## 1.

## Introduction

Absorption and fluorescence light microscopy are largely equivalent to the spatial mapping of molecular dipole transitions at high resolution. Most molecules feature a linear transition dipole moment **μ** in which case the transition rate is proportional to
|**μ⋅E**|^{2},
whereby
**E**≡(E_{x},E_{y},E_{z})
defines the electric field in the focus.^{1} Since the rate depends on the mutual orientation of the two vectors, controlling the orientation of the field **E** is highly desirable. This is readily accomplished in a spectrometer with two orthogonal, low aperture lenses; this principle has also been successfully applied in confocal imaging.^{2}
^{3} However, when flat sample mounting conditions and high collection efficiency are required an orthogonal lens arrangement is impossible. In this case, the transverse nature of light precludes a strong component
E_{z}
along the optic axis. As the
μ_{z}
component is less accessible, molecules whose transition dipoles are chiefly oriented in the *z* direction will absorb less. Moreover, the lack of a *z*-polarized diffraction maximum prevents the measurement of the full orientation of **μ** at high spatial resolution, which is of particular concern in single molecule spectroscopy.^{4}
^{5}
^{6} Significant longitudinal fields are created in surface-bound near-field optical microscopy,^{7} but the results obtained by this method are entangled with proximity effects. In high-resolution far-field microscopy, the convenient determination of **μ** is regarded as difficult.^{8}

Electromagnetic focusing theory^{9} reveals that the spherical curvature of the wave front gives rise to a longitudinal component
E_{z},
which received some attention in high aperture confocal^{10}
^{11} and 4Pi-microscopy theory.^{12} However, in these works
E_{z}
never was of primary importance because the orientation in the focal maximum is that of the incoming wave. Catalyzed by the observation of strong longitudinal fields in near-field optics and the desire to establish the three-dimensional (3D) orientation of the transition dipole of a molecule, longitudinal orientations in focused light fields recently became of strong interest to single-molecule spectroscopy and near-field optics.^{13}
^{14} In an interesting study, free-space modes with radially polarized fields^{15}
^{16} were calculated, and it was shown that these fields are accompanied by significant *z*-polarized components. In this arrangement, however, the *z*-polarized field was still of the order of its *x* and *y* counterparts. Comparatively strong *x* and *y* components are not disturbing when “imaging” spatially *isolated* point objects such as individual single molecules, because the difference between their spatial structure facilitates the spatial separation of their individual contribution. In other words, the *x, y* and *z* components produce characteristic spatial patterns of fluorescence that can be readily distinguished.

In recent calculations it was shown that parabolic mirrors also lead to major longitudinal fields because of their strong amplitudes at high focusing angles; the application of a confocal pinhole was considered in these studies,^{17} as was also the potential application of this scheme for 3D imaging. However, the definition of an imaging mode producing a measured signal that is predominantly due to the *z*-polarized field has not been tackled. The *z*-polarized microscope (ZPM) proposed herein is technically robust and employs only one or two binary phase plates and a polarizer. Engineering the effective point-spread function (PSF)^{18} with a suitable detection PSF creates a microscope with a signal *primarily* stemming from molecules with strong
μ_{z}.

## 2.

## Concept and Results

Our theoretical studies of ZPM are performed by calculating the focal field with the Huygens–Fresnel principle^{19}

## (1)

$$\mathbf{\text{E}}(x,y,z)=A{\int}_{0}^{\alpha}{\int}_{0}^{2\pi}P(\varphi )\sqrt{\text{cos}\text{}\theta}\text{}{\mathbf{\text{E}}}^{\prime}(\theta ,\varphi ){e}^{\mathit{iks}(\theta ,\varphi )}\times \text{sin}\text{}\theta \text{}d\theta \text{}d\varphi ,$$*A*a constant, and the angles (θ,ϕ) point on the Gaussian reference sphere. If the field is linearly polarized in the

*x*direction, the field

**E**

^{′}(θ,ϕ) across the wave front is

## (2)

$${\mathbf{\text{E}}}^{\prime}(\theta ,\varphi )=\left(\begin{array}{c}{\text{sin}}^{2}\text{}\varphi +{\text{cos}}^{2}\text{}\phi \text{}\text{cos}\text{}\theta \\ \hfill \text{sin}\text{}\varphi \text{}\text{cos}\text{}\varphi (\text{cos}\text{}\theta -1)\hfill \\ \hfill \text{cos}\text{}\varphi \text{}\text{sin}\text{}\theta \hfill \end{array}\right)\text{.}$$*x,y,z*); $\rho =\sqrt{{x}^{2}+{y}^{2}}$ and ϕ

_{P}=arctan(y/x). The term

*f*denotes the focal length of the system. P(ϕ) is a phase function across the entrance pupil, which we seek to modify to enhance E

_{z}.

In regular focusing,
E_{z}
vanishes along the optic axis because the *z* components that are symmetric with respect to the optic axis are reverted in sign. Figure 1 shows an arrangement generating a focal electric field with a predominant
E_{z}
component. An *x*-polarized, plane wave front passes a phase plate inducing a phase shift of π between the upper
(x>0)
and lower
(x<0)
half of the beam, so that
P(ϕ)=sign(cos φ).
The beam then passes an annulus covering a centered circular area of 90 of the total aperture, thus ensuring that the illumination occurs exclusively with rays that are strongly bent with respect to the axis. Due to the phase step, the *z*-oriented components are now in phase at the optic axis, whereas those of the *x*- and *y*-oriented components vanish for the same reason. Evidently, a semiaperture angle
α→90°
would produce the strongest *z*-polarized field; however, in this study we confine ourselves to
α=67.3°
which corresponds to an oil immersion lens of 1.4 numerical aperture. We elect a wavelength
λ=500 nm;
results for other wavelengths can be gained by simple scaling.

Figure 2 shows the intensity PSF
|**E**|^{2}≡|(E_{x},E_{y},E_{z})|^{2}
when introducing the above phase plate. By comparing
|E_{x}(x,y,z=0)|^{2}
and
|E_{y}(x,y,z=0)|^{2}
with
|E_{z}(x,y,z=0)|^{2},
shown in (a), (b), and (c), respectively, we contrast the intensity distribution of the transverse components with that of the longitudinal component in a quadrant of the focal plane. They fundamentally differ from the distribution of a regular high-aperture lens:
E_{x}
and
E_{y}
vanish at the focal point and their main peak is shifted off axis. The calculations show that
|E_{y}|^{2}⩽8.24
and
|E_{x}|^{2}⩽76.5.
Molecules with a substantial
μ_{x,y}
are excited when slightly offset from the focal point, which is not desired. However, the main maximum is indeed *z* polarized [Figure 2(c)] with a full width half maximum (FWHM) of 110 nm in the *x* and a FWHM of 160 nm in the *y* direction. It is accompanied by higher order side maxima with relative intensities of 51.8 and 20.8 in the *x* direction. Clearly, the insertion of a simple phase plate produces a pronounced main maximum that is entirely polarized in the *z* direction.

An immediate benefit is that the relative strengths of the components
μ_{x,y,z}
of a molecule can now be quantified by creating distinct field orientations in the diffraction maximum. For this purpose one subsequently records a set of three “*xy* images” of a molecule with focal spots of *x, y*, and *z* polarization, respectively; the *x*- and *y*-polarized PSFs are readily created by removing the phase plate and rotating the field with a retarder. Since the molecule is much smaller than the focal dimensions, the center of symmetry of the *xy* images coincides with the focal center, in which the molecule experiences a field that is exclusively *x, y*, or *z* polarized. One has to take into account that, due to the different distribution, in the *z*-polarized PSF the intensity at the focal point is 0.701 times smaller than for *x* and *y* polarization, so that the fluorescence generated by this PSF should be multiplied by 1.42. The full orientation of a linear molecular transition dipole can be determined with high spatial definition.

So far, our study applies to conventional and scanning microscopy. Although the maximal intensity of the *z* component is stronger by a factor 1.31 than that of its *x*-polarized counterpart, Figure 2 reveals that the latter is still strongly pronounced. In addition, the *z*-polarized field features significant lobes. To create a microscope with a predominant longitudinal field contribution, we now confocalize the microscope in a special manner. We implement a detection path that is similar to that for excitation (Figure 1) but rotate the phase plate by 90° and insert an analyzer pointing in the *y* direction. Another distinct difference to the illumination path is that we do not employ an annulus. By denoting the emitted field with **F** we obtain for the contribution probability of an arbitrarily oriented molecule with fixed transition dipoles

## (3)

$$\begin{array}{ccc}h(x,y,z)\hfill & \hfill =\hfill & {C}_{1}\cdot {\int}_{0}^{\pi}{\int}_{0}^{2\pi}{|\mathbf{\text{E}}\cdot \mathbf{\mu}|}^{2}{|\mathbf{\text{F}}\cdot \mathbf{\mu}|}^{2}\text{}\text{sin}\text{}{\theta}_{d}\text{}d{\theta}_{d}\text{}d{\phi}_{d}\hfill \\ \hfill & \hfill =\hfill & {C}_{2}\cdot ({h}_{x}+{h}_{y}+{h}_{z}+{\displaystyle \sum _{ij}}{h}_{\mathit{ij}})\hfill \end{array}$$**μ**≡(cos φ

_{d}sin θ

_{d},sin φ

_{d}sin θ

_{d},cos θ

_{d}). ℜ{ } denotes the real part. Importantly, the rotated phase plate causes strong contributions from E

_{x}to match low values F

_{y}but E

_{z}and F

_{z}to coincide favorably in space (see Figure 2), so that contributions of the longitudinal field to the signal, h

_{z}, will be emphasized. The “mixed contributions” h

_{ij}are calculated to be negligible. The integral is performed over all dipole orientations in space, so that Eq. (3) gives the probability of an indefinitely oriented molecule to contribute to the signal. Therefore, the expression in Eq. (3) is the effective PSF of a ZPM.

The values of
h_{x,y,z}
in the focal plane are displayed for
λ=500 nm
in Figures 3(a), 3(b), and 3(c), respectively. A typical 8–10 difference in wavelength induced by the Stokes shift has been neglected for simplicity. The FWHM is 100 and 136 nm in the *x* and the *y* direction, respectively. The fact that the FWHM is sharper than in regular confocal microscopy (144 nm in *x* and 128 nm in *y* direction) stems from the fact that marginal rays are employed for illumination, as in annular aperture confocal microscopy. The *yz* section of
h_{z}
[Figure 3(d)] demonstrates that the confocalization with a full circular aperture compensates for the axial elongation due to the annular illumination. As a result, the main focal maximum of ZPM is confined to the region around the focus. In Figure 4 we compare the profiles
h_{z}(x,0,0)
with
h_{x}(x,0,0)
revealing that the contribution from the longitudinal excitation field dominates the signal in the detector.

## 3.

## Discussion and Conclusion

An advantage of ZPM is the mapping of fluorescence molecules in an object whose transition dipoles are lined up along the optic axis, such as fluorophores in inner or outer membranes of cells. In conjunction with images taken with regular (*x* or *y*) polarized fields, molecular orientation relaxation should be measurable in all directions. If the quantum efficiency of the fluorophore is known, ZPM can be used for quantitative polarization spectrometry and the determination of absolute values of
μ_{z}.
As
μ_{x}
and
μ_{y}
are found by operating the microscope in a conventional manner, one can establish the orientation and the total magnitude of **μ** at microscopic scale. The scheme does not require any interferometry so that ZPM should be facile to implement in a regular confocal microscope, where it would emphasize the contribution of *z*-oriented molecules. An important advantage of ZPM over other arrangements with significant longitudinal field components is that its effective PSF is engineered in such a way that the contributions of the undesired transverse components are weakened by the orthogonal orientation of the illumination and detection PSF. This makes ZPM particularly interesting for the imaging of the *z*-polarized features in randomly oriented agglomerations of molecules, as is the case in biomedical microscopy.

The arbitrary change of the field orientation will be equally important to nonlinear interactions of the field with the molecule, as the orientation of the focal field to the first and second order hyperpolarizibility tensors is crucial for the effective generation of nonlinearly induced scattering or fluorescence signals. Therefore, we anticipate that ZPM will also be relevant to multiphoton absorption, higher harmonics, as well as coherent-anti-Stokes Raman scattering imaging spectroscopy at high spatial resolution.

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