1. Introduction

Fluorescence microscopy is an indispensable technique in imaging biological samples and novel materials. The technique became successful with the use of fluorescent nanoemitters such as semiconductor quantum dots (QDs) [1]. However, most of the optical properties of these nanoemitters strongly depend on their relative position in the sample plane, 3D orientation, and quantum yield (QY). Determining these parameters are crucial. In this paper, we demonstrate a universal method of obtaining the position, orientation, and relative quantum yield ratio (RQYR) of a fluorescent nanoemitter by combining laser scanning microscopy (LSM) and polarization measurement. The proposed methodology is applicable, but not limited, to experimental conditions involving fluorescent nanoemitters located at the interface of two media. We used ZnS coated CdSe quantum dot (CdSe/ZnS QD) fluorescent nanoemitters in the air-glass interface to validate our process. These QDs have isotropic 3D absorption dipole moment [2,3], allowing us to disregard the variation of absorption from one nanoemitter to another.

CdSe/ZnS QDs are among the widely used semiconducting fluorescent nanoemitters because of their strong emission, compatibility to biological materials, and ability to absorb excitation energy in any orientation due to its isotropic 3D absorption dipole moment [2,3]. Several techniques have been used to determine the CdSe/ZnS QDs orientation in the sample plane such as defocused imaging [4–6] and polarization measurements [7–9]. Defocused imaging utilizes defocused emission patterns on the pixels of the camera. Hence, it requires sophisticated image analysis to determine the orientation [4–6]. In polarization measurement, the analyzer dependence of the fluorescence emission determines the nanoemitter orientation [7–9]. The position of the nanoemitter on the sample plane must be established before the orientation. LSM is a versatile technique to scan and locate the nanoemitters. Sick et al. determined the location and orientation of a 1D dipole in a single molecule by obtaining the absorption/emission image profile [10]. CdSe/ZnS QD presents a challenge because of the differences in the absorption and emission dipole moment. The absorption occurs on the isotropic 3D dipole moment that includes a non-emitting “dark axis” parallel to the c-axis [2,3], as shown in Fig. 1(a). The emission emanates from a 2D degenerate dipole moment, often called the “bright plane”, located at the plane perpendicular to the nanocrystal c-axis as shown in Fig. 1(b) [2,11,12].

 

Fig. 1 Model used for (a) absorption and (b) emission dipole moment of CdSe/ZnS QD. The red dashed arrow in both (a) and (b) represents the QD c-axis aligned to the dark axis. The green solid arrow in (b) represents the degenerate dipole moment (bright plane) while the green dashed arrow is the projection of the bright plane on the sample plane.

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Aside from structural factors, intrinsic transition dipole moments, and excitation parameters, fluorescent emission is also dependent on the nanoemitter quantum yield (QY) defined as the ratio of the emitted photons to the absorbed photons. It is sensitive to the local environment since it can affect the zero-phonon line of the emitter [13]. The QY and excitation cross section dependence to the local environment has been observed in other works [14,15]. The QY of the nanoemitters can be modified by changing its local environment such as using sharp tip in near field scanning optical microscopy [16,17] and placing metallic nanoparticle beside the nanoemitter [18–20]. However, there might be unwanted transfer of energy resulting to quenching because of the presence of sharp tip and metallic nanoparticle. Investigating the position, orientation, and QY of the nanoemitters through non-invasive optical methods can improve our understanding the behavior of isolated nanoemitters.

Absolute measurement of QY requires the use of an integrating sphere [21]. The nanoemitters are prepared in a solution form placed in a cuvette for absorption and emission measurements [22]. This approach is challenging in single nanoemitter measurement level. We need to carefully quantify the photons absorbed and emitted by the nanoemitter. A microscope collects a fraction of the total light emitted by the nanoemitter, and the relative quantum yield or RQY can be measured using a standard or a reference sample [21–24]. The reference and the experimental sample must be carefully prepared in the same way and tested in an identical experimental configuration. In experiments and investigations involving production of uniform nanoemitters, the actual value of the QY may not be necessary, as long as they have the same QY. Aside from that, if the nanoemitters are produced with identical QY, then the discrepancy in the experimentally determined QY can be used to investigate the effects of local environment of the nanoemitter.

In this work, we introduce an easy and straight-forward alternative method to compare the QY of the nanoemitters without using the reference sample. We define relative quantum yield ratio (RQYR) as RQY of a nanoemitter with respect to a basis nanoemitter. We employed an all-optical and non-invasive LSM and polarization measurement to determine the position, orientation, and RQYR of a nanoemitter (CdSe/ZnS QD) in an air-glass interface. LSM with different excitation polarization are used to determine the position and intensity of the QD on the sample plane before conducting polarization measurement to determine the 3D orientation. The RQYRs of the QDs were calculated using the intensity and orientation obtained from LSM and polarization measurement.

2. Materials and methods

To prepare the CdSe/ZnS samples, the surface of a coverslip (thickness ~150 µm) was cleaned by reactive ion etching and rinsed in deionized water via sonication. It is then submerged in a 4% 3-Aminopropyltriethoxysilane (APTS) solution and rinsed by ethanol. CdSe/ZnS QD (Invitrogen Qdot) in phosphate buffer solution (PBS) was prepared (10 µL QD: 10 mL PBS) and a small amount of QD solution (~50 µL) is dispersed on coverslip before rinsing with distilled water.

Figure 2(a) shows the schematic diagram of the experimental set-up used for LSM image acquisition and polarization measurements. Linearly polarized laser light (λ = 532 nm) passes through a polarizer, half wave plate, and beam expander. A polarization converter (Z-pol, Nanophoton) enables selective generation of azimuthal and radial polarization by setting the appropriate polarization direction of the incoming linearly polarized light [25]. Removal of the polarization converter from the optical path generated linearly polarized excitation light. The conditioned beam was then directed to an inverted confocal microscope equipped with a high NA oil immersion objective lens (Nikon, 100×, NA = 1.49) that focuses laser light onto the sample mounted on a XYZ translation PZT stage. Fluorescence emission was collected by the same objective lens and passed through polarization detection section consisting of a half-wave plate and analyzer. The excitation signal is removed from the collected emission by a long pass edge filter (532 nm RazorEdge) prior to detection by a spectrometer (focal length = 300 mm, grating = 600 g/mm) equipped with a liquid nitrogen cooled charged coupling device (CCD) camera for spectral acquisition and an avalanche photodiode (APD) for LSM.

 

Fig. 2 (a) Schematic diagram of the experimental set-up. (P – polarizer, HWP – halfwave plate, BE – beam expander, Z-pol – polarization converter, OL – objective lens, XYZ-PZT – sample stage, NPBS – nonpolarizing beam splitter, A – analyzer, EF – long pass edge filter and L – lens) (b) Sample fluorescence signal of an individual QD nanoemitter..

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During the LSM scans, the wavelength response of the spectrometer was centered at 550 nm to coincide with the peak emission of the individual QD nanoemitter as seen in Fig. 2(b). A 5.0 x 5.0 μm² area was stage scanned and 40 nm pixel size with 30 ms observation time per pixel was used to obtain LSM image profile. An isolated QD was located for polarization measurement. Laser power was maintained at ~ 60 nW at the sample plane during polarization measurement to prevent photo-bleaching. Polarization measurements were performed by utilizing a fixed analyzer polarization directed into the spectrometer while rotating the scattered light polarization via a half-wave plate positioned prior to the analyzer. This arrangement circumvents the polarization dependence of the grating diffraction efficiency of the spectrometer. Fluorescence intensity signal were scanned at 10° increments averaged by 100 data points per angle with 30 ms acquisition time per data point.

3. Results and discussion

The LSM image profiles of the QDs shown in Figs. 3(a) to 3(c) were obtained using azimuthal, radial, and linear (along y – axis) polarization excitation, respectively. The inset in each figure corresponds to the computed electric field intensity distribution at the focus spot of the azimuthally, radially and linearly polarized light excitation [26,27]. The LSM image profile has the same spatial intensity distribution as the excitation electric field. It is donut-shaped image for azimuthal polarization, bright-centered ring for radial polarization and elliptical pattern for linear polarization. Appendix A provides a comprehensive discussion about the calculated electric field intensity distribution at the focus spot. More than 10 QDs are visible in the scan area and are separated within micron range.

 

Fig. 3 LSM image profile of the QD’s using (a) azimuthal, (b) radial and (c) linear polarization excitation (along y – axis). The numbering of the QDs in (a) applies to (b) and (c) as well. (d), (e) and (f) corresponds to sample intensity line scans of QD5 comparing experimental and calculated intensity cross – section of the LSM image profile with azimuthal, radial and linear light excitation (along x – and y – axis).

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We observe that the shape of LSM image profiles are similar from one QD to another even though the nanoemitters are randomly oriented. The similarity of the shape is a result of the 3D absorption dipole moment of CdSe/ZnS QD as depicted in Fig. 1(a). The projected absorption dipole moment along x –, y –, and z – axis remains the same for different QD orientation. The fluorescence emission rate, being proportional to ${\left|\overrightarrow{\mu }\cdot {\overrightarrow{E}}_{exc}\right|}^2$ ($\overrightarrow{\mu }$ is absorption dipole moment and ${\overrightarrow{E}}_{exc}$ the excitation electric field) [28,29], provides the same rate for every dark axis orientation. Effectively, LSM image profiles has the same shape as the excitation electric field distribution.

Figures 3(d) to 3(f) are the sample intensity line scans of QD5 (shown in Fig. 3(a)) excited with azimuthally, radially, and linearly polarized excitation, respectively. The experimental intensity line scans are in good agreement with simulated LSM image profile line scans. Similarly, an intensity line scan using polarization parallel to x- axis for the case of linearly polarized light agrees with the simulated LSM image profile line scan. Due to the similar intensity images of the QDs, the resulting LSM image profile is not enough to obtain the QD orientations. Appendix B presents a detailed discussion regarding this. Nevertheless, the LSM image profile can be used to determine the position and intensity emitted by the QD. Since a coverslip (with QDs at the top) was placed in an XYZ-PZT stage to control the location and depth QDs, the electric field intensity at the focus spot in z-position is the same for all QDs. The LSM can be applied to determine any sample depth by Z-PZT for other random fluorescent nanoemitter samples.

After locating the QDs, polarization measurements were conducted to determine the QD orientation. The results of the polarization measurements of the 11 selected QDs are shown in Fig. (4). Each measurement is normalized to the highest intensity for each QD and fitted with the function given by

 

Fig. 4 Normalized intensity of 11 QDs selected on the LSM image profile at different analyzer angle with fittings.

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We found that intensity modulation varies with each QD. For example, QD2 and QD10 exhibits higher modulation than QD1 and QD6. The phase of the intensity modulation also varies for each QD. QD2 and QD10 exhibits a maximum at 68° and 123° respectively. The bright plane of the QD (Fig. 1(b)) located to a plane perpendicular to the dark axis explains the differences in the modulation and phase. The dark axis is parallel to the nanocrystal c – axis and is therefore an indicator of the QD 3D orientation [9]. If the dark axis is oriented at some nonzero angles Θ and Φ, the projection of the bright plane along x – and y – axis (dashed green arrow in Fig. 1(b)) are not the same [2]. It results to modulation in the intensity and phase difference from polarization measurement that can be used to determine the out-of-plane (Θ) and in-plane orientation (Φ) of the QD relative to the sample plane [2,8,9].

The reconstructed dark axis orientation in comparison to the LSM image profile is shown in Fig. 5(a) using the calculated out-of-plane (Θ) and in-plane (Φ) orientation of the QD’s. We did not observe any dark axis oriented exactly parallel and perpendicular to the optical axis. The observed out-of-plane orientation of the 11 QDs are distributed from 14°–84° as shown in the histogram in Fig. 5(b). This preference in this angular range can be a result of geometrical structure of CdSe/ZnS QD [8]. The QDs used in the experiment have an elongated structure with faceted surface on both ends which could lead to preferential orientation in this angular range [8].

 

Fig. 5 (a) QD dark axis orientation (right) in relation to the LSM image profile (left). (b) Histogram of the out-of-plane orientation of the 11 QDs observed.

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The intensity (time-integrated fluorescence emission rate) of LSM image profiles of the QD as detected in the experiment is a product of the collection efficiency (CE) of the objective lens, QY of the nanoemitter, and the interaction of the absorption dipole moment with the excitation electric field. It can be expressed by the equation

Our results from the polarization measurement enabled us to determine the QD orientation and calculate the CE. Table 1 shows the CE for each QD. In general, a higher CE leads to a brighter LSM image profile. We observed QDs with high CE but with relatively dark LSM image profiles; QD5 with CE of 0.82140 and QD10 with CE of 0.8348. Although QD10 has a higher CE compared to QD5, QD5 is brighter than QD10. We have ruled out intensity changes due to our controlled experimental configuration. Therefore, the QY is not the same for all the QD observed in the LSM image profile.

Table 1. CE, maximum intensity from LSM image profile (radial and linear) and relative quantum yield ratio (radial and linear) of the QDs. QD11 is the basis QD.

View Table

Equation (3) allows us to determine QY after computing the CE. However, the measurement of the QY relies on the electric field interacting with the absorption dipole moment and the calibration of the power emitted by the dipole to the intensity measured by the APD. On the other hand, measurement of RQY depends on the QY of a nanoemitter in terms of the QY of a reference nanoemitter (QYR). It is convenient to compare the QY of one nanoemitter to another without a reference sample. We define the RQYR_i of the ith QD as the RQY_i of the ith QD compared to the basis QD within the sample area. This is given by the equation

The observed maximum intensity in the LSM image profile using radially polarized light is lower than linearly polarized light because of lowered transmission from the polarization converter. However, the calculated RQYR for radially polarized light is not always lower than with linearly polarized light. 7/10 QD’s have higher radial RQYR than the linear RQYR. We also observe that the radial and linear RQYRs are within 10% - 70% and 10% - 50% of the basis QD respectively. The QY of each QD differ with one another irrespective of the excitation polarization, implying that the differences lie on the properties of QDs itself. This comparison is based on the 550 nm region of the emission as detected by the APD. This analysis can be extended by incorporating the full fluorescence spectrum of the QD.

The deviation in the RQYR of the QDs can be a result of several processes in its core-shell structure and local environment. The QD samples used has a polymer coating around the core-shell structure to protect the QD from mechanical degradation and facilitate molecular conjugating in aqueous solution [29]. Our experiment was performed in ambient condition. Hence, the occurrence of QD photo-oxidation is possible. Oxygen interaction can happen by oxygen diffusion through the polymer layer or exposure of core-shell structure by mechanical defects [31]. Oxygen acts as an electron acceptor [32], changing the energy transfer mechanism in the QD. Changes in the QD’s immediate environment can modify the local electric field due to changes in refractive index [33].

The intrinsic structural differences between the observed QDs can also contribute to the observed changes in the RQY. For QD’s with small diameter, the absorption cross section decreases [34] and the quantum efficiency of the functionalized QD increases [35]. It effectively changes the fluorescence emission. The lattice defects in the QD can also create non-radiative recombination pathways, causing a decrease in fluorescence intensity [31]. In these possibilities, the effective number of photons absorbed and/or emitted changes, causing discrepancies in the RQYR values.

4. Conclusion

In conclusion, we presented a universal method of determining the position and 3D orientation of an individual fluorescent nanoemitter, and proposed a new approach to relate the QY of one nanoemitter to another using RQYR. Our proposal is validated by using CdSe/ZnS QD as the nanoemitter. We first perform LSM to determine both the location and intensity of the QD emission, and conduct polarization measurement to determine the QD’s 3D orientation. From the measured intensity and extracted CE based on QD’s orientation, we determined the RQYR of QDs relative to a basis QD. The results, in general, demonstrate that a relatively simple and non-invasive optical technique can be used to obtain more local information about fluorescent nanoemitter by combining the results from LSM and polarization measurements.

Appendix A Electric field distribution at the tightly focused spot with linear, radial, and azimuthal excitation polarization

The absorption and emission properties of nanoemitter depends strongly with the excitation electric field. If an incident polarized light is tightly focused on the interface (x – y plane, z = 0) of medium 1 (air) and medium 2 (glass) using a high numerical aperture (NA) objective lens, the electric field vector [26,29] for a linearly, radially, and azimuthally polarized light are given by

(6)$${\overrightarrow{E}}_{lin}=\frac{1}{2}\left(F_0(\theta )+F_2(\theta )\frac{x^2-y^2}{x^2+y^2}\right)\widehat{x}+\frac{1}{2}\left(F_2(\theta )\frac{2xy}{x^2+y^2}\right)\widehat{y}+i{F}_1(\theta )\frac{x}{\sqrt{x^2+y^2}}\widehat{z}$$

(7)$${\overrightarrow{E}}_{rad}=i{G}_0(\theta )\frac{x}{\sqrt{x^2+y^2}}\widehat{x}+i{G}_0(\theta )\frac{y}{\sqrt{x^2+y^2}}\widehat{y}+-G_1(\theta )\widehat{z}$$

(8)$${\overrightarrow{E}}_{azi}=H(\theta )\frac{y}{\sqrt{x^2+y^2}}\widehat{x}+H(\theta )\frac{x}{\sqrt{x^2+y^2}}\widehat{y}$$

The quantities $F_0(\theta ),F_1(\theta ),F_2(\theta ),G_0(\theta ),G_1(\theta )$ and $H(\theta )$ [26,29] are given by

(9)$$\begin{array}{l}{F}_0(\theta )={\displaystyle {\int }_0^{\theta NA}\sqrt{\cos \theta }}\sin \theta J_0\left(-k_x\sqrt{x^2+y^2}\right)\left(t_p(\theta )\sqrt{1-{\left(\frac{n_1}{n_2}\sin \theta \right)}^2}+t_s(\theta )\right)d\theta \\ F_0(\theta )={\displaystyle {\int }_0^{\theta NA}\sqrt{\cos \theta }}{\sin }^2\theta J_1\left(-k_x\sqrt{x^2+y^2}\right)\left(\frac{n_1}{n_2}{t}_p(\theta )\right)d\theta \\ F_0(\theta )={\displaystyle {\int }_0^{\theta NA}\sqrt{\cos \theta }}\sin \theta J_2\left(-k_x\sqrt{x^2+y^2}\right)\left(t_s(\theta )-t_p(\theta )\sqrt{1-{\left(\frac{n_1}{n_2}\sin \theta \right)}^2}\right)d\theta \end{array}$$

(10)$$\begin{array}{l}{G}_0(\theta )={\displaystyle {\int }_0^{\theta NA}\sqrt{\cos \theta }}\sin \theta J_1\left(-k_x\sqrt{x^2+y^2}\right)\left(t_p(\theta )\sqrt{1-{\left(\frac{n_1}{n_2}\sin \theta \right)}^2}+\right)d\theta \\ G_0(\theta )={\displaystyle {\int }_0^{\theta NA}{(\cos \theta )}^{1/2}}{\sin }^2\theta J_0\left(-k_x\sqrt{x^2+y^2}\right)\frac{n_1}{n_2}{t}_p(\theta )d\theta \end{array}$$

(11)$$H(\theta )={\displaystyle {\int }_0^{\theta NA}\sqrt{\cos \theta }}\sin \theta J_1\left(-k_x\sqrt{x^2+y^2}\right)t_p(\theta )d\theta$$

The electric field vectors are functions of index of refraction of medium 1 and 2, spatial coordinates, angle covered by the NA of the objective lens, angle between the observation point and the optical axis (θ), and the wavenumber of the electromagnetic wave. t_s(θ) and t_p(θ) are the Fresnel transmission coefficients in s – and p – polarization respectively, and J_n’s are Bessel Function of order n. The electric field vectors are based on the interference of incident, reflected, and diffracted electric field [26,27].

Figure 6 shows the corresponding intensity distribution of different incident electric field component at the focus spot for azimuthal, radial and linear polarization excitation. Intensity distribution for electric field component along x –, y –, and z – axis and excitation polarization are different to one another. Focused radially and linearly polarized light has longitudinal component due to the high NA of the objective lens [14,26]. Because of this, the in-plane and out-of-plane component of the absorption dipole moment (ADM) can be excited. The intensity distribution of all the electric field components (last column) shows a donut-shaped pattern for azimuthally polarized light, bright-centered ring for a radially polarized light, and a bright elliptical shape for a linearly polarized light.

 

Fig. 6 Intensity distribution of different electric field components at the tightly - focused spot in the air – glass interface with azimuthally, radially and linearly polarized light excitation (N.A. = 1.49).

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Appendix B Absorption and emission image profile

Absorption image profile is an image describing how the incident excitation electric field interacts with ADM. If the ADM is parallel to the polarization of incident electric field, there will be maximum absorption by the molecule. In this work, we define the absorption image profile to be the intensity distribution governed by the equation $I_{abs}={\left|{\overrightarrow{\mu }}_{abs}\cdot {\overrightarrow{E}}_{exc}\right|}^2$, with ${\overrightarrow{\mu }}_{abs}$ as the orientation of ADM and ${\overrightarrow{E}}_{exc}$ the excitation electric field at the tightly focused spot. When a molecule with an arbitrary ADM is illuminated, the out-of-plane component will be excited by the longitudinal component of the incident electric field while the in-plane component will be excited by the transverse component of the incident electric field. For the case of CdSe/ZnS QDs, the ADM is isotropic. The QD ADM component along the x – y – and z – axis is the same at any QD orientation. Because of this, the shape and magnitude of the intensity distribution is constant at any QD orientation, as seen in Fig. 7(a). The dark region corresponds to maximum absorption while the white region corresponds to the minimum absorption.

 

Fig. 7 (a) Absorption image profile of CdSe/ZnS QD for azimuthally (top row), radially (middle row) and linearly (bottom row) polarized light excitation at varying dark-axis orientation. (b) Emission image profile of QD placed in an interface at varying QD orientation.

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Generally, the emission from the dipole is detected, not the absorption. However, the emission is highly dependent on the absorption process. We define the emission image profile of a nanoemitter to be the time-integrated fluorescence emission rate images [28,29] as collected by the objective lens, written as

(12)$$I_{ems}=(CE)\cdot (QY)\cdot {\left|{\overrightarrow{\mu }}_{abs}\cdot {\overrightarrow{E}}_{exc}\right|}^2$$

where CE is the collection efficiency (discussed appendix 4) and QY is the quantum yield. Effectively, the emission image profile will look like the absorption image profile. For CdSe/ZnS QD, this results to unchanging shape of emission image profile at different dark axis orientation as seen in Fig. 7(b). This is observed experimentally in the work of Chizhik et al. [3]. Because of this, emission image profile cannot be used to determine the 3D orientation of CdSe/ZnS QD.

Appendix C Analyzer dependence of the dipole emission

Polarization measurement takes advantage of the polarized emission from the dipole passing through an analyzer to determine the nanoemitter orientation [2,8]. The electric field component of the dipole emission parallel/antiparallel to the analyzer will be detected while the perpendicular component will be minimized. Intensity modulation as a function of the analyzer can then provide orientation information about the nanoemitter. Fig. 8 illustrates the model used in describing the analyzer dependence of the dipole emission.

 

Fig. 8 Schematic representation of the model used to describe analyzer dependence of the dipole emission

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The electric field from the dipole emission located within the interface of two media with refractive index n₁ and n₂ [29,36,37] is given by

(13)$$\overrightarrow{E}=(E_{\theta }\cos \theta \cos \phi -E_{\varphi }\sin \phi )\widehat{x}+(E_{\theta }\cos \theta \sin \phi +E_{\varphi }\cos \phi )\widehat{y}+(-E_{\theta }\sin \theta )\widehat{z}$$

The E_θ and E_φ are given by

(14)$$E_{\theta }=f(r)\left[C_j^2({\mu }_x\cos \phi +{\mu }_y\sin \phi )\cos \theta -C_j^1{\mu }_z\sin \theta \right]$$

(15)$$E_{\phi }=f(r)\left[-C_j^3\left({\mu }_x\sin \phi -{\mu }_y\cos \phi \right)\right]$$

with f(r) a complex function of the distance from the dipole to the observation point [29,36,37]. The dipole orientation is given by

(16)$$\overrightarrow{\mu }=\sin \Theta \cos \Phi \widehat{x}+\sin \Theta \sin \Phi \widehat{y}+\cos \Theta \widehat{z}$$

where Θ is the out-of-plane orientation and Φ the in-plane orientation. For the case of CdSe/ZnS QD, the emission is described by the “bright plane” [2,11,12], which is a combination of 1D dipole emitter oriented parallel to sample plane (Θ₁ = 90°, Φ₁ = Φ + 90°) and another dipole oriented at (Θ₂ = Θ + 90°, Φ₂ = Φ) [4]. Equivalently, the 2D dipole emitter orientation is dictated by the orientation of the dark axis given by (Θ, Φ).

The subscript j ∈ {1,2} in Eqs (14) and (15) corresponds to the electric field emission above and below the air – glass interface respectively and the coefficients $C_{j=1,2}^{i=1,2,3}$ are listed as

(17)$$C_1^1=1+r_p(\theta )$$

(18)$$C_1^2=1+r_p(\theta )$$

(19)$$C_1^3=1+r_s(\theta )$$

(20)$$C_2^1={\left(\frac{n_2}{n_1}\right)}^2\frac{\cos \theta }{s_z(\theta )}{t}_p(\theta )$$

(21)$$C_2^2=\frac{n_2}{n_1}{t}_p(\theta )$$

(22)$$C_2^3=\frac{n_2}{n_1}\frac{\cos \theta }{s_z(\theta )}{t}_s(\theta )$$

where k is the wavenumber of the radiation and the function s_z(θ) is equal to ${\left(\left(\frac{n_1}{n_2}\right)-{\sin }^2\theta \right)}^{1/2}$,which is real for subcritical angle and is imaginary for supercritical angle.

The analyzer can be represented mathematically by

(23)$$\overrightarrow{A}=\cos \alpha \widehat{x}+\sin \alpha \widehat{y}$$

Effectively, the intensity at a given analyzer angle [8] can be obtained using the equation

(24)$$I={\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^{{\Theta }_{NA}}{\left|\overrightarrow{E}\cdot \overrightarrow{A}(\alpha )\right|}^{2}A(\theta )r^2}}\sin \theta d\theta d\phi .$$

with apodization factor A(θ) taken into consideration. The in-plane and the out-of-plane orientation of the nanoemitter be extracted from the phase and intensity modulation by the analyzer.

Figure 9 illustrates the intensity dependence of CdSe/ZnS QD emission as a function of the dark axis orientation and the in-plane orientation set at Φ = 0. Each out-of-plane orientation has unique intensity modulation with analyzer angle. Intensity has no modulation for Θ = 0°, corresponding to dark axis aligned along the optical axis. This absence of modulation is a result of bright plane having a circular projection along the sample plane with analyzer allowing equal amount of electric field all throughout the angle. The modulation increases as the dark axis angle increases because of deviation from the circular projection of bright plane in the sample plane. This leads to unequal projection of the bright plane along the x – and y – axis. There is unique value of modulation for every dark axis orientation and is therefore an indication of the out-of-plane orientation.

 

Fig. 9 Intensity of QD emission as a function of dark axis orientation and analyzer angle.

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The minimum value of intensity modulation at a given out-of-plane orientation of the dark axis is obtained when the analyzer angle is parallel / antiparallel to the in-plane orientation of the dark axis. The maximum is obtained when the in-plane orientation is perpendicular to the analyzer angle. Because of this, we can use the minimum value of the intensity modulation with the analyzer angle to determine the in-plane orientation.

Appendix D Collection efficiency

Fluorescence emission will occur after the nanoemitter is excited. However, only part of the total power radiated by the nanoemitter is collected by the objective lens. We define collection efficiency (CE) as the capacity of the objective lens to collect the fluorescence emission from the nanoemitter. It is the ratio of the total power collected by the objective lens to the total power emitted by nanoemitter as defined by Eq.(4). If the nanoemitter is represented by a dipole emission, the power can be expressed from the fluorescence electric field using the Poynting vector

(25)$$\overrightarrow{S}=\frac{1}{2}\mathrm{Re}(\overrightarrow{E}\times {\overrightarrow{H}}^{*})$$

and

(26)$$P=r^2〈\overrightarrow{S}〉\cdot \widehat{e_r}$$

where $\widehat{e_r}$ is the unit vector in radial direction [29].

The $P_{coll},P_{{}_{ems}}^{\uparrow }$ and $P_{{}_{ems}}^{\downarrow }$ in Eq. (4) can be expressed as the integral of the angular power distribution $P_{coll},P_{{}_{ems}}^{\uparrow }$ and $P_{{}_{ems}}^{\downarrow }$ given by the equations

(27)$$\begin{array}{l}{P}_{coll}={\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^{{\theta }_{NA}}{p}_{coll}^{\downarrow }}}(\Theta ,\Phi ,\theta ,\phi )\sin \theta d\theta d\phi ,\\ P_{ems}^{\uparrow }={\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_{\pi /2}^{\pi }{p}_{ems}^{\uparrow }}}(\Theta ,\Phi ,\theta ,\phi )\sin \theta d\theta d\phi ,\\ P_{ems}^{\downarrow }={\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^{\pi /2}{p}_{ems}^{\downarrow }}}(\Theta ,\Phi ,\theta ,\phi )\sin \theta d\theta d\phi .\end{array}$$

For air – glass interface, the $P_{{}_{ems}}^{\downarrow }$ will be composed of dipole emission above and below the critical angle [29,38]. $P_{{}_{coll}}^{{}^{\downarrow }}$ is dependent to the NA of the objective lens. For θ_NA greater than the critical angle, the objective lens can collect emission above the critical angle, increasing the collection efficiency.

The $P_{coll},P_{{}_{ems}}^{\uparrow }$ and $P_{{}_{ems}}^{\downarrow }$ are dependent on the dipole orientation, hence, CE is also dependent to the dipole orientation. In a homogeneous media, the angular power distribution of a dipole with 1D EDM is

(28)$$p=\frac{3}{8\pi }{\left[\cos \Theta \sin \theta +\sin \Theta \cos (\phi -\Phi )\cos \theta \right]}^2+\frac{3}{8\pi }{\left[\sin \Theta \cos (\phi -\Phi )\right]}^2$$

where the 1st and 2nd term is angular power distribution in p – and s – polarization respectively [4,38]. If the dipole is placed between 2 dielectric media (air-glass), the angular power distribution $P_{{}_{ems}}^{\downarrow }$ is composed of emission below $\left(p_{\theta <\theta c}\right)$ and above $\left(p_{\theta >\theta c}\right)$ the critical angle (θ_c), given by

(29)$$\begin{array}{l}{p}_{\theta <\theta c}=\frac{3}{2\pi }\{\frac{n^3{\cos }^2\theta {\left[\cos \Theta \sin {\theta }_1+\sin \Theta \cos (\phi -\Phi )\cos {\theta }_1\right]}^2}{n^2-1}\\ +\frac{n^3{\cos }^2\theta {\sin }^2\Theta {\sin }^2(\phi -\Phi )}{{(\cos {\theta }_1+n\cos \theta )}^2}\}\end{array}$$

and

(30)$$\begin{array}{l}{p}_{\theta <\theta c}=\left(\frac{3}{2\pi }\frac{n^3{\cos }^2\theta }{n^2-1}\right)[\frac{n^2{\cos }^2\Theta {\sin }^2\theta }{(n^2-1){\sin }^2\theta -1}\\ +\frac{{\sin }^2\Theta {\cos }^2(\phi -\Phi )(n^2{\sin }^2\theta -1)}{(n^2-1){\sin }^2\theta -1}+{\sin }^2\Theta {\sin }^2(\phi -\Phi )]\end{array}$$

with $n=n_2/n_1>1$ and $n_1\sin {\theta }_1=n_2\sin \theta$ [38]. Effectively, the P_coll will be given by

(31)$$\begin{array}{l}{P}_{coll}={\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^{\theta c}{p}_{\theta <\theta c}(\Theta ,\Phi ,\theta ,\phi )}}\sin \theta d\theta d\phi \\ +{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_{{\theta }_c}^{{\Theta }_{NA}}{P}_{\theta >\theta c}}}(\Theta ,\Phi ,\theta ,\phi )\sin \theta d\theta d\phi .\end{array}$$

For CdSe/ZnS QD, the fluorescence emission originates from a degenerate 2D EDM or bright plane oriented perpendicular to the dark axis [2,11,12].

The plot of the collection efficiency (NA = 1.49) of the CdSe/ZnS QD emission against the QD dark-axis out-of-plane orientation is shown in Fig. 10. A collection efficiency of ~ 80% has been observed which is due to the high collection angle from the objective lens and presence of interface from the difference in the refractive index of glass and air.

 

Fig. 10 Collection efficiency of the CdSe/ZnS QD on an air – glass interface using high NA objective lens.

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Results showed that about 81.6% of the emission is collected if the QD out-of-plane orientation is parallel to the optical axis, while 84.3% of the QD emission is collected if the out-of-plane orientation is parallel to the interface. This difference in the CE influences the intensity collected by the objective lens and must be taken into consideration whenever QY investigation using objective lens is used.

Acknowledgment

The experiment and analytical part of the research has been partially done in the Near-Field NanoPhotonics Research Lab of RIKEN during the internship of LLD and RBJ.

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