Employing interference patterns for illumination has been shown to reduce the focal volume in fluorescence microscopy. For example, the 4Pi technique employs two interfering laser beams and significantly decreases the focal volume, as compared to conventional microscopy. We study theoretically the effect of using multiple interfering laser beams on the focal volume. In realistic setups with three or four beams, the focal volume is about half of that from the 4Pi case. This improvement reaches a limit quickly as more beams are added, and for the idealized case of an infinite number of beams the focal volume is rather close to the three- or four-beam cases. Thus, our study suggests a limit for the possible reduction of the focal volume in a purely optical far-field setup.
© 2009 Optical Society of America
Until recently, the resolution of optical far-field microscopy was limited to the size λ/(2nsinα) (or 200–300 nm in practice), where λ is the wavelength of the light used in the microscope, n is the refraction index, and α is the half-aperture angle of the light focused by the lens. This barrier, known as the Abbe diffraction limit, has been overcome with the emergence of approaches that employ structured illumination, or, in other words, interference of the illumination light [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. For instance, 4Pi microscopy  provides a resolution of 100 nm by employing the interference of two focused counterpropagating coherent laser beams at the focal spot to illuminate the sample, while a conventional microscope uses only one beam. Microscopies with more than two interfering beams have also been proposed, such as the theta-microscopy  and MOM . The MIAM technique , although not employing interference, combines images from four tetrahedrally arranged objectives focused on the same sample, an approach that has been shown to increase the resolution significantly in comparison with the case of a single objective.
The structured illumination microscopy techniques are essentially using purely optical principles. These techniques reached the resolution of 100 nm, but improving the resolution further has been difficult, if not impossible. Fortunately, the advances in fluorescence photophysics made further breakthroughs in the available resolution possible. Approaches for superresolution fluorescence microscopy have been invented  that take advantage of photophysical effects (e.g., photobleaching or photoswitching), resulting in such techniques as STED , SSIM , FPALM , PALM , or STORM . These methods furnish the imaging resolution of 20–30 nm, by far superseding the performance of their purely optical counterparts.
Besides imaging, the optical microscope can be used also as a probe for the dynamics of cellular components, most notably in the fluorescence microphotolysis (FM) or fluorescence correlation spectroscopy (FCS) modes. These approaches are largely non-invasive and, thus, allow one to detect molecular diffusion rates, binding, and other phenomena in living cells. The smallest volumes that FM or FCS can sample are determined by the distribution of light in the focus of the microscope, described by the point spread function (PSF); generally, the higher resolution is associated with a smaller focal volume. Although superresolution imaging techniques provide a nanoscale resolution, they may not necessarily provide a better sensitivity for FM or FCS. Proposed schemes for PALM, FPALM, and STORM employ wide-field techniques not compatible with FM or FCS. STED has been used for FCS and provided an improved sensitivity , but most probably it will not be suitable for improving sensitivity of FM. The reason is that STED achieves superior resolution by applying photobleaching to volumes much larger than that defined by the nominal imaging resolution and, thus, the effective volume sampled by FM measurements is not reduced in comparison with conventional microscopy. On the other hand, the structured illumination techniques employ purely optical principles to obtain a narrower PSF, providing an ideal probe for FM or FCS measurements. Indeed, FM and FCS measurements have been recently performed using the 4Pi microscope , with the 4Pi effective volume being about half of that of the conventional confocal microscope.
Thus, a class of important applications may not be able to take advantage of the improvement brought about by the fluorescence superresolution techniques. One then resorts to purely optical microscopy methods to reduce the microscope focal volume, and it is essential to understand what are the limits of the reduction in such case. We theoretically investigate here the principle limits for confining illumination using ideas similar to those behind 4Pi microscopy, i.e., employing multiple interfering coherent laser beams for illumination. While microscopies with two or more interfering beams have been proposed [1, 3, 12], to our knowledge, different numbers of the beams have not been sampled systematically, and corresponding PSF effective volumes have not been considered. Likewise, the question of what is the limit of light confinement with purely optical approaches has not been addressed. We investigate this question and find that the realistically achievable limit is reached already for three or four interfering beams. The factor of two decrease in focal volume for this limit (over 4Pi microscopy) promises a significant improvement for FM and FCS applications . On the other hand, our findings suggest a fundamental physical barrier for further improvements, defining the best resolution that can ever be achieved through multiple-beam microscopy.
We consider setups where multiple coherent laser beams are focused at the same spot. This is achieved by using multiple objectives, i.e., one objective for one beam . Resulting PSFs, h(r→), are computed. The PSF resolution can be defined as the full-width-half-maximum (FWHM) of the PSF, and the effective volume is defined  as
Schematics of the considered arrangements are shown in Fig. 1. The 4Pi setup features two beams of equal phase, and we extend this arrangement to 3, 4, or more beams, with collinear polarization (which decreases the illumination spot), and with all beams being in the same plane. We also consider setups with out-of-plane beams added, such as the case of 6 beams in Fig. 1. Two limiting cases are considered. The first is an infinite number of beams of the same polarization, with all wave vectors being in the same plane (“Inf. beams, 2D”). The second case is an infinite number of beams converging from all directions in space, representing the most massive focusing possible. The latter arrangement can be realized in many different ways, depending on the choice of polarization and phase for each beam. We consider the arrangement in which all possible rotations, from angle 0 to π, of the plane with “Inf. beams, 2D” are summed up (see Fig. 1).
We assume the same half-aperture angle α for each beam in the same setup (larger α leads to a narrower illumination spot). Theoretically, α ∈ [0,90°], but technically α is usually limited to < 70°, although higher values, e.g., α = 74°, have been reported recently . For many beams, α has to be limited to even smaller values due to the spatial restrictions, e.g., to 60° for 3 beams in one plane. For the limiting cases “Inf. beams, 2D” and “Inf. beams, 3D”, the value of a has to be zero, which would lead to a divergence of the illumination spot. Below, we formally set α ≤ 90° for these arrangements, as an idealized and abstract case of the most narrow illumination idealistically possible. Thus, the 3-, 4-, or 6-beam setups are realistic, as they can be in principle realized with technically available lens parameters (e.g., using α = 60° in the 3-beam case or α = 45° for 4- and 6-beam cases). In practice, the values of a for these setups may need to be somewhat smaller, so that the objective edges for each beam are not overlapping. The values α = 60° and α = 45° are thus the maximal possible values for the considered setups. The cases with an infinite number of beams are unrealistic in the sense that a is set to values incompatible with the setup geometry, but the consideration of such idealized cases gives us an estimate for the absolute limit of the light focusing in a multi-beam setup.
The PSFs are obtained as follows. Consider a light beam traveling in the - z direction and focused by a lens, resulting in the spherical wavefront converging around r→ = 0. The electric field [26, 27] is E→(r) = -i[I 0(r→) +I 2(r→)cos2ϕ,I 2(r→)sin2ϕ, -2i I 1(r→)cosϕ], where
f 0(0) = 1 + cosθ, f 1(θ) = sinθ, f 2(θ) = 1 - cosθ. Here, ϕ is the angle between the plane of oscillation of E→(r→) and the plane of the objective lens (we assume the most common case of ϕ = 90°); J 0, 1,2 are Bessel functions of the first kind. For a beam with an arbitrary direction k→, we can define a coordinate frame (x′,y′,z′), such that the beam is traveling in the - z′ direction, and the electric field is described by the above formula with r→ replaced by r→′. Then, it is simple to obtain E→(r→) for this beam in the original coordinate frame (x,y,z) through rotation of all vectors by the angle defined by the direction and polarization of the beam. For arbitrary beams i, i= 1,2, …,N, we obtain the total electric field E→tot(r→) =ℑN i-1 E→i→(r→) and PSF h(r→) = ∣E→tot(r→)2nexc (nexc-photon excitation is used; we set nexc = 2 as commonly used in 4Pi microscopy). We assume that all beams are of the same intensity and are coherent. Phases of all beams are matched in such a way that, at the common focus of all objectives, electric field vectors of all beams are always of the same magnitude (meaning that collinear beams have identical phases).
The PSFs (see Fig. 2) are obtained using the software fmtool , or, for an infinite number of beams, by numerical integration of E→(r→) from single beams; visualization is done with VMD . We describe only illumination PSF here (not combined with the detection PSF), since it is the illumination PSF that determines diffusion-bleaching dynamics on the sample in FM or FCS experiments . The distances are measured in λ/n. Typical values of λ and n are, e.g., λ = 910 nm (the high value is due to the two-photon excitation) and n = 1.46. For the 1-beam case, we assume one-photon excitation and a wavelength of λ/2, where λ is the wavelength used for all other cases. Using two-photon excitation for the 1-beam case decreases the PSF size by the factor ~ √2, but, due to the two-fold increase of the wavelength, the size actually increases by ~ √2. For other cases, two-photon excitation leads to a smaller PSF volume than that for one-photon excitation, due to the constructive interference of multiple beams.
The 4Pi PSF is narrower in the z-direction than that of the 1-beam setup, and its volume is further decreased by using 3, 4, or more beams (Fig. 2). However, as the number of beams grows, one has to employ smaller a. For example, if all beams are in the same plane, the maximal a is 60° for 3 beams, 45° for 4, 30° for 6, etc., and the PSF elongates in the x-direction as more beams are used (can be noticed already for 3 vs. 4 beams). A similar effect is observed if out-of-plane beams are added (such as 6 beams in Fig. 2). We found that the PSF volume for setups with >4 beams in one plane or >6 in 3D is larger than that for 4- or 6-beam cases. Even if an idealized value α = 70° is used, the PSF size is not decreased much beyond the 3-or 4-beam case. For example, 60 beams in Fig. 2 correspond to an overlap of three sets of 20 beams uniformly distributed in the x,y-, y,z-, and x,z-planes. The resulting PSF does not have side lobes or other outstanding features, but its size is not significantly reduced in comparison with the case of 3 beams. Likewise, the PSFs for an infinite number of beams with α = 70° are similar in size to the 3-beam PSF with α = 60°.
Effective volumes V eff are plotted in Fig. 3. The 4Pi Veff is about 1/2 of that for the 1-beam microscope. Another decrease by a factor of 2 results from employing 3, 4, or 6 beams. Even with α = 60° for 3 beams or α = 45° for 4 beams in one plane, or 6 beams in 3D, V eff in these cases is about 1/2 of Veff of the 4Pi PSF with α = 70°. For 3 beams at α = 60°, V eff ≈ 0.24 (λ/n)3 vs. V eff ≈ 0.4 (λ/n)3 for the 4Pi at α = 70°. For an infinite number of beams in 2D or 3D, V eff is not reduced much further than in the case of 3 beams, even if an idealized and unrealistic value of α = 90° is employed, resulting in V eff ≈ 0.14 (λ/n)3. Thus, the realistic setups with 3, 4, or 6 beams allow for values of V eff which are about twice smaller than the best 4Pi values, and that are already very close to the minimal theoretically possible, but probably practically inaccessible, limit of light focusing.
Although V eff is decreased by using more than 2 beams, the imaging resolution cannot be improved beyond that of the 4Pi microscope, as suggested by the FWHM values in Table 1 (see also Fig. 4). The resolution along x- and y-directions can be somewhat improved by using 4 or an infinite number of beams, but in the z-direction, the 4Pi FWHM is the smallest. Interestingly, the FWHM of the 3-beam PSF is greater than that of the 4Pi PSF in each dimension, while V eff is smaller for the 3-beam case (the same is true for “Inf. beams, 3D” vs. “Inf. beams, C;2D”). This is because the side lobes or exterior rings present in the 4Pi and “Inf. beams, 2D” PSFs contribute significantly to the value of V eff, even though the central part of the PSF is relatively narrow.
Let us now estimate the limit for light focusing when multiple beams are employed, assuming the “Inf. beams, 3D” setup (Fig. 1). The electric field is the result of interference of electric fields from each beam, given by Eq. (2). The case of “Inf. beams, 3D” is obtained by first taking the “Inf. beams, 2D” case, where beams are converging uniformly in one plane towards the focal point, and rotating this plane by π. The resulting electric field is therefore obtained through double integration of the electric field for one beam: one integration is over the polar angle φ (from 0 to 2π) corresponding to the in-plane rotation for “Inf. beams, 2D”, and the other is over the azimuthal angle χ (from 0 to π). The integration can be performed independently for each of the functions I 0, 1,2 in Eq. (2), since the expression for the electric field is linear over these functions. Let us use spherical coordinates: x = rsinθrcosϕr, y = rsinθrsinθr, z = rcosθr. Then, contributions to the total electric field of “Inf. beams, 3D” at point r→ = (x,y,z) from functions I 0,1,2 are proportional to
where g(φ) = cos φ for the x-component of the electric field, and g(φ) = sinφ for the y-component; the z-component is zero. Note that θ and θr, as well as ϕr and φ, are different variables. The integration over φ, originally from 0 to 2π, can be replaced by an integration from - ϕr to π - ϕr due to the symmetry of the “Inf. beams, 2D” case, where, for each beam, there is always another beam traveling in the opposite direction in the same plane (see Fig. 1).
The PSF that one obtains from the electric field given by the expression for A 0,1,2(r→) above is not, in general, spherically symmetric. However, we saw for the numerically constructed PSFs for the “Inf. beams, 3D” case that the asymmetry in those PSFs is not exceptionally high. Also, we would like to enforce the condition of spherical symmetry on A 0,1,2(r→) to obtain a simple analytical approximation for the PSF in the case of an infinite number of beams in 3D. If spherical symmetry applies, A 0,1,2 (r→) has to assume the same value for any values of ϕr and θr, i.e., we can choose any fixed value to substitute for these variables. Since we are looking for the PSF with the smallest effective volume, we should choose ϕr and θr that correspond to the narrowest PSF. From the computationally obtained PSFs for the “Inf. beams, 3D” case, these are ϕr = 0 and θr = 0 (or x = y = 0, see Figs. 2 and 4, and Table 1). The expression for A 0,1,2(r→) in this case becomes 0 for the x-component; the y-component is
The factor 2 appeared here as a result of the integration, .
Thus, we obtain a simple, spherically symmetric form for the electric field, which corresponds to the narrowest PSF for the “Inf. beams, 3D” case. Only the y-component of E→(r→), Ey(r→) ∝ A 0(r→) - A 2(r→), is not zero. Using numerical integration, we found A 0 >> A 2 for all values of α in the relevant interval. We further notice that the narrowest A 0(r) is obtained if one sets θ = α and χ = 0 for the functions in the integral. Therefore, we neglect A 2 and approximate E→(r→) as
We found that in fact A 0 (r) is almost the same (within a few percent deviation) for all values of α, and that an accurate representation of A 0(r) (with any α) is given by Eq. (5) if a is set to ≈ 60° in this equation (a reasonable agreement is found if α is between ≈ 60° and 90°). Then, the PSF (nexc = 2) is
Its resolution (FWHM) is ~ 0.3λ/n (cf. Table 1). This approximation is valid only for the central peak of the PSF, because the function in Eq. (5) features side lobes that are missing in most of the space around the central peak of the real “Inf. beams, 3D” PSF (Fig. 2). If the side lobes are ignored, the estimate for Veff based on Eq. (5) is V 1 ≈ 0.069 (λ/n)3. However, the real PSF is about 1.5 times wider in the y- than in the z- or x-directions, and the estimate for V eff with r replaced by 1.5r is V 2 ≈ 0.231 (λ/n)3. Using these values, we estimate V eff= (V 2 1 V 2)1/3, which gives V eff ≈ 0.1 (λ/n)3, close to the computed value (Fig. 3).
The light confinement in the focus of a microscope can be improved in comparison with the 1-beam and 4Pi cases, if one uses ≥3 beams. For 3 or 4 beams, V eff can be decreased two-fold in comparison with the 4Pi V eff, which itself is ≈ 1/2 of the V eff for the conventional 1-beam microscope. With more than 3 or 4 beams, further decrease in V eff is not significant. In the limiting case of an infinite number of beams, V eff is decreased by another 30 to 50 %, but this is achieved only if one employs an abstract and idealized assumption of arbitrary α. The smallest possible V eff achieved under this assumption is ≈ 1/2 of that in the 3-beam case. Thus, the 3-beam setup provides the smallest value of V eff for realistic α, and this value is close to the theoretical limit.
Note that the 3-, 4-, or 6-beam setups are realistic, even though some assumptions (e.g., α = 60°) may be difficult to realize in practice, and the actual focal volume of such setup may be somewhat larger than in theory. The setups with three, four, or six symmetrically focused illumination beams have been constructed [3, 12, 15], demonstrating a reduction in the size of the illumination spot when compared with conventional microscopy. The reported performance has been similar to or somewhat improved over the 4Pi case, and our theoretical study suggests that in principle such setups can decrease the focal volume down to one half of the 4Pi microscope focal volume, which would be a significant improvement for FM and FCS methods. The consideration of the case with an infinite number of beams convinces one that only a factor of two of a further improvement is possible in the best case, which is, however, based on a strongly idealized assumption. This finding shows that there is a fundamental physical limit, similar to the Abbe diffraction limit for single beams, for decreasing the illumination spot in a microscope employing structured illumination.
Remarkably, the resolution of the 4Pi microscope in one (z) dimension is the highest one among the systems studied. Techniques such as STED , SSIM , STORM , FPALM , and PALM  provide better resolution only by using photophysical effects. Thus, even the maximal light confinement using purely optical principles, as considered here, cannot compete with these techniques for imaging resolution. However, purely optical light confinement is necessary for applications like FM and FCS, and in such cases the smaller V eff furnished by, e.g., 3-beam setup, might be useful.
We thank John Stone, Kirby Vandivort, Jana Hüve, Martin Kahms, and Reiner Peters for useful discussions. The support is from the National Science Foundation (grant PHY0822613), and the L. S. Edelheit fellowship (A. A.).
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