Elliptical mirror based imaging systems permit aperture angles greater than π/2 to be realized. It is therefore possible to collect part or all of both the forward and backward diffractive fields emitted from single molecules. In this paper we derive rigorous formulae for the image intensity when the single molecule is modeled as a dipole emitter. It is found in theory that the point spread function can be 2.44 times narrower at full-width-half-maximum in the axial direction when using an elliptical mirror with the maximum aperture angle of 2π/3 as compared with a parabolic mirror system with the aperture angle of π/2 whereas the side lobe level is increased by only 0.21% when the dipole is oriented along Z axis.
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There is a great deal of interest in increasing the resolution of far-field optical microscopy to the nano-scale and a number of approaches have been proposed [1, 2]. However, whichever approach is chosen the collection of high frequency information is fundamental in order to achieve high resolution. The aperture angle of the imaging objective is therefore a crucial factor in determining the resolution. Traditional refractive objectives have a maximum theoretical aperture angle of π/2, although this limit is rarely achieved in practice. Any method to increase this angle is likely to prove important. Another reason for wishing to expand the aperture angle is to increase the signal measured from a given small volume of the specimen. This is especially important in single molecule and live cell imaging. As pointed out in Ref [3, 4], a resolution increase by a factor of 2 requires four times more signal to image a single plane and eight times more if a 3D volume is to be sampled, and as a result, improved image quality can sometimes be obtained by gathering a larger fraction of the emitted light rather than simply by exposing the specimen with increased illumination.
Synthetic aperture imaging is a technique for aperture expansion which usually involves replacing a single high numerical aperture (NA) lens by a group of low numerical aperture lenses [5, 6]. However, this does not strictly lead to an expansion in solid angle and the maximum aperture angle is still under π/2. A recent application of synthetic aperture techniques was concerned with the 3D imaging of living cells in translational motion without the need of axial scanning of objective lens, rather than for NA expansion . The 4Pi-confocal fluorescence microscopy is often described in terms of an increased solid angle of illumination and collection. As calculations demonstrate an axial resolution up to four times that of a confocal microscope can be achieved in the case of two-photon excitation [8, 9]. In fact, the major contribution to this increased resolution is caused by the interference of waves from the two counter-positioned objective lenses. However, the maximum aperture angle of the individual lens is still below π/2, and hence high frequency information is not captured as discussed previously.
The use of parabolic mirrors have been proposed in confocal microscopy and spectroscopy in order to enlarge the NA to a value close to 1.0  and since they are also free from chromatic aberration . However, in terms of aperture angle there is no fundamental change in parabolic mirror based imaging (PMBI) as compared to refractive objectives since the NA is still close to unity and the diffractive field which can be captured is still restricted to the forward field.
In this paper, therefore, we will discuss the use of elliptical mirror based imaging (EMBI) since such elements can have an aperture angle greater than π/2. We will base our discussion on the imaging of single molecules which we will model as dipole emitters. When excited these molecules emit both forward and backward propagating fields and the use of an imaging element with a collection angle greater than π/2 permits part of the backward propagating field to be collected. We will compare the performance of elliptical mirror based systems with parabolic mirror based systems. It is worth noting that in comparison to the use of parabolic mirror with plane wave illumination the point source illumination of an elliptical mirror system has advantage in terms of aberration control when using a large mirror in conjunction with stage scanning, since there is no need to fabricate an additional large collimator as is needed in the parabolic mirror case.
2 Model of dipole emitter imaging
Since our analysis will be concerned with aperture angles greater than π/2 it is necessary to restrict ourselves to the imaging of transparent specimens where both forward and backward diffractive fields exist. Single molecule imaging is a particularly important example since not only is this of great interest in a number of application in biology , biochemistry , material sciences  and quantum optics [15, 16], but also because it naturally requires us to include polarization effects in our analysis. We shall model the single molecule as a dipole emitter.
In order to simulate an imaging in an elliptical mirror based system, a dipole emitter is located in vicinity of first geometrical focus F1 as shown in Fig. 1 . The far-field diffraction generated by the dipole emitter can be expressed as [17, 18]
As shown in Fig. 1, the field generated by the dipole emitter at F1 which propagates within the angle, α, can be detected at F2. Compared with the conventional definition of an open aperture, the part of the diffractive field corresponding to α≤π/2 can be defined as the forward diffractive field since the propagation is in the positive direction of the Z axis, whereas the remaining part of the field, α>π/2, is defined as the backward diffractive field. This backward diffractive field cannot be collected by traditional refractive lenses. s0 and s1 are the unit vectors describing the rays incident on and reflected from the elliptical mirror, g0 and g1 are the unit vectors of corresponding electric fields perpendicular to s0 and s1. s0, s1, g0 and g1 are all in current meridional plane ZON and may be written as.
In Eqs. (2)-(5), unit vectors i, j and k are defined for coordinate axes X, Y and Z respectively. The electric field, e1, immediately after reflection may be resolved into components in the perpendicular planes, g1 and g1 × s1, and can hence be written as
For the sake of brevity, energy loss caused by mirror reflection and absorption has been neglected. The time independent part of the electric field in the vicinity of focal point F2 may now be expressed as
The refractive index of medium nm in the vicinity of first geometrical focus F1is taken to be unity for the case of free space focusing. rm is the vector from focus F1 to the location of the molecule, rp is the distance vector from focal point F2 to the dipole emitter image, and rp can be written as
The apodization factor of an elliptical mirror l(θ) has previously been derived in Ref . In the calculations which follow the dipole moment p is described in terms of unit vectors i, j and k corresponding to the orientation along axes X, Y and Z. αmax and θmax may be geometrically from the elliptical mirror parameters a, b and c. The parameters of the elliptical mirror to be used need to be optimized before actual design and fabrication. The relationship between object and image aperture angles can be worked out using α = arctan[(z + c)tanθ/(z-c)], α∈[0, π/2) and α = π-arctan[(z + c)tanθ/(c-z)], α∈[π/2, αmax]. The distance between point M and F1 along Z is denoted using z, z = [ab2(1 + tan2θ)1/2-a2ctan2θ]/(a2tan2θ + b2).
3 Dipole orientation and 3D imaging
In conventional and confocal microscopy, a molecule is generally modeled as a radiating electric dipole [16, 20, 21]. The polarization of dipole emitters is known to effect the 3D point spread function (PSF) and hence the 3D resolution of nano or sub-micro scale observations.
In order to understand how the dipole orientation affects the 3D point spread function when the aperture angle becomes greater than π/2, the cases when the dipole is oriented along the X,Y and Z directions respectively will now be discussed. The influence of the illumination excitation is not considered in this paper which rather concentrates on the basic performance of the EMBI when part of the backward diffractive field is detected.
The aperture angle of traditional refracting objectives is usually far below π/2. For example the aperture angle of a 1.4 NA oil immersion lens is only 0.76 (π/2). The PMBI system is, to our knowledge, one of the few cases where aperture angles close to π/2 have been reported and supported by experiments [10, 22]. We therefore choose to compare these systems with EMBI to highlight the significance of the ability of to collect part of the backward propagating field.
We emphasize that Eq. (9) is used for calculations in the EMBI case whereas Eq. (7) of Ref  is used for the PMBI case. We also note that the magnification of the EMBI system was chosen to be four together with values of a = 500 mm and b = 300 mm in order to make the performance comparable with the system described in Ref .
3.1 Dipole orientation in the X and Y directions
The full width at half maximum (FWHM) is widely used as a measure of spatial resolution. However, side lobe levels are also important and so we will also consider the amplitude of the main side lobe (ASL) relative to the peak amplitude of main lobe. This later comparator is likely to be important in terms of noise. A lower ASL indicates a higher signal-noise ratio in real imaging.
To illustrate the performance of EMBI when the aperture angle is greater than π/2, the dipole is firstly assumed be oriented in the X direction. The corresponding distribution of electric field, |E|2 in the imaging space are plotted along axes X (ϕ = 0), Y (ϕ = π/2) and Z in Fig. 2 (a) , 2(b) and 2(c), respectively. The aperture angle in the PMBI case is taken to be π/2 whereas a value of 2π/3 = 1.33 (π/2) can be used in the EMBI case. The value of 2π/3 in EMBI is 1.75 times the aperture angle of a 1.4 NA oil immersion lens which is 0.76 (π/2).
As shown in Fig. 2(a), both the ASLs of EMBI and PMBI are far below 0.1%, and the PSF of PMBI is 1.05 times that of EMBI at FWHM. But as shown in Fig. 2(b), the diffractive spot size of EMBI is about 67% smaller than that of PMBI whereas the ASL is 3.34% and 2.03%, respectively. These two Figs. indicate the asymmetry of the electric field distribution in high aperture imaging. Both the spots are compressed in direction Y when the dipole polarization is along X and the size of PSF is seen to be narrower along the axis perpendicular to the dipole origination. The case of a dipole polarization oriented in the Y direction, p = j, does not need to be discussed separately since the distribution of |Efe|2 along Y in the focal plane is contained in Fig. 2 simply changing the definition of axes X and Y.
The distributions of |Efe|2 along Z are plotted in Fig. 2(c). The FWHM of PMBI and EMBI are 6.9λ and 3.14λ with the corresponding ASL levels of 7.69% and 14.04% respectively. The diffractive spot of PMBI in direction Z is 2.2 times wider than the spot size in the EMBI case, but its ASL is about 54.7% of the ASL in EMBI.
3.2 Dipole orientation in the Z direction
Figure 3(a) shows that the expected on-axis dip in the PMBI case is almost half the maximum value of the PSF. The behavior in the EMBI case is in stark contrast with a considerably reduced central dip. The ASL of EMBI is 0.36% which is about 3.89 times below that of PMBI case. In comparison with the distributions drawn in Fig. 2(a) and 2(b), we see that the spot size in lateral direction is larger when the orientation of the dipole emitter is changed from p = i to p = k.
But in Fig. 3(b), we see that the axial spot size of EMBI is dramatically compressed when compared with that of PMBI. This FWHM of EMBI is 2.44 times narrower than that of PMBI, whereas the ASL is relatively unchanged, 3.09% and 2.88%, respectively. The side lobe of EMBI is raised only by 0.21%. This result underlines the importance of detecting in the backward diffractive field to achieve high axial resolution.
It has traditionally been the case, in both refractive and PMBI systems, that an aperture angle of π/2 has been considered to be the limiting case. This has restricted analyses and experiments to consider only the forward diffracted wave. The influence of the backward wave has not previously been discussed. Our analysis of EMBI systems with an aperture angle greater than π/2 has highlighted the significance of the backward wave in reducing the FWHM of the focal spot albeit together with a slight increase in the side lobe level. Further the increased solid angle leads to increased signal levels.
4 Off-axis imaging
All our considerations so far have been concerned with on-axis imaging, rm = 0. In order to compare the two systems for off-axis imaging we elect, somewhat arbitrarily to display, in Fig. 4 and Fig. 5 , the cube root of |Efe|2 to highlight the changes on high order side lobes. Again the maximum aperture angle αmax of PMBI and EMBI are π/2 and 2π/3, respectively. Sub-Figs. (a), (b) and (c) relate to the PMBI case while sub-Figs. (d), (e) and (f) correspond to EMBI. For the sake of brevity, dipole orientations along the X, Y or Z axes are denoted by p = i, p = j or p = k in Fig. 4 and Fig. 5.
As shown in Fig. 4, the off-axis aberration of a transverse defocused dipole emitter appears as a coma-like aberration in both the PMBI and EMBI cases. In the case of p = i and p = j, in Fig. 4(a), 4(b), 4(d) and 4(e), the focused spot in PMBI has a better shape than that of EMBI. This kind of degeneration agrees with the basic rule of optical imaging that an enlarged aperture system is generally more sensitive to aberration. But the focused spot in the EMBI case is still single peaked and is relatively sharp compared with the hollow distributed diffractive spot in PMBI when p = k. The high order side lobes shown in Fig. 4(f) degrade faster than those in Fig. 4(c).
When defocus is considered, rm = (0,0,λ), the symmetry in each lateral imaging PSF is maintained very well as shown in Fig. 5. Figure 5(a) and 5(d) can, of course, be obtained from Fig. 5(b) and 5(e) through a 90° rotation since a pure longitudinal defocused dipole emitter does not affect the transverse component ratio in a focal plane. A small dark central peak in Fig. 5(c) can be identified by comparing with the bright central peak in Fig. 5(f). Clearly, the main difference between PMBI and EMBI in Fig. 5 is the central peaks. The amplitude of the central peak is reduced in the case of imaging a defocused dipole emitter at rm = (0, 0, λ). However, the central lobe in EMBI remains a single peaked distribution as shown in Fig. 5(d), 5(e), and 5(f) although the amplitudes of central peaks are below the maximum amplitudes of side lobes. It can be seen from the results that the sensitivity of longitudinal defocus aberration in EMBI is lower than that of PMBI when the aperture angle is over π/2 and the backward diffractive field is collected.
An EMBI with aperture angle greater than π/2 has been proposed in principle for use in high resolution, high photon efficiency 3D imaging. Rigorous formulae have been derived for the images of single molecules modeled as simple dipole emitters. The effects of collecting both the forward and backward radiating fields have been considered as has the case of off-axis imaging.
It is found through simulations that the PSF of EMBI can be made narrower as compared with PMBI, although the precise factor depends on the orientation of dipole. One of the main results is that the side lobe of EMBI is raised by only 0.21% whereas the PSF is compressed 2.44 times at FWHM when the orientation of dipole polarization is along Z axis. This observation illustrates the importance of collecting both the backward and forward diffractive field in optical microscopy.
The authors thank National Nature Science Foundation of China (Grant No. 50905048) for its fund.
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