1. Introduction

Point spread function (PSF) engineering using pupil-filters is currently an active area of research [1–7]. This paper is written to call attention to the fact that the use of pupil filters alone in PSF engineering cannot increase the fundamental achievable resolution for a large class of linear, shift-invariant, far-field optical instruments. The class of optical instruments encompassed includes coherent or incoherent widefield imaging, confocal transmission, reflection or fluorescence imaging, two-photon fluorescence microscopy [8] and 4Pi fluorescence microscopy [9]. A full three-dimensional, vectorial, frequency-domain analysis is used to show the addition of pupil-plane filters will not improve the resolution, as no additional frequency-domain components of the object are passed by the instrument. However, other benefits are possible from pupil filters — a confocal fluorescence example is used to demonstrate benefits in terms of signal-to-noise ratio and imaging with a finite-sized confocal pinhole.

2. Transfer function analysis

This section uses a construction first presented by McCutchen [10] to show that a true resolution increase cannot be achieved through the use of pupil-plane filters. McCutchen’s result has also been used and generalized by a number of other authors, e.g. [11–17].

In a three-dimensional, linear, shift-invariant system, the object o(r) (r is the spatial vector (x,y, z)) can be related to the system output d(r) through a convolution with the point spread function h(r) or through a Fourier domain multiplication with a transfer function.

Here capitalization represents the Fourier transform and k is the wavenumber vector (k_x,k_y,k_z). The support of the transfer function H(k) (i.e. the set of k values for which H(k) is non-zero) has a very strong relationship to the achievable resolution of the system, as it determines which spatial frequencies in the object are passed.

In a far-field imaging system, the form of the point spread function will be intimately tied to the imaging properties of the lens(es) used. A method of calculating the three-dimensional, vectorial distribution of a large-aperture lens’ diffraction pattern in the vicinity of the focal point is given in [18]. This distribution, known as the amplitude PSF, can then be used to construct the overall PSF h(r) and the corresponding transfer function can be calculated by taking the Fourier transform (see [19] for examples). The transfer function can also be calculated directly. The Fourier domain representation of the scalar amplitude PSF is given in [10]. This representation can then be used to build transfer functions for various instruments entirely in the Fourier domain [11, 12]. More recently, these results have been extended to vectorial optics [13–15].

 

Fig. 1. Graphical representation of the three-dimensional Fourier transform of the amplitude PSF of an aplanatic lens. The maximum collection angle for this plot is 65°.

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Constructing the transfer function directly allows insight to be gained about its properties. Of particular interest is the region over which the transfer function is non-zero — i.e. its support. The support of the transfer function determines which spatial frequencies of the object are passed by the system and hence which may be recovered in the image. This ultimately determines the maximum achievable resolution. Examples of this type of analysis can be seen in [16, 17].

Let the Fourier transform of the amplitude PSF of a lens be given by P(k). This function is non-zero only on a very specific region. Let the x and y coordinates be in the plane of focus (the lateral dimension), z along the optic axis (the axial dimension) and the nominal focus at r=0. In the frequency domain let $\theta ={\tan }^{-1}(\sqrt{k_x^2+k_y^2}⁄k_z)$ — this is the angle from the k_z axis. The support of P(k) is given as follows.

In the above equation n is the refractive index of the medium, λ is the free-space wavelength and NA is the lens’ numerical aperture. The constraint on the magnitude of k shows that the amplitude PSF is constructed only from plane waves with a spatial frequency corresponding to the wavelength used. The second constraint (on θ) restricts the directions of propagation that can be used. Thus S_P is a spherical cap where the radius of the sphere depends on the wavelength used and the angular extent depends on the maximum collection angle of the lens. A graphical representation of this is shown in Fig. 1.

The function P(k) is a delta sheet that is non-zero only on S_P and weighted by some apodization function A(k) (the apodization function is scalar if a scalar analysis is used and a three-dimensional vector function if vector optics are used). For an aplanatic lens, conservation of energy leads to an apodization function of the form $\sqrt{\cos \left(\theta \right)}$ [18], as shown in Fig. 1. The apodization function can be modified to account for aberrations or defocus, in which case it may be necessary to use a complex A(k). The effect of pupil-plane filters is to change the apodization function. The pupil-filter effect may consist of a change in transmittance, in which case the magnitude of A(k) will be affected and/or a change in the phase of A(k). An example of changing the transmittance is considered later in this paper. The Toraldo concept [20] is an example of a technique that only modifies the phase and has recently been the subject of further investigation [4, 5].

The amplitude PSF is represented in the Fourier domain by P(k) and this is the transfer function for widefield coherent imaging. The PSF for widefield incoherent imaging is simply the intensity of the amplitude PSF. Basic Fourier transform properties mean that the widefield incoherent transfer function is the autocorrelation of P(k). The support of the transfer function is therefore given by the range of offsets over which two P(k) functions will overlap. Similar arguments can be made for confocal imaging systems [11]. For the incoherent confocal case the PSF is the product of two widefield incoherent PSFs and thus the transfer function is the convolution of two autocorrelations of P(k) functions. The transfer function of a 4Pi system can also be constructed directly in the Fourier domain [12]. In this case the result is the convolution of two autocorrelations of [P(k)+P(-k)] functions. Changing the apodization function(s) won’t change the support of these transfer functions. The resolution of these systems increases from widefield to confocal to 4Pi due to differences in the optical setup, which result in transfer function constructions with increased support. The addition of pupil-plane filters changes only the magnitude and/or phase of the apodization function(s) and not the construction methods. Since the support depends only on the construction method and the support of the P(k) functions, the pupil-filters do not increase the transfer function support.

This type of Fourier-domain transfer function construction can be carried out for coherent or incoherent widefield imaging, confocal transmission, reflection or fluorescence imaging, two-photon fluorescence microscopy and 4Pi fluorescence microscopy. Thus a change in the apodization function(s) will not change the support of any of these transfer functions. This implies that the addition of pupil filters will not change the maximum achievable resolution. More generally, basic Fourier transform properties can be used to show that any PSF constructed by sum, multiplication, intensity, rotation, convolution and/or shift (defocus) operations on a set of amplitude PSFs will have a transfer-function support that is unchanged by the addition of pupil-plane filters. This is because the support of the overall transfer function depends only upon the support of the constituent P(k) functions and not the values of the apodization functions. Physically, the range of spatial frequencies passed by the system is influenced only by the spatial frequencies collected by the lenses and not their relative strengths. An important caveat to this result is that within the calculated support the transfer function may have zero crossings. In this case pupil-plane filters may be able to shift the zero crossings and thus allow previously unobservable components of the object to be imaged. An example of this effect is provided later in the paper. Several modern instruments rely on non-linear optical phenomena [21, 22] and will not be included in the class of instruments to which this paper is applicable.

3. Performance in the presence of noise

While pupil-plane filters cannot increase the transfer function support, they do have utility in boosting the signal-to-noise ratio (SNR). This point and the fact that the transfer function support cannot be enlarged will be demonstrated here with a confocal fluorescence example from [6].

The point spread function of a confocal fluorescence microscope depends on the two amplitude PSFs present in the system — the excitation amplitude PSF p_ex(r) and the detection amplitude PSF p_det (r). The overall PSF can be written as follows.

 

Fig. 2. Lateral and axial profiles along the axes for the confocal fluorescence point spread functions. One system is aplanatic while the other uses a shaded-ring pupil filter for the excitation. Note that the lateral profiles are too similar to distinguish.

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In the Fourier domain Eq. 3 becomes the following.

Here ⋆ represents the autocorrelation operation and ∗ represents convolution.

The PSF for a confocal fluorescence microscope was calculated using the same method as [19] (which is based on the vector optics results presented in [18]). The physical parameters of the system were chosen to match those used in the shaded-ring pupil filter example of Fig. 2 in [6] — specifically, the excitation wavelength is 350 nm, the detection wavelength is 440 nm and the maximum collection angle is 64.5° (this corresponds to a numerical aperture of 1.2 in water). The PSF for the shaded-ring system was also calculated. Lateral ($\sqrt{x^2+y^2}$) and axial (z) sections (through the origin) of these two point spread functions are shown in Fig. 2. The images have been scaled so that the excitation powers are equal which means that equal photobleaching would occur using either system.

In Fig. 2 it can be seen that the pupil-plane filter has changed the shape of the PSF. As claimed in [6], the effect has been to reduce the axial extent of the point spread function. This axial narrowing of the shaded-ring system’s PSF suggests an improved axial resolution. However, when the transfer functions are calculated it becomes apparent that the Fourier domain support has not changed. Lateral and axial sections of the transfer functions on a semilogarithmic scale are plotted in Fig. 3. They were calculated by taking the Fourier transform of the PSFs of Fig. 2 and normalizing so that the maximum value of the aplanatic-lens transfer function was 1. The maximum extents of these transfer functions were calculated theoretically [23] to be 3.88 µm^-1 axially and 12.3 µm^-1 laterally. The cutoff frequencies for both systems are equal and agree with the theoretical figures (the low values outside the theoretical support are below calculation precision).

In Fig. 3 it can be seen that axial frequencies near the cutoff are passed more strongly by the shaded-ring-filter system (the maximum ratio between the two transfer functions is approximately 3.5). This effect could be synthesized in the aplanatic data by applying a linear post-processing filter to boost the high axial frequencies. Thus the effect of the pupil filters can be regarded as a physical implementation of an edge-enhancing high-boost filter and not true superresolution. The noise level in the two systems will be comparable as the overall signal strength is largely determined by the low-frequency values of the transfer functions, which are similar. Since the high-frequency information is more strongly passed in the pupil-filter system, the SNR will be improved at these frequencies. Thus for the high axial frequencies, the SNR is improved by the application of this shaded-ring pupil filter. It worth noting that the linear processing operation performed by the inclusion of pupil-plane filters does not require the collection of a complete data set, as computational Fourier-domain processing methods do. This physical implementation of linear processing may also have advantages over computational methods in coherent systems where phase information can be lost at detection.

 

Fig. 3. Lateral and axial profiles along the axes for the confocal fluorescence transfer functions. One system is aplanatic while the other uses a shaded-ring pupil filter for the excitation.

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4. Detector aperture effects and modifying zero-crossing positions

In this section it is noted that there are scenarios in which a transfer function may have zero crossings within its calculated support. In these cases pupil-plane filters may be used to shift the zero crossings and thus, ideally, change the spatial frequency components of the object that can be observed in the image. This is demonstrated for completeness and is not meant to suggest that pupil filters are a practical way of addressing the problem of zero-crossings in a transfer function.

In the previous confocal example, the transfer function was strictly positive over the whole support. When the effects of aberrations or a finite-sized detection pinhole are considered, this positivity property is not necessarily retained [23]. The effect of the pinhole is to multiply the detection component of the transfer function by the Fourier transform of the pinhole. The Fourier transform of the circular pinhole used here does have negative values. When the resulting detection component of the transfer function is convolved with the excitation component (see Eq. 4) the result may also be negative or zero for particular k values. In [23] it is shown that for larger pinholes the transfer function will become negative in certain areas and will thus have zero-crossings within the support. At these zero crossings the corresponding spatial frequency is unobservable and in the vicinity of the zero crossings the SNR is very poor. Using a pupil filter in the excitation will give different values to the excitation component of the transfer function, so that when convolved with the detection component the zero crossings may no longer occur in the same position.

 

Fig. 4. Axial profiles along the z axis for the confocal fluorescence transfer functions when a large detector pinhole is used. One system is aplanatic while the other uses a shaded-ring pupil filter for the excitation.

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In Fig. 4 the axial profile of the transfer functions are shown when a large circular detector pinhole of radius 600 nm (when projected into object coordinates) is added to the previous example. The transfer functions are again normalized so that the aplanatic transfer function has a maximum value of 1. The plots are now on a linear scale and a detail of the zero crossings is shown. In Fig. 4 it can be seen that the excitation pupil filter has had the effect of moving the zero crossings on the axial axis (the lateral changes are not significant). This means that there are spatial frequency components of the object that are visible in the pupil-filter data but cannot be seen in the aplanatic data (and vice versa). It should be noted that a similar zero-crossing shifting could be achieved by simply using a pinhole of a different size.

5. Conclusions

For a considerable class of optical instruments the use of pupil-plane filters cannot increase the support of the system’s transfer function, which implies that they offer no fundamental increase in achievable resolution despite suggestive spatial domain results. The reasoning behind this result is evident in the derivations contained in the microscopy transfer-function literature cited here. Benefits of pupil filters were shown by example. Specifically, pupil filters can be used to improve the imaging of high spatial frequencies (which correspond to fine detail) in the presence of noise. They may also be used to shift the zero crossings in a transfer function and thus change the unobservable spatial frequencies within the transfer function support. Pupil-filter PSF engineering also has many practical applications not discussed here (e.g. increasing the depth of focus in two-dimensional microscopy).