Digital holography (DH) is an attractive measuring optical technique in the fields of engineering and science due to its remarkable accuracy and efficiency. The holograms are recorded by an interferometer and reconstructed by numerical methods such as Fresnel transform, convolution approach, and angular spectrum. Because harmful coherent noise often arises when long coherent lengths are used, bright femtosecond pulse light with ultrashort coherent length may be the solution to reduce both spurious and speckle noises. Since the usual DH uses a visible light, it is difficult to visualize 3D internal structure of visibly opaque objects due to their limited penetration depth. The terahertz (THz) radiation has a good penetration capability; thus, 3D visualization of both surface shape and internal structure in visibly opaque object can be achieved via THz-DH technique.
- digital holography
- numerical reconstruction
- 3D surface metrology
- femtosecond DH
- terahertz DH
Various optical techniques have been developed for measuring 3D shape from a single position. The height distribution of the surface of the object is encoded into a deformed fringe pattern, and then shape is directly decoded by one of those optical techniques. DH is considered one of those optical techniques that have received much interest for surface characterization. DH has been heavily developed over recent years because of newly available high-resolution charge-coupled device (CCD) cameras and advances in digital and automated image processing techniques. Performing Fourier transforms and spectral filtering without the need for additional optical components for reconstruction has given advantages of digital holography over conventional optical holography. DH enables the extraction of both amplitude and phase information of objects in real time with high resolution. DH technique is less sensitive to external perturbations and has long-term stability in object measurement. DH has merits of wide applications covering particles, living cells, and 3D profiling and tracking of micro-objects or nano-objects. In this chapter, we present recent developments in DH techniques carried out by the author. In Section 2, the principle of holography with highlights on three numerical reconstruction methods, namely, Fresnel approximation, convolution approach, and angular spectrum, is explained. In Section 3, the impact of slightly imperfect collimation of the reference wave in off-axis DH is presented. In Section 4, low-coherence, off-axis digital holographic microscope (DHM) by the use of femtosecond pulse light for measuring fine structure of in vitro sliced sandwiched biological sarcomere sample is described. In Section 5, off-axis terahertz DH using continuous-wave radiation generated from cascade laser source for testing a letter T from paper is described. Section 6 gives concluding discussions and remarks.
2. The principle of holography
The word holography is derived from the Greek words “holos,” which means whole or entire, and graphein, which means to write. Holography is a method that records and reconstructs both irradiance of each point in an image and the direction in which the wave is propagating at that point. Holography consists of two procedures: recording as shown in Figure 1(a) and reconstruction as shown in Figure 1(b). Because of the development of digital recording process for recording and computer technology for numerical reconstruction, the optical holography has been replaced by digital holography. The idea of numerical reconstruction was proposed by Goodman and Lawrence . In 1993, Schnars and Juptner  used a CCD camera to record a hologram and performed numerical reconstruction in order to reconstruct this digital hologram.
2.1. Numerical reconstruction in digital holography
In digital holography, the CCD or Complementary Metal-Oxide Semiconductor (CMOS) camera captures the image and transfers it to the computer. This image is saved digitally as a digital hologram. This hologram is digitally accessed and numerically reconstructed by a virtual reference wave, which effectively simulates the reference wave used in the process of recording. The speed of reconstruction procedure depends on the implementation of the numerical reconstruction algorithm and the speed of the computer processing. Because the reference wave has to be generated virtually in the computer, a plane wave or a spherical wave is usually used in the recording process. Figure 1 shows the typical setup of digital holography and the reconstruction with virtual reference wave. Let us consider the coordinate system as shown in Figure 2; then the diffraction by the aperture or hologram in the distance of
If the reference wave is set up to be nominally normal incident to the hologram, then the diffracted light is approximated by the Fresnel-Kirchhoff integral as
And the reconstructed phase is
where Re denotes the real part and Im denotes the imaginary part. The calculated diffraction pattern is the complex amplitude at a distance
where is conjugate to the original reference wave
2.1.1. Reconstruction by the Fresnel approximation
In digital holography, the values of the coordinates
the effect of it and the terms after it are negligible, and they can be removed. Thus the distance
This equation is known as the Fresnel approximation or Fresnel transformation due to its mathematical similarity with the Fourier transform.
The intensity is calculated by squaring
And the phase is calculated by
Therefore Eq. (8) turns into
Because the maximum frequency is determined by the sampling interval in the spatial domain according to the theory of the Fourier transform, the relationships among
Eq. (14) is known as the discrete Fresnel transform. The matrix
2.1.2. Reconstruction by the convolution approach
where the impulse response
Eq. (15) can be regarded as a convolution and the convolution theorem can be applied. The convolution theorem states that the Fourier transform of the convolution of two functions is equal to the product of the Fourier transforms of the individual functions. In other words, the convolution of two functions in the spatial domain can be easily obtained through the multiplication of them in another domain, namely, spatial frequency domain.
Applying the convolution theorem to Eq. (4), it is converted to
Eq. (17) includes two forward Fourier transformations and one inverse Fourier transformation, all of which can be practically implemented via the FFT algorithm. Both Fresnel and convolution methods are practically implemented via the FFT algorithm. In the Fresnel approximation, only a forward FFT is performed. However, two or three FFTs are performed in the convolution approach. In the convolution approach, the pixel sizes in the reconstructed image are equal to that of the hologram. It would seem that a higher resolution could be achieved if a CCD or CMOS device with a smaller pixel size was used in the recording process. For applications that detect very small objects, the convolution approach has more advantages and is more accurate than the Fresnel approximation algorithm.
2.1.3. Reconstruction by the angular spectrum method
Both the Fresnel approximation and the convolution approach suffer the same limitation, i.e., that the object under observation must be placed farther away than some minimum distance. If it is placed inside this distance, the spatial frequency of the detector is too low and aliasing occurs. This minimum distance is given :
where . The reconstructed complex wave field at any plane perpendicular to the propagating
The resolution of the reconstructed images from the angular spectrum method is the same as that in the hologram plane, which means that the pixel size does not vary with changes of wavelength or reconstruction distance.
3. Impact of imperfect collimation of the reference wave in off-axis DH
In digital holography, the simulation of the plane wave is essential for performing a numerical reconstruction. Conventionally, a shear interferometer is used for producing perfect collimation and hence producing a perfect plane wave. In Section 3, we show that using a slightly imperfect plane wave in digital holography experiments is acceptable . We experimentally proved that by using the Mickelson interferometer, no influence of imperfect collimation of the reference wave in an off-axis digital holography exists, as has been previously claimed. We applied perfect and imperfect collimations to three different surface (flat, spherical, and step height) shapes for height inspection, and the results were almost in good agreement. The samples being tested were mounted in the Michelson interferometer one by one, as shown in Figure 3. A laser diode beam passed through a collimating lens of focal lens 100 mm and expanded. The beam splitter splits the collimated beam into two equal beams: one for reference (optical flat of
The slightly imperfect collimations were seven equidistance displacements of the collimating lens with 1.5 mm from the perfect collimation and between two successive displacements. The off-axis holograms for the samples under test at perfect and slightly imperfect plane waves were captured and then reconstructed. Details of the reconstruction process are explained in reference . The reconstructed phase for perfect Figure 4(a) and slightly imperfect collimations Figure 4(2–8) and the height line profile for both along the central
The wrapping phase for the spherical surface for perfect collimations and slightly imperfect collimations is shown in Figure 5(a) and
For the third object (step height surface), the off-axis holograms at the perfect and slightly imperfect collimations were captured and reconstructed. The height line profiles along the central
As seen from Figures 4–6, for flat, spherical, and step surfaces, the measured height values of the three tested surfaces were almost consistent at perfect and imperfect collimations. Very small variations may be observed due to noise, which is commonly observed in interferometry measurements. We claim that the variations may be due to the mechanical imperfection of the collimating lens as shown in Figure 7. We claim that when the lens mounting was ideally displaced as shown in Figure 7(a) at position 2 (slightly imperfect collimation), the beams converge toward the reference and the object. The convergent beams would then be canceled out and subsequently have no impact on the height variations. However, it is hard to achieve ideal mounting mechanically. Thus we expect that the effect of nonideal mounting as shown in Figure 7(b) may be the reason of small height variations as shown in Figure 7(b).
4. Low coherence, off-axis DHM for in vitro sandwiched biological samples investigation
In digital holographic microscopy (DHM), optical sources with long coherent lengths such as He-Ne laser have been widely used to feature the sample. Because of the high degree of coherence of the He-Ne laser light, harmful coherent noise often arises. This noise affects the quality of the holograms and hence leads to error in phase measurement. The larger the phase noise is, the lower the measurement precision will be. The harmful coherent noise is mainly classified into two types. The first is the random diffraction patterns (speckle noise) due to scattered light. The second is the formation of unwanted interference fringes (spurious noise) due to stray light. The spurious noise is formed when light reflected or scattered from various surfaces in the optical path is coherent with the main beam. The amplitude of the scattered light as adds vectorially to the amplitude of the main beam, resulting in a phase error Δφ as illustrated in Figure 8(a). Some practical solutions such as introducing a wedged beam splitter, a rotating diffuser, and antireflection coating to the optical surfaces, in the optical system setup, were proposed to minimize the unwanted coherent noise. Although these practical solutions are effective and may circumvent to suppress the coherent noise to some amount, they have some drawbacks in terms of blurring the fringe visibility and hindering the fringe formation in DHM, which require perfect alignment. Optical sources with short coherent lengths such as LEDs were proposed in order to avoid the harmful coherent noise. However, the limited coherence length of LED and its insufficient brightness hinder its application in an off-axis DHM, since just a limited number of interference fringes with poor visibility appear in the field of view (FOV). In Section 4, we present an off-axis DHM configuration using bright femtosecond pulse light with ultrashort coherent length, which makes possible to feature sandwiched biological samples with no coherent noise (speckle and spurious) in the reconstructed object wave .
A typical configuration of the sandwiched biological sample is shown in Figure 8(b), where the specimen is mounted in between two thin glass plates to avoid dehydration. Investigating such biological samples using conventional DHM with long coherent He-Ne laser light is challenging because of existence of the harmful coherent noise. Photograph of the investigated sample taken by the proposed DHM system (see Figure 9) with blocking the reference arm is shown in Figure 8(c). The DHM experimental setup is schematically shown in Figure 9. The configuration is comprised of two parts: generation of femtosecond pulse light in near-infrared region and a Mach-Zehnder interferometer in transmission. A mode-locked Er-doped fiber laser light (center wavelength
The output power was sufficient to illuminate the sample and produce off-axis holograms with high contrast in the entire field of the CCD camera. The coherence length of the SHG pulse light was 30 μm. The SHG beam was expanded to a diameter of 20 mm by a telescope system. In the Mach-Zehnder off-axis setup in transmission, a pair of non-polarized beam splitters (NPBS1 and NPBS2) was used to separate the two SHG beams into reference (
The off-axis digital hologram recorded by the CCD camera was reconstructed using convolution-based Fresnel method. Three spectra were obtained when 2D-FFT was implemented to the off-axis hologram as shown in Figure 10. Only one filtered spectrum from the three spectra in 2D-FFT is used. The inverse 2D-FFT was applied after filtering out the spectrum, and the calculation result gives a complex object wave (amplitude and phase). The obtained complex amplitude was multiplied by the digital reference wave
The obtained off-axis hologram was reconstructed with the same procedure of Figure 10 and the final amplitude and phase obtained at
The phase profiles have been measured at different locations, and the average phase profile of the proposed method was found to be in good agreement with nominal values of the sarcomere depth. The contrast (axial resolution) of the proposed method is estimated from Figure 12 to be two times better than the contrast of He-Ne phase-based result. It is noted that the mismatching in the peaks of the phase profiles shown in Figure 12 is due to difference in magnifications of the captured off-axis holograms of both He-Ne and femtosecond pulse light. To enhance the differentiation of the sarcomere structure within the reconstructed amplitude map, the 3D reconstructed amplitude maps of both He-Ne laser light and femtosecond pulse light may be displayed in a false color representation as shown in Figure 13(a) and (b), respectively.
It is noted that these 3D reconstructed amplitude maps were flipped upside down to see the sarcomeres from different views. In Figure 13(a), existence of coherent noise in the background image makes it difficult to visualize the structure of the sarcomeres. On the other hand, Figure 13(b) shows the high quality and contrast of structure detail on the sarcomeres and provides accurate profile edges of hexagonal shape of the sarcomeres. This makes our technique preferable in featuring such sandwiched biological samples, which is quite difficult to investigate using atomic force microscope (AFM).
Our deductions were verified by applications of apodization technique , to estimate the coherent noise level in the reconstructed amplitude maps of the off-axis holograms generated by both He-Ne and femtosecond pulse light, respectively. Apodization is the same topic as windowing in signal processing. The transmission of the apodized aperture function is completely transparent in the large central part of the profile. At the edges, the transmission varies from zero to unity following a curve defined by a cubic spline interpolation. The 2D transmission size of (480×480) pixels of the apodized aperture function is multiplied with the off-axis holograms (480×480) pixels of both He-Ne laser light and femtosecond pulse light, respectively. Normalization of intensity distribution of four sarcomeres at the middle of the off-axis hologram of the He-Ne laser light before and after application of apodization is shown in the left side of Figure 14. Normalization of intensities distribution of four sarcomeres at the middle of the off-axis hologram of the femtosecond laser light before and after application of apodization is shown in the right side of Figure 14.
As seen from the left figure, the variation in intensities before and after application of apodization function indicates that there is a coherent noise in the reconstructed amplitude of He-Ne off-axis hologram. Such coherent noise is totally disappearing in the right figure, whereas no variation in intensities before and after application of apodization. This confirms that the reconstructed amplitude of femtosecond pulse light off-axis hologram is free from coherent noise. Due to the short coherence length of the pulsed light, only fraction of hologram shows high-contrast fringe, resulting in a good reconstruction in this region as shown in Figure 13(b) which is corresponding to the off-axis hologram of Figure 11(d). The contrast of the fringes in Figure 11(d) reduces from center to two sides diagonally, which indicates the zero-path-difference point is nearly in the middle of the sensor. To obtain a high-brightness full-field image, we can move the center of the fringes diagonally and collect holograms with different zero path difference. Twelve off-axis holograms were generated diagonally by varying the optical path length of the interferometer to cover a sarcomere sample of around 45×45 micrometers in the field of view (FOV) with high-contrast fringes at different regions in the image. The full-field image can be obtained by adding multiple such single reconstructions on the intensity basis as shown in Figure 15(a)–(l). The synthesized image map is shown in Figure 15(l). Because of the holograms’ shifting process, the reference and object beam will have some phase shifts when the second hologram is recorded. When the second hologram is subtracted from the first hologram, the zero order is removed. Subtractions among successive holograms were superimposed using a direct linear addition method to constitute a full-field, high-quality synthesized hologram as shown in Figure 9. The intensity of the synthesized hologram
5. Off-axis terahertz DH using continuous-wave THZ radiation
Terahertz (THz) radiation is an electromagnetic radiation lying between the microwave and infrared portions of the spectrum as shown in Figure 16. THz radiation can be produced by many techniques such as quantum cascade lasers (QCLs). QCLs are semiconductor heterostructures that can emit continuous-wave (CW) THz radiation . QCLs are electron-only devices that exploit transitions between conduction band states. The conduction band offsets between neighboring materials in the superlattice create a series of quantum wells and barriers. The most widely used QCL designs are based on GaAs/AlGaAs superlattices. The word cascading means that one electron can produce many photons in the superlattice periods. A significant advantage of THZ radiation is that it can easily inspect sealed packages for contrasting metal and plastic objects, testing pharmaceutical tablets for integrity, detecting skin cancers, etc.
In Section 5, we present the usage of THz radiation for 3D surface characterization via a digital holography (DH) technique . Since the usual DH uses a visible light, it is difficult to visualize 3D internal structure of visibly opaque objects due to their limited penetration depth. The compelling advantage of THz radiation is that it has a good penetration capability; thus, 3D visualization of both surface shape and internal structure in visibly opaque object can be achieved . We constructed off-axis THz digital holography (THz-DH) system equipped with CW-THz radiation generated by QCL, and the THz digital hologram is captured by a THz camera.
Figure 17 illustrates a schematic diagram of off-axis THz-DH system. An optical chopper (OC) is positioned in front of the QCL to reduce the noise. The radiation was collimated by a Teflon lens (f = 300 mm). The collimated THz beam of diameter around 60 mm is divided into a signal THz beam and a reference one by a silicon beam splitter (BS). Figure 18(a) shows 2D intensity distribution of the THz beam without the sample. Figure 18(b) shows the dark frame captured by the THz camera when there is no illumination. The signal THz beam passed through a sample (here, the sample is a letter T from paper), while a reference THz beam is reflected by a mirror (M). Then, these two THz beams were spatially overlapped at a certain off-axis angle to generate 2D fringe of THz beams. Finally, the off-axis THz digital hologram is captured by a THz camera as shown in Figure 18(c). The off-axis THz hologram has been reconstructed with angular spectrum method to extract both amplitude and phase. Figure 18(d) shows the 3D reconstructed phase of the sample.
In conclusion, we have presented the recent developments of digital holography techniques for surface characterization. In this chapter, the principle of digital holography with focus on numerical reconstruction algorithms is presented. Also, influence of slightly imperfect collimation of the reference wave in off-axis DH is discussed. Finally, we have described two different DH techniques for surface characterization: the first technique by using short coherent length, namely, high-brightness DHM, and the second technique by using long coherent length, namely, THz-DH. Experimental results are presented to verify the principles.
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