We explore the diffraction limited focusing and confocal imaging properties of a high NA parabolic mirror for confocal imaging and spectroscopy of nanoparticles and single molecules. Vector field calculations of the electric fields near focus for both linear and radially polarized illumination are discussed and show that the optical field can be similar tightly focused as in the case of a high NA objective lens. Furthermore they show that a high NA parabolic mirror allows an easy orientation of the polarization of the illuminating light in all spatial directions. The simulation of confocal imaging of single molecules is discussed and yields, that the use of radially polarized excitation light gives an easy access to their orientations.
© Optical Society of America
Parabolic mirrors with small numerical aperture (NA) are very common in astronomical telescope design and telecommunications. High NA parabolic mirrors have been used as efficient light collectors in single molecule spectroscopy [1–5], however, for imaging they have been avoided due to their poor off-axis imaging properties. The classical imaging devices for microscopy and imaging are objective lenses, because of their wide field of view and excellent resolution resulting from the high degree of correction and the high achievable NA. Parabolic mirrors focus a parallel beam, travelling along the optical axis, to one point without any aberrations in the geometrical optics approximation and in this sense they are ideal focusing devices. The drawback is that slight deviations of the incident beam from the optical axis or from parallelism give rise to huge aberrations for a high NA mirror resulting in a very small field of view.
Confocal microscopy is a point to point imaging technique and hence has to correct aberrations only for one single object point. This purpose is perfectly solved by parabolic mirrors in a stage scanning microscope where the focus rests on the optical axis. In addition it is fairly easy to achieve a high aperture without chromatic aberrations since the deflecting surface can be made highly reflective over a large spectral range.
In this work we investigate in the first section theoretically the focusing properties of a parabolic mirror with high NA and compare them with an aplanatic objective lens for linear and as well for radially polarized illumination. In a radially polarized beam the electric field vector is everywhere oriented radially with respect to the optical axis . We show that high aperture parabolic mirrors illuminated with radially polarized light have the property that they produce a large longitudinal component of the electric field in the focus which is approximately fourteen times stronger then the transversal components also present near the focus. This property is especially interesting for all applications where a strong longitudinal field component is desirable, e. g. (1) for strong local field enhancement with a sharp metallic tip in near-field optics [7, 8], or (2) for polarization selective spectroscopy of nano-particles or single molecules in three spatial dimensions.
In the second section the confocal imaging of a dipole emitter, such as a single molecule, is discussed likewise for linear as well as for radially polarized excitation light. The use of radially polarized light and a reduced illumination aperture yields a distribution of longitudinal and in-plane components of the electric field which allows an easy determination of the emission dipole orientation.
In this section the properties of a parabolic mirror as focusing element are discussed for linear and radial polarized incident light. The basic machinery by which polarized beams in focal regions can be analyzed was described by Richards and Wolf  using a decomposition of the incident field into an angular spectrum of plane waves. The adaptations for radially polarized beams are discussed in reference . Basic work concerning the diffraction limited focusing with parabolic mirrors was published by Ignatovsky in 1920  and numerical calculations for a linear polarized plane wave by Sheppard  and with more detail by Barakat  and recently for mirrors with arbitrary openings, particularly with openings larger then π, by Varga and Török [13, 14].
We compare the parabolic mirror with an objective lens and emphasize the peculiarities of the mirror. Fig. 1 shows a parabolic mirror and an aplanatic objective lens both with a geometrical aperture of sinθ=1.0. Note that the focal length of the objective lens is twice that of the parabolic mirror. Hence the rays at the rim of the beam in both cases hit the focal point under an angle of incidence θ of π/2. The parabolic mirror reflects a ray incident at a given distance r from the optical axis to a large angle θm . The same ray is refracted by the objective lens to reach the focal point at a smaller angle θo . This means, due to energy conservation, that in a parabolic mirror much of the incident energy reaches the focus under high angles while the contrary is valid for an objective lens. This has noticeable consequences as will be seen later.
A second difference is, that the reflection at the mirror surface causes a phase jump of π to the component of the electric field which is polarized perpendicular to the plane of incidence (s) while the refraction by the objective lens does not. Note also the inverted sign of the p-polarized x,y-component: it points towards the optical axis for the objective and away from it for the mirror. Both together lead to inverted x,y-components for the mirror, which has only minor consequences. Furthermore, it can be seen that field components along the mirror axis arise in the focal region due to the high aperture.
We will give a short summary of the equations needed for the calculation of the optical field near the focus of a parabolic mirror, especially we will mention the differences to the case of an aplanatic lens as described in references  and  respectively. At optical frequencies, only the electric field is responsible for the interaction with matter (scattering, fluorescence, excitation, polarization, etc.). Hence only the electric fields will be discussed.
Fig. 1 shows the geometry of the problem with the used coordinate system. Starting point of the calculations is the angular spectrum representation of the incident field. The time independent part of the electric field at a point P (in polar coordinates rP , θP and φP ) near the focus can then be written as (neglecting a constant phase term due to the total path length)
 with the wave number k (in the medium), the focal length f, the solid angle Ω, the electric field at the focal sphere E f , which is calculated by geometrical optics rules, a unit vector in the direction of propagation s, the angles θ and φ as shown in Fig. 1. Complete derivations for an aplanatic lens are given in references [6, 9]. The analogue derivation for a parabolic mirror yields the electric field at a point P near the focus of an ideally conducting parabolic mirror (see section 5.).
a) For light which is linear polarized along the x-axis the field components are
The integrals I 0,l, I 1,l and I 2,l are defined as
where the J n are Bessel functions of the first kind of order n. α1 is the opening angle of the mirror, α0 the smallest angle of acceptance not hidden by the sample, θ the angle θm as shown in Fig. 1 and l0 (θ)=N exp(-r 2(θ)/) the electrical field amplitude of the illuminating laser beam. The normalization constant N is used to compare the objective lens and the mirror quantitatively. r is the distance from the optical axis as shown in Fig. 1 and w0 is the waist radius of the illuminating Gaussian beam.
The first difference to the equations by Richards and Wolf  is the apodization factor 2/(1+cosθ) instead of the cos½ θ. This factor is due to energy conservation. All energy flowing through a surface element in the parallel illuminating beam must flow through the corresponding surface element on the focal sphere fm . A derivation can be found in . The second difference is the positive sign for ex and ey that originates from the phase jump at the mirror surface. The z component arises from the reflection of components parallel to the plane of incidence (p-polarized) only and hence has no phase jump at all. The resulting effect is that the direction of rotation of the electric field vector in a plane perpendicular to the x, y-plane is inverted (see Fig. 4). A third difference is the negative exponent, which reflects just the inverted direction of propagation of light in the negative z direction.
b) We discuss the results for radially polarized illumination. Theoretical aspects concerning inhomogeneously polarized light, like radially polarized light, are discussed in references [6, 17–21], their generation in references [22–28], the simple generation from linear polarized light  should be mentioned specially. For this case we find the following equations for the electric field at a point P near the focus:
Er is the radial component and Ea the azimuthal component showing the radial symmetry. The integrals are here defined as:
with the field distribution l0 (θ)=N exp(-r 2(θ)/) J1(2 r(θ)/w0 ) of a Bessel-Gauss beam.
The same differences between an objective lens  and a parabolic mirror are observed as mentioned above except that we have only parallel (p-polarized) components at the reflection in the case of radially polarized light and hence no phase jump.
2.2. Numerical field calculations
The results presented here are all calculated for the special case of a parabolic mirror with a numerical aperture of 1.515 (oil, index of refraction n=1.518), corresponding to a half opening angle of 86.4°, which is experimentally easily feasible. Such a mirror can be realized by a glas paraboloid with a high reflection coating on the curved surface. For the calculations a focal length f=4.5 mm was used and a central spot of 3.0 mm in diameter of the incident beam is blocked by a sample, corresponding to α0 =18.9°. Although the results for an objective lens are well known, we present them here for direct comparison with the field distribution in the focal region of the parabolic mirror. In the case of the objective lens the same focal length f of 4.5 mm was used and a numerical aperture of 1.40 (oil, n=1.518) corresponding to a half opening angle of 67.3°, which is a typical value for an objective lens with high NA.
2.2.1. Linear polarization
The illuminating beam has a Gaussian beam profile with a beam waist w0 of 3/2 times the pupil radius. It is beyond the scope of this paper to discuss the optimum value of the beam waist.
Fig. 2a) and Fig. 2e) show contour plots of constant |E|2 in the x, y-plane of a focused Gaussian beam polarized along the x-direction for an aplanatic objective lens and a parabolic mirror respectively. A logarithmic scale is used with a factor of two between adjacent contour levels. It can be seen that, (1) due to the higher aperture of the mirror system the size of the focal spot is slightly reduced but the distortion is more pronounced, (2) the energy is more spread out in the x, y-plane of the mirror. This fact is caused by the field incident under high angles. It forms a standing wave in the x, y-plane as Fig. 3e) and Fig. 4 show as well. And (3) the y and z components are enhanced by the mirror as Fig. 2c), g) and Fig. 2d), h) show. This fact is disturbing in conventional polarization microscopy. For single molecule detection however this might be an advantage as will be seen later.
Fig. 3 shows sections of |E|2 parallel to the optical axis. While in the case of the objective lens the depth of the focus has a full width at half maximum (FWHM) of 1.51 λ/n, it is smaller in the mirror case namely FWHM=1.00 λ/n. The reason is again found in the energy incident from the high angles. Most of the energy is incident on a cone with a half opening angle of about 30° in the objective lens while this angle is about 50° for the mirror (see also Fig. 4). The interference region of this cone-shaped beam is either elongated in the optical axis, as for the objective lens, or more spread out in the x, y-plane, as for the mirror.
Fig. 4 shows a movie of the time evolution of the electric field near the focus of a Gaussian beam, which is linearly polarized along the x-axis, formed by an objective lens (a) and a parabolic mirror (b). Contour plots of |Re(E(t))|2 in the x,z-plane, where Re denotes the real part, are overlaid by the orientations of the field vectors at several points.
The formation of the interference of the intensity from the angles which contribute most of the energy appears very clearly in this movie. In the case of the objective lens the wavefronts shown in Fig. 4 has been measured by Schrader and Hell .
2.2.2. Radial polarization
For radially polarized illumination a Bessel-Gauss beam with a beam waist w0 =3/2×pupil radius was used.
Fig. 5 and Fig. 6 compare contour plots of |E|2 for radial polarization for an objective lens and a parabolic mirror in the x, y-plane and perpendicular to it. In the case of radial polarization major differences can be seen: The central spot is very sharp in the parabolic mirror and rather blurred for the objective lens. By comparing the radial and longitudinal components the effect is obvious: The radial components are formed by the intensity incident under small angles θ while the longitudinal components emerge from the large angles (see Fig. 6b), c), e) and f)). Due to the higher NA and the high concentration of energy in the large angles the parabolic mirror produces predominantly longitudinal components, while the objective lens produces longitudinal and radial components in comparable amounts. The blurred spot in the objective lens is formed by the superposition of the central longitudinal and the annular radial components. We observe here as well a reduced longitudinal size of the focal region in the mirror (mirror: FWHM=1.18 λ/n, objective lens: FWHM=1.65 λ/n). The ratio between the longitudinally and laterally polarized intensities is 14.3 in the parabolic mirror while it is only 3.50 in the objective lens.
To draw a quantitative comparison between objective lens and parabolic mirror the total intensities of the parts of the illuminating beams which are focused, have to be normalized. In Fig. 7 the intensities are normalized to 1 mW. The parabolic mirror shows a slightly larger intensity of the electric field for linear polarized light and a strongly enhanced intensity at the center for radially polarized light as compared to an aplanatic objective lens.
2.3. Alignment and coma
The drawback of a parabolic mirror is that it shows strong coma if it is not illuminated exactly along its optical axis. A slight misalignment of the propagation direction of the illuminating beam causes already a strong distortion of the field distribution near the focus. Hence a proper alignment of the illuminating beam is critical for the use of a parabolic mirror. Fig. 8 shows movies of |E|2 in the focal region of a parabolic mirror illuminated with radially polarized light with a wavelength of 500 nm for different angles of incidence. From image to image the angle of incidence is increased by 0.00318° starting with 0°. Calculations for linear polarized light are comparable. To calculate the field in the focal region an additional phase term is included into the integrals of equation 5. This phase term takes the tilted wave front into account. The integration is performed over the focal sphere f m, as proposed by Sheppard . A deviation of the angle of incidence by more than about 0.006° causes a severe asymmetry of the field and reduces the maximum intensity by more then a factor of two. Hence the propagation direction of the illuminating beam has to be aligned to the optical axis of the mirror better than 0.006°. This accuracy can be reached by a steering mirror equipped with micrometer adjustment screws.
3. Single molecule imaging
The observation of single molecules and the determination of their orientation earns growing interest [31–34]. We analyze the detection of a dipole emitter in a homogeneous environment through a telecentric optical system consisting of a parabolic mirror and an aplanatic objective lens with small NA. A detector with limited detection area, as the commonly used avalanche photo diodes, measures the intensity in the image focus. Telecentric systems have been studied by Sheppard . A comparable study for an objective lens instead of a parabolic mirror has also been published by Sheppard  and recently by Enderlein .
The major difference to the objective lens is here again the apodization factor of the mirror and the phase jump of π for the perpendicular components at the reflection as well as the aberrations for an emitter located outside the geometrical focus. The electric far-field caused by a dipole emitter can be written as
in SI-units, where ε0 is the electric permittivity of vacuum, k the wave number, p is the dipole moment and sm a unit vector pointing in the direction of observation, r the distance from the dipole to the point of observation, ω the angular frequency and t the time . Transversal components (s-polarized, indexed ⊥) suffer the phase jump of π while parallel components (indexed ‖) are deflected by the mirror and the objective lens. Using equations 1 and 6 and neglecting a constant phase term due to the total path length, the time independent part of the electric field in the secondary focus can be written as
where f o and f m are the focal length of mirror and objective lens respectively. A schematic representation of the geometry with the angles θm and θo is shown in Fig. 9. k is the vacuum wave number, nm the refractive index of the mirror material, r m is the vector from the geometrical focus of the mirror to the location of the molecule and r o the vector from the geometrical focus of the objective lens to the point of observation. The unit vectors e ⊥, e m,‖, e o,‖, sm and so are defined as follows:
2/(1+cosθm ) is the apodization factor of the mirror, cos½ θo that of the lens. The term in square brackets describes the orientation of the electric field on the focal sphere of the lens, the exponential factor accounts for the phase corrections for points outside the mirror- and lens-focus and sin θo dθo d φ is the solid angle element. Using the magnification M=fo /fm , θm and the integration limits θ o, min and θ o, max can be given by
where NAm is the numerical aperture of the mirror, nm the refractive index and rsamp the radius of the central part of the mirror which is covered by the sample holder.
In the case of an objective lens as collection optics the linear relation sinθm =M sinθo is valid. So a shift of the molecule out of the optical axis can be simulated by just moving the image of the in-focus dipole by -M times this shift. This can be seen by analyzing the exponential term of equation 7. The relation connecting sinθm and sinθo in the case of a parabolic mirror as collector is strongly nonlinear in M (see equation 9). The images of off-axis lying molecules will hence be distorted. The longitudinal magnification as well does not equal M 2, as it would in the paraxial approximation. A molecule located on the optical axis, but slightly moved out of the geometrical focus, does not produce the same wave front in the parallel beam as it would by passing through a thin lens.
For calculating the intensity measured by a detector positioned in the geometrical focus of the lens while scanning the dipole, equation 7 must be integrated over the detection area. The ratio of the detected to the emitted intensity as a function of the emitters position yields the collection efficiency function (CEF). When the numerical aperture of the image forming objective lens is small, the electric and magnetic fields in the focal region are almost perpendicular to each other and their absolute values have a nearly constant ratio. Hence the energy flow 1/(2µ0 ) Re(E×B*) is proportional to E·E* (the asterisk denotes complex conjugation and µ0 is the vacuum permeability) and it is sufficient to know the electric field in the focus. The detected intensity is thus proportional to the spatial integral ∫ A E·E*da over the detector area A. Since the field distribution in the detector plane depends strongly on the position of the emitter in the focal region, the field distribution must be calculated for each position of the dipole to obtain the CEF. To reduce computational effort symmetry considerations can be taken into account.
The fluorescence rate R(r m) of a dipole emitter located at r m can be written as
at sufficiently low intensities, where saturation effects can be neglected . E(r m) is the electric field vector of the excitation field at the position of the emitter, p its absorption dipole moment and c a constant. For the following calculations it is assumed, that the absorption and emission dipole moments are equal and the fluorescence quantum efficiency is unity. The confocal image of the dipole emitter is obtained by multiplication of the detection efficiency function with the fluorescence rate caused by an illuminating electrical field.
3.2. Image distortion and collection efficiency
All numerical results are calculated for a parabolic mirror with focal length 4.5 mm, numerical aperture 1.515 (oil, refractive index=1.518) and a shadowed center of 3 mm in diameter. The objective lens has a focal length of 225 mm yielding a magnification M of 50. The illuminated aperture of the objective lens is small, NA=0.0375, and hence the approximation of the energy flow by c·E·E* is still good (where c is a constant).
Fig. 10 shows images of |E|2 of a dipole emitter in the focal region of a parabolic mirror as seen in the plane of the detector. The dipole lies in the focal point in Fig. 10a) - c) and is moved by one wavelength away from the optical axis in Fig. 10d) - f). Depending on the dipole orientation different images are obtained (see also ). The symmetry of the field distribution is reduced when the dipole is moved away from the focus. Hence, in contrast to an aplanatic objective lens, a high NA parabolic mirror has one focal point instead of a focal plane. Note as well the similarity of the intensity (and also polarization) distribution of a dipole oriented along the optical axis (Fig. 10c)) to that of a radially polarized beam.
The black circles in Fig. 10a) and c) have a diameter of 1.22 λ/NAo corresponding to the diameter of the Airy disk produced by a plane wave in paraxial approximation. While for a dipole oriented in the x, y-plane almost all energy of the main maximum is collected within this circle it is not for a longitudinally oriented dipole. To collect light from longitudinally oriented dipoles with high efficiency the diameter of the detector has to be increased slightly at the expense of a slightly reduced resolution. For the following results a factor of 1.2 was used. The resulting detection efficiency is 0.39 for a longitudinally oriented dipole and 0.37 for a dipole oriented perpendicular to the optical axis. In the case of the objective lens (as described in paragraph 2.2.) as collector the detection efficiency reduces to about 0.30 for the dipole in the x, y-plane and to 0.16 for the longitudinally oriented one, due to the smaller aperture.
Fig. 11 shows contour plots of the collection efficiency function (CEF) for different orientations of a dipole emitter scanned either in the x, y-plane, a), or in the x, z-plane, b), of a parabolic mirror collector. Only orientations of the dipole axis with an azimuth angle φ between 0 and 90 degree are shown. Orientations between 90 and 360 degree can be obtained by symmetry considerations. Note that a linear intensity scale is used in these figures.
The reason for the asymmetry of the collection efficiency for the in-plane scan (Fig. 11a) is given by the asymmetry of the corresponding distribution of energy (see Fig. 10) due to the sin2 θ direction dependence of the dipole radiation. An elongation of the main maximum in x-direction causes an elongation of the CEF in y-direction for instance. The point symmetry in the x, z-scan can be understood by the geometrical optics image inversion: light on the left side in front of the focus is found on the right side behind focus (Fig. 11b), θ=45°, φ=0°, 45°).
Due to this asymmetries the resolution limit is not a constant but depends on the relative distance and orientation of two dipole emitters. The shape of the CEF shows a low distortion at the half maximum value. Hence it is reasonable to describe the resolution in terms of the full width at half maximum (FWHM). The CEF of an in-plane emitter has a lateral FWHM of 1.11 λ/n in the direction of the dipole axis, 1.04 λ/n perpendicular to it and a longitudinal FWHM of 1.46 λ/n. For a longitudinally oriented dipole the in-plane FWHM is increased to 1.20λ/n while the longitudinal FWHM is decreased to 1.20 λ/n.
3.3. Confocal imaging
Combining the collection efficiency with a given excitation, according to equation 10, scanning confocal optical images of single fluorescent molecules can be calculated. Fig. 12 shows results for linear, a) and b), and radial, c) and d), polarization. The molecule is either scanned in the x,y-plane, as in a) and c) or in the x, z-plane, as in b) and d). In the case of linear polarized excitation along the x axis every orientation of the dipole produces a different confocal image, by which its orientation can be identified. However the intensities vary by up to about two orders of magnitude which makes it very difficult to detect all orientations of emitters simultaneously at a reasonable signal to noise level. The depth resolution of the mirror setup is good. The full width at half maximum is 0.80 λ/n while the lateral FWHM of a dipole lying along the polarization direction is 0.43 λ/n and 0.35 λ/n, respectively.
The intensity patterns in the images of an emitter which is radially excited allow the determination of its orientation. But the intensities of the lateral and longitudinal fields differ also by one order of magnitude. By reducing the illumination aperture of the mirror this ratio can be reduced to obtain comparable intensities. Fig. 12c) and d) show results calculated with the illumination aperture reduced to NAm =1.21 and the ratio between entrance pupil and beam waist of the Bessel-Gauss beam kept constant. The ratio of the in-plane polarized intensity to the longitudinal one is 0.44 in this case. By reducing the numerical aperture a slight reduction of the resolution must be accepted but in return the orientation of the dipole emitter can be easily determined from the intensity pattern from an image. In order to determine the polar angle θ of the dipole the patterns must be analyzed quantitatively.
It has been shown, that a parabolic mirror has a good depth and lateral resolution due to the high numerical aperture which can be easily reached. However, for generating a clean focus, the alignment of the illumination beam is critical due to strong coma.
The use of a concave parabolic mirror with high NA could also have advantages in fempto second spectroscopy due to the absence of chromatic aberration.
By using radially polarized light strong and well localized longitudinal components of the electric field of the light can be produced with only weak disturbing lateral fields. This feature might be very useful for some Near-field Scanning Optical Microscopy (NSOM) applications, where a sharp tip is placed in a laser focus for further confining the optical field. If the polarization of the laser is along the tip axis a very strong field enhancement occurs at the tip end which allows imaging of the sample with super resolution . For this application a concave mirror must be used to place a tip in the focus.
The longitudinal component in combination with linear polarized light along two perpendicular directions yields a platform for investigation of nanometer sized particles or single molecules with optical fields in all three spatial dimensions.
The use of a parabolic mirror for single molecule detection is mainly suitable because of its high numerical aperture which allows the detection of arbitrary oriented dipole emitters, such as single molecules, with a constant high detection efficiency. For determining the orientation of a dipole emitter the use of radially polarized excitation with an optimized illumination NA is best suited.
Calculation of the diffraction integrals
For calculating the electrical field on the focal sphere fm the rules of ray optics are used. According to Fig. 1 the unit vector pointing in the direction of propagation s, the p-polarized and s-polarized electrical field components after reflection, E p and E s , and the vector r P pointing from the geometrical focus of the mirror to the point of observation can be written as:
in Cartesian coordinates, with the field components before reflection being denoted as E 0,p and E 0,s. In addition the power density of the incident field increases towards the focal region. This factor can be calculated using the fact, that the energy flowing through a differential surface perpendicular to a ray must be constant on its path to the focus. A derivation of this so called apodization factor is given in . In the case of a parabolic mirror it is 2/(1+cosθ).
For linear polarized light we have E 0,p=cosφ l0 (θ) and E 0,s=-sinφ l0 (θ) with the field strength of the incident beam l0 (θ) we find equations 2 and 3. For radially polarized light the incident beam has no s-polarized components. The resulting field on the focal sphere fm is (neglecting a constant phase factor due to the path length)
The evaluation of the scalar product s·r p in the exponential term of equation 1 yields
We are much obliged to Alexander F. Nowikow from the St.Petersburg State Technical University, St. Petersburg, Russia for sending us copies of the Ignatovsky paper.
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