Aberration correction in coherence imaging microscopy using an image inverting interferometer

Daniel Weigel; Armin Kiessling; Richard Kowarschik

doi:10.1364/OE.23.020505

1. Introduction

Whenever optical imaging is applied, problems of aberrations occur that can reduce the quality of the image significantly. The causes of these aberrations are diverse, often depending on the concrete field of application. In microscopy, for instance, spherical aberrations can result from the coverslip or from refractive index differences between the specimen and the immersion liquid [1–4]. As these aberrations depend on both the refractive index differences and the depth of the plane observed, they can only be effectively reduced by employing adaptive optics [5–7]. Furthermore, variations of the refractive index in the specimen itself can result in phase aberrations that can affect the image significantly [5–8].

In astronomy, similar limitations can be found due to the turbulent atmosphere causing refractive index fluctuations. In this way, the resolution ability is de facto not restricted by diffraction limits but by prevailing seeing conditions [9]. With the emergence of adaptive optics in the late 1980s, however, these restrictions could be largely overcome [10]. Already in the early 1970s, it was suggested to measure the mutual coherence function of astronomical objects by means of a rotational shear interferometer [11]. If the amplitude and the phase of the complex coherence function can be gained, the intensity distribution of the object can be reconstructed. With the help of this method, known as interferometric imaging, it is possible to reduce the influence of the turbulent atmosphere by averaging many short-time exposures [12–16].

During the past 40 years, numerous studies about similar methods have been published that can all be attributed to the acquisition of the complex coherence function and the cross-spectral density, respectively. A major part of these examines the possibility of reconstructing the three-dimensional structure of spatially incoherent object distributions [17–22]. In this context, this technique is therefore also referred to as coherence imaging or (in)coherence holography.

It is furthermore possible to reconstruct images with an infinite depth of field from a reduced subset of the complex coherence function [23–25]. This is particularly interesting for applications in microscopy requiring high numerical apertures while at the same time requiring the specimen to be imaged completely sharp in depth. Possible applications would, for instance, be the manufacturing control, the inspection of integrated circuit boards, microfluidic flow cytometry, or stereophotogrammetric three-dimensional shape measurements.

Wagadarikar et al. presented an approach to correct stationary wavefront errors by means of coherence imaging [26]. For this purpose, they used an iterative two-step alternating minimization algorithm based on the so-called compressive sampling [27]. If the object structure is sparse, the algorithm converges to a local minimum and both the corrected reconstruction of the object distribution and the aberration function can be obtained. While the requirement of a sparse object distribution in microscopy should be fulfilled in most cases, the fact that only a local and not the global minimum of the minimization problem can be found could lead to problems in practice. One of the consequences can be remaining aberrations or artifacts.

This suggestion therefore seems useful if it is not possible to determine the wavefront errors by a reference measurement. However, examples from microscopy using adaptive optics show that, in many situations, wavefront distortions can be directly determined by measuring so-called guide stars (e.g. fluorescent nanobeads) [28, 29]. We thus suggest an application of this approach to the coherence imaging microscopy. As we have already demonstrated, it is appropriate to measure the coherence function with the help of an image inverting interferometer in the pupil of the microscope objective [25]. Furthermore, this approach has the advantage that aberrations which are symmetric to the optical axis do not disturb the reconstruction of the object structure. This is a great benefit as these form a major part of the aberrations (e.g. astigmatism, spherical aberration) in practice. Moreover, we prove that it is possible to determine remaining wavefront errors by the system itself and to correct them in a purely numerical way.

2. Preliminary consideration

The functional principle of coherence imaging microscopy has already been derived in detail [25]. To discuss the possibility of aberration correction and to introduce all variables and parameters, the main issues of the derivation will be delineated in the following. Note that constant prefactors are omitted for reasons of clarity and comprehensibility.

It is useful to study the propagation of light from the object plane towards the microscope objective first. Here, the object plane is located in a distance d = f + Δz from the front principal plane of the microscope objective, with f denoting its focal length and Δz being the displacement from the focal plane. Figure 1 shows this schematically by means of one single point. In the context of the paraxial approximation, the propagation into the entrance pupil is expressed as a Fresnel transform, which can be formulated as a Fourier transform of the product of the object amplitude distribution U₀(ρ, t) and the quadratic phase function $h (ρ) = exp (i \frac{k}{2 d} ρ^{2})$ :

U_{L} (r, Δ z, t) = \iint U_{0} (ρ, t) exp (i \frac{k}{2 d} {(r - ρ)}^{2}) d ξ d η

= \frac{exp (i \frac{k}{2 d} r^{2})}{i λ d} ℱ {U_{0} (ρ, t) \cdot h (ρ)} (r, Δ z, t) .

The lateral coordinates in the object space are described by the position vector ρ = (ξ, η) and those in the plane of the lens by r = (x, y). As we assume a spatially incoherent luminescent object or a spatially incoherent illumination, the complex amplitude distribution in the object plane is given by

U_{0} (ρ, t) = \sqrt{I (ρ)} exp (i ϕ (ρ), t)

. Here, I(ρ) represents the intensity distribution of the object structure and ϕ(ρ, t) is a phase function that statistically varies in space and time [30]. Note that the dependency on time t is restricted to the description of the temporally varying phase function; however, U₀ is assumed to be a stationary amplitude distribution. As is customary, our notation denotes the complex unit by i and the wavenumber by

k = \frac{2 π}{λ}

, with λ being the wave length.

Fig. 1 Schematic illustration of the light propagation from the object point to the lens.

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The amplitude distribution U′_L(r, Δz, t) in the exit pupil of the microscope objective can be described as a multiplication of U_L(r, Δz, t) by a quadratic phase function and by the pupil function P(r, R) with the radius R:

U_{L}^{'} (r, Δ z, t) = U_{L} (r, Δ z, t) exp (- i \frac{k}{2 f} r^{2}) P (r, R)

As shown in [25], it is possible to reconstruct the intensity distribution I by the coherence function Γ measured in the exit pupil of the microscope objective. Therefore, Γ can be measured, for instance, by means of an image inverting interferometer (III). This is a rotational shearing interferometer using a shear angle of 180°. Hence, with respect to the inversion axis r_ia = (0, 0), opposite points of U′_L(r, Δz, t) are correlated at the exits of this interferometer. The III provides the greatest possible correlation distance Δr = 2r at a given size of the pupil and hence the maximal information content of the object structure [25].

Figure 2 shows the scheme of such a coherence imaging system based on an III. The exit pupil of the microscope objective (MO) is located in the front focal plane of lens L₁ and the detector is located in the back focal plane of lens L₂. The relay lenses L_1/2 have a distance of f_L1 + f_L2, whereby the amplitude distribution U′_L(r, Δz, t) is reproduced on the detector. Without loss of generality, we assume a magnification M = f_L2/f_L1 of 1. We therefore continue to use the position vector r in order to describe the coordinates of the detector plane. Due to the III between the relay lenses, the amplitude distributions U′_L(r, Δz, t) and U′_L(−r, Δz, t) interfere on the detector.

Fig. 2 Schematic sketch of the proposed coherence imaging microscope.

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As a result, both a bias intensity I₀(r) and an interference term I_IF(r) can be observed. The latter corresponds to the real part of the complex coherence function Γ in the plane of the exit pupil [25]. Under quasi-monochromatic conditions, Γ can be obtained, for instance, by capturing interference patterns at no less than three different phase differences Δψ [31, 32]. If the quasi-monochromatic approximation cannot be met, there is the possibility of determining Γ at the desired wavelength λ using approaches from Fourier spectroscopy [25, 33, 34].

The measurable complex coherence function can be expressed by Eq. (4). With respect to the inversion axis, opposite points from U′_L(r, Δz, t) are correlated with each other for that purpose. Consequently, Γ is a function of the correlation distance Δr = 2r.

Γ (Δ r) = {〈 U_{L}^{'} (r, Δ z, t) \cdot U_{L}^{' *} (- r, Δ z, t) 〉}_{t}

= P (r, R) P^{*} (- r, R) {〈 U_{L} (r, Δ z, t) U_{L}^{*} (- r, Δ z, t) 〉}_{t}

Inserting Eq. (3) in Eq. (4) leads to the loss of the quadratic phase term caused by the lens. Furthermore, the factor

exp (i \frac{k}{2 d} r^{2})

[Eq. (2)] vanishes due to the multiplication of U_L by

U_{L}^{*}

. This causes a loss of the depth information, as already discussed in detail in [25]. As a consequence, a quasi-infinite depth of field can be gained.

Calculating the temporal average 〈·〉_t and utilizing the assumed spatial incoherence, Eq. (5) can be transferred into Eq. (6) [25]. Therefore, Γ in the pupil plane corresponds to the Fourier transform (ℱ) of the object distribution I, multiplied by P(r, R)P^* (−r, R).

Γ (Δ r) = P (r, R) P^{*} (- r, R) \cdot ℱ {\frac{1}{λ^{2} d^{2}} I (\frac{ρ}{2})} (Δ r)

Γ thus represents the spatial frequency spectrum of the reconstructible object distribution I_R. Discussing the spatial frequency spectrum, it is useful to transform Δr into the spatial frequencies

ν = \frac{Δ r}{λ d}

. In the case of a circular exit pupil with radius R, a cutoff-frequency

ν_{max} = \frac{2 R}{λ d}

occurs. Hence, for structures in the focal plane of the microscope objective, the cutoff-frequency becomes

ν_{max} = \frac{2 NA}{λ}

. Due to the factor 2 in the definition of the spatial frequencies ν, the substitution ρ′= ρ/2 suggests itself and Eq. (6) results in:

Γ (ν) = P (ν, ν_{max}) P^{*} (- ν, ν_{max}) \cdot ℱ {\frac{1}{λ^{2} d^{2}} I (ρ^{'})} (ν)

= {OTF}_{IF} (ν, ν_{max}) \cdot \tilde{I} (ν) .

The term P(ν, ν_max)P^* (−ν, ν_max) limits the spatial frequency spectrum of the object distribution

\tilde{I} (ν) = ℱ {\frac{1}{λ^{2} d^{2}} I (ρ^{'})} (ν)

and can therefore be considered as the optical transfer function OTF_IF of this interferometric system. Finally, the reconstructed image can be gained by an inverse Fourier transform of Γ(ν). Therefore, I_R can be written as the convolution (⊗) of the interferometric point spread function PSF_IF with the intensity distribution of the object I:

I_{R} (ρ^{'}) = ℱ^{- 1} {{OTF}_{IF} (ν, ν_{max}) \cdot \tilde{I} (ν)}

= {PSF}_{IF} (ρ^{'}) \otimes I (ρ^{'})

3. Optical transfer function

In general, the pupil function P(ν, ν_max) must be considered as complex and can be separated into the modulus p(ν, ν_max) and the phase function Φ(ν):

P (ν, ν_{max}) = p (ν, ν_{max}) exp (i \frac{2 π}{λ} Φ (ν)) .

While p(ν, ν_max) influences the modulus of the incident light, Φ(ν) modifies its phase. Hence, Φ(ν) corresponds to the phase aberrations within the exit pupil and can be denoted as aberration function [30]. If we assume conventional microscope objectives with circular exit pupils and negligibly small absorption effects, p(ν, ν_max) corresponds to a disc function, which is defined as:

p (ν, ν_{max}) = {\begin{matrix} 1 & | ν | / ν_{max} \leq 1 \\ 0 & | ν | / ν_{max} > 1 . \end{matrix}

In this case, the modulation transfer function (MTF) and therefore the modulus of the OTF_IF is also a disc function, which leads to a constant transfer of spatial frequencies up to the cutoff-frequency. Whereas ν_max is the same in conventional incoherent imaging, high spatial frequencies are transferred more efficiently using the method presented. As a result, an improved two-point resolution and a better image contrast can be observed. Obviously, the strength of the transferred spatial frequencies is not influenced by phase aberrations. This fact is important for the aberration correction, as will be shown later. Φ(ν) only leads to a modulation of the phase transfer function (PTF), being the argument of the OTF_IF. Introducing the phase difference ΔΦ(ν) = Φ(ν) − Φ(−ν), the OTF_IF can be written as:

{OTF}_{IF} (ν, ν_{max}) = P (ν, ν_{max}) P^{*} (- ν, ν_{max})

= p (ν, ν_{max}) exp (i \frac{2 π}{λ} Δ Φ (ν)) .

As we will prove, aberrations symmetric to the optical axis do not influence the imaging quality of the proposed technique. This feature applies for position-dependent and position-independent aberrations. Especially for biological specimens, this could be an huge benefit. Here, differences between the refractive index of the specimens and the immersion fluid lead to depth-dependent spherical aberrations. In this paper, we want to concentrate on the completely isoplanatic case. We therefore assume that the aberration function Φ is spatially invariant.

3.1. Uncorrected system

For a better discussion, it is useful to write the aberration function Φ(ν) as the weighted infinite sum of Zernike polynomials [35–37]. They are a normalized, complete and orthogonal system of polynomials by which any aberration in the exit pupil can be described. These polynomials are defined on the unit circle and are, due to their structure, often expressed in polar coordinates. We therefore choose the substitutions ν̄ = |ν|/ν_max, ψ = atan(ν_y/ν_x) in order to express the Zernike polynomials, yielding:

Φ (ν) = \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} c_{n}^{m} Z_{n}^{m} (\bar{ν}, ψ) .

Here,

c_{n}^{m}

are the so-called Zernike coefficients, which weight the respective Zernike polynomials

Z_{n}^{m}

. In the chosen notation, the value of

c_{n}^{m}

also represents the root mean square error (RMSE), which is caused on the wavefront by the respective aberration. The Zernike polynomials can be described as the product of the normalization factor

N_{n}^{m}

with a radial polynomial

R_{n}^{| m |}

and an angle-dependent term:

Z_{n}^{m} (\bar{ν}, ψ) = {\begin{matrix} N_{n}^{m} R_{n}^{| m |} (\bar{ν}) cos (m ψ) & m \geq 0 \\ - N_{n}^{n} R_{n}^{| m |} (\bar{ν}) sin (m ψ) & m < 0 . \end{matrix}

Here applies that n − |m| ≥ 0 and n − |m| have to be even. The index n indicates the highest power in the radial polynomial

R_{n}^{| m |}

[Eq. (17)], thereby corresponding to the radial order of the Zernike polynomials. Index m, on the other hand, is referred to as the azimuthal order of the polynomials. It defines the frequency of the sine and cosine function in Eq. (16), which is why m is also called azimuthal frequency. δ_m0 in Eq. (18) represents the Kronecker delta function and is 1 for m = 0 and otherwise 0.

R_{n}^{| m |} (\bar{ν}) = \sum_{l = 0}^{(n - | m |) / 2} \frac{{(- 1)}^{l} (n - l)!}{l! (\frac{n + | m |}{2} - l)! (\frac{n - | m |}{2} - l)!} {\bar{ν}}^{n - 2 l}

N_{n}^{m} = \sqrt{\frac{2 (n + 1)}{1 + δ_{m 0}}}

Table 1 presents an overview of the Zernike polynomials up to the fourth radial order. Obviously, all Zernike polynomials with even m, and thereby all even radial orders n, feature the symmetry property $Z_{n, even}^{m} (\bar{ν}, ψ) = Z_{n, even}^{m} (\bar{ν}, ψ + π)$ . However, for odd radial orders n, the Zernike polynomials are point-symmetric to the optical axis; therefore applies $Z_{n, odd}^{m} (\bar{ν}, ψ) = - Z_{n, odd}^{m} (\bar{ν}, ψ + π)$ . So, the phase difference ΔΦ(ν) = Φ(ν) − Φ(−ν) can be expressed as the sum of all odd radial orders of the Zernike polynomials:

Δ Φ (ν) = 2 \sum_{\begin{array}{c} n = 0, \\ odd \end{array}}^{\infty} \sum_{m = - n}^{n} c_{n}^{m} Z_{n}^{m} (\bar{ν}, ψ)

In contrast to conventional incoherent imaging systems, aberrations of even radial orders (e.g. astigmatism, defocus, or spherical aberration) therefore do not influence the reconstruction of the object structure. Note that this is a special feature of the III. Using other interferometers instead, for instance radial shearing interferometers [22] or rotational shearing interferometers with shear angles ≠ 180° [24], larger subsets of aberrations may disturb the imaging process.

Table 1. Overview of the Zernike polynomials up to the fourth radial order. Polynomials with even radial orders are marked gray.

View Table

3.2. Aberration correction

If the system shows aberrations of odd radial orders n limiting the quality of the reconstruction in a significant way, the method introduced here offers a simple possibility of correcting these aberrations purely numerically without worsening the signal-to-noise ratio (SNR).

Due to the fact that the experimentally measured Γ corresponds to the complex object spectrum, it is possible to correct all phase errors in the isoplanatic case by numerically multiplying Γ by a complex correction function. With regard to Eq. (14), it is obvious that this correction function has to be $exp (- i \frac{2 π}{λ} Δ Φ (ν))$ .

ΔΦ can be easily measured by the coherence imaging system itself. It is a further advantage of this approach that there is no need of additional optics or hardware to measure the phase errors influencing the imaging quality. To obtain the experimentally measured ΔΦ, from now on denoted with ΔΦ_exp, it is necessary to image a defined reference wavefront through the optical system, including the III, onto the detector. Using a spherical wave for that purpose is most appropriate as it is especially easy to realize and to analyze. In this case, the simple relation ΔΦ_exp = λ · arg(Γ)/(2π) applies. In practice, a spherical wave can be easily achieved, for instance, by focusing a plane wave into the object plane of the detection microscope objective. If this is not possible or if, additionally, aberrations occurring deep inside the specimen shall be corrected, there is the possibility of preparing the specimen with reflecting and fluorescent sub-diffraction microspheres [29, 38, 39], respectively. Using the inverse of the so measured aberration function as a correction phase function, it is possible to gain the aberration-corrected object spectrum Γ_corr(ν) and, therefore, a diffraction-limited imaging:

Γ_{corr} (ν) = exp (- i \frac{2 π}{λ} Δ Φ_{exp} (ν, ν_{max})) \cdot {OTF}_{IF} (ν, ν_{max}) \cdot \tilde{I} (ν)

{= OTF}_{IF, corr} (ν, ν_{max}) \cdot \tilde{I} (ν)

Note that in the case of a circular exit pupil OTF_IF,corr (ν, ν_max) = p(ν, ν_max). Due to the disc-shaped OTF_IF, the PSF_IF of the aberration-corrected coherence imaging system is a Bessel-sinc function given by Eq. (22) [25]. As a result, the full width at half maximum (FWHM) of the PSF_IF is 32% smaller than the FWHM of the conventional PSF (Airy disc), which leads to a slightly improved two-point resolution and an enhanced image contrast. In contrast to the Airy disc, the Bessel-sinc function also has side lobes with negative values up to −0.13 (normalized to 1). This has to be considered while interpreting and illustrating the reconstructed images.

{PSF}_{IF} (ρ^{'}) = \frac{J_{1} (2 ρ^{'})}{2 | (ρ^{'}) |}

In practice, the measurement of ΔΦ_exp might be compromised by different influences. For instance, using a laser source, impurities on the optical surfaces like dust particles or scratches, could cause coherent artifacts. Therefore, it is advisable to decompose ΔΦ_exp into the Zernike polynomial system. The value of the Zernike coefficients

c_{n}^{m}

for real optical aberrations typically decreases rapidly for higher n; hence, it is sufficient to take only the first n_max orders into account. If n_max is chosen properly, all appreciable aberrations can be considered, and influences of the usually high frequent disturbances are avoided. The resulting phase function ΔΦ_fit can be calculated by:

Δ Φ_{fit} (ν, ν_{max}) = 2 \sum_{n = 0}^{n_{max}} \sum_{m = - n}^{n} c_{n}^{m} Z_{n}^{m} (\bar{ν}, ψ) .

The difference of ΔΦ_fit and ΔΦ_exp corresponds to the residual phase error ΔΦ_res. In this, the mentioned disturbances as well as aberrations of higher radial orders are included.

4. Experimental setup

In order to prove the numerical aberration correction capability of the proposed technique, we used the setup shown in Fig. 3. It is a similar setup like for the investigation of the infinite depth of field of such a coherence imaging microscope [25]. Due to practical reasons we decided to use a transmitted light microscope as the basic setup. Therefore, the object plane (OP) is illuminated by a microscope objective (MO_ill, NA = 0.40 / 20×). For reasons of clarity, the illumination system (Köhler illumination) is not illustrated here. To calibrate the III and for the reference measurements we use a helium neon laser (HeNe, λ = 632.8nm). Due to the required spatial incoherence, an LED (center wavelength λ̄ ≈ 633nm) is applied for the illumination of the extended object (resolution test chart). Light, diffracted by the specimen, illuminates the exit pupil of the detection microscope objective (MO_det), which is imaged onto the detection plane through the III by means of the relay lenses L₁ (f₁ = 300mm) and L_2a/b (f_2a/b = 105mm). As we only want to prove the principle, a microscope objective with a low numerical aperture (NA) of 0.07 and a magnification of 2.5× was used for practical purposes. The CCD₁ camera at the constructive exit of the III is located in the back focal plane of L_2a. In this way, the superposition of the exit pupil with its spatially inverted copy can be observed. The distance of the CCD₂ camera at the destructive exit of the III is chosen in a way that it is located in a plane conjugated to the object plane. Therefore, by blocking the light in one of the interferometer arms, this exit is used to acquire the conventional image of the specimen including all aberrations of the complete imaging system, being the microscope and the respective other arm of the III. The III consists of a Mach-Zehnder interferometer, which is supplemented by two Dove prisms (D_1/2), one in each arm. They are rotated with respect to each other by 90°, whereby the amplitude distributions in both arms of the interferometer are rotated with respect to each other by 180°. This corresponds to the required spatial inversion [25]. Between MO_det and L₁ there is a beam splitter BS₀, by which a conventional control image can be observed on the CCD₀ camera using the tube lens L_T. Hence, the aberrations arising within the III have no influence on this image.

Fig. 3 Sketch of the experimental setup. The III is marked by the dashed box.

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In the first part of the experiment, the aberrations inherent in the coherence imaging system are examined. For that purpose, the complete system consisting of the microscope objective and the III must be considered. The aberrations of the microscope objective used were comparably low; therefore, to demonstrate the ability of the suggested method, large Dove prisms (length of the base: 81.3mm, aperture 20.0mm × 20.0mm) were applied for image inversion. The so induced large wavefront errors are primarily determined by astigmatism, coma, and trefoil. Note that the so caused aberrations depend, strictly speaking, on the viewing angle leading to non-isoplanatic aberrations. Thus, during the following experiments, a small field of view (FOV) is used and therefore, the aberrations can be assumed to be isoplanatic.

In order to determine the effective wavefront errors ΔΦ_exp, a collimated HeNe laser beam is focused onto the object plane of the illumination microscope objective. The spherical waves emanating from the focus point are collected by the detection microscope objective MO_det. Its exit pupil is then, as described, imaged through the III onto the CCD₁. Due to the temporal coherence of the light of the laser, at least three interference patters with different phase differences Δψ are sufficient for a complete determination of Γ. For this purpose, we used the algorithm described in [32]. As discussed above, using a spherical wave, the argument of the so measured Γ is equivalent to $\frac{2 π}{λ} Δ Φ_{exp}$ . After determining these aberrations, the test chart was set onto the object plane and illuminated by a red LED using a spectral filter (peak transmission at λ = 633nm, Δλ ≈ 10nm). As the quasi-monochromatic condition was not satisfied, we employed a method using approaches from the Fourier spectroscopy to gain Γ [25, 33, 34]. Therefore, one mirror of the interferometer was shifted with the help of a piezo actuator (P). While varying the optical path difference by ±1.9μm, 43 single interferograms were acquired.

In the second part of the experiment, we generated additional aberrations by positioning a glass slide (GS) between object plane and microscope objective. A transparent adhesive film on the slide produces distinct random wavefront errors, reducing the imaging quality of the microscope to a considerable extent. Figure 4 shows a scheme of both the measuring of the aberration function ΔΦ_exp [Fig. 4(a)] and the imaging of the test chart [Fig. 4(b)].

Fig. 4 (a) Scheme of the reference measurement (determination of the aberrations) and (b) of the actual measurement of the resolution test chart.

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5. Results & discussion

5.1. System-inherent aberrations

Figure 5(a)–5(c) shows the three interferograms measured in order to determine ΔΦ_exp and ΔΦ_fit, respectively. The interferences, encoding the effective phase aberrations of the whole imaging system, are obviously disturbed by high-frequent ring structures and other coherent artifacts. They are assumedly caused by diffraction at dust particles within the optical path. Correspondingly, these effects can also be found in $Δ Φ_{exp} = \frac{λ}{2 π} arg (Γ)$ [Fig. 5(d)]. To dispose of these artifacts, the λ -periodic ΔΦ_exp was unwrapped [40,41] and then decomposed into Zernike polynomials up to the 10th radial order. As a result, ΔΦ_fit [Fig. 5(e)] and the Zernike coefficients $c_{n}^{m}$ , respectively, could be calculated. As already mentioned, the Dove prisms inside of the interferometer cause the major part of the aberrations and as they are different in both arms, aberrations of an even order can be observed as well. As expected, the Dove prisms mainly cause astigmatisms ( $c_{2}^{- 2} = 0.22 λ$ , $c_{2}^{+ 2} = 1.08 λ$ ); however, the trefoil ( $c_{3}^{- 3} = - 0.06 λ$ , $c_{3}^{+ 3} = - 0.09 λ$ ) and coma ( $c_{3}^{+ 1} = - 0.05 λ$ ) also contribute to the total wavefront error significantly. Comparing ΔΦ_exp, ΔΦ_fit and the residual phase errors ΔΦ_res [Fig. 5(d)–5(f)], it is evident that ΔΦ_res is dominated by the coherent disturbances, and ΔΦ_fit contains all relevant wavefront errors. Hence, n_max = 10 is a reasonable choice for our setup.

Fig. 5 From the interferograms at three different phase differences (a)–(c), ΔΦ_exp (d) can be gained. By unwrapping ΔΦ_exp and decomposing it into Zernike polynomials up to the 10^th radial order, ΔΦ_fit can be gained (e). The remaining errors ΔΦ_res are shown in (f). Note that for a better illustration, (d)–(f) represent the modulo of the respective functions.

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If ΔΦ_exp is interpreted as the effective part of the distorted wavefront, ΔΦ_res can be regarded as the aberration-corrected wavefront, containing only aberrations and disturbances which could not be corrected by ΔΦ_fit. Accordingly, it is possible to quantify the wavefront correction by the peak-to-valley (PV) wavefront error and the RMSE of the wavefront, respectively. Whereas ΔΦ_fit shows a very large PV wavefront error of 6.20 λ, that of ΔΦ_res is 0.96 λ, still appearing to be relatively large. But as it can be seen from Fig. 5(f), only very small parts of ΔΦ_res lead to this large PV error. The RMSE, with its higher significance concerning the imaging quality of an optical system, decreases from 1.17 λ to 0.04 λ, which corresponds to an improvement by a factor of 29.3.

5.1.1. Point spread function

Due to the fact that the numerical aperture of MO_ill was much larger than the one of MO_det, the laser beam, focused onto the object plane, can be treated as a point-like source of light. Therefore, it is appropriate to use the setup shown in Fig. 4(a) (for now without the glass slide) also to measure the point spread function (PSF) of the conventional and the coherence imaging system. In Fig. 6, the images of this point source, reconstructed from Γ, are compared to these gained in a conventional way. Figure 6(a) shows the nearly diffraction-limited conventional control image observed by means of the tube lens L_T on the CCD₀ [Fig. (2)]. Next to it, Figs. 6(b) and 6(c) show the point images acquired by the CCD₂ behind the III. To gain these images after the passage of light through the two arms of the interferometer, the light passing the respective other arm was blocked. As a result of the large Dove prisms, strong astigmatic distortions can be observed in both point images. Figure 6(d) shows the reconstruction of the point-like source by means of the Fourier transform of Γ. Here, the differences of the wavefront errors (ΔΦ) from both arms of the interferometer lead to an astigmatic distortion as well. However, if Γ is multiplied by the correction factor $exp (- i \frac{2 π}{λ} Δ Φ_{fit})$ , all aberrations of the imaging system up to the 10^th order are corrected. Therefore, we will call this procedure system correction from now on. The resulting point image is shown in Fig. 6(e). Due to the disc-shaped MTF, its FWHM is even smaller by 32 % than the FWHM of the conventional point image [Fig. 6(a)]. The comparison to the theoretically expected point image without wavefront errors [Fig. 6(f)] indicates that the remaining wavefront errors are practically not relevant.

Fig. 6 Illustration of the point images. (a) shows the conventional control image before the III. The conventional images on the CCD₂, which were observed after passing through arms 1 and 2, respectively, are shown in (b) and (c). In (d) the reconstruction of the point source without the phase correction can be seen, whereas in (e) the phase correction was applied. For comparison, (f) shows a simulation of the diffraction-limited point image.

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5.1.2. Two-point structures

In order to show that this correction also works with more complex structures, a test chart with different two-point structures was examined, using the setup shown in Fig. 4(b) but without the glass slide between the test chart and the detection microscope objective. By means of lithography, pinholes with a diameter of 1μm were edged into the chrome substrate on a glass slide. They are arranged in pairs with distances from 1.5μm to 4.4μm. As the diffraction-limited conventional PSF have a radius of 0.61 λ /NA ≈ 5.5μm (λ = 633nm, NA= 0.07), the pinholes can be treated as point-like.

Figure 7(a) shows the conventional control image (CCD₀) of the two-point structures without the influence of the III. Due to the size of the PSF, even the points with a distance of 4.4 μm (top, left) can hardly be resolved. Figure 7(b) shows the two-point structures after being imaged through arm 1 of the interferometer onto CCD₂. The pairs are heavily blurred due to the wavefront errors emerging within the interferometer. However, the system aberrations also lead to heavy blurring effects in the reconstruction of the object structure with the help of Γ. As a result, the point pairs cannot be recognized in Fig. 7(c), either. Applying the system correction, and therefore multiplying Γ by the correction factor $exp (- i \frac{2 π}{λ} Δ Φ_{fit})$ , these aberrations can be corrected almost completely, as shown in Fig. 7(d). The resolution of the point pairs is even better than in Fig. 7(a). In order to illustrate the improved resolution, cross-sections through the first and through the second column of Fig. 7(a) and Fig. 7(d) are shown in Fig. 6(e) and Fig. 7(f). The dip between the conventionally imaged point pairs with a distance of 4.4 μm is comparable to the interferometrically imaged ones with a distance of 3.4 μm. From this, an enhanced two-point resolution of about 23 % can be estimated.

Fig. 7 Illustration of the resolution of two-point objects with different distances. This is a comparison of the conventional image without the III (a), to the conventional image behind the III (b) as well as to the reconstruction of the structures by means of Γ without (c) and with correction of the aberrations (d). (e) shows the cross-section through the first column of two-point structures and (f) the cross-section through the second column.

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We want to point out that the wavefront errors induced by the III can be clearly minimized by other realizations of interferometers. For instance, by applying plane mirrors for image inversion [42], the large astigmatisms could be reduced considerably.

5.2. Additionally induced aberrations

In this section, we will examine additionally large phase errors emerging between the object plane and the microscope objective. For this purpose, a glass slide carrying an adhesive film was set between the microscope objective and the object plane (Fig. 4). By the position of the glass slide, different random wavefront errors could be generated. Two of them will be discussed in the following, denoted as wavefront error 1 (WFE 1) and 2 (WFE 2). Like in the preceding section, the phase correction functions ΔΦ_fit,1/2, corresponding to the respective wavefront errors, were determined with the help of the HeNe laser first [Fig. 4(a)]. Hereafter, the test chart was acquired using the Köhler illumination (LED) [Fig. 4(b)]. Note that besides the two random wavefront errors caused by the adhesive film, ΔΦ_fit,1/2 also includes the aberrations ΔΦ_fit of the imaging system (microscope objective and III), measured in section 5.1. So, by subtracting ΔΦ_fit from ΔΦ_fit,1/2 it is possible to calculate the phase distortions that have emerged between the microscope objective and the object plane.

The results of the these two measurement series are summarized in Fig. 8. Both examples show in an impressive way how well the correction of aberration works. When comparing the images, note that different imaging scales were chosen for a better impression, but the white scale bars are always 20μm. While the upper half shows the measurements for WFE 1, the lower half shows that for WFE 2. Here, the first lines present the point images, which were measured using the focused HeNe laser beam, as shown in Fig. 4(a).

Fig. 8 Influence of two random wavefront errors on the point images and the associated MTFs and the PTFs, respectively. Besides the conventional images, three reconstructions by means of Γ are shown. The latter represent the uncorrected, the system-corrected, and the fully corrected images. To verify that especially aberrations of odd radial orders influence the imaging process, the Zernike coefficients $c_{n}^{m}$ of the system-corrected wavefront are plotted. Additionally, images of an 2D object under the respective conditions are shown.

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The conventional point image observed behind the interferometer (arm 1) is shown on the left [Figs. 8(a) and 8(n)]. All aberrations of the imaging system, including the adhesive film, the Dove prisms, and the beam splitters of the interferometer contribute to these images. Due to the large wavefront errors in both cases, the conventional images are strongly blurred in different ways and cannot be identified as a single point source any more.

To the right, there are the reconstructions of the point source by means of the Γ [Figs. 8(b) and 8(k)]. They contain both the aberrations caused by the adhesive film and the system-inherent aberrations as well. Comparing the uncorrected reconstructions of the point source [Figs. 8(b) and 8(o)] to each other, it is apparent that the differences are significantly smaller here than in the conventional case. This already indicates that the even-order aberrations caused by the adhesive film do not have an effect on the image reconstructions and therefore, these images are dominated by the system’s aberrations and the remaining odd-order aberrations.

Figures 8(c) and 8(p) illustrate the reconstructions gained from Γ after correcting the system-inherent aberrations, which were measured in section 5.1. Therefore, Γ was multiplied by $exp (- i \frac{2 π}{λ} Δ Φ_{fit})$ and then the Fourier transform was applied. Hence, only the phase aberrations caused by the adhesive film take effect on this reconstruction. It can be seen that the system-corrected images are already more point-like but there are still numerous distortions arising from the adhesive film.

On the right, the fully corrected point images are shown [Figs. 8(d) and 8(q)], for which ΔΦ_fit,1/2 was subtracted from the phase of the measured Γ before the Fourier transform. Whereas the point image in Fig. 8(d) can be treated as quasi diffraction limited, Fig. 8(q) shows distinct asymmetric structures in the side lobes. Furthermore, the peak intensity declines from 0.87 (WFE 1) to 0.81 (WFE 2), each normalized to the peak intensity of a diffraction limited point image.

Figures 8(e) and 8(r) show the conventional MTFs achieved numerically by the Fourier transform of the conventional point images. Both images illustrate that the MTFs are clearly different from 0 only in a small area around the center. This leads to the major part of the object information being irreversibly lost due to the conventional imaging.

In contrast, this loss of information does not occur with the technique suggested here. The MTF of the proposed method is, also in the presence of phase aberrations, constant up to the cutoff-frequency. Hence, Figs. 8(f)–8(h) and Figs. 8(s)–8(u) show the PTF, being more significant in this case. Whereas Figs. 8(f) and 8(s) show the uncorrected phase of Γ, corresponding to the point images Figs. 8(b) and 8(o), Figs. 8(g) and 8(t) illustrate the effects of the numerical system correction. In this way, the RMSE of the wavefronts could be reduced by a factor of about 3.7 (WFE 1) and 2.6 (WFE 2). So, the major part of the aberrations is extinguished and the pure influence of the adhesive film can be observed. The phase distributions feature a distinct point symmetry referring to the domination of aberrations with odd radial orders. To clarify this property of the suggested technique, the Zernike coefficients $c_{n}^{m}$ of these system-corrected wavefronts are plotted in Figs. 8(i) and 8(v). Note that the piston $c_{0}^{0}$ and the tilts $c_{1}^{\pm 1}$ were omitted, as they do not influence the imaging quality. At WFE 1, the RMSE of all odd-order aberrations with 0.12 λ is nearly twice as large as the RMSE of all even-order aberrations being 0.07 λ. At the WFE 2, the difference with 0.38 λ for the odd and 0.13 λ for the even orders is even more pronounced.

We assume that there are two main reasons for the remaining even-order aberrations. Due to the additional aberrations caused by the adhesive film, the light passes through the interferometer in a slightly different way than in the first part of the experiment. Because of the thick Dove prisms and the beam splitters, we can assume them to cause slightly different phase errors in both arms. As these emerge within the III, ΔΦ may contain some even-order aberrations as well. Furthermore, the aberration function ΔΦ_exp can only be decomposed by a finite number of Zernike polynomials. This can lead to mismatches, which can be increased by influences such as coherent artifacts or image noise.

The fully corrected PTFs [Figs. 8(h) and 8(u)] only have an RMSE of 0.06 λ and 0.09 λ, corresponding to the high quality of the respective point images. While Fig. 8(h) exhibits deviations of a flat phase distribution only in four small areas, Fig. 8(u) shows deviations over the whole measured area. These are partly fine-structured and could therefore only be reduced by considering higher-order Zernike polynomials.

Finally, the fourth line of Fig. 8 presents the images of an extended object structure. Here, we chose a grid whose 180 lp/mm (line pairs per millimeter) were close to the system’s cutoff-frequency of 221 lp/mm. Next to the grid, there is the lettering “180”. Figures. 8(j) and 8(w) correspond to the conventional wide-field images behind the interferometer (arm 1), whereas the three other images in these lines correspond to the uncorrected, the system-corrected, and the fully corrected reconstruction based on Γ. Whereas the two-dimensional test structure is completely unrecognizable in both cases when imaged conventionally [Figs. 8(j) and 8(w)], the images reconstructed by the Fourier transform of Γ [Figs. 8(k) and 8(x)] already make it possible to recognize its rough structure. This is due to the fact that even-order aberrations (e.g. defocus, astigmatism) do not disturb the reconstructions in coherence imaging microscopy. After subtracting the system errors, both the grid structure and the lettering can be seen clearly in Figs. 8(l) and 8(y). In the case of WFE 1, the full correction leads to a quasi diffraction-limited image of the grid [Fig. 8(m)]. However, remaining slight distortions can be observed applying WFE 2 [Fig. 8(z)]. They probably arise from position-dependent tilts that cannot be corrected with only one reference point source. This implies the limits of the isoplanatic approximation considered here.

6. Conclusion

We have shown that it is possible to gain almost diffraction-limited images of microscopic structures in spite of large aberrations by means of coherence imaging microscopy. As expected, after subtracting large system aberrations, particularly odd-order aberrations contributed to the reduction of the image quality. These phase errors, occurring between the microscope objective and the specimen, could be corrected nearly completely with the help of a reference measurement. For this purpose, a laser beam focused onto the object plane was used, serving as reference point source. As has been shown, this method known from adaptive optics yields excellent results. Position-dependent odd-order aberrations, however, cannot be corrected by these means. It is possible to overcome this limitation by inserting several guide stars into the specimen or by conducting a reference measurement with the help of an array from the point. In this way, each part of the FOV could be corrected by its associated correction function.

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n	m	$Z_{n}^{m} (\bar{ν}, ψ)$	Name
0	0	1	piston
1	−1	2ν̄ sin(ψ)	y-tilt
1	+1	2ν̄ cos(ψ)	x-tilt
2	−2	$\sqrt{6} {\bar{ν}}^{2} sin (2 ψ)$	astigmatism (45°)
2	0	$\sqrt{3} (2 {\bar{ν}}^{2} - 1)$	defocus
2	+2	$\sqrt{6} {\bar{ν}}^{2} cos (2 ψ)$	astigmatism (0°)
3	−3	$\sqrt{8} {\bar{ν}}^{3} sin (3 ψ)$	trefoil (30°)
3	−1	$\sqrt{8} (3 {\bar{ν}}^{3} - 2 \bar{ν}) sin (ψ)$	coma (90°)
3	+1	$\sqrt{8} (3 {\bar{ν}}^{3} - 2 \bar{ν}) cos (ψ)$	coma (0°)
3	+3	$\sqrt{8} {\bar{ν}}^{3} cos (3 ψ)$	trefoil (0°)
4	−4	$\sqrt{10} {\bar{ν}}^{4} sin (4 ψ)$	quadrafoil (22:5°)
4	−2	$\sqrt{10} (4 {\bar{ν}}^{4} - 3 {\bar{ν}}^{2}) sin (2 ψ)$	secondary astigmatism (45°)
4	0	$\sqrt{5} (6 {\bar{ν}}^{4} - 6 {\bar{ν}}^{2} + 1)$	spherical aberration
4	+2	$\sqrt{10} (4 {\bar{ν}}^{4} - 3 {\bar{ν}}^{2}) cos (2 ψ)$	secondary astigmatism (0°)
4	+4	$\sqrt{10} {\bar{ν}}^{4} cos (4 ψ)$	quadrafoil (0°)
⋮	⋮	⋮	⋮

Aberration correction in coherence imaging microscopy using an image inverting interferometer

Abstract

1. Introduction

2. Preliminary consideration

3. Optical transfer function

3.1. Uncorrected system

3.2. Aberration correction

4. Experimental setup

5. Results & discussion

5.1. System-inherent aberrations

5.1.1. Point spread function

5.1.2. Two-point structures

5.2. Additionally induced aberrations

6. Conclusion

References and links

Cited By

Figures (8)

Tables (1)

Equations (23)

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