## Abstract

We theoretically investigate the use of spatial light modulators (SLMs) for transformation of the collected fluorescence field in a high numerical aperture confocal microscope, for improved molecular orientation determination in single-molecule spectroscopy. The electric vector field in the back aperture of the microscope objective is calculated using the Weyl representation and taking into account components emitted at angles above the critical angle of the coverglass-immersion fluid interface. The coherently imaged fluorescence undergoes spatially-dependent phase and polarization transformation by the SLMs, before it passes to a polarization beamsplitter, and is subsequently focused onto two pinholes and single-photon detectors.

©2008 Optical Society of America

## 1. Introduction

The study of fluorescent molecules near surfaces by optical methods has a long history [1, 2]. Past research has concerned both the efficient optical excitation of the molecule and its anisotropic fluorescence emission. Also, formulas governing the transformation of a polarized beam of light on imaging through an aplanatic optical system have been presented in a seminal paper [3]. In more recent work, radially and azimuthally polarized beams with high-numerical aperture (NA) optical systems have been studied and used to achieve superresolution and to manipulate the polarization of the focused optical field [4–6]. The application of liquid crystal spatial light modulators to adjust the spatial profile of the polarization of the incident laser field and thereby to control the direction of the electric field at the focus was proposed [7], and simulations of the intensity pattern at the microscope image plane that results from a single-molecule dipole at given orientations excited by such polarization-engineered beams have been presented [8]. Defocused imaging in a wide-field optical microscope was used to infer single-molecule orientations without polarization optics [9] and wave-optical modeling of the imaging was presented [10]. Also, the dipole-emission intensity-pattern in the back aperture of the objective has been used for determination of the molecule emission-dipole orientation by fitting the observed image of the intensity profile with theoretical calculations [11]. Theoretical comparison of the fluorescence collection efficiency of a surface-bound molecule using a parabolic mirror and high-NA microscope objective have emphasized the importance of the so-called “forbidden light” emitted at angles above the critical angle in single-molecule detection [12].

In this work, we extend these previous studies to consider spatially-dependent phase and polarization manipulation of the collected electric field-amplitude components of a single photon that is emitted from a single-molecule dipole near an interface. Following the spatially-dependent manipulation, the field-amplitude contributions are split into two beams by a polarization beamsplitter and then coherently summed by focusing each beam to a point detector. Without the manipulation, the optical field collected by a high-NA objective will contain contributions emitted in different angles that have different polarizations and phases. Thus the superposition of these contributions is in general elliptically polarized and each photon then has a statistical probability to be registered at each of the two detectors. Appropriate engineering of the collected electric field can recover an electric field with pure linear polarization from a molecular dipole of any chosen orientation so that every photon falls to the same detector. Thus polarization pupil engineering of collected fluorescence would provide improved fidelity in fluorescence polarization anisotropy measurements with high-NA optical collection.

In Section 2 we describe briefly the angular distribution and polarization of light emitted by a dipole in the vicinity of the coverglass-immersion fluid interface. The proposed optical system to manipulate the spatial profile of phase and polarization of the collected optical field is described in Section 3. A mathematical description and computational illustrations of the electric field in the exit pupil of the single-molecule microscope are presented in Section 4. The phase and polarization transformations, which may be implemented by a phase retarder and polarization rotator, are derived in Sections 5 and 6, respectively. Section 7 discusses polarization transformations due to focusing of the light onto the detectors and shows how the detected signal is calculated. In Section 8, we demonstrate that the proposed polarization engineering of the collected single-molecule emission leads to unambiguous determination of molecule orientation and maximization of the degree of polarization for the targeted dipole orientation. Section 9 summarizes the advantages of the proposed polarization engineering technique for low photon budget single-molecule spectroscopy.

## 2. Dipole emission near a planar interface

The optical field collected with a low numerical aperture lens from a dipole oriented laterally (perpendicular to the optical axis) is approximately linearly polarized, as shown in Fig. 1(a). However, as shown in Section 4, for a high-NA lens, the collected field varies across the pupil of the lens and in the marginal zone it is elliptically polarized. A high-NA lens also collects longitudinal components of emission, and for a dipole at focus oriented longitudinally (along the optical axis), the collected field is radially polarized, as shown in Fig. 1(b). If the dipole is located within a fraction of a wavelength *λ* from a planar interface, such as a microscope cover-glass, then emission occurs preferentially into the interface at angles above the critical angle, *θ _{cr}*, as shown in Fig. 1(c). If collected, this so-called forbidden light is dominant and is in general elliptically polarized. Overall, the field varies in phase and polarization across the pupil of the lens, no matter what the orientation of the emitting dipole may be. However, spatial light modulators may be used to transform the phase and polarization of the field so that it becomes spatially homogeneous and linear, as described in Section 3.

## 3. Polarization engineering of collected light

An optical system for implementing the polarization and phase engineering in the back aperture of the objective is presented in Fig. 2. The phase and polarization of the collected light is to be manipulated so that for any chosen targeted dipole orientation (TDO) every photon will fall to the same detector while for any other dipole orientation (ODO) photons will fall to both of the two detectors. The emission of the dipole P, located a small distance above the coverglass COG, which is at the focal point of a high-NA objective OBJ, is collected by the objective into a collimated beam. (The axial locations of subsequent components are hence non-critical.) The beam passes to a polarizing beamsplitter PBS1, which directs the electric field components with vertical polarization (parallel to the *η*-axis) into one arm (*η*-arm) of an interferometer and with horizontal polarization (parallel to the *ξ*-axis) into the other arm (*ξ*-arm). For the TDO, a spatial light modulator (SLM) that acts as a spatially-dependent phase retarder RET adjusts the phase of the optical field components in the *η*-arm with respect to those in the *ξ*-arm, so that when the two beams are combined at the polarizing beam combiner PBC the resulting field is linearly polarized throughout the cross section of the beam. Generally, for any ODO, the beam in the marginal zone of the exit pupil will remain elliptically polarized. The beam then passes to a second SLM that acts as a spatially-dependent polarization rotator ROT so that for the TDO the resulting beam becomes linearly polarized in the same direction throughout the cross section of the beam. Sections 5 and 6 derive the spatially-dependent profiles of the phase retarder and of the polarization rotator that are required in order to convert the input spatially varying polarization of the optical field into a profile that is uniformly linearly polarized along a chosen direction for any chosen TDO.

The goal is to transform the optical field from the TDO into a uniformly linearly polarized beam that will be fully transmitted through the second polarizing beamsplitter PBS2. Thus each photon detected from a molecule with orientation equal to the TDO will be registered at DET_{ξ} with 100% certainty. On the other hand, the transformed optical field resulting from a molecule with any ODO will be distributed between the two detectors DET_{ξ} and DET_{η}, so that each photon has a statistical probability for being registered at either detector. However, it will be seen (in Figs. 8 and 9) that the probability for a photon to be registered at DET_{ξ} is a smooth monotonic function of the deviation between the molecule orientation and the TDO, and hence polarization engineering provides a means for measuring the angle between any single-molecule orientation and the TDO. Single molecules interacting with a surface often have a preferred orientation due to their shape and binding forces. The SLMs can be adjusted prior to a measurement, so the TDO is chosen to be aligned with this expected molecule orientation, or even in an iterative manner during a measurement, so that the TDO becomes aligned with the actual orientation. Also, screening of many single molecules to find those that are aligned with a fixed TDO, or observation of fluctuations of a single molecule orientation about some TDO will be possible.

Optical systems other than those presented in Fig. 2 could also be implemented to effect the same transformations. For single-molecule spectroscopy applications, the goal is to transform the spatial profile of the polarization for the TDO into a uniformly linearly polarized beam without attenuation of any part of the optical field, so that the single-photon detection efficiency remains high.

## 4. Electric field in back aperture

To determine the required spatially-dependent phase retardation and polarization rotation operations, we need to know the electric field distribution in the back aperture (BA) of the microscope objective. The mathematical formulations for the electric field of the dipole emission collected in the BA of an aplanatic objective are given in several papers [3, 11, 12]. Here we follow Ref. [11]. Figure 3 introduces the notation.

The time independent part of the optical field in the BA, **E**(*ξ*,*η*) is composed of vector components along the local *p*, *s* coordinate system:

From Ref. [11], the complex amplitudes of the *p*- and *s*-polarized components of the field, for emission from a dipole located a small distance *δ*<*λ* above the interface, may be expressed as

where

and the phase shift due to distance from the interface is

with *k _{i}* = 2

*πn*/

_{i}*λ*, where

*λ*is the vacuum wavelength. The quotient of $\sqrt{\mathrm{cos}\theta}$ at the end of Eqs. (2) and (3) accounts for the conservation of energy of the beam as it is refracted by an aplanatic lens [3]. In Eqs. (4) and (5),

*t*and

^{p}*t*are the Fresnel transmission coefficients for

^{s}*p*- and

*s*-polarized light [13], given by

These coefficients are complex for *θ*>*θ _{cr}*. Note that emission characteristics of a dipole far from an interface can be deduced by setting

*n*=

_{i}*n*in the above equations.

To express the optical field in the BA in terms of the global *ξ*,*η* coordinate system, with unit vectors *ê _{ξ}* and

*ê*, the following equations are inserted into Eq. (1):

_{η}where

Also, the angle *θ* in Eqs. (2) and (3) is given as

where *f* is the effective focal length of the objective, and *θ* is limited by the numerical aperture *NA* of the objective and the refractive index *n* to *θ*<*θ*
_{max} = sin^{-1}(*NA*/*n*). When Eqs. (9–12) are inserted into Eqs. (1), (2), and (3), the optical field in the BA plane will be expressed in terms of the global coordinates:

where *E _{ξ}* and

*E*denote the magnitudes and

_{η}*φ*and

_{ξ}*φ*the phases of the components of the field along the

_{η}*ê*and

_{ξ}*ê*unit vector directions.

_{η}The polarization of the electric field is elliptical if the phase difference *ε* between the *η* and *ξ* components, in the range [0,π), differs from 0, where

Thus for for *θ*<*θ _{cr}*, the electric field polarization is linear, but with a direction that varies with

*θ*,

*ϕ*. However for

*θ*>

*θ*, the field is in general elliptically polarized, as the Fresnel coefficients

_{cr}*t*and

^{p}*t*are complex with different phases for the incident evanescent waves. The electric field polarization is linear throughout the whole back aperture, albeit with a direction that depends on

^{s}*ϕ*, only for one special case, that for which the dipole is perpendicular to the surface (

*θ*= 0). In this case, the field in the BA is radially (

*p*-) polarized, as shown in Fig. 1(b).

A program in MATLAB was developed to compute the electric field distribution in the BA by insertion of Eqs. (2), (3), (9), and (10) into Eq. (1). The magnitude of the field amplitude, the polarization, and the phase difference between the components of the electric field, *ε* given by Eq. (14), are shown in Fig. 4 for three selected orientations of the dipole. The crucial role of the forbidden light collection is apparent in these figures from the large magnitude of the field for angles *θ*>*θ _{cr}*. All results shown are for an oil immersion objective of numerical aperture NA=1.4 with a coverglass refractive index of

*n*= 1.52, an emission wavelength of 532 nm, and with the molecule embedded in a medium with refractive index

*n*= 1.33.

_{i}## 5. Polarization engineering by phase retardation

The role of the phase retarder RET is to change the phase *φ _{η}* (

*ξ*,

*η*) of the beam in the

*η*branch to make the combined beam leaving the polarizing beam combiner PBC linearly polarized for the targeted dipole orientation (TDO), which is at Θ

_{T},Φ

_{T}. This may be accomplished by the following transformation:

The transformation retards the phase *φ _{η}* (

*ξ*,

*η*) at the back aperture points where the electric field

**E**(

*ξ*,

*η*) is elliptically polarized. The phase retardation to be programmed onto the spatial light modulator RET is given by Eq. (14) and is in the range [0,π).

The action of the phase retarder on the beams for other dipole orientations (ODO) at Θ,Φ, may be described by the following formula:

The electric field leaving the PBC will read

This electric-field polarization is linear (albeit with different directions) at all points of the back aperture for the TDO; however for any ODO it may be (and in most cases is) elliptically polarized for the back aperture points of *θ*>*θ _{cr}*.

## 6. Polarization rotator

The polarization rotator ROT makes the polarization of the TDO electric field components collinear with the *ξ* axis. The beam polarized parallel to the *ξ* axis will be fully transmitted through the polarizing beamsplitter PBS2 and focused on detector DET*ξ*. Moreover, the polarization transformation by polarization rotator ROT ensures that the instantaneous electric field vectors in the whole exit pupil are directed in the same (positive) *ξ* direction to support a constructive coherent addition of the electric field components parallel to the detector surface upon focusing by a low NA lens. The rotated electric field **E″**
_{TDO} is described by the following expression:

where

and the angle *φ _{rot}* (

*ξ*,

*η*) is given by the formula:

The angle *φ _{rot}* is positive for counterclockwise rotation of the vector. The necessary phase shift to be implemented by ROT is in the [0,2

*π*) range.

Polarization of the electric field components emitted by ODO dipoles may not be collinear with the *ξ* axis after passing the polarization rotator ROT, moreover they may be elliptically polarized for the back aperture points corresponding to *θ*>*θ _{cr}*. Their electric vector is described by the following general formula:

Figure 5(a) shows that the transformed field from a dipole with orientation Θ,Φ, equal to that of a TDO (in this case with Θ_{T} = 30°, Φ_{T} = 45°, but with any choice of orientation possible), is linearly polarized along the *ξ* axis throughout the BA, so that it would pass PBS2 and fall entirely on DET_{ξ}, whereas Figs 5(b) and 5(c) show two examples in which the same transformation yields inhomogeneous polarization profiles for ODOs, which would result in a statistical division of photons between the DET_{ξ} and DET_{η}.

## 7. Focusing of beam onto detector

In order to calculate the probability of photon detection at each detector, we evaluate the *z*-component of time-averaged Poynting vector from the electric and magnetic field components at each of the detectors. To this end, we first compute the polarization components of the electric field **E**
_{t} (*ξ*,*η*) transmitted through the focusing lens (LENS in Fig. 2) of each of detectors DET_{ξ} and DET_{η} and projected onto a reference spherical surface with the vertex point in the exit pupil of lens and the sphere center at the detector. The transformed field **E**
_{t}(*ξ*,*η*) is the plane wave spectrum of the focused field at the detector, thus the focused field can be computed from the weighted Fourier transform of the aperture field **E**
_{t}(*ξ*,*η*) [14].

To compute a change of the electric-field polarization upon transmission through the focusing lens, the incident field is decomposed into *p* - and *s* - polarization components:

$${E}_{s}^{{\mathrm{DET}}_{\xi}}(\xi ,\eta )=-{E}_{\xi}^{\u2033}(\xi ,\eta )\mathrm{sin}{\phi}_{\mathrm{ps},\xi \eta},$$

$${E}_{s}^{{\mathrm{DET}}_{\eta}}(\xi ,\eta )={E}_{\eta}^{\u2033}(\xi ,\eta )\mathrm{cos}{\phi}_{\mathrm{ps},\xi \eta},$$

where *φ _{ps,ξη}*(

*ξ*,

*η*) is the angle between the

*p*axis of the local

*p*,

*s*coordinate system and the

*ξ*axis of the

*ξ*,

*η*coordinate system. A positive angle

*φ*(

_{ps,ξη}*ξ*,

*η*) describes counterclockwise rotation of the coordinate system. The

*s*-polarization component does not change its orientation on passing through the lens; however, the vector of the

*p*-polarization component is rotated by

*θ*, as shown in Fig. 3. The

_{t}*ξ*and

*η*components of the electric field after passing the lens are described by the following formulas

$${E}_{\mathrm{t\eta}}^{{\mathrm{DET}}_{\xi}}(\xi ,\eta )={E}_{p}^{{\mathrm{DET}}_{\xi}}(\xi ,\eta )\mathrm{sin}{\phi}_{\mathrm{ps},\xi \eta}\mathrm{cos}{\theta}_{t}+{E}_{s}^{{\mathrm{DET}}_{\xi}}(\xi ,\eta )\mathrm{cos}{\phi}_{\mathrm{ps},\xi \eta},$$

$${E}_{\mathrm{t\eta}}^{{\mathrm{DET}}_{\eta}}(\xi ,\eta )={E}_{p}^{{\mathrm{DET}}_{\eta}}(\xi ,\eta )\mathrm{sin}{\phi}_{\mathrm{ps},\xi \eta}\mathrm{cos}{\theta}_{t}+{E}_{s}^{{\mathrm{DET}}_{\eta}}(\xi ,\eta )\mathrm{cos}{\phi}_{\mathrm{ps},\xi \eta}.$$

The electric field at a given point (*ξ*,*η*) in the back aperture is proportional to the spatial frequency (-*k _{tξ}*/2

*π*,-

*k*/2

_{tη}*π*) component of the spectrum of the electric field at the detector. The magnetic field vector (

*H*(

_{tξ}*ξ*,

*η*),

*H*(

_{tη}*ξ*,

*η*)) for each spatial frequency is calculated from the electric field for that spatial frequency (which may be interpreted as a plane wave) and the wave impedance of the medium. The electric and magnetic fields in the detector plane are calculated using the inverse Fourier transform representation of the Debye diffraction integral [14]:

where *f* is the focal length, *λ*
_{0} is the wavelength, *k _{x}* and

*k*are the wave vector components. The time averaged

_{y}*z*-component of the Poynting vector is then evaluated using the formula

Formulas (26–28) hold for both detectors. Superscripts *DET _{ξ}* and

*DET*were omitted to simplify the notation.

_{η}## 8. Detection of molecule orientation

We define the signed degree of polarization (*SDP*) as a rational measure to infer dipole orientation Θ,Φ

where *P _{ξ}* and

*P*are the normalized and time averaged

_{η}*z*-components of the Poynting vectors integrated over the detector surface for DET

_{ξ}and DET

_{η}, respectively

$${P}_{\eta}(\Theta ,\Phi )=\frac{1}{{P}_{n}}\underset{{D}_{\eta}}{\int}{S}_{z\eta}\left(x,y;\Theta ,\Phi \mid {\Theta}_{T},{\Phi}_{T}\right)d\sigma ,$$

Here, the Poynting vector *S* at each detector plane is that for a dipole with orientation on Θ,Φ, whose field has been engineered by SLMs set for a TDO with orientation Θ_{T}, Φ_{T}. Also, the normalization factor *P _{n}* is the integral over the detector DET

_{ξ}surface of the time averaged

*z*-components of the Poynting vectors computed for a dipole oriented longitudinally (Θ=0,Φ=0) and for no polarization engineering employed:

The value of *P _{n}* would be the same if DET

_{η}would be chosen in the above definition due to radial symmetry of the longitudinal dipole emission.

The advantages of the presented polarization engineering technique may be demonstrated by a comparison with a simple polarization beamsplitting in the back aperture and focusing the polarized beams onto two detectors DET_{ξ} and DET_{η}. For this benchmark technique, many and very different orientations give us the same signals on the detectors. For example, the signals *P _{ξ}* and

*P*for the TDO of Θ

_{η}_{T}= 30°,Φ

_{T}= 45° cannot be distinguished from those of many other angles Θ, Φ, as shown in Fig. 6. This makes it impossible to solve the inverse problem: infer the orientation of the dipole from the measurements. Figure 7 shows the

*SDP*for the simple beamsplitting technique. We can observe that a single value of the

*SDP*corresponds to a large range of (Θ,Φ) values located on the magenta-colored lines (at Φ=±45°) in Fig. 7(b). Thus without polarization engineering, it is impossible to determine Θ when Φ=±45°, yet in many applications, the angle of inclination Θ is particularly of interest. The signal on the detectors and the degree of polarization corresponding to the TDO are marked by a blue dot in Figs. 6 to 9.

The proposed polarization engineering technique directs all photons for the TDO to the detector DET_{ξ}, as shown in Fig. 8. Thus the *SDP* is equal to 1 for this dipole orientation, i.e., the solution of the inverse problem is unambiguous. A small change in the dipole orientation in the vicinity of the TDO results in a small change in the SDP, as shown in Fig. 9. The SDP is a smooth monotonic function in the vicinity of the TDO, so that the polarization engineering enables a stable solution of the inverse problem, i.e., a measure of the SDP can determine the deviation of the dipole orientation Θ, Φ from that of the TDO Θ_{T}, Φ_{T} within a range of Θ, Φ values around the TDO. The contours around the peak in Fig. 9(a) or 9(b) give an indication of the deviation of the dipole orientation Θ, Φ from that of the TDO Θ_{T}, Φ_{T} when the measured SDP is less than 1. For example, if the SDP equals 0.9, then the dipole orientation is determined to be about 25° from that of the TDO.

For single-molecule experiments with low photon counts, the precision of the SDP may be limited by shot noise, particularly if the measured SDP differs from 1. For statistically efficient analysis of such measurements, maximum-likelihood (ML) methods can be used to determine the range of possible values that the orientation Θ, Φ could be, within a given statistical confidence. With the SLMs set for detection of a given TDO Θ_{T}, Φ_{T}, if the dipole orientation is actually Θ, Φ, the probability that a photon falls to DET_{ξ} is given by

where *B _{ξ}* and

*B*are the probabilities for background at each detector. The above probability is inserted into the binomial distribution to determine the probability for

_{η}*n*,

_{ξ}*n*photon counts if the orientation were Θ,Φ

_{η}This function is normalized to find the probability density function Pdf that the orientation is within sinΘ*d*Θ*d*Φ of Θ, Φ:

A contour plot of this probability density function gives the confidence intervals that the orientation is within a given range of values.

The above results were computed for infinitely large detectors. The proposed polarization engineering technique provides an additional advantage for confocal detection through a pinhole or small detectors. Uniform orientation of linear polarization in the focusing lens aperture leads to an on-axis constructive interference in the focal region of a low-NA focusing lens and a high detection efficiency. Contrary to this, a variation in orientation of the linear polarization leads to destructive on-axis interference and focusing of photons outside a small detector.

## 9. Conclusions

In this paper we proposed a polarization engineering technique implemented in the back aperture of a high-NA microscope to simultaneously perform single-molecule spectroscopy and unambiguous single-molecule orientation detection. The technique applies spatially-dependent phase and polarization manipulation of the collected electric field-amplitude components of a single photon that is emitted from a dipole near an interface, for example, with the use of spatial light modulators. The field-amplitude contributions in the back aperture are split into two beams by a polarization beamsplitter and the phase of one of these beams is manipulated in a spatially-dependent manner to enable conversion of the elliptical polarization of the forbidden light into a field with linear polarization but spatially varying orientation. Both beams are combined, and then a spatially-dependent polarization rotation is implemented followed by a second polarizing beamsplitting operation. The obtained optical fields are coherently summed by focusing each beam to a detector.

Without the manipulation, the optical field collected by a high-NA objective will contain contributions emitted in different angles that have different polarizations and phases, so that the superposition of these contributions is in general elliptically polarized. Appropriate engineering of the collected electric field can recover an electric field with uniform linear polarization from a molecular dipole of any chosen targeted orientation. All photons emitted by a molecule with the emission dipole orientation of interest will be collected onto one of the two detectors. The polarization and phase engineering makes determination of the molecule orientation unambiguous and improves the statistical fidelity of single-molecule spectroscopy measurements. On the other hand, photons emitted by a molecule at some other dipole orientation will be statistically distributed between both detectors. However, the polarization and phase engineering enables a measure of the deviation of the molecule from the targeted orientation, as the fractions of photons that fall on each detector are smooth monotonic functions of the deviation. It is worth to mention, that the method is not based on an image recording which is difficult at low photon levels and requires image recording and processing hardware. Our further work will include implementation of the theoretically demonstrated polarization engineering using subwavelength diffractive optical elements.

## Acknowledgments

This project was supported by the Center for Laser Applications at the University of Tennessee Space Institute and DARPA grant W911NF-07-1-0046.

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