circularly polarized and linearly polarized light, respectively.

An alternative approach would be to move the microsphere toward the lens, i.e. increasing d. At d = 7R, 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 #233013 - $15.00 USD © 2015 OSA


x(μm)

Electric Intensity in Logarithm Scale 5 4 4 3 3 2 1 2 0 −1 1 −2 0 −3 −4 −1 −5 −5 −4 −3 −2 −1 0 1 2 3 4 5 z(μm)

a) λ = 397.44 nm (|c43 | = 23.6364).

x(μm)

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. 7(a), 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]. Electric Intensity in Logarithm Scale 5 6 4 3 4 2 1 0 2 −1 −2 0 −3 −4 −2 −5 −5 −4 −3 −2 −1 0 1 2 3 4 5 z(μm)

b) λ = 439.44 nm (|c43 | = 330.6155).

Fig. 8. Resonance with a converging wave illumination and d = 7R.

We note that this FWHM is close to λ /(2ns ), where ns is the index of refraction for the interior of the microsphere, although it is not directly dependent on the wavelength; λ /(2ns ) is 136 nm and 150.5 nm for 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 /(2ns ) in the λ = 439.44 nm case. In this resonant case, the effective NA of the incident beam is approximately equal to ns which is attributed to the dominant contribution of the surface wave. 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 doublelobed 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 (|d43 | = 280.258), as shown in Fig. 9(b). Cross sections of the electric intensities at the focus along the xˆ direction are shown in Fig. 7(b) for incident circularly polarized plane waves as well as plane waves linearly polarized in the xˆ and yˆ directions. The FWHM along the xˆ direction for circular polarization is 124.8 nm, while

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for linear polarization, the FWHMs are 172.2 nm and 98.3 nm for polarization along xˆ and y, ˆ 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 9c) 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.

a) λ = 439.44 nm.

b) λ = 445.51 nm.

4

x 10

2

8

1

6

0

4

−1

2

−2

x(μm)

x(μm)

Electric Intensity

5

x 10 2 1.5 1 0.5

Electric Intensity 3 300 2 1 200 0 −1 100 −2 −3 −3 −2 −1 0 1 2 3 4 z(μm)

x(μm)

−3 −3 −2 −1 0 1 2 3 4 z(μm)

Electric Intensity 3 2 1 0 −1 −2 −3 −3 −2 −1 0 1 2 3 4 z(μm)

3

c) λ = 403.07 nm.

Fig. 9. A plane wave illumination with a circular polarization.

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. 7(a). 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 x-dipole ˆ on the surface of the microsphere (d = R and R = 2.37 µm) at two different wavelengths— #233013 - $15.00 USD © 2015 OSA


0.03

|p1Ml|

0.02 0.01 0

1.5

|p1El|

10

20 30 Multipole Order l

40

|P1El|, |P1Ml|

|P1El|, |P1Ml|

0.04

|d43|=280.258

|p1Ml|

0.5 0

45

a) Multipole coefficients with off-resonance.

1

|p1El|

10

20 30 Multipole Order l

40

45

b) Multipole coefficients with resonance.

Fig. 10. Multipole coefficients with a) λ = 450 nm and b) λ = 445.5127 nm.

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 f1 = 2.25 mm, and thus k f1 ≈ 3 × 104 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.

a) No sphere.

b) Sphere in off-resonance.

c) Sphere in resonance.

Fig. 11. Images of two dipoles spaced 109 nm apart with a) λ = 445.5127 nm, b) λ = 450 nm, and c) λ = 445.5127 nm.

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. 11a), 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 Δλ for efficiently exciting a resonant TE mode as follows: Δλ =

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−2λ0

(1) (n2s − 1)k0 R Hˆ l (k0 R) − Jˆl (k0 R)

2 .

(29)


In our case, we substitute ns = 1.46, λ0 = 0.445 512 7 µm, R = 2.37 µm, k0 × R = (2π /λ0 ) × (2.37) = 33.4247, and l = 43 into Eq. (29) and obtain Δλ ≈ 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 Δλ ≈ 4 × 10−4 nm (Δ f = 350MHz) [42]. We also note that the microsphere in resonance transfers energy to the far-field region more efficiently. The maximum intensity in Fig.11(c) 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 |d48 | = 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 ns /(2 − ns ) ≈ 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 offresonant 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. 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 sampleillumination 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. NRFCRP10-2012-04).

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