Advantages and disadvantages of several digital holography reconstruction techniques.
Abstract
In this Chapter, we discuss the latest advances in digital holography (DH) and digital holographic microscopy (DHM). Specifically, we study the different setup configurations such as single and multiwavelength approaches in reflection and transmission modes and the reconstruction algorithms used. We also propose two novel telecentric recording configurations for single and multiwavelength digital holographic microscopy (TMWDHM) systems. Brief theory and results are shown for each of the experimental setups discussed. The advantages and disadvantages of the different configurations will be studied in details. Typical configuration features are, ease of phase reconstruction, speed, vertical measurement range without phase ambiguity, difficulty in applying optical and numerical postprocessing aberration compensation methods. Aberrations can be due to: (a) misalignment, (b) multiwavelength method resulting in Chromatic aberrations, (c) the MO resulting in parabolic phase curvature, (d) the angle of the reference beam resulting in linear phase distortions, and (e) different optical components used in the setup, such as spherical aberration, astigmatism, coma, and distortion. We conclude that telecentric configuration eliminates the need of extensive digital automatic aberration compensation or the need for a second hologram’s phase to be used to obtain the object phase map through subtraction. We also conclude that without a telecentric setup and even with postprocessing a residual phase remains to perturb the measurement. Finally, a custom developed userfriendly graphical user interface (GUI) software is employed to automate the reconstruction processes for all configurations.
Keywords
 digital holography
 multiwavelength digital holography
1. Introduction to digital holography (DH)
Digital holograms are generated by recording the interference pattern of two mutually coherent beams. These two beams are the object beam and the reference beam and the recording medium is usually a CCD [1]. The digital hologram recorded on the CCD due to the interference of the object beam EO and the reference beam ER is given by
where the * notation denotes the complex conjugate. Traditionally, in analog holography the reconstruction is performed by illuminating a holographic film by the conjugate of the reference beam
1.1 Numerical reconstruction by discrete Fresnel transformation
The Fresnel Transform is based on the Fresnel approximation to the HuygensFresnel diffraction integral, and under the paraxial approximation, i.e.,
where
The image resolution given by Eq. (3) is considered to be “naturally scaled,” such that the value of
A reflection type Fresnel DH setup based upon the MachZehnder interferometer is schematically shown in Figure 2(a). Light from a Laser source is divided into two parts with a beam splitter. One of the beams forms the reference, while the other is reflected off the object, then both interfere on a CCD camera to form a Fresnel hologram. Figure 2(b) shows a Michelson type setup [8, 9]. An example of a recorded hologram of a Newport Logo recorded using an Argon laser @ 496.5 nm and its reconstruction using Fresnel transform method are shown in Figure 3(a) and (b), respectively.
1.2 Numerical reconstruction by the convolution approach
Since the diffracted field at a distance z = d from the hologram can be expressed as
the convolution approach can be written as:
where the
where
1.3 Numerical reconstruction by the angular spectrum approach
In Fourier space and across any plane the various spatial Fourier components of the complex field distribution of a monochromatic wave can be considered as plane waves traveling in different directions away from that plane. The field amplitude at any other point can be calculated by adding the weighted contributions of these plane waves, taking into account of the phase shifts they have undergone during propagation [1]. Similar to the convolution approach above, the angular spectrum approach is based on direct application of the propagation of the angular spectrum of the field in the hologram plane. Accordingly, we define the angular spectrum of the field
where
Therefore, the reconstructed field at a distance z = d becomes:
which is similar to Eq. (5) above.
Table 1 shows the advantages and disadvantages of the different reconstruction techniques discussed in this section.
Technique  Advantages  Disadvantages 

Fresnel 


Convolution 


Angular spectrum 


2. Digital holographic microscopy (DHM)
DHM is usually applied to determine 3D shapes of small objects, with height excursions on the order of microns (or phase excursions on the order of a few radians). Since small objects are involved, a microscope objective (MO) is often used to zoom onto a small area of the object to enhance the transverse resolution. Holograms of microscopic objects recorded with DHM setups can be numerically reconstructed in amplitude and phase using the same DH reconstruction techniques discussed in Section 1. The phase aberrations due to the MO and the tilt from the reference beam have to be corrected to obtain the topographic profile or the phase map of the object [10, 11, 12]. Figure 4(a) and (b) show a Michelson DHM in reflection and transmission configurations, respectively.
For a reflective object on a reflective surface, the height profile on the sample surface is simply proportional to the reconstructed phase distribution
For a transmissive phase object on a reflective surface, its thickness can be calculated as:
where
For a transmissive phase object on a transmissive surface or between transmissive surfaces, the phase change (optical thickness) can be calculated as:
As stated above, in DHM we introduce a MO to increase the spatial resolution which was computed according to Eq. (3). Due to the magnification ‘M’ introduced by the MO the pixel size in the image plane,
which is simply the magnification predicted by geometric imaging. This is intuitively understood by realizing that the holographic recording is now simply a recording of the geometrically magnified virtual image located at distance d as shown in Figure 5. Thus, the pixel resolution is automatically scaled accordingly. We can enhance the transverse resolution approximately to be equal to the diffraction limit 0.61λ/N.A of the MO, where N.A is the numerical aperture of the MO.
The complete reconstruction algorithm is governed by the equation [9, 10, 11, 12, 13, 14]:
where
Aberration compensation can be performed manually using a phase mask
where
Consider the transmission setup shown in Figure 4(b). A USAF 1951 resolution chart target is used as an object. The resolution of a USAF resolution chart is documented as:
where
3. Multiwavelength digital holography (MWDH)
It is well known that one main application of holographic interferometry (HI) is the generation of a fringe pattern corresponding to contours of constant elevation with respect to a reference plane [2]. These contour fringes can be used to determine the shape of a macroscopic or microscopic threedimensional object.
There exist three main techniques to create holographic contour interferograms: (a) The twoilluminationpoint method, (b) the tworefractiveindex technique, which is generally not practical because we have to change the refractive index of the medium where the object is located, and (c) Multiwavelength method, which was adopted in this section [6, 8, 11]. For large height profiles (larger than several microns) 2D topography using single wavelength holographic approach is not appropriate since phase unwrapping has limitations especially for sharp edge variations. As shown in Figure 7, the axial displacement of an image recorded with wavelength λ _{1} and reconstructed with another wavelength λ _{2}, with respect to the image recorded and reconstructed with λ _{2}, is [2, 6, 15, 16, 17, 18, 19, 20, 21, 22, 23].
This means that the phase shift depends on the distance z between the object and the hologram plane. The height jump between two adjacent fringes in the reconstructed image is
where Λ is known as the synthetic wavelength. For larger deformations along the z direction, the phase changes could be hundreds of multiples of 2π. Such large fringe densities may lead to difficulty in determining the object phase using the single wavelength technique. However, multiwavelength illumination encodes the object height in terms of 2π multiples of the synthetic wavelength, which is generally much longer than either fundamental wavelength. This allows larger object deformations to be measured by multiwavelength illumination as if illuminated by the single wavelength method, where the “single” wavelength is now given by the synthetic wavelength Λ. Typically, synthetic wavelengths can range from few microns to 10’s of microns [16, 18]. The topographic resolution is typically on the order of 1/100 of Λ and the vertical measurement range can reach several Λ’s by employing phase unwrapping for heights larger than Λ [24]. However, much longer or shorter synthetic wavelengths to measure millimeterscale features can also be performed. Figure 8 shows the advantages of using MWDH to extend the vertical measurement range without phase ambiguity [6]. MWDH may be used to quantify surface topography and displacement measurements for both fixed objects and timevarying objects [17, 18]. It is worth noting that DHM has a lot of applications in living cells [25, 26, 27, 28, 29, 30], neural science [31], tissue analysis [32], particle tracking [33, 34, 35, 36], and MEMS analysis [37, 38, 39].
In multiwavelength DH, both holograms are reconstructed separately at the correct fundamental wavelengths, λ _{1} or λ _{2}. From the resulting reconstructed complex amplitudes
The synthetic wavelength phase image is now calculated directly by pixelwise subtraction of the fundamental wavelength hologram phases
This phase map is equivalent to the phase distribution of a hologram recorded with the synthetic wavelength
At normal incidence, a 2π phase jump corresponds to a height step of Λ/2, and the change in longitudinal distance or height
Note that the transverse resolution is the same as in DH namely,
respectively.
One important detail that must be considered when applying the two wavelengths technique is pixel matching. Recall from Eq. (3) that the pixel resolution Δξ of each hologram is dependent upon the fundamental recording wavelength (λ _{1} or λ _{2}). In order for the reconstruction to be successful, the subtraction described by Eq. (18) must be performed on a pixelbypixel basis, in which the pixel sizes match between each hologram (i.e. Δξ _{1} = Δξ _{2}). This can be accomplished by zeropadding the holograms to alter the numerical resolution according to the following procedure: One hologram is zeropadded prior to reconstruction such that its value of Δξ matches that of the second hologram. The second hologram is then either zeropadded after reconstruction, or the first hologram (which is now larger) is cropped, such that the total sizes of each image are again equal. If it is assumed that λ1 > λ2 then the degree of padding applied to both the λ1 hologram prereconstruction and the λ2 hologram postreconstruction is
where pad size is the number of zero elements to be added symmetrically to each edge of the hologram matrix, rounded to the nearest integer value.
3.1 Experimental results for MWDH with and without spatial heterodyning (MWDHSH)
An example of a 3D profile setup using the MWDH technique is shown in Figure 10. Since the two holograms are recorded sequentially for each wavelength, this technique needs two sequential CCD recordings (i.e. two “shots”). Obviously, this “twoshot” method will not work for dynamic objects. Figure 11(a) shows the Newport logo test object while Figure 11(b) shows the reconstructed hologram of the Newport Logo at one of the wavelengths used (λ_{1} = 496.5 nm). Figure 11(c) shows the wrapped phase and Figure 11(d) shows the unwrapped 3D surface profile.
Spatial heterodyning technique has the ability to capture both wavelength measurements in a single composite holographic exposure [6, 9, 21]. This is accomplished by introducing a different angular tilt to the λ _{1} and λ _{2} reference beams. These angular tilts in the spatial domain introduce linear phase shifts in the frequency domain of the recorded composite hologram. When reconstructed, the different phase shifts result in spatially separated object locations in the image that each correspond to their respective λ _{1} and λ _{2} recordings. One of these reconstructed images is cropped and digitally overlaid upon the other to perform the required phase subtraction. A typical recording configuration using MWDHSH method is shown in Figure 12. Since the two holograms are recorded for each wavelength at the same time using spatial heterodyning this technique needs only one CCD exposure (i.e. “oneshot”) [6, 9, 21, 22]. This method is well suited for dynamic objects which change relatively quickly and only limited by the integration time of the CCD. Figure 13 shows the reconstructed Newport logo test object. The reconstructed image resolution
In order to align the two phase images, a block matching algorithm (BMA) is used. After cropping the two reconstructed holograms it is necessary to slide the reference image over the target image looking for best correlation, this is shown in Figure 14. Given the typically rapid variation in object phase, BMA algorithms can only match to within ½ pixel. Hence, BMA matching will generally underperform the twoshot method. After aligning the images, the phase difference is calculated by phase subtraction similar to the two shot technique. The example shown in Figure 15 is for synthetic wavelength
An alternative method of matching the two images is to introduce a phase “tilt” to either one, or both holograms during reconstruction which causes lateral shifts in the position of each image. This is typically referred to as introducing a phase mask,
3.2 Experimental results for multiwavelength DHM (MWDHM) with and without spatial heterodyning (MWDHMSH)
In this section, we show an example of using the MWDH technique using a microscopy setup similar to the single wavelength DHM shown in Figure 4. Thus, the technique would be abbreviated as (MWDHM). A series of microscale objects have been custom fabricated for this experiment as shown in Figure 16(a) and (b). Figure 17(a) shows a MWDHM setup with achromatic optics and can be operated in either the “oneshot” or “twoshot” Michelson configuration. Figure 17(b) is a photograph of the object consisting of 4 bars of photoresist (See element in red circle in Figure 16(a)), each 50 μm wide, on a silicon wafer substrate. Figure 17(c) is the intensity reconstruction of the λ _{1} hologram only. Figure 17(d) shows the wrapped phase difference between λ _{1} and λ _{2} reconstructions. Figure 17(e) shows the unwrapped phase (Ref. [24]) with the residual MO quadratic phase curvature and Figure 17(f) is the 3D topogram after removal of the MO phase using Eq. (13). The relevant reconstruction parameters are: d = 22.7 cm, λ _{1} = 632.8 nm, λ _{2} = 488.0 nm, Λ = 2.13 μm, Δx = 6.7 μm, and N = 1024, and M = 2.75. Note that the phase rings are due to a slight mismatch in collimation between the λ1 and λ2 beams, which causes circularly symmetric phase beating since at least one wavefront is not well collimated. This situation arises often in physical lab setups in which both beams are coaxially aligned and filtered using the same pinhole prior to using a single collimation lens. Chromatic dispersion will prevent both wavelengths from being collimated simultaneously, unless an achromatic lens is used.
Here we show an example using the MWDHM technique with the microscopy setup of Figure 17(a), operated in the spatial heterodyne configuration (MWDHMSH). The object is a set of 3 rectangular photoresist bars, each 75 μm wide, on a silicon wafer (See element in black circle in Figure 16(a)). In this case, the object is simultaneously illuminated by two wavelengths at normal incidence and only a single composite hologram is recorded by the CCD (i.e. “oneshot”). The single hologram is reconstructed twice, one at each fundamental wavelength, and the blockmatch algorithm is used to align the images prior to phase subtraction. Figure 18(a) shows the intensity reconstruction of the λ _{1} hologram only, with the region of interest circled, Figure 18(b) shows the wrapped phase difference between λ _{1} and λ _{2} reconstructions after block matching and phase subtraction, while Figure 18(c) shows the unwrapped phase with the residual MO quadratic phase curvature, and Figure 18(d) is the 3D topogram after removal of the MO phase and correction of phase errors. The relevant reconstruction parameters are: d = 23 cm, λ _{1} = 632.8 nm, λ _{2} = 488 nm, Λ = 2.13 μm, Δx = 6.7 μm, N = 1024, and M = 2.75.
4. Theoretical background of telecentric systems
In a conventional lens systems the magnification changes with object position change, the image has distortion, perspective errors, image resolution loss along the field depth, and edge position uncertainty due to object border lighting geometry. However, a telecentric system, such as the one shown in Figure 19, provides nearly constant magnification, virtually eliminates perspective angle error (Object with large depth will not appear tilted), and eliminates radial and tangential distortion. In a bitelecentric system, both the entrance pupil (EP) and exit pupil (XP) are located at infinity. Given that double telecentric systems are afocal, shifting either the image or object does not affect magnification.
As shown in Sections 2 and 3, traditional DHM systems record a digital hologram using a MO. The object phase recovered from digital reconstruction using the Fresnel transform suffers from a parabolic phase factor introduced by the MO. The phase of the MO is superposed over the object phase, often obscuring it. Also, the phase tilt introduced by the reference beam results in linear fringes with high frequency that also obscure the real phase of the object. Numerical techniques as well as optical configurations are usually employed to compensate for both the parabolic phase curvature and the phase tilt. One wellknown technique discussed in Section 2 is based on phase mask during reconstruction, which requires knowledge of the setup parameters [13, 14, 40, 41, 42]. If the object parameters are unknown a twostep method is used, in which the hologram of a flat reference surface is initially recorded, and upon reconstruction it is subtracted from that of the hologram of the real object [43]. In this section, we adopt two telecentric configurations in reflection and transmission modes to remove optically, instead of numerically, the phase curvature due to MO [44, 45, 46]. This telecentric setup can be used in a single wavelength or multiwavelength DHM configurations. It is worth noting that while operating in the nontelecentric mode, a posteriori numerical methods will not eliminate the phase aberration completely, as it depends on sample location in the field of view (FOV) [45].
In traditional DHM, the recorded wavefront on the CCD includes the interference of the reference wavefront and the total object wavefront. The total object phase consists of the defocused object phase on the image plane as well as the spherical (quadratic in paraxial approximation) phase due to propagation of the object wave from the image plane to the CCD. The object phase is expressed as [11, 46]:
where R is the radius of curvature of the spherical curvature.
Typical multiwavelength DHM setups using telecentric configurations in reflection and transmission modes are shown in Figure 20(a) and (b), respectively. In each setup, the telecentric system is formed by employing two achromatic lenses and an aperture stop similar to Figure 19. The achromatic lenses are crucial to eliminate achromatic aberration due to the use of multiwavelength illumination. The telecentric system is set in an afocal configuration, where the back focal plane of L_{1} coincides with the front focal plane of L_{2}
Hence, the 3D amplitude distribution in the image space will be a scaled defocused replica of the 3D amplitude distribution of the object space due to the convolution with the PSF of the lens system. For each wavelength
where,
4.1 Experimental results for single wave telecentric DHM (TDHM)
Figure 21 shows a custom developed userfriendly graphical user interface (GUI) for the single wave reflection Telecentric DHM (TDHM) setup similar to that shown in Figure 20(a). The target object is shown in Figure 16(b).
The MATLAB GUI is connected to a Lumenra LU120M CCD camera using a USB cable. The GUI is equipped with all the parameters needed to adapt to different CCD camera pixel size, laser wavelength, reconstruction distance, reflection vs. transmission mode. In this example, the laser wavelengths used is
The telecentric technique has a lot of advantages compared to a standard DHM system since the reconstruction parameters in a standard DHM are hard to obtain and need to be measured precisely to obtain the 3D phase information.
4.2 Experimental results for telecentric multiwavelength DHM (TMWDHM)
Figure 22 shows the GUI for the reflection configuration shown in Figure 20(a). The target in this experiment is a transmissive object (PMMA) on a reflective Si background (See element in blue circle in Figure 16(a)). The laser wavelengths used are
5. Simulation of coherent speckle on phase and intensity
Due to the use of coherent optical sources, the recorded holograms and reconstructed fields contain coherent speckle patterns, as seen in the inset in Figure 23(a). Speckle is produced by the coherent interference of a set of wavefronts. Mutual interference occurs when coherence is lost, where coherence is defined as the wavefront having constant phase at each frequency. A wellknown mechanism for incoherence is optical roughness; when illuminated with monochromatic light the reflected (or scattered) wave consist of the contribution from many scattering points. Different scattering areas or small highlights on the object emit spherical wavelets which combine and interfere coherently resulting in a complex interference pattern known as speckle (Ref. [48, 49, 50, 51, 52, 53, 54]). This speckle generation mechanism also applies to transmission (scattering) through an optically rough phase object.
Due to variable phase shifts produced as the wavefront propagates through an optically rough object, the field leaving the object has a corrugated structure of interference. In addition, the presence of an optical diffuser before the object (which consists of small thickness variations) in transmissive configuration, has the same effect as a rough surface in reflective imaging. In this section, we seek to demonstrate an accurate representation of speckle in transmissive imaging through nearly transparent samples, valid for biological imaging applications. To simulate speckle, we consider the complex phasor amplitude
where the total phase is the sum of the phase derived from the height profile
As an example, we consider a USAF resolution target with a maximum thickness of 10 microns and random height deviations of 1 micron (10% of total height and 1.6λ for red light) due to roughness, imaged through a telecentric holographic configuration with 3x magnification. Figure 24 shows probability density functions computed using the phase reconstruction of simulated speckle patterns; the real and imaginary components of the complex speckle field (A,B) are i.i.d. Gaussian random variables, such that magnitude and intensity (C,D) are Rayleigh and
In a typical experiment speckle can be reduced using diversity in polarization, space, frequency, or time (Ref. [49]). One of the time domain techniques is through rotating a diffuser or by using a liquid crystal based electronic speckle reducer (Ref. [55]). Another technique is to average multiple holograms or reconstructions recorded by varying the optical path length of the reference beam relative to the object beam (Ref. [56]). Figure 25 show the reconstructed height profile averaged over increasing phase reconstruction frames, where the initial roughness distributions are assumed to be statistically independent from frame to frame due to the varying optical path length difference between the object and reference beam. Figure 26 shows the standard deviation of the phase and height profile contribution of simulated speckle as a function of increasing averaging frames. As expected, the standard deviation decreases as
In this section, we have demonstrated that the distributions of the simulated speckle phase and intensity are consistent with theory and observations in the limit when the optical roughness is large relative to the optical wavelength. In addition, we have shown that the reduction of speckle standard deviation associated with averaging is as expected. While we have demonstrated an accurate and robust numerical representation of optical speckle patterns in holographic imaging, we do not seek to address speckle mitigation techniques in detail. Our goal is to mimic experimentally recorded and reconstructed holograms for realistic machine learning training not to mitigate speckle, as shown in Section 6 below. In future work we seek to explore the sensitivity of speckle statistics to the roughness of the object relative to the optical wavelength.
6. Conclusion
In this Chapter, we developed the theory, the reconstruction algorithms, and discussed the different experimental configurations for digital holography and digital holographic microscopy. We also showed typical experimental setups for single and multiwavelength configurations. We concluded that single wavelength setups are used for heights that do not exceed few microns while multiwavelengthbased setups are used for heights that can reach 100’s of microns depending on the synthetic wavelength used. We also discussed in details the two shot versus the one shot MWDH setup. Although hologram reconstruction using oneshot setup needs an extra digital correlation step, it is very well suited for dynamic objects which change relatively quickly. We also discussed briefly how Zernike polynomials are used to cancel the residual phase due to the different aberrations in the optical system. We also discussed the theory and experimental setups of novel reflection as well as transmission telecentric digital holographic microscopy configurations. The setup optically removes, without the need of any postprocessing, the parabolic phase distortion caused by the microscope objective which is present in a traditional multiwavelength digital holographic microscope. Without a telecentric setup and even with postprocessing a residual phase remains to perturb the measurement. The telecentric technique has a major advantage since the reconstruction parameters needed and hard to obtain in a standard DHM do not need to be measured precisely to obtain the 3D phase information. Finally, a custom developed userfriendly GUI was employed to automate the recording and reconstruction process.