We demonstrate in experiment that the resolution of a conventional light microscope can be enhanced by 26% with the help of an image inverting interferometer. In order to prove this statement, we measured the point spread function of the system as well as the resolution of two-point objects. Additionally, the contrast transmission function of the interferometric setup was measured and compared to the results gained with a conventional wide-field microscope. Using the interferometric system, the spatial frequencies near the cutoff-frequency were far better transmitted than by the conventional microscope. Finally, we demonstrate the improved resolution with the help of images of two-dimensional structures.
© 2011 OSA
Today, confocal fluorescence microscopy (CFM) is a technique often applied in medical and biological science. The CFM has a limited resolution, as do all other optical methods. The lateral resolution of these conventional systems exceed Abbe’s limit  only insignificantly. During the last years, other methods with a lower limit of resolution have appeared. Non scanning methods are for instance STORM  and SIM . STORM uses the statistical activation of fluorescence markers and a calculation of the core area of the visible Airy disks thus gain a resolution of up to 25 nm . Another possibility of improving resolution is provided by SIM. Here, spatial frequencies higher than the cutoff-frequency are transformed to lower areas with the help of structured illumination. This facilitates a resolution of around 50 nm . In order to gain an image with SIM, multiple exposures (shift and rotation of the illumination structure) have to be recorded.
Scanning methods are for example 4Pi [6, 7] and STED [8, 9] systems. Using a STED microscope, the area containing fluorescence molecules is significantly reduced by stimulated emission. Hence, a lateral resolution of 25 nm can be achieved . Interference effects enable an axial resolution of up to 80 nm capitalized by 4Pi microscopes . Nevertheless, the lateral resolution of a 4Pi microscope is not increased compared to a CFM.
Sandeau et al. [12–14] suggested a method to increase the lateral resolution of 4Pi microscopes. It is improved by utilizing the spatial incoherence of the fluorescent sample with the help of an image inverting interferometer (III). A similar proposal was introduced by Wicker and Heintzmann using CFMs . They demonstrated a possibility to enhance lateral resolution of CFMs up to 30% in theory. The resolution limit of such systems cannot reach the limit of some of the systems mentioned above, but CFMs are commonly used systems. It is possible to upgrade existing systems with this technique because the III can be put directly onto the exit of such microscopes without changing their setup. Furthermore, it is possible to combine an III to other kinds of scanning imaging systems using spatial incoherent light.
After giving evidence that an improved resolution is probable [15, 16], we were the first team to demonstrate an enhanced two-point resolution with a conventional wide-field microscope . The enhancement of the resolution was limited to 12% due to the used setup. We will demonstrate in this article that it is possible to increase the resolution to a maximum of 26% by modifying the applied setup. Additionally, other measurements have been done characterizing the system and exploring its options. For practical purposes, a low aperture objective lens was used to demonstrate the feasibility of the system. Furthermore, we present the first two-dimensional images of practical samples which could be recorded with this method. Instead of a CFM, we use a conventional wide-field microscope (in scanning mode).
2.1. The interferometric point spread function
An image inverting interferometer (III) is a two-beam interferometer with an image inverting optics (e.g. a telescopic 4-f-system) included in one of the arms. Hence, there are two exits where the inverted and the non-inverted amplitude distribution mix. If the setup is aligned, both a constructive and a destructive superposition can be observed at the exits. The point spread function (PSF) of the setup can be calculated by examining the transmission of a single point through the full system (Fig. 1). A source point at the position r0 is imaged to the position r′0 as well as to −r′0 because of the III between object and image plane, with r′0 = −Mr. The coordinates in object room are given by r = (x,y) and the coordinates in image space are r′(x′,y′). M denotes the magnification of the system. As a result, one gets two coherent amplitude distributions on opposite sites of the optical axis (ro.a. = (0,0)) given by the systems amplitude point spread function (APSF) h(r′). Possible wavefront deformations created in the arms of the interferometer can be considered by eiφi(r′) in the inverting arm respectively eiφni(r′) in the non-inverting arm. The intensity distribution on the detectors is determined by Eq. (1).
The first two terms of Eq. (1) identify the Airy disks. They are identical at both exits and can be eliminated by a subtraction of the intensity distribution of both exits. Hence, the differential image only consists of the interference structure. In this image, an area P with the optical axis as center is chosen as the area of integration. The result is a signal which depends only on the position of the source point projected onto the image plane r′0. Because the signal characterizes the transmission of a point by the system, it can be described as an interferometric point spread function IPSFpc(r′0) (see Eq. (3)). The index pc identifies the considered wavefront deformation in the arms of the interferometer.
An integration over the full plane of the difference image (P → ∞) results in the IPSF (Eq. (4), Fig. 2) calculated by Wicker and Heintzmann  if the wavefront deformations are equal in both arms or do not exist (Δφ(r⃗′) ≡ 0).Fig. 2). Two points may be 32% closer to each other than by using a conventional microscope considering the Rayleigh criterion (Eq. (5)). The dip in the intensity distribution between the maxima down to 74% could be measured.
If the wavefront deformations differ, the radius of integration providing the best IPSFpc will be finite . This resulting function will be broader than its corresponding BSinc-function in any case.
The interferometric image of the object can be calculated by a convolution of the IPSF with the distribution of the object (Eq. (6)) if the object consists of source points incoherent to each other, or if the illumination of the object is incoherent. Due to the integration over the image plane, only information of the point on the optical axis is available. Hence, the object has to be scanned in order to obtain a complete image.
2.2. The interferometric modulation transfer function
Apart from the PSF an optical system can be characterized by the optical transfer function (OTF), which describes the blurring of a point in space. The modulation transfer function (MTF) is the absolute value of the OTF. It describes the transfer of spatial frequencies from the object plane into the image plane.17]. Eq. (8)). The cutoff frequency fmax is identical to the cutoff frequency of the incoherent MTF. Nevertheless, all frequencies up to fmax are transferred with a constant value. Hence, it is not possible to transmit interferometrically higher spatial frequencies than in incoherent imaging, but fine structures are better to detect because they are transmitted with a higher contrast. As soon as the IPSF differs from the BSinc-function, the IMTF will not be a cylindrical function anymore.
Figure 3 shows the setup that has been used. The light source is imaged by an objective lens (10x / 0.25) onto the object plane to guarantee a fully incoherent illumination of the object plane. As a light source an LED with λ̄ =516 nm and a bandwidth of 16 nm is used. The objective lens OL (3.2x / 0.10) images the object plane OP onto the focal plane of the entrance lens L1 of the Mach-Zehnder interferometer. We use an objective lens with a low numerical aperture because the setup is easier to handle, and we mainly want to demonstrate the functional principle as well as the feasibility of the setup. Additionally, the requirements to the test objects are lower. High numerical aperture lenses will be used in future as well as fluorescent specimens. Behind the beamsplitter BS, half of the intensity passes the non-inverting arm. In contrast, the amplitude distribution passing the inverting arm of the interferometer is inverted spatially by a mirror telescope to avoid chromatic aberrations. The exit lenses L2a and L2b image the intermediate image onto the CCD-cameras at the exits of the interferometer. Due to different focal lengths of the lenses L1 and L2a/b, the intermediate image is magnified by a factor of 7.2. Post-processing (subtraction of the images and integration) is done by a PC. The integrated difference image contains the information of the point on the optical axis in the object plane only. Therefore, it is necessary to scan the specimen to get the whole interferometric image. In this setup it is realized by a piezo positioning system.
Except for the image inversion the wavefront in the focal exit plane of the telescope in the inverting arm is identical to the wavefront at the focal input plane, because the mirror telescope is built as a 4-f-setup. This results in differing wavefront curvatures since the propagation in free space leads to another wavefront than the imaging with the telescope. It can be described as a Fresnel diffraction resulting in a phase difference Δφ ∼ |r′|2. As described in  the integration area P must have a finite size to gain the narrowest IPSFpc(r′0) possible. For that reason it was useful to use CCD-cameras instead of detectors without a spatial resolution. With the help of the camera images it was possible to vary the integration area to get the smallest IPSF. In our setup, the best radius of integration consists of 13 camera pixels (ca. 96.2μm).
In recent works, we were able to measure an IPSF with a 12% narrower FWHM compared to the same system yet without image inversion . The resolution could be increased by 10% by using the Rayleigh criterion and different two-point objects. We were able to demonstrate that the increase in resolution is smaller than the expected one because of the mentioned phase differences Δφ. To reduce the influence of these, the systems pre-magnification was altered from 6.1x to 3.2x. Hence, the Airy disks on the CCD-cameras own only about half the size they had before. However, this does not influence the cosine-term (Eq. (3)) which modulates the resulting interference pattern because it is caused by the propagation of light inside the interferometer only. Therefore, the influence of the cosine-term to the interfering area near to the optical axis could be strongly reduced compared to previous measurements. This leads to an IPSF which is more narrow and converges to the expected BSinc-function.
The PSF and the IPSF were measured with a pinhole of the size 1.00μm, which was etched into a chromed glass plate. This illuminated pinhole is used as a point-like light source. This is feasible because the diameter of the pinhole is nearly two times smaller than the FWHM of the theoretically expected IPSF. Calculations evinced that the influence of the spatial stretch of the pinholes is negligible. Two of these pinholes are applied at different distances to measure the two-point resolution of the system. With the help of different gratings the IMTF of the used image inverting microscope could be recorded in detail and compared to theoretical calculations. Furthermore, to verify the benefit of this method with practical specimens we imaged the logo of the FSU Jena and a cluster of polystyrene beads.
4.1. Point spread function
Figure 4 shows the measured IPSFpc compared to the Airy-disk of the system. Also a theoretical calculation of the expected IPSFpc is plotted considering the properties (wavefront curvatures, radius of integration, ...) of the used setup. The FWHM of the point spread is reduced by 24% from 2.55μm in the conventional case to 1.93μm in the interferometric one. The distance between the first zeros is reduced by 48% to 3.36μm and the distance of the first minima decreases by 30% to 4.51μm. The measured values correspond to the theory (see Eq. (3)) in good approximation (see Fig. 4, Table 1). Deviations appear in the intensity for values higher than the first minima. This is most likely caused by an idealized quadratic wavefront deformation in the theoretical model. Aberrations might appear due to optical components - especially by the image inverting telescope - which are not included in the theory. There is a strong indication that the higher order deformations are small compared to the quadratic ones because the significant deviations from theory appear at the side lobes for the first time.
4.2. Two-point resolution
We use the Rayleigh criterion (Eq. (5)) to characterize the two-point resolution of the system. The intensity decreases between the images of two points to 74% if the imaging is ideal and diffraction limited, and if the maximum intensity of one image is located at the first minimum of the other. Hence, a resolution of this setup (without the image inversion) up to 3.15μm could be expected. We measured two pinholes with a distance of 3.40μm and two with a distance of 3.20μm, resulting in a dip to 70% and respectively 77% of the highest intensity value. So, a distribution similar to the Rayleigh criterion can be calculated to 3.29μm. At a 2.40μm distance between two pinholes, the curve is dipped to 80% using the interferometric setup. The intensity decreases further to 70% with a two-point object of 2.5 μm distance. Hence, the resolution is increased by 26% using the III.
A cross-section of the image of a two-point object with a point pitch of 3.2μm can be found in Fig. 5(a). The black line illustrates the intensity distribution of the conventional image, whereas the distribution using the III is represented by the red line. The dip to 77% with the conventional system drops down to 20% with the interferometer.
The conventional microscope cannot resolve a two-point object with a distance of 2.40μm between the two points (Fig. 5(b)) in contrast to the interferometric system. Here, the signal dips to 80% of its maximum value, so that the two points can be resolved clearly.
4.3. (I)MTF recording
Additionally, we characterized the system by the measured MTF. For that purpose, rectangular gratings representing different spatial base frequencies were scanned. Afterwards, the contrast in the images was measured. The cutoff frequency of an incoherent imaging system can be calculated by . Hence, a cutoff frequency of fMax = 388 Lp/mm can be found for an objective lens with a NA of 0.10 assuming an average wavelength of λ̄ = 0.516μm. For this reason, 17 gratings were recorded with base frequencies f0 between 40 Lp/mm and 400 Lp/mm. The measured gratings have to be fitted by a Fourier series to gain the contrast of the fundamental mode as well as the contrast of existing harmonics because a rectangular grating can be represented as an infinite series of spatial frequencies.
Figure 6(a) presents the cross-sections through a rectangle with the base frequency of f0 = 71 Lp/mm. In the interferometric case, the contrast of the base frequency is significantly higher with 0.82 compared with 0.72 in the classical image. In the interferometric curve, overshoots exist in the bar of the grating. The slopes of the function are stronger than in the conventional case. This is a strong evidence that higher spatial frequencies are transferred better by the III. The increase in contrast of spatial frequencies near the cutoff frequency is clearly visible at the grating with 360 Lp/mm (Fig. 6(b)). A contrast of 0.21 can be detected with the III while no modulation can be measured in the conventional image.
Figure 7 illustrates the measured MTFs in the conventional and in the interferometric case. It shows the contrasts of the fundamental modes as well the contrasts of the higher harmonics. All in all, the values of the conventional measurement are near the theoretical curve, although they are 5% below theory in major parts. The deviations are most likely caused by the non-ideal objective lens of the microscope, which results in a non-diffraction limited imaging. The values of the harmonics fit with the curve of the fundamental mode as expected.
The theoretical graph of the interferometric measurement was calculated by Eq. (7) as the Fourier transform of the theoretical IPSFpc (Eq. (3)). Qualitatively, the distribution fits with the measured values. A dip of the lower spatial frequencies under the value of one is predicted, as well as a contrast amplification for middle spatial frequencies. The area of the strongest contrast amplification at 230 Lp/mm is also described correctly. The IMTFs maximum value of 1.21 (theory) cannot be gained by the highest experimental value of 1.16. Quantitatively, there are significant deviations of the IMTF compared to theory. The location of the local contrast minimum is not depicted exactly at the calculated location either. The differences between measurement and theory amount up to 50%.
Two causes might result in the insufficient correlation of the measured and the calculated IMTF. At first, the calculation was based on an ideal PSF. Apparently, this is an approximation. The measured point spread differs from the theoretical expectations as well as the measured MTF. The conclusion is that the APSF is not identical to the calculated BSinc-function either. Secondly, only quadratic wavefront deformations are assumed in the theoretical model. Additional wavefront deformations caused by the optical elements and the image inverting system were not considered.
4.4. Imaging of complex object structures
The system has been characterized by the PSF, the resolution of two points and the MTF in the sections above. Now, the potential of the system will be illustrated by the recording of a complex object’s structure. The first sample is the logo of the university etched into a chromium layer.
Figure 8 enables a direct comparison of a detail (36.75μm x 29.75μm) of the sample conventionally (b) and interferometrically (c) measured. Besides an increased contrast in the interferometric image, more details are visible than in the conventional image. At the etches and at fine structures overshoots or negative image values appear, caused by the structure of the IPSF. For human beings, this results in an unfamiliar impression of the image, although it is possible to compensate such effects via post-processing.
The red dashed line in Fig. 8 marks the area that is illustrated as cross-section in Fig. 9. In this figure, finer structures are visible in the interferometric image than in the conventional one. Especially for distances bigger than 4 μm an enhanced contrast can be observed. The actual enhanced resolution can be noticed in the area of 3μm to 4μm. Here, structures can be resolved that are not visible in the classical image.
The second sample were polystyrene beads with a diameter of 1.10μm on a glass carrier. The beads can be considered as point-like objects due to their small dimensions compared to the (I)PSF. Figure 10 illustrates a number of such beads recorded conventionally (a) and interferometrically (b). An enhanced contrast is clearly observable as well as an enhanced resolution. Two clusters are visible surrounded by single beads. In the conventional image, no inner structure can be resolved, in contrast to the interferometric image. Single beads are visible inside the clusters.
Three cross-sections are presented in Fig. 11. They proceed in direction of the marked lines out of Fig. 10. Figure 11(a) is a cross-section of both clusters, where clear structures are visible in the interferometric image that are not resolved in the classical image.
The FWHM of a point spread is again measured by the diameter an image of a bead (Fig. 11(b)). Compared to the conventional image, the FWHM decreases from 2.70μm to 2.21μm by 21%. This result fits in approximation with the measurement of the pinhole (see section 4.1). There are two beads with a distance of 2.82μm in the observed area. Using the classical microscope, there was no dip observable in the cross-section (Fig. 11(c)). Only a bar is visible. Interferometrically measured, there is a significant increase of the signal, resulting in two resolved points.
We were able to proof that the resolution can be improved by 26% by using an image inverting interferometer compared to a conventional wide-field microscope. The distance of the first two minima of the IPSFpc was reduced by 29.8% compared to the PSF of an equivalent wide-field microscope, and the distance between the two first zeros decreased even by 47.7%. After measuring the two-point resolution, the IMTF was determined by rectangular gratings of different periods. During these measurements, significant quantitative differences appeared in relation to the theory despite a good qualitative correlation of the drawn graph with the theoretical curve. They can be explained by non-perfect optics as well as by additional wavefront deformations due to the interferometer.
Furthermore, we could present images of two-dimensional object structures for the first time. It was shown that the contrast and the resolution of the images could be significantly improved by using the III. For example, details like the inner structure of a cluster of polystyrene beads, which were not resolvable with the classical wide-field microscope became visible.
In a previous work we reached an increase of the two-point resolution of 12% only . The huge difference to the theoretically predicted 32% were caused by wavefront deformations in the III. We were able to reduce their influence by modifying our experimental setup. The resolution could possibly be increased even further by minimizing the differing wavefront aberrations in the inverting and the non-inverting arms of the interferometer. This reduction can for instance be realized by a three dimensional setup of the interferometer. In such a system the inversion is performed only by plane mirrors. Hence, the spherical deformation of the wavefront is identical in both arms due to equal optical paths. Such a system was already discussed theoretically by Wicker et al. . As soon as these optimizations are completed, lenses with high numerical apertures can be used. Finally, the step to a confocal setup with fluorescent samples can be performed.
References and links
1. E. Abbe, Die Lehre von der Bildentstehung im Mikroskop (Friedrich Vieweg & Sohn, 1910).
5. M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution,”Proc. Natl. Acad. Sci. USA (PNAS) 102, 13081–13086 (2005). [CrossRef]
6. S. W. Hell and E. H. K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A 9, 2159–2166 (1992). [CrossRef]
7. J. Bewersdorf, A. Egner, and S. W. Hell, “4Pi Microscopy,” in Handbook of Biological Confocal Microscopy, 3rd ed., J.B. Pawley, eds. (SpringerScience+Business Media, New York, 2006), pp. 561–570. [CrossRef]
8. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19, 780–782 (1994). [CrossRef] [PubMed]
9. T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. USA (PNAS) 97, 8206–8210 (2000). [CrossRef]
10. G. Donnert, J. Keller, C. A. Wurm, S. O. Rizzoli, V. Westphal, A. Schoenle, R. Jahn, S. Jakobs, C. Eggeling, and S. W. Hell, “Two-color far-field fluorescence nanoscopy,” Biophys. J. 92, L67–69L (2007). [CrossRef] [PubMed]
11. H. Gugel, J. Bewersdorf, S. Jakobs, J. Engelhardt, R. Storz, and S. W. Hell, “Cooperative 4Pi excitation and detection yields sevenfold sharper optical sections in live-cell microscopy,” Biophys. J. 87, 4146–4152 (2004). [CrossRef] [PubMed]
12. N. Sandeau and H. Giovannini, “Increasing the lateral resolution of 4pi fluorescence microscopes,” J. Opt. Soc. Am. A 23, 1089–1095 (2006). [CrossRef]
13. H. Rigneault, N. Sandeau, and H. Giovannini, “Interferometric confocal microscope,” World Patent n°: WO/2007/141409 (2007).
14. N. Sandeau, L. Wawrezinieck, P. Ferrand, H. Giovannini, and H. Rigneault, “Increasing the lateral resolution of scanning microscopes by a factor of two using 2-Image microscopy,” J. Eur. Opt. Soc. Rap. Public. , 4 (2009), p. 09040. [CrossRef]
16. D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Investigation of the impulse response of an image inversion interferometer,” Opt. Commun. 283, 368–372 (2010). [CrossRef]
17. D. Weigel, H. Babovsky, A. Kiessling, and R. Kowarschik, “Investigation of the resolution ability of an image inversion interferometer,” Opt. Commun. 284, 2273–2277 (2011). [CrossRef]
19. J. W. Goodmann, Introduction to Fourier Optics (McGraw-Hill Classic Textbook Reissue, 1988).