1. Introduction

Modern fluorescence microscopy provides outstanding spatio-temporal resolution, sensitivity and selectivity for investigating biological samples at the sub-micrometer scale. Due to these properties, it has become the method of choice in life science and biomedical research. Besides impressive improvements in image resolution and contrast, the need for quantitative measurements becomes more and more important. Fluorescence fluctuation spectroscopy (FFS) with correlation analysis (FCS) [1, 2], intensity distribution analysis (FIDA, PCH) [3, 4] or Förster resonant energy transfer (FRET) [5] addresses these demands. For these quantitative measurements, knowledge of the optical point spread function is required. This corresponds to the brightness profile of the detected fluorescence, which demands the combined calculation of the excitation focus field, the response of fluorescent markers and finally the detection efficiency. This allows accurately modeling and predicting the measured fluorescence signal as recently shown in detail by Enderlein et al. [6]. The physical concepts of focusing electromagneticwaves and detecting dipole waves are well-known and described for instance by Török, Higdon and Enderlein [7, 8, 9, 10]. In analogy to the classical Debye formulation [11] and the seminal work byWolf and Richards [12, 13], we recently reformulated the calculation of the electromagnetic field in the focus of high numerical aperture objectives based on a Fourier or chirp z transform [14] and achieved unprecedented calculation speed and flexibility. These calculations yet describe the excitation fields and thereby the excitation rates of fluorophores within these foci. In this work, we revisit the induced fluorescence emission based on the classical reciprocity principle in order to calculate accurately the detection efficiency within the same rapid and flexible framework. Our calculation accounts for the anisotropic distribution of the emitted fluorescence and yields the absolute detection efficiency through normalization with the total radiated power of the fluorophore [10].

Without any restriction, we assume that the biological sample is contained in water. The induced fluorescence response can be caused by auto-fluorescence or specific biomarkers. Based on a classical description, the fluorophore is described as a dipole absorbing photons at the excitation wavelength λ_ex and responding by the emission of fluorescence within a wavelength range λ_fl, where λ_fl≳λ_ex due to the Stokes shift. Under these general assumptions, the total fluorescence process can be described by:

All these quantities vary in general with the position r⃗=(x, y, z) of the fluorophore and the orientation Ω=(Θ,φ) of its absorption and emission dipoles upon absorption and emission, respectively, as well as with the wavelength of excitation and emission [15]. The calculation of the excitation field (1) is described in a former publication [14] and is used as a conceptual framework for the field calculations. The average excitation cross-section σ_ex (2) as provided by the manufacturer is typically in the order of a few Ų. The general fluorophore response (3) is well described by B. Valeur [16] for instance. The specific photophysical response of Cy5 was investigated by Widengren and Schwille [17], whereas Eggeling et al. [18] recently reviewed the photochemical response of Rhodamine 6G. In this work, we focus on the calculation of the emission rate (4), the emitted field (5) and the detection efficiency (6). As a final demonstration, we calculate the fluorescence detection at a glass–water interface and compare the detection yields for total internal reflection fluorescence (TIRF) microscopy in an epi- and a trans-illumination configuration.

Throughout this paper, we omit the time dependency exp(-iωt). ω=2πc ₀/λ is the angular frequency, c ₀ the speed of light and λ the wavelength in free space. Further we assume a relative magnetic permeability µ_r=1 for all materials. Therefore, the refraction index n is given by the dielectric permittivity ∊_r=n ². In order to avoid sign ambiguities in k ² _z=k ²-k ² _xy, we keep all wavevectors k⃗ pointing towards the positive z-axis with a proper sign choice for k_z.

2. Emission close to planar interfaces

This section describes the interaction of a fluorophore with planar layers and derives the emission rate R_fl(r⃗,Ω), where r⃗ is the position of the fluorophore and Ω=(Θ,φ) its orientation as shown in Fig. 1. The fluorophore is modeled as a point dipole with fixed dipole moment µ⃗. The dissipated power is given in a semi-classical picture by the dipole-light interaction. In a quantum-mechanical description, the dissipated power is analyzed with the transition probabilities and results in equivalent formulae, where the decay rate Γ of an electronic state defines the overall rate of all de-excitation channels from that state. In general, Γ contains a radiative decay Γ_em and a non-radiative decay Γ_nr and is simply the sum of both.

 

Fig. 1. Dipole µ⃗ located at r⃗ ₀ above the first interface. n ₁ is the refraction index of the upper half-space (z>0) around the dipole. n_m is the refraction index of the lower half-space (z<-d) and n_i the refraction indices of the intermediate layers (-d<z<0).

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Γ_em is associated with electromagnetic radiation in, and interaction with, the environment; whereas Γ_nr stands for any non-electromagnetic dissipation, i.e. an intrinsic relaxation. The presence of a layered structure affects the radiative decay Γ_em of the fluorophore because it modifies the local density of states (LDOS). For instance, a high index medium in the near field of the fluorophore increases the LDOS at the fluorophore position. In consequence, Γ_em increases, or equivalently the lifetime ${{\tau }_S}_1$ of the first excited singlet state S ₁ is shortened, because more radiation modes are available. On the other hand, Γ_nr is assumed to depend only on the microenvironment of the fluorophore, e.g. it is not affected by the position and orientation dependent LDOS but rather by adsorption to a surface. In the following, the dissipated power P of a point dipole with fixed dipole moment µ⃗ is calculated for two cases:

First, case (a) is calculated as reference for obtaining the radiative enhancement factor γ(r⃗,Ω) in case (b). Because Γ_em=P/h̄ω, the radiative enhancement γ(r⃗,Ω)=Γ_em(r⃗,Ω)/Γ_em,∞ is reproduced by the ratio P(r⃗,Ω)/P _∞, where the subscript 1 indicates the unperturbed case (a). In this context, we should keep in mind that the dissipated power describes the total power emitted via the dipole field, i.e. photon emission as well as radiative loss.

In the late seventies, Lukosz calculated the emission of electric and magnetic dipoles near a planar dielectric interface [19, 20]. In the eighties, Burghardt and Thompson [21] and Hellen and Axelrod [28] refined the calculation for TIRF microscopy. Recently, Novotny [23] revisited the theory for calculating the light field of interacting dipolar particles; and Mertz [24] unified the description of a classical dipole near a dielectric interface with a simple input–output formalism based on the Lorentz reciprocity theorem. The following description relies on the general ideas given by Ford and Webber [25]. This description is particularly advantageous because only the electric field at the dipole position is required for calculating the dissipated power, which circumvents any confusion in case of absorbing media (i.e. metals) in the vicinity of the dipole. According to Fig. 1, the total dissipated power of a point dipole at position r⃗ ₀ is given by

where µ⃗ is the dipole moment and E⃗(r⃗) the electric field radiated by this dipole. The dipole is described as a current source j⃗(r⃗)=-iωµ⃗δ(r⃗-r⃗ ₀) in medium 1 and the radiated field has to fulfill the Maxwell equations.

Using time harmonic fields and a plane wave expansion

the Maxwell Eq. (3) read as

Solving for the electric field E⃗_k propagating along the wavevector k⃗ yields

Substituting this expression in (4) and splitting the lateral and axial integration, an integral representation of the radiated field is obtained.

The k_z integral is a contour integral in the complex plane, which can be evaluated using the complex residues at the first order poles k_z=±k _1z. With k⃗ ₁=k⃗ _xy+k⃗ _1z(z-z ₀)/|z-z ₀|, the electric field is then

The first term does not contribute to the dissipated power because it is real valued. The second term is imaginary only for k_xy<k ₁, which is the far field radiation domain in medium 1. Therefore, the dissipated power is obtained with (2) and

The electric field is now described by the coupling between the dipolemoment µ⃗ and the electric field E⃗_k, i.e. by a projection of µ⃗ onto E⃗_k. Integrating over all directions of k⃗_xy yields

and evaluates to the well-known Larmor formula, e.g. the dissipated power is given by

 

Fig. 2. Coupling of the dipole moment µ⃗ with the electric fields. E_d represents the direct dipole field, E_r the reflected field and E_t the transmitted field. s⃗ is perpendicular to the incidence plane, whereas p⃗, q⃗ and o⃗ are parallel to the incidence plane.

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In the second case (b) where the dipole is near to planar interfaces, the radiation towards the interfaces is partially reflected and interferes with the direct radiation E⃗_d of the dipole as outlined in Fig. 2. The calculation involves the reflection coefficients r^p,s _1m at the interfaces n ₁→n_m for p-and s-polarized fields E⃗ ^p,s _k. Therefore, the field in Eq. (8) is separated in the p-and s-polarized components using the vector identity

where p⃗=k_xye⃗_z+k _1z e⃗ _xy and s⃗=k ₁ e⃗ _xy×e⃗_z with the unit vectors e⃗_xy=k⃗_xy/k_xy and e⃗_z=(0,0,1). Hence, the first term gives the p-polarized component E⃗^p_k and the second the s-polarized component E⃗^s_k. Using this in Eq. (8), the field propagating towards the interfaces is

The total field in the region 0<z<z ₀ is this downward propagating dipole field plus an upward propagating reflected field. Upon reflection, the p-polarized component becomes proportional to q⃗=k_xye⃗_z-k _1z e⃗_xy. Therefore, the total field in this region is

where r ^p,s _1m are the reflection coefficients on the structure n ₁ to n_m for p- and s-polarizations. The power dissipated by the dipole in the presence of the planar interfaces is obtained by inserting Eq. (14) in Eq. (2). Integrating over all directions of k⃗_xy, the dissipated power for a dipole at position r⃗ is

The dissipated power in case (b) can be rewritten as P _∞ plus a contribution ΔP_z(r⃗,Ω) from µ⃗_z perpendicular to the structure and a contribution ΔP_xy(r⃗,Ω) from µ⃗_xy parallel to the interfaces. That is P(r⃗,Ω)=P _∞+ΔP_z(r⃗,Ω)+ΔP_xy(r⃗,Ω) with

Figure 3 shows results for a horizontal and a vertical dipole near an air–glass and a water–glass interface, respectively. The dissipated power is significantly enhanced if the dipole-interface distance is less than about λ_fl/5. For the horizontal dipole, the power enhancement shows an oscillation caused by the interferences between the direct and the reflected field in medium 1 (air, water). For the vertical dipole, the enhancement simply decreases with increasing distance (approximately exponentially). In addition, the enhancement is significantly stronger for the vertical dipole, which reflects the stronger coupling of the p-polarized near field in medium 1 to waves propagating at super-critical angles in the denser medium 2 (glass).

 

Fig. 3. Enhanced power dissipated by a vertical and a horizontal dipole near an interface.

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Now, the interaction of a fluorophore with planar interfaces can be calculated. Taking into account that |µ⃗_z|=|µ⃗|cosΘ and |µ⃗_xy|=|µ⃗|sinΘ, the radiative enhancement factor γ(r⃗,Ω) is given by 1+ΔP_z(r⃗,Ω)/P _∞+ΔP_xy(r⃗,Ω)/P _∞.

In the limit z→∞, the integral vanishes which corresponds to the homogeneous case (a).

In general, the presence of the interface(s) in the vicinity of the fluorophore modifies its photophysical parameters and its emission pattern. For instance, the fluorescence quantumyield increases to

where q_fl is the fluorescence quantumyield in case (a) and q′_fl in case (b). Because the radiative enhancement is anisotropic, the fluorophore emits more frequently when its dipole moment is oriented vertically to the glass–water/air interface. In summary, the vicinity of a planar structure affects the rate of the electromagnetic emissions of the fluorophore. This influence manifests itself as a modification of the excited state lifetimes as well as the transition probabilities. In the general case, the fluorescence emission rate R_fl is given by

where ${\tau }_{S_1}$ is the lifetime of the excited singlet state S ₁ and ${P\prime }_{S_1}$ its occupation probability. But we should keep in mind that R_fl(r⃗,Ω) is the radiation rate into different channels. This radiation is either transmitted to the far field in medium 1 or m, coupled to a wave-guide mode or surface plasmon, or absorbed in the structure. If the fluorophore approaches a metal for instance, its emission rate will significantly increase as well as q′_fl does. But this increased emission will be counter-balanced by an increased energy dissipation, i.e. due to electron-hole excitations in the metal. For a fluorophore–metal distance ≲20nm, the energy loss usually becomes so dominant that the observable fluorescence intensity is effectively lower than that far from the metal. These and many more effects are well described in literature (c.f. [26, 27, 28, 29] for example). Here, we concentrate on the calculation of the absolute detection efficiency. For that purpose, the emitted field is calculated next and then propagated to the pinhole.

3. Emitted dipole field

In this section, the far field emission is calculated in view of describing the collection efficiency by the microscope objective. For this purpose, the radiated far field is described as a plane wave spectrum according to Eq. (4). Analogous to Eq. (8), (13) and (14), it is given by the dipole field superimposed by the reflected field in medium 1, both propagating towards a collection optics at z>z ₀.

The far field spectrum in medium m is given by the transmitted field propagating downwards. The Fresnel transmission coefficients through the structure are given by t _1m=t _m1 k _1z/k_mz and the field spectrum is

where d is the total thickness of all layers and o⃗=k_xy e⃗_z+k_mz e⃗_xy.

 

Fig. 4. Radiated angular power density ∝|E⃗_k|² for a horizontal dipole along the x-axis (arrow). The dipole is located at the glass–water interface.

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Figure 4 shows the far field spectrum emitted by a horizontal dipole along the x-axis. In case (a), i.e. no interface, the dipole radiates homogeneously around its axis (thin lines outline the xz and yz cross-sections). In case (b), i.e. on a glass–water interface, the dipole radiates mainly into two lobes in the yz-plane (thick lines). The maximum radiation indicates the critical angle for total internal reflection at the interface. The total radiated power increases by less than 10%, but more than 69% of the radiation is directed into the glass. Compared with case (a), the radiated power is substantially increased in the glass whereas it is decreased by about 33% in the water.

The emitted field E⃗_fl(r⃗,Ω) is calculated from these far field spectra of a dipole at position r⃗ with orientation Ω. This representation as a k⃗ spectrum of the p- and s-polarized components is required anyhow for calculating the propagation to the detector in the next section. We would like to emphasize that it is important to consider the fluorophore as a fixed power dipole, whose power is imposed by the current emission rate. Therefore, we could require P_fl(r⃗,Ω)=R_fl(r⃗,Ω)h̄ω_fl for normalizing the dipole moment |µ⃗(r⃗,Ω)| right here. Instead, we include this normalization in the detection efficiency when dividing the detected power by P_fl(r⃗,Ω) for obtaining the detection efficiency Q_fl(r⃗,Ω).

4. Detection efficiency

The detection efficiency is considered as the complementary part of the excitation field. We define it by the probability of receiving a photon in the detection aperture (pinhole) if this photon was emitted at position r⃗ by a fluorophore with dipole orientation Ω. Figure 5 summarizes the calculation of the detection efficiency Q_fl(r⃗,Ω), which is accomplished by calculating the ratio q_dT_flP_p(r⃗,Ω)/P_fl(r⃗,Ω) with P_p the power transmitted through the pinhole. q_d is the quantum yield of the detector [30] and T_fl the transmission efficiency of the filter set [31], both at the fluorescence wavelength λ_fl (not shown).

 

Fig. 5. Calculation of the electromagnetic field in the pinhole plane P. The objective and the tube lens are represented by their principal planes (thin lines; refraction loci ≡ principal planes), the object focus F _o, the aperture A in the back-focal plane of the objective, and the image focus F _i.

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P_p(r⃗,Ω) is obtained by integrating the intensity falling on the pinhole. This requires calculating the electromagnetic field in the pinhole plane P. Using the superscripts p and s for the p-and s-polarized components, the calculation of the field in the pinhole is subdivided into three distinct steps:

Step 1 The fluorescence emission is collected by the high NA objective.

The emitted field E⃗_k in the direction of the wavevector k⃗ is calculated based on Eq. (21) or (22). The field E⃗_o collected by the objective is essentially E⃗_k, but the phase is referenced to the object focus F _o. Reversing the calculation of the excitation field in [14] leads to the field E⃗_a in the objective aperture A (the reciprocity in optics was recently reviewed by Potton [32]). Considering medium n_m as the immersion medium allows us to identify n_t=n_m and k_t=k_m. Recall that f and R are the focal length and the aperture radius of the objective with numerical aperture NA. Then, Eq. (1,8) in [14] read as

The electric field arriving at the aperture A is therefore

The transmission coefficients t_at given by Eq. (21,22) in [14] were calculated from the aperture A to the immersion. For the reverse direction, they are given by t_ta=n_tt_at (A is in air).

Figure 6(a) shows the electric field |E⃗_a| in the apertureA of a 1.45 NA oil immersion objective observing a dipole in the focus at the cover slip–sample interface. For a x-oriented dipole (left), the field is relatively homogeneous at sub-critical angles (NA<1.33). At super-critical angles, it exhibits a significant increase in particular perpendicular to the dipole axis, e.g. along the y-axis. Along the x-axis, the field vanishes right at the critical angle. The vertical dipole (right) emits a rotationally symmetric field (p-polarized), which is particularly strong at super-critical angles.

 

Fig. 6. Electric fields in the aperture of a 100×1.45 NA oil immersion objective (a) and at the pinhole (b) for a dipole at the cover glass–sample (water) interface emitting at a wavelength of λ_fl=525nm. The circle in (b) indicates a pinhole of 50µm in diameter. The left half-pictures show the field of the horizontal dipole. The field of the vertical dipole is shown in the right half-pictures.

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Step 2 The fluorescence emission is propagated to the tube lens.

Because the field distribution in A can be described as paraxial and the propagation distance z_p-z_a≳120mm, the Fresnel approximation for this free space propagation can be applied. In general, the Fresnel approximation is valid for

This is a sufficient condition, which would demand a propagation distance z_p-z_a≳500mm. However, if the main contribution of the field E⃗_t at point (x_p, y_p, z_p) comes from points (x_a, y_a, z_a) close to (x_a, y_a, z_p), i.e. for paraxial fields, the Fresnel approximation is also valid for smaller propagation distances. This is achieved with the pseudo-paraxial Fresnel transformation for removing the wave front tilt [33] and an equivalent transform [34, 35] for reducing the wave front curvature.

In a first order approximation, this entire step can be skipped by setting the field E⃗_t on the tube lens equal to the field E⃗_a at the objective.

Step 3 The fluorescence emission is focused onto the pinhole.

The field E⃗_p near the image focus F _i in the pinhole plane is calculated with the Debye diffraction integral following the method by Leutenegger et al. [14]. If the focusing angles are small, the Fraunhofer approximation may be used.

Figure 6(b) shows the electric field |E⃗_p| at the pinhole P. The image of the horizontal dipole resembles a deformed Airy pattern, whereas the field of the vertical dipole is strongest in a ring around the axis (donut with small z-polarized component in the center).

Integration of the intensity $I_p=\sqrt{\frac{{∊}_0}{{\mu }_0}}\frac{{\mid {\overrightarrow{E}}_p\mid }^2}{2}$ over the clear aperture of the pinhole and normalization by the emitted power P_fl(r⃗,Ω) yields finally the detection efficiency Q_fl(r⃗,Ω) for an arbitrary position r⃗ and orientation Ω of the fluorophore. If we assume random orientation with a uniform probability P(Ω)=1/4π, the average detection efficiency is calculated as

In general, the energy flux through the aperture is given by integrating the normal component of the Poynting vector 〈S⃗〉=〈E⃗×H⃗〉 over the aperture area (see e.g. Eq. (21) by Enderlein et al. [6]). As we consider standard microscopes, the image NA is in general less than 0.05 [36], which yields a relative error 1-cosNA<0.2% if I_p is taken instead of 〈S⃗〉.

5. Results

Figure 7(a) shows the average detection efficiency for a 100×1.45 NA oil immersion objective observing randomly oriented fluorophores emitting at a wavelength λ_fl=525nm near the glass–sample interface. The projected pinhole diameter is 0.5µm on the interface, which results in a hemi ellipsoidal detection volume of 0.7µm base diameter and 0.5µm axial extension. Assuming q_dT_fl=1 [37], the peak detection efficiency at the interface is about 24%, which is 2–3× better than with the following 1.20 NA water immersion objective.

 

Fig. 7. Detection efficiency (through the cover glass) of isotropically oriented fluorophores achieved with two immersion objectives focused on the cover glass–water interface. The iso-surfaces show the efficiencies Q_fl(r⃗)=e ^-1…-4 Q_fl(0) in the sample. The oil immersion objective (a) has a peak detection efficiency of ≈24% and the water immersion objective (b) of ≈14%.

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The detection efficiency of a 40×1.20 NA water immersion objective at identical conditions is shown in Fig. 7(b). The projected pinhole diameter is 0.6µm on the interface and the peak efficiency is about 14%(65nm in the sample, ≈13% at the interface). The detection volume has a base diameter of 0.8µm and an axial extension of 0.8µm into the sample. No super-critical light is collected and the water immersion matches the refraction index of the sample, which results in a detection volume extending significantly deeper into the sample than with the oil immersion objective.

Figure 8(a) compares the detection efficiency for the outlined cases whereas Fig. 8(b) compares the performance of the water immersion objective for epi- and trans-illumination, that is for collection from the glass side and, respectively, the sample side. The detection efficiency with trans-illumination shows the typical undulations due to interferences between the direct and the reflected radiation. The 1.20 NA objective collects up to about 15% at a distance of 120nm from the interface, whereas it collects ≲12% on the interface. Such a trans-illumination configuration is often used in combinationwith TIRF (c.f. also Lieto et al. [38]), but is of advantage only if the observed fluorophores are at a distance of about λ_fl/5 from the glass interface.

 

Fig. 8. Detection efficiency of isotropically oriented fluorophores for immersion objectives focused on the cover glass–sample interface. White dotted lines outline an efficiency of 20% and 15%, white solid lines of 10%, black solid lines of 5%, dotted lines of 2% and 1% and ticked lines of 0.5%, respectively.

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6. Conclusions

We presented a complete framework for accurately calculating the detection efficiency of fluorescence emission at interfaces. Our calculation accounts for the spatial distribution of the emitted fluorescence and yields the absolute detection efficiency defined as the probability of collecting an emitted photon. We presented examples for fluorescence detection at a glass–water interface and compared the detection yields for total internal reflection fluorescence microscopy in epi- and trans-illumination. These calculations clearly show the superiority of the 1.45 NA objective for epi-TIRF, firstly because it outperforms the detection efficiency as well as the resolution of the 1.20 NA objective and, secondly, because it is able to provide an evanescent illumination without further equipment. We recently introduced epi-TIRF for fluorescence correlation spectroscopy (FCS) and showed experimentally a 2–3× increase of the molecular brightness [39, 40], which we attributed to the high detection efficiency in combination with an increased emission rate and an enhanced intensity in the evanescent excitation field. Our experimental findings fully support the presented calculations. Together, the focus field and detection efficiency calculations foster the understanding of existing concepts in fluorescence microscopy, but proved to be an indispensable element for quantitative microscopic molecular investigations.