Interferometric resolution improvement for confocal microscopes

Kai Wicker; Rainer Heintzmann

doi:10.1364/OE.15.012206

1. Introduction

Fluorescence confocal microscopy has become an indispensable tool of modern biology, allowing the imaging of fluorescent specimen with high lateral as well as axial resolution. As an advancement of the basic confocal microscope the 4Pi-microscope allows axial resolution as small as 100 nm [1]. Sandeau et al. proposed the 4Pi’-microscope to further increase the lateral resolution [2, 3]. It is a modification of the 4Pi-microscope that contains an image inversion system in one of the microscope’s arms. We propose a method of improving the lateral resolution that does not require the separate arms of the 4Pi-microscope but can instead be applied to regular confocal microscopes [4]. While working on this project a similar scheme [5, 6] by Sandeau et al. was brought to our attention.

To achieve this resolution improvement, an interferometer is placed in the detection pathway of the microscope, as shown for the Mach-Zehnder interferometer in Fig. 1. The image from the microscope is split at the first beam splitter. It is then inverted in one of the interferometer’s arms before being recombined at the second beam splitter. After passing through optional pinholes, both the interferometer’s constructive (I ₊) and destructive (I _-) output intensities are measured. For the case of light emitted by a fluorophore on the optical axis, the inverted amplitude image is identical to the non-inverted one. Therefore all light will be collected in the constructive output, while the destructive output remains dark. If, however, the fluorophore is far off axis, the inverted and non-inverted amplitude images will have hardly any spatial overlap and cannot interfere, therefore leading to equal intensities in both I ₊ and I _-. At intermediate distances destructive interference can further decrease the I ₊ signal.

Fig. 1. Lateral resolution can be improved using an interferometer. Light coming from a microscope is split, inverted in one arm of the interferometer and recombined.

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This general bias of on-axis light being detected preferably in I ₊ leads to the aforementioned lateral resolution improvement.

2. Analysis of the interferometric detection PSF without a pinhole

For the case of the 4Pi’-microscope this resolution enhancing effect has been analysed by means of geometrical optics [2] and numerical calculations of the optical transfer function (OTF) [7]. Below we want to derive an analytical interpretation for the case of interferometric detection without a pinhole.

2.1. Lateral resolution of the detection PSF

We consider a two dimensional optical system with an arbitrary amplitude point spread function (APSF) a(r⃗), r⃗=(x,y) and its corresponding amplitude transfer function (ATF) ã(k⃗), where the tilde denotes the Fourier transform. For a point source of constant brightness (we do not consider illumination in this analysis) displaced by a distance d⃗ from the optical axis (r⃗=0) the combined APSFs in the constructive (g ₊) and destructive (g _-) channel respectively are

g_{\pm} (\vec{r}, \vec{d}) = 1 ⁄ 2 (a (\vec{r} - \vec{d}) \pm a (\vec{r} + \vec{d}))

where ⊗ denotes the convolution operation. In Fourier space this can be rewritten as

{\tilde{g}}_{\pm} (\vec{k}, \vec{d}) = 1 / 2 \tilde{a} (\vec{k}) (\exp (ι \vec{k} \cdot \vec{d}) \pm \exp (- ι \vec{k} \cdot \vec{d})),

where k⃗=(k _x,k _y) is the spatial wave vector and $ι = \sqrt{- 1}$ . Without a pinhole the intensity at the detector in dependence on displacement d⃗ is

I_{\pm} (\vec{d}) = \int {\int_{- \infty}^{\infty} | g_{\pm} (\vec{r}, \vec{d}) |}^{2} d x d y .

Using Parseval’s theorem, which states that the Fourier transform is unitary or in other words that the integrated spectral energy equals the integrated spatial energy, the above equation can be written as

\begin{array}{l} I_{\pm} (\vec{d}) = \int {\int_{- \infty}^{\infty} | {\tilde{g}}_{\pm} (\vec{k}, \vec{d}) |}^{2} d k_{x} d k_{y} \\ = \frac{1}{4} \int \int_{- \infty}^{\infty} {| \tilde{a} (\vec{k}) |}^{2} (2 \pm e^{ι 2 \vec{k} \cdot \vec{d}} \pm e^{- ι 2 \vec{k} \cdot \vec{d}}) d k_{x} d k_{y} \\ I_{\pm} (\vec{d}) = \frac{1}{2} I_{0} \pm \frac{1}{16} \int \int_{- \infty}^{\infty} | \tilde{a} (\frac{\vec{k}'}{2}) |^{2} e^{ι \vec{k}' \cdot \vec{d}} d k_{x}^{'} d k_{y}^{'} \pm \frac{1}{16} \int \int_{- \infty}^{\infty} | \tilde{a} (- \frac{\vec{k} ″}{2}) |^{2} e^{ι \vec{k} ″ \cdot \vec{d}} d k_{x}^{″} d k_{y}^{"}, \end{array}

where we used the substitutions k⃗^′=2k⃗ and k⃗^″=-2k⃗ and I ₀ is the total integrated intensity in both outputs. The last two terms of can easily be identified as inverse Fourier transforms. A Fourier transform of I _± thus gives us the OTF

{\tilde{I}}_{\pm} (\vec{k}) = 2 π (\frac{I_{0}}{2} δ (\vec{k}) \pm \frac{1}{16} [{∣ \tilde{a} (\vec{k} ⁄ 2) ∣}^{2} + {∣ \tilde{a} (- \vec{k} ⁄ 2) ∣}^{2}]) .

This interferometric OTF contains the absolute square of the original ATF magnified to twice its original size, as indicated by the argument k⃗/2. This is an improvement even over the non-interferometric wide field OTF, h̃(k⃗)=ã(k⃗)⊗ã*(-k⃗), which is an auto-correlation of the original ATF. It is however worth mentioning that for constant illumination and without a pinhole the confocal detection PSF is constant and therefore has no resolution at all, whereas the wide field resolution is only achieved for a closed pinhole.

As an example we look at the conventional wide field ATF in scalar theory at low numerical aperture (NA):

$\tilde{a} (\vec{k}) = {\begin{matrix} \sqrt{I_{0}} ⁄ (\sqrt{π} k_{c}), ∣ \vec{k} ∣ \leq k_{c} \\ 0, ∣ \vec{k} ∣ > k_{c} . \end{matrix}$

For the proposed setup this gives us an interferometric OTF

{\tilde{I}}_{\pm} (\vec{k}) = {\begin{matrix} π I_{0} δ (\vec{k}) \pm \frac{I_{0}}{4 k_{c}^{2}}, ∣ \vec{k} ∣ \leq 2 k_{c} \\ 0, ∣ \vec{k} ∣ > 2 k_{c} \end{matrix}

and a corresponding interferometric PSF

I_{\pm} (r) = I_{0} (1 ⁄ 2 \pm J_{1} (2 k_{c} r) ⁄ (2 k_{c} r)) .

where J₁ is the first order Bessel function of the first kind.

Figure 2(b) shows the interferometric detection OTF Ĩ₊ in comparison to the corresponding wide field OTF h̃. The resolution improvement is obvious: while Ĩ_± has the same support as h̃ it does not fall off towards the edge of the support region, therefore significantly enhancing the higher frequency components.

Note that the above scalar derivation holds for each vector component of the electric field. The vector theory intensity PSF can be calculated as the sum of the individual intensities of each component.

Fig. 2. Comparison of PSFs (a) and OTFs (b). The resolution improvement is strongest for the difference signal ΔI=I ₊-I _-. The interferometric OTFs do not fall off towards the edge of the support region, therefore enhancing high frequency components. Note that the interferometric signals were calculated for detection without pinhole, in which case the detection PSF (OTF) of a confocal system would be constant (a δ-peak) and not contribute to the overall resolution at all.

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2.2. Detection PSF without pinhole is independent of axial position z

In order to analyse the out-of-focus behaviour of interferometric detection without a pinhole we need to multiply ã(k⃗_xy) with the z-dependant free space ATF, $\tilde{o} ({\vec{k}}_{xy}, z) = \exp (- ι z \sqrt{{(2 π ⁄ λ)}^{2} - {\vec{k}}_{xy}^{2}})$ [8], where k⃗_xy=(k _x,k _y) is the spatial wave vector in the xy-plane only. As this only affects the phases in Fourier space but not the spatial frequency intensity spectrum, the detected integrated intensity and therefore the detection PSF I _± and OTF Ĩ_± remains independent of the axial position z, a fact that proves useful in extended focus imaging (see section 5).

3. Minimising the constant background

The δ-peak in Eq. (2) is responsible for the offset of I ₀/2 in the detection PSF I _± (Eq. (3)) and ensures positivity of I _± (see Fig. 2). However it is also responsible for unwanted contribution to the signal by light sources far away from the optical axis. There are several ways to reduce the impact of this constant term in the PSF.

3.1. Localised illumination

The contribution of the δ-peak in the OTF can be reduced by combining interferometric detection with localised illumination (e.g. confocal or two-photon [9]). As long as the illumination OTF does not have a δ-peak, the combined OTF, which is the convolution of the detection OTF with the illumination OTF, does not exhibit a δ-peak either. Interferometrically enhancing high frequencies in the detection OTF results in an improved frequency response in the combined OTF.

3.2. Detection pinholes

Introducing detection pinholes reduces unwanted contributions caused by the constant offset in the detection PSF. Additionally, the pinholes also block out-of-focus light and therefore allow confocal sectioning. At the same time the relative significance of interferometric detection on the lateral resolution will be reduced for small pinholes: for the case of a closed pinhole the PSF I ₊ resulting from a symmetric APSF a is identical to the regular confocal PSF, while I- remains zero and no interferometric improvement in resolution is achieved.

3.3. Multiple use of the interferometer

The constant offset in Eq. (3) can in principle be eliminated through multiple use of interferometers. Light in the constructive output can be subjected to another pass through the same or a similar interferometer. However, applying the inversion operation twice results in the identity operation 1̂. Again using image inversion in a second interferometer would therefore always lead to constructive output only. A different operation has thus to be used in order to reduce the offset. When using a second interferometer, an image rotation of 90° would reduce the offset to I ₀/4. As long as the operation R̂ applied on the image fulfills the conditions R̂ⁿ≠1̂ and R̂ⁿ a(0)=a(0), ∀n ∈ N, multiple use of the interferometer can make the constant offset arbitrarily small while preserving the on-axis performance.

3.4. Subtraction of signals

The PSFs of the two interferometer outputs differ only in the sign of the non-constant terms. Subtracting the two signals yields the difference signal ΔI=I ₊-I _- from which the offset has been removed. However, one has to consider that this ΔI will result in some negative values and that there may be signal to noise issues, as the δ-peak/offset will still contribute to noise despite not contributing to the signal. High photon numbers or finite samples as well as the methods mentioned above will reduce the influence of this noise. When using successive interferometers, a weighted subtraction or more advanced methods such as weighted averaging or combined deconvolution should be applied [10].

4. Simulations for confocal systems

Using MATLAB (The MathWorks, MA, USA) together with the DIPimage toolbox (Quantitative Imaging Group, TU Delft, The Netherlands) we have simulated the effect of interferometric detection in combination with confocal illumination and detection pinholes. The simulations were done for circularly polarised light using high NA vector theory [11]. Interference was calculated individually for each electric field vector component. Parameters used for these simulations were: excitation wavelength λ _e=488 nm, detection wavelength λ_d=525 nm, numerical aperture NA=1.2, refractive index n=1.33, pinhole size 1 Airy disc. In order to distinguish the resulting PSFs from the detection-only PSFs we refer to the simulated PSFs as C (confocal), C _± (interferometric with pinhole) and ΔC=C ₊-C _- (difference in interferometric outputs).

4.1. Point spread functions

Figure 3 shows the resulting PSFs for a pinhole of the size of one Airy disc. Figure 3(a) shows the constructive output C ₊. While the axial resolution on axis is the same as for the confocal case C (Fig. 3(c)), the lateral resolution improvement is evident, with a full width at half maximum (FWHM) of about 168 nm for C+ as compared to 218 nm for C. Subtracting the destructive output signal C- (Fig. 3(b)) from C ₊ results in the difference signal ΔC shown in Fig. 3(d). This eliminates the δ-peak in the OTF Eq. (2) and therefore the offset of I ₀/2 in Eq. (3), thus further improving the resolution, with a FWHM of about 135 nm. Note that this subtraction results in some negative values (red lines) of -2.9%.

Fig. 3. PSFs for the constructive (a) and destructive (b) interferometer outputs C _±, for the confocal case without an interferometer C (c) and the difference in signal in the two interferometric outputs ΔC (d). Interferometric detection yields an improvement in lateral resolution: Lateral FWHM are 218 nm for C, 168 nm for C+ and 135 nm for ΔC. ΔC exhibits small negative values of about -2.9%. For simulation parameters see section 4. In all cases the pinhole has the size of one Airy disc.
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4.2. Sectioning capability

The axial extension of the interferometric PSF (with detection pinhole and confocal illumination) is governed by the pinhole size and not improved beyond that of the corresponding confocal PSF. However, the sectioning capability is still improved: Fig. 4 shows the simulated signal generated by a homogenous fluorescent plane perpendicular to the optical axis for a conventional confocal microscope C _plane, as recorded in the constructive interferometric output C _plane,+ and for the difference in constructive and destructive signal ΔC _plane. While for large distances from the focus all curves exhibit the same falloff proportional to z ^-2, the interferometric curves C _plane,+ and ΔC _plane drop off faster for small distances from the focus, leading to a FWHM of about 617 nm for C _plane, 594 nm for C _plane,+ and 550 nm for ΔC _plane.

Fig. 4. Capability of the various methods for sectioning fluorescent planes. (a) The interferometer has an increased sectioning capability, with the difference signal surpassing the constructive output. (b) The logarithmic plot shows a z ^-2-dependance far away from the focal plane for all methods. However, the intensities of the interferometric measurements fall off more quickly close to the focus.
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5. Extended focus imaging

As the in-plane detection PSF without a pinhole (I _±) derived in section 2 has no z-dependence, interferometric detection without a pinhole seems an ideal candidate for realising extended focus imaging with high resolution.

A sample can be illuminated with a z-independent excitation PSF, such as a Bessel beam [12] for scalar theory:

I_{B} (r, z) \propto J_{0}^{2} (k_{B} r),

where J ₀ is the zeroth order Bessel function of the first kind. The large side lobes of the Bessel beam lead to excessive blurring for this type of imaging (Fig. 5, 6(b), 7(b)). However, instead of collecting all light emitted by the sample, we can combine this illumination with the interferometric detection scheme (PSF from Eq. (3)). An instrument could thus have a scanning Bessel beam and the proposed partial image inversion interferometric detection with no (or a very large) pinhole. We get as a resulting extended focus PSF for the respective interferometer outputs

C_{ef, \pm} (r, z) = I_{B} (r, z) \cdot I_{\pm} (r, z) \propto J_{0}^{2} (k_{B} r) (1 ⁄ 2 \pm \frac{J_{1} (2 k_{c} r)}{2 k_{c} r}) .

Using the same parameters and full vector theory as in section 4, we simulated PSFs for extended focus imaging (Fig. 5). The Bessel beam used for illumination was simulated for an annular aperture with a width of 0.5% of the NA, positioned at maximum spatial frequency. The advantage in suppressing off-axis light in the difference signal is clearly visible.

Fig. 5. Radial plots of the extended focus PSFs. The simulation used circular polarisation and high NA vector theory. It is visible that the signal for Bessel beam excitation with integral detection decays much slower than for the difference term, for which it quickly reaches zero when the emitter is off axis. The FWHMs of the respective PSFs are 199 nm (integral detection), 146 nm (constructive) and 116 nm (difference).
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To look at the imaging performance of these point spread functions, a synthetic sample was constructed consisting of a wagon wheel noodle, a solid and a hollow sphere and a point-like structure (Fig. 6(a)). Maximum photon numbers of 15,000 per pixel, as typical for widefield data, were assumed in the simulated images. The image size corresponds to 6µm. Figure 6(b) shows the result for Bessel beam excitation and integrating detection without the interferometric setup. Details are blurred and the point-like structure is hardly visible. The image simulated for the constructive channel (Fig. 6(c)) already shows a slight improvement in contrast. Subtraction of the image recorded in the destructive channel (Fig. 6(d)) yields the difference image (Fig. 6(e)), which has strikingly improved contrast and resolution and clearly shows the point-like structure. Figure 6(f) shows a contrast enhanced difference image, in which negative values were clipped.

In order to illustrate the usability for the imaging of biological samples we also simulated extended focus images based on a 4Pi data set of an actin and tubulin cytoskeleton of a cardiac fibroblast (Fig. 7).

The blue channel was simulated for an excitation wavelength of λ _e=390 nm and an detection wavelength of λ _d=440 nm, whereas the green channel again used the parameters of section 4. The sample data used is shown in Fig. 7(a). The image recorded in the constructive output (Fig. 7(c)) shows little improvement over the image one would get when using Bessel beam excitation and detecting integrated emission (Fig. 7(b)). However, subtracting the image recorded in the destructive output (not shown) yields an image with significant increase in detail, contrast and resolution (Fig. 7(d)).

6. Conclusions

Interferometric detection with image inversion in one beam path can be used to improve the lateral resolution of scanning fluorescence microscopes such as confocal, 4Pi or extended focus imaging microscopes. Especially subtracting the signals recorded in the two interferometer outputs yields a significant increase in resolution. Applying weighted averaging in Fourier space or a combined deconvolution [10] can further enhance the final image. This is especially valuable for the two-dimensional images acquired with extended focus imaging. These approaches as well as image subtraction could also be used to increase the quality of regular 4Pi and 4Pi’ images.

Fig. 6. Simulated extended focus images using scanning Bessel beam excitation. The synthetic sample (a) consists of a wagon wheel noodle, a solid and a hollow sphere and a point-like structure. The scalebar indicates a length of 2 µm. (c) and (d) are the constructive (I ₊) and destructive (I _-) outputs respectively. The constructive image already exhibits a slight improvement in contrast over the non-interferometric integrating detection (b). The difference (e) of the two interferometric outputs ∆I shows a significant improvement in performance. (f) shows a contrast enhanced version of (e), where negative values were clipped. Maximum photon numbers of 15,000 per pixel were assumed in all simulations.
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At equal lateral resolution the interferometric detection would allow for larger pinholes than confocal detection, increasing the photon detection efficiency substantially. However, increasing the pinhole size will decrease the axial resolution. For extended focus imaging, where no detection pinholes are used, our method yields a significant improvement in resolution and contrast, making it potentially interesting for optical tomography.

For the 4Pi’ microscope the optical axis defined by the image inversion does not coincide with the moving optical axis defined by the scanned beam, which requires the 4Pi’ to be operated as a stage scanning system. An interferometer as proposed in this paper could be added to the detection pathway of most point scanning microscopes. When placed after the descanning mechanisms, it will also work for beam scanning systems. However, a major challenge remains in building a stable achromatic image inversion interferometer which also should not suffer from polarisation problems.

Multiple use of the interferometer can reduce the constant offset in the detection PSF to nearly zero, further improving the resolution and eliminating potential signal-to-noise problems.

Fig. 7. Simulated extended focus images using scanning Bessel beam excitation. (a) shows a sum projection image of the sample data used for the simulation, a 4Pi data set of a cardiac fibroblast. (b) shows the simulated image for non-interferometric integrating detection. While the constructive (I ₊) image, (c), only exhibits a slight improvement in contrast, the difference (d) of the two interferometric outputs ∆I (negative values clipped) shows a significant improvement in performance. Maximum photon numbers of 15,000 per pixel were assumed. The scalebar in (a) shows a length of 2 µm. We thank Elisabeth Ehler for the cardiac fibroblast sample.
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References and links

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