## Abstract

Surface plasmon-coupled emission microscopy (SPCEM) was proposed as a high sensitivity technique that makes use of a thin layer of metal deposited on glass slides to efficiently excite fluorophores and to collect the emission light. However, the theoretical aspect of SPCEM imaging has not been well studied. We propose a model for SPCEM and show, through theoretical analysis and empirical results, that the point spread function of SPCEM is irregular and has an annular-like structure, significantly different from the familiar point spread function of the conventional wide-field microscopy. This result is due to the highly polarized and anisotropic emission caused by the metal layer.

© 2007 Optical Society of America

## 1. Introduction

Total internal reflection fluorescence microscopy (TIRFM) is a widely used imaging technique that is useful in probing the structure and dynamics of the basal surfaces of cells due to its excellent surface selectivity and ability to suppress background fluorescence [1]. Only fluorophores located within about 100 nm of the surface are excited by the evanescent field, while those located further away from the surface are not efficiently excited by the exponentially decaying evanescent wave. TIRFM has been used in the studies of many cellular processes, such as actin dynamics [2], exocytosis [3] and receptor-ligand binding [4].

Recently, an imaging technique that combines surface plasmon-coupled emission (SPCE) with the TIRF configuration was used in the imaging of muscle fibrils to study the dynamics of interaction between actin and myosin cross-bridges [5]. The surface plasmon-coupled emission method was first proposed as a high sensitivity and efficient fluorescence detection method by Lakowicz et al. [6, 7, 8]. They showed, through imaging fluorophores near to a silver-coated glass substrate, that the fluorescence emission was highly directional and this increased the collection efficiency to nearly 50%, as well as contributed to low background noise. SPCE has found a number of applications in areas such as biotechnology and biological measurements.

For example, Malicka et al. demonstrated the use of SPCE in DNA hybridization measurements [9], while Borejdo et al. extended fluorescence correlation spectroscopy with SPCE to improve its signal-to-noise ratio [10].

In surface plasmon-coupled emission experiments, the imaging optics typically reside
on the same side as the excitation optics. A hemicylindrical or hemispherical prism
in the Kretschmann-Raether configuration is used to induce surface plasmons on the
metal layer which excite the samples [11]. The excited fluorophores then emit light that is coupled
back through the metal layer and is collected by the detection optics to form the
final image. In the recent 2D imaging experiments by Borejdo *et al*. [5], the prism has been replaced by a high NA TIRF objective
which is used for both excitation and collection purposes. This configuration of
using SPCE with TIRFM will henceforth be called surface plasmon-coupled emission
microscopy (SPCEM).

To our knowledge, there has not been any theoretical study on the image formation process of SPCEM, especially with regards to its point spread function (PSF) characteristics. In this paper, we propose a theoretical model for SPCEM and derive its theoretical point spread function within a 4f optical system. We show that the anisotropic emission of SPCE results in an annular-like PSF characteristic. We then compare the theoretical results with experimental data and show that our theoretical results agree with the observed data.

## 2. Theory

#### 2.1. *E*_{x}citation of dipole in the object space

_{x}

In SPCEM, dipoles on the metal-coated glass slide are excited by a
*p*-polarized incident plane wave as shown in Fig. 1. *n*
_{1} denotes the
refractive index of the object space, *n*
_{2} is the
refractive index of the metal layer and *n*
_{3} is the
refractive index of the glass slide and immersion medium. The dipole is placed
at a distance *d* from the metal surface.
*k*
_{3,inc} is the wave number
of the incident plane wave in medium 3 intersecting the interface at an angle of
θ_{i} from the surface normal, and is given as
*k*
_{3,inc} =
2π*n*
_{3}/λ_{inc}, where λ_{inc} is the wavelength of the incident light in free space.

As shown in Fig. 1, the optical system used in SPCEM to excite the
fluorophores by surface plasmons is essentially the same as that of a TIRF
microscope [1]. The excitation field **E _{i}** of
SPCEM is simply given by

$$\phantom{\rule{1.2em}{0ex}}\times \mathrm{exp}\left({\mathrm{ik}}_{3,\mathrm{inc}}x\mathrm{sin}{\theta}_{i}\right)\mathrm{exp}\left({k}_{3,\mathrm{inc}}z\sqrt{{\mathrm{sin}}^{2}{\theta}_{i}-{\left(\frac{{n}_{1}}{{n}_{3}}\right)}^{2}}\right)$$

where **e _{x}** and

**e**denote the unit vectors in the

_{z}*x*and

*z*direction, and τ

_{p,inc}is the three-layer Fresnel coefficient [11] for p-polarized light given by

where *t*
^{32}
* _{p}* and

*t*

^{21}

*are the Fresnel coefficients for each of the respective two-layer interfaces and θ*

_{p}_{2}is the angle in the metal layer given by Snell’s law.

#### 2.2. Electric field in the image space

Figure 2 shows the typical setup of a 4f system for SPCEM
imaging. The point dipole **p** = μ→
exp(*iωt*) at a distance *d* away
from the first interface is excited by the field calculated in Eq. (1) and emits light at a different wavelength λ
than the excitation wavelength λ* _{inc}*.

Region 2 is a thin metal layer of thickness *t* with an index of
refraction *n*
_{2}. A high NA objective is placed in
medium 3 and the tube lens in medium 4.

Due to the highly polarized emission of SPCE and the high NA objective used, a full vectorial formulation will be derived for the model shown in Fig. 2, since a scalar treatment of the model does not take into account these effects and is inadequate here. Using the vectorial Debye integral of Richards and Wolf [12], the field in the image space of the tube lens is given by

$$\phantom{\rule{.2em}{0ex}}\times \mathrm{exp}\left({\mathrm{ik}}_{4}z\mathrm{cos}{\theta}_{4}\right)\mathrm{exp}\left(i\Phi \right){d\theta}_{4}d\varphi $$

where the coordinates of a point in the image space are given by
(*r*,φ, *z*),
*k*
_{4} is the wave number in medium 4,
θ_{4} is the polar angle of the wave vector in medium 4
and ϕ is the azimuthal angle of the wave vector. It is assumed that
as the wave propagates through the lenses, the meridional plane of the
propagation remains constant. Φ denotes the wave front aberration
function which is a measure of the path difference between the wave front at the
exit aperture of the tube lens and the wave front of an ideal spherical wave [13].

To obtain the electric strength vector **E**´_{4} in
Eq. (3), we start by considering the field emitted by the dipole
in medium 1 which is subsequently transmitted into medium 3, and then propagate
the waves through the 4f system into medium 4. We note that although Torok
solved the problem of the propagation of dipole waves through dielectric
interfaces [14], we do not use his approach here, primarily because it
was assumed that the distance of the dipole from the first interface is large
and consequently, the contribution of dipole evanescent waves was ignored. On
the contrary, when considering SPCEM, it is important to treat the case where
the fluorophores are close to the metal surface, in the order of nanometers
scale, which implies that the near-field emission cannot be ignored. Therefore,
we follow the procedure given by Arnoldus and Foley [15] to obtain the field in medium 3. We note that the field
in medium 3 had also been derived by Hellen and Axelrod [16] in a similar way.

Figure 3 shows the axis convention that is used for
obtaining the expression for the dipole radiation in medium 3. The complex
amplitude of the dipole source field in an infinite medium with index of
refraction *n*
_{1} is given by

where *v _{i}* =
(

*n*

^{2}

_{i}-α

^{2})

^{1/2}and α=

*k*∥/

*k*

_{0}as introduced in [15], and

where **k**
_{∥} denotes the surface wave vector. In
Eq. (4), the field in medium 1 is effectively represented by an
angular spectrum of plane waves, where the limits of integration indicate that
both evanescent waves and travelling waves are taken into consideration.

To obtain the field in medium 3, it is then necessary to decompose plane waves
which are propagating in the positive *z* direction into their
*p* and *s* components, so that we can make
use of the Fresnel transmission coefficient for the three-layer system. First,
we write down the unit vectors for a cylindrical coordinate system whose radial
and axial component is contained in both the plane of incidence of a particular
plane wave and the meridional plane as the waves propagates through the system.

Then, the unit vector for the *p* component in medium 1 and medium
3 are,

respectively, and **k _{3}** is the wave vector of the
transmitted waves in medium 3 given by

Decomposing the plane waves in Eq. (4) into their *p* and *s*
components and using the Fresnel transmission coefficient for the three-layer
system, the complex amplitude of the field in medium 3 (*z*
> t) can be expressed in terms of these unit vectors and we arrive at
a similar expression as Arnoldus and Foley

where

denotes the electric field strength vector of the plane waves in medium 3 and the Fresnel transmission coefficients have a similar form to Eq. (2)

Equation 7 describes the field in medium 3 due to the radiating dipole in medium 1. As the waves propagate to the objective lens, the wave front aberration function that is incident at the entrance pupil of the objective lens relative to its Gaussian focus can be obtained as

Next, in order to propagate these plane waves to medium 4, we make use of the
generalized Jones matrix which was introduced by Torok *et al* [17, 18, 19] to express **E´ _{4}** in
terms of E´

_{3}. We do not take into account the reflection and transmission coefficients of the lenses, as they are irrelevant here and complicate the equations unnecessarily. Hence,

where *A*(θ_{3},θ_{4}) is
the apodization factor for a system that satisfies Abbe’s sine
condition [12], and is given by

**L _{3}** and

**L**is a rotation matrix that rotates about the axis defined by

_{4}**e**

_{s},

and **M** is a transformation matrix that transforms the
ρ*sz* coordinate system to the
*r*φ*z* coordinate system, and is
given by

where φ is the azimuthal observation angle and ϕ is the azimuthal angle of the wave vector.

Finally, substituting Eq. (11) into Eq. (3), and replacing the dipole orientation vector
**p** with (μ sin θ_{d}
cosϕ_{d},μsinθ_{d}
sinϕ_{d} ,μ cosθ_{d}) we
obtain

$${E}_{4,\phi}=-\frac{{\mathrm{ik}}_{4}}{2}\left\{\mu \mathrm{sin}{\theta}_{d}\mathrm{sin}\left({\varphi}_{d}-\phi \right)\right[{K}_{0}^{I}-{K}_{2}^{I}\left]\right\}$$

$${E}_{4,z}=-\frac{{\mathrm{ik}}_{4}}{2}\left\{\mu \mathrm{sin}{\theta}_{d}\mathrm{cos}\left({\varphi}_{d}-\phi \right)[2i{K}_{1}^{\mathrm{II}}\right]-\mu \mathrm{cos}{\theta}_{d}\left[2i{K}_{0}^{\mathrm{II}}\right]\}$$

where

$${K}_{1}^{I}={\int}_{0}^{\sigma}\sqrt{\frac{\mathrm{cos}{\theta}_{4}}{\mathrm{cos}{\theta}_{3}}}{\mathrm{sin}\theta}_{4}{\tau}_{p}\mathrm{sin}{\theta}_{1}\mathrm{cos}{\theta}_{4}{J}_{1}\left({k}_{4}r\mathrm{sin}{\theta}_{4}\right)\times \mathrm{exp}\left({\mathrm{ik}}_{4}z\mathrm{cos}{\theta}_{4}\right)\phantom{\rule{.2em}{0ex}}\mathrm{exp}\left(i\Phi \right){d\theta}_{4}$$

$${K}_{2}^{I}={\int}_{0}^{\sigma}\sqrt{\frac{\mathrm{cos}{\theta}_{4}}{\mathrm{cos}{\theta}_{3}}}{\mathrm{sin}\theta}_{4}\left({\tau}_{s}-{\tau}_{p}\mathrm{cos}{\theta}_{1}\mathrm{cos}{\theta}_{4}\right){J}_{2}\left({k}_{4}r\mathrm{sin}{\theta}_{4}\right)\times \mathrm{exp}\left({\mathrm{ik}}_{4}z\mathrm{cos}{\theta}_{4}\right)\phantom{\rule{.2em}{0ex}}\mathrm{exp}\left(i\Phi \right){d\theta}_{4}$$

$${K}_{1}^{\mathrm{II}}={\int}_{0}^{\sigma}\sqrt{\frac{\mathrm{cos}{\theta}_{4}}{\mathrm{cos}{\theta}_{3}}}{\mathrm{sin}\theta}_{4}{\tau}_{p}\mathrm{sin}{\theta}_{1}\mathrm{cos}{\theta}_{4}{J}_{0}\left({k}_{4}r\mathrm{sin}{\theta}_{4}\right)\times \mathrm{exp}\left({\mathrm{ik}}_{4}z\mathrm{cos}{\theta}_{4}\right)\phantom{\rule{.2em}{0ex}}\mathrm{exp}\left(i\Phi \right){d\theta}_{4}$$

$${K}_{1}^{\mathrm{II}}={\int}_{0}^{\sigma}\sqrt{\frac{\mathrm{cos}{\theta}_{4}}{\mathrm{cos}{\theta}_{3}}}{\mathrm{sin}\theta}_{4}{\tau}_{p}\mathrm{cos}{\theta}_{1}\mathrm{sin}{\theta}_{4}{J}_{1}\left({k}_{4}r\mathrm{sin}{\theta}_{4}\right)\times \mathrm{exp}\left({\mathrm{ik}}_{4}z\mathrm{cos}{\theta}_{4}\right)\phantom{\rule{.2em}{0ex}}\mathrm{exp}\left(i\Phi \right){d\theta}_{4}$$

and *J _{n}*(∙) denotes the Bessel function of
the first kind of order

*n*. The wave front aberration function Φ is given in Eq. (10). The collection semi-angle σ is obtained by using the sine condition for an aplanatic system

where *mag* is the overall magnification of the 4f system.

Equations15 and 16 describe the electric field in the image space (Fig. 2). It is interesting to see that these expressions,
which were derived by starting from Arnoldus and Foley’s
representation, resemble the expressions derived by Torok *et
al*. [18, 19] which is for the case of imaging a dielectric scatterer
in a 4f system placed in a homogenous medium. The main difference for our
expressions is the presence of the Fresnel transmission coefficients as well as
the non-zero wave aberration function that is caused by the refractive index
mismatch as the dipole waves traverse across the metal interface. In fact, if we
place the dipole at the origin and remove the metal interface by setting
*d* and *t* to zero, and setting
*n*
_{3} to be equal to
*n*
_{4}, the expressions in Eqs. (15) and (16) will reduce to the same form as the expressions in [18, 19].

#### 2.3. Intensity in image space

The intensity that is detected in the image space is

where the normalization factor *P _{T}* is the total power
emitted by the dipole in the presence of a metal interface [20, 16]. For a dipole which is perpendicularly oriented to the
metal interface, θ

_{d}= 0°, the intensity is rotationally symmetric

On the other hand, the image of a dipole which has an orientation parallel to the
metal interface along the *x* direction is not rotationally
symmetric, that is, it has a dependence on φ

In practice, since an ensemble of fluorophores is imaged in a given sample, it is more meaningful to find an expression for the average intensity distribution for an ensemble of dipole orientations. Therefore in this paper, we treat the first two cases that were discussed in [19] and [21]. For the first case, the dipole is excited by the full incident field and the electric dipole moment of the emission is randomly oriented. Since the fluorophore can assume many different orientations during its emission lifetime, Eq. (17) is integrated over all polar and azimuthal angles, and thus the intensity in the image space is given by

In the second case, the dipole is excited by the component of the incident field that is aligned to it. The total intensity is then obtained by averaging the intensity over all possible dipole orientations.

*I*
_{1} corresponds to the case where the dipole rotates
and aligns with the incident field and gets excited by the full incident field.
The emission dipole moment, however, is uniformly distributed over all polar and
azimuthal angles because the dipole may rotate arbitrarily during emission. On
the other hand, *I*
_{2} corresponds to the case where
each dipole is fixed and is only excited by the component of the incident field
that is aligned with it. It represents the intensity of an ensemble of randomly
oriented dipoles which are fixed between excitation and emission. In a real
situation, for fluorophores in a solution the detected intensity could be
somewhere in between these two limiting cases.

#### 2.4. Adding a linear polarizer

Linear polarizers can be added to the model, for example by placing them in between the objective lens and the tube lens. This can be accomplished by modifying Eq. (11) into

where

transforms from the ρ*sz* coordinate system into
cartesian coordinates, and **P** is the polarization matrix. For a
linear polarizer in the *x* and in the *y*
direction, their polarization matrices are given respectively by

For both of these cases, the expressions for the field and intensity in the image space are derived by first substituting Eq. (22) into Eq. (3), and then simplifying the results. This process is similar to the derivation of Eqs. (15), (16), (17), (20) and (21) above.

## 3. Results and discussion

Equations in the previous section were evaluated numerically with the following
values. An excitation beam with a wavelength of 532 nm is directed at 45°
to the metal interface of thickness 50 nm. Gold, which has a complex refractive
index of 0.32+2.83*i* [22] is used as the metal layer. The incident light is coupled
into surface plasmons which then excite the dipole. The excited dipole emits at a
wavelength of 560 nm and the emission is collected by a high NA objective with
NA1.45 and 60X magnification. Table 1 shows the values used for the simulation. These
values were chosen to match the actual experimental conditions which will be
described later.

Figure 4 shows the distribution of the electric field
components on the image plane for a dipole oriented perpendicular to the metal
interface which was calculated using Eq. (18), while Fig. 5 shows the distribution for a dipole parallel to the
interface in the *x* direction calculated with Eq. (19). Comparing Fig. 4(c) to Figs. 4(a),(b) and Fig. 5(c) to Figs. 5(a),(b) for both the perpendicular and parallel dipoles, it can
be seen that the *E _{z}* component of the field is negligible
relative to the

*E*and

_{x}*E*components. Furthermore for a perpendicular dipole (Fig. 4), there exists symmetry in the distribution of the

_{y}*E*and

_{x}*E*field components and both components have the same peak amplitudes. Such symmetry can be expected for a perpendicular dipole because the axial symmetry would result in field patterns which are radial when projected onto the image plane. In contrast, for a dipole parallel to the metal interface and aligned in the

_{y}*x*direction (Fig. 5), the

*E*component dominates over the

_{x}*E*and

_{y}*E*components. Furthermore, symmetry does not exist between the

_{z}*E*and

_{x}*E*components. Comparing both figures, it can also be seen that the field on the image plane due to a perpendicular dipole is almost two times stronger than that of a parallel dipole.

_{y}Next, experiments were carried out to obtain the experimental point spread function of SPCEM in order to compare with the theoretical results. Figure 6 shows the experimental setup that was used. A continuous wave laser source with a wavelength of 532 nm (Verdi-10, Coherent) was used to excite subdiffraction-limited 40 nm fluorescent beads (Molecular Probes, OR) that were placed on a cover slip with a 50 nm gold-coated layer. The excitation angle was tuned to the surface plasmon resonant angle of 45°, so that the fluorescent beads were efficiently excited by the surface plasmons induced on the metal surface. A high NA objective lens (Olympus Plan Apo NA1.45 60X) was used to collect the emission from the excited fluorophores with a peak emission wavelength of 560 nm. A barrier filter (HQ545LP, Chroma, Rockingham, VT) was used to block the scattered excitation light. The emission light exiting from the tube lens was then magnified 16 times and the image was captured on an intensified CCD (Pentamax, Princeton Instrument, Trenton, NJ). The same image was captured 10 times and averaged to reduce the background noise. The detailed setup can be found in [23] except that only one beam was used in this experiment.

Figure 7(a) and Fig. 7(b) compares the simulated image of the point spread function of SPCEM calculated using Eq. (21) with the actual point spread function obtained from the experiments. Comparison of the two figures show a high degree of similarity between the calculated PSF and the experimental PSF in terms of their morphology. The figure also shows that the point spread function of SPCEM has a rotationally symmetric and annular-like characteristic, with multiple rings extending out with decreasing intensity. Interestingly, the PSF has a “valley” near the center which is not seen in the PSFs of conventional wide-field microscopy and confocal microscopy. It is also noted that the experimental PSF has rings which are slightly more spread out and wider than the theoretical case. This is likely due to the fact that the fluorescent bead has a finite diameter, whereas in the theoretical case a point dipole was assumed.

To further validate our theoretical model, we also compared the experimental and
theoretical results when a linear polarizer was introduced into the system. Using
the same experimental setup as in Fig. 6, a linear polarizer (LP-VIS100, Thorlabs) was added
between lens 3 and lens 4, and the PSF was captured on the intensified CCD. For the
theoretical model, the polarizer was added between the objective lens and the tube
lens (see Eq. (22)). Figure 7(c) and Fig. 7(d) show the theoretical and experimental PSFs when a
linear polarizer oriented in the *x* direction was added (the PSF for
a *y* oriented linear polarizer is not shown since it is symmetrical
to that of a *x* oriented linear polarizer). Both the theoretical and
experimental PSFs are similar in terms of their morphology. As in the previous case,
the experimental results show that the lobes are slightly wider than the theoretical
results.

A more quantitative comparison is shown in Fig. 8, which plots the cross-sectional profiles of the theoretical and experimental PSFs. In Fig. 8(a) where there is no linear polarizer added, the locations of the first peaks for both the theoretical and experimental PSFs differ by approximately 20 nm, while the second peaks for the experimental PSF are further away from those of the calculated PSF by roughly 60 nm. The widths of the peaks for the experimental PSF are also wider than those of the theoretical PSF. In Fig. 8(b) where a linear polarizer was added to the emission path, the first peaks for the experimental PSF are separated by 35 nm away from the first peaks of the calculated PSF and their second peaks are located approximately 80 nm away. For both of the cases in Fig. 8, these differences between the calculated and experimental PSFs could be due to a few reasons. Firstly, as mentioned earlier, the actual beads have a finite diameter of 40 nm whereas the model assumed an infinitesimally small point dipole. Secondly, the effective numerical aperture of the high NA objective may actually be lower than its nominal value of NA 1.45 due to strong attenuation of the marginal rays [24]. According to our calculations for an effective numerical aperture of NA 1.2, the difference between the locations of the first peaks would differ by 15 nm and the second peaks by 30 nm, which is roughly a two-fold reduction in the errors. Other reasons for the differences could be due to aberrations of the relay optics and misalignment of the optical setup.

In general, the theoretical results are in agreement with the experimental results, which suggest that our theoretical model is applicable for SPCEM. It has been shown that the point spread function of SPCEM departs from the familiar PSFs of conventional wide-field microscopy and confocal microscopy. Instead, the PSF of SPCEM resembles a rotationally symmetric annular-like structure having multiple rings. This has a few implications. Firstly, compared with conventional wide-field microscopy where the point spread function typically assumes an airy-disc structure and the resolution of a microscope is defined using the Rayleigh criterion, it is not so clear in this case how the resolution should be defined and what is its relation to wavelength and numerical aperture. An intuitive way would be to define the resolution of SPCEM by disregarding the “valley” and considering the full-width at half-maximum (FWHM) of the resulting structure. A future study that may clarify this issue is to quantify the modulation transfer function of this optical system. This would enable us to compare the resolution of SPCEM with other microscopy techniques. Furthermore, the irregular point spread function of SPCEM suggests that we must proceed with caution when interpreting images captured using this method. It is entirely possible the annular-like PSF may introduce artifacts in images that ultimately lead to misinterpretation. Therefore, more studies need to be carried out to understand the implications of such an irregular point spread function.

## 4. Conclusion

Surface plasmon-coupled emission microscopy, or SPCEM, is a relatively new imaging
technique that potentially promises better detection sensitivity and higher
signal-to-noise ratio than conventional TIRF imaging. Although a recent paper showed
theoretical studies that suggest that the sensitivity of SPCEM is actually reduced
due to the metal layer, this has yet to be shown experimentally as the theory did
not take into account surface roughness [25]. Nonetheless, Boredjo *et al*. showed that
the small detection volume of SPCEM makes it possible to follow the rotational
motion of actin molecules during muscle contraction [5]. To date however, the image formation process of this
imaging method has not been studied and is not well understood.

In this paper, we proposed a model for SPCEM and derived its point spread function within a 4f optical system. It was also shown that the point spread function has a rotationally symmetric annular-like structure and a “valley” in the central region. In addition, the theoretical results were supported with empirical data, both with and without the addition of a linear polarizer. With this insight into the point spread function characteristic of SPCEM, additional work remains in understanding the implications of such an irregular point spread function. Further investigation should also focus on the compensation for this irregular point spread function (the central “valley”) to improve image resolution. This could be done via point spread function engineering approaches, or through designing specific deconvolution algorithms for SPCEM.

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