We present a theoretical formalism for three dimensional (3D) imaging properties of digital holographic microscopy (DHM). Through frequency analysis and visualization of its 3D optical transfer function, an assessment of the imaging behavior of DHM is given. The results are compared with those from other types of interference microscopy. Digital holographic microscopy does not result in true 3D imaging. The main advantage of holographic microscopy lies in its quick acquisition of a single 2D image. Full 3D imaging can be obtained with DHM using a broad-band source or tomographic reconstruction.
©2007 Optical Society of America
Holographic imaging was invented by Gabor  around 60 years ago, and the unique wave-front reconstruction imaging technique has since then been improved in various aspects. Recent advancements in CCD electronics and computer resources has meant that the traditionally optical reconstruction process can be performed digitally . In particular, microscopy based on digital holography (DHM) has proved to be a convenient method in phase contrast imaging for samples in biological and biomedical studies [3–5].
DHM is a coherent imaging system, and its advantage lies in the instantaneous and quantitative acquisition of both amplitude and the phase information from the reconstruction of the wave-front. Imaging of phase distributions with high spatial resolution can be used to determine refractive index variations as well as the thickness of the specimen. This is very attractive for studies of live specimens where stains may not be appropriate. The ability to detect very minute phase variations also allows quantitative phase imaging to reveal the three-dimensional (3D) characteristics of cells and tissues and their structures in depth which, in turn, may have many potential implications in medical diagnosis.
Various experimental setups have been made in modern DHM systems to extract phase, as well as 3D information, from different types of phase objects. In order to overcome the limitation of CCD pixel size, the phase-shifting technique of interferometry was applied to holography , where a minimum of three holograms need to be captured while the relative phase difference between reference and object beam are changed stepwise. Recent advances in DHM research focuses both on off-axis [3, 4] and on-axis  interference configurations for detection of refractive index, shape, and other information that is useful in biomedical applications. Through managing of quantitative phase information, the wave front curvature inherent in the DHM system can be compensated for , and it is also feasible to achieve high resolutions using a lens of small numerical aperture (NA) through Fourier holography and synthetic aperture techniques [8, 9].
The rapid advancements and vast applications of this technique indicate a need to understand the fundamental imaging behavior of DHM and how it is different to other alternative coherent phase imaging methods such as interference or confocal microscopy. Using three-dimensional scalar theory, Chmelik  compared the image performances in high-aperture interferometric, holographic, and heterodyne microscopes. But in his paper, DHM was implemented in a low coherence, parallel confocal mode, which is not the conventional setup of DHM, and hence his conclusion cannot be taken as the overall imaging behavior representative of single-shot DHM.
We demonstrate here that DHM in its basic single exposure configuration is significantly different to other types of interference or confocal microscopes in its three-dimensional image formation properties. Our analysis compares the coherent imaging techniques using the spatial cutoffs in their 3D optical transfer functions. For the basic “one-shot” setup of DHM, its 3D coherent transfer function (CTF) gives more insights into the overall imaging properties. Finally the case of multi-spectral illumination and tomography in DHM is considered.
2. Review of optical transfer function theory
Three-dimensional imaging analysis was initiated by the work of McCutchen , who considered the process of 3D focusing by a lens. He showed that by calculating the 3D Fourier transform (FT) of a 3D pupil, which is the cap of a sphere of radius 1/λ, also called the Ewald sphere in x-ray diffraction theory, the relevant 3D amplitude distribution in the focal region of a lens can be obtained. For a lens of semi-angular aperture α, the cap of the sphere subtends an angle α from its center. Although originally proposed as a scalar theory, this approach can be extended to the vectorial case .
It is often regarded that holography results in a 3D image. Wolf  showed that holographic imaging can be described using the concept of 3D coherent transfer function (CTF). He considered in-line transmission holography, and showed that the 3D spatial frequencies reproduced in a holographic image lie on the surface of a sphere, also of radius 1/λ, that passes through the origin of reciprocal space (as shown in Fig. 1). An incident plane wave denoted as k1 vector is diffracted by an object, which can be considered as a grating, producing a scattered wave in the direction of vector k2. Vector k represents the characteristic wave vectors of the grating, and the two end points of k lie on a cap of a sphere due to the conservation of momentum. The range of angles detected by the holographic medium truncate the spherical shell. The shift of the sphere can be explained by considering Bragg diffraction by a grating vector of the object spectrum. Hence the center of the sphere is shifted by the k1 vector of the illuminating wave. The case of reflection holographic imaging was later considered by Sheppard , and illustrated in Fig. 1 (b). The spatial frequency cutoff range in transverse and axial directions are sin(α/λ) and 2 sin2(α/2)/λ, respectively, and labeled as such in the figure. It can therefore be seen that in holographic imaging, the 3D information is limited to spatial frequencies that lie on a spherical shell, and is far from complete. This is compatible with the view that a 2D hologram does not have sufficient capacity to store complete 3D information.
Frieden  introduced the concept of the 3D optical transfer function (OTF) for an incoherent system. The object’s spectrum, multiplied by the OTF, produces the image’s spectrum. The 3D FT of the intensity point spread function for a conventional microscope is equal to the 3D OTF. Thus, 3D OTF of a confocal microscope that operates in fluorescence mode can be derived from the 3D FT of its intensity point spread function. On the other hand, confocal reflection microscopy is a coherent imaging technique, so it can also be described in terms of a 3D CTF .
The 3D FT of the image for an interference microscope consists of four parts; the reference beam corresponds to a δ-function at the origin; the ordinary image transforms to a region around the origin; the interference terms transform to two regions each of which contain the object amplitude information (as shown in Fig. 2).
Optical coherence tomography (OCT)  is a form of confocal interference imaging  with the addition of a coherence gate. The overall CTF is thus given by an integral over the spectral components, increasing the region of support. Usually in OCT, the numerical aperture is small so that the axial resolution resulting from the confocal effect is negligible compared with that from the coherence effect. Thus, the transverse CTF for a particular wavelength is identical to the 2D OTF for an incoherent system. The overall form of the OCT CTF has been illustrated elsewhere .
Figure 2 compares the spatial frequency cutoff for conventional interference microscopy (such as coherence probe microscopy (CPM) ) versus confocal interference microscopy (such as OCT). The difference in geometry of the two diagrams is the result of the different imaging characteristics embedded in these two types of systems; the ordinary image is conventional in the first case but is confocal in the latter. The maximum NA in air required for the interference image to be separated from the object image is 0.943 (which corresponds to cos α= 1/3) in the conventional case, while it is reduced to 0.866 (cosα = 1/2) in the confocal case.
In comparison, the working principle of a holographic imaging system is also based on interference, which results in four terms as well after Fourier transformation as shown in Fig. 3. Compared to Fig. 2, the spatial frequency support in the interference region is significantly reduced. If a single image is recorded, the interference terms can be extracted using an off-axis reference beam, but if a through-focus series is acquired, the interference terms can be separated using an axial FT.
3. DHM analysis
3.1. Basic setup
The basic configuration of a DHM, which is shown in Fig. 4, is based on a Mach-Zehnder interferometer. Fig. 4 (a) shows simplified single-exposure DHM in transmission mode, while Fig. 4 (b) shows DHM in reflection mode. The collimated laser beam is divided by a beam splitter (represented in the figure as BS1). The 3D microscopic biological sample is illuminated by one beam, and a microscope objective (MO) collects the transmitted or reflected light and forms the object wave (O). This object wave interferes in an on-axis configuration (an off-axis configuration is also possible and the derivation is similar) with the reference wave (R) to produce a hologram intensity that is recorded by a CCD camera.
The complex field amplitude recorded on the CCD is given by,
where O(r) is the complex field amplitude generated by the object beam, R(r) is the complex field amplitude of the reference beam, and the asterisk (∗) denotes the complex conjugation. The information content in the wave-front from the 3D biological sample is contained in the interference terms.
3.2. Coherent transfer function (CTF)
Several methods can be used to derive the 3D CTFs . Here we apply a direct 3D FT to the 3D amplitude point spread function (APSF). The part of interest is only the interference term where object amplitude is embedded. Our attention is focused on the scalar theory but including the spherical converging wave-front instead of paraxial (paraboloidal) approximation. In addition, an apodization effect satisfying the sine condition for practical lens design is also considered.
As shown in section 3.1, a DHM system is based on a Mach-Zehnder interferometer, but it has strong distinctions from a conventional interference microscopy setup. Sheppard and Wilson  analyzed the imaging effects originated from both object and reference optical paths in a Mach-Zehnder interferometer. However, because of the use of collimated beams and the lack of condenser, reference beam wavefront plays a trivial role in the overall imaging of DHM after reconstruction. Hence the theoretical imaging model used here consists of an aberration-free microscope objective (MO) in first order optics which has magnification M, numerical aperture NA and a maximum subtended half-angle α. For an optical system of a high-aperture circular lens [22,23], the 3D amplitude point spread function (APSF) in the image space can then be derived as (with a scaling term omitted):
where P(θ) is the pupil function or apodization function related to the angle of convergence of the light ray, J 0 is the zeroth order Bessel function of the first kind, i = √-1, and v and u are defined as optical coordinates of transverse and axial directions:
where r and z are radial and axial coordinates in the image space, k = 2π/λ, and the NA is equal to sinα 0. We define normalized spatial frequencies m,n,s in the x,y,z directions, and cylindrical transverse spatial frequency as:
Performing a 3D inverse FT on Eq.2 gives the 3D CTF in object space,
The 3D CTF for holographic microscopy can thus be derived in consideration of the axial shift, and normalizing l and s by 1/λ for both transmission and reflection:
In both of the above cases, only monochromatic light is considered and the CTF can be represented by the cap of a sphere, which is consistent with Wolf’s theory . In the aplanatic case, so that 2D transverse space invariance can be obtained, the apodization function is , and the CTF becomes:
where ∓ accounts for transmission and reflection system respectively. The CTF is illustrated in Fig. 5. Due to the quasi-monochromatic light source, an incremental band of wavelength Δk is taken into account which results in a thickness of the shells in reciprocal wavelength space. It can be seen that there is less weighting on larger l values due to the aplanatic factor.
3.3. Broadband DHM
The integral effect of coherence gating from low coherence imaging increases the support in spatial frequency cut-offs and, hence, more spatial-frequencies are included as illustrated in Fig. 6. If we assume polychromatic light of Gaussian spectral density with mean wave number k 0 and FWHM defined by A, then the CTF for DHM with polychromatic light source is represented by:
Substituting Eq. 7 into Eq. 8, for a system satisfying the sine condition, we have the CTF for both transmission and reflection cases with different regions of support for s and l given by:
Assuming a Gaussian spectral density has a problem that it predicts negative frequency components if the spread is large, then an alternative expression for the spectral density called the Pearson Type III distribution given by
can be applied.
In Eq. 10, the mean wave number is α+ρβ. β and ρ are parameters that are greater than zero, and the distribution is predicted for a positive frequency range only if α is bigger than zero, thus k ∈ [α,∞). For a white light source with coherence length of a few microns, the low-coherence holographic reflection CTFs are shown in Fig. 7. The CTF that results from Pearson spectral density is slightly more skewed than the case of the Gaussian spectrum.
As the coherence length of a broad-band source is limited, this may create difficulties in using an off-axis reference beam. However, the 3D image can be recorded as a through-focus series with a broad-band source with an on-axis reference beam . The imaging properties would then be improved by using a condenser lens, making the system equivalent to CPM.
3.4. Holographic tomography
In order to obtain high-quality 3D images, the limited spatial frequency content in single-exposure DHM needs to be increased. In the previous section, low temporal coherence was considered; however, another method is through tomography.
In Fig. 8, it is demonstrated that through the rotation of the illumination or the object, more spatial frequency content is covered, and when enough projections are taken and interpolated, the spatial content becomes a volume, which can be reconstructed as in optical diffraction tomography (ODT) [25,26].
The quantitative phase information derived from DHM can give optical thickness or surface height information, sometimes called 2½D, but does not result in true 3D imaging, as it has very limited spatial frequency support. The main advantage of holographic microscopy lies in its quick acquisition of this 2½D information in a single 2D image. It is also possible to use Fresnel propagation principle to digitally adjust the focus of the image once it is reconstructed from a hologram. However, acquiring full 3D information from a single 2D image using one wavelength does not seem feasible. It is sometimes claimed that a whole volume of 3D information is obtained with single-wavelength DHM, but this is not true, as more than often only topographic view as a result of optical path length difference is observed. It has been shown in this paper using optical transfer function theories, that single wavelength DHM has very limited spatial frequency content, especially in the axial direction. What is sometimes claimed as a 3D effect in single-wavelength DHM comes mostly from digital focusing in the axial direction, which is different from effect of optical sectioning, and only the latter can be regarded as true 3D imaging as it has the capacity to resolve information layer by layer in depth.
To improve resolution for 3D imaging in DHM, it is possible to combine coherence gating from a broad-band source as it allows the CTF to be spread as a result of the spectral content, or to implement diffraction tomography together with holographic acquisition, however, the advantage of quick acquisition in single-shot configuration is then lost.
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